introduction to the theory of the early universe: hot big bang theory

DMITRY S GORBUNOV VALERY A RUBAKOV

INTRODUCTION TO

THE THEORY OF THE

EARLY UNIVERSEHot Big Bang Theory

INTRODUCTION TO

THE THEORY OF THE


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N E W J E R S E Y L O N D O N S I N G A P O R E B E I J I N G S H A N G H A I H O N G K O N G TA I P E I C H E N N A I

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INTRODUCTION TO

THE THEORY OF THE


DMITRY S GORBUNOVInstitute for Nuclear Research of the Russian Academy of Sciences

VALERY A RUBAKOVInstitute for Nuclear Research of the Russian Academy of Sciences & Moscow State University

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INTRODUCTION TO THE THEORY OF THE EARLY UNIVERSEHot Big Bang Theory

To Olesya and Elvira

Preface

It is clear by now that there is deep interconnection between cosmology and particlephysics, between macro- and micro-worlds. This book is written precisely from thisperspective. We present here the results on the homogeneous and isotropic Universeat the hot stage of its evolution and at subsequent stages. This part of cosmology isoften dubbed as the Hot Big Bang theory. In the accompanying book we study thetheory of cosmological perturbations (inhomogeneities in the Universe), inflationarytheory and theory of post-inflationary reheating.

This book grew from the lecture course which is being taught for a number ofyears at the Department of Quantum Statistics and Field Theory of the PhysicsFaculty of the Lomonosov Moscow State University. This course is aimed at under-graduate students specializing in theoretical physics. We decided, however, to add anumber of more advanced Chapters and Sections which we mark by asterisks. Thereason is that there are problems in cosmology (nature of dark matter and darkenergy, mechanism of the matter-antimatter asymmetry generation, etc.) whichhave not found their compelling solutions yet. Most of the additional material dealswith hypotheses on these problems that at the moment compete with each other.

Knowledge of material taught in general physics courses is in principle sufficientfor reading the main Chapters of this book. So, the main Chapters must be under-standable by undergraduate students. The necessary material on General Relativityand particle physics is collected in Appendices which, of course, do not pretend togive comprehensive account of these areas of physics. On the other hand, some partslabeled by asterisks make use of the methods of classical and quantum field theoryas well as nonequilibrium statistical mechanics, so basic knowledge of these methodsis required for reading these parts.

Literature on cosmology is huge, and presenting systematic and comprehensivebibliography would be way out of the scope of this book. To orient the reader, inthe end of this book we give a list of monographs and reviews where the issues wetouch upon are considered in detail. Certainly, this list is by no means complete. Weoccasionally refer to original literature, especially in those places where we presentconcrete results without detailed derivation.

vii

viii Preface

Both observational cosmology and experimental particle physics develop veryfast. Observational and experimental data we quote, the results of their compilationsand fits (values of the cosmological parameters, limits on masses and couplings ofhypothetical particles, etc.) will most probably get more precise even before thisbook is published. This drawback can be corrected, e.g., by using the regularlyupdated material of Particle Data Group at http://pdg.lbl.gov/.

We would like to thank our colleagues from the Institute for Nuclear Researchof the Russian Academy of Sciences F. L. Bezrukov, S. V. Demidov, V. A. Kuzmin,D. G. Levkov, M. V. Libanov, E. Y. Nugaev, G. I. Rubtsov, D. V. Semikoz,P. G. Tinyakov, I. I. Tkachev and S. V. Troitsky for participation in the prepa-ration of the lecture course and numerous helpful discussions and comments.Our special thanks are to S. L. Dubovsky who participated in writing this bookat an early stage. We are deeply indebted to V. S. Berezinsky, A. Boyarsky,A. D. Dolgov, D. I. Kazakov, S. Y. Khlebnikov, V. F. Mukhanov, I. D. Novikov,K. A. Postnov, M. V. Sazhin, M. E. Shaposhnikov, A. Y. Smirnov, A. A. Starobinsky,R. A. Sunyaev, A. N. Tavkhelidze, O. V. Verkhodanov, A. Vilenkin, M. B. Voloshinand M. I. Vysotsky for many useful comments and criticism.

Contents

Preface vii

1. Cosmology: A Preview 1

1.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Universe Today . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Homogeneity and isotropy . . . . . . . . . . . . . . . . . . 31.2.2 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Age of the Universe and size of its observable part . . . . . 71.2.4 Spatial flatness . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.5 “Warm” Universe . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Energy Balance in the Present Universe . . . . . . . . . . . . . . . 131.4 Future of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . 191.5 Universe in the Past . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5.1 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . 201.5.2 Big Bang Nucleosynthesis (BBN) . . . . . . . . . . . . . . 211.5.3 Neutrino decoupling . . . . . . . . . . . . . . . . . . . . . . 221.5.4 Cosmological phase transitions . . . . . . . . . . . . . . . . 231.5.5 Generation of baryon asymmetry . . . . . . . . . . . . . . 241.5.6 Generation of dark matter . . . . . . . . . . . . . . . . . . 25

1.6 Structure Formation in the Universe . . . . . . . . . . . . . . . . . 251.7 Inflationary Epoch . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2. Homogeneous Isotropic Universe 29

2.1 Homogeneous Isotropic Spaces . . . . . . . . . . . . . . . . . . . . 292.2 Friedmann–Lemaıtre–Robertson–Walker Metric . . . . . . . . . . . 322.3 Redshift. Hubble Law . . . . . . . . . . . . . . . . . . . . . . . . . 342.4 Slowing Down of Relative Motion . . . . . . . . . . . . . . . . . . . 382.5 Gases of Free Particles in Expanding Universe . . . . . . . . . . . . 40

3. Dynamics of Cosmological Expansion 45

3.1 Friedmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 45

ix

x Contents

3.2 Sample Cosmological Solutions. Age of the Universe. CosmologicalHorizon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.1 Non-relativistic matter (“dust”) . . . . . . . . . . . . . . . 493.2.2 Relativistic matter (“radiation”) . . . . . . . . . . . . . . . 523.2.3 Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2.4 General barotropic equation of state p = wρ . . . . . . . . 56

3.3 Solutions with Recollapse . . . . . . . . . . . . . . . . . . . . . . . 57

4. ΛCDM: Cosmological Model with Dark Matter and Dark Energy 61

4.1 Composition of the Present Universe . . . . . . . . . . . . . . . . . 614.2 General Properties of Cosmological Evolution . . . . . . . . . . . . 644.3 Transition from Deceleration to Acceleration . . . . . . . . . . . . 654.4 Transition from Radiation Domination to Matter Domination . . . 664.5 Present Age of the Universe and Horizon Size . . . . . . . . . . . . 684.6 Brightness-Redshift Relation for Distant Standard Candles . . . . 714.7 Angular Sizes of Distant Objects . . . . . . . . . . . . . . . . . . . 794.8 ∗Quintessence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.8.1 Evolution of scalar field in expanding Universe . . . . . . . 814.8.2 Accelerated cosmological expansion due to scalar field . . . 854.8.3 Tracker field . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5. Thermodynamics in Expanding Universe 91

5.1 Distribution Functions for Bosons and Fermions . . . . . . . . . . . 915.2 Entropy in Expanding Universe. Baryon-to-Photon Ratio . . . . . 985.3 ∗Models with Intermediate Matter Dominated Stage:

Entropy Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.4 ∗Inequilibrium Processes . . . . . . . . . . . . . . . . . . . . . . . . 105

6. Recombination 111

6.1 Recombination Temperature in Equilibrium Approximation . . . . 1116.2 Photon Last Scattering in Real Universe . . . . . . . . . . . . . . . 1166.3 ∗Kinetic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.4 Horizon at Recombination and its Present Angular Size. Spatial

Flatness of the Universe . . . . . . . . . . . . . . . . . . . . . . . . 127

7. Relic Neutrinos 133

7.1 Neutrino Freeze-Out Temperature . . . . . . . . . . . . . . . . . . 1337.2 Effective Neutrino Temperature. Cosmological Bound

on Neutrino Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.3 ∗Sterile Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

8. Big Bang Nucleosynthesis 143

8.1 Neutron Freeze-Out. Neutron-Proton Ratio . . . . . . . . . . . . . 143

Contents xi

8.2 Beginning of Nucleosynthesis. Direction of Nuclear Reactions.Primordial 4He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8.3 Kinetics of Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . 1518.3.1 Neutron burning, p + n → D + γ . . . . . . . . . . . . . . . 1528.3.2 Deuterium burning . . . . . . . . . . . . . . . . . . . . . . 1538.3.3 ∗Primordial 3He and 3H . . . . . . . . . . . . . . . . . . . 1568.3.4 ∗Production and burning of the heaviest elements in

primordial plasma . . . . . . . . . . . . . . . . . . . . . . . 1588.4 Comparison of Theory with Observations . . . . . . . . . . . . . . 160

9. Dark Matter 165

9.1 Cold, Hot and Warm Dark Matter . . . . . . . . . . . . . . . . . . 1659.2 Freeze-Out of Heavy Relic . . . . . . . . . . . . . . . . . . . . . . . 1689.3 Weakly Interacting Massive Particles, WIMPs . . . . . . . . . . . . 1729.4 ∗Other Applications of the Results of Section 9.2 . . . . . . . . . . 177

9.4.1 Residual baryon densityin baryon-symmetric Universe . . . . . . . . . . . . . . . . 177

9.4.2 Heavy neutrino . . . . . . . . . . . . . . . . . . . . . . . . 1789.5 Dark Matter Candidates in Particle Physics . . . . . . . . . . . . . 1799.6 ∗Stable Particles in Supersymmetric Models . . . . . . . . . . . . . 179

9.6.1 Neutralino . . . . . . . . . . . . . . . . . . . . . . . . . . . 1829.6.2 Sneutrino . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1939.6.3 Gravitino . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

9.7 ∗Other Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . 2049.7.1 Axions and other long-lived particles . . . . . . . . . . . . 2049.7.2 Superheavy relic particles . . . . . . . . . . . . . . . . . . . 215

10. Phase Transitions in the Early Universe 217

10.1 Order of Phase Transition . . . . . . . . . . . . . . . . . . . . . . . 21910.2 Effective Potential in One-Loop Approximation . . . . . . . . . . . 22810.3 Infrared Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

11. Generation of Baryon Asymmetry 243

11.1 Necessary Conditions for Baryogenesis . . . . . . . . . . . . . . . . 24311.2 Baryon and Lepton Number Violation in Particle Interactions . . . 247

11.2.1 Electroweak mechanism . . . . . . . . . . . . . . . . . . . . 24711.2.2 Baryon number violation in Grand Unified Theories . . . . 25311.2.3 Violation of lepton numbers and Majorana masses

of neutrino . . . . . . . . . . . . . . . . . . . . . . . . . . . 26111.3 Asymmetry Generation in Particle Decays . . . . . . . . . . . . . . 26311.4 Baryon Asymmetry and Neutrino Masses: Leptogenesis . . . . . . 271

xii Contents

11.5 Electroweak Baryogenesis . . . . . . . . . . . . . . . . . . . . . . . 27711.5.1 Departure from thermal equilibrium . . . . . . . . . . . . . 27811.5.2 ∗Thick wall baryogenesis . . . . . . . . . . . . . . . . . . . 28011.5.3 ∗Thin wall case . . . . . . . . . . . . . . . . . . . . . . . . 284

11.6 ∗Affleck–Dine Mechanism . . . . . . . . . . . . . . . . . . . . . . . 29111.6.1 Scalar fields carrying baryon number . . . . . . . . . . . . 29111.6.2 Asymmetry generation . . . . . . . . . . . . . . . . . . . . 293

11.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 298

12. Topological Defects and Solitons in the Universe 301

12.1 Production of Topological Defects in the Early Universe . . . . . . 30212.2 ∗’t Hooft–Polyakov Monopoles . . . . . . . . . . . . . . . . . . . . 303

12.2.1 Magnetic monopoles in gauge theories . . . . . . . . . . . . 30312.2.2 Kibble mechanism . . . . . . . . . . . . . . . . . . . . . . . 30712.2.3 Residual abundance: the monopole problem . . . . . . . . 308

12.3 ∗Cosmic Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31012.3.1 String solutions . . . . . . . . . . . . . . . . . . . . . . . . 31012.3.2 Gas of cosmic strings . . . . . . . . . . . . . . . . . . . . . 31512.3.3 Deficit angle . . . . . . . . . . . . . . . . . . . . . . . . . . 31712.3.4 Strings in the Universe . . . . . . . . . . . . . . . . . . . . 322

12.4 ∗Domain Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32612.5 ∗Textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32912.6 ∗Hybrid Topological Defects . . . . . . . . . . . . . . . . . . . . . . 33212.7 ∗Non-topological Solitons: Q-balls . . . . . . . . . . . . . . . . . . 333

12.7.1 Two-field model . . . . . . . . . . . . . . . . . . . . . . . . 33312.7.2 Models with flat directions . . . . . . . . . . . . . . . . . . 338

13. Color Pages 347

Appendix A Elements of General Relativity 355

A.1 Tensors in Curved Space-Time . . . . . . . . . . . . . . . . . . . . 355A.2 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 359A.3 Riemann Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364A.4 Gravitational Field Equations . . . . . . . . . . . . . . . . . . . . . 368A.5 Conformally Related Metrics . . . . . . . . . . . . . . . . . . . . . 371A.6 Interaction of Matter with Gravitational Field.

Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . 375A.7 Particle Motion in Gravitational Field . . . . . . . . . . . . . . . . 380A.8 Newtonian Limit in General Relativity . . . . . . . . . . . . . . . . 382A.9 Linearized Einstein Equations about Minkowski Background . . . 384A.10 Macroscopic Energy-Momentum Tensor . . . . . . . . . . . . . . . 385A.11 Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . 386

Contents xiii

Appendix B Standard Model of Particle Physics 389

B.1 Field Content and Lagrangian . . . . . . . . . . . . . . . . . . . . . 389B.2 Global Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 399B.3 C-, P-, T-Transformations . . . . . . . . . . . . . . . . . . . . . . . 401B.4 Quark Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402B.5 Effective Fermi Theory . . . . . . . . . . . . . . . . . . . . . . . . . 407B.6 Peculiarities of Strong Interactions . . . . . . . . . . . . . . . . . . 408B.7 The Effective Number of Degrees of Freedom

in the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . 409

Appendix C Neutrino Oscillations 411

C.1 Oscillations and Mixing . . . . . . . . . . . . . . . . . . . . . . . . 411C.1.1 Vacuum oscillations . . . . . . . . . . . . . . . . . . . . . . 411C.1.2 Three-neutrino oscillations in special cases . . . . . . . . . 415C.1.3 Mikheev–Smirnov–Wolfenstein effect . . . . . . . . . . . . 417

C.2 Experimental Discoveries . . . . . . . . . . . . . . . . . . . . . . . 419C.2.1 Solar neutrino and KamLAND . . . . . . . . . . . . . . . . 419C.2.2 Atmospheric neutrino, K2K and MINOS . . . . . . . . . . 426C.2.3 CHOOZ: limit on |Ue3| . . . . . . . . . . . . . . . . . . . . 428

C.3 Oscillation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 428C.4 Dirac and Majorana Masses. Sterile Neutrino . . . . . . . . . . . . 431C.5 Search for Neutrino Masses . . . . . . . . . . . . . . . . . . . . . . 436

Appendix D Quantum Field Theory at Finite Temperature 439

D.1 Bosonic Fields: Euclidean Time and Periodic BoundaryConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

D.2 Fermionic Fields: Antiperiodic Boundary Conditions . . . . . . . . 443D.3 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 447D.4 One-Loop Effective Potential . . . . . . . . . . . . . . . . . . . . . 449D.5 Debye Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

Books and Reviews 457

Bibliography 463

Index 471

Chapter 1

Cosmology: A Preview

The purpose of this Chapter is to give a preview of the field which we consider inthis and the accompanying book. The presentation here is at the qualitative level,and is by no means complete. Our purpose is to show the place of one or anothertopic within the entire area of cosmology.

Before proceeding, let us introduce units and conventions that we use throughoutthis book.

1.1 Units

We mostly use the “natural” system of units in which the Planck constant, speedof light and the Boltzmann constant are equal to 1,

� = c = kB = 1.

Then the mass M , energy E and temperature T have the same dimension (since[E] = [mc2], [E] = [kBT ]). A convenient unit of mass and energy is 1 eV or 1GeV =109 eV; the proton mass is then equal to mp = 0.938GeV, and 1K is approximately10−13 GeV. Time t and length l in the natural system have dimension M−1 (since[E] = [�ω], [ω] = [t−1] and [l] = [ct]), with 1 GeV−1 ∼ 10−14 cm and 1GeV−1 ∼10−24 s. We give the coefficients relating various units in Tables 1.1 and 1.2.

Problem 1.1. Check the relations given in Tables 1.1 and 1.2. What are 1 Volt (V),1 Gauss (G), 1 Hertz (Hz) and 1 Angstrom (A) in natural system of units?

In natural system of units, the Newton gravity constant G has dimension M−2.This follows from the formula for the gravitational potential energy V = −Gm1m2

r ,since [V ] = M , [r−1] = M . It is convenient to introduce the Planck mass MPl inthe following way,

G = M−2Pl .

Numerically

MPl = 1.2 · 1019 GeV, (1.1)

1

2 Cosmology: A Preview

Table 1.1 Conversion of natural units into CGS units.

Energy 1 GeV = 1.6 · 10−3 erg

Mass 1GeV = 1.8 · 10−24 g

Temperature 1GeV = 1.2 · 1013 K

Length 1GeV−1 = 2.0 · 10−14 cm

Time 1GeV−1 = 6.6 · 10−25 s

Particle number density 1GeV3 = 1.3 · 1041 cm−3

Energy density 1GeV4 = 2.1 · 1038 erg · cm−3

Mass density 1GeV4 = 2.3 · 1017 g · cm−3

Table 1.2 Conversion of CGS units into natural units.

Energy 1 erg = 6.2 · 102 GeV

Mass 1 g = 5.6 · 1023 GeV

Temperature 1K = 8.6 · 10−14 GeV

Length 1 cm = 5.1 · 1013 GeV−1

Time 1 s = 1.5 · 1024 GeV−1

Particle number density 1 cm−3 = 7.7 · 10−42 GeV3

Energy density 1 erg · cm−3 = 4.8 · 10−39 GeV4

Mass density 1 g · cm−3 = 4.3 · 10−18 GeV4

and the Planck length, time and mass are

lPl =1

MPl

= 1.6 · 10−33 cm,

tPl =1

MPl

= 5.4 · 10−44 s, (1.2)

MPl = 2.2 · 10−5 g.

The gravitational interactions are weak precisely because MPl is large.

Problem 1.2. Check the relations (1.1) and (1.2).

Problem 1.3. What is the ratio of gravitational interaction energy to Coulombenergy for two protons?

The traditional unit of length in cosmology is Megaparsec,

1 Mpc = 3.1 · 1024 cm.

Let us also introduce a convention which we use in this book. The subscript 0denotes present values of quantities which can depend on time. As an example, ρ(t)denotes the energy density in the Universe as a function of time, while ρ0 ≡ ρ(t0)is always its present value.

There are several units of length that are used in astronomy, depending on sizes ofobjects and length scales considered. Besides the metric system, in use are

astronomical unit (a.u.), which is the average distance from the Earth to the Sun,

1 a.u. = 1.5 · 1013 cm;

1.2. The Universe Today 3

light year, the distance that a photon travels in one year,

1 year = 3.16 · 107 s, 1 light year = 3 · 1010 cm

s· 3.16 · 107 s = 0.95 · 1018 cm;

parsec (pc) — distance from which an object of size 1 a.u. is seen at angle 1 arc second,

1 pc = 2.1 · 105 a.u. = 3.3 light year = 3.1 · 1018 cm.

To illustrate the hierarchy of spatial scales in the Universe, let us give the distances tovarious objects expressed in the above units.

10 a.u. is the average distance to Saturn, 30 a.u. is the same for Pluto, 100 a.u. is theestimate of maximum distance which can be reached by solar wind (particles emitted bythe Sun). 100 a.u. is also the estimate of the maximum distance to cosmic probes (Pioneer10, Voyager 1, Voyager 2). Further out is the Oort cloud, the source of the most distantcomets, which is at the distance of 104 − 105 a.u. ∼ 0.1 − 1 pc.

The nearest stars — Proxima and Alpha Centauri — are at 1.3 pc from the Sun.The distance to Arcturus and Capella is more than 10 pc, the distances to Canopus andBetelgeuse are about 100 pc and 200 pc, respectively; Crab Nebula — the remnant ofsupernova seen by naked eye — is 2 kpc away from us.

The next point on the scale of distances is 8 kpc. This is the distance from the Sun tothe center of our Galaxy. Our Galaxy is of spiral type, the diameter of its disc is about30 kpc and the thickness of the disc is about 250 pc. The distance to the nearest dwarfgalaxies, satellites of our Galaxy, is about 30 kpc. Fifteen of these satellites are known; thelargest of them — Large and Small Magellanic Clouds — are 50 kpc away. Search for new,dimmer satellite dwarfs is underway; we note in this regard that only eight of Milky Waysatellites were known by 1994.

The mass density of the usual matter in usual (not dwarf) galaxies is about 105 higherthan the average over the Universe.

The nearest usual galaxy — the spiral galaxy M31 in Andromeda constellation —is 800 kpc away from the Milky Way. Despite the large distance, it occupies a sizeablearea on the celestial sphere: its angular size is larger than that of the Moon! Anothernearby galaxy is in Triangulum constellation. Our Galaxy together with Andromeda andTriangulum galaxies, their satellites and other 35 smaller galaxies constitute the LocalGroup, the gravitationally bound object consisting of about 50 galaxies.

The next scale in this ladder is the size of clusters of galaxies, which is 1–3 Mpc.Rich clusters contain thousands of galaxies. The mass density in clusters exceeds theaverage density over the Universe by a factor of a hundred and even sometimes a thousand.The distance to the center of the nearest cluster, which is in the Virgo constellation, isabout 15 Mpc. Its angular size is about 5 degrees. Clusters of galaxies are the largestgravitationally bound systems in the Universe.

1.2 The Universe Today

We begin our preview with the brief discussion of the properties of the presentUniverse (more precisely, of its observable part).

1.2.1 Homogeneity and isotropy

The Universe is homogeneous and isotropic at large spatial scales. The sizes ofthe largest structures in the Universe — superclusters of galaxies and gigantic


voids — reach1 tens of Megaparsec. At larger scales all parts of the Universe lookthe same (homogeneity). Likewise, there are no special directions in the Universe(isotropy). These facts are well established by deep galaxy surveys which collecteddata from millions of galaxies.

About 20 superclusters are known by now. The Local Group belongs to a superclusterwith the center in the direction of Virgo constellation. The size of this supercluster isabout 30 Mpc, and besides the Virgo cluster and Local Group it contains about a hundredgroups and clusters of galaxies. Superclusters are rather loose: the density of galaxies inthem is only twice higher than the average in the Universe. The nearest to Virgo is thesupercluster in Hydra and Centaurus constellations; its distance to Virgo supercluster isabout half a hundred Megaparsec.

The largest catalog of galaxies and quasars up to date is the freely available catalogof SDSS [2] (Sloan Digital Sky Survey). This catalog is the result of the analysis of thedata collected during almost 8 years of operation of a dedicated telescope, 2.5 metersin diameter, which is capable of measuring simultaneously spectra of 640 astrophysical

objects in 5 optical bandpasses (photon wavelengths λ = 3800 − 9200 A). The catalogincludes more than 230 million celestial objects. Measurements of spectra of more than930 thousand galaxies and more than 120 thousand quasars resulted in the creation of a3-dimensional map covering a large part of the visible Universe. Its area exceeds a quarterof the sky. There are other catalogs which cover smaller parts of the Universe (see, e.g.,Ref. [3] for the next-to-largest catalog based on the 2dF Galaxy Redshift Survey).

The early SDSS results are illustrated in Fig. 13.1 in color pages, where positions of 40thousand galaxies and 4 thousand quasars are presented. The covered part of the celestialsphere has the area of 500 squared degrees. Recognizable are clusters of galaxies and voids.Isotropy and homogeneity of the Universe are restored at spatial scales of order 100 Mpcand larger. Color of each dot refers to the type of the astrophysical object. The dominationof one type over others is, generally speaking, caused by peculiarities of structure formationand evolution. Thus, what one observes is partially pictured in time rather than in space.

Indeed, from the distance of 1.5 Gpc, where the distribution of bright red ellipticalgalaxies (red dots in Fig. 13.1) is at maximum, light travels to the Earth for about 5billion years. At that epoch, the Universe was different (for instance, there was no Solarsystem yet). One more reason for choosing objects of a certain type is the finite sensitivityof a telescope. In particular, only highly luminous objects can be detected at the largestdistances, while the highest-luminosity, continuously shining objects in the Universe arequasars.

1.2.2 Expansion

The Universe expands: the distances between galaxies increase.2 Loosely speaking,the space, being always homogeneous and isotropic, stretches out.

To describe this expansion, one introduces the scale factor a(t) which grows intime. The distance between two far away objects in the Universe is proportional toa(t) and the number density of particles decreases as a−3(t). The rate of the cosmo-logical expansion, i.e., the relative growth of distances in unit time, is characterized

1This is a somewhat loose statement: the galaxy correlation function falls off as power-law at

large separations.2Of course, this does not apply to galaxies that are gravitationally bound to each other in clusters.


by the Hubble parameter,

H(t) =a(t)a(t)

.

Hereafter dot denotes the derivative with respect to the cosmic time t. The Hubbleparameter depends on time; its present value, according to our convention, isdenoted by H0.

The expansion of the Universe gives rise also to the growth of the wavelengthof a photon emitted in distant past. Like other distances, the photon wavelengthincreases proportionally to a(t); the photon experiences redshift. This redshift z isdetermined by the ratio of photon wavelengths at absorption and emission,

λab

λem≡ 1 + z. (1.3)

Clearly, this ratio depends on the moment of the emission (assuming that the photonis detected today on the Earth), i.e., on the distance to the source. Redshift isa directly measurable quantity: the wavelength at emission is determined by thephysics of the emission process (e.g., by energies of an excited and the ground stateof an atom), while λab is the measured wavelength. Thus, one identifies the systemof emission (or absorption) lines and determines how much they are shifted to thered spectral region, and in this way one measures the redshift.

In reality, the identification of lines makes use of patterns which are characteristicof particular objects, see Fig. 1.1, Ref. [5]. If the spectrum contains absorption dips,as in Fig. 1.1, then the object whose redshift is being measured is between theemitter and observer.3 The peaks in the spectrum — emission lines — mean thatthe object is an emitter itself.

For z � 1, the distance to the source r and the redshift are related by theHubble law

z = H0r, z � 1, (1.4)

At larger z the redshift-distance relation is more complicated, as we discuss in detailsin this book.

The determination of absolute distances to far away sources is a complicatedproblem. One of the methods is to measure the photon flux from a source whoseabsolute luminosity is assumed to be known. These sources are sometimes calledstandard candles.

Systematic uncertainties in the determination of H0 are not particularly wellknown and are presumably rather large. We note in this regard that the value ofthe Hubble constant as determined by Hubble in 1929 was 550 km/(s ·Mpc). Themodern determinations give [7]

H0 = (70.5 ± 1.3)km

s · Mpc. (1.5)

3Photons of definite wavelengths experience resonant absorption by atoms and ions, with subse-

quent isotropic emission. This leads to the loss of photons reaching the observer.


Fig. 1.1 Absorption lines of distant galaxies [5]. The upper panel shows the measurement of the

differential energy flux from a far away galaxy (z = 2.0841). The vertical lines show the position of

atomic lines whose identification has been used to measure redshift. The spectra of nearer galaxies

have more pronounced dips. The plot with the spectra of these galaxies, shifted to comoving frame,

is shown in the lower panel.

Problem 1.4. Relate the dimensionless redshift to distance expressed in Mpc.

Let us comment on the traditional unit for the Hubble parameter used in (1.5).A naive interpretation of the Hubble law (1.4) is that the redshift is caused bythe radial motion of galaxies from the Earth with velocities proportional to thedistances,

v = H0r, v � 1. (1.6)

Then the redshift (1.4) is interpreted as the longitudinal Doppler effect (at v � c,i.e., v � 1 in natural units, the Doppler shift equals to z = v). According to this


interpretation, the dimension of the Hubble parameter H0 is [velocity/distance]. Westress, however, that the interpretation of the cosmological redshift in terms of theDoppler effect is unnecessary, and often inadequate. The right way is to use therelation (1.4) as it is.

Problem 1.5. Consider a system of many particles in Newtonian mechanics. Showthat it is spatially homogeneous and isotropic if and only if the density of the par-ticles is constant over space, and the relative velocity of each pair of particles i andj is related to the distance between them by the “Hubble law”

vij = H0rij ,

where H0 is independent of spatial coordinates.

The quantity H0 is usually parameterized in the following way,

H0 = h · 100km

s · Mpc, (1.7)

where h is a dimensionless parameter of order one (see (1.5)),

h = 0.705± 0.013.

We use the value h = 0.705 in numerical estimates throughout this book.

One type of objects used for measuring the Hubble parameter is Cepheids, stars ofvariable brightness whose variability is related to absolute luminosity in a known way.This relationship is measured by observing Cepheids in compact systems like MagellanicClouds. Since Cepheids in one and the same system are, to good approximation, at thesame distance from us, the ratio of their visible brightness to absolute luminosity is thesame for every star. The periods of Cepheid pulsations range from a day to tens of days,and during this period the brightness varies within an order of magnitude. The resultsof observations show that there is indeed a well-defined relation between the period andluminosity: the longer the period, the brighter the star. Hence, Cepheids serve as standardcandles.

Cepheids are giants and super-giants, so they are visible at large distances from ourGalaxy. By measuring their spectra, one finds redshift of each of them, and by mea-suring the period of pulsations one obtains the absolute luminosity and hence the distance.Using these data, one measures the Hubble constant in (1.4). Figure 1.2 shows the Hubblediagram — redshift-distance relation — obtained in this way [10].

Besides Cepheids, there are other objects used as standard candles. These include, inparticular, supernovae of type Ia.

1.2.3 Age of the Universe and size of its observable part

The Hubble parameter in fact has dimension [t−1], so the present Universe is char-acterized by the time scale

H−10 =

1h· 1100

s · Mpckm

= h−1 · 3 · 1017 s

(1.8)= h−1 · 1010 yrs ≈ 1.4 · 1010 yrs


Fig. 1.2 Hubble diagram for Cepheids [10]. Solid line shows the Hubble law with the Hubble

constant H0 = 75 km/(s ·Mpc), as determined from these observations. The dashed lines show the

uncertainty in the determination of the Hubble parameter.

and the scale of distances

H−10 = h−1 · 3000 Mpc ≈ 4.3 · 103 Mpc. (1.9)

Roughly speaking, all distances in the Universe will become twice larger in about 10billion years; galaxies at distance of order 3Gpc from us move away with velocitiescomparable to the speed of light. We will see that the time scale H−1

0 gives theorder of magnitude estimate for the age of the Universe, and the distance scale H−1

0

is roughly the size of its observable part. We will discuss the notions of the ageand size of the observable part in the course of presentation, and here we point outthat bold extrapolation of the cosmological evolution back in time (made accordingto the equations of classical General Relativity) leads to the notion of the BigBang, the moment at which the classical evolution begins. Then the age of theUniverse is the time passed from the Big Bang, and the size of the observablepart (horizon size) is the distance travelled by signals emitted at the Big Bangand moving with the speed of light. We note in passing that the actual size of ourUniverse is larger, and most probably much larger than the horizon size; the spatialsize of the Universe may be infinite in General Relativity.

Irrespective of the cosmological data, there exist observational lower bounds on theage of the Universe t0. Various independent methods give similar bounds at the level

t0 � 13 billion years = 1.3 · 1010 yrs. (1.10)

One of these methods makes use of the distribution of luminosities of white dwarfs. Whitedwarfs are compact stars of high density, whose masses are similar to the solar mass. Theyslowly cool down and get dimmer. There are white dwarfs of various luminosities in theGalaxy, but the number of them sharply drops off below a certain luminosity. This means


that there is a maximum age of white dwarfs, which, of course, should be smaller than theage of the Universe. This maximum age is found from the energy balance of a white dwarf(see, e.g., Ref. [12]). In this way the bound like (1.10) is obtained.

Other methods include the studies of the radioactive element abundances in the Earthcore, in meteorites (see, e.g., Ref. [13]), and in the metal-poor4 stars (e.g., Ref. [14]),the comparison (e.g., Ref. [15]) of the stellar evolution curve for main-sequence starson the Herzsprung–Russel diagram (luminosity-color or brightness-temperature) with theabundance of the oldest stars in metal-poor globular clusters,5 the analysis of relaxationprocesses in stellar clusters, measurement of the abundance of hot gas in clusters ofgalaxies, etc.

1.2.4 Spatial flatness

Homogeneity and isotropy of the Universe do not imply, generally speaking, that ateach moment of time the 3-dimensional space is Euclidean, i.e., that the Universe haszero spatial curvature. Besides the 3-plane (3-dimensional Euclidean space), thereare two homogeneous and isotropic spaces, 3-sphere (positive spatial curvature) and3-hyperboloid (negative curvature). A fundamental observational result of recentyears is the fact that the spatial curvature of our Universe is very small, if not exactlyzero. Our 3-dimensional space is thus Euclidean to a very good approximation. Wewill repeatedly get back to this statement, both for quantifying it and for explainingwhich observational data set bounds on the spatial curvature. We only note herethat the main source of these bounds is the study of the temperature anisotropy ofthe Cosmic Microwave Background (CMB), and that at the qualitative level, thesebounds mean that the radius of spatial curvature is much greater than the size ofthe observable part of the Universe, i.e., much greater than H−1

0 .

We note here that CMB data are also consistent with the trivial spatial topology. Ifour Universe had compact topology (e.g., topology of 3-torus) and its size were of the orderof the Hubble length, CMB temperature anisotropy would show a certain regular pattern.Such a pattern is absent in measured anisotropy, see Ref. [11].

1.2.5 “Warm” Universe

The present Universe is filled with Cosmic Microwave Background (CMB), gas ofnon-interacting photons, which was predicted by the hot Big Bang theory and dis-covered in 1964. The number density of CMB photons is about 400 per cubic cen-timeter. The energy distribution of these photons has thermal, Planckian spectrum.This is shown in Fig. 1.3 [16]. According to the analysis of Ref. [17], the present

CMB temperature is

T0 = 2.726± 0.001 K. (1.11)

4The term “metals” in astrophysics is used for all elements heavier than helium.5Globular clusters are structures of sizes of order 30 pc inside galaxies; they can contain hundreds

of thousand and even millions of stars.


10−17

10−18

10−19

10−20

10−21

10−22

101 100 1000

10 1.0 0.1Wavelength (cm)

Frequency (GHz)

FIRASDMRUBCLBL-ItalyPrincetonCyanogen

COBE satelliteCOBE satellitesounding rocketWhite Mt. & South Poleground & balloonoptical

2.726 K blackbody

I ν (

W m

−2 s

r−1 H

z−1)

Fig. 1.3 The measured CMB spectrum [16]. Dashed line shows the black-body, Planckian

spectrum.

The temperature of photons coming from different directions on celestial sphere isthe same at the level of better than 10−4 (modulo dipole component, see below);this is yet another evidence for homogeneity and isotropy of the Universe.

Still, the temperature does depend on the direction in the sky. The angularanisotropy of the CMB temperature has been measured, as shown in Fig. 1.4 [9](see Fig. 13.2 on color pages). It is of order δT/T0 ∼ 10−4 − 10−5.

We will repeatedly come back to CMB anisotropy and polarization, since, onthe one hand, they encode a lot of information about the present and early Universeand, on the other hand, they can be measured with high precision.

Let us note that the existence of CMB means that there is special reference framein our Universe: this is the frame in which the gas of photons is at rest. Solar systemmoves with respect to this frame towards Hydra constellation. The velocity of thismotion determines the dipole component of the measured CMB anisotropy [18],

δTdipole = 3.346 mK. (1.12)

Problem 1.6. Making use of the value of the dipole component, estimate thevelocity of motion of the Solar system with respect to CMB.

Problem 1.7. Estimate the seasonal modulation of the CMB anisotropy caused bythe motion of the Earth around the Sun.


Fig. 1.4 WMAP data [9]: angular anisotropy of CMB temperature, i.e., variation of the temper-

ature of photons coming from different directions in the sky, see Fig. 13.2 for color version. The

average temperature and dipole component are subtracted. The observed variation of temperature

is at the level of δT ∼ 100 μK, i.e., δT/T0 ∼ 10−4 − 10−5.

The present Universe is transparent to the CMB photons:6 their mean free pathwell exceeds the horizon size H−1

0 . This was not the case in the early Universe, whenphotons actively interacted with matter.

Problem 1.8. Greisen–Zatsepin–Kuzmin effect [20, 21]. Interaction of pho-ton with proton at sufficiently high energies may lead to the absorption of photon andcreation of π-meson. Let the cross section of the latter process in the center-of-massframe be (in fact, this is a pretty reasonable approximation for this problem)

σ ={

0 at√

s < mΔ

0.5 mb at√

s > mΔ

where√

s is the total energy of photon and proton, mΔ = 1200MeV (Δ-resonancemass), 1 mb = 10−27 cm2.

Find mean free path of a proton in the present Universe with respect to thisprocess as a function of proton energy. At what distance from the source does protonlose 2/3 of its energy? Ignore all photons (e.g., emitted by stars), except for CMB.

Since the CMB temperature T depends on the direction n on celestial sphere, it isconvenient to perform its decomposition over spherical harmonics Ylm(n). The latter

6This is not completely true in some regions of the Universe. As an example, photons scatter

off hot gas (T ∼ 10 keV) in clusters of galaxies and gain some energy. Thus, CMB is warmer in

the directions towards clusters. This is called the Sunyaev–Zeldovich effect [19]. It is small but

measurable at the current precision of observations.


Fig. 1.5 CMB temperature anisotropy measured by various instruments [8], see Fig. 13.3 for color

version. The theoretical curve is the best fit of the ΛCDM model (see Chapter 4) to the WMAP

data; this fit is in agreement with other experiments as well.

form the basis of functions on a sphere, and the decomposition is the closest analogof the Fourier decomposition. The temperature fluctuation δT in the direction n isconveniently defined as

δT (n) = T (n) − T0 − δTdipole

and its decomposition is

δT (n) =∑l,m

al,mYlm(n),

where the coefficients al,m obey a∗l,m = (−1)mal,−m, so that temperature is real. The

multipoles l correspond to fluctuations of characteristic angular size π/l. The currentmeasurements are capable of studying angular scales ranging from the largest onesto less than 0.1◦ (l ∼ 1000, see Fig. 1.5 [8] and Fig. 13.3 on color pages).

The observational data are consistent with the property that temperature fluc-tuations δT (n) are Gaussian random field, i.e., that the coefficients al,m are statis-tically independent for different l and m,

〈al,ma∗l′,m′〉 = Clmδll′δmm′ , (1.13)

where brackets mean averaging over an ensemble of Universes like ours. The coef-ficients Clm do not depend on m in isotropic Universe, Clm = Cl. They determinethe correlation of temperature fluctuations in different directions,

〈δT (n1)δT (n2)〉 =∑

l

2l + 14π

ClPl(cos θ),

where Pl are the Legendre polynomials, functions of the angle θ between the vectorsn1 and n2. In particular, the temperature fluctuation is

〈δT 2〉 =∑

l

2l + 14π

Cl ≈∫

l(l + 1)2π

Cl d ln l.

1.3. Energy Balance in the Present Universe 13

Thus, the quantity l(l+1)Cl

2π determines the contribution to the fluctuation of adecimal interval of multipoles. It is this quantity that is shown in Fig. 1.5.

It is important that the measurement of the CMB anisotropy gives not just anumber, but a large set of data, the values of Cl for different l. This set is determinedby numerous parameters of the present and early Universe, hence its measurementprovides a lot of cosmological information. Additional information comes from themeasurement of CMB polarization.

1.3 Energy Balance in the Present Universe

A dimensional estimate of the energy density in the Universe may be obtainedin the following way. Given the energy density ρ0, the density of “gravitationalcharge” is of order Gρ0. Since the dynamics of the Universe is governed by gravity,the “charge” Gρ0 must somehow be related to the present expansion rate. The“charge” has dimension of M2; the same dimension as H2

0 . This suggests that ρ0 ∼H2

0G−1 = M2Pl

H20 . Indeed, we will see that the present energy density in spatially

flat Universe is given by

ρc =38π

H20M2

P l.

With precision better than 2% this is the energy density in our Universe today.7

Numerically

ρc = 1.05 · h2 · 10−5 GeVcm3

≈ 0.52 · 10−5 GeVcm3

. (1.14)

According to the data of cosmological observations which we will discuss in duecourse, the contribution of baryons (protons, nuclei) into the total present energydensity is8 about 4.6%,

ΩB ≡ ρB

ρc= 0.046.

The contribution of relic neutrinos of all types is even smaller; cosmological boundsobtained by different methods are in the range (see Chapter 7)

Ων ≡∑

ρνi

ρc< 0.004− 0.02, (1.15)

where sum runs over the three species of neutrinos νe, νμ, ντ and anti-neutrinos νe,νμ, ντ . We stress that there is still no cosmological evidence for the neutrino mass;it is rather likely that the neutrino contribution is quite a bit smaller than the righthand side of Eq. (1.15). Other known stable particles give negligible contributionto the present total energy density. Thus, the dominating material in the presentUniverse is something unknown.

7These 2% have to do with observationally allowed effect of spatial curvature.8Note that only 10% of baryons are in stars. Most of baryons are in hot gas; this follows from

data on clusters of galaxies.


There is strongest possible evidence that this “something unknown” consists oftwo fractions, one of which is capable of clustering, and another is not. The formercomponent is called “dark matter”. Its contribution to the energy density is about20%.

We will discuss the results (Big Bang Nucleosynthesis, CMB anisotropy,structure formation) which show that dark matter cannot consist of known par-ticles. Most probably it is made of new stable particles which were non-relativisticin very distant past and remain non-relativistic today (cold, or possibly warm darkmatter). This is one of a few experimental evidences for New Physics beyond theStandard Model of particle physics. Direct detection of dark matter particles is anextremely important, and yet unsolved problem of particle physics.

According to current viewpoint, the rest of energy in the present Universe, about75%, is homogeneously spread over space. This is not matter consisting of someunknown particles, but rather unconventional form of energy of vacuum type. It iscalled by different names: dark energy, vacuum-like matter, quintessence, cosmo-logical constant, Λ-term. We will use the term “dark energy” and will use the terms“quintessence” and “cosmological constant” for dark energy with specific properties:in the case of cosmological constant the energy density does not depend on time,while for quintessence weak dependence, instead, exists.

It is not excluded that observational data which are quoted as showing thepresence of dark energy can be explained in an alternative way. One possibility isthat gravity deviates from General Relativity at cosmological distance and timescales. There is theoretical activity in the latter direction indeed, but it is out ofthe scope of this book to discuss it in any detail. We will assume throughout thatgravitational interactions are described by General Relativity.

We will further discuss dark energy and observations leading to this notion in duecourse. Here we mention the property that unlike the energy (mass) density of non-relativistic particles which decays as a−3(t) as the Universe expands, dark energydensity either does not depend on time at all, or depends on time very weakly. Hence,at some stage of the cosmological evolution dark energy starts to dominate. Thetransition from matter dominated to dark energy dominated expansion occurred inour Universe at z 0.5.

Density of baryons and dark matter in clusters of galaxies is determined by variousmethods of measurement of the gravitational potential, i.e., total mass distribution. As anexample, the left panel of Fig. 1.6 (see Fig. 13.4 on color pages) shows mass distributionin a cluster, obtained by the method of gravitational lensing [22]. The idea is that lightrays from galaxies residing behind the cluster get bent by the gravitational field of thecluster. This gives rise to multiple, distorted images of the distant galaxies9 (see the rightpanel of Fig. 1.6). Hence, this method enables one to measure the gravitational potentialin a cluster irrespectively of the sort of matter producing it. The result is that visiblematter, whose density can be determined independently, makes rather small fraction oftotal mass; most of the mass is due to dark matter. The latter is clustered; its density is

9This is called strong lensing, as opposed to weak lensing which only affects the intensity of light.


Fig. 1.6 Cluster of galaxies CL0024 + 1654 [22], see Fig. 13.4 for color version.

inhomogeneous over the Universe. Assuming that the ratio of dark and visible mass in theUniverse is the same as in clusters of galaxies,10 one finds that the mass density of baryonsand dark matter together constitutes about 25% of the total energy,

ρM ≈ 0.25ρc. (1.16)

Gravitational lensing is by no means the only way to measure the gravitationalpotential in clusters of galaxies. In particular, X-ray observations show that most of baryonsin clusters are in hot, ionized intergalactic gas. The total mass of baryons in the gas exceedsthe mass of baryons in luminous matter by an order of magnitude. As a matter of fact,X-rays are produced by electrons. So, the observations give the spatial distributions oftheir number density ne(r) and temperature T (r). Assuming spherical symmetry, oneobtains the spatial distribution of total mass density ρ(r) from the equation of hydrostaticequilibrium,

dP

dR= −μne(R)mp

GM(R)

R2, M(R) = 4π

Z R

0

ρ(r)r2dr, (1.17)

where μne(R) is the number density of baryons in the gas, and μ is determined by itschemical composition (the gas is electrically neutral, so the numbers of baryons and elec-trons are the same up to factor μ). The pressure P is mostly due to electrons, and itis related to temperature in the usual way, P (R) = ne(R)Te(R). All quantities in (1.17)except for M(R) are obtained from observations, so the mass M(R) is uniquely determined.Again, this method gives the results consistent with (1.16), see, e.g., Ref. [23].

The same conclusion follows from the study of the motion of galaxies and their groupsin clusters. Assuming that the relaxation processes for galaxies have been over, one makesuse of the virial theorem to infer the mass of a cluster,

3M〈υ2r〉 = G

M2

R. (1.18)

Here M and R are the mass and size of a cluster, and 〈υ2r〉1/2 is the dispersion of projections

of velocities on the line of sight. The latter is determined by the analysis of the Doppler

10This is not an innocent assumption, since most of galaxies are not in clusters; conversely, clusters

host only about 10% of galaxies and probably about 10% of dark matter.


Fig. 1.7 Observation of “Bullet cluster” 1E0657-558, two colliding clusters of galaxies [24], see

Fig. 13.5 for color version. Lines show gravitational equipotential surfaces, the bright regions in

the right panel are regions of hot baryon gas.

effect in the spectra of either entire cluster or individual galaxies. The masses obtainedfrom the virial theorem (1.18) by far exceed the sum of masses of individual galaxies inclusters (even including dark galactic haloes); this means that most of the mass is due todark matter which is distributed smoothly over the cluster.

It is known for a long time that nearby galaxy groups and clusters move towardsVirgo constellation. Assuming that this motion is due to the gravitational attraction bythe central cluster of galaxies, one estimates its mass. This estimate again shows that themass of galaxies is too small, so this Virgo cluster contains dark matter.

A particularly convincing argument for dark matter in clusters of galaxies comes fromthe observation of clusters just after their collision. The result [24] is shown in Fig. 1.7(see Fig. 13.5 on color pages). Bright colors in the right panel show the distribution ofhot gas whose X-ray emission has been observed by Chandra telescope. This gas containsabout 90% of all baryons in both clusters, while galaxies add the remaining 10%. Thegravitational potential is measured via gravitational lensing; the distribution of galaxiesfollows the gravitational potential. It is clear that the gravitational potential is not at allproduced by baryons; rather, its source is dark matter. Dark matter and galaxies hereare collisionless: they passed through each other and move away from each other with theoriginal speed. Hot gas loses its velocity due to collisions, and it lags behind dark matter.

Dark matter also explains the rotational velocities of stars, gas clouds, globular clustersand satellite dwarf galaxies at the periphery of galaxies. Under the assumption of circularmotion, the dependence of velocity v(R) on distance R from galactic center follows fromNewton’s law,

v(R) =

rG

M(R)

R, M(R) = 4π

Z R

0

ρ(r)r2dr,

where ρ(r) is the mass density. Observationally, v(R) = constant sufficiently far away fromthe center, see Fig. 1.8 [25], while the contribution of luminous matter to ρ would lead to

v(R) ∝ 1/√

R. This is explained by assuming that luminous matter is embedded into darkcloud of larger size — galactic halo.

Clustered dark matter could in principle consist of the usual particles, baryons andelectrons, if they were contained in dark objects like neutron stars and brown dwarfs. Theseare dim, dense objects of small sizes. To cope with observations, the latter should be presentnot only in the disc of our Galaxy but also in the halo. Similar distribution should becharacteristic to other galaxies. The density of these objects can be found observationally.Moving across the line of sight between the Earth and a star (say, from a nearby dwarf


Fig. 1.8 Rotation curve of a galaxy NGS 6503 [25]. Different curves show the contributions of

three major components of matter to the gravitational potential.

galaxy) these objects would serve as gravitational lenses. Some evidence of lensing of thistype (microlensing) has indeed been observed [26], but it is by far not as frequent asnecessary for explaining whole of dark matter in the halo [26, 27]. Furthermore, many ofcandidate compact objects are not suitable for other reasons. As an example, neutron starsare remnants of supernovae. The latter are main sources of oxygen, silicon and other heavyelements. The abundances of these elements in galaxies are well known. In this way thenumber of supernova explosions, and hence neutron stars, is estimated, and this numberis definitely too small.

There are several observational results showing the existence of dark energy. We alreadymentioned that the Universe is spatially flat to high precision: the present total energydensity coincides with the critical density ρc to better than 2%. On the other hand, theestimate for the density of clustered energy is given by (1.16), which is considerably smallerthan ρc. The rest of the energy density is attributed to dark energy.

An independent argument for dark energy is as follows. We will see that the expansionrate of the Universe now and in the past depends on its energy content. The expansionrate determines in turn the relationship between redshift and visible brightness for distant“standard candles”. This relationship has been measured by making use of supernovae Ia(SNe Ia) which have been argued to have the properties of standard candles.11 Already thefirst data [28, 29] have shown that distant supernovae are relatively dimmer than nearbyones. This is interpreted as evidence for the accelerated expansion of the Universe todayand in recent past. In General Relativity, accelerated expansion is possible only if there isdark energy whose density weakly depends on time (or does not depend on time at all).

There are several other, independent arguments based, in particular, on the estimateof the age of the Universe, structure formation, cluster abundance, CMB anisotropy. Allof them point to the existence of dark energy whose density today is at the level of 0.75ρc.

11Nearby SNe Ia show empirical relation between the absolute luminosity at maximum brightness

and the time behavior of the light emission: brighter supernovae shine longer. Assuming the same

relation for distant supernovae, their absolute luminosities are inferred from the durations of their

bursts.


The hope is that future observations will shed light on the nature and properties of thisenergy component in our Universe.

One of the candidates to dark energy is vacuum. Particle physics theories oftenignore vacuum energy, as it serves merely as a reference point for the energy, whileone is interested in masses and energies of particles — excitations about vacuum.Completely different situation occurs in General Relativity: vacuum energy, like anyother form of energy, gravitates. If gravitational fields are not extremely strong,vacuum is the same everywhere anytime, so its energy density is constant in spaceand time. Hence, vacuum energy does not cluster, so it is indeed an excellentcandidate for dark energy. The problem, though, is that the energy density hasdimension M4, and one would expect offhand that the value of vacuum energydensity would be of the order of the fourth power of the mass scale of fundamentalinteractions. These scales are 1 GeV for strong interactions, 100GeV for electroweakinteractions and MPl ∼ 1019 GeV for gravitational interactions themselves. Thus,one would estimate the corresponding contributions to vacuum energy as follows,

ρvac ∼ 1 GeV4, strong interactions

∼ 108 GeV4, electroweak interactions

∼ 1076 GeV4, gravitational interactions (1.19)

Any of these estimates exceeds by many orders of magnitude the actual dark energydensity

ρΛ ∼ ρc ∼ 10−5 GeVcm3

∼ 10−46 GeV4. (1.20)

This is a serious problem for theoretical physics which is dubbed cosmological con-stant problem: it is a mystery that the vacuum energy density is so small compared tothe estimates (1.19), and even greater mystery that it is different from zero (if darkenergy is vacuum energy indeed). Without exaggeration, this is one of the majorproblems, if not the major problem, of fundamental physics. There are many puzzlesand coincidences here, which require fine tuning of parameters of different nature.Too large vacuum energy density (but still many orders of magnitude below the par-ticle physics scales) would be incompatible with our existence: large and positivevacuum energy would lead to very fast cosmological expansion totally suppressingformation of galaxies and stars, while the Universe with large and negative vacuumenergy would recollapse long before any structure would form. A coincidence callingfor explanation is that the three different energy components — dark energy, darkmatter and baryons — are of the same order of magnitude in the present Universe(“Why now?”). These components have different origins, so a priori they would givecontributions of different orders of magnitude.

Let us stress that vacuum is by no means the only dark energy candidate dis-cussed in literature; we will consider some other candidates later on in this book.

1.4. Future of the Universe 19

1.4 Future of the Universe

The future of the Universe is mostly determined by its geometry and the propertiesof dark energy.

We will see that the contribution of the spatial curvature into effective energydensity is inversely proportional to the scale factor squared, a−2. So, if spatial cur-vature is non-zero, it will sooner or later start to dominate over energy density ofnon-relativistic matter which decays like a−3.

Hence, in the long run, the competition will be between the spatial curvatureand dark energy. If the latter is time-dependent and will relax to zero sufficientlyrapidly in future, then the expansion of Universe with positive curvature (closedmodel) will slow down, then terminate, and eventually the Universe will recollapseto singularity. The Universe with negative spatial curvature will expand forever,though its expansion will slow down. Clusters of galaxies will move away from eachother to larger and larger distances. The same would happen to galaxies not boundto clusters. All systems that are not gravitationally bound will disappear. The sameproperties hold for spatially flat Universe with dark energy relaxed to zero (but theexpansion in that case will be even slower).

If dark energy density does not depend on time, like in the case of vacuumenergy,12 or depends on time weakly, then the dominant player will be dark energy.Positive dark energy will lead to exponential expansion; the Universe will expandforever with (almost) constant acceleration.

One cannot exclude also the possibility that the dark energy will become negativein distant future. In that case the dark energy will slow the expansion down, andin the end the Universe will recollapse to singularity.

We stress that it is in principle impossible to predict the ultimate fate of theUniverse on the basis of cosmological observations only. These observations enableone, generally speaking, to figure out the dependence (or independence) of darkenergy density on time in the past, but the behavior of dark energy in the futurecan only be hypothesized. To predict the distant future of our Universe, one wouldneed to know the nature of dark energy (or, more generally, the precise reason forthe accelerated expansion of the Universe at present). Whether and how such aknowledge can be obtained is hard to tell. Nevertheless, one can extrapolate, withreasonable confidence, the future evolution of the Universe in 10–20 billion years.During this time, the Universe will expand at rate comparable to the present Hubblerate.

Yet another possibility discussed in literature is that the dark energy density will growin future. If this growth will be sufficiently fast, the Universe will end up in Big Rip:in finite time the scale factor will become infinite, interactions between particles will beinsufficient to keep them in bound states, and all bound systems, including atoms and

12We do not discuss here the possibility of the cosmological phase transition which would change

the energy balance and thus have major effect on the evolution of the Universe.


nuclei, will disintegrate and point-like particles will move away from each other to infinitedistance.

1.5 Universe in the Past

The very fact that the Universe expands clearly implies that it was denser andwarmer in the past. On the basis of General Relativity and standard thermody-namics we will see that matter had higher and higher temperature and density atearlier and earlier epochs, and that at most stages it was in thermal equilibrium.Hot Big Bang theory is precisely the theory of such a Universe. Going back in time,and, accordingly, up in temperature, we find a number of particularly important“moments” (better to say, more or less lengthy periods) in the cosmological evo-lution, see Fig. 1.9. Let us briefly discuss some of them.

1.5.1 Recombination

At relatively low temperatures the usual matter in the Universe was in the stateof neutral gas (mainly hydrogen). At earlier stage, i.e., at higher temperatures, thebinding energy was insufficient for keeping electrons in atoms, and the matter was in

inflationary

Fig. 1.9 Stages of the evolution of the Universe.

1.5. Universe in the Past 21

the state of baryon-electron-photon plasma. The temperature of recombination —the transition from plasma to gas — is determined, very crudely speaking, by thebinding energy in hydrogen atom, 13.6 eV. We will see, however, that recombinationoccurred at somewhat lower temperature, T ∼ 0.3 eV. This is a very importantepoch: before it photons actively scattered off electrons in the plasma, while afterrecombination the neutral gas was transparent to photons.13 Hence, CMB that wesee today comes from the recombination epoch, and it carries information about theproperties of the Universe at the epoch when its temperature was about 0.26 eV ≈3000K and age about 370 thousand years.

We have already pointed out that the high degree of CMB isotropy shows thatthe Universe was highly homogeneous at recombination, the density perturbationsδρ/ρ were comparable to temperature fluctuations and were roughly of order 10−5.Nevertheless, these perturbations in the end have given rise to structures: first stars,then galaxies, then clusters of galaxies.

In fact, the optical depth (scattering probability) for photons after recombination isdifferent from zero and is equal to τ � 0.06 − 0.12. The reason is the reionization in theUniverse which begins at the time of active formation of the first stars, z ∼ 10.

The fact that hydrogen in the Universe is almost completely ionized (nH/np < 10−5)at z ≤ 6, has been known for a long time from the observation of hydrogen emission linesof quasars: if the light from quasars traveled through hydrogen, it would get completelyabsorbed. Indeed, even though the light from quasars gets redshifted, there would beenough hydrogen atoms with appropriate velocities to absorb it.

Evidence for the early reionization, z ∼ 10, comes from CMB anisotropy and polar-ization. In particular, the polarization at large angular scales is due to scattering of relicphotons off free electrons at that time. The number of electrons required to explain theoptical depth τ � 0.06 − 0.12 corresponds to complete ionization of cosmic hydrogen atz � 8 − 13 (or partial ionization at somewhat earlier time).

1.5.2 Big Bang Nucleosynthesis (BBN)

Another important epoch in the cosmological evolution occurs at much higher tem-peratures, whose order of magnitude is determined, crudely speaking, by the scaleof binding energies in nuclei, i.e., 1–10MeV. Again, the actual temperatures aresomewhat smaller; the reason is discussed in Chapter 8. In any case, at high tem-peratures protons and neutrons were free in cosmic plasma, but after the Universecooled down due to expansion, neutrons have been captured into nuclei. As a result,besides hydrogen, there are light nuclei in primordial plasma: mostly helium-4 (themost tightly bound light nucleus) and also small amount of deuterium, helium-3and lithium-7; heavier elements were not synthesized in the early Universe.14 This

13In fact, three consecutive “events” happened at temperatures 0.3 − 0.2 eV: recombination —

formation of hydrogen atoms, last scattering of photons and freeze out of electrons (termination

of their integration into hydrogen atom).14Heavy elements are produced during stellar evolution. In particular, one of the important ele-

ments in the nucleosynthesis chain, carbon, is synthesized in the fusion of three 4He-nuclei. This


epoch of Big Bang Nucleosynthesis (BBN) is the earliest epoch studied directly sofar: the calculation of the light element abundances makes use of General Relativityand known microscopic physics (physics of nuclei and weak interactions), whilemeasurements of these primordial abundances, though difficult, are quite precise bynow.

Good agreement between BBN theory and observations is one of the cornerstonesof the theory of the early Universe. Let us stress that BBN epoch lasted from about1 to 300 seconds after the Big Bang, the relevant temperatures range from 1 MeVto 50 keV.

The difficulty in measuring primordial element abundances is that most of the baryonicmaterial in the Universe has been reprocessed in stars, and its composition is differentfrom that of primordial plasma. Nevertheless, it is possible to find places in the Universewhere matter can be claimed, with good confidence, to have not been reprocessed, and itscomposition coincides with the primordial composition.

1.5.3 Neutrino decoupling

While photons last scattered at temperature 0.26 eV, neutrino interactions withcosmic plasma, as we will see, terminated at temperature 2–3MeV. Before that,neutrinos were in thermal equilibrium with the rest of matter, and after that,they freely propagate through the Universe. We will calculate the temperature andnumber density of neutrinos, and here we note that they are of the same order ofmagnitude as the temperature and number density of photons. Unfortunately, directdetection of relic neutrinos is a very difficult, and maybe even unsolvable problem.

Problem 1.9. If there exist neutrinos of ultra-high energies in Nature, they mayscatter off relic neutrinos. The neutrino-neutrino cross section in the StandardModel of particle physics is very small. Its maximum value, σνν = 0.15 μb =1.5 · 10−31 cm2, is reached at center-of-mass energy

√s ≈ MZ ≈ 90GeV when neu-

trinos pair-annihilate through the resonant Z-boson production. The observation ofphotons produced in Z-decay chains would be indirect evidence for the existence ofrelic neutrinos.

Find the mean free path of ultra-high energy neutrino in the present Universewith respect to the above process. Does one expect a cutoff in the spectrum of ultra-high energy neutrinos, similar to the GZK cutoff in the spectrum of ultra-high energyprotons (see Problem 1.8)?

The role of neutrinos in the present Universe is not particularly important.However, the neutrino density in the early Universe is an important parameter

process is possible only at very high densities reached in stellar interiors after hydrogen has been

burned out. All other elements are synthesized from carbon. Relatively light elements, including

iron, are produced in thermonuclear reactions in stars; heavier elements are synthesized as a result

of neutron capture in stars and supernova explosions, and some elements are produced, presumably,

in the processes of proton or positron capture.

1.5. Universe in the Past 23

of BBN theory. Primordial nucleosynthesis occurs in expanding Universe, and theneutrino component affects the expansion rate and hence cooling rate of plasmaat the time of BBN. The latter is important for inequilibrium processes of thelight element synthesis. The success of BBN theory gives decisive evidence for theexistence of relic neutrinos.

Relic neutrinos play some role in structure formation; they affect CMB angularspectrum too. In this and the accompanying book, we will study relic neutrinos andtheir effects in some details.

1.5.4 Cosmological phase transitions

Going further back in time, we come to the epochs which have not been directlyprobed by observations so far. Hence, we have to make more or less reasonableextrapolations. It is likely that the history of the hot Universe goes back to temper-atures of the order of hundreds GeV and quite possibly to even higher temperatures.At so high temperatures there are epochs of interest, at least from theoretical view-point. Some of them can be loosely called epochs of phase transitions.

— Transition15 from quark-gluon matter to hadronic matter. Its temperatureis determined by the energy scale of strong interactions and is about 200 MeV. Atmuch higher temperatures quarks and gluons behave as individual particles (ratherstrongly interacting towards the transition epoch), while at lower temperatures theyare confined in colorless bound states, hadrons. At the same time (or almost thesame time) there was the transition associated with chiral symmetry breaking.

— Electroweak transition [30–33]. Simplifying the situation we can say that attemperatures above 100 GeV (energy scale of electroweak interactions), the Higgscondensate is absent, and W - and Z-bosons have zero masses. The present phasewith broken electroweak symmetry, Higgs condensate and massive W - and Z-bosonsis the result of the electroweak transition16 that occurred at temperature of order100GeV.

— Grand Unified transition. There are hints towards Grand Unification, thehypothesis that at energies and temperatures above 1016 GeV there is no distinctionbetween strong, weak and electromagnetic interactions: these interactions are unifiedinto a single force. If so, and if these temperatures existed in the Universe, then attemperature of Grand Unification TGUT ∼ 1016 GeV there was the correspondingphase transition. We note, however, that maximum temperature in the Universe maywell be below TGUT , so the phase of Grand Unification may not exist in the earlyUniverse. This is the case in many models of inflation: there, the reheat temperatureis lower than TGUT .

15This is likely a smooth crossover rather than phase transition proper.16In fact, the situation is somewhat more complicated: an order parameter is absent in electroweak

theory (at least within the Standard Model of particle physics), so the phase transition can be

absent. Indeed, given the experimental bound on the Higgs boson mass, there is smooth crossover

in the Standard Model instead of the phase transition.


1.5.5 Generation of baryon asymmetry

The present Universe contains baryons (protons, neutrons) and practically noantibaryons. Quantitative measure of the baryon abundance is the ratio of baryonand photon number densities. The studies of BBN and CMB give

ηB ≡ nB

nγ= 6.2 · 10−10 (1.21)

with precision of about 3%. The baryon number is conserved at sufficiently lowenergies and temperatures, and we will see that in the early Universe nB/nγ is ofthe same order of magnitude as given in (1.21). Thus, baryon asymmetry ηB is oneof the most important parameters of cosmology.

At temperatures of hundreds MeV and higher, there were a lot of quarks andantiquarks in the cosmic plasma, which were continuously pair-created and anni-hilated. Thus, unlike in the present Universe, there were almost as many particleswith negative baryon number (antiquarks) as those with positive baryon number(quarks). Simple thermodynamical arguments, to be given in this book, show thatthe number of quark-antiquark pairs at high temperatures is about the same as thenumber of photons, so the baryon asymmetry can be understood as determiningthe following ratio

nq − nq

nq + nq∼ ηB ∼ 10−10. (1.22)

Here nq and nq are number densities of quarks and antiquarks, respectively. We seethat in the early Universe, there existed one uncompensated quark per 10 billionof quark-antiquark pairs. It is this tiny excess that is responsible for the existenceof baryonic matter in the present Universe: as the Universe expanded and cooleddown, antiquarks annihilated with quarks, while uncompensated quarks remainedthere and in the end formed protons and neutrons.

One of the problems of cosmology is to explain the very existence of the baryonasymmetry [34, 35], as well as to understand its value (1.21). It is extremely implau-sible that the small excess of quarks over antiquarks (1.22) existed in the Universefrom the very beginning, i.e., that it is one of the initial data of the cosmologicalevolution; it is much more reasonable to think that the Universe “in the beginning”was baryon-symmetric. The same conclusion comes from inflationary theory. Asym-metry (1.22) was generated in the course of the cosmological evolution due to pro-cesses with baryon number non-conservation. We will discuss possible mechanismsof generation of this asymmetry, but we stress right away that today there is nounique answer to the question of its origin. Here we note only that baryon asym-metry was generated most probably at very high temperatures, at least 100GeV andmaybe much higher, although its generation at lower temperatures is not completelyexcluded.

The problem of the baryon asymmetry cannot be solved within the StandardModel of particle physics. This is another cosmological hint towards New Physicsbeyond the Standard Model.

1.6. Structure Formation in the Universe 25

1.5.6 Generation of dark matter

Which particles make non-baryonic clustered dark matter is not known experimen-tally. One expects that these are stable or almost stable particles that do not existin the Standard Model of particle physics. Hence, the very existence of dark matteris a very strong argument for incompleteness of the Standard Model. This makes thedetection and experimental study of the dark matter particle extremely interestingand important. On the other hand, the lack of experimental information on the prop-erties of these particles makes it impossible to give a unique answer to the question ofthe mechanism of the dark matter generation in the early Universe. We will discussvarious dark matter particle candidates in this book, and here we make one remark.We will see that hypothetical stable particles of mass in GeV–TeV range, whoseannihilation cross section is comparable to weak cross sections do not completelyannihilate in the course of the cosmological evolution, and that their resulting massdensity in the present Universe is naturally of the order of the critical density ρc.Therefore, these particles are natural dark matter candidates, especially becausethey exist in some extensions of the Standard Model, including Minimal Super-symmetric Standard Model and its generalizations. Particles which we have brieflydescribed are called WIMPs (weakly interacting massive particles). Their freeze-out, i.e., termination of annihilation, occurred at temperature somewhat below theirmass, i.e., T ∼ 1 − 100GeV.

Of course, there are many other dark matter particle candidates besides WIMPs.These include axion, gravitino, sterile neutrino, etc. Some of them are discussed inthis book.

1.6 Structure Formation in the Universe

We discussed in previous Sections the most important stages of the cosmologicalevolution. Each of them, be it nucleosynthesis or recombination, has finite timeduration. There is, however, a process in the Universe that began at a very earlyepoch and continues at present. This process is formation of structures — firststars, galaxies, clusters of galaxies, superclusters. The order we put them here isnot random: smaller objects get formed earlier.

The theory describing structure formation is based on the Jeans instability,the gravitational instability of matter density perturbations. It should be assumed,of course, that the perturbations have already existed at the very early stage ofthe cosmological evolution, even though they were very small in amplitude. TheHot Big Bang theory cannot explain the very existence of the primordial pertur-bations, let alone predict their properties. Generation of primordial perturbationsneeds additional mechanisms, the most plausible of which exists in inflationarytheory. Amazingly, the inflationary mechanism is consistent with all cosmologicaldata.


The source of primordial perturbations is not particularly relevant to the theoryof structure formation. Rapid growth of density perturbations occurs at that stageof the cosmological expansion when the dominating energy component is mass ofnon-relativistic matter. Transition to this stage happened 80 thousand years afterBig Bang; before that the Universe was so hot that the dominant componentwas relativistic (“radiation”). The epoch of this transition is called radiation-matter equality. At that time the density perturbations had small amplitude,δρ/ρ ∼ 10−3−10−5. Regions of higher density are sources of gravitational potential,they attract surrounding matter, and the density in these regions becomes evenhigher. This is precisely the physical reason for gravitational instability.17 Once theoverdensity is large enough, the overdense region becomes gravitationally boundand starts living its own life; in particular, the size of this region does not growdespite the expansion of the Universe. Instead, gravitational interaction within thisregion leads to its collapse to an object of much smaller size. In this way protostarsand protogalaxies get formed. First stars were formed at z ∼ 10 and somewhatearlier, and first galaxies somewhat later.

The mass of an object — galaxy, cluster of galaxies — is determined by thesize of the primordial overdense region. So, the number densities of galaxies andclusters and their mass distribution reflect the spectrum of primordial perturbations.Existing observational data on structures are consistent with the simplest “flat”primordial spectrum, called Harrison–Zeldovich spectrum. The defining property ofthe latter is its scale invariance: in a certain sense, perturbations of different sizeshave one and the same amplitude.

Perturbations existing in cosmic medium at recombination give rise to CMBtemperature anisotropy and polarization. Hence, the primordial spectrum can bedetermined also from the CMB observations. Notably, the spectra found from CMBand structures are in good agreement with each other.

Structure formation gives yet another argument for the presence of dark matter:without dark matter, density perturbations would start to grow after recombinationonly, and by now they would not have developed into structures yet. Furthermore,it follows from the theory of structure formation that the major part of dark mattermust be cold (or warm), i.e., it must consist of particles that became non-relativisticat a very early epoch. If a large fraction of dark matter were hot, i.e., consistedof particles remaining relativistic until late times, then the formation of objectsof relatively small size would be suppressed, in contradiction to observations. Animportant example of hot dark matter is neutrino of mass mν ∼ 1 − 10−3 eV. Thissuggests that cosmology is capable of setting bounds on neutrino masses.

We discuss in the accompanying book the structure formation and the role playedin it by different components of cosmic matter.

17This mechanism does not work for relativistic particles, since weak gravitational field cannot

keep them inside an overdense region.

1.7. Inflationary Epoch 27

1.7 Inflationary Epoch

The Hot Big Bang theory has its problems. Part of them is due to the fact that thistheory needs very special initial conditions, otherwise it would grossly fail to describethe early and present Universe. We discuss the problem of initial conditions in theaccompanying book, and here we give one example showing what sort of problemswe are talking about.

As the Universe is “warm”, it can be characterized by its entropy. Entropydensity is of order of photon number density; in the present Universe

s ∼ 103 cm−3.

Thus, the estimate for the entropy in the observable part of the Universe, whosesize is R0 ∼ 104 Mpc ∼ 1028 cm, is

S ∼ sR30 ∼ 1088.

This huge dimensionless number is one of the properties of our Universe. Why doesthe Universe have such a large entropy? The Hot Big Bang theory does not answerthis question, since entropy is (almost) conserved during the hot stage. The hugeentropy has to be introduced into the Hot Big Bang theory “by hands”, as one ofthe initial data. This uncomfortable situation is called entropy problem. There areseveral problems of this sort in the Hot Big Bang theory; at qualitative level all ofthem boil down to the fact that this theory cannot explain why our Universe is solarge, warm, spatially flat, homogeneous and isotropic.

Another problem of the Hot Big Bang theory has to do with primordial pertur-bations. At the hot stage, the Universe was not completely homogeneous, densityperturbations in it were of order δρ/ρ ∼ 10−5. As we mentioned, the Hot Big Bangtheory does not contain a mechanism for their generation, the perturbations mustalso be introduced “by hands”, as initial data.

Both of these classes of problems find elegant solutions in inflationary theory.According to this theory, the hot cosmological epoch was preceded by the epochof exponential expansion (inflation). During the inflationary epoch, initially smallregion of the Universe (whose spatial size was comparable, say, to the Planck lengthlPl) inflated to very large size, typically many order of magnitudes larger than thesize of the part of the Universe we see today. This in the end explains flatness,homogeneity and isotropy of the observable part of the Universe. Due to the expo-nential character of expansion, the duration of the inflationary epoch may be short:the first class of problems of the Hot Big Bang theory is solved provided that theduration is greater than (50 − 70)H−1

infl where Hinfl is the Hubble parameter atinflation, say Hinfl ∼ 10−6MPl (Hinfl may be quite a bit smaller). In that case theminimum duration of inflation is of order 108tPl ∼ 10−35 s. It is likely that inflationlasted much longer, but in any case it is plausible that we deal with microscopictime scale.


The inflationary regime occurs if the energy density in the Universe depends ontime very weakly. Energy density of conventional matter — gases of particles —does not have this property. Therefore, models of inflation make use of hypotheticalnew field(s).18 Under certain conditions this new field — inflaton — is spatiallyhomogeneous and changes slowly in time in inflating region. Its potential energychanges slowly too, and this gives rise to the inflationary expansion regime.

At some moment of time the conditions for inflationary expansion get violated,and inflation terminates. A new epoch — post-inflationary reheating — sets in, atwhich the inflaton energy is tranferred to the energy of conventional matter. As aresult, the Universe heats up to very high temperature, and the Hot Big Bang epochbegins. Reheating occurs with entropy generation, which provides the solution tothe entropy problem.

Inflationary theory was originally proposed as a solution to the first class ofproblems, but it soon became clear that it also solves the problem of primordialperturbations. The original source of perturbations are vacuum fluctuations ofquantum field(s), in the simplest version, vacuum fluctuations of the inflatonfield itself. These fluctuations get strongly enhanced at inflationary stage due tostrong time-dependence of the gravitational field in the Universe. At the end ofinflation they are reprocessed into density perturbations of conventional matter. Theamplitude of primordial perturbations depends on unknown parameters of a model,but their spectrum (dependence on wavelength) is uniquely calculable within agiven model of inflation. Notably, most models predict almost flat (almost Harrison–Zeldovich) spectrum, in gross agreement with observational data. However, a typicalprediction is a slight tilt in the spectrum, which can be discovered by cosmologicalobservations, and, in fact, some models of inflation have already been ruled out.

Another prediction of inflationary models is the existence of primordial gravi-tational waves. They are also generated at inflationary stage from vacuum fluctua-tions, now of the gravitational field. In some models the amplitudes of gravity wavesof wavelengths comparable to the present Hubble size are of order 10−5−10−6. Thesegravity waves would affect CMB temperature anisotropy and polarization. Theseeffects have not been discovered yet, but they may be observable in future. Discoveryof primordial gravity waves will not only be a strong argument in favor of inflation,but also will enable us to determine its fundamental parameter, inflationary Hubblescale Hinfl.

Presently, inflationary theory is well developed. We study various aspects of thistheory in the accompanying book.

To end our preview we note that it misses some of the topics which we study inthis book. We hope, nevertheless, that this preview makes the content of the bookreasonably clear.

18Another possibility is to have extra terms in the action for gravitational field. This case is often

equivalent to the introduction of new field(s).

Chapter 2

Homogeneous Isotropic Universe

In this book we use the basic notations and equations of General Relativity, seeAppendix A. The notations and conventions are summarized in Sec. A.11.

2.1 Homogeneous Isotropic Spaces

To a very good approximation, our Universe is homogeneous and isotropic at suf-ficiently large scales. This means that at given moment of time, the geometry ofspace is the geometry of homogeneous and isotropic manifold. There are only threesuch manifolds1 (up to overall scale): 3-dimensional sphere, 3-dimensional Euclideanspace (3-plane) and 3-dimensional hyperboloid. The geometry of 3-dimensionalsphere is best understood by imagining that it is embedded into (fictitious)4-dimensional Euclidean space and writing equation of 3-sphere in the standardform,

(y1)2 + (y2)2 + (y3)2 + (y4)2 = R2,

where ya (a = 1, . . . , 4) are coordinates of the 4-dimensional Euclidean space andR is the radius of the 3-sphere. Let us introduce three angles χ, θ and φ, so that

y1 = R cosχ,

y2 = R sinχ cos θ,(2.1)

y3 = R sinχ sin θ cosφ,

y4 = R sinχ sin θ sin φ.

Then the distance between two points on the 3-sphere with coordinates (χ, θ, φ)and (χ + dχ, θ + dθ, φ + dφ) is

dl2 = (dy1)2 + (dy2)2 + (dy3)2 + (dy4)2

= R2{[d(cosχ)]2 + [d(sin χ cos θ)]2 + [d(sin χ sin θ cosφ)]2 (2.2)

+ [d(sin χ sin θ sin φ)]2}.1It is important that the 3-dimensional metric has Euclidean signature, i.e., it can locally be cast

into the standard form dl2 = (dx1)2 + (dx2)2 + (dx3)2.

29

30 Homogeneous Isotropic Universe

After simple calculation we find that the metric of the 3-sphere has the followingform,

3-sphere: dl2 = R2[dχ2 + sin2 χ(dθ2 + sin2 θdφ2)]. (2.3)

Note that no trace of the fictitious 4-dimensional Euclidean space is left in thisformula, as should be the case.

In analogy to 3-sphere, 3-dimensional hyperboloid is conveniently described byits embedding into fictitious 4-dimensional Minkowski space with metric

ds2 = −(dy1)2 + (dy2)2 + (dy3)2 + (dy4)2,

while the equation for the hyperboloid is

(y1)2 − (y2)2 − (y3)2 − (y4)2 = R2. (2.4)

We are interested in the one connected component y1 > 0.

Problem 2.1. Show that the hyperboloid is indeed a homogeneous and isotropicspace. Hint: Begin with defining what precisely is a homogeneous and isotropic space.

Coordinates on 3-hyperboloid can be introduced in analogy to (2.1):

y1 = R coshχ,

y2 = R sinhχ cos θ,

y3 = R sinhχ sin θ cosφ,

y4 = R sinhχ sin θ sin φ.

The calculation of the distance between two points on the hyperboloid is analogousto (2.2). It gives

3-hyperboloid: dl2 = R2[dχ2 + sinh2χ(dθ2 + sin2 θdφ2)]. (2.5)

For completeness, let us also write the metric of 3-plane (3-dimensional Euclideanspace)

3-plane: dl2 = (dx1)2 + (dx2)2 + (dx3)2. (2.6)

One of the properties of homogeneous and isotropic spaces is that all covariantgeometrical quantities are expressed through the metric tensor γij and, possibly,tensors δj

i and Eijk , the latter existing in any Riemannian space, see Appendix A(here i, j = 1, 2, 3; we denote the metric tensor of 3-dimensional space by γij todistinguish it from the metric tensor gμν of 4-dimensional space-time). Furthermore,the coefficients do not depend on coordinates. In particular, the Riemann tensor isequal to

(3)Rijkl =κ

R2(γikγjl − γilγjk), (2.7)

2.1. Homogeneous Isotropic Spaces 31

where we introduced the parameter κ = 0,±1, which distinguishes 3-plane, 3-sphereand 3-hyperboloid,

κ =

⎧⎪⎨⎪⎩

+1, 3-sphere

0, 3-plane

−1, 3-hyperboloid

(2.8)

It follows from (2.7) that the Ricci tensor is(3)Rij = 2

κ

R2γij , (2.9)

and the curvature scalar is constant in space and equal to 6κR−2.

Problem 2.2. Obtain the relations (2.7) and (2.9) by direct calculation.

Metrics of 3-sphere, 3-hyperboloid and 3-plane can be written in a unified form.To this end, let us first note that by introducing the coordinate ρ = Rχ on thesphere and hyperboloid, and spherical coordinates (ρ, θ, φ) on 3-plane, the metrics(2.3), (2.5) and (2.6) can be written as follows,

dl2 = dρ2 + r2(ρ)(dθ2 + sin2 θdφ2), (2.10)

where

r(ρ) =

⎧⎪⎨⎪⎩

R sin (ρ/R), 3-sphere

ρ, 3-plane

R sinh(ρ/R), 3-hyperboloid

(2.11)

The interpretation of quantities entering (2.10) is obvious: ρ is the geodesic(shortest) distance from the origin to a point with coordinates (ρ, θ, φ), whiler(ρ) determines the area of two-dimensional sphere at distance ρ from the origin,S = 4πr2(ρ). It is also clear that an interval of length l at distance ρ from the originis seen from the origin at angle

Δθ =l

r(ρ).

Instead of ρ, one can choose r as the radial coordinate. With this choice we obtain,e.g., for hyperboloid,

dρ2 =dr2

cosh2( ρR )

=dr2

1 + r2

R2

,

so that the metrics of the three spaces take the following form,

dl2 =dr2

1 − κr2

R2

+ r2(dθ2 + sin2 θdφ2). (2.12)

where the parameter κ is the same as in (2.8). Note that the coordinates (r, θ, φ)cover only half of the 3-sphere: the region 0 ≤ r < R is part of the 3-sphere extendingfrom the origin (pole) to the 2-dimensional surface of maximum area (equator ofthe 3-sphere). This is the reason for the coordinate singularity in the metric (2.12)at r = R for κ = 1.


2.2 Friedmann–Lemaıtre–Robertson–Walker Metric

Expanding homogeneous isotropic Universe is described by the metric

ds2 = dt2 − a2(t)γijdxidxj , (2.13)

where γij(x) is the metric of unit 3-sphere (metric (2.3) with R = 1), unit3-hyperboloid or 3-plane. Metric (2.13) is called the Friedmann–Lemaıtre–Robertson–Walker metric (FLRW). One distinguishes closed Universe (the space is3-sphere, κ = +1), open and flat Universe (the space is 3-hyperboloid and 3-plane,κ = −1 and κ = 0, respectively). For closed and open Universe, the scale factora(t) at given moment of time is the radius of spatial curvature. On the other hand,for spatially flat Universe the scale factor itself does not have physical significance,since at a particular moment of time it may be set equal to any number (say, unity)by rescaling the spatial coordinates. What has the physical meaning in the flat Uni-verse is the ratio of scale factors at different times, a(t1)/a(t2), and, in particular,the Hubble parameter

H(t) =a(t)a(t)

.

Note that in the cases of closed and open Universe the spatial coordinates xi

entering (2.13) are dimensionless, while the scale factor has dimension of length.On the other hand, in the case of spatially flat Universe it is convenient to assignthe dimension of length to the coordinates xi, while treating the scale factor asdimensionless quantity.

The metric of homogeneous and isotropic Universe has the form (2.13) in acertain reference frame. This frame is singled out by the fact that space is the sameeverywhere at each moment of time. Furthermore, this frame is comoving: worldlines of particles which are at rest with respect to this frame are geodesic, i.e., theseparticles are free. Before showing that, let us point out that for these particles onehas ds2 = dt2, i.e., the time coordinate t has the meaning of proper time of particlesat rest. In the present Universe, these particles may be thought of as galaxies, if onedisregards their local (peculiar) motion caused by gravitational potentials (whichare due to, e.g., nearby galaxies).

Let us show that the world line xi = const obeys the geodesic equation inmetric (2.13),

duμ

ds+ Γμ

νλuνuλ = 0, (2.14)

where uμ is 4-velocity (see Appendix A). To this end, let us calculate the Christoffelsymbols,

Γμνλ =

12gμσ(∂νgλσ + ∂λgνσ − ∂σgνλ). (2.15)

Non-zero components of the metric tensor are

g00 = 1, gij = −a2(t)γij(x),

2.2. Friedmann–Lemaıtre–Robertson–Walker Metric 33

while for the inverse tensor we have

g00 = 1, gij = − 1a2(t)

γij(x).

It is clear that

Γ000 = 0, Γ0

0i = 0, Γi00 = 0

(in the expression in parenthesis in (2.15), at least two indices are equal to zero,but g0i = 0, ∂μg00 = 0). For Γi

0j we have

Γi0j =

12gik∂0gjk =

a

aδij. (2.16)

The remaining Christoffel symbols are also straightforwardly calculated,

Γ0ij = aaγij , (2.17)

Γijk = (3)Γi

jk, (2.18)

where (3)Γijk are the Christoffel symbols for metric γij .

Let us now turn to Eq. (2.14). The only non-vanishing component of the4-velocity uμ = dxμ/ds of a particle at rest is

u0 =dx0

ds=

dt

dt= 1.

Equation (2.14) is obviously satisfied, since duμ/ds = 0 and Γμ00 = 0 for any μ.

Thus, the world lines of particles which are at rest in our reference frame are indeedgeodesic.

Problem 2.3. If one chooses a time coordinate which does not coincide with propertime of particles at rest, FLRW metric takes the form

ds2 = N2(t)dt2 − a2(t)γijdxidxj .

Show by direct calculation that in this metric too, world lines of particles at restare geodesic (this is of course obvious, since these lines are the same lines as in thetext).

To end this Section, we note that for both closed and open models, the spatialcurvature can often be neglected, so that one can use the spatially flat metric

γij = δij . (2.19)

This is certainly possible for processes at spatial scales much smaller than the cur-vature radius a(t). We already mentioned in Chapter 1 that our Universe is spa-tially flat to a very good approximation, both in the past and at present. Hence,the approximation (2.19) is actually very good for all scales. We will quantify thisstatement in what follows.


2.3 Redshift. Hubble Law

The scale factor a(t) grows in time, and the distances between points of fixed spatialcoordinates xi grow too — the Universe expands. Because of that, the wavelengthof a photon emitted long ago by a distant source increases as the photon movestowards an observer, i.e., the photon wavelength gets redshifted. To descibe thisphenomenon, let us write the action for free electromagnetic field in space-timewith metric gμν(x):

S = −14

∫d4x

√−ggμνgλρFμλFνρ, (2.20)

where, as usual,

Fμν = ∇μAν −∇νAμ = ∂μAν − ∂νAμ,

and Aμ is the electromagnetic vector-potential. The action (2.20) is the simplestcovariant generalization of the action of Maxwell’s electrodynamics; the factor

√−g

ensures the invariance of 4-volume (see Appendix A). One could in principle add tothe action (2.20) other invariant terms vanishing in the Minkowski limit, like

δS =α

M2Pl

∫d4x

√−ggμνRλρFμλFνρ (2.21)

where Rλρ is the Ricci tensor, α is a dimensionless constant, and the factor M−2Pl

is included on dimensional grounds. However, terms like (2.21) are negligibly smallcompared to (2.20) if the space-time curvature is small compared to the Planckvalue |Rμν | � M2

Pl, and the constant α is not extremely large. Therefore, at allclassical stages of the evolution of the Universe, when its parameters are far fromPlanckian, freely propagating photons are described by the action (2.20).

Consider now the propagation of a photon in the homogeneous isotropic Uni-verse. Realistically, the photon wavelength is small compared to the spatial cur-vature radius, even if the Universe is open or closed. Therefore, the Universe canbe considered spatially flat, and one can use the metric

ds2 = dt2 − a2(t)δijdxidxj . (2.22)

It is convenient to use conformal time η instead of cosmic time t. The former isdefined by

dt = adη, (2.23)

i.e.,

η =∫

dt

a(t).

In terms of this new time coordinate the metric is

ds2 = a2(η)[dη2 − δijdxidxj ]. (2.24)

2.3. Redshift. Hubble Law 35

In other words,

gμν = a2(η)ημν , (2.25)

where ημν is the Minkowski metric. The metric (2.25) differs from the Minkowskimetric by overall time-dependent rescaling, i.e., the metric of homogeneous isotropicUniverse has conformally flat form in coordinates (η, xi). In these coordinates onehas

gμν = a−2ημν ,√−g = a4.

By substituting these expressions into (2.20) we obtain that in coordinates (η, xi)the action of electromagnetic field coincides with

S = −14

∫d4x ημνηλρFμλFνρ. (2.26)

This property is inherent in a theory of massless vector field; in theories of otherfields, the action in conformal coordinates (η, xi) in general does not reduce to theflat space-time action. In this regard, free electromagnetic fields is called conformal,while other fields, generally speaking, are not conformal.

It follows from (2.26) that solutions to equations for the free electromagneticfield in the Universe with metric (2.22) (or, equivalently, with metric (2.24)) aresuperpositions of plane waves

A(α)μ = e(α)

μ eikη−ikx,

where k is a constant vector (called coordinate momentum, or conformal momen-tum), k = |k|, and e

(α)μ are the standard polarization vectors of a photon, α = 1, 2.

We stress that k is not the physical momentum of a photon, and k is not the physicalfrequency, since dx and dη are not physical distance and physical time interval. Thequantity Δx = 2π/k is the coordinate wavelength of a photon, while the physicalwavelength at time t according to (2.22) is

λ(t) = a(t)Δx = 2πa(t)k

. (2.27)

Similarly, Δη = 2π/k is the period of electromagnetic wave in conformal time, whileaccording to (2.23), the period in physical time2 t is

T = a(t)Δη = 2πa(t)k

. (2.28)

Hence, the physical momentum p and physical frequency of a photon at time t are

p(t) =k

a(t), ω(t) =

k

a(t). (2.29)

In expanding Universe, the scale factor a(t) increases in time, the physical wave-length (2.27) grows, while the physical momentum and frequency decrease: they get

2Note that we assume here that the period T is much smaller than the time scale of the evolution

of the scale factor. This is of course true at all cosmological epochs of interest.


redshifted. If a photon was emitted at time ti with given wavelength λi (e.g., inthe process of transition of hydrogen atom from excited to ground state), then itswavelength at the Earth is equal to

λ0 = λia0

a(ti)≡ λi[1 + z(ti)]. (2.30)

As usual, the index 0 refers to the present value of the relevant quantity.The quantity

z(t) =a0

a(t)− 1 (2.31)

is called redshift. The further is the object emitting photons, the longer thesephotons travel through the Universe to the Earth, the smaller is a(ti): objects atlarger distances have larger redshifts. The redshift is a directly measurable quantity;its measurement boils down to the indentification of an emission (or absorption)lines and determination of their actual wavelengths, see Chapter 1.

Let us stress that formulas (2.30) and (2.31) are general; they are valid at all z.For not so distant objects, the difference (t0 − ti) — the photon travel time —

is not very large, and we can expand in (t0 − ti),

a(ti) = a0 − a(t0)(t0 − ti)

In terms of the present value of the Hubble parameter H0 = H(t0) one obtains

a(ti) = a0[1 − H0(t0 − ti)].

Hence, to the linear order in (t0 − ti) one has

z(ti) = H0(t0 − ti).

Finally, the travel time is equal to the distance to the emitter, up to corrections oforder (t0 − ti)2,

r = t0 − ti.

In this way one obtains the Hubble law,

z = H0r, z � 1. (2.32)

When deriving it, we assumed that (t0 − ti) is not large; this corresponds to smallredshift, as we indicated in Eq. (2.32).

The Hubble parameter H0 is one of the fundamental cosmological parameters ofthe present Universe. We have already discussed its importance in Chapter 1. Herewe recall that the measured value is given by (1.5); for the definition of the relatedparameter h see (1.7), while the associated time and distance scales are presentedin (1.9) and (1.9).

To end this Section, let us make the following comment. Our derivation ofEqs. (2.27)–(2.29) and, accordingly, Eq. (2.30) was based on the fact that elec-tromagnetic field is conformal, i.e., in coordinates (η, xi) its action reduces to the

2.3. Redshift. Hubble Law 37

action in Minkowski space-time. However, these formulas are of general character;they are valid for all massless particles.

Consider as an example a massless scalar field with action

S =12

∫d4x

√−ggμν∂μφ∂νφ. (2.33)

In conformal coordinates (η, xi), the explicit form of this action in space-time withmetric (2.24) is

S =12

∫d3xdη a2(η)ημν∂μφ∂νφ (2.34)

The scalar field with action (2.33) is not conformal: the action (2.34) does notreduce to the flat space-time action by any change of variables. By varying (2.34)with respect to φ we obtain the equation for the scalar field in metric (2.24),

1a2

∂η(a2∂ηφ) − ∂i∂iφ = 0. (2.35)

(summation over i = 1, 2, 3 is assumed; ∂i∂i is Laplacian in flat 3-dimensionalspace). In the first place, let us notice that the operator in the left hand side of thisequation — covariant d’Alembertian — does not depend on xi, so the solution canbe written as a linear combination of 3-dimensional plane waves,

φ =1

a(η)f(η)e−ikx.

The factor a−1(η) is introduced for convenience; due to this factor, Eq. (2.35)becomes the equation for f(η) without first-order time derivative,

∂2ηf − ∂2

ηa

af + k2f = 0. (2.36)

If the expansion rate of the Universe and its time derivative are small compared tothe frequency, the second term in Eq. (2.36) can be neglected, and the solutions tothe scalar field equation are superpositions of plane waves

φ =1

a(η)eikη−ikx. (2.37)

Coordinate frequency and momentum, k and k, are again independent of time,which again leads to Eqs. (2.27)–(2.29), as promised. Note that the factor a−1(η) inthe solution (2.37) compensates for the factor a2(η) in the Lagrangian in (2.33); inother words, the action for the field f(x, η) = a(η)φ(x, η) reduces to the flat space-time action of massless scalar field up to corrections containing time derivatives ofthe scale factor.

Problem 2.4. Give quantitative formulation of conditions under which the secondterm in Eq. (2.36) is small compared to the third term. Express these conditions interms of the physical frequency, Hubble parameter and its derivative with respect tophysical time t.


Problem 2.5. Consider photons and massless scalar particles (actions (2.20) and(2.33), respectively). Show that the relation (2.31) between redshift and scale factorremains valid in open and closed Universes, provided that the wavelength λ(t) issmall compared to the radius of spatial curvature a(t) at all times between emissionand absorption.

2.4 Slowing Down of Relative Motion

Physical momenta of massive free particles also decrease as

p =k

a(t), (2.38)

where k is time-independent coordinate momentum. To see this, let us consider thegeodesic equation

duμ

ds+ Γμ

νλuνuλ = 0, (2.39)

for 4-velocity of a free particle,

uμ =dxμ

ds

(see Appendix A). Let us note right away that ui are not the physical values ofspatial components of 4-velocity: since the physical distances dX i are related tocoordinate distances by dX i = a(t)dxi, the physical components are

U i =dX i

ds= a(t)ui.

The physical momenta are expressed through the physical velocities in the usual way,

pi = mU i,

while the usual 3-velocities

vi =dXi

dt

are related to U i by

U i =vi

√1 − v2

. (2.40)

Indeed, the 4-velocities uμ obey the relation gμνuμuν = 1. In metric (2.25) the latterrelation is

(u0)2 − a2uiui = 1

(summation over i is assumed), or(dt

ds

)2

− (U i)2

= 1.

2.4. Slowing Down of Relative Motion 39

Therefore,

vi =dX i

dt=

dXi

ds

ds

dt=

U i

√1 + U2

,

which gives precisely Eq. (2.40).Let us come back to the geodesic equation (2.39). For metric (2.22), the only non-

vanishing components of connection are given by (2.16) and (2.17). Thus, spatialcomponents of Eq. (2.39) (μ = i = 1, 2, 3) read

dui

ds+ Γi

0ju0uj + Γi

j0uju0 = 0,

or

dui

ds+ 2

a

a

dt

dsui = 0.

In terms of the physical components the latter equation is

dU i

dt= − a

aU i.

Hence, velocities of free particles decrease in time as

U i =consta(t)

,

i.e., the law (2.38) is indeed valid.Thus, the velocities of free particles with respect to comoving frame decrease in

expanding Universe; particles gradually “freeze in”. In particular, if at early cosmo-logical stages massive particles were relativistic, they become non-relativistic at laterstages. This behavior is characteristic of neutrino, if its mass is well above 10−4 eV.

To conclude this Section, let us give another derivation of Eq. (2.38). Thisderivation has to do with solutions to massive field equations in expanding Uni-verse. Consider as an example the theory of massive scalar field with action

S =∫

d4x√−g

(12gμν∂μφ∂νφ − m2

2φ2

).

In metric (2.24) this action has the form

S =∫

d3xdη

(12a2ημν∂μφ∂νφ − m2a4

2φ2

).

Equation obtained from this action by varying with respect to φ is the Klein–Gordonequation in expanding Universe (in conformal coordinates)

1a2

∂η(a2∂ηφ) − ∂i∂iφ + m2a2φ = 0. (2.41)


This equation again does not contain spatial coordinates explicitly, so its solutionsare again superpositions of plane waves

φ =1

a(η)f(η)e−ikx,

where k is independent of time. The physical wavelength of a particle is thus givenby (2.27), so its physical momentum indeed redshifts according to (2.38).

Problem 2.6. Show that for slowly varying scale factor, solutions to Eq. (2.41) aresuperpositions of

φ =1

a(η)√

Ω(η)ei

R η Ω(η)dηe−ikx · (1 + O(∂ηa))

where Ω(η) =√

k2 + m2a2(η). Thus, coordinate frequency (derivative of theexponent with respect to conformal time) equals Ω(η), while the physical frequency is

ω(η) =Ω(η)a(η)

=√

p2 + m2,

as should be the case in relativistic physics.

2.5 Gases of Free Particles in Expanding Universe

Let us consider the behavior of gases of stable non-interacting particles in a homo-geneous isotropic Universe. We will discuss local properties like number density,so we again neglect the spatial curvature and use metric (2.22). The gas of par-ticles is characterized by the distribution function f(X,p), so that f(X,p)d3Xd3pis the number of particles in physical volume element d3X in interval of physicalmomenta d3p. The distribution in general depends on time and is not one of theequilibrium distribution functions. We will consider homogeneous gases, whose dis-tribution functions are independent of coordinates and depend only on momenta(and time).

We have seen in previous Sections that coordinate momenta of free particles kare independent of time. The coordinate volume d3x is also constant in time, so thedistribution function written in terms of coordinate momentum is time-independent,

f(k) = const.

The number of particles in an element of comoving phase space is also independentof time,

f(k)d3xd3k = const.

The comoving phase space volume coincides with the physical one,

d3xd3k = d3(ax)d3

(ka

)= d3Xd3p.

2.5. Gases of Free Particles in Expanding Universe 41

Hence, the time dependence of the distribution function of free particles is entirelydetermined by the redshift of momenta,

f(p, t) = f(k) = f [a(t) · p].

If the distribution function is known at some moment of time and equal to fi(p),then at later times it is equal to

f(p, t) = fi

(a(t)ai

p)

. (2.42)

We stress that, generally speaking, this formula is valid for free particles only.We will see that the known particles, as well as (most likely) dark matter par-

ticles actively interacted with each other in the early Universe, so that they werein thermal equilibrium. At some moment of time (different moment for differenttypes of particles) the Universe expanded to the extent that the density and tem-perature became sufficiently small, and interactions between particles switched off.At that moment the particles had thermal distribution function, and after that thedistribution function evolved as given by3 (2.42).

Let us consider in more details two limiting cases. We begin with massless par-ticles; the most interesting among them are photons. At the time ti when theirinteraction with matter switches off, they have Planckian distribution function. Thelatter depends only on the ratio of momentum and temperature at that time, |p|

Ti:

fi(p) = fPl

( |p|Ti

)=

1(2π)3

1e|p|/Ti − 1

. (2.43)

For massless fermions this would be the Fermi–Dirac distribution at zero mass (seeChapter 5), which also depends only on the ratio |p|

Ti. According to (2.42), the

distribution function at later times is

f(p, t) = fPl

(a(t)|p|aiTi

)= fPl

( |p|Teff (t)

),

where

Teff (t) =ai

a(t)Ti. (2.44)

Hence, the distribution function always has the equilibrium shape despite the factthat photons are not in thermal equilibrium. It follows from (2.44) that relic photonshave Planckian spectrum with effective temperature decreasing in time according to

Teff (t) ∝ 1a(t)

. (2.45)

3This statement is not exact for various reasons. One is that there are gravitational potentials

in our Universe which are produced by galaxies, clusters of galaxies, etc. This fact is to good

approximation irrelevant for relic photons, but it is very relevant for massive particles. The latter

obeyed Eq. (2.42) at sufficiently early cosmological stages only, when the inhomogeneities in the

Universe were small.


We will see in what follows that the same behavior (with reservations) is character-istic also of the distribution function of photons in the early Universe, when photonsare in thermal equilibrium with matter.

It goes without saying that the relation (2.45) is valid for other masslessparticles, including fermions (if there are massless particles in Nature besidesphotons; gravitons is entirely different story). Importantly, the effective temper-ature decreases according to (2.45) from the time at which these particles get out ofthermal equilibrium with cosmic medium (time of decoupling). At later times theireffective temperature may be different from the photon temperature. We encounterthis situation in Chapter 7. If particles have small but non-zero mass m and at thetime of decoupling were relativistic, their spectrum remains thermal, and effectivetemperature falls as a−1(t) as long as Teff � m. At later times the spectrum is nolonger thermal. Indeed, the distribution function is always Planckian, i.e., it has theform (2.43) with temperature Teff (t) ∝ a−1(t), while at Teff � m the equilibriumdistribution is the Maxwell–Boltzmann distribution. Relic particles of this sort arecalled “hot dark matter”; it is hot in the sense that the particles decouple beingrelativistic and hence they always have massless distribution function in momenta.4

An important example of hot dark matter is relic neutrino.

Problem 2.7. Consider particles of mass m that decouple at T � m. Show thattheir distribution in momenta is given by (2.43) at all times afterwards, includinglate times when Teff � m.

Let us now consider another limiting case of particles which decouple beingnon-relativistic. At the time of decoupling they have the Maxwell–Boltzmann dis-tribution function (see Chapter 5)

f(p) =1

(2π)3exp

(−m − μi

Ti

)exp

(− p2

2mTi

),

where Ti and μi are the temperature and chemical potential at that time. Accordingto (2.42), the distribution function at later times is equal to

f(p) =1

(2π)3exp

(−m − μi

Ti

)exp

(− a2(t)p2

2ma2i Ti

).

It can again be written in the Maxwell–Boltzmann form,

f(p, t) =1

(2π)3exp

(−m − μeff

Teff

)exp

(− p2

2mTeff

), (2.46)

where

Teff (t) =(

ai

a(t)

)2

Ti,

4The same property holds for “warm” dark matter; the distinction between hot and warm dark

matter is discussed in Sec. 9.1.

2.5. Gases of Free Particles in Expanding Universe 43

and effective chemical potentail μeff is determined by the relation

m − μeff (t)Teff

=m − μi

Ti.

Equation (2.46) means that the distribution function still has the equilibrium form5

(although particles are no longer in thermal equilibrium), and the effective temper-ature changes as

Teff (t) ∝ 1a2(t)

.

It decreases faster than in the case of massless particles. Dark matter particles whichwere non-relativistic at their decoupling are called cold dark matter. As we pointedout in Chapter 1, cold (or possibly warm) dark matter makes about 20% of totalenergy density today.

5See footnote 3 in this Chapter.

Chapter 3

Dynamics of Cosmological Expansion

3.1 Friedmann Equation

The law of the cosmological expansion, i.e., the dependence of the scale factor a ontime, is determined by the Einstein equations (see Appendix A),

Rμν − 12gμνR = 8πGTμν .

Let us find their explicit form for homogeneous and isotropic metric (2.13). Webegin with the calculation of the Ricci tensor,

Rμν = ∂λΓλμν − ∂μΓλ

νλ + ΓλμνΓσ

λσ − ΓλμσΓσ

λν . (3.1)

The non-vanishing components of the Christoffel symbols are given by (2.16)–(2.18).These formulas imply, in particular, that

Γμ0μ =

a

aδii = 3

a

a, Γμ

iμ=(3)Γjij .

Let us first calculate R00. Since Γμ00 = 0, the only contributions come from the

second and fourth terms in (3.1), and we obtain

R00 = −∂0Γλ0λ − Γλ

0σΓσ0λ = −∂0Γλ

0λ − Γi0jΓ

j0i

= −∂0

(3a

a

)−(

a

a

)2

δijδ

ji = −3∂0

(a

a

)− 3

(a

a

)2

.

Finally,

R00 = −3a

a. (3.2)

Let us now turn to the mixed components R0i. Keeping only non-vanishingChristoffel symbols in (3.1), we write

R0i = ∂jΓj0i − ∂0Γλ

iλ + Γj0iΓ

λjλ − Γk

0jΓjik. (3.3)

This expression in fact is equal to zero, since Γj0i are independent of spatial coordi-

nates, Γλiλ=(3)Γj

ij are calculated with static metric γij and hence are independent

45

46 Dynamics of Cosmological Expansion

of time and Γj0k ∝ δj

k, which leads to the cancellation of the last two terms in (3.3).Hence,

R0i = 0. (3.4)

This should have been expected, since R0i transform as components of a 3-vectorunder spatial rotations, and there is no special direction in isotropic space.

Let us finally calculate the spatial components Rij . We again keep non-vanishingChristoffel symbols in (3.1) and write

Rij = (∂0Γ0ij + ∂kΓk

ij) − ∂iΓλjλ

+ (Γ0ijΓ

σ0σ + Γk

ijΓσkσ) − (Γ0

ikΓkj0 + Γk

i0Γ0jk + Γk

ilΓljk), (3.5)

where we collected in parentheses the terms that come from each of the four termsin (3.1). Taking into account (2.18), we combine into the Ricci tensor (3)Rij thoseterms in (3.5) which contain spatial derivatives only; the tensor (3)Rij is calculatedwith the 3-dimensional metric γij . Four other terms are calculated directly, and weobtain

Rij = ∂0(aa)γij + aaγij · 3 a

a− aaγik · a

aδkj − a

aδki aaγjk + (3)Rij .

Finally,

Rij = (aa + 2a2 + 2κ)γij , (3.6)

where we made use of relation (2.9).Let us now use Eqs. (3.2), (3.4) and (3.6), to find the scalar curvature,

R = gμνRμν = g00R00 + gijRij = R00 − 1a2

γijRij .

Since γijγij = 3, we have

R = −6(

a

a+

a2

a2+

κ

a2

).

As a result, the 00-component of the left hand side of the Einstein equations takesthe simple form,

R00 − 12g00R = 3

(a2

a2+

κ

a2

). (3.7)

The other components of the Einstein tensor Gμν ≡ Rμν − 12gμνR are calculated in

a similar way.Let us now turn to the right hand side of the Einstein equations. At cosmological

epochs of interest to us here, matter in the Universe can be described macroscop-ically: it can be considered as homogeneous fluid with energy density ρ(t) andpressure p(t). This fluid as a whole is at rest with respect to comoving referenceframe, so the only non-zero component of its 4-velocity is u0. From the relationgμνuμuν = 1 we have

u0 = 1, u0 = 1.

3.1. Friedmann Equation 47

Hence, the 00-component of energy-momentum tensor is (see Appendix A)

T00 = (p + ρ)u0u0 − g00p = ρ. (3.8)

Combining (3.7) and (3.8) we conclude that the 00-component of the Einstein equa-tions has the following form for homogeneous and isotropic Universe,(

a

a

)2

=8π

3Gρ − κ

a2. (3.9)

This is called Friedmann equation; it relates the rate of the cosmological expansion(the Hubble parameter H = a/a) to the total energy density ρ and spatial curvature.

The Friedmann equation has to be supplemented with one more equation, sinceEq. (3.9) contains two unknown functions of time, a(t) and ρ(t). To obtain theadditional equation, it is convenient to consider the covariant conservation of theenergy-momentum tensor (see Appendix A)

∇μT μν = 0.

Setting ν = 0, we write

∇μT μ0 ≡ ∂μT μ0 + ΓμμσT σ0 + Γ0

μσT μσ = 0. (3.10)

Non-vanishing components of the energy-momentum tensor in the comovingframe are

T 00 = g00g00T00 = ρ, (3.11)

T ij = gikgjlTkl = gikgjl(−gklp) =1a2

γijp, (3.12)

Here we made use of the fact that the spatial components of 4-velocity are zero incomoving frame, so that Tij = (p + ρ)uiuj − pgij = −pgij . Note that γij entering(3.12) are time-independent components of the matrix inverse to γij . We againuse the expressions for non-zero Christoffel symbols, Eqs. (2.16)–(2.18), and writeEq. (3.10) in the following explicit form,

ρ + 3a

a(ρ + p) = 0. (3.13)

To close the system of equations determining the dynamics of homogeneous andisotropic Universe, one has to specify the equation of state of matter,

p = p(ρ). (3.14)

The latter equation is not a consequence of equations of General Relativity. Theequation of state is determined by matter content in the Universe. For non-relativistic particles one has p = 0, while p = 1

3ρ for relativistic particles andp = −ρ for vacuum, see Sec. 3.2.

Problem 3.1. Find the equation of state for the scalar field with Lagrangian

L = −V0

√1 − ∂μφ∂μφ, V0 > 0,

assuming that the field φ is classical and spatially homogeneous (depends on timeonly).


Equations (3.9), (3.13) and (3.14) completely determine dynamics of the cosmo-logical expansion. Let us make two remarks concerning Eqs. (3.13) and (3.14). First,if there exist several types of matter in the Universe which do not interact with eachother, then the energy-momentum tensor of each type of matter obeys the covariantconservation equation independently. Therefore, Eqs. (3.13) and (3.14) must be sat-isfied by every type of matter separately. On the other hand, the energy density ρ inthe Friedmann equation (3.9) is the sum of energy densities of all types of matter.Second, if matter in the Universe is in thermal equilibrium at zero chemical poten-tials, then Eq. (3.13) has simple interpretation. It can be written in the followingform,

dρ

p + ρ= −3d(ln a). (3.15)

The left hand side of this equation coincides with d(ln s) where s is the entropydensity, see Chapter 5. Hence, Eq. (3.13) reduces to the relation

sa3 = const,

which means the entropy conservation in comoving volume. In other words, due tothe expansion of the Universe, entropy density decreases as the inverse element ofthe comoving volume, s ∝ a−3. We see in Sec. 5.2 that the same interpretationholds for systems at non-vanishing chemical potentials too.

To conclude this Section, let us make two comments. First, we note that whenobtaining the equations governing the dynamics of the cosmological evolution wemade use of only one Einstein equation, R00 − 1

2g00R = 8πGT00, and only oneof the covariant conservation equations, ∇μT μ0 = 0. One can show that otherEinstein equations and covariant conservation equations are identically satisfied forsolutions to Eqs. (3.9) and (3.13). Nevertheless, let us write for future referencethe equation that follows from the ij-components of the Einstein equations. It isclear from (3.6) that ij-components of the Einstein tensor are proportional to γij .Furthermore, the ij-components of the energy-momentum tensor Tij = −p gij arealso proportional to γij . So, all ij-components of the Einstein equations are reducedto single Raychaudhuri equation,

2a

a+

a2

a2= −8πGp − κ

a2. (3.16)

We do not use this equation in this book; it can be viewed as a consequence of theFriedmann equation (3.9) and covariant conservation of energy.

The second comment is that the assumption that matter filling the Universe isideal fluid is in fact unnecessary for deriving Eqs. (3.9), (3.13) as well as Eq. (3.16).The cosmological model we discuss is self-consistent if matter is homogeneous andisotropic. The isotropy means that the energy-momentum tensor necessarily hasthe diagonal structure, T00 = ρ, Tij = −p gij , where the parameters ρ and p areenergy density and effective pressure. Homogeneity requires that these parameters

3.2. Sample Cosmological Solutions. Age of the Universe. Cosmological Horizon. 49

are functions of time only, ρ = ρ(t), p = p(t). With this understanding, the analysisof this Section applies to the general case.

Problem 3.2. Prove that equations

R0i − 12g0iR = 8πGT0i, Rij − 1

2gijR = 8πGTij , ∇μT μi = 0

are identically satisfied in the case of homogeneous and isotropic Universe, providedthat the following equations are satisfied,

R00 − 12g00R = 8πGT00, ∇μT μ0 = 0.

Do not assume in the proof that the Universe is filled with ideal fluid; it is importantonly that matter is homogeneous and isotropic.

3.2 Sample Cosmological Solutions. Age of the Universe.Cosmological Horizon.

Before discussing the realistic model of our Universe, let us present several examplesof cosmological solutions. In this Section we mostly consider a spatially flat model,

κ = 0.

This is a very good approximation to the real Universe; we will see in what followsthat the term κ/a2 in the Friedmann equation (3.9), if any, is small compared tothe first term in the right hand side at both present and early epochs.

In the spatially flat model, the Friedmann equation takes the following form,(a

a

)2

=8π

3Gρ. (3.17)

Simple solutions described in this Section are obtained for the Universe filled withone type of matter. Then Eqs. (3.13), (3.14) (or, equivalently, (3.14) and (3.15))determine the energy density as a function of the scale factor, ρ = ρ(a), and thenthe dependence of the scale factor on time is found from Eq. (3.17). Let us recall (seeSec. 2.2) that in the case of spatially flat Universe, physically meaningful quantity isthe ratio of scale factors taken at different times, rather than the scale factor itself.Therefore, one expects that a solution a(t) will contain an arbitrary multiplicativeconstant. Also, Eqs. (3.13) and (3.17) are invariant under time translation, so asolution will contain another arbitrary constant, “the beginning of time”.

3.2.1 Non-relativistic matter (“dust”)

We begin with the model of the Universe filled with non-relativistic matter whoseequation of state is

p = 0.


We obtain from Eq. (3.15)

ρ =consta3

. (3.18)

Everywhere in this Section const denotes an arbitrary positive constant (which isgenerally different in different formulas). Relation (3.18) has simple interpretation:number density of particles n decreases as the comoving volume increases, anda3n = const, meaning the conservation of the total number of particles. Since theenergy density is given by ρ = mn, where m is mass of a particle, the energy densitybehaves in the same way, i.e., ρa3 = const.

With account of (3.18), Eq. (3.17) takes the following form,(a

a

)2

=consta3

and has the solution

a(t) = const · (t − ts)2/3, (3.19)

where ts is an arbitrary constant. The Universe expands, and its expansion decel-erates, a < 0. The energy density is

ρ(t) =const

(t − ts)2. (3.20)

The solution (3.19), (3.20) is singular at t = ts: at that moment the scale factor isequal to zero (all distances are vanishingly small), and the energy density is infinite.This is an example of the cosmological singularity, the moment of Big Bang. Wewill see that many other cosmological solutions start from the singularity. Clearly,it does not make sense to extrapolate the classical evolution back to the Big Bangsingularity. What we learn is that it is quite possible that the Universe started off itsevolution from a state in which the energy density was very high (say, comparableto the Planck energy density ρPl ∼ M4

Pl ∼ 1076 GeV4), and for which classical lawsof physics were not applicable.

In what follows we will count time from the cosmological singularity (if a solutionhas this singularity). For the solution (3.19), (3.20) this means that

ts = 0.

Then t is the age of the Universe. Making use of (3.19) we relate it to the Hubbleparameter,

H(t) =a

a(t) =

23t

. (3.21)

Making use of Eq. (3.17) once again, we find

ρ =3

8πGH2 =

16πG

1t2

. (3.22)

Equalities (3.21) and (3.22) relate the physical quantities; they do not contain arbi-trary parameters.


If most of the evolution of our Universe occurred in the regime of domination ofnon-relativistic matter, the age of the Universe would be given by (3.21), i.e.,

t0 =2

3H0. (3.23)

Making use of (1.9), we would obtain

t0 = h−1 · 0.65 · 1010 yrs = 0.93 · 1010 yrs (h = 0.7). (3.24)

This value, even with an account of the uncertainty in the determination of H0

(see (1.5)), would contradict independent bounds on the age of the Universe, t0 ≥1.3 · 1010 yrs, which we mentioned in Chapter 1. We will see that the situationbecomes comfortable if the expansion of the Universe today is determined mostlyby dark energy.

Let us make use of the solution (3.19) to introduce another important notion,the cosmological horizon. Imagine signals emitted at the moment of Big Bang andsince then traveling with the speed of light. We are interested in the distance lH(t)that such a signal travels from the point of its emission by the time t. The physicalmeaning of lH(t) is that it is equal to the size of the region (in infinite Universe!)causally connected by the time t: an observer living at the time t cannot know inprinciple what has happened outside the sphere of radius lH(t). This sphere is calledthe cosmological horizon, and lH(t) is the size of the observable part of the Universeat the time t. Clearly lH increases in time; the horizon opens up.

Let us point out that another term for the cosmological horizon introduced hereis “particle horizon”. This is to be distinguished from “event horizon” which isdiscussed in Sec. 3.2.3.

For calculating lH(t), it is convenient to make use of conformal time η, seeSec. 2.3. Light-like geodesics obeying ds2 = 0 in metric (2.24) are described by theequation

|dx| = dη.

Therefore, the coordinate size of the horizon at time t is equal to η(t), and itsphysical size is

lH(t) = a(t)η(t) = a(t)∫ t

0

dt′

a(t′)(3.25)

For solution (3.19) we have

lH(t) = 3t =2

H(t). (3.26)

If our Universe were matter dominated, the size of the horizon today would beequal to

lH,0 =2

H0.


We find numerically from (1.9) that

lH,0 = h−1 · 6000 Mpc

= 0.85 · 104 Mpc = 2.6 · 1028 cm (h = 0.705). (3.27)

Yet another property of the horizon is the subject of the following problem.

Problem 3.3. Show that signals emitted at the distance lH(t) come to an observerat time t with infinite redshift.

To conclude, in models with the cosmological horizon, the region in the Universewhich is observable in principle, has a finite size even if the Universe itself is infinite.

3.2.2 Relativistic matter (“radiation”)

If the energy density in the Universe is due to relativistic matter, the equation ofstate is (see Chapter 5)

p =13ρ.

In this case Eq. (3.15) gives

ρ =consta4

(3.28)

This behavior differs from (3.18) due to the fact that the expansion of the Universeleads not only to the dilution of the number density of particles, n ∝ a−3, but alsoto redshift of the energy of each particle, ω ∝ a−1 (the latter properties are validfor both non-interacting relativistic particles, cf. (2.29) and particles in thermalequilibrium, see Sec. 5.1).

Equation (3.17) becomes (a

a

)2

=consta4

and has the solution

a(t) = const · t1/2

(we again set the parameter ts equal to zero). The properties of this solution areanalogous to those of the solution (3.19), (3.20): the Universe undergoes deceleratedexpansion; the moment t = 0 is the moment of the Big Bang singularity; the age isinversely proportional to the Hubble parameter (cf. (3.21))

H ≡ a

a=

12t

,

and the energy density is inversely proportional to the age squared (cf. (3.22)),

ρ =3

8πGH2 =

332πG

1t2

.


The horizon size is again finite,

lH = a(t)∫ t

0

dt′

a(t′)= 2t =

1H(t)

. (3.29)

It is useful to relate the expansion rate (the Hubble parameter) to temperature,assuming thermal equilibrium between all types of particles and neglecting chemicalpotentials. At temperature T the energy density is (see Chapter 5)

ρ =π2

30g∗T 4, (3.30)

where

g∗ =∑

b

gb +78

∑f

gf

is the effective number of degrees of freedom. Here summation runs over bosonic(b) and fermionic (f) particle species of masses smaller than T , gb and gf are thenumbers of spin degrees of freedom of the boson b and fermion f . Using G = M−2

Pl

(see Sec. 1.1) we write the relation (3.17) in the following form,

H =T 2

M∗Pl

, (3.31)

where

M∗Pl =

√90

8π3g∗MPl =

11.66

√g∗

MPl (3.32)

is the reduced Planck mass. We will repeatedly use the relation (3.31) keeping inmind that the parameter M ∗

Pl depends on the effective number of degrees of freedomg∗ and hence on temperature (since a particle of mass m contributes to g∗ only atT � m). This dependence, however, is rather weak, and when discussing the earlyUniverse at certain stages of its evolution one can often treat M∗

Pl as a constant.By comparing (3.28) with (3.30) we observe that the temperature of relativistic

matter in thermal equilibrium is inversely proportional to the scale factor (up toslightly varying factor depending on g∗),

T (t) ≈ consta(t)

. (3.33)

Let us recall that the same relation (which in that case is exact) holds for theeffective temperature of a gas of non-interacting relativistic particles, see Sec. 2.5.Finally, it is useful to note that Eqs. (3.31) and (3.33) imply

T

T≈ −H = − T 2

M∗Pl

. (3.34)

The latter two relations, (3.33) and (3.34), are exact at those periods of the cosmo-logical evolution in which the effective number of degrees of freedom g∗ does notchange.


3.2.3 Vacuum

In flat space-time, vacuum is the same in all inertial reference frames. It may havenon-zero energy density, and the Lorentz-invariance dictates the form of its energy-momentum tensor,

Tμν = ρvacημν . (3.35)

The vacuum energy density is equal to T00 = ρvac, while pressure, defined by Tij =−pηij , is

p = −ρvac.

Thus, vacuum has an exotic equation of state p = −ρ; for positive energy density,the vacuum pressure is negative.

In the curved space-time of not very high curvature, the expression (3.35)remains valid in any locally-Lorentz frame, and in an arbitrary frame one has

Tμν = ρvacgμν . (3.36)

Here ρvac is a constant in space and time; in principle it should be calculable ina complete theory of elementary particles and their interactions. Up to now, nocompelling calculation of the vacuum energy density has been made, and this is oneof the major problems of fundamental physics.

Time-independent ρvac is consistent with Eq. (3.13), which for p = −ρ givesρ = 0. This is in fact obvious: Eq. (3.13) is a consequence of the covariant con-servation of the energy-momentum tensor, while the tensor (3.36) obeys covariantconservation law for ρvac = const due to the fact that ∇μgλρ = 0 (see Appendix A).

One can look at the right hand side of the Einstein equations with Tμν = const ·gμν from slightly different prospective. General covariance allows, to add to theEinstein–Hilbert action SG of General Relativity, the term

SΛ = −Λ∫ √−gd4x.

By varying the action (SG + SΛ) with respect to metric one obtains, in the absenceof matter, the following equations (see Appendix A),

Rμν − 12gμνR − 8πGΛgμν = 0.

They are exactly the same as the Einstein equations with energy-momentumtensor (3.36), with identification Λ = ρvac. Historically this route was the first,and the parameter Λ is often called cosmological constant for historical reasons.The difference between the cosmological constant and vacuum energy density ispurely philological, at least at the current level of understanding of this issue.

The solution to the Friedmann equation (3.17) with ρ = const = ρvac is

a = const · eHdSt, (3.37)


where the Hubble parameter

HdS =

√8π

3Gρvac

is independent of time. The space-time with metric

ds2 = dt2 − e2HdStdx2 (3.38)

is called de Sitter space.1 It is the space-time of constant curvature.

Problem 3.4. Show that de Sitter metric obeys (cf. (2.7))

Rμνλρ = −H2dS(gμλgνρ − gμρgνλ).

Unlike in previous examples of cosmological solutions, the Universe undergoesaccelerated expansion, a > 0. Furthermore, de Sitter space does not have the initialsingularity: even though a(t) tends to zero as t → −∞, the metric can be madenon-singular at t → −∞ by a coordinate transformation.

Problem 3.5. Consider a fictitious flat five-dimensional space with metric

ds2 = (dy0)2 − (dy1)2 − (dy2)2 − (dy3)2 − (dy4)2.

Consider a hyperboloid embedded into this space according to equation

(y0)2 − (y1)2 − (y2)2 − (y3)2 − (y4)2 = −H−2 = const.

It is clear that this hyperboloid does not have singularities. Let us choose the coor-dinates (t, xi), i = 1, 2, 3 on this hyperboloid, such that

y0 = −H−1sinhHt − H

2x2eHt,

yi = xieHt,

y4 = H−1coshHt − H

2x2eHt.

(3.39)

Show that with this choice of coordinates, the metric induced on the hyperboloidfrom the five-dimensional space coincides with (3.38). What part of the hyperboloidis covered by the coordinates (t,x)?

In the case of de Sitter space, the cosmological (particle) horizon is absent. The“beginning of time” is shifted to t = −∞. So, instead of Eq. (3.25) we have for thehorizon size,

lH(t) = a(t)∫ t

−∞

dt′

a(t′).

1The coordinates (t, x) cover only half of the de Sitter space, see [36] and the problem in this

Section.


We obtain from (3.37) that

lH(t) = eHdSt

∫ t

−∞dt′e−HdSt′ = ∞,

which precisely means that the particle horizon is absent.For spaces like de Sitter, another notion of horizon is introduced. It is quite

different from the particle horizon discussed in Sec. 3.2.1. Namely, let there be anobserver at the point x = 0 at the moment of time t. Let us ask what is the size ofthe region from which signals emitted at that moment of time will ever reach theobserver (which will stay at the point x = 0) in arbitrary distant future. Since thetime-like geodesics obey |dx| = dη, the coordinate size of this region is

η(t → ∞) − η(t) =∫ ∞

t

dt′

a(t′),

and its physical size at time t equals

ldS = a(t)∫ ∞

t

dt′

a(t′)=

1HdS

. (3.40)

The observer will never know about events that happen at a given moment of timeat distances exceeding ldS = H−1

dS ; this is the meaning of the de Sitter horizon. It isalso called event horizon.

Problem 3.6. Show that the solutions of Secs. 3.2.1 and 3.2.2 do not have eventhorizons.

3.2.4 General barotropic equation of state p = wρ

Let us continue with the brief discussion of a model with the matter equation ofstate

p = wρ,

where w is a constant larger than −1. Non-relativistic and relativistic matter cor-respond to w = 0 and w = 1/3, respectively. Models with negative (and, generally,time-dependent) w attract considerable attention in recent years; matter with thiseffective equation of state has different names: quintessence, time-dependent Λ-term,etc.

At w > −1 the solution to Eq. (3.13) is

ρ =const

a3(1+w). (3.41)

We find from Eq. (3.17) that

a = const · tα,

where

α =23

11 + w

.

3.3. Solutions with Recollapse 57

The parameter α is positive; there is the cosmological singularity at t = 0. Theenergy density behaves as

ρ =const

t2;

it tends to infinity as t → 0. Since

a = const · α(α − 1)tα−2,

the expansion of the Universe decelerates (a < 0) for α < 1 and accelerates atα > 1. In terms of the equation-of-state parameter w we have

(a) w > −13

: deceleration

(b) w < −13

: acceleration

Note that in the open Universe dominated by negative spatial curvature, theeffective energy density is equal to ρ = const/a2 (see Eq. (3.9) with κ = −1).This would mean that w = −1/3 (see (3.41)). In that case the expansion wouldneither accelerate nor decelerate, a = 0.

The cases (a) and (b) differ also in the following respect. The models from theclass (a) have cosmological (particle) horizon and do not have event horizon, whilethe situation is reverse for models from the class (b). Indeed, the particle horizonexists if the integral ∫ t

0

dt′

a(t′)

converges (see (3.25)). For α < 1 (i.e., w > − 13 ) this integral converges indeed, while

for α > 1 (i.e., w < − 13 ) the integral diverges at the lower limit of integration. In the

latter case the particle horizon gets shifted to spatial infinity. The existence of eventhorizon is determined by the convergence properties of the integral (see (3.40))∫ ∞

t

dt′

a(t′).

It diverges at the upper limit of integration for α < 1 (event horizon is absent) andconverges for α > 1 (event horizon exists).

Problem 3.7. Is it possible in expanding Universe with equation of state p = p(ρ)to evolve from the regime (p+ ρ) > 0 to the regime (p+ ρ) < 0 without violating thereal-valuedness of the sound speed us defined as u2

s = ∂p/∂ρ?

3.3 Solutions with Recollapse

For completeness, we briefly discuss in this Section the homogeneous and isotropicsolutions, in which expansion of the Universe is followed by contraction (recollapse).This situation occurs if the right hand side of the Friedmann equation (3.9) contains


both positive and negative terms, and if positive terms decrease with growing scalefactor faster than the absolute values of negative terms. Examples of possibly inter-esting negative contributions are the contribution of curvature in the closed model(κ = +1) and the dark energy contribution. Regarding the latter, we know that itis positive at the present epoch, but we cannot exclude that it depends on time andwill become negative in distant future.

As an example, let us consider the closed cosmological model with non-relativistic matter. Making use of (3.18) we write the Friedmann equation(

a

a

)2

=am

a3− 1

a2, (3.42)

where the parameter am is determined by the total mass of matter in the Universe.At a � am the Universe expands in the same way as in the spatially flat case(Sec. 3.2.1). The expansion terminates when

a = am;

at that time the right hand side of (3.42) vanishes.

Problem 3.8. Find the relation between am and the total mass in the closedUniverse. What would be the maximum size of the Universe having 1 kg of non-relativistic matter?

The explicit solution has a simple form in terms of the conformal time η definedby dt = adη (see (2.23)). The Friedmann Eq. (3.42) then takes the form

1a4

(da

dη

)2

=am

a3− 1

a2

and its solution is

a = am sin2 η

2. (3.43)

The expansion begins from singularity at η = 0, the Universe has maximum size atη = π, and at η = 2π it collapses back to the singularity. The relation between thephysical and conformal time is

t =∫

a(η)dη =am

2(η − sin η). (3.44)

Thus, the total lifetime and the maximum size are related by ttot = π · am.

Problem 3.9. Show that at a � am the solution (3.43), (3.44) indeed coincideswith the spatially flat solution (3.19).

Similar situation occurs in the case when the expansion of the Universe ter-minates due to negative Λ-term. The Universe exists for finite time between itsemergence from and recollapse to the singularity.

Problem 3.10. Find the law of evolution a = a(t) of the spatially flat Uni-verse (κ = 0) with negative, time-independent cosmological constant, assuming that

3.3. Solutions with Recollapse 59

matter in the Universe has equation of state p = 0. Find the total lifetime. Hint:Make use of the Friedmann equation in physical time.

Problem 3.11. Consider the Universe filled with matter whose equation of state isthat of Chaplygin gas [37],

p = −A

ρ. (3.45)

(1) Find the dependence of the Hubble parameter on the scale factor.(2) Find the law of evolution a = a(t) at small and large scale factors in all three

cases, κ = 0,±1.(3) Find the complete evolution a = a(t) in the case of spatially flat Universe.(4) What values of κ admit static solutions to the Einstein equations?(5) What can be said about the future of the Universe, if it is known that at some

moment of time the expansion of the Universe accelerates? Consider all threecases, κ = 0,±1.

(6) Consider a theory of a scalar field with action

Sφ =∫

d4x√−g

[12gμν∂μφ∂νφ − V (φ)

].

In the case of spatially flat Universe, find the scalar potential V (φ) for which thespatially homogeneous solution describes the same evolution as in the case (3), andthe relation between the energy density and pressure has the form (3.45).

Chapter 4

ΛCDM: Cosmological Modelwith Dark Matter and Dark Energy

4.1 Composition of the Present Universe

Cosmological solutions studied in Sec. 3.2 are not realistic. Energy density in thepresent Universe is due to non-relativistic matter (baryons and dark matter, as wellas those neutrino species whose mass is considerably larger than 5 K ∼10−3 eV),relativistic matter (photons and light neutrino of mass less than 10−4 eV, if sucha neutrino exists, see Appendix C) and dark energy. In general, the Universe mayhave non-vanishing spatial curvature. Therefore, all these contributions have to beincluded into the right hand side of the Friedmann equation (3.9), and this equationtakes the form

H2 ≡(

a

a

)2

=8π

3G(ρM + ρrad + ρΛ + ρcurv), (4.1)

where ρM , ρrad, ρΛ are energy densities of non-relativistic matter, relativistic matter(“radiation”) and dark energy, respectively, and by definition

8π

3Gρcurv = − κ

a2(4.2)

is the contribution due to spatial curvature. Let us introduce the critical densityρc by

ρc ≡ 38πG

H20 . (4.3)

We stress that we always use the notion of critical density only in application to thepresent Universe; for us ρc is the time-independent quantity. Its meaning is that ifthe total energy density in the present Universe, ρM,0 +ρrad,0+ρΛ,0, equals preciselyρc, then the Universe is spatially flat (since in this case ρcurv = 0 and κ = 0). Thenumerical value of the critical energy density is given in (1.14). We note that theaverage energy density in the present Universe is fairly small: it is equivalent to 5proton masses per cubic meter.

Let us introduce the parameters,

ΩM =ρM,0

ρc, Ωrad =

ρrad,0

ρc, ΩΛ =

ρΛ,0

ρc, Ωcurv =

ρcurv,0

ρc. (4.4)

61

62 ΛCDM: Cosmological Model with Dark Matter and Dark Energy

Again, these parameters refer to the present Universe only, and by definition theydo not depend on time. It follows from (4.1) and (4.3) that∑

i

Ωi ≡ ΩM + Ωrad + ΩΛ + Ωcurv = 1. (4.5)

The parameters Ωi are equal to relative contributions of different sorts of energy, andalso of spatial curvature, into the right hand side of the Friedmann equation (4.1)at the present epoch. Ω’s make one of the most important sets of cosmologicalparameters.

It is rather straightforward to estimate the relative contribution of relativisticparticles Ωrad. It comes mainly from relic photons of temperature T0 = 2.726K.According to the Stefan–Boltzmann law, their energy density is (see also Chapter 5)

ργ,0 = 2π2

30T 4

0 ,

where the factor 2 is due to two photon polarizations. Numerically,

ργ,0 = 2.6 · 10−10 GeVcm3

,

and

Ωγ = 2.5 · 10−5h−2 = 5.0 · 10−5, h = 0.705. (4.6)

If there exist massless or light neutrinos (with mν � 5 K∼ 10−3 eV), their contri-bution is somewhat smaller than Ωγ. Hence

Ωrad � 10−4. (4.7)

Because of this, the effect of relativistic particles on cosmological expansion is neg-ligible today and in the late Universe.

We mentioned in Chapter 1 that observations of CMB anisotropy imply thatthe spatial curvature of the Universe is either zero or very small. Quantitatively,there is the bound on Ωcurv,

|Ωcurv| < 0.02. (4.8)

We will discuss in what follows how this bound is obtained, and will simply use itfor the time being.

There are several independent observational methods to determine ΩM and ΩΛ.We discuss some of them in this and the accompanying book, while others are justmentioned (see also Chapter 1). We quote here the current values,

ΩM ≈ 0.27, ΩΛ ≈ 0.73 (4.9)

with precision of about 5%. Hence, the present expansion of the Universe is deter-mined to large extent by dark energy, and to less extent by non-relativistic matter.

The present energy density of non-relativistic matter is the sum of mass densitiesof baryons and dark matter,

ΩM = ΩB + ΩDM ,

4.1. Composition of the Present Universe 63

with

ΩB = 0.046

and

ΩDM = 0.23.

At least two types of neutrinos have masses exceeding the temperature in the presentUniverse, see Appendix C. Neutrinos of these types are non-relativistic today. Wediscuss neutrino contribution Ων in Chapter 7, and here we assume for simplicitythat it is negligible as compared to ΩM . The number of electrons equals the numberof protons by electric neutrality, so that1

Ωe ≈ me

mp· ΩB 2.5 · 10−5.

Thus, the main contribution into ΩM comes from dark matter. As we discuss inChapter 9, dark matter is very likely cold.

Spatially flat cosmological model with non-relativistic cold dark matter and darkenergy with parameters close to (4.9) will be called2 ΛCDM model. In what followswe will specify this model further, by adding more cosmological parameters. Let usmake one qualification right now: unless we state the opposite, we will assume, inthe framework of ΛCDM that ρΛ is independent of time (cosmological constant ≡vacuum energy density).

ΛCDM model is consistent with the entire set of cosmological data. This doesnot necessarily mean, of course, that this model is exact, or that no alternativemodels are possible. In any case, ΛCDM serves as an important reference pointamong numerous cosmological models.

The relative contributions Ωi to the right hand side of the Friedmannequation (4.1) are characteristic of the present epoch only, since ρrad, ρM , ρΛ andρcurv behave in time in different ways, namely ρrad ∝ a−4 (see (3.28)), ρM ∝ a−3

(see (3.18)), ρcurv ∝ a−2 (see (4.2)), and, according to our assumption, ρΛ is inde-pendent of time. Thus, the Friedmann equation in the ΛCDM model can be writtenas follows,(

a

a

)2

=8π

3Gρc

[ΩM

(a0

a

)3

+ Ωrad

(a0

a

)4

+ ΩΛ + Ωcurv

(a0

a

)2]

. (4.10)

A subtlety here is that the number of relativistic species, as well as the numberof non-relativistic ones, is different at different cosmological epochs. In particular,neutrinos of mass mν � 5 K are non-relativistic today, but they were relativistic atearly stages of the evolution. This subtlety is not very important for this Chapter,

1The first equality here is approximate, since there exist neutrons bound into nuclei and con-

tributing to ΩB. This is not important for our purposes here.2In general, ΛCDM often denotes wider class of models. We will use this term in a narrow sense

given in the text. This model is also dubbed in literature as “concordance model”.


but one should keep in mind that the Friedmann equation in the form (4.10) mustbe used with care.

Another remark concerns dark energy. It cannot be excluded that ρΛ in factdepends on time. In particular, one can consider dark energy with equation of statepΛ = wΛρΛ with wΛ �= −1; in that case its energy density evolves with scale factoraccording to power law (3.41). Observational data show that wΛ belongs to theinterval −1.2 � wΛ � −0.8. Most of the conclusions of this Chapter remain validfor such a dark energy, although formulas get more cumbersome for wΛ �= −1.It is worth stressing that the question about time-dependence of ρΛ is extremelyimportant both for cosmology and particle physics, since it is directly related to thenature of dark energy: if ρΛ = const, then the dark energy is cosmological constant,while time-dependence of ρΛ would imply the existence of a new form of matter inNature (e.g., fairly exotic scalar field dubbed quintessence).

4.2 General Properties of Cosmological Evolution

Let us discuss, first at the qualitative level, which contributions to the right handside of the Friedmann equation are most relevant at different cosmological epochs.In the first place, the contribution due to curvature has never dominated. Indeed,it follows from (4.8) and (4.9) that this contribution, if any, is presently smallcompared to contributions of both non-relativistic matter and dark energy. In thepast, the contribution of non-relativistic matter was enhanced with respect to thatof curvature by a factor a0/a = 1 + z, so that curvature was even less important. Ifthe dark energy does not depend on time, the curvature contribution will be smallin future as well: the spatial curvature will decrease as 1/a2, while ρΛ stays constant.

Talking about future, let us note that all terms in the right hand side ofEq. (4.10), except for ΩΛ, decrease as a grows. Thus, the expansion rate of theUniverse will be determined by dark energy, and the behavior of the scale factorwill tend to the exponential law, (3.37) with ρvac ≡ ρΛ = ρcΩΛ. Of course, thisconclusion holds for time-independent ρΛ only. It is not known whether the latterassumption is valid, so distant future of the Universe cannot be reliably predicted,cf. Sec. 1.4.

Problem 4.1. Give examples of the evolution of dark energy leading to scenariosfor the future of the Universe outlined in Sec. 1.4. In particular, find the boundson maximum size and lifetime of the closed Universe assuming that dark energyinstantaneously switches off right after the present epoch. Hint: Use the bound (4.8)and the values (4.9).

We will be interested in the past of our Universe. Equation (4.10) shows thatthe major contribution to the present energy density comes from dark energy. Thiscontribution, ρΛ, has become important only recently. Before that, there was long

4.3. Transition from Deceleration to Acceleration 65

period of domination of non-relativistic matter (“matter domination”). At evenearlier times, at sufficiently small a, relativistic matter was prevailing (“radiationdomination”). Within the concepts developed so far in this book, the radiationdominated epoch began right from the cosmological singularity. We discuss in theaccompanying book the problems with this picture and study how these problemsare solved in inflationary theory. Here we concentrate on the Hot Big Bang theory,i.e., consider post-inflationary evolution of the Universe.

“Moments” when the regimes of evolution change are of considerable interest.Let us discuss them in some details.

4.3 Transition from Deceleration to Acceleration

We neglect the contributions due to relativistic matter and curvature and writeEq. (4.10) as follows,

a2 =8π

3Gρc

(ΩMa3

0

a+ ΩΛa2

).

This gives for acceleration

a = a4π

3Gρc

(2ΩΛ − ΩM

(a0

a

)3)

.

Since 2ΩΛ > ΩM and hence a > 0, the expansion accelerates at the present epoch.In the past, at sufficiently large z ≡ a0/a−1, the expansion was decelerating, a < 0.The transition from deceleration to acceleration occurred at(

a0

aac

)3

=2ΩΛ

ΩM

,

i.e., at

zac =(

2ΩΛ

ΩM

)1/3

− 1.

For ΩM = 0.27, ΩΛ = 0.73, we find numerically

zac ≈ 0.76.

Thus, transition from deceleration to acceleration occurred in the Universe prettylate.

Problem 4.2. At what z did the contributions to energy density due to non-relativistic matter and dark energy become equal to each other?

Since the dependence ρM ∝ a−3 is quite strong, while ρΛ does not dependon a, the transition occurred rather abruptly. Before the transition, the Universeexpanded as a ∝ t2/3 (matter dominated epoch, see Sec. 3.2.1).

Problem 4.3. At what z does the transition from deceleration to acceleration occurfor dark energy with equation of state p = wρ, w = const? For what value of the


parameter w this transition would occur now? Give numerical estimate using thevalues (4.9).

4.4 Transition from Radiation Domination to Matter Domination

As we have already pointed out, in Hot Big Bang theory the earliest epoch isradiation domination. The “moment” of the transition from radiation domination tomatter domination (“equality”) is very important from the viewpoint of the growthof density perturbations: we show in the accompanying book that perturbationsbehave in a very different way at these two epochs.

Crude estimate for the moment of equality is obtained from Eq. (4.10) andestimates (4.6) and (4.9). Neglecting dark energy and curvature at this moment, wefind that the contributions of radiation and non-relativistic matter were equal at

zeq + 1 =a0

aeq∼ ΩM

Ωrad∼ 104. (4.11)

At that time the temperature in the Universe had the following order of magnitude,

Teq = T0(1 + zeq) ∼ 104 K ∼ 1 eV. (4.12)

Thus, equality occurred at rather distant past.The estimates (4.11) and (4.12) have to be refined. At temperature of order 1 eV,

not only photons, but most likely also all three species of neutrinos are relativistic(see Appendix C). We see in Chapter 7 that neutrinos do not interact betweenthemselves and with the rest of cosmic matter at this temperature. According toSec. 2.5, their distribution functions are thermal nevertheless. We show in Chapter 7that the effective temperature of neutrinos is

Tν =(

411

)1/3

Tγ, (4.13)

where Tγ is the photon temperature. Everywhere in this book we identify the tem-perature in the Universe with the photon temperature, so that

Tγ ≡ T.

The energy density of relativistic neutrinos is given by the appropriately modifiedStefan–Boltzmann law (see Chapter 5),

ρν = 3 · 2 · 78

π2

30T 4

ν , (4.14)

where the factor 3 corresponds to three neutrino species, factor 2 is due to theexistence of both neutrino and anti-neutrino, one polarization each, and factor 7/8accounts for the Fermi statistics. Thus, the energy density of relativistic matter atthe epoch of interest is

ρrad = ργ + ρν =

[2 +

214

(411

)4/3]

π2

30T 4, (4.15)

4.4. Transition from Radiation Domination to Matter Domination 67

where the first and second terms in square brackets come from photons and threeneutrino species, respectively. Hence,

ρrad = 1.68ργ = 1.68(a0

a

)4

Ωγρc . (4.16)

The energy density of non-relativistic matter is still given by

ρM =(a0

a

)3

ΩMρc. (4.17)

We find that equality between relativistic and non-relativistic componentshappens at

1 + zeq =a0

aeq= 0.6

ΩM

Ωγ,

and making use of (4.6) we obtain

1 + zeq = 2.4 · 104 ΩMh2, (4.18)

which for ΩM = 0.27 and h = 0.705 gives

1 + zeq = 3.2 · 103. (4.19)

At that time, temperature is

Teq = (1 + zeq)T0 = 5.6 ΩMh2 eV (4.20)

Teq = 0.76 eV for h = 0.705, ΩM = 0.27 (4.21)

The expressions (4.18)–(4.21) refine the estimates (4.11), (4.12) with account ofthree light neutrino species.

Let us estimate the age of the Universe at equality. To simplify the calculation,we neglect non-relativistic matter before equality and note that during most of theevolution only photons and neutrinos were relativistic (the lightest of other particles,electrons and positrons, are relativistic at temperatures T � me = 0.5 MeV only).Therefore, we use formulas of Sec. 3.2.2 with effective number of degrees of freedomg∗ obtained by comparing (3.30) with (4.15):

g∗ = 2 +214

(411

)4/3

= 3.36. (4.22)

Making use of (3.29) and (3.31) we estimate the age as

teq =1

2Heq=

M∗Pl

2T 2eq

, (4.23)

where, as before, M∗Pl = MPl/1.66

√g∗. Using (4.20) and (4.22) we obtain for h =

0.705, ΩM = 0.27

teq = 3.5 · 1036 GeV−1 = 2.3 · 1012s = 73 thousand yrs. (4.24)

This time is of course much shorter than the present age of the Universe, t0 ≈ 14billion years.


To conclude this Section we note that the definition of equality ρM = ρrad

is unambiguous. However, from the viewpoint of the cosmological evolution, thetransition from radiation domination to matter domination is actually not a welldefined moment in the history of the Universe, but the process whose duration iscomparable to the Hubble time at that epoch, H−1

eq (in other words, comparable tothe age teq). The ratio between energy densities of non-relativistic and relativisticmatter depends on scale factor rather weakly, ρM/ρrad ∝ a, so this ratio does notchange very much in Hubble time. Thus, the jump from the expansion law a ∝ t1/2

to a ∝ t2/3 at t = teq would be only a crude approximation. The relation (4.23)between the temperature Teq and age teq is approximate as well, as in our derivationof (4.23) we neglected non-relativistic matter at t < teq.

Problem 4.4. Neglecting dark energy and spatial curvature and assuming that theeffective number of relativistic degrees of freedom determining the energy density isthe constant given by (4.22), find exact law of evolution a = a(t) at temperatures oforder Teq. Find the age of the Universe at the moment when its temperature equalsto Teq given by (4.20). Show that at h = 0.705 and ΩM = 0.27 the numerical valueis teq = 57 thousand years, which refines the estimate (4.24).

4.5 Present Age of the Universe and Horizon Size

The fact that the cosmological expansion for fairly long time has been affected bydark energy leads to different values of the present age of the Universe and presenthorizon size as compared to the estimates (3.24) and (3.27). To calculate them, weneglect spatial curvature and relativistic matter: as we discussed in Sec. 4.1, thecurvature is negligible at all epochs, while the relativistic matter is relevant at earlytimes only, t � teq. Then Eq. (4.10) becomes(

a

a

)2

= H20

[ΩM

(a0

a

)3

+ ΩΛ

], (4.25)

where we made use of (4.3). Here

ΩM + ΩΛ = 1, ΩΛ > 0. (4.26)

The solution to Eq. (4.25) is

a(t) = a0

(ΩM

ΩΛ

)1/3 [sinh

(32

√ΩΛH0t

)]2/3

. (4.27)

It is clear from this formula that the law of matter dominated expansion, a ∝ t2/3,is restored at early times, while at late times the scale factor grows exponentially,as should be the case.

The present age of the Universe is now found from the following equation,(ΩM

ΩΛ

)1/3 [sinh

(32

√ΩΛH0t0

)]2/3

= 1.

4.5. Present Age of the Universe and Horizon Size 69

It is given by

t0 =2

3√

ΩΛ

1H0

Arsinh√

ΩΛ

ΩM

. (4.28)

At ΩΛ → 0 and ΩM → 1 we restore formula (3.24). For positive ΩΛ the age exceeds2/(3H0). To see this, we plot the scale factor as function of time for matter dom-inated model (ΩΛ = 0) and ΛCDM model (ΩΛ > 0) in such a way that they aretangential to each other (values of both a and a coincide) at the present time, i.e.,at a = a0 (equality of derivatives corresponds to one and the same value of thepresent Hubble parameter H0 = (a/a)0). Since the Friedmann equation for realisticmodel is given by (4.25) and the right hand side for matter dominated model equalsH2

0 (a0/a)3, the time derivative a is greater for matter dominated model at everya < a0, so we come to the plot shown in Fig. 4.1.

The distance along the horizontal axis from the point of singularity a = 0 to thepoint a = a0 is precisely the present age; it is clear that the age is larger for theUniverse with ΩΛ > 0. We obtain from (4.28)

t0 = 1.38 · 1010 yrs for ΩM = 0.27, ΩΛ = 0.73, h = 0.705.

This age does not contradict the independent bounds discussed in Chapter 1. It isthe presence of dark energy that removes the contradiction between the age of theUniverse, calculated by using the measured value of the present Hubble parameter,and the bounds on this aged obtained by other methods.

Problem 4.5. Consider open model without dark energy (this model in fact isexcluded by CMB data), in which ΩM �= 0, Ωcurv �= 0, ΩΛ = 0 and ΩM + Ωcurv = 1.

Fig. 4.1 Evolution of a = a(t) for spatially flat models.


Find the present age of the Universe at given value of H0. Calculate the numericalvalue for ΩM ≈ 0.3 (estimated from the studies of clusters of galaxies) and h = 0.7.

Problem 4.6. Find the present age of the Universe for dark energy with equationof state p = wρ, w = const. Give numerical estimates for w = −1.1 and w = −0.9with ΩM = 0.27, ΩΛ = 0.73.

The discussion of the cosmological horizon in ΛCDM model is not so instructive.Nevertheless, let us make the estimate. According to general formula (3.25), thepresent horizon size is

lH,0 = a0

∫ t0

0

dt

a(t).

Since a(t) ∝ t2/3 at small t (see (4.27)), this integral is convergent at the lower limit,i.e., the cosmological horizon size is finite. For given H0 it is larger than the value2/H0 obtained for the flat model with matter but without dark energy. Numerically,for ΩM = 0.27, ΩΛ = 0.73, the estimate is

lH,0 =2

H0· 1.7 = 14.4 Gpc, h = 0.705. (4.29)

Problem 4.7. Show that the horizon size is greater than 2/H0 in the model withmatter and positive dark energy. Confirm the numerical value (4.29).

To end this Section, let us make the following remark. The bound (4.8) for Ωcurv

together with the estimate (4.29) can be used to show that there are many regionsof size lH,0 outside our horizon. Let us recall in this regard that in the Hot BigBang theory these regions are causally disconnected. In any case, we cannot obtainany information on what is going on in these regions; as an example, relic photonstraveled the distance slightly smaller than lH,0 since the recombination epoch.

Clearly, there is infinite number of regions in question in open and flat models,so we are talking about closed Universe, 3-sphere.3 It follows from the definitions(4.2), (4.3) and (4.4) that the radius of this sphere a0 is related to Ωcurv as follows,

1a20

= H20 |Ωcurv|. (4.30)

Comparing this with (4.29) and using the bound (4.8) we find

a0

lH,0=

13.4√|Ωcurv|

> 2.1.

Thus, the radius of the Universe is larger than the horizon size. This becomes evenmore striking if we calculate the number of regions like ours. The latter is equal to

3We assume here that the Universe is homogeneous and isotropic both inside and outside our

horizon.

4.6. Brightness-Redshift Relation for Distant Standard Candles 71

the ratio of the volume of the 3-sphere 2π2a30 and the volume of a region of radius

lH,0:

N ≈ 2π2a30

(4π/3)l3H,0

= 4.7(

a0

lH,0

)3

> 43. (4.31)

Hence, observational data show that we see less than 2% of the whole Universe.In the accompanying book, we give theoretical arguments suggesting that Ωcurv

is many orders of magnitude smaller than the observational bound (4.8), i.e., thenumber of regions outside our horizon is by many orders of magnitude larger thanthe bound (4.31).

4.6 Brightness-Redshift Relation for Distant Standard Candles

Let us discuss one of the important methods of determination of cosmologicalparameters, such as ΩM , ΩΛ and Ωcurv. This method is also capable of discrim-inating between the cosmological constant (vacuum energy) and time-dependentforms of dark energy. The method makes use of the simultaneous measurement ofredshift z and visible brightness of “standard candles” which are at distances com-parable to the horizon size. These standard candles are very luminous objects whoseabsolute luminosity is assumed to be known. Among the objects used as standardcandles4 are supernovae of type Ia (SNe Ia).

Let us find the relation between redshift and visible brightness of a source withabsolute luminosity (energy emitted in unit time) L. Although the analysis thatfollows (but not concrete results!) is straightforwardly generalized to time-dependentdark energy, let us assume, for the time being, that dark energy density ρΛ doesnot depend on time. It is useful to keep spatial curvature not equal to zero andignore the bound (4.8). Let us choose for definiteness open cosmological model withκ = −1 and Ωcurv > 0. The flat model is obtained in the limit Ωcurv → 0, or,equivalently, a0 → ∞, see (4.30).

Let us use the form (2.10) of the metric,

ds2 = dt2 − a2(t)[dχ2 + sinh2 χ(dθ2 + sin2 θdφ2)]. (4.32)

As usual, the coordinate distance betwen the source emitting light at time ti andobserver at the Earth at time t0 equals

χ =∫ t0

ti

dt

a(t). (4.33)

4As everywhere in this book, we do not discuss astrophysical and observational aspects of this

problem. In particular, we leave aside the issue about the nature of SNe Ia, the question of why

they are candidates for standard candles, etc.


Let us first find the relation between the coordinate distance and redshift z of thesource. To this end, we use the Friedmann equation in the form (4.10), and neglectradiation. Changing the integration variable in (4.33) from t to

z(t) =a0

a(t)− 1

we obtain

χ =∫ z

0

dz′

a0(a/a)(z′).

Using Eq. (4.10) we cast this integral into

χ(z) =∫ z

0

dz′

a0H0

1√ΩM(z′ + 1)3 + ΩΛ + Ωcurv(z′ + 1)2

(4.34)

This integral cannot be evaluated analytically, but its numerical calculation is easy.According to (4.32), the physical area of the sphere crossed by photons today is

S(z) = 4πr2(z) , (4.35)

where

r(z) = a0 sinh χ(z). (4.36)

The number of photons crossing unit surface at the observer’s position is inverselyproportional to S, while the energy of each photon differs from the energy atemission by the redshift factor (1 + z)−1. The same factor additionally enters theexpression for the number of photons crossing unit surface in unit time, since thetime intervals for the source and observer differ by factor (1 + z)−1. The latterpoint can be understood as follows. In conformal coordinates (η,x) photons behavein the same way as in Minkowski space, see Sec. 2.3. Therefore, in these coordinatesthe time intervals between emission and absorption of two photons are equal toeach other, dηe = dη0. This gives the relation between the physical time intervals,dt0 = (1 + z)dte.

Thus, the visible brightness (energy flux at observer’s position) equals

J =L

(1 + z)2S(z). (4.37)

This is the desired expression for the brightness-redshift relation for a source whoseabsolute luminosity L is assumed to be known.

Let us introduce photometric distance rph in such a way that the relation betweenL and J is formally the same as in Minkowski space-time,

J =L

4πr2ph

.

We find from (4.37)

rph = (1 + z) · r(z), (4.38)

where r(z) is given by (4.36).


At first sight, the relation (4.37) involves five cosmological parameters: H0, a0,ΩM , ΩΛ and Ωcurv. In fact, the number of independent parameters is three in viewof the relations (see (4.2) and (4.4), (4.5))

ΩM + ΩΛ + Ωcurv = 1 (4.39)

and

|Ωcurv| =1

a20H

20

. (4.40)

Note that at z � 1, one can neglect z′ in the integrand in (4.34), then χ(z) =z/(a0H0) and r(z) = a0χ(z), so we get back to the Hubble law r(z) = H−1

0 z.In that case, the leading order expression for brightness coincides with the usualformula

J =L

4πr2(z), z � 1.

Let us now discuss the general case. It is clear from (4.34)–(4.40) that the threeindependent parameters enter the brightness-redshift relation in a non-trivial way.In principle, all of them may be determined by measurements in a wide range of z.This is illustrated in Fig. 4.2.

To understand what is shown in Fig. 4.2 we note that the dependence on cos-mological parameters enters the formula (4.37) through the function r(z). If wemeasure r(z) in Hubble units H−1

0 , then

H0r(z) =1√

Ωcurv

sinh χ(z), (4.41)

Fig. 4.2 Dependence of H0r(z) on redshift z for various cosmological models.


Fig. 4.3 Degeneracy in the parameter space (ΩM , ΩΛ). The cases (ΩM = 0, ΩΛ = 0.55) and

(ΩM = 0.6, ΩΛ = 0.85) correspond to open and closed models, respectively, with Ωcurv = 1 −ΩM − ΩΛ. Note that the horizontal scale here is logarithmic, unlike in Fig. 4.2.

χ(z) =∫ z

0

√Ωcurvdz′√

ΩM(1 + z′)3 + ΩΛ + Ωcurv(1 + z′)2.

Hence, the right hand side of (4.41) does not explicitly depend on H0; it is thisquantity that is shown in Fig. 4.2.

Let us first discuss black and dark-gray curves in Fig. 4.2, which correspond tospatially flat Universes with: ΩM = 0.27, ΩΛ = 0.73 (black curve) and ΩM = 1,ΩΛ = 0 (dark-gray curve). We obtain formulas for the flat model from the generalexpression (4.36) by taking the limit a0 → ∞, Ωcurv → 0. In this way we get

r(z) =1

H0

∫ z

0

dz′√ΩM(z′ + 1)3 + ΩΛ

, Ωcurv = 0,

with ΩM + ΩΛ = 1. It is clear that the larger is ΩΛ (and, accordingly, the smaller isΩM), the faster the function r(z) grows with z; distant supernovae are dimmer inΛCDM model as compared to the flat model without dark energy. It is this propertythat have been observed [28, 29]. Figure 4.4 shows one of the early sets of data [38],while Fig. 4.5 illustrates uniqueness of the interpretation in terms of dark energywith enlarged dataset [39].

Now, black and light gray lines in Fig. 4.2 correspond to ΛCDM model andopen model without the cosmological constant. They differ already at moderate z.Therefore, the model with ΩΛ = 0, Ωcurv = 0.73 is also inconsistent with data.Overall, data on SNe 1a reject models without dark energy, see Figs. 4.4–4.6.


34

36

38

40

42

44

ΩM=0.28, ΩΛ=0.72

ΩM=0.20, ΩΛ=0.00

ΩM=1.00, ΩΛ=0.00

m-M

(m

ag)

MLCS

0.01 0.10 1.00z

-1.0

-0.5

0.0

0.5

1.0

Δ(m

-M)

(mag

)

Fig. 4.4 Hubble diagram for SNe Ia: early data [38]. Upper panel shows the brightness distribution

of supernovae (appropriately corrected). Lower panel illustrates the incompatibility of observations

to the CDM model with spatial curvature (ΩM = 0.2, ΩΛ = 0, Ωcurv = 0.8, dotted line) and flat

CDM model (ΩM = 1, ΩΛ = 0, Ωcurv = 0, dashed line). Black line is the prediction of the ΛCDM

model with ΩM = 0.28, ΩΛ = 0.72, Ωcurv = 0.0 which is consistent with the data. The notation

on vertical axis is related to brightness measure in astronomy, apparent magnitude. The difference

(m−M) is related to photometric distance by m−M = 5 log10(rph/Mpc)+25. The larger (m−M)

the dimmer the object.

This is probably the strongest argument for dark energy. We stress, however,that there are other, independent arguments. Namely, we mentioned already theargument based on the extrapolation of the mass estimates of clusters of galaxiesto the whole Universe (giving ΩM ≈ 0.3), together with CMB bound on spatialcurvature. We also presented the argument based on the age of the Universe. Otherarguments come from the analysis of CMB and large scale structure; some of themare discussed in the accompanying book.

Let us now turn to Fig. 4.3. We present it here to illustrate the degeneracyin parameters: models with very different parameters give very similar results atmoderate z; this range of z is of particular interest, since objects at large z are dim,and hence difficult to observe. To see what is going on, let us find the first correction


-1.0

-0.5

0.0

0.5

1.0

Δ(m

-M)

(mag

)

HST DiscoveredGround Discovered

0.0 0.5 1.0 1.5 2.0z

-0.5

0.0

0.5

Δ(m

-M)

(mag

)

ΩM=1.0, ΩΛ=0.0

high-z gray dust (+ΩM=1.0)Evolution ~ z, (+ΩM=1.0)

Empty (Ω=0)ΩM=0.27, ΩΛ=0.73"replenishing" gray Dust

Fig. 4.5 Plots [39] illustrating different possible interpretations of observations of SNe Ia. They

show the deviation from the prediction of a model of curved empty Universe (ΩM = ΩΛ =

0, Ωcurv = 1; recall that such a Universe expands at constant speed, a = 0). Among cosmo-

logical models, considered are flat ΛCDM (ΩM = 0.27, ΩΛ = 0.73) and flat CDM model without

dark energy (ΩM = 1, ΩΛ = 0). Model with evolution is the model in which absolute lumi-

nosity of SNe 1a decreases like z−1 (in flat CDM cosmology). Models with non-standard inter-

galactic medium are models with “dust” absorbing the supernovae light; the “dust” densities are

ρ(z) = ρ0(1 + z)α, where α = 3 (dash-dotted line) and α = 3 at z < 0.5 while α = 0 at other

z (thin black line). Data at upper panel are from the Hubble telescope (bullets) and terrestrial

telescopes (diamonds). At the lower panel these data are combined and binned in z. We note that

independent observations of SNe Ia [40] give results consistent with those shown here.

to the Hubble law at small z. We make use of the relation ΩΛ = 1 − ΩM − Ωcurv

and write to quadratic order in z

χ(z) =1

a0H0

[z − z2

4(3ΩM + 2Ωcurv)

].

To this order r(z) = a0χ(z) + O(z3), so

r(z) =1

H0

[z − z2

4(3ΩM + 2Ωcurv)

]. (4.42)

The second term in the right hand side is precisely the correction we are concernedabout. It is clear from (4.42) that this correction depends only on the combination(3ΩM +2Ωcurv) or, in terms of ΩM and ΩΛ, on the combination (ΩM −2ΩΛ), but noton ΩM and ΩΛ separately. It is this property that is responsible for the degeneracy atsmall z. To study the degeneracy at moderate z one has to use higher order terms inz in Eq. (4.42). It turns out that the terms depending on the combination (ΩM−2ΩΛ)


Fig. 4.6 Regions in the plane of cosmological parameters (ΩM , ΩΛ), consistent with observa-

tional data on SNe Ia [41]. Dotted lines show contours obtained with proper estimates of possible

systematics.

partially cancel at the orders z2 and z3 in the interesting range of parameters.Uncompensated contribution of order z3 depends on another linear combination,(2ΩM − ΩΛ). Hence, the observations studying the expansion law at moderate z

are sensitive to the latter combination. The data show that this combination isclose to zero, while there is approximate degeneracy along the orthogonal linearcombination. Therefore, the allowed region of parameters is stretched along the line2ΩM − ΩΛ = 0. This is seen in Fig 4.6 [41].

Problem 4.8. Show that to the order z3 the degeneracy in parameters is removed,

i.e., r(z) given by (4.36) depends in a non-trivial way on all three parametersH0, ΩM , ΩΛ. Show, nevertheless, that in the interesting region of cosmologicalparameters there remains approximate degeneracy at moderate z along the line2ΩM − ΩΛ = 0.


There is nothing surprising in the degeneracy in parameters. It is clear that thelowest in z correction to the Hubble law can depend only on the present values ofthe Hubble parameter and acceleration parameter. We define the latter as follows,5

q0 =1

H20

(a

a

)0

(4.43)

By measuring one parameter q0 one can determine only one combination of ΩM andΩΛ, hence the degeneracy.

Regarding Fig. 4.6, let us make a general comment. We often show regions in aparameter plane allowed by one or another data. If we do not state the opposite,three regions embedded into each other correspond to regions allowed at 1σ, 2σ and3σ confidence level (assuming Gaussian distribution for relevant quantity), i.e., atthe confidence level of 68.3%, 95.4% and 99.7%.

Problem 4.9. Find the lowest in z correction to the Hubble law, i.e., the functionr(z) at quadratic level in z, in terms of H0 and q0. Do not use the Friedmannequation. Show that after using the Friedmann equation, this expression coincideswith (4.42).

The approximate degeneracy in parameters makes it difficult to determine ΩM ,ΩΛ and Ωcurv by studying standard candles only. On the other hand, CMB tem-perature anisotropy gives strong bound on Ωcurv: |Ωcurv| < 0.02. Making use ofthis bound, one can determine ΩM and ΩΛ from the observations of SNe Ia. This isshown in Fig. 4.6, where the line Ωcurv = 0 is denoted as Ωtot = 1; it is seen thatsupernovae observations give (0.23 < ΩM < 0.39, 0.77 > ΩΛ > 0.61) at 95%CL.

To conclude, the observations of SNe Ia together with CMB measurements areone of the main sources of information on dark energy. The fit to existing cosmo-logical data gives the following determination [7],

ΩM = 0.273± 0.014, ΩΛ = 0.726 ± 0.015,

at 68% CL.Similar observations with better precision and statistics and at larger z will most

likely make it possible to find the dependence of dark energy density on time (orestablish strong bounds on this dependence). The existing data are consistent withtime-independent dark energy (i.e., dark energy equation of state p = −ρ), andunder the assumption of time-independence of the parameter w of the dark energyequation of state p = wρ, these data give (see Fig. 4.7 [41] and Fig. 13.6 on colorpages)

−1.1 < w < −0.9. (4.44)

Refining this result is a very important task for future observations.

5It is traditional to use deceleration parameter, which differs by sign from the acceleration

parameter (4.43). We think that using deceleration parameter for accelerating Universe does not

make much sense.

4.7. Angular Sizes of Distant Objects 79

Problem 4.10. Generalize formulas of this Section to the case of dark energy withequation of state p = wρ, w = const. Taking Ωcurv = 0 and ΩM = 0.27, draw plotsof r(z) for w = −2, w = −1.5, w = −1, w = −0.75 and w = −0.5. Making use ofFig. 4.6 show that existing data are indeed capable of determining w at the level ofaccuracy given in (4.44).

4.7 Angular Sizes of Distant Objects

An important characteristic of an extended object (e.g., galaxy) is its angular size.In this regard, a useful concept is angular diameter distance Da that relates theabsolute size of an object d to angle Δθ at which this object is seen today,

d = Da(z) · Δθ,

where z is the redshift of the object. To find the expression for Da(z) let us recallagain that in conformal coordinates photons behave in the same way as in Minkowskispace-time. Therefore, the coordinate size of an object is related to its coordinatedistance from us χ and angular size Δθ by the relation

dconf = sinhχ · Δθ.

The physical size of the object emitting photons at time ti is equal to

d = a(ti)dconf =a(ti)a0

· a0 sinh χ · Δθ.

0.0 0.1 0.2 0.3 0.4 0.5-1.5

-1.0

-0.5

0.0

m

w

SNe

BAO

CMB

Fig. 4.7 Regions in the plane of parameters (ΩM , w) allowed (for the flat Universe) by observa-

tions of CMB anisotropy, by large scale structures (BAO) and by SNe Ia data [41], see Fig. 13.6

for color version. The intersection region corresponds to the combined analysis of all these data.


0 1 2 3 4 5 6

0.1

0.2

0.3

0.4

Fig. 4.8 Angular diameter distance as function of redshift in ΛCDM model (ΩM = 0.27, ΩΛ =

0.73, Ωcurv = 0).

Making use of (4.36) we obtain

Da(z) =1

1 + zr(z),

where r(z) is given by formulas of Sec. 4.6.Angular diameter distance increases with z relatively slowly at moderate z, see

Fig. 4.8. On the other hand, photometric distance (4.38) rapidly increases with z, sogalaxies become more and more dim. At large z, the large distance from the Earthto the galaxies shows up not in the smallness of their angular sizes but in their lowsurface brightness (visible brightness of a region of unit angular size).

4.8 ∗Quintessence

Cosmological constant (vacuum energy density) is not the only possible reason ofthe accelerated expansion of the Universe at the present epoch. Nature of darkenergy is one of the major problems of contemporary natural science. No wonder,numerous hypotheses have been put forward in this regard. One of these hypothesesis the existence of “quintessence”, spatially homogeneous field whose energy playsthe role of dark energy [42–44]. Unlike the cosmological constant, quintessence is adynamical field, and its energy density depends on time. In terms of the effectiveequation of state p = wρ this means that w �= −1 with, generally speaking, time-dependent w.

Quintessence is often (but not always) associated with a scalar field. We considerone class of models of this sort later in this Section, but before that we study generalproperties of scalar field dynamics in expanding Universe. These results will be

4.8. ∗Quintessence 81

useful also in our discussions of other topics. Understanding the dynamics of scalarfield is particularly important in the context of inflationary theory.

4.8.1 Evolution of scalar field in expanding Universe

Let us consider a theory of real scalar field with action

S =∫

d4x√−gL =

∫d4x

√−g

[12gμν∂μϕ∂νϕ − V (ϕ)

], (4.45)

where V (ϕ) is the scalar potential. Let us stick to spatially flat Universe whosemetric has standard Friedmann–Lemaıtre–Robertson–Walker form (2.22) with scalefactor a(t) being a known function of time.

The equation for the scalar field is obtained, as usual, by varying the action (4.45)with respect to ϕ, and has the form

1√−g∂μ(

√−ggμν∂νϕ) = −∂V

∂ϕ. (4.46)

Let us specify to spatially homogeneous (independent of coordinates) scalar fieldϕ(t) in the background metric (2.22). In that case, Eq. (4.46) reduces to

ϕ + 3Hϕ = −∂V

∂ϕ. (4.47)

Equation (4.47) formally coincides with the equation for the classical mechanics of a“particle” with coordinate ϕ in the “potential” V (ϕ), which experiences friction withtime-dependent friction coefficient H . Depending on the strengths of the drivingforce and friction, two regimes are possible: (1) fast roll regime when Hϕ � V ′

(prime denotes the derivative with respect to ϕ), friction is weak, and the “particle”rapidly rolls down to the minimum of the potential V (ϕ); (2) slow roll regime whenfriction is strong, and the “particle” barely moves. Let us begin with the secondregime. In that case

Hϕ ∼ V ′. (4.48)

During the Hubble time H−1 the field changes by

δϕ ∼ ϕH−1 ∼ V ′

H2.

This change is small compared to the field itself, δϕ � ϕ, provided that

V ′

ϕ� H2. (4.49)

For power-law potentials like m2ϕ2 or λϕ4 one has V ′ ∼ V ϕ−1, so the conditionfor slow roll is

V

ϕ2� H2. (4.50)


Thus, in the case of power-law potential, the condition (4.50) is necessary for slowroll, at which the value of the field remains practically constant during the cosmo-logical evolution. Once this condition is violated, the field rapidly rolls down to theminimum of V (ϕ).

Problem 4.11. The condition (4.50) is not sufficient for slow roll. The secondnecessary condition is that the first term on the left hand side of Eq. (4.47) does notexceed the second term, ϕ � Hϕ. The latter condition ensures that velocity does notchange much in one Hubble time, δϕ ∼ ϕH−1 � ϕ, so the velocity remains smallduring the evolution. Find the second slow roll condition in terms of the potentialV (ϕ) and its derivatives, and H(t) and its derivatives. Simplify this condition in thecase of power-law dependence of a(t) on time (like a ∝ t2/3, matter domination).

Problem 4.12. Show that for the power-law dependence of the scale factor on time,a(t) = tα, α > 1/3, the general solution to (4.47) in the approximation V ′ = const is

ϕ = ϕi + C · (t2 − t2i ) + d ·[1 −

(tit

)3α−1]

, (4.51)

where ϕi is the initial value of the field, ϕi = ϕ(ti). In the case ϕ(ti) � H(ti)ϕ(ti),find the constants C and d in terms of ϕi and ϕ(ti). Show that in this case thethird term in (4.51) is always small. Find the values of t at which the second termin (4.51) is small compared to the initial value ϕi and show that for a power-lawpotential this time is in accordance to (4.50), and in general case to (4.49). Showthat in this time interval the relation (4.48), as well as ϕ ∼ Hϕ, hold.

Although we will not need in this Section to know the behavior of the scalarfield near the minimum of the scalar potential, let us consider how ϕ(t) approachesthis minimum. Let the minimum of V (ϕ) be at ϕ = 0, and near the minimum thepotential be

V (ϕ) =m2

2ϕ2. (4.52)

Then Eq. (4.47) near the minimum has the form

ϕ + 3a

aϕ + m2ϕ = 0. (4.53)

To analyze equations of this sort, it is convenient to get rid of the friction termproportional to ϕ and cast the equation into the form of the oscillator equation withtime-dependent frequency (cf. end of Sec. 2.3). In our case the change of variablesis given by

ϕ(t) =1

a3/2(t)· χ(t),

where χ is a new unknown function which obeys

χ +(

m2 − 32

a

a− 3

4a2

a2

)χ = 0. (4.54)


Note thata

a∼ a2

a2= H2.

It then follows from (4.54) that at m2 � H2 the mass term is negligible. In thatcase the potential (4.52) obeys the condition (4.50), and we come back to the slowroll regime.

Problem 4.13. Obtain the results of the previous problem starting from Eq. (4.54).

We are now interested in the opposite regime, m2 � H2. In this case, we canneglect terms of order H2 in parenthesis in (4.54), so the solution has the formχ = const · cos(mt + β) where β is an arbitrary phase. Hence, at m2 � H2 the fieldrelaxes to the minimum in the following way,

ϕ(t) = ϕ∗ · cos(mt + β)a3/2(t)

, (4.55)

where ϕ∗ is some constant. The field oscillates near the minimum with decreasingamplitude. Note that the relative change of the amplitude in one period of oscilla-tions is of order

a

a· m−1 ∼ H

m,

so it is small in the case under study, m � H .The solution (4.55) can be found also from energy considerations. In the case of

free massive scalar field, the action (4.45) in the Universe with metric (2.22) takesthe form

S =∫

d4x · a3 ·(

12ϕ2 − 1

2a2(∂iϕ)2 − m2

2ϕ2

).

If the scale factor were independent of time, the homogeneous field would haveconserved energy,

E =∫

d3x · a3 ·(

12ϕ2 +

m2

2ϕ2

).

The solution for the field equation would oscillate with frequency m, i.e., ϕ ∝cos(mt + β). If the scale factor slowly (adiabatically) grows in time, the oscillatorybehavior of solutions persists, and energy is still (approximately) conserved, as weknow from classical mechanics. The latter property means that

a3(t)(

12ϕ2 +

m2

2ϕ2

)= const. (4.56)

This tells that the amplitude of oscillations decreases as a−3/2, i.e., we come backto the solution (4.55). Note that the approximate conservation law (4.56) can beinterpreted as (approximate) energy conservation in comoving volume. Note alsothat for fast changing a(t), there is no conservation of energy at all; energy isnot conserved in time-dependent backgrounds (in this case a(t)). It does not make


sense to talk about energy (including energy of the gravitational field) of the entireUniverse; there is no such an integral of motion in General Relativity.6

To end this Section, let us find the energy-momentum tensor of the scalar fieldin different regimes. The general expression for energy-momentum tensor in thetheory with action (4.45) is

Tμν =2√−g

δS

δgμν= ∂μϕ∂νϕ − gμνL. (4.57)

For homogeneous scalar field and potential V (ϕ) we have for non-zero componentsin the locally-Lorentz frame (i.e., setting the local metric gμν in (4.57) equal to ημν)

T00 =12ϕ2 + V (ϕ) ≡ ρϕ, Tij =

(12ϕ2 − V (ϕ)

)δij ≡ pϕδij . (4.58)

The estimate (4.48) gives in the slow roll regime

ϕ2 ∼ V ′2

H2� V ′ · ϕ, (4.59)

where we used (4.49) to obtain the inequality. Power-law potentials obey V ′ϕ ∼ V ,so (4.59) gives

ϕ2 � V.

Hence, in the slow roll regime

ρϕ ≈ −pϕ ≈ V (ϕ), (4.60)

i.e., the equation of state is approximately the same as for vacuum, p ≈ −ρ (althoughit is clear from (4.58) that one always has p > −ρ).

To discuss the regime of fast oscillations about the minimum of the scalarpotential, we use (4.52) and (4.55) and obtain

T00 =m2ϕ2

∗2

· 1a3(t)

, Tij = −m2ϕ2∗

21a3

cos(2mt + 2β) · δij ,

where we again used inequality H � m. Thus, energy density and pressure, averagedover a period of oscillations, are

T00 ≡ ρϕ =m2ϕ2

∗2

· 1a3(t)

, Tij ≡ pϕ · δij = 0. (4.61)

We conclude that average energy-momentum tensor of the coherent scalar fieldoscillations coincides with the energy-momentum tensor of non-relativistic matter:pressure is zero, while energy density decreases as a−3. As we already mentioned,the latter property reflects the energy conservation in comoving volume, see (4.56).

From the quantum field theory viewpoint, homogeneous oscillating scalarfield (4.55) is to be viewed as a collection of free particles of mass m, all at zero

6There are certain cases in which one can define total energy (including energy of the gravitational

field) in the framework of General Relativity. One special case is asymptotically flat space-time.

So, the notion of energy (mass) of a gravitating body, away from which space-time is Minkowskian,

is well-defined.


spatial momentum (at rest). The number density of these particles is given by

n =ρ

m=

12mϕ2

∗ ·1

a3(t).

As any number density, n(t) decreases as a−3. The fact that pressure vanishes hasnatural interpretation in this picture: particles at rest do not produce any pressure.

4.8.2 Accelerated cosmological expansion due to scalar field

Accelerated expansion of the Universe at the present epoch may be explainedby introducing scalar field ϕ (quintessence) with action (4.45) and choosing thepotential V (ϕ) and the present value of the field ϕ in such a way that the evolutionof the field occurs in the slow roll regime. Furthermore, one should assume thatthe field ϕ is spatially homogeneous; initial data of the latter type are natural inthe framework of inflationary theory, so the latter assumption is not particularlystrong.

The effective equation of state in the slow roll regime is pϕ ≈ −ρϕ, see (4.60).Thus, the Universe indeed undergoes accelerated expansion, if the scalar field givesdominant contribution into the energy density. Let us find the conditions underwhich the slow roll regime indeed holds. Let us assume for definiteness that nearthe present value of ϕ the scalar potential may be approximated by power law,V (ϕ) ∝ ϕk where |k| is not very large. Then the slow roll condition takes theform (4.50). Since energy density in the Universe is mostly due to the scalar field,the Friedmann equation is

H2 =8π

3M2Pl

ρϕ =8π

3M2Pl

V (ϕ), (4.62)

where we used (4.60). By combining (4.50) and (4.62) we obtain

ϕ � MPl. (4.63)

This is the form of the slow roll condition for power-law potentials and the Universedominated by the scalar field itself.

Despite the fact that the value of the scalar field is extremely large, the value ofthe potential V (ϕ) must be very small,

V (ϕ) ∼ ρc

(more precisely, V (ϕ) = 0.73ρc). This means for power-law potential (and also formore general potentials) that it must be extremely flat. As an example, in the caseV (ϕ) = m2

2 ϕ2 the mass must be extremely small,

m �√

ρc

MPl∼ 10−33 eV,


while in the case V (ϕ) = λϕ4 the coupling must be tiny,

λ � 10−122.

Thus, the quintessence idea works for very exotic scalar potentials only.

Problem 4.14. Show that for power-law potentials the condition (4.63) ensuresϕ � Hϕ. Thus, the second slow roll condition (see problem 4.11) is satisfied auto-matically.

Quintessence models can, at least in principle, be tested by observations. In thesemodels, the relation p = −ρ for dark energy is approximate. Dark energy densitydepends on time, so the cosmological expansion is different from that in the modelwith cosmological constant.

Problem 4.15. In quintessence model with potential V (ϕ) = m2

2 ϕ2, find the presentvalue of the dark energy equation of state parameter w as function of the presentvalue ϕ(t0) = ϕ0. Choose the present value ϕ0 in such a way that w0 = 0.9, andfind w(z) as function of redshift at 2 > z > 0. Take the values Ωϕ ≡ ΩΛ = 0.73,

ΩM = 0.27 and make use of the fact that the scalar field changes slowly at thepresent epoch.

Problem 4.16. Under conditions of the previous problem, and with w0 = 0.9 findnumerically the photometric distance rph(z) as function of redshift (see Sec. 4.6).Draw a plot analogous to Fig. 4.2 and compare it to the plot with ΩM = 0.27,

ΩΛ = 0.73 and time-independent ρΛ.

Finally, let us note that the future of the Universe in quintessence models is,generally speaking, different from the future of the Universe with cosmological con-stant, see also discussion in Sec. 1.4. This difference is particularly strong in modelswhere the field ϕ(t) rolls down to negative values of the potential V (ϕ): in that caseexpansion will eventually terminate, and the Universe will recollapse to singularity.

Problem 4.17. Assuming that today ϕ = ϕ0 � MPl, study the future of the Uni-verse in a model with potential V (ϕ) = m2

2 ϕ2 + εϕ, where |ε| � m2MPl is a smallparameter, and m is such that V (ϕ0) ∼ ρc. Hint: Use the results of Sec. 4.8.1.

We stress that quintessence models do not, generally speaking, solve the cos-mological constant problem. They only divide this problem into two parts: one isthe question of why the “true” cosmological constant (vacuum energy) is zero, andanother is the question of why the energy density of quintessence is so small. Thefirst question cannot in principle be answered in quintessence models; the second hasto do with naturalness of the scalar potential. Furthermore, there is a new questionin quintessence models: what mechanism ensures the “correct” present value of thescalar field? A possible answer to the latter question is given in models with suitablescalar potentials [43–47]. A subclass of these are “tracker” models [44, 47] whichwe now discuss.


4.8.3 Tracker field

Let us consider a subclass of quintessence models, where the potential has the form7

V (ϕ) =Mn+4

nϕn,

where M is a parameter of dimension of mass, and n > 2 is a numerical parameter.At radiation or matter domination, when a(t) ∝ tα, the field equation (4.47) takesthe form

ϕ +3α

tϕ − Mn+4

ϕn+1= 0. (4.64)

It has special, “tracker” solution

ϕ(tr)(t) = CM1+νtν , (4.65)

where

ν =2

n + 2, (4.66)

and C = C(n, α) is determined from Eq. (4.64).

Problem 4.18. Show that (4.65) is indeed a solution to Eq. (4.64). Find C(n, α)entering (4.65).

Solution (4.65) is a power-law attractor: if at initial time ti the field ϕi is smallerthan solution (4.65) taken at the same time, ϕi < ϕ(tr)(ti), then the driving force

F(ϕ) ≡ −V ′(ϕ) =Mn+4

ϕn+1

is larger than the driving force for the solution (4.65),

F(ϕ(t)) > F(ϕ(tr)(t)).

Hence, the solution with ϕi < ϕ(tr)(ti) catches up the tracker solution (4.65). Con-versely, a solution with ϕi > ϕ(tr)(ti) rolls slower than the tracker solution, so theformer also approaches the latter in the course of evolution. This precisely meansthat the solution (4.65) is an attractor: in sufficiently wide range of initial data,solutions tend to (4.65) at late times; the evolution of the field ϕ is basically inde-pendent of initial data and is described by the tracker solution.

The solution (4.65) obeys (we omit superscript (tr) in what follows)

ϕ2 ∼ V (ϕ) ∝ 1t2−2ν

.

It is clear, first, that the regime of evolution of the field is not the slow roll regime.Second, energy density ρϕ (see (4.58)) decreases in time slower than the energydensity of the dominant matter (relativistic or non-relativistic): the latter decreases

7This potential is rather exotic from particle physics viewpoint.


as H2 ∝ t−2. The relative contribution of the tracker field into total energy densityincreases in time as t2ν . Third, since a ∝ tα, we have

ρϕ ∝ 1

a2−2ν

α

.

Making use of (3.41) we find the following expression for the equation of stateparameter wϕ entering pϕ = wϕρϕ,

wϕ = −1 +23

1 − ν

α.

This gives, with account of (4.66) and the results of Sec. 3.2.4,

wϕ = wn

n + 2− 2

n + 2, (4.67)

where w is the parameter of equation of state of the dominant matter (w = 13 and

w = 0 for relativistic and non-relativistic matter, respectively). Thus, equation ofstate for tracker field depends on the equation of state of dominant matter. This isthe origin of the term “tracker field”. Note that at large n the equation of state ofthe tracker field is close to that of dominant matter, wϕ ≈ w.

Problem 4.19. By calculating pressure and energy density according to (4.58)derive (4.67) in an alternative way.

Problem 4.20. Show that solution (4.65) obeys V ′′ ∝ H2 at both radiation andmatter domination. This is another reason to use the term “tracker field”.

Solution (4.65) is valid at times when energy density of the tracker field is smallerthan matter energy density. The relative contribution of tracker energy into totalenergy density grows in time, and tracker field starts to dominate at some point.After that, the solution (4.65) is no longer valid. Let us find the value of the fieldat the time when it just starts to dominate. At that time

V (ϕ) ∼ ϕ2 ∼ ϕ2

t2,

i.e.,

ρϕ ∼ ϕ2

t2.

Matter energy density at that time is

ρM =38π

M2PlH

2(t) ∼ M2Pl

t2.

Requiring ρϕ ∼ ρM we find that the tracker field starts to dominate when

ϕ ∼ MPl.

Let us note in parenthesis that the latter condition also implies that the initial valueof the tracker field may be fairly arbitrary, but should obey ϕi � MPl.


At later times the field ϕ(t) increases, and fairly soon the relation (4.63) startsto hold. The evolution of ϕ enters the slow roll regime, and the expansion of theUniverse enters the regime of acceleration.

Clearly, the cosmological evolution has rather special character at the transitionperiod from matter domination to tracker field domination. Therefore, cosmologicalobservations may well be capable to confirm or falsify the model. Existing dataexclude neither cosmological constant nor tracker field; they are consistent withmany other quintessence models too.

Problem 4.21. Estimate, for various n, at what values of M the tracker field modelis consistent with known facts about the accelerated cosmological expansion.

Chapter 5

Thermodynamics inExpanding Universe

5.1 Distribution Functions for Bosons and Fermions

We consider in most Chapters of this book processes that occur in expandingUniverse filled with various species of interacting particles. As we will see, the ratesof interactions between these particles are often much higher than the expansionrate of the Universe, so the cosmic medium is in thermal equilibrium at any momentof time. So, we will need a number of basic relations and formulas from equilibriumthermodynamics. We collect them in this Chapter. We note here that as a rule,the most interesting periods in the cosmological evolution are those when one oranother reaction goes out of equilibrium (“freezes out”). The laws of equilibriumthermodynamics are still useful in such a situation for semi-quantitative analysis,as they enable us to estimate the time of departure from equilibrium and determinethe direction of inequilibrium processes.

The thermodynamical description of a system with various particle species ismade in terms of chemical potential μ for each type of particles. If there is a reaction

A1 + A2 + · · · + An ↔ B1 + B2 + · · · + Bn′ , (5.1)

where Ai, Bj denote types of particles, then the chemical potentials in thermalequilibrium1 obey

μA1 + μA2 + · · · + μAn = μB1 + μB2 + · · · + μBn′ . (5.2)

In particular, any reaction involving charged particles may occur with emission ofa photon (e.g., there is an inelastic scattering reaction e−e− → e−e−γ). Therefore,it follows from Eq. (5.2) that photon has zero chemical potential,

μγ = 0.

As another application of (5.2), let us consider the annihilation process

e+ + e− ↔ 2γ.

1More precisely, equality (5.2) holds when the system is in chemical equilibrium with respect to

the reaction (5.1), i.e., when the rate of this reaction is higher than the rate at which external

parameters evolve (higher than the expansion rate in our case).

91

92 Thermodynamics in Expanding Universe

Since μγ = 0, Eq. (5.2) gives

μe− + μe+ = 0.

Clearly, the same relation is valid for any other particles and their antiparticles, sincethey can also annihilate into photons. Thus, chemical potential of an antiparticleequals to that of a particle taken with opposite sign.

A convenient way to deal with all relevant reactions in thermalized system with variousparticle species is to introduce chemical potentials μi to conserved quantum numbers Q(i)

only. Q(i) should be independent of each other and should form a complete set of conservedquantum numbers. The chemical potential for particle of type A is then

μA =X

i

μiQ(i)A , (5.3)

where Q(i)A are quantum numbers carried by particle A. Say, at temperatures 200 MeV

T 100 GeV, conserved quantum numbers are baryon number B, lepton numbers Le,Lμ, Lτ and electric charge Q (see Appendix B, color of quarks and gluons is irrelevant

for us here), i.e., the complete set is Q(i) = B, Le, Lμ, Lτ , Q. In this case the chemicalpotential of, say, u-quark (baryon number 1/3, electric charge 2/3) is

μu =1

3μB +

2

3μQ,

while for d-quark, electron and electron neutrino we have

μd =1

3μB − 1

3μQ, μe = μLe − μQ, μνe = μLe .

Relations of the type (5.2) hold automatically for all reactions; an example is a weakprocess

u + e− → d + νe.

Problem 5.1. Show that the relations (5.2) hold automatically if chemical potentials are

given by (5.3), and charges Q(i) are conserved in all reactions (5.1), i.e.,

Q(i)A1

+ · · ·Q(i)An

= Q(i)B1

+ · · ·Q(i)Bn′ .

Now, the chemical potentials to the charges Q(i) can be found if charge densities nQ(i)

are known. Indeed, the number density of each particle species nA is a function of μA, sothe system of equations X

A

Q(i)A nA = nQ(i)

together with (5.3) determines2 all μi in terms of nQ(i) .

Particle interactions in cosmic plasma are often fairly weak; we will quantifythis statement in appropriate places. In this case the equilibrium distributions inspatial momenta p in locally Lorentz frame are given by distribution functions ofBose- and Fermi-gases,

f(p) =1

(2π)31

e(E(p)−μ)/T ∓ 1. (5.4)

2It is important here that quantum numbers Q(i) are independent and form a complete set.

5.1. Distribution Functions for Bosons and Fermions 93

Here

E(p) =√

p2 + m2 (5.5)

is energy of a particle of mass m, T is the temperature. Minus sign in (5.4) refersto bosons, plus sign to fermions.

Sometimes it is possible to disregard ±1 in the denominator in (5.4), then thedistribution has the universal Maxwell–Boltzmann form,

f(p) =1

(2π)3e−(E(p)−μ)/T . (5.6)

This form describes, in particular, low density gas of non-relativistic particles, forwhich m � T , (m − μ) � T . In the latter case

f(p) =1

(2π)3e

μ−mT · e− p2

2mT (5.7)

Upon integrating the distribution function over momenta one obtains thenumber density of particle species i,

ni = gi

∫f(p)d3p = 4πgi

∫f(E)

√E2 − m2

i EdE. (5.8)

When obtaining the second relation, we integrated over angles and used the factthat

EdE = |p|d|p|, (5.9)

following from (5.5). The factor gi in (5.8) is the number of spin states (degrees offreedom). As an example, photons, electrons and positrons have gγ = ge− = ge+ = 2,for neutrinos and antineutrinos gν = gν = 1, while the overall number of degreesof freedom of W+-, W−-, Z0-bosons and the Higgs boson in the Standard Modelat T � 100GeV is equal to 10: massive W+-, W−-, Z0-bosons have 3 polarizationseach, while the Higgs boson adds one degree of freedom.3

It follows from the form of the distribution function (5.4) that the difference ofnumber densities of particles and their antiparticles depends on chemical potential.These differences are typically very small at high enough temperatures as comparedto number densities themselves. As an example, at temperatures T � 1 GeV theasymmetry between quarks and antiquarks is of order 10−10, see Sec. 1.5.5. Therelative difference of electron and positron numbers is of the same order: the Uni-verse is electrically neutral, so the net positive charge of quarks is compensated bythe negative charge of electrons. Hence, chemical potentials are indeed very smallin the early Universe and can often be neglected.

Energy density ρi for particle species i is given by the following integral

ρi = gi

∫f(p)E(p)d3p = 4πgi

∫f(E)

√E2 − m2

i E2dE, (5.10)

where we again integrated over angles and used (5.9).

3We show in Chapter 10 that electroweak symmetry is restored at high temperatures, and W - and

Z-bosons are massless. So, another way of counting the degrees of freedom is more adequate: two

degrees of freedom due to each of the W+-, W−-, Z0-bosons and four due to the complex Higgs

doublet.


To find the expression for pressure, let us consider surface element ΔS orthogonalto the third axis. The number of particles of momenta between p and p+dp hittingthis surface during time Δt is

Δn = vzf(p)d3pΔSΔt,

where

vz =pz

E> 0

is the third component of velocity. Upon elastic scattering off the surface, particlewhose third component of momentum is pz transfers to the surface momentum

Δpz = 2pz.

Pressure is equal to the ratio of the total transfered momentum to the time Δt andarea ΔS, so we have

pi = gi

∫pz>0

2p2

z

Ef(p)d3p

(5.11)

=4πgi

3

∫ ∞

0

|p|4d|p|E(p)

f(p) =4πgi

3

∫ ∞

0

f(E)(E2 − m2i )

3/2dE,

where we made use of the fact that only half of particles move towards the surface,and that on average

〈p2z〉 =

13〈p2〉 =

13〈E2 − m2〉.

due to spatial isotropy.Let us now study the expressions for number density, energy density and pressure

in physically interesting limiting cases. We begin with gas of relativistic particles,

T � mi

at zero chemical potential,

μi = 0.

Expression (5.10) gives then the Stefan–Boltzmann law,

ρi =gi

2π2

∫E3

eE/T ∓ 1dE =

⎧⎪⎪⎨⎪⎪⎩

giπ2

30T 4 − Bose

78gi

π2

30T 4 − Fermi

(5.12)

The calculation of the integrals used in this Chapter is given, e.g., in the book[48]; we collect the relevant formulas in the end of this Section. As one shouldhave expected on dimensional grounds, the energy density is equal to T 4 up to anumerical coefficient; the contribution of fermions differs from that of bosons by afactor 7/8. Note that relativistic particles are conveniently characterized by their


helicity — spin projection onto the momentum direction; the parameter gi is thenthe number of helicity states.

If there are various relativistic particles of one and the same temperature T , andchemical potentials are negligible, the energy density of the relativistic component is

ρ = g∗π2

30T 4, (5.13)

where

g∗ =∑

bosonswith m � T

gi +78

∑fermionswith m � T

gi (5.14)

is the effective number of degrees of freedom.The expression (5.11) for pressure has a simple form in the relativistic case,

pi =gi

6π2

∫E3

eE/T ∓ 1dE =

ρi

3. (5.15)

Thus, equation of state of relativistic matter is

p =13ρ.

Finally, the expression (5.8) for number density gives in the relativistic case

ni =gi

2π2

∫E2

eE/T ∓ 1dE =

⎧⎪⎪⎨⎪⎪⎩

giζ(3)π2

T 3 − Bose

34gi

ζ(3)π2

T 3 − Fermi

(5.16a)

(5.16b)

Numerical value for zeta-function here is

ζ(3) ≈ 1.2.

Making use of (5.12), (5.16) we find the average energy per particle,

〈E〉 =ρi

ni≈{

2.70 T − Bose

3.15 T − Fermi(5.17)

As a simple example of application of these formulas, let us estimate themaximum temperature at which relativistic cosmic matter is in thermal equilibriumwith respect to electromagnetic interactions. We consider here the high temperaturecase, T � 1MeV, when electrons are relativistic. Relevant processes are Comptonscattering, e+e−-annihilation (Fig. 5.1), etc. Amplitudes of these processes are pro-portional to α ≡ e2/(4π), and the cross sections to α2. Together with dimensionalarguments, this gives for the rate (inverse of time between scatterings of a singleparticle with other particles in the medium)

Γ ∼ α2T.

The processes are in thermal equilibrium if this rate exceeds the cosmologicalexpansion rate,

Γ � H(T ).


Fig. 5.1 Feynman diagrams for Compton scattering and e+e−-annihilation.

We know from Sec. 3.2.2 that

H(T ) =T 2

M∗Pl

.

Hence, the cosmic plasma is in thermal equilibrium at

T � α2M∗Pl ∼ 1014 GeV.

Very similar but not exactly the same estimate is valid for weak and strong (QCD)interactions. Thus, the cosmic medium is indeed in thermal equilibrium during longepoch of the cosmological evolution. As we mentioned above, most interesting arein fact inequilibrium phenomena; we discuss many of them in this book.

Let us now consider the dilute gas of non-relativistic particles, whose distributionfunction is given by (5.6). In this case the expression for particle number density is

ni = gi

(miT

2π

)3/2

eμi−mi

T , (5.18)

while energy and pressure are

ρi = mini +32niTi (5.19)

and

pi = Tni � ρi. (5.20)

As anticipated, non-relativistic gas has equation of state p = 0 up to corrections oforder O(T/m).

Problem 5.2. Show that at temperature exceeding the mass and chemical potential,T � m, μ, the difference of the number densities of particles and their antiparticlesof a given helicity is

Δn = μT 2

3− Bose (5.21)

and

Δn = μT 2

6− Fermi. (5.22)

Hints: (1) Recall that chemical potentials of particles and their antiparticles differby sign only; (2) Take linear in μ term in the integral (5.8) with m = 0, integrateby parts and use formulas presented in the end of this Section.


Problem 5.3. Find the differences of the number densities of all relativisticStandard Model particles and their antiparticles in the cosmic medium at tem-perature T = 400MeV for given densities of baryon and lepton numbers nB ,

nLe , nLμ , nLτ in the realistic case nB , nLe , nLμ , nLτ � T 3. Hints: (1) At thistemperature, relativistic are u-, d-, s-quarks, gluons, photons, electrons, muonsand all neutrino species; other Standard Model particles are non-relativistic;(2) neutrinos have one helicity, while other fermions have two; (3) quarks havethree color states, gluons have eight.

Let us give numerical values of useful integrals.

(1) Z ∞

0

dz

ez + 1= log 2.

(2) For positive integer nZ ∞

0

z2n−1

ez − 1dz =

(2π)2n

4nBn,

Z ∞

0

z2n−1

ez + 1dz =

22n−1 − 1

2nπ2nBn,

where Bn are Bernoulli numbers,

B1 =1

6, B2 =

1

30, B3 =

1

42, . . .

(3) For arbitrary x

Z ∞

0

zx−1

ez − 1dz = Γ(x)ζ(x),

Z ∞

0

zx−1

ez + 1dz = (1 − 21−x)Γ(x)ζ(x),

where ζ(x) is the Riemann zeta-function, some of whose values are

ζ(3) = 1.202, ζ(5) = 1.037, ζ(3/2) = 2.612, ζ(5/2) = 1.341.

Recall that for positive integer n

Γ(n) = (n − 1)!

The values of gamma-function Γ(x) at half-integer x can be found by making use of thevalue

Γ(1/2) =√

π

and the relation Γ(1 + x) = xΓ(x).


5.2 Entropy in Expanding Universe. Baryon-to-Photon Ratio

One of the main thermodynamical characteristics of a system is its entropy. Hence,it is useful to discuss entropy of cosmic medium in the expanding Universe. Recallthat entropy in thermodynamics enters the first law. In the general case of variablenumber of particles the latter has the form

dE = TdS − pdV +∑

i

μidNi, (5.23)

where S is entropy of the system and subscript i labels particle species. This sub-script and the corresponding summation will be omitted wherever possible.

Energy E and number of particles are extensive quantities (proportional tovolume of the system), while temperature and pressure are local characteristicsindependent of the volume. In accordance to the first law (5.23) entropy is extensivequantity. It is convenient to introduce densities

ρ ≡ E

V, n ≡ N

V, s ≡ S

V. (5.24)

Then we come to the following form of the first law,

(Ts− p − ρ + μn) dV + (Tds− dρ + μdn)V = 0. (5.25)

This relation is valid for both the entire system and any of its part. Using it for aregion of constant volume inside the system we obtain

Tds = dρ − μdn.

We now use (5.25) for the entire system and find for entropy density

s =p + ρ − μn

T.

An important example is relativistic matter at vanishing chemical potentials, forwhich

s =p + ρ

T. (5.26)

Making use of the latter relation and expressions of the previous Section for ρ andp we obtain the following expression for the contribution of i-th particle species,

si =43

ρi

T=

⎧⎪⎪⎨⎪⎪⎩

gi2π2

45T 3 − Bose

78gi

2π2

45T 3 − Fermi

(5.27)

By comparing these expressions to formulas (5.16) for number density, we see thatentropy and number of particles differ in relativistic case by a numerical factor oforder 1 only.

To obtain the expression for entropy in non-relativistic case, we make use of(5.19) and (5.20) and write

si =52ni +

mi − μi

Tni.

5.2. Entropy in Expanding Universe. Baryon-to-Photon Ratio 99

The chemical potential is related to the number density by Eq. (5.18), so we findfinally

si = ni

{52

+ log

[gi

ni

(miT

2π

)3/2]}

.

Number density of non-relativistic particles is always small in cosmic medium; attemperatures T � 100MeV, when protons and neutrons are non-relativistic, theirnumber density is estimated as nB ∼ 10−9nγ , see (1.21). The same estimate appliesto electrons which are non-relativistic at T � 0.5MeV. Thus, non-relativistic con-tribution to entropy is negligible, so the total entropy is given by

s = g∗2π2

45T 3, (5.28)

where the effective number of relativistic degrees of freedom is defined by (5.14).One of the key properties of entropy making it a very useful quantity is the

second law of thermodynamics. According to this law, entropy of any closed systemcan only increase, and it stays constant for equilibrium evolution, i.e., slow evolutionduring which the system always remains in thermal equilibrium. Let us see how thislaw works in expanding Universe, assuming that the evolution of cosmic mediumis close to equilibrium. To this end, we come back to the relation (5.23) and recallthat it is adequate to introduce chemical potentials to conserved quantum numbers,rather than to individual particle species; thus, dNi in (5.23) should be understoodas differentials of conserved quantum numbers. With this qualification, we apply therelation (5.23) to the comoving volume, V = a3. Then the conservation of quantumnumbers in comoving volume gives dNi = 0, so that

TdS

dt≡ T

d(a3s)dt

= (p + ρ)dV + V dρ = a3

[(p + ρ) · 3 a

a+ ρ

]= 0. (5.29)

where we used the covariant conservation of energy in expanding Universe,Eq. (3.13). We see that total entropy in comoving volume is conserved,

sa3 = const. (5.30)

As we have already mentioned in Sec. 3.1, the covariant conservation of energy hassimple interpretation in terms of the entropy conservation in comoving volume.

Entropy conservation is used for introducing quantitative, time-independentcharacteristics of asymmetries of conserved quantum numbers. In particular, aswe discussed in Chapter 1, there is baryon asymmetry in the Universe, as there areprotons and neutrons but none of their antiparticles today. At temperatures belowT ∼ 100GeV, there are no processes violating baryon number, so it is conserved incomoving volume,4

(nB − nB)a3 = const, (5.31)

4In some scenarios, baryon asymmetry is generated at temperatures below 100 GeV, see, e.g.,

Sec. 11.6. We do not consider this possibility here.


where nB and nB are number densities of baryons and antibaryons. By comparing(5.30) and (5.31) we see that ratio

ΔB =nB − nB

s(5.32)

is a time-independent characteristic of the baryon asymmetry.Let us give an estimate for the baryon asymmetry in the early Universe. When

discussing the Universe at relatively low temperatures (T � 1MeV), one tradi-tionally uses baryon-to-photon ratio

ηB =nB

nγ,

where nγ is the photon number density. At these temperatures ηB is independent oftime, and it differs from ΔB by a numerical factor of order 1. The latter factor is dueto both the difference between nγ and photon entropy sγ, see (5.16) and (5.28), andthe neutrino contribution to entropy, see Chapter 7. At T � 1MeV (but, strictlyspeaking, at temperatures exceeding the neutrino masses) entropy density is

s =2π2

45g∗,0T

3, (5.33)

where

g∗,0 = 2 +78· 2 · 3 · 4

11=

4311

is the effective number of relativistic degrees of freedom after neutrino decoupling.5

The first and second terms here are due to photons and neutrinos, respectively(effective temperature of neutrinos is related to photon temperature Tγ ≡ T by(4.13); the origin of the factor 7

8 · 2 · 3 is the same as in (4.14)). It is useful for whatfollows to note that the present value of entropy density is6

s0 = 2.9 · 103 cm−3. (5.34)

Thus, we have

ΔB =nγ

sηB =

2ζ(3)π2

2π2

45 g∗,0

· ηB = 0.14 · ηB. (5.35)

The value of baryon-to-photon ratio obtained from studies of BBN and CMB isquoted in (1.21). With this value we have

ΔB = 0.88 · 10−10.

As we discussed in the previous Section, this means, in particular, that baryonchemical potential is very small at T � 200MeV.

5Note the difference between the parameters g∗,0 and g∗, the latter entering (4.22).6There is a subtlety here, since photons and neutrinos are non-interacting in the present Universe.

This is not very relevant, however, since the distributions of these particles in momenta are thermal

and relativistic, see Sec. 2.5. To be more precise, one should consider the right hand side of (5.33) as

the definition of s in the recent and present Universe. This definition is useful precisely because sa3

remains constant throughout the history of the Universe, unless there are inequilibrium processes.

5.3. ∗Models with Intermediate Matter Dominated Stage: Entropy Generation 101

Problem 5.4. Estimate the value of the chemical potential for u-quark at temper-ature 1 GeV.

To conclude this Section we note that the baryon-to-photon ratio is directlyrelated to the relative energy density of baryons ΩB = ρB/ρc = mBnB/ρc, seeChapter 4. The present number of photons calculated according to (5.16a) is

nγ = 2ζ(3)π2

T 30 = 411 cm−3.

Making use of the value (1.14) of critical density, we find from (1.21)

ΩBh2 0.023, (5.36)

and hence ΩB 0.046, the value quoted in Sec. 4.1.

Problem 5.5. Estimate the contribution of electrons and protons into total energyat T = 50 keV and T = 1 eV. Hint: Recall that electrons and protons are non-relativistic at these temperatures. Neglect the presence of light nuclei in the plasma.

5.3 ∗Models with Intermediate Matter Dominated Stage: EntropyGeneration

Let us consider a possibility that at some early epoch, before relatively late radi-ation domination, the expansion of the Universe was matter dominated. There areat least two scenarios for how that can happen. One of them assumes that there existnew heavy particles G in Nature whose lifetime τ is large enough, and that in thebeginning of the Hot Big Bang epoch the expansion is radiation dominated, but thecosmic medium contains an admixture of G-particles. G-particles are non-relativisticat temperature below their mass. As the Universe expands, energy density of therelativistic component decreases as a−4, while energy density of non-relativisticG-particles decreases as a−3. Therefore, at a certain moment of time t∗ theexpansion becomes dominated by non-relativistic G-particles, and the intermediatematter dominated stage begins. The main assumption here is that

t∗ � τ ≡ Γ−1, (5.37)

where Γ is the total decay width of G-particle. During later evolution of the Universe,an important process is G-particle decay into light particles, say, quarks, leptonsand other Standard Model particles. These rapidly (faster than in one Hubble time)thermalize and become part of the relativistic component. We will see that theepoch of G-particle domination ends up soon after the expansion rate slows downto H ∼ Γ: at about that time, all G-particles decay away, the Universe becomesfilled with hot relativistic matter and the radiation domination resumes. We notethat thermal equilibrium is strongly violated in the time interval t∗ < t < Γ−1: theconcentration of G-particles is very far from equilibrium. This property is of coursea consequence of (5.37); it gives rise to strong entropy generation in the Universe.


This scenario is realized in some models with heavy gravitino, see Sec. 9.6.3,hence our notation for the new heavy particles. Another example is unstable sterileneutrino.

The second scenario makes use of the observation that the role of non-relativisticmatter in cosmology may be played by massive spatially homogeneous scalar fieldoscillating near the minimum of its scalar potential, see Sec. 4.8.1. As we discussedin Sec. 4.8.1, this field can be viewed as a collection of large number of particlesat rest. It is assumed that before the Hot Big Bang epoch, all energy is due tothis field, and its oscillations begin at time t∗ (before that the field evolves in theslow roll regime, see Sec. 4.8.1). Hence, there is no relativistic matter at all at timet∗. One assumes further that the decay width of the scalar particles again obeyst∗ � Γ−1, and that these decays is the main mechanism of the decay of the scalarfield oscillations. In fact, the latter assumption is pretty strong: there are alternativemechanisms leading to the decay of coherent oscillations of scalar fields, which areoften more efficient. We discuss some of these mechanisms in the accompanyingbook, and here we will proceed under the above assumption. Then, like in the firstscenario, the Universe becomes radiation dominated when H(t) ∼ Γ.

The analysis of the two scenarios goes in parallel up to some point. Let ρM andρrad be energy densities of non-relativistic and relativistic component, respectively.Let us assume for simplicity that the effective number of relativistic degrees offreedom g∗ does not change in time (the opposite case is the subject of the problemin the end of this Section). The number of heavy particles in comoving volume(nMa3) decreases only due to their decays, hence

d

dt(nMa3) = −ΓnMa3.

Since ρM ∝ nM , we find

ρM + 3HρM = −ΓρM . (5.38)

Energy density of relativistic particles decreases due to the cosmological expansion,since their number density gets diluted and energy of each particle gets redshifted.However, energy is injected into this component due to decays of heavy particles.Hence, equation for ρrad has the form

ρrad + 4Hρrad = ΓρM . (5.39)

The third equation that closes the system of evolution equations is the Friedmannequation

H2 =8π

3G(ρM + ρrad). (5.40)

Our purpose is to study the solutions to the system (5.38)–(5.40).

Problem 5.6. Making use of Eqs. (5.38) and (5.39) show that the total energy andpressure (sums of energies and pressures of the two components) obey the covariantconservation equation ρtot = −3H(ρtot + ptot).

5.3. ∗Models with Intermediate Matter Dominated Stage: Entropy Generation 103

In the first place, the solution to (5.38) is

ρM (t) =consta3(T )

e−Γt. (5.41)

At both radiation and matter domination one has t ∼ H−1, so the energy density ofnon-relativistic component is small at late times, when Γ � H(t). In this regime theright hand side of Eq. (5.39) can be neglected, and the energy density of relativisticmatter falls as a−4. This is power-law rather than exponential decrease, hence atΓ � H(t) one has ρrad � ρM . Thus, the expansion is indeed radiation dominatedat late times. The time of transition from matter domination tMD→RD is roughlyestimated as

Γ ∼ H(tMD→RD). (5.42)

The above reasoning does not exclude the possibility that the right hand side of(5.42) contains a logarithmically large factor, so that the transition occurs somewhatlater; neither it excludes earlier transition, at time when Γ � H . We will see thatneither of this possibilities is realized.

Let us consider solutions to Eqs. (5.38)–(5.40) after the beginning of matterdominated stage, t > t∗, but at sufficiently early times when Γ � H(t). In otherwords, we are interested in the time interval

t∗ < t � Γ−1.

Let us assume that the expansion is matter dominated in the whole interval; we willjustify this assumption a posteriori. At these times the exponential factor in (5.41)is nearly equal to 1, i.e., the loss of particles due to their decays is negligible. Hence,the evolution of the Universe is described by formulas of Sec. 3.2.1; in particular,a ∝ t2/3 and

H =23t

, ρM =3

8πGH2 =

16πGt2

.

By substituting the latter expressions into (5.39), we obtain the equation

ρrad +83t

ρrad =Γ

6πG

1t2

.

Its general solution is

ρrad =Γ

10πG

1t

+C

t8/3, (5.43)

where C is an arbitrary constant. This constant is determined by initial data andit is different in the two scenarios outlined above.

Let us begin with the first of them. We will see that at the time t∗ of thetransition to matter domination, the first term in (5.43) is small, and the propertythat ρrad(t∗) ρM(t∗) gives

C

t8/3∗

16πGt2∗

.


Hence, the behavior of ρrad at t � t∗ is

ρrad =Γ

10πG

1t

+1

6πG

t2/3∗

t8/3≡ ρgen + ρinit. (5.44)

The first term here is due to injection of energy through decays of heavy particles,while the second is the energy density of the relativistic matter existing in theUniverse initially. Note that the first term decreases in time relatively slowly, whilethe second term decays as a−4, as should be the case. Our basic assumption Γ � t−1

∗implies that the first, induced term is indeed small at t = t∗; it starts to dominate att t

2/5∗ Γ−3/5 � Γ−1, i.e., long before the transition back to radiation domination.

At the latter time, efficient generation of entropy begins.Let us point out that ρrad, and hence the temperature of relativistic component,

monotonically decreases in time. This may be a surprise: one might naively expectthat maximum temperature in the Universe is reached at H(t) ∼ Γ, when most ofheavy particles decay. The above analysis shows that the smallness of the relativenumber of heavy particle decays at early times is compensated for by high numberdensity of these particles, so the contribution of the decays into ρrad is not small atsmall t.

Problem 5.7. Show that temperature of the hot component monotonically decreasesat H(t) � Γ as well.

In the second scenario, ρrad = 0 at t = t∗, which gives

ρrad =Γ

10πG

(1t− t

5/3∗

t8/3

). (5.45)

Energy density of hot matter rapidly (during time of order t∗) increases from zeroto its maximum value7

ρrad,max ∼ Γ10πGt∗

,

and then decreases as t−1. The maximum temperature is reached soon after thebeginning of matter dominated stage.

In both scenarios one has ρrad � ρM at times t � Γ−1. The energy density ofhot component approaches that of heavy particles as t approaches tMD→RD ∼ Γ−1,i.e., the time when H ∼ Γ. This means that the relation (5.42) is valid without largelogarithms. Right after the transition to radiation domination one can use formulasof Sec. 3.2.2 for estimates, so the relation (5.42) can be written as

Γ ∼ T 2MD→RD

M∗Pl

.

7Since the reheating is fast, the approximation of instantaneous beginning of matter dominated

stage is not adequate. For this reason, the numerical coefficient in (5.45) depends on the details of

evolution at the beginning of that stage.

5.4. ∗Inequilibrium Processes 105

It is known from the analysis of Big Bang Nucleosynthesis (see Chapter 8) that theexpansion was radiation dominated at temperatures of at least TBBN ∼ 1MeV, sothere is a bound TMD→RD > TBBN . This bound gives the bound on the lifetime ofheavy particles,

τ = Γ−1 � M∗PL(TBBN )T 2BBN

1 s. (5.46)

This bound is important, e.g., for some models with heavy gravitino, see Sec. 9.6.3.Decays of heavy particles is a strong source of entropy in the above scenarios.

In the first of them, entropy density at the time of the transition back to radiationdomination is mostly due to these decays, and its value is sgen ∼ ρ

3/4gen(t = Γ−1).

If there were no heavy particles, the entropy density would be equal to sinit ∼ρ3/4init(t = Γ−1), see (5.44). This gives the entropy growth factor

sgen

sinit∼ 1√

Γt∗.

Clearly, this factor is large for t∗ � Γ−1 i.e., entropy generation is very effective.In the second scenario, there is no entropy in the beginning at all, so all entropycomes from decay processes.

Problem 5.8. Generalize the analysis of this Section to the case when effectivenumber of relativistic degrees of freedom g∗ changes in time. Hint: Instead ofEq. (5.39), derive and use similar equation for entropy of the hot component.

5.4 ∗Inequilibrium Processes

We often encounter in this book a situation when most processes in cosmic mediumare fast, but there is one or several slow processes whose low rate keeps the mediumout of complete thermal equilibrium.

Let us give a typical (but by no means unique) example of such a situation.Let there exist new long living heavy particle X (say, of mass mX � 100GeV),and the only processes by which the number of X-particles and their antiparticlesX in comoving volume can change are their pair creation and annihilation. Letus assume that scattering of X-particles off the Standard Model particles (quarks,leptons, photons, etc.) occurs at sufficiently high rate. At temperature T exceedingthe mass of X-particles mX, creation and annihilation of X-particles are typicallyalso rapid, so they have equilibrium abundance. Thus, the system is in completethermal equilibrium at T � mX. Let us consider symmetric medium with equalnumber of particles X and their antiparticles X. At T � mX , the number densityof X-particles exponentially decays as temperature decreases, and the annihilationof X-particles occurs less and less often simply because the probability for anX-particle to meet its antiparticle becomes smaller and smaller. This is an exampleof the situation we are going to consider: at T � mX all processes except for anni-hilation and creation of X-X-pairs are fast, while the latter processes are slow, and


the abundances nX and nX = nX may differ from the equilibrium abundance. Inthis case the medium still has well-defined temperature and numbers of particles(including numbers of X and X themselves). The distribution functions are givenby formulas of Sec. 5.1; the fact that the system is out of thermal equilibrium isreflected by the property that μX �= 0, and μX = μX for symmetric matter (recallthat μX = 0 for symmetric matter in equilibrium).

Leaving aside for a while the cosmological expansion, let us discuss how thesystem relaxes to thermal equilibrium in these cases. In general, as the systemrelaxes, its effective temperature and number densities of all particles evolve in time.Temperature changes, if the relaxation occurs with heat release; this is the case inour example if the X-particle abundance is higher than in equilibrium, nX > neq

X ,so the relaxation proceeds via annihilation (rather than creation) of X-X-pairs.Still, at any moment of time, temperature and chemical potentials take well-definedvalues. So, at any time the system has a certain free energy8 F (T, Ni). Let us recallthat free energy is related to the number of states (partition function) by

Z = e−F/T . (5.47)

where we assume that the system has finite, albeit large, volume. As the systemapproaches thermal equilibrium, free energy tends to its minimum, and partitionfunction to its maximum.

The system relaxes to thermal equilibrium due to “direct” slow process (in ourexample this is X-X-annihilation; we assume for definiteness that nX > neq

X ). Letus denote the rate of these processes (number of their occurrences per unit time perunit volume) by9 Γ+. There are of course inverse processes whose rate is denotedby Γ− (these are processes of X-X-pair creation in our example). Let us find therelation between Γ+ and Γ−.

As a result of a single direct process the system makes a transition from a statewith smaller partition function Z−, and hence larger free energy F−, to the statewith larger partition function Z+ and smaller free energy F+. Let i− be one ofmicroscopic states10 before the direct process, and j+ be one of microscopic statesthat can appear after the direct process. The probability that the system is in theinitial state i− is

P (i−) =e−E(i−)/T

Z−,

where E(i−) is the energy of the state i− (it is important here that the system is inthermal equilibrium with respect to all processes except for the one we consider). Let

8It is often more convenient to use Grand potential rather than free energy, considering chemical

potentials rather than particle numbers as the thermodynamical parameters. In this Section,

however, we will discuss states with fixed number of particles. The formula (5.47) is valid pre-

cisely for these states.9The notation has to do with the fact that the direct process increases the value of the partition

function; this process is thermodynamically favored.10As an example, in classical mechanics this is a state in which position and velocity of every

particle are well-defined.


γ(i− → j+) be the probability of the process i− → j+. Then the total probabilityof the direct process in the system is

Γ+ =∑

i−,j+

P (i−) · γ(i− → j+) =1

Z−

∑i−,j+

e−E(i−)/T γ(i− → j+).

Similarly, the total probability of an inverse process is

Γ− =1

Z+

∑i−,j+

e−E(j+)/T γ(j+ → i−).

The probabilities of the process i− → j+ and the inverse process j+ → i− are equalto each other, γ(j+ → i−) = γ(i− → j+). Making use of this fact, and also theenergy conservation, E(i−) = E(j+), we obtain

Γ+

Γ−=

Z+

Z−= e(F+−F−)/T ,

and finally

Γ+

Γ−= e−ΔF/T , (5.48)

where ΔF is the difference of free energies after and before the direct process;according to our definitions, ΔF < 0. In what follows, Γ+ and Γ− denote thenumber of direct and inverse processes occurring per unit time per unit volume.Clearly, the formula (5.48) is valid for these quantities too. The relation (5.48) isknown as the general formula of detailed balance.

In our example, we have

ΔF =∂F

∂NX

ΔNX +∂F

∂NX

ΔNX ,

where ΔNX = ΔNX = −1 is the change in the number of X- and X-particles in asingle direct process (annihilation). Hence,

ΔF = −2μX, (5.49)

where we made use of the facts that ∂F/∂NX = μX and μX = μX .The relation (5.48) shows, in the first place, that free energy indeed decreases

during relaxation: Γ+ > Γ− for ΔF < 0. Now, if |ΔF | � T then Γ+ � Γ−, sothe inverse processes are negligible. Conversely, if |ΔF | � T then the deviationfrom thermal equilibrium is small, (Γ+ − Γ−) � Γ±, so direct and inverse processesoccur at almost equal rates. In the latter case, the rates are approximately the sameas in equilibrium Γ+ ≈ Γ− ≈ Γeq, so the relaxation rate is given by

Γ+ − Γ− = −ΔF

TΓeq, |ΔF | � T. (5.50)


In the situations we discuss, it is often possible to introduce local (independentof volume V ) parameter characterizing the medium, n = N/V . In our example thisis the number density of X-particles, while N is the total number of X-particles inthe system. The quantity N does not change in fast processes and changes by ΔN

and (−ΔN) in each direct and inverse process. Still leaving aside the expansion ofthe Universe, let us write down the equation for n(t),

dn

dt= ΔN · (Γ+ − Γ−) (5.51)

(recall that Γ± are the rates per unit volume). The free energy as function of N isat minimum in thermal equilibrium, so

(ΔF )eq =(

∂F

∂N

)eq

· ΔN = 0.

Near thermal equilibrium, ΔF is proportional to the deviation of n from its equi-librium value neq, hence

ΔF = −α (n − neq) ,

where α is a positive parameter. Equation (5.51) then takes the form

dn

dt=

α

TΔNΓeq (n − neq) , (5.52)

Since n does not depend on time in thermal equilibrium,dneq

dt= 0,

Equation (5.52) can be written as

d (n − neq)dt

=α

TΔNΓeq (n − neq) . (5.53)

It is now clear that the relaxation to thermal equilibrium has exponential char-acter.11

In our example of symmetric medium, the abundance of particles X and X atT � mX is given by (5.18) with non-vanishing μX = μX, hence

nX = nX = eμX/T neqX .

Thus, near thermal equilibrium, i.e., at μX

T � 1, Eq. (5.49) gives

ΔF = −2μX = −2TnX − neq

X

neqX

.

We see that α = 2Tneq

Xand, since ΔN = −1, the relaxation to thermal equilibrium is

described by the equation

d (nX − neqX )

dt= −2

Γeq

neqX

(nX − neqX

) . (5.54)

11Note that the right hand side of Eq. (5.53) is always negative: for n > neq the direct process

decreases n, i.e., ΔN < 0, and vice versa.


Finally, let us note that the equilibrium rate of annihilation processes per unitvolume is

Γeq = ΓeqX · neq

X ,

where ΓX,eq = τ−1X is the lifetime of X-particle in the equilibrated medium. We get

d (nX − neqX )

dt= −2Γeq

X (nX − neqX ) .

which is the Boltzmann equation for the number density in our example.

The latter result can be obtained also in elementary way. Let nX and nX be numberdensities of particles and antiparticles. Then the probability of annihilation of a givenparticle X with some antiparticle X per unit time is

ΓX = σann · nX · vX ,

where vX is X-particle velocity, and σann is the annihilation cross section (during timeτX = Γ−1

X a particle of cross section σann meets precisely one antiparticle X). Hence, theannihilation probability per unit time per unit volume is

Γ+ = ΓX · nX = σann · nX · vX · nX = Γeq nX

neqX

nX

neqX

. (5.55)

The probability of the inverse process of X-X-pair creation is independent of the actualnumber of X-particles in the medium, so it is given by

Γ− = Γeq.

Hence

d (nX − neqX )

dt= Γeq

„1 − nXnX

neqX

neqX

«.

In symmetric medium, nX = nX and neqX

= neqX , so for |nX − neq

X | neqX we recover

Eq. (5.54).

Equation (5.51) is no longer valid in expanding Universe. As the Universeexpands, the parameters like number density decrease as a−3 even if slow processesare switched off. So, it makes sense to consider the quantity N in comoving volume,N ∝ na3. Instead of (5.51), we now have the following equation,

d(na3

)dt

= ΔN · (Γ+ − Γ−) · a3, (5.56)

where Γ+ and Γ− are still the rates of slow processes per unit physical volume,which in general obey the relation (5.48). Clearly, Γ+ and Γ− depend on time,since temperature and number densities decrease in time. Near thermal equilibrium,Eq. (5.52) is generalized to

d(na3

)dt

=α

TΔN · Γeq (n − neq) · a3. (5.57)

Equation (5.56) is the Boltzmann equation for the number density in expandingUniverse. We will repeatedly encounter equations like this in the course of ourstudy.

Chapter 6

Recombination

6.1 Recombination Temperature in Equilibrium Approximation

At temperatures exceeding the binding energy of electrons in atoms, the matterin the Universe was in the phase of plasma consisting of electrons, photons andbaryons. As we discuss in Chapter 8, at temperatures below a few dozen keVthe baryon component was predominantly protons (about 75% of total mass) and4He nuclei (about 25% of total mass). As the Universe cools down, it becomesthermodynamically favourable for baryons and electrons to combine into atoms.This cosmological epoch is called recombination. Before recombination, the plasmawas opaque because of photon scattering off free electrons, while after recom-bination relic photons propagate freely. These are the CMB photons we seetoday.

We are interested in recombination of hydrogen, since recombination of heliumoccurs earlier and has very little effect on properties of photon-electron-protonplasma afterwards. The recombination of helium is the subject of Problem 6.5.

We begin our analysis by determining the temperature at which formation ofatoms becomes thermodynamically favourable. We will assume temporarily thatthe expansion of the Universe is adiabatically slow. Under this assumption, themedium is always in thermal equilibrium. In particular, there is chemical equilibriumbetween its components. This is not an adequate approximation: recombination isnot an equilibrium process [49, 50], it is slightly delayed. We consider inequilibriumrecombination in Sec. 6.2, and here we restrict ourselves to the complete thermalequilibrium. In this way we will understand the relevant range of temperatures andcosmological times. We will also obtain the equation of chemical equilibrium (Sahaequation) which is used in this and other Chapters. We stress again that one shouldbear in mind that the picture we consider in this Section is not directly relevant forrecombination in the real Universe.

One could naively expect that the temperature of equilibrium recombination issimilar to the binding energy of electron in hydrogen atom. This is not the case,however: the recombination occurs at much lower temperature. The physical reasonis as follows. At low electron and proton densities, the recombination of a given

111

112 Recombination

electron with some proton occurs in time1 τ+, inversely proportional to the numberdensity of protons τ+ ∝ n−1

B . The newly formed hydrogen atom is ionized by aphoton whose energy exceeds the binding energy of hydrogen atom, ΔH . Thereexist thermal photons of these energies even at T � ΔH , though their number isexponentially small. Hence, the lifetime of a hydrogen atom in medium is finite,albeit exponentially long, τ− ∝ eΔH/T . Recombination becomes effective when

τ+ ∼ τ−, (6.1)

which indeed gives T � ΔH for small nB (and small electron velocities which enterτ+). One could find the equilibrium recombination temperature by elaborating onthis kinetic approach (see problem below). It is simpler, however, to make use ofthermodynamical approach which we now present.

We will see that the temperatures of interest are about 0.3 eV. At these temper-atures and in chemical equilibrium, electrons and protons are non-relativistic, whilemost hydrogen atoms are in the ground state (see Problem 6.6). The equilibriumnumber densities of electrons, protons and hydrogen atoms in the ground state are

ne = ge

(meT

2π

)3/2

e(μe−me)/T , (6.2)

np = gp

(mpT

2π

)3/2

e(μp−mp)/T , (6.3)

nH = gH

(mHT

2π

)3/2

e(μH−mH)/T . (6.4)

These formulas involve yet unknown chemical potentials. The number of spin statesof proton and electron equals ge = gp = 2, while for hydrogen atom gH = 4.

Problem 6.1. Show that gH = 4.

The equilibrium recombination temperature T eqr is determined from the relation

np(T eqr ) nH(T eq

r ). (6.5)

To find T eqr , we need three more equations that complete the system of equations on

number densities and chemical potentials at given temperature. One is the baryonnumber conservation,

np + nH = nB. (6.6)

Here and in the next Section (and only in these Sections) nB denotes the baryonnumber density without helium (for total baryon number density we use the notationntot

B ),

nB(T ) = ηBnγ(T ), ηB = 0.75ηB. (6.7)

1Superscripts + and − refer to direct and inverse processes — recombination and ionization,

respectively.

6.1. Recombination Temperature in Equilibrium Approximation 113

The baryon-to-photon ratio is given by (1.21), so that

ηB = 4.6 · 10−10, (6.8)

while the number density of photons nγ(T ) is a known function of temperature, see(5.16a). Another equation,

μp + μe = μH , (6.9)

follows from the assumption of thermal equilibrium for the reaction

p + e ↔ H + γ. (6.10)

The last equation expresses electric neutrality of the Universe,

np = ne. (6.11)

Thus, we have six equations, (6.2), (6.3), (6.4), (6.6), (6.9) and (6.11), determiningthree number densities ne, np, nH and three chemical potentials μe, μp, μH at giventemperature T . The quickest way to simplify this system of equations is to multiplyEqs. (6.2) and (6.3):

npne = gpge

(mpT

2π

)3/2(meT

2π

)3/2

e(μp+μe−mp−me)/T .

Now, using (6.4), (6.9) and (6.11) we get rid of chemical potentials and write thelatter equation in the form

n2p =

(meT

2π

)3/2

nHe−ΔH/T , (6.12)

where

ΔH ≡ mp + me − mH = 13.6 eV

is the binding energy of hydrogen atom, and we neglected the difference between mp

and mH in the pre-exponential factor in (6.12). Together with Eq. (6.6), equation(6.12) determines the equilibrium proton and hydrogen densities np and nH.

It is convenient to introduce dimensionless ratios

Xp ≡ np

nB

, XH ≡ nH

nB

.

Then the baryon number conservation equation is

Xp + XH = 1. (6.13)

Using Eq. (6.12) we express XH through Xp, and then Eq. (6.13) becomes theequation for the only variable Xp,

Xp + nBX2p

(2π

meT

)3/2

eΔH

T = 1. (6.14)

Equation (6.14) (as well as Eq. (6.12)) is known as Saha equation. It is convenientto express nB through the baryon-to-photon ratio ηB by making use of (6.7) and

114 Recombination

then use the formula (5.16a) for photon number density nγ(T ). This leads to theequation containing dimensionless quantities only,

Xp +2ζ(3)π2

ηB

(2πT

me

)3/2

X2pe

ΔHT = 1. (6.15)

The second term here is the equilibrium number density of hydrogen atomsexpressed in terms of Xp,

XH =2ζ(3)π2

ηB

(2πT

me

)3/2

X2pe

ΔHT . (6.16)

We now see explicitly that recombination occurs at temperatures well below thebinding energy ΔH . Indeed, unless the exponential factor in (6.16) is very large, theequilibrium proton abundance Xp is close to 1, while the hydrogen abundance XH issmall. The suppression is due to both smallness of the baryon-to-photon ratio ηB andthe fact that electrons are non-relativistic at temperatures of interest, T/me � 1.In chemical equilibrium, recombination begins when Xp ∼ 1 and XH ∼ 1 at thesame time. At that time the suppression by pre-exponential factors in (6.16) iscompensated for by the exponential factor eΔH/T . This occurs when ΔH/T is muchgreater than 1,

ΔH

T eqr

− log

[2ζ(3)π2

ηB

(2πT eq

r

me

)3/2]

. (6.17)

The latter equation is obtained by substituting XH(T eqr ) ∼ Xp(T eq

r ) ∼ 1 intoEq. (6.16). Equation (6.17) determines the equilibrium recombination temperature.

It is useful at this point to discuss solutions of equations of the type (6.17). Thisequation belongs to the class of equations of the following form,

x = log (Axα) . (6.18)

In our case the unknown x is

x =ΔH

T eqr

,

while parameters A and α are given by

α =32, A =

√π

4√

2ζ(3)

(me

ΔH

)3/2

η−1B .

It is important that

log A � 1. (6.19)

In the leading logarithmic approximation, i.e., treating log A as a large parameter,we can set x = 1 in the right hand side of Eq. (6.18) and immediately obtain thesolution

x ≈ log A. (6.20)

Problem 6.2. The solution x can be written as

x = (1 + ε) logA.

6.1. Recombination Temperature in Equilibrium Approximation 115

Find the correction ε to the first non-trivial order in (log A)−1 and show that it isindeed small provided that α ∼ 1.

Coming back to recombination, we obtain in the leading logarithmic approxi-mation

T eqr ≈ ΔH

log

[ √π

4√

2ζ(3)

(me

ΔH

)3/2

η−1B

] ≈ 0.38 eV. (6.21)

The numerical solution of (6.17) with the effective value ηB presented in (6.8) givesT eq

r = 0.33 eV. The corresponding redshift is zr ≈ 1400.By comparing the equilibrium recombination temperature (6.21) to the tem-

perature at equality (4.21) we see that recombination occurs at matter-dominatedstage. Let us find the age of the Universe at T eq

r = 0.33 eV. To this end, we use therelation (3.22) valid at matter domination and write

teqr =

23H−1(teq

r ) =[

M2Pl

6πρM (T eqr )

]1/2

.

The energy density of non-relativistic matter (including dark matter) is

ρM(T ) =ΩM

ΩB

· mp · ntotB

(T ).

The total baryon number density is ntotB = ηB · nγ(T ). Making use of (5.16a) we

obtain finally

teqr =

[π

12ζ(3)ΩB

ΩM

M2Pl

ηB (T eqr )3 mp

]1/2

. (6.22)

Numerically teqr 340 thousand years, while the Hubble time is

H−1(T eqr ) 510 thousand years (6.23)

for T eqr = 0.33 eV, ΩB = 0.046, ΩM = 0.27 and ηB = 6.2 · 10−10.

Later on the proton abundance is small, and we can neglect the first term in theleft hand side of Eq. (6.15). It is then clear that the equilibrium proton abundanceexponentially decreases as the Universe cools down, Xp ∝ e−ΔH/2T . More precisely

neqp = neq

e = n1/2B

(meT

2π

)3/4

e−ΔH/2T . (6.24)

This formula will be useful in what follows.

Problem 6.3. Find chemical potentials μe, μp, μH in equilibrium at temperatureT eq

r = 0.33 eV. Compare them with masses of electron, proton and hydrogen atom.

Problem 6.4. Give alternative derivation of the equilibrium recombination tem-perature based on kinetic approach and the relation (6.1).

Problem 6.5. Find equilibrium recombination temperature for helium. Hint: Donot forget that helium nucleus has to catch two electrons to make a neutral heliumatom.

116 Recombination

Problem 6.6. Find relative equilibrium abundances of hydrogen atoms at 2s- and2p-levels at temperature T eq

r = 0.33 eV. Disregard higher levels in this calculation.

6.2 Photon Last Scattering in Real Universe

In the previous Section we defined the “moment” of recombination as the time atwhich the equilibrium abundances of free protons and hydrogen atoms are equal.In fact, this is not quite the “moment” of primary interest for cosmology. Reallyinteresting is the epoch at which photons scatter for the last time, and after whichrelic photons propagate freely through the Universe. We are talking here aboutCompton scattering of photons off free electrons,

γe → γe. (6.25)

The cross section of this process is determined by the diagrams of quantum electro-dynamics (QED) shown in Fig. 6.1. We are interested in temperatures, and hencephoton frequencies, much lower than the electron mass. In this case the Comptoncross section reduces to the Thomson cross section known from classical electro-dynamics (the study of Compton scattering in QED can be found, e.g., in thebook [51]; the derivation of the Thomson cross section in classical theory is givenin the book [53]). The Thomson cross section is

σT =8π

3α2

m2e

≈ 0.67 · 10−24 cm2. (6.26)

The mean free time of photon with respect to the Compton scattering is

τγ =1

σT · ne(T ), (6.27)

where ne is the number density of free electrons. At the time when the abundances offree protons (and hence free electrons) and hydrogen atoms are equal, np = ne nH ,the electron number density is

ne nB

2=

ζ(3)π2

ηBT 3.

At T = 0.33 eV this is equal to

ne 260 cm−3. (6.28)

Fig. 6.1 Feynman diagrams for Compton scattering.

6.2. Photon Last Scattering in Real Universe 117

The photon mean free time (6.27) in this situation equals τγ 1.9 · 1011 s 6000yrs.This time is much smaller than the Hubble time (6.23). Hence, photon last scatteringhappens somewhat later, at smaller density of free electrons. We will see that thetemperature at last scattering is Tr = 0.26 eV, and the purpose of this Section isprecisely to calculate this temperature.

At temperatures of interest to us, there is kinetic equilibrium between photons,electrons, and also protons. We will show that in Sec. 6.3. On the contrary, chemicalequilibrium does not hold. The latter property can be understood by noticing thatthere are very few thermal photons capable of ionizing hydrogen from the groundstate, i.e., thermal photons with energies ω > ΔH . Indeed, the number density ofphotons of energies above a given value ω is (hereafter the superscript eq refers tothermal, equilibrium quantities)

neqγ (ω) =

∫ ∞

ω

F eqγ (ω′)dω′, (6.29)

where

F eqγ (ω) = 8πω2fPl(ω)

and fPl(ω) is the Planck distribution (that is the Bose–Einstein distribution (5.4)for zero mass and two polarization states). At ω � T we have

F eqγ (ω) =

ω2

π2e−ω/T , (6.30)

and

neqγ (ω) =

ω2T

π2e−ω/T .

For ω = ΔH and T = Tr = 0.26 eV this number density is neqγ (ΔH) 1.1 ·

10−8 cm−3. This is much less than the number density of baryons. So small numberof energetic photons is insufficient for maintaining chemical equilibrium. We willsee in the end of this Section that there exist more numerous non-thermal photonswith ω > ΔH ; this fact will unequivocally prove that the reaction of recombinationto the ground state is out of thermal equilibrium.

Reactions of transition to the ground state 1s from excited states are also out ofequilibrium. We will see that later on. On the other hand, chemical equilibrium doeshold for free electrons and protons and hydrogen atoms at excited levels. Indeed, thenumber density of photons capable of ionizing, say, 2s- and 2p-levels is neq

γ (Δ2s) 8 · 107 cm−3, where Δ2s = ΔH/4 is the binding energy of 2s- and 2p-levels. Thisis much greater than the baryon number density, which serves as a hint towardsthermal equilibrium: even if all electrons and protons recombined into 2s- and 2p-levels, the photon spectrum in the region ω ∼ Δ2s would not change, and wouldremain thermal. Later on we will further discuss the conditions for this partialchemical equilibrium, see Problem 6.8.

So, the system is in thermal equilibrium with respect to all reactions exceptfor the reactions of production and destruction of hydrogen 1s-atoms. It is useful

118 Recombination

to note at this point that if 1s-atoms were not produced at all, the abundancesof excited atoms would be small, and practically all electrons would be free atT = Tr = 0.26 eV.

Problem 6.7. Prove the last statement. Hint: Make use of the formulas of Sec. 6.1.

The last remark shows that electrons are mostly either free or bound in 1s-atoms. Therefore, the key processes are production and destruction of 1s-atoms.In what follows, we will take into account 1s-, 2s- and 2p-levels only, and ignorethe higher levels. This is actually a reasonable approximation, since the equilibriumratio of 3s-levels to 2s is

n3s

n2s e−(Δ2s−Δ3s)/T 0.7 · 10−3. (6.31)

Hereafter n2s, n3s, etc., denote number densities of atoms at the correspondinglevels; the atoms themselves will be denoted simply as 1s, 2s, etc. When writing(6.31) we used Δ3s = ΔH/9.

Within our approximation, 1s-atoms can be produced and disappear in thefollowing reactions,

e + p ↔ 1s + γ, (6.32)

2s ↔ 1s + 2γ, (6.33)

2p ↔ 1s + γ. (6.34)

Let us stress that 2s → 1s transition is a two-photon process, while transitionsfrom continuum and from 2p-state occur by emitting one photon. The 2p → 1s

transition produces photon in the first line of Lyman series, Ly-α.Let us begin our analysis from the transitions from continuum (6.32). These turn

out to be basically irrelevant (!). In the situation of interest, n1s � ne, a photonemitted in the process e + p → 1s + γ ionizes a “neighboring” 1s-atom beforelosing its energy in collisions with electrons. Indeed, the ionization cross sectionnear threshold for 1s-atom is [51]

σ1s =29π2

3e4

1αm2

e

= 6.3 · 10−18 cm2,

and it is much larger than the Thomson cross section (6.26). Also, we will see inthe end of this Section that photon redshift caused by the cosmological expansionis also inefficient for the reactions (6.32). Hence, the processes (6.32) effectively donot change the numbers of free electrons and 1s-atoms.

Let us now consider the processes (6.33). Unlike in recombination of freeelectron and proton, there are two photons produced in the direct process, so theirappearance in the medium does not trigger the inverse reaction. The width of thedirect process 2s → 1s + 2γ is [54]

Γ2s ≈ 8.2 s−1.


This width is small, so the direct process is slow in the sense that 2s-atom ismore likely to get ionized by a thermal photon than to experience this transition,see Problem 6.8. This means that 2s → 1s transitions are unimportant from theviewpoint of the abundance of 2s-atoms relative to electrons, and there is indeedpartial thermal equilibrium in the medium, as we discussed above.

Problem 6.8. Show that at T > 2500K the probability that 2s-atom gets ionizedin the reaction 2s + γth → e + p is higher than the probability that it becomes1s-state due to the process 2s → 1s + 2γ. Here γth denotes thermal photon. Hint:Use ionization cross section near threshold

σ2s =216π2

3e8

1αm2

e

=27

e4σ1s ≈ 2.34 σ1s.

We note that the temperature 2500K is lower than Tr = 0.26 eV = 3000K, so theapproximation of partial equilibrium is indeed valid at photon last scattering epoch.We note also that the typical time scales t � Γ−1

2s are much smaller than the timescale of the cosmological expansion, so the expansion does not spoil the equilibriumfor processes e + p ↔ 2s + γ.

Since 2s-atoms are in equilibrium with free electrons and protons, they obey theSaha relation (cf. Sec. 6.1)

nenp = n2e = n2s ·

(meT

2π

)3/2

· e−Δ2sT , (6.35)

The transition rate to 1s-level via 2s → 1s channel is determined by the number of2s-atoms and the width of 2s-1s transition, so the rate at which free electrons leakinto 1s-states via this channel is(

dne

dt

)2s→1s

= −Γ2sn2s. (6.36)

Here we neglect the inverse process which converts 1s-atom into 2s-state viaabsorption of two photons. We justify this approximation in the end of this Section.

Finally, let us discuss the processes (6.34). Similarly to free electron recom-bination, Ly-α photon emitted in the direct process has high probability to getabsorbed by “neighboring” 1s-atom in the inverse process, leaving no change inthe medium. However, redshift of photons due to the cosmological expansion isimportant in the case (6.34). The point is that there are many Ly-α photons inthe medium. Some of them get redshifted away from the Ly-α line and no longerparticipate in the inverse process 1s+γ → 2p. This results in the overall increase ofthe number of 1s-atoms and decrease of the number of free electrons and protons.We will consider this effect at quantitative level later on, and here we simply quotethat the corresponding effective rate is numerically similar to the estimate (6.36).Hence, the number density of free electrons decays according to

dne

dt= −Γeff (T )n2s, (6.37)

120 Recombination

where Γeff (T ) mildly depends on temperature, see Eq. (6.49), and Γeff (Tr) 2 Γ2s.Making use of (6.35) we now write Eq. (6.37) as the equation for the number

density of free electrons,

dne

dt= −Γeff ·

(2π

Tme

)3/2

· eΔ2sT · n2

e.

Neglecting the time-dependence of the pre-exponential factor and changing variablefrom t to 1/T according to (3.34) we find the solution

1ne

=Γeff T

Δ2sH·(

2π

Tme

)3/2

· eΔ2sT + const.

The first term here grows very fast, so the integration constant is irrelevant at t ∼ tr.Recalling that Δ2s = ΔH/4, we obtain finally

ne(T ) =ΔHH

4Γeff T·(

Tme

2π

)3/2

· e−ΔH4T . (6.38)

Clearly, the result of non-equilibrium analysis is quite different from the equilibriumformula (6.24), as both exponential and pre-exponential factors are different. Toavoid confusion we recall that the formula (6.38) is valid at T > 2500 K = 0.22 eV.

Problem 6.9. Show that at recombination, when ne/nB � 1, the inequilibriumdensity (6.38) always exceeds the equilibrium one (6.24). Consider only temperaturesT > 2500K, for which the above analysis is valid.

Let us come back to the photon last scattering. The average number of collisionsof a photon with free electrons from time t to the present time (the latter may betreated as infinity for this purpose) is

N(t) =∫ ∞

t

σT nedt′.

Let us define the time of last scattering as the time since which photon scattersonce, N(tr) = 1. Changing the integration variable to temperature, we have for thetemperature of last scattering Tr,

1 =∫ Tr

0

σT nedT ′

HT ′ =4Tr

H(Tr)ΔHσT ne(Tr); (6.39)

when evaluating the integral we made use of the strong dependence of the exponentin integrand on temperature, see (6.38). This gives the equation for Tr,(

Trme

2π

)3/2

· e−ΔH4Tr =

Γeff

σT.


Interestingly, Tr depends on cosmological parameters very weakly; such a depen-dence exists in Γeff (T ) only. The result of numerical solution is

Tr = 0.26 eV.

At the last scattering epoch, the Hubble parameter is

H(Tr) 4.5 Mpc−1, (6.40)

and the age of the Universe is tr = (2/3)H−1 = 480 thousand years (with thecosmological parameters we use throughout this book). This estimate is refined inProblem 6.11.

Problem 6.10. Obtain the estimate (6.40).

Problem 6.11. Find the Hubble rate at temperature Tr = 0.26 eV taking intoaccount the contribution of radiation into the energy density. With the latterrefinement, show that the age of the Universe at this temperature is actually

tr = 3.7 · 105 years. (6.41)

The number density of free electrons at last scattering is obtained from (6.39),

ne(tr) 33 cm−3. (6.42)

This is considerably smaller than the electron number density at the beginning ofrecombination, see (6.28).

Let us make a remark here. Of course, recombination does not happen instan-taneously: the electron abundance (6.38) smoothly decreases in time. Yet the mainchange occurs in a small interval of temperatures,

ΔT � Tr.

This property is due to fast evolution of the exponential factor in (6.38). The intervalΔT can be defined by the requirement that the exponential factor in (6.38) is e timeslarger and smaller than the central value at the ends of this interval, i.e.,∣∣∣∣ ΔH

4(Tr ± ΔT )− ΔH

4Tr

∣∣∣∣ = 1.

This givesΔT

Tr≈ 4Tr

ΔH

∼ 0.1.

Thus, the number density of free electrons falls considerably when the temperaturedecreases by several percent. In other words, the process of recombination occursduring a fraction of Hubble time, Δt � H−1(tr).

Let us note that as we have seen, photons emitted at recombination do not,generally speaking, thermalize; instead, they keep their high energies. As a result,the CMB spectrum deviates from the Planck shape in the high-frequency region.This effect, however, turns out to be small, see book [55] for details.

Problem 6.12. Show that the mean free time of a photon with respect to scat-tering off neutral atoms exceeds the Hubble time at recombination and afterwards.

122 Recombination

This means that the presence of neutral hydrogen and helium does not affect CMBphotons.

Let us consider the contribution of processes (6.34) into the production rate of 1s-atoms, or, equivalently, the rate of disappearance of free electrons. This contribution is dueto the photon redshift caused by the cosmological expansion. The decrease of the numberof Ly-α photons created in the direct reaction and participating in the inverse reaction isgiven by the number of photons leaving Ly-α line, i.e., crossing the low-frequency boundaryof this line in unit time,

dnγ(ω∗)dt

= Fγ(ω∗)dω

dt(ω∗) = −ω∗HFγ(ω∗), (6.43)

where nγ(ω) is the number density of photons with energy greater than ω, Fγ(ω) =dn(ω)/dω is the photon spectral density (note that the equilibrium values of these quan-tities enter (6.29)), while ω∗ is the lower boundary of the Ly-α line; we will see that theprecise value of ω∗ is irrelevant for the calculation. Photons of energy below ω∗ cannotexcite 1s-state into 2p, so the effective increase of the number of 1s-atoms and decrease ofthe number of free electrons are given by

dn1s

dt= −dne

dt= −dnγ(ω∗)

dt. (6.44)

Since the Universe expands slowly, very few photons exit the Ly-α line, so the directand inverse processes are to very good approximation in thermal equilibrium. Thus, thespectral density Fγ(ω) can be calculated in the approximation of equal rates of direct andinverse processes,

Γ2pϕ(ω)n2pdω = C(ω)ϕ(ω)Fγ(ω)n1sdω. (6.45)

The left hand side of this formula gives the number of photons emitted in the process2p → 1s + γ per unit time per unit volume in the frequency interval (ω, ω + dω), Γ2p isthe width of 2p-level, ϕ(ω) is the shape of Ly-α line. The right hand side is the absorptionrate of photons by 1s-atoms. We used the fact that this rate is proportional to the photonspectral density and to the number density of 1s-atoms, so that the coefficient C dependson ω only. It is given by

C(ω) =π2Γ2p

ω2. (6.46)

Note that A(ω) = Γ2pϕ(ω) and B(ω) = C(ω)ϕ(ω)/ω are nothing but the Einstein coeffi-cients known from the theory of radiation, and the equality (6.46) is the Einstein relation,see the book [51].

Problem 6.13. Derive the relation (6.46). Hint: Make use of the fact that equality (6.45)is valid in thermal equilibrium and consider the low temperature regime T ω.

The relations (6.45), (6.46) give

Fγ(ω) =ω2

π2

n2p

n1s. (6.47)

This is a smooth function of ω, so one can set ω∗ = ΔH − Δ2p = 3ΔH/4 in (6.43). Also,most protons are bound in 1s-atoms at last scattering epoch, so we can take n1s ≈ nB .Making use of Eqs. (6.43) and (6.44) we find that the decrease of the number of freeelectrons due to reactions (6.34) is„

dne

dt

«2p→1s

= − H

nBπ2

„3ΔH

4

«3

n2p. (6.48)


Finally, at partial thermal equilibrium we have n2p = 3 n2s according to the degeneracy ofthe levels. Adding (6.48) to (6.36) we obtain the formula (6.37) with

Γeff = Γ2s + 3H

nBπ2

„3ΔH

4

«3

(6.49)

At T = Tr = 0.26 eV and nB = 4.6 · 10−10nγ the second term here equals 8.2 s−1, so thatΓeff (Tr) = 16.4 s−1 = 2Γ1s. By the end of recombination, about half of electrons get to1s-level via 2s → 1s transitions, while another half of electrons get there via Ly-α channel.This is consistent with the complete analysis of Ref. [52].

An analogous phenomenon exists also in the case of the reactions (6.32). In that caseinstead of (6.45) we have

A(ω)n2e = σI(ω)Fγ(ω)n1s. (6.50)

The rate of the direct reaction of recombination of electron and proton is proportional tonpne = n2

e which is reflected in the left hand side of (6.50). The rate of inverse reaction isproportional to the number of 1s-atoms, ionization cross section σI and photon spectraldensity at energy ω, where ω ≥ ΔH . Let us recall that the direct and inverse rates are equalin thermal equilibrium, and write the equation for the coefficient A(ω) (this is nothing butdetailed balance equation),

A(ω) (neqe )2 = σI (ω)F eq

γ (ω)neq1s.

We express A(ω) from this equation and substitute it into (6.50). This gives for the spectraldensity at ω ≥ ΔH

Fγ(ω) =n2

eneq1s

(neqe )2n1s

F eqγ (ω). (6.51)

Redshift of photons away from the region ω > ΔH is given by (6.43) with ω∗ = ΔH ; thesame formula gives the decrease of the number of free electrons due to the processes (6.32).The ratio of the latter contribution to that of Ly-α processes is given by the ratio of (6.51)and (6.47) (modulo the ratio of photon energies, ΔH/(ΔH − Δ2p) = 4/3),

(dne/dt)ep→1s

(dne/dt)2p→1s� F eq

γ (ΔH)neq1s

(neqe )2

π2

Δ2H

„Tme

2π

«3/2

· e−ΔH4T

where we inserted n2p = 3 n2s, made use of (6.35) and omitted factor of order 1. Let usnow make use of (6.30), (6.24) and set n1s ≈ nB . We obtain finally

(dne/dt)ep→1s

(dne/dt)2p→1s� e−

ΔH4T .

This ratio is small, which means that the processes (6.32) are irrelevant indeed.We note in passing that in our case of delayed recombination the number of free

electrons ne is considerably larger than its equilibrium value neqe , while the number of 1s-

atoms is smaller than in equilibrium. Therefore, the actual spectral density (6.51) exceedsconsiderably the equilibrium spectral density at ω ≥ ΔH , as we have claimed in the text.

Finally, let us justify the approximation which neglects the process inverse to 2s →1s + 2γ. The inverse process occurs via absorption of two thermal photons. Its rate isproportional to the number of 1s-atoms, and in thermal equilibrium it must be equal tothe rate of direct process (6.36). This immediately gives„

dne

dt

«1s→2s

= Γ2sneq2s

n1s

neq1s

= Γ2sn1se−(ΔH−Δ2s)/T .

This process is negligible as compared to the direct one when its rate is smaller than therate (6.36), i.e., for

n1se−(ΔH−Δ2s)/T n2s. (6.52)

124 Recombination

n2s can be found from (6.35), (6.38), while the estimate for n1s is n1s ≈ nB = ηBnγ .Inserting the numerical values, one obtains that (6.52) indeed holds.

Problem 6.14. Make the numerical estimate sketched here.

6.3 ∗Kinetic Equilibrium

Our previous calculations were made under the assumption of kinetic equilibriumat recombination epoch. Let us check that this assumption is valid. The point isto show that the rate of energy exchange between different components of plasma(photons, electrons and protons) is faster than the cosmological expansion rate.This will mean that the distribution functions of all particles have the equilibriumshapes (6.2)–(6.4) with one and the same temperature.

We have seen in Sec. 2.5 that in the absence of interactions between photons andnon-relativistic particles, the distributions of all these particles would have thermalform. However, the effective temperature of photons would decrease in time slowerthan that of electrons and protons. Thus, we have to check that energy transferfrom photons to electrons and protons is sufficiently fast.

Let us begin with electrons. They get energy from photons via Compton scat-tering process (6.25) that occurs with Thomson cross section (6.26). The timebetween two subsequent collisions of a given electron with photons is τ = 1/(σTnγ).We are interested, however, in the energy transfer. So, we have to find the time τE inwhich an electron obtains kinetic energy of order T due to the Compton scattering.To estimate this time, we note that the typical energy transfer in a collision of aslow electron with a low energy photon is actually suppressed. To see this, let uswrite down the 4-momentum conservation equation in the following form,

pe + pγ − p′γ = p′e,

where pe, pγ are 4-momenta of incident electron and proton, while primed quantitiesrefer to final particles. The square of this equality gives

me(ω − ω′) − |pe|(ω cos θ1 − ω′ cos θ2) − ωω′(1 − cos θ) = 0, (6.53)

where pe is 3-momentum of the initial electron, ω, ω′ are initial and final photonfrequencies, θ1, θ2 are angles between the initial electron momentum and momentaof initial and final photon, pγ and p′

γ , respectively, while θ is the photon scatteringangle, see Fig. 6.2. If electron is initially at rest, the second term in Eq. (6.53)vanishes, so the typical energy transfer is

ΔE = ω′ − ω ∼ ω2

me. (6.54)

From this we would conclude that the number of collisions after which the electronhas energy of order T is

N ∼ T

ΔE∼ me

T. (6.55)

6.3. ∗Kinetic Equilibrium 125

Fig. 6.2 Kinematics of Compton scattering.

However, we are interested in the situation where electron kinetic energy Ee beforethe collision is comparable to the photon frequency. In this situation the electron3-momentum is

|pe| =√

2Eeme ∼ √ωm.

Hence, the third term in Eq. (6.53) is negligible, and we obtain the estimate of theenergy transfer in a single collision,

ΔE ∼ ω

√ω

m. (6.56)

We note, however, that unlike the third term in (6.53), the second term can haveany sign: the angles θ1 and θ2 take random values. Therefore, we use the analogywith random walk and estimate the number of scattering events needed to heat upa moving electron,

N ∼(

T

ΔE

)2

∼ me

T,

This coincides with the estimate (6.55) obtained for electron originally at rest.2 Inthis way we obtain the time of electron heating,

τE(T ) ∼ Nτe(T ) ∼ me

TσT nγ(T ). (6.57)

Using numerical values of parameters in this formula we find that at recombination

τE(Tr) 1.8 · 108 s ≈ 5.5 years,

which is much smaller than the Hubble time (6.41). Thus, energy transfer betweenphotons and electrons is efficient, so the two fractions have the same temperature.

Let us now discuss heating of protons. The direct interaction of proton withphotons is irrelevant, since the corresponding time for protons is given by Eq. (6.57)but with mp substituted for me. Since the Thomson cross section is proportional to

2Note that for fast electron, the second term in (6.53) gives rise to systematic decrease of electron

energy, since the number of collisions with photons moving towards electron is larger than that

with photons “chasing” electron. This is the mechanism of thermalization of fast electron in photon

medium.

126 Recombination

m−2e , the time for protons is larger by a factor (mp/me)

3; this time is larger thanthe Hubble time (6.41).

Energy transfer to protons occurs due to elastic scattering of electrons offprotons. The non-relativistic cross section of the latter process in the proton restframe is given by the Rutherford formula,

dσR =πα2

m2ev

4

sin θ

sin4 θ/2dθ, (6.58)

where v is the electron velocity and θ is the scattering angle. The differential crosssection (6.58) has power law singularity dθ/θ3 at small scattering angle, so the totalcross section diverges, and the mean free time is formally zero. However, we areagain interested in energy transfer time τE rather than mean free time. To obtainits estimate we note that the energy transfer to proton initially at rest is

ΔE = 2me

mpEe(1 − cos θ), (6.59)

where Ee is the electron energy.

Problem 6.15. Obtain the formula (6.59).

Like in the case of Compton scattering, energy transfer to moving proton has asign-indefinite contribution which is suppressed by

√me/mp rather than by me/mp

as in (6.59). However, upon averaging over angles this term again has the same effectas the term (6.59), so we stick to (6.59).

Making use of (6.58) and (6.59) we obtain the estimate

τE ∼(

nev

∫dθ

dσR

dθ

ΔE (θ)E

)−1

=(

8neπα2

mpmev3

∫dθ ctg

θ

2

)−1

. (6.60)

The integral in the right hand side of (6.60) is still divergent at small angles, but nowthe divergence is logarithmic. This softening of the divergence is natural: the energytransfer is small at small collision angle. To obtain finite value for the integralin (6.60) we recall that the Coulomb interaction in plasma experiences the Debyescreening at distance [48]

rD =(

T

4πneα

)1/2

.

Hence, the integral in (6.60) must be cut off at the scattering angle correspondingto impact parameter equal to rD ,

θD ∼ α

mev2rD

. (6.61)

Problem 6.16. Making use of the Rutherford formula (6.58), obtain the estimate(6.61) for θD.

6.4. Horizon at Recombination and its Present Angular Size. 127

The leading contribution to the integral in (6.60) comes from small scattering anglesθ ∼ θD, so we obtain the energy transfer time

τE(T ) ∼ mpmev3

16πne (T )α2 log θ−1D

∼ mpme

16πne (T )α2 log (6TrD/α)

(3T

me

)3/2

. (6.62)

Numerically,τE(Tr) ∼ 3 · 104 s,

which is a very short time compared to the Hubble time at recombination (6.41).Hence, kinetic equilibrium between electrons and protons is an excellent approxi-mation.

6.4 Horizon at Recombination and its Present Angular Size.Spatial Flatness of the Universe

CMB photons, which last scattered at recombination, give us a photographic pictureof the Universe at temperature Tr = 0.26 eV and redshift zr = 1100. The character-istic length scale at recombination epoch is the horizon size at that time, lH,r. Thisscale is of interest for two reasons. First, in the Hot Big Bang theory, regions sepa-rated at recombination epoch by distance exceeding lH,r were causally disconnectedat that epoch. Second, this scale is imprinted in the properties of CMB photons. Weshow in accompanying book that CMB anisotropy and polarization have featuresin their spectra whose angular sizes are determined, roughly speaking, by the angleat which the horizon at recombination is seen today. In this Section we give prelim-inary estimate of this angle and discuss its dependence on cosmological parameters.Our purpose here is to give a general idea of how cosmological parameters are deter-mined from CMB measurements and other observations; this is discussed in moredetails in the accompanying book.

In fact, more relevant for CMB is the distance traveled by sound waves in the cosmicplasma from Big Bang to recombination epoch. In complete analogy to (3.25) this distanceis

lSH,r = a(tr)

Z tr

0

us(t)dt

a(t)(6.63)

where

us =

s∂pBγ

∂ρBγ

is the sound velocity in the baryon-electron-photon plasma (hence the notation for pressureand energy density of this plasma). The sound velocity depends on the baryon density,hence on time. Still, since the baryon-to-photon ratio is well known from the analysis ofBig Bang Nucleosynthesis (see Chapter 8) and from the measurements of other features ofCMB, there is basically one-to-one correspondence between the sound horizon lSH,r andparticle horizon lH,r. In this Section we will consider particle horizon lH,r , having in mindthis qualification.

Problem 6.17. Find sound velocity in the baryon-electron-photon plasma before recombi-nation as function of temperature, assuming that the baryon-to-photon ratio is known.

128 Recombination

As we have noticed already, recombination occurs at matter dominated epoch.So, we estimate the horizon size at recombination by making use of Eq. (3.26); thisestimate is straightforward to refine, see Problem 6.21. We write

lH,r =2

Hr

where Hr ≡ H(tr) is the Hubble parameter at recombination. Further analysis canbe done in the same way as the calculation of the age of the Universe tr that leadus to (6.22). However, we make use here of a different, though equivalent approach.Let us find Hr from the Friedmann equation,

H2r =

8π

3GρM (tr).

Recall that ρM(tr) = ρM,0 (a0/ar)3 = ρM,0(1 + zr)3, with ρM,0 = ρcΩM , see

Chapter 4. By combining these formulas with the definition of ρc (see (4.3)), wefind

lH,r =2

H0

√ΩM

1(1 + zr)3/2

. (6.64)

Problem 6.18. Show that the results (6.64) and (6.22) coincide, if one recalls thatlH,r = 3tr in the approximation of matter domination. Hence, the dependence ofexpression (6.64) on parameters of the present Universe is elusive.

The length interval (6.64) has stretched (1 + zr) times since recombination, soits present size is

lH,r(t0) =2

H0

√ΩM

1√1 + zr

.

Clearly, this size is about√

1 + zr 30 times3 smaller than the present horizon size(4.29); in other words, the Universe visible at present contains about (1 + zr)3/2 ∼3 · 104 regions that were causally disconnected by recombination (of course, this istrue in the framework of the hot Big Bang theory only). Nevertheless, these regionswere exactly the same, we know that from both CMB observations and galaxysurveys. How did it happen that despite the absence of causal contact the differentregions have exactly the same properties? This question cannot be answered withinthe hot Big Bang theory; this is a problem for this theory called horizon problem.The horizon problem finds its elegant solution in inflationary theory.

Let us calculate the present angular size of a region whose spatial size at recom-bination was equal to lH,r. Like in Sec. 4.6, we do not assume that the Universe isspatially flat: as we discuss in this Section at the qualitative level (and quantitativelyin the accompanying book), it is the calculation of this sort and its comparison withCMB data that lead to the result that the spatial curvature is close to zero. Weassume for simplicity that dark energy density ρΛ is constant in time (cosmologicalconstant); the generalization to time-dependent dark energy is straightforward.

3More precise estimate gives 50 instead of 30, see Problem 6.21.


Like in Sec. 4.6, let us choose the open cosmological model (κ = −1, Ωcurv > 0),whose metric is given by (2.10). The last scattering surface is at the coordinatedistance

χr =∫ t0

tr

dt

a(t); (6.65)

CMB photons traveled precisely this distance since recombination. This coordinatedistance is still given by Eq. (4.34), where we have to set z = zr. Since zr � 1, theintegration in (4.34) can be extended to zr = ∞, which corresponds to the limittr → 0 in the integral (6.65). In physical terms this means that we neglect thedifference between the distance that photon traveled since recombination and thepresent horizon size. Hence,

χr χH,0 =∫ ∞

0

dz

a0H0

1√ΩM(1 + z)3 + ΩΛ + Ωcurv(1 + z)2

, (6.66)

which is the coordinate size of the present horizon. Making use of the results ofSec. 4.7, we write the angular size,

Δθr =lH,r

ra(zr),

where ra(zr) = (1 + zr)−1 · a0 · sinhχr is the angular diameter distance to recombi-nation. We finally obtain

Δθr =2√

ΩMa0H0 sinh χr

1√zr + 1

. (6.67)

When discussing this formula, one should bear in mind the relations (4.39) and(4.40).

We note here that formula (6.67) must be corrected, since the horizon at recom-bination does not quite coincide with (6.64) because of the presence of radiation inthe Universe. This is irrelevant for the discussion that follows, so we proceed with(6.67).

Let us begin with the hypothetical case of spatially flat Universe without darkenergy, Ωcurv = 0, ΩΛ = 0. Spatially flat Universe corresponds to the limit a0 → ∞,which gives

Δθr =1√

zr + 1, Ωcurv = ΩΛ = 0. (6.68)

Of course, this result would be easier to obtain by using directly the formulas forthe Universe filled with matter given in Sec. 3.2.1.

Problem 6.19. Obtain the formula (6.68) directly in the model of spatially flatUniverse filled with non-relativistic matter.

For zr = 1100 the formula (6.68) gives Δθr = 0.03, or Δθr = 1.7◦. A manifes-tation of the horizon problem is that photons coming from directions separated by

130 Recombination

more than 2◦ were emitted in causally disconnected regions, and yet they have thesame temperature within 0.01%.

Continuing the discussion of formula (6.67) we recall that the angle4 Δθr deter-mines the angular scale of the features in the CMB angular spectrum. Hence, theangle Δθr is a measurable quantity. It is therefore of interest to consider the depen-dence of Δθr on cosmological parameters. Since zr is known, there are only tworelevant parameters:5 due to relations (4.39) and (4.40) this pair may be chosenas (ΩM , Ωcurv). To see which parameter is more relevant (in the sense that thedependence on it is stronger), let us first consider the case of spatially flat Universe,Ωcurv = 0. Unlike in the case (6.68), we now have ΩΛ �= 0 and therefore ΩM �= 1.We take the limit a0 → ∞, and the expression (6.67) becomes

Δθr =1√

zr + 11

I(ΩM), Ωcurv = 0, (6.69)

where

I =√

ΩM

2

∫ ∞

0

dz√ΩM(z + 1)3 + ΩΛ

, (6.70)

and ΩΛ = 1 − ΩM . Changing the variable to (1 + z) = y−2 we write this integral inthe following form,

I =∫ 1

0

dy√1 +

ΩΛ

ΩM

y6

. (6.71)

It is clear that if ΩM is not very small, this integral depends on ΩΛ/ΩM ratherweakly. At ΩM = 1, ΩΛ = 0 it is equal to 1, while for ΩM = 0.27, ΩΛ = 0.73 its valueis 0.90. We conclude that the dependence of the angle Δθr on the distribution ofenergy between ΩM and ΩΛ is rather weak.

On the other hand, the angle Δθr depends on Ωcurv quite strongly. To see this,let us consider the hypothetical case ΩΛ = 0, when ΩM + Ωcurv = 1. In that casethe change of variables (1 + z) = y−2 gives the analytic expression for the integral(6.66), namely

χr = 2Arsinh√

Ωcurv

ΩM

, ΩΛ = 0,

where we made use of (4.40). The angle Δθr is then given by

Δθr =1√

zr + 11√

1 + Ωcurv/ΩM

.

4More precisely, similar angle determined by the sound horizon, see (6.63); we talk about the

angle Δθr for definiteness, while the discussion applies to the observable angle as well.5In the realistic case, there is some dependence on other parameters, but one of these param-

eters, baryon-to-photon ratio, is known from other observations while dependence on others (e.g.,

neutrino masses) is weak.


Fig. 6.3 Early bounds on cosmological parameters from the analysis [56] of the CMB anisotropy

data together with SNe Ia observations. Regarding contours, see comment to Fig. 4.6 given before

Problem 4.9.

0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

FlatBAO

CMB

SNe

No Big Bang

Fig. 6.4 The region on the plane (ΩM , ΩΛ) allowed by CMB, large scale structure distribution

(BAO) and SNe Ia data [41]. The contours correspond to 1 σ, 2 σ and 3 σ confidence levels.

132 Recombination

Clearly, at Ωcurv ∼ ΩM this result is quite different from the results (6.68) and(6.69) which are valid for the flat Universe. It becomes clear that the measurementof Δθr (more precisely, angular scales related to it) gives strong bound on the spatialcurvature of the Universe.

Finally, let us consider the case when Ωcurv is small compared to both ΩM andΩΛ, while ΩM is roughly comparable to ΩΛ. Then one can neglect Ωcurv in thesquare root in (6.66) (this corresponds to small contribution of curvature into theFriedmann equation, see the discussion in Sec. 4.2), and using (4.40) we obtain

χr = 2√

Ωcurv

ΩM

I(ΩM , ΩΛ),

where I is the same integral (6.70) or (6.71), but ΩΛ is no longer equal to (1−ΩM).Using (6.67) again, we find

Δθr 1√zr + 1

2√

Ωcurv/ΩM

sinh(2√

Ωcurv/ΩMI) . (6.72)

The spatial curvature shows up here through the fact that the denominator containshyperbolic sine rather than linear function; the angle at which a given interval isseen in hyperboloid is smaller than the angle in Euclidean space. The dependence ofthe right hand side of (6.72) on Ωcurv/ΩM is rather strong, unlike the dependence onΩΛ/ΩM . This reiterates our conclusion that Δθr is particularly sensitive to spatialcurvature. Let us note that we again encounter the degeneracy in parameters, aswe did in Sec. 4.6, but now less important parameter is ΩΛ/ΩM .

The first measurents of the CMB angular anisotropy at relatively small angularscales (about an angular degree) already gave rise to the conclusion that the spatialcurvature of our Universe is small. Together with the SNe Ia observations they leadto the ΛCDM model as the working hypothesis (see Fig. 6.3). Later data are not(yet?) in contradiction with the ΛCDM model, they rather pin down its parametersfurther. This is illustrated in Fig. 6.4.

Problem 6.20. Calculate the integral (6.66) numerically and draw the lines ofconstant Δθr on the plane (ΩΛ, ΩM). Compare with Fig. 6.3. Hint: Use the relations(4.39), (4.40).

Problem 6.21. Refine the formula (6.64) by taking into account the contributionof radiation into the energy density in the Universe. Hint: Make use of the resultsof Problem 4.4.

Chapter 7

Relic Neutrinos

The earlier the period in the history of the Universe, the higher the temperatureand density of matter. Hence, those processes which are too weak to be relevant inthe present Universe, are important at earlier stages and may leave imprints in theUniverse we see today.

We have discussed the phenomenon of this sort in Chapter 6 in the context ofrelic photons. Here we move further back in time and consider other light particles,neutrinos.

7.1 Neutrino Freeze-Out Temperature

Let us estimate the temperature at which neutrino interactions between themselvesand with cosmic plasma switch off (“freeze-out temperature”). We will see that thistemperature is of the order of a few MeV. At this epoch, electrons and positronsare still relativistic, and their number density is given by (5.16b). On the otherhand, baryons are non-relativistic, so their abundance is suppressed by a factor oforder ηB relative to the abundance of e+e−-pairs. Hence, as far as neutrino freeze-out is concerned, relevant processes are neutrino scattering off electrons, positronsand themselves and neutrino-antineutrino annihilation into e+e−-pair and neutrino-antineutrino pair of different type, as well as inverse processes. In all these processes,particles are relativistic at temperatures of interest.

We will not need the precise value of neutrino freeze-out temperature in whatfollows. So, a dimensional estimate for the cross sections of the above processesis sufficient for our purposes. Neutrino participate in weak interactions only (seeAppendix C). At energies of interest, the cross sections are proportional to theFermi constant squared G2

F , whereGF = 1.17 · 10−5 GeV−2.

On dimensional grounds, we immediately obtain for the cross section of any of theabove processes at E > me,

σν ∼ G2F E2,

where E is a typical collision energy, E ∼ T .

133

134 Relic Neutrinos

The mean free time of neutrino is given, as usual, by

τν =1

〈σνnv〉 , (7.1)

where v is the relative velocity of neutrino and a particle it collides with, and n isthe number density of the latter particles. In our case all particles are relativistic,the number density is given by (5.16), i.e., n ∼ T 3 and v 1. In this way we cometo the estimate of the mean free time,

τν ∼ 1G2

F T 5. (7.2)

We now compare τν with the Hubble time (see (3.31))

H−1 =M∗

Pl

T 2. (7.3)

We see that as the Universe cools down, τν increases faster than H−1. Thus, inaccordance with our expectation, at early stages the mean free time is shorter thanthe Hubble time, and neutrinos are in thermal equilibrium with matter. Indeed, thenumber of neutrino collisions since time t is estimated as

N(t) ∼∫ ∞

t

dt′

τν(t′)=∫ ∞

t

dt′

t′t′

τν(t′)∼ t

τν(t)∼ 1

H(t)τν(t),

where we made use of the fact thatt

τν(t)∼ 1

H(t)τν(t)

rapidly decreases in time. If N(t) � 1 then neutrinos are in thermal equilibrium,while for N(t) � 1 they are non-interacting. Hence, neutrino interactions switchoff at

τν(T ) ∼ H−1(T ).

It follows from (7.2) and (7.3) that this happens at freeze-out temperature

Tν,f ∼(

1G2

F M∗Pl

)1/3

∼ 2 − 3 MeV.

Problem 7.1. Estimate the age of the Universe at neutrino freeze-out.

Thus, at freeze-out temperature Tν,f neutrinos interacted for the last time. Sincethen they freely propagate through the Universe and their number in comovingvolume does not change. The very assumption that there existed temperatures ofthe order of a few MeV leads to the conclusion that the present Universe contains thegas of relic neutrinos analogous to the gas of relic photons, CMB. We will see thatthe latter assumption is strongly supported by Big Bang Nucleosynthesis, so thereis no doubt that relic neutrinos indeed exist.

7.2. Effective Neutrino Temperature. Cosmological Bound on Neutrino Mass 135

7.2 Effective Neutrino Temperature. Cosmological Boundon Neutrino Mass

It follows from the results of Sec. 2.5 that neutrinos after freeze-out are still describedby relativistic distribution function with present effective temperature

Tν,0 = Tν,fa(tν)a(t0)

=Tν,f

1 + zν, (7.4)

where zν is redshift at neutrino freeze-out. At the time of freeze-out, neutrino tem-perature equals to that of photons. After freeze-out, the photon temperature alsodecreases due to the cosmological expansion. However, at neutrino freeze-out, thecosmic plasma contained also a lot of relativistic electrons and positrons. As thetemperature drops below the electron mass, electrons and positrons annihilate away,injecting energy into the photon component. Due to this process, the photon temper-ature becomes higher than the effective neutrino temperature. This effect is quan-titatively described by making use of entropy conservation of the electron-photoncomponent in comoving volume,

g∗(T )a3T 3 = const, (7.5)

where g∗(T ) is the effective number of relativistic degrees of freedom in the electron-photon plasma, see (5.28) and (5.30). Right after neutrino freeze-out, this plasmaconsisted of relativistic electrons, positrons and photons, which gives

g∗(Tν,f ) = 2 +78(2 + 2) =

112

,

After e+e−-annihilation, the entropy is due to photons only, g∗ = 2, and we arriveat the following result for temperatures well below electron mass (including presentepoch),

Tγ,0

Tν,0=(

g∗(Tν,f )g∗(T0)

)1/3

=(

114

)1/3

1.4. (7.6)

We conclude that the present neutrino temperature is1

Tν(t0) 1.95 K. (7.7)

Making use of (5.16) we find the present number density of each species of neutrinostogether with its antineutrinos,

nν,0 =34· 2 · ζ(3)

π2T 3

ν (t0) 112 cm−3. (7.8)

Problem 7.2. Let us make a wrong assumption that there is no Z0-boson in Nature,while W± bosons exist. Furthermore, let us neglect loop processes and neutrinooscillations (which is also wrong). What are the relic abundances of neutrinos ofdifferent types?

1Let us recall that massive neutrinos today have relativistic distribution in momenta with effective

temperature (7.6). Their abundance is independent of their mass and is given by (5.16).

136 Relic Neutrinos

Direct detection of relic neutrinos appears to be practically impossible, in viewof extremely small cross section of low energy neutrino interactions and tiny energyrelease.

Problem 7.3. Assuming that neutrinos are massless, estimate the mass of adetector required for having one relic neutrino interaction per year.

If neutrinos were massless, their contribution to the present energy density wouldbe small. Making use of (5.12) we would find for each type of neutrino plus itsantineutrino,

Ων = 278

π2

30T 4

ν,0

ρc≈ 10−5,

so neutrinos would make very little effect on the expansion of the Universe. It isworth noting that this was not always the case: neutrino contribution was sizeable atradiation domination, and, indeed, Big Bang Nucleosynthesis imposes strong boundon the number of neutrino species, as we discuss in Chapter 8.

The situation is quite different for neutrinos of mass mν > Tν,0. Their presentenergy density is

ρν,0 = mνnν,0,

and the relative contribution to the total energy density is

Ων =ρν,0

ρc≈( mν

1 eV

)· 0.01 h−2. (7.9)

Let us require that the neutrino energy density does not exceed the total energydensity of non-relativistic matter. Recalling that there are three types of neutrinos,we obtain the cosmological bound on the sum of their masses [60],∑

i

mνi < 100 · h2ΩM eV (7.10)

With conservative bound ΩM < 0.4 and h = 0.7, we obtain∑i

mνi < 20 eV.

For long time similar bound was the strongest bound on the masses of μ- and τ -neutrino. Presently one can combine the direct limit on electron neutrino mass [1],

mνe < 2 eV,

with the results of neutrino oscillation experiments. The latter show that the differ-ences of masses squared Δm2 between νe, νμ and ντ are small, Δm2 � 5 ·10−3 eV2.So, all types of neutrinos have low masses,

mν < 2 eV, (7.11)

see Appendix C. The bound (7.11) and the relation (7.9) show that the contributionof all types of neutrinos into energy density is not very large,∑

i

Ωνi < 0.12. (7.12)

7.2. Effective Neutrino Temperature. Cosmological Bound on Neutrino Mass 137

Nevertheless, by comparing this to the relative energy density of all non-relativisticmatter, ΩM ≈ 0.24, we see that the bound (7.12) per se allows neutrinos to besubstantial part of dark matter. However, the data on cosmic structures and CMBexclude the relative contribution of neutrinos at the level∑

i

Ωνih2 < 0.002 − 0.01, (7.13)

depending on details of the analysis. This corresponds to the bound on the sum ofneutrino masses [7, 61] ∑

i

mνi < 0.2 − 1.0 eV

and rules out neutrino as dark matter candidate. We discuss the origin of the bound(7.13) in the accompanying book.

To end this Section we note that the above results have been obtained under theassumption that there is essentially no asymmetry between neutrinos and antineu-trinos in the Universe. In other words, we assumed that the neutrino chemicalpotential is close to zero. This assumption is quite plausible, especially in viewof the observation that electroweak processes at T � 100 GeV equalize, modulofactor of order 1, lepton and baryon asymmetries (see Chapter 11), and that thebaryon asymmetry is very small, ηB ∼ 10−9. Nevertheless, one cannot rule out com-pletely the possibility of substantial neutrino-antineutrino asymmetry. In that case,neutrino oscillation data (in particular, lower bound on the mass of the heaviestneutrino, mν > matm ≈ 0.05 eV), together with the bound (7.13) can be used forplacing the bound on the lepton asymmetry in the present and early Universe. Thisis not a particularly strong bound, and we leave it for the problem.

Problem 7.4. Making use of the neutrino oscillation results (Appendix C) andthe bound (7.13), obtain the bound on neutrino asymmetry in the present Universedefined as

ΔL,0 =∑

i(nνi − n νi)s

,

where s is entropy density. Translate this bound into the bound on lepton asymmetryin the early Universe, whose definition is

ΔL =∑

i nL,i

s,

where nL,i = (nνi −nνi)+(nli −nli) is the number density of each lepton flavor andli denotes charged leptons, l1 = e−, l2 = μ−, l3 = τ−. The lepton asymmetry ΔL

is time-independent provided that there are no lepton number violating processes incosmic plasma.

Stronger bound on the lepton asymmetry is obtained from Big Bang Nucleo-synthesis. In Sec. 8.1 we obtain the bound on the chemical potential of electron

138 Relic Neutrinos

neutrino at temperature of order 1 MeV at the level |μνe/T | < 0.05. Precise bound(at 95% confidence level) is

−0.023 <μνe

T< 0.014, T ∼ 1 MeV. (7.14)

In fact, this bound applies to all types of neutrinos, since neutrino oscillationsequalize neutrino abundances before neutrino freeze-out. Indeeed, the typical timefor neutrino oscillations at energy E is (see Appendix C)

tosc π4E

Δm2.

At E 3T , T 3 MeV and even for the smallest value Δm2sol 8 · 10−5 eV2 this

time is

tosc 5 · 10−4 s,

which is much shorter than the Hubble time at freeze-out, H−1(T 3 MeV) ∼ 0.1 s.Note that we neglected matter effects in our estimate; this is indeed a reasonableapproximation at temperature of about 3 MeV (see [57, 58] for details). Hence, atneutrino freeze-out

μνe = μνμ = μντ , T ∼ 3 MeV.

This means that the bound (7.14) indeed applies to all types of neutrinos.Neutrinos excess over antineutrinos is given by (5.22), and the neutrino number

density itself by (5.16b). So, for each type of neutrinos we have

nν − nν

nν + nν=

π2

9ζ(3)μν

T.

This asymmetry persists until the present time, so we find from (7.14) that for eachtype of neutrinos

|nν − nν |nν + nν

< 0.06.

We conclude that the excess of neutrinos over antineutrinos (or vice versa) is verysmall in our Universe.

7.3 ∗Sterile Neutrinos

The existence of neutrino oscillations points towards incompleteness of the StandardModel of particle physics (see Appendix C for details). Some extensions of theStandard Model include new particles, sterile neutrinos. These may be described asMajorana fermions which mix with conventional neutrinos. They are sterile in thesense that the new fields do not interact with gauge fields of the Standard Model;in particular, they do not directly participate in weak interactions. Conventionalneutrinos, interacting with W - and Z-bosons, are called active in this context. Thetotal number of neutrino states is then equal to 3 + Ns where Ns is the number of

7.3. ∗Sterile Neutrinos 139

sterile neutrino species. We will restrict ourselves to the case Ns = 1 for definiteness.We note here that experimental data on neutrino oscillations, except possibly fordata from LSND and MiniBOONE, do not require the existence of sterile neutrino(see discussion in Sec. C.3). However, such an extension of the Standard Models isquite reasonable and worth discussing.

We are going to consider relatively light sterile neutrino, ms � 100 MeV. Inthe simplest models [62–64] the creation of sterile neutrino states |νs〉 in the earlyUniverse occurs due to their mixing with active neutrinos |να〉, α = e, μ, τ . In theapproximation of mixing between two states only, we have

|να〉 = cos θα|ν1〉 + sin θα|ν2〉, |νs〉 = − sin θα|ν1〉 + cos θα|ν2〉,where |να〉 and |νs〉 are active and sterile neutrino states, |ν1〉 and |ν2〉 are masseigenstates of masses m1 < m2, and θα is the vacuum mixing angle between sterileand active neutrino. Let us assume that mixing is weak, θα � 1, so that the heavystate is mostly sterile neutrino |ν2〉 ≈ |νs〉. In this situation the mass of the heavystate is naturally called the sterile neutrino mass, m2 ≡ ms. Let us assume that thesterile neutrino mass is large compared to the mass of active neutrino, ms � m1.All these assumptions are very natural from particle physics viewpoint.

The calculation of probability of oscillation να ↔ νs is done in the same way asin the case of oscillation between different types of active neutrinos (see Sec. C.1).For relativistic neutrino, Eν � ms, the probability of the transition να → νs intime t in vacuo is

P (να → νs) = sin2 2θα · sin2

(t

2tvacα

),

tvacα =

2Eν

Δm2, Δm2 = m2

s − m21 m2

s.

(7.15)

Cosmic plasma effects, however, are quite important, especially at high tempera-tures. The plasma affects the propagation of active neutrino |να〉, so the Hamiltonianin the active-sterile basis (|να〉, |νs〉) is

H = U · diag(

m21

2Eν,

m22

2Eν

)· U† + Vint, (7.16)

where the mixing matrix U and matrix Vint describing matter effects are

U =

(cos θα sin θα

− sin θα cos θα

), Vint =

(Vαα 0

0 0

).

The quantity Vαα can be calculated for να = νe, νμ, ντ by making use of the methodsof quantum field theory at finite temperatures. These methods are presented inAppendix D. Assuming the absence of lepton asymmetry, one finds that the leadingcontributions to Vαα cancel out. Most important are the contributions emerging dueto the momentum dependence of the W -boson propagator. These are suppressedby a factor T 2/M2

W ∝ T 2GF sin2 θW /α. At interesting temperatures, T ∼ 100 MeV

140 Relic Neutrinos

there are no τ -leptons and few muons in the medium, but there are numerous rela-tivistic electrons and positrons. For this reason, Vαα are quite different for differenttypes of neutrinos. One has the following expression [65] for ντ ,

Vττ =14π

45αsin2 θW cos2 θW · G2

F T 4 · Eν ≈ 25 · G2F T 4 · Eν ,

The numerical coefficient is 3.5 larger for electron neutrino, while the coefficient formuon neutrino is between 1 and 3.5 depending on the ratio of temperature to muonmass.

By diagonalizing the Hamiltonian (7.16) one obtains effective masses and mixingangles of neutrinos in medium, which are different from those in vacuo. As a result,the oscillation probability is given by the formula similar to (7.15) but with differentmixing angle θmat

α and period of oscillations tmatα ,

P (να → νs) = sin2 2θmatα · sin2

(t

2tmatα

),

tmatα =

tvacα√

sin2 2θα + (cos 2θα − Vαα · tvacα )2

,

sin 2θmatα =

tmatα

tvacα

· sin 2θα,

(7.17)

where tvacα is the oscillation time in vacuo at energy Eν ∼ T . At interesting tem-

peratures, when Eν ∼ T ∼ 100 MeV, and sterile neutrino masses m1 � ms � T ,we have the following estimate for the time of oscillations in the medium,

tmatα = min(tvac

α , V −1αα ) = min(2Tm−2

s , 0.04 · T−5 · G−2F ). (7.18)

The typical oscillation time is not only smaller than the Hubble time H−1(T ), butalso smaller than the typical time of weak interactions τν given by (7.2). Scatteringof active neutrino leads to the collapse of its wave function. Hence, during time τν

active and sterile neutrinos oscillate, and at the moment of scattering the coherenceis destroyed.

Thus, every active neutrino να oscillates into sterile neutrino νs many timesbefore it collides with a particle in plasma. Therefore, the probability that aftertime τν this neutrino produces the sterile neutrino is

〈P (να → νs)〉 =14· sin2 2θmat

α , (7.19)

where we performed averaging of (7.17) over several oscillation periods (see [59]regarding the factor 1/4). In other words, every active neutrino exits the plasmaas sterile neutrino in time τν with probability (7.19). At temperatures T � 3 MeV,when active neutrinos are in thermal equilibrium with plasma, this process does notreduce the number of active neutrinos but it leads to production of sterile neutrinos.The production rate per unit volume is

τ−1ν · 〈P (να → νs)〉 · nνα .

7.3. ∗Sterile Neutrinos 141

This gives for the density of sterile neutrinos nνs (assuming that they do not decayand neglecting the inverse process of their oscillation into active neutrinos)

dnνs

dt+ 3Hnνs = τ−1

ν · 〈P (να → νs)〉 · nνα , (7.20)

where the second term in the left hand side is due to the cosmological expansion.Equation (7.20) is conveniently written as the equation for the sterile-to-active ratio,

d(nνs/nνα)d ln T

= −〈P (να → νs)〉H(T )τν

, (7.21)

where we use the argument T instead of t and disregard the temperature-dependenceof the effective number of degrees of freedom g∗. The temperature-dependence of themixing angle in matter θmat

α is strong, so the right hand side of Eq. (7.21) behavesas T 3 at low temperatures and as T−7 at high temperatures. This implies that theproduction of sterile neutrinos takes place mostly in a narrow interval around somecritical temperature T∗. The latter is determined from the requirement that theright hand side of (7.21) is at maximum, i.e., the two expressions in parenthesisin (7.18) are of the same order of magnitude. This gives

T∗ ∼(

ms

5GF

)1/3

200 MeV ·( ms

1 keV

)1/3

.

As the sterile neutrino production rate has power-law dependence on temperature,their number density is estimated as

nνs(T∗)nνα(T∗)

∼ sin2 2θα

H(T∗) · τν(T∗)∼ T 3

∗M∗PlG

2F · sin2 2θα

∼ 10−2 ·( ms

1 keV

)·(

sin 2θα

10−4

)2

. (7.22)

The number of sterile neutrinos in comoving volume remains constant afterwards,so does the ratio nνs/nνα . Making use of the expression (7.8) for the number densityof active neutrinos, we find from (7.22) the estimate for the present contribution ofsterile neutrinos into energy density,

Ωνs 0.2 ·(

sin 2θα

10−4

)2

·( mν

1 keV

)2

. (7.23)

Accurate calculations show that this estimate is reasonably precise. Thus, sterileneutrinos of mass mν � 1 keV and small mixing angle θα � 10−4 serve as darkmatter candidate. At ms ∼ 1 − 10 keV this would be warm dark matter, whilesterile neutrinos of masses ms � 1 keV are too hot to be dark matter particles (seediscussion in Sec. 9.1). Let us note that sterile neutrino in this type of models isunstable against decay into active neutrino and photon. This leads to bounds onparameters of sterile neutrino, as the model should not contradict measurementsof cosmic photon flux at energies 1 − 10 keV. These bounds are so strong that

142 Relic Neutrinos

they make questionable the above mechanism of generation of sterile neutrino darkmatter [64].

We note that our estimate for sterile-to-active ratio (7.22) coincides with theratio of the rate of sterile neutrino production to the expansion rate entering (7.21)at temperature T = T∗ when this ratio is at maximum. This implies that sterile neu-trinos are never in thermal equilibrium with plasma, provided that the ratio (7.22)is much smaller than 1.

To conclude this Section we note that more complicated models have othermechanisms of production of light sterile neutrinos in the early Universe, besidesthe mechanism we described above. An example is given by decays of heavy par-ticles. Hence, our estimate (7.23) is, generally speaking, a lower bound on the sterileneutrino abundance.

Chapter 8

Big Bang Nucleosynthesis

The earliest epoch in the hot Universe which has been tested observationally is theBig Bang Nucleosynthesis epoch (BBN). As we will see, it begins at temperatureof about 1 MeV and lasts until the temperature drops to a few dozen keV. At thistime, neutrons in cosmic medium combine with protons into light nuclei, mostlyhelium-4 (4He = α-particle) with small but measurable admixture of deuterium(D≡2H), helium-3 (3He) and lithium-7 (7Li). The main thermonuclear reactions ofBBN are listed in the beginning of Sec. 8.3. We will see that at relevant temperaturesneutrons are less abundant than protons; “extra” protons remain free and in theend form hydrogen atoms.

Measurements of the primordial chemical composition not only confirm thetheory of the hot Universe, but also provide the determination of an importantcosmological parameter, baryon-to-photon ratio ηB . Furthermore, BBN constraintsmodels pretending to extend the Standard Model of particle physics.

Precise calculation of light element abundances produced at the BBN epoch isa complicated task. An appropriate tool here is the numerical analysis of kineticequations with account of numerous thermonuclear reactions, see, e.g., Ref. [72].In this Chapter, like in many other Chapters of this book, we limit ourselves byorder-of-magnitude estimates having in mind our main purpose: to discuss physicsof processes in the early Universe and explain, at qualitative level, the dependenceof the results on cosmological parameters.

8.1 Neutron Freeze-Out. Neutron-Proton Ratio

The first stage of BBN is neutron freeze-out. We will see in a moment that it occursat temperature of about 1MeV, which is still too high for light nuclei production.

In the early Universe, neutrons are produced and destroyed in weak interactionprocesses

p + e ↔ n + νe, (8.1)

and crossing processes. The energies relevant here are the mass difference betweenneutron and proton,

Δm ≡ mn − mp = 1.3 MeV

143

144 Big Bang Nucleosynthesis

and electron mass me = 0.5MeV. Let us assume for simplicity that temperature ishigh enough,

T � Δm, me. (8.2)

Then, like in Chapter 7, the mean free time of a neutron with respect to the process(8.1) can be estimated on dimensional grounds,

τn = Γ−1n , Γn = CnG2

F T 5, (8.3)

where Cn is a constant of order 1. The processes (8.1) and crossing processes ter-minate when the time τn becomes of the order of the Hubble time, i.e.,

Γn(T ) ∼ H(T ) =T 2

M∗Pl

. (8.4)

As before,

M∗Pl =

MPl

1.66√

g∗, (8.5)

and the effective number of relativistic degrees of freedom is

g∗ = 2 +78· 4 +

78· 2 · Nν . (8.6)

The first and the second terms here are due to photons and electrons/positrons,while the third term comes from neutrinos. We temporarily denote the number ofneutrino species by Nν , the actual value is Nν = 3. Let us recall here (see Chapter 7)that at temperatures T > me, i.e., before electron-positron annihilation, neutrinoshave the same temperature as photons.

Equations (8.3) and (8.4) determine the neutron freeze-out temperature,

Tn =1

(CnM∗PlG

2F )1/3

. (8.7)

The constant Cn entering (8.3) is known: the process (8.1) originates from thefour-fermion vertex shown in Fig. 8.1(a); the same vertex describes neutron decay,Fig. 8.1(b). Hence, the constant Cn is determined from neutron lifetime; numerically,Cn = 1.2. Thus, freeze-out temperature (8.7) does not contain unknown parameters.It depends, however, on the number of light neutrino species, see (8.5) and (8.6).

Recalling that the Fermi constant is GF = 1.17 · 10−5 GeV (see Sec. B.5) andg∗ = 43/4 for Nν = 3 we find numerically

Tn ≈ 1.4 MeV. (8.8)

Fig. 8.1 Feynman diagrams for processes n + νe ↔ p + e (a) and n → peνe (b).

8.1. Neutron Freeze-Out. Neutron-Proton Ratio 145

It is worth noting that our assumption (8.2) is not well justified, so more accuratecalculation is needed. We quote the result below.

It is amazing that neutron freeze-out temperature and the mass difference Δm

almost coincide. This property would not hold if the masses of u- and d-quarks, orthe Fermi constant, or Newton’s gravity constant were different from what they are.This is one of the coincidences which makes the chemical composition of primordialplasma interesting at all. Due to this coincidence, the neutron abundance at freeze-out is rather large, so the abundances of light nuclei are considerable in the end:if it turned out that Δm � Tn, then the neutron abundance at freeze-out wouldbe exponentially small, nn ∝ exp(−Δm/Tn), see (8.13). On the other hand, forΔm � Tn neutrons and protons would be equally abundant at freeze-out, so all ofthem would end up in 4He nuclei and the primordial plasma would lack hydrogen —such a hydrogen-free Universe would hardly be habitable.

Let us estimate the neutron abundance after freeze-out. With reasonableaccuracy, it is equal to the neutron abundance just before freeze-out. It is usefulfor what follows to write the general formula for the number density of particlesof type A (protons, neutrons, light nuclei) in chemical equilibrium at temperatureT � mA (see Chapter 5):

nA = gA

(mAT

2π

)3/2

e(μA−mA)/T , (8.9)

where μA is chemical potential for particle A. We now apply this formula for protonsand neutrons and make use of the fact that reaction (8.1) is in equilibrium just beforefreeze-out, hence μp + μe = μn + μν , i.e.,

μn = μp + μe − μν . (8.10)

Electrons and neutrinos are relativistic, and we have from (5.22)

ne− − ne+ ∼ μeT2, (8.11)

so thatμe

T∼ ne− − ne+

T 3.

The difference between number densities of electrons and positrons is equal to thenumber density of protons by electric neutrality,

ne− − ne+ = np,

while np/T 3 is of the order of the baryon-to-photon ratio, np/T 3 ∼ ηB ∼ 10−9. Weconclude that the chemical potential for electrons is negligibly small,

μe

T∼ 10−9.

Let us assume here that lepton asymmetry of the Universe is small (we will discussthe opposite situation in the end of this Section), i.e.,

nν − nν � nν + nν ∼ T 3.


Then μν/T is also negligible, and we obtain from (8.10) that

μn = μp. (8.12)

It then follows from (8.9) that neutron-proton ratio at freeze-out isnn

np= e−(mn−mp)/Tn ≡ e−Δm/Tn (8.13)

(we used the fact that both proton and neutron have two spin states, and setmn = mp in the pre-exponential factors).

Neutron-proton ratio (8.13) is roughly of order 1, so one needs to calculate thefreeze-out neutron abundance with good precision. The result is [69, 70]

nn

np= 0.18, (8.14)

which corresponds to the effective freeze-out temperature Tn 0.75MeV, cf. (8.8).Note that the neutron-proton ratio depends on the number of light neutrino speciesNν . More generally, the dependence is on the effective number of relativistic degreesof freedom in the plasma at T ∼ 1MeV. On the other hand, there is practically nodependence on other cosmological parameters.

Let us calculate the age of the Universe at neutron freeze-out. According to(3.29) we have

t =1

2H(Tn)=

M∗Pl

2T 2n

.

For Tn = 0.75MeV and Nν = 3 we get numerically

t = 1.1 s.

Thus, BBN theory deals with the cosmological epoch beginning at one second afterthe Big Bang.

To conclude this Section, we recall that the above calculation has been doneunder the assumption of negligible lepton asymmetry. If this assumption is notvalid, then instead of (8.12) we have μn = μp −μνe , and instead of (8.13) we obtain

nn

np= exp

(−Δm

Tn− μνe

Tn

).

The neutron-proton ratio in the end determines the 4He abundance (see Sec. 8.2),

n4He ∝ nn

np.

By comparing BBN theory predictions with observations one finds that the deviationof the ratio nn/np from the standard value should not be large,∣∣∣∣Δ

(nn

np

)∣∣∣∣ � 0.05.

This leads to the bound on chemical potential for electron neutrino,∣∣∣μνe

T

∣∣∣ � 0.05.

More accurate bound is given in (7.14).

8.2. Beginning of Nucleosynthesis. Direction of Nuclear Reactions. Primordial 4He 147

8.2 Beginning of Nucleosynthesis. Direction of Nuclear Reactions.Primordial 4He

The chain of thermonuclear reactions in the early Universe begins with deuteriumproduction in the reaction

p + n → D + γ. (8.15)

This reaction is denoted in nuclear physics as

p(n, γ)D. (8.16)

Other nuclear reactions are denoted in a similar way. To calculate the temper-ature at which this reaction begins, we use the following approach. We assumethat the reaction (8.15) is fast, so there is chemical equilibrium between deuterium,protons and neutrons. We also switch off other thermonuclear reactions. Under theseassumptions we calculate equilibrium abundance of deuterium at temperature T .This abundance is small at high temperatures, and we conclude that deuterium isnot produced. The physics behind this phenomenon is that the deuterium nuclei,that are created in neutron-proton collisions, dissociate very fast back into neu-trons and protons by absorbing hard photons from the tail of thermal distribution.The deuterium production begins when its equilibrium abundance calculated underabove assumptions becomes comparable to abundances of neutrons and protons(the latter are of the same order of magnitude, see (8.14)).

This “equilibrium” approach enables us to determine direction of thermonuclearreactions. Whether or not these reactions are actually fast depends on their crosssections and the cosmological expansion rate. In fact, the latter is rather high,and chemical equilibrium is not reached. Because of the latter property, the relicabundances of D, 3He, 7Li are not negligibly small: in chemical equilibrium, allneutrons would end up in 4He, the most tightly bound light nucleus.

Continuing with the equilibrium approach, let us write (8.9) as the Sahaequation. Because of thermonuclear reactions, chemical potentials of neutrons andprotons are different. The formula (8.9) gives for neutrons and protons

nn = 2(

mpT

2π

)3/2

e(μn−mn)/T , (8.17)

np = 2(

mpT

2π

)3/2

e(μp−mp)/T , (8.18)

where we neglected mass difference of neutron and proton in pre-exponential factors.If a nucleus of atomic weight A and charge Z is in chemical equilibrium,1 then itschemical potential is

μA = μp · Z + μn · (A − Z) .

1Hereafter subscript A labels nucleus (A, Z); to simplify formulas, we do not use more accurate

notations like μA,Z .


Indeed, chemical equilibrium means that there is a chain of fast reactions leadingto production of a nucleus (A, Z) from Z protons and (A − Z) neutrons.

Proceeding as in Sec. 6.1, we obtain from (8.9), (8.17), (8.18) that

nA = nZp nA−Z

n 2−AgAA3/2

(2π

mpT

) 32 (A−1)

eΔA/T , (8.19)

where we set mA = Amp in pre-exponential factor and introduced the bindingenergy of nucleus (A, Z),

ΔA = Zmp + (A − Z)mn − mA.

Let us also introduce dimensionless ratio of the number of baryons in nuclei (A, Z)to the total number of baryons,

XA =AnA

nB

.

Then Eq. (8.19) is written in the form of the Saha equation,

XA = XZp XA−Z

n nA−1B 2−AgAA5/2

(2π

mpT

) 32 (A−1)

eΔA/T .

The number of baryons is

nB = ηB · nγ = ηB · 2ζ(3)π2

T 3 = 0.24ηBT 3.

So, we have finally

XA = XZp XA−Z

n 2−AgAA5/2ηA−1B

(2.5T

mp

) 32 (A−1)

eΔA/T . (8.20)

Like in Sec. 6.1, the right hand side of this equation contains small factorηA−1

B (T/mp)32 (A−1), so the equilibrium abundance of nuclei becomes sizeable at

T � ΔA only, i.e., when the temperature becomes much smaller than nuclearbinding energy.

Nucleosynthesis begins when production of deuterium becomes thermodynam-ically favored, i.e., when XD becomes roughly of order 1 (recall that creation ofheavier nuclei is assumed to be switched off). Since at that time Xp, Xn ∼ 1, wefind from (8.20), modulo factor of order 1,

XD(TNS ) ∼ ηB

(2.5TNS

mp

)3/2

eΔD/TNS ∼ 1, (8.21)

where TNS is nucleosynthesis temperature, ΔD = 2.23MeV, and we used A = 2,Z = 1 for deuterium. This gives for ηB = 6.2 · 10−10 the value TNS ≈ 65 keV. Moreaccurate estimate is [69]

TNS ≈ 80 keV.

Hence, under the assumption of fast rate of deuterium production, the nucleosyn-thesis occurs at temperatures T � 80 keV. Note that TNS depends on ηB weakly(logarithmically).

8.2. Beginning of Nucleosynthesis. Direction of Nuclear Reactions. Primordial 4He 149

In fact, at these temperatures thermodynamically favored is the production of4He. Let us see this, making use of the equilibrium approach. If practically allneutrons at T ≈ TNS are indeed in 4He, then Eq. (8.20) must give X4He ∼ 1, whileabundances of all other light nuclei, including free neutrons, must be small. Let uswrite Eq. (8.20) for 4He (A = 4, Z = 2, gA = 4),

X4He = X2pX2

n · 8η3B

(2.5T

mp

)9/2

eΔ4He/T .

There are more protons than neutrons in plasma (see (8.14)), so extra protons giveXp ∼ 1. Again omitting factors of order 1, we express the neutron abundance interms of X4He,

Xn = X1/24Heη

−3/2B

(2.5T

mp

)−9/4

e−Δ4He/2T . (8.22)

By substituting (8.22) into (8.20), and taking X4He ∼ 1 we find for other nuclei,modulo factors of order 1,

XA =

[ηB ·

(2.5T

mp

)3/2] 3

2 Z− 12 A−1

eΔA−Δ4He(A−Z)/2

T

107.4(A+2−3Z) eΔA−Δ4He(A−Z)/2

T , (8.23)

where numerical value of the pre-exponential factor is given for ηB = 6.2 · 10−10,T = 80keV. Note that the sign in the exponent depends on binding energy perneutron, ΔA/(A−Z). Among light nuclei, this quantity is the largest for 4He, thatis why 4He is predominantly produced in the early Universe.

Binding energies of relevant stable or almost stable light nuclei are given inTable 8.1. Making use of these data, we find from (8.23) the following estimatesat T = 80keV: Xn ∼ 10−55, XD ∼ 10−57, X3H ∼ 10−93, X3He ∼ 10−42, X6Li ∼10−64, X7Li ∼ 10−94, X7Be ∼ 10−49, X8B ∼ 10−63. Thus, at T = TNS equilibriumabundances of light nuclei are much smaller than that of 4He.

Table 8.1 Binding energies of some stable or almost stable nuclei (MeV)

Z Nucleus ΔA ΔA/A ΔA/(A − Z)

1 2H ≡ D 2.23 1.11 2.233H ≡ T 8.48 2.83 4.24

2 3He 7.72 2.57 7.724He ≡ α 28.30 7.75 14.15

3 6Li 31.99 5.33 10.667Li 39.24 5.61 9.81

4 7Be 37.60 5.37 12.53

5 8B 37.73 4.71 12.58

6 12C 92.2 7.68 15.37


Let us make two points here. First, applying (8.23) to 12C we would get X12C �1. This means that if carbon could be produced, our assumption about dominationof 4He would be wrong: almost all neutrons would end up in 12C (and heaviernuclei). However, one can see from Table 8.1 that 12C cannot be produced in two-body reactions involving lighter stable nuclei:2 fusion of two 6Li isotopes is veryseldom because of tiny abundance of this isotope (this isotope is produced only infusion of helium-3 and tritium, and this process is strongly suppressed as comparedto production of helium-4 in collision of the same nuclei); elements with A = 5 andA = 8 cannot be produced in two-body reactions of abundant nuclei. This is whynucleosynthesis chain does not reach carbon in the early Universe. Thermonuclearreactions proceed towards production of 4He.

Second, if matter in the Universe were in chemical equilibrium with respect toproduction of 4He at T > 80keV, the nucleosynthesis would occur at higher tem-peratures. This is because binding energy of 4He is greater than that of deuterium.However, 4He is produced by deuterium burning, rather than directly from protonsand neutrons. Hence, it is the production of deuterium that determines the nucle-osynthesis temperature (“deuterium bottleneck”). In this sense the nucleosynthesisis delayed in the early Universe.

Problem 8.1. Find nucleosynthesis temperature in a hypothetical case of fast pro-duction of 4He directly from protons and neutrons.

Let us now find the age of the Universe at the epoch of thermonuclear reactions,i.e., at TNS ≈ 80 keV. According to (3.29) we have

tNS =1

2H(TNS)=

M∗Pl

2T 2NS

. (8.24)

The expansion rate is determined by photons and neutrinos, and the latter arealready frozen out. So, we have

M∗Pl =

MPl

1.66√

g∗, where g∗ = 2 +

78· 2 · Nν ·

(411

)4/3

. (8.25)

The age of the Universe at T = 80keV is, therefore (for Nν = 3),

tNS ≈ 200 s ≈ 3.3 min.

Using this estimate we now calculate the primordial abundance of helium-4. Wewill see in Sec. 8.3 that thermonuclear reactions proceed rapidly at T TNS . AfterBBN, most neutrons are collected in helium-4, so the abundance of helium-4 is halfof that for neutrons,

n4He(TNS ) =12nn(TNS ).

2Three-body reactions are practically absent in the early Universe since element abundances are

suppressed by ηB ∼ 10−9, i.e., are very small. Element 12C is produced in stars from 4He precisely

in three-body reactions (understood in a broad sense).

8.3. Kinetics of Nucleosynthesis 151

The latter, in turn, is related to proton abundance as follows,nn(TNS )np(TNS )

=nn(Tn)e−tNS/τn

np(Tn) + nn(Tn)(1 − e−tNS/τn

) ≈ 0.14, (8.26)

where we modified the relation (8.14) by taking into account neutron lifetime, τn ≈886 s. As a result, we obtain the mass fraction of 4He,

X4He =m4He · n4He(TNS )

mp[np(TNS ) + nn(TNS )]=

2np(TNS )nn(TNS ) + 1

≈ 25%. (8.27)

We note that this mass fraction depends both on nucleosynthesis time tNS and onneutron freeze-out temperature Tn (see (8.13)). Both of these quantities depend inturn on the effective number of relativistic degrees of freedom in primordial plasma.Therefore, the determination of the primordial helium abundance enables one toobtain the bounds on the number densities of new relativistic particles at BBNepoch, which are often given in terms of effective number of neutrino species (see(8.6) and (8.25)). The result is that this number should not deviate much fromNν = 3:

ΔNν,eff ≤ 1, T ∼ 1 MeV (8.28)This gives rise to constraints on models beyond the Standard Model which includenew light particles.

Problem 8.2. Find the lowest possible freeze-out temperature of hypotheticalmassless particles, assuming 50% uncertainty in Nν,eff . Do the same for uncer-tainty of 10%.

Problem 8.3. Making use of (8.5)–(8.7), (8.13) and (8.24)–(8.26) show that everynew type of neutrinos gives rise to the correction to primordial helium abundanceat the level of 5%.

An important uncertainty in the theoretical prediction of primordial heliumabundance is due to uncertainty in measured neutron lifetime. The latter is relevantfor both the temperature Tn (see (8.7), (8.13), (8.14)) and the number of neutronsremaining in the plasma at time tNS (see (8.26)).

8.3 Kinetics of Nucleosynthesis

We have seen in the previous Section that nuclear reactions proceed towards pro-duction of helium-4. In this Section we discuss the rates of the most relevant reac-tions and estimate the residual abundances of other light elements.

The direction of nuclear reactions that we found in the previous Section suggeststo divide these reactions into several categories:

(1) p(n, γ)D, production of deuterium, initial stage.(2) D(p, γ)3He, D(D,n)3He, D(D,p)T, 3He(n, p)T, preliminary reactions preparing

material for 4He production.


(3) T(D,n)4He, 3He(D,p)4He, production of 4He.(4) T(α, γ)7Li, 3He(α, γ)7Be, 7Be(n, p)7Li, production of the heaviest elements.(5) 7Li(p, α)4He, burning of 7Li.

We note that the reaction rates are proportional to abundances of colliding nuclei,so among all possible reactions the most relevant are those involving at least one ofthe abundant nuclei, i.e., p, n, D, 4He.

Let us consider these reactions in turn, with the purpose to estimate theirrates in the early Universe. By comparing these rates with the expansion rate atnucleosynthesis,

H(TNS = 80 keV) = 2.5 · 10−3 s−1,

we will find the residual inequilibrium abundances. These are of course much higherthan abundances that would be present in thermal equilibrium.

8.3.1 Neutron burning, p + n → D + γ

As we have seen, deuterium production becomes thermodynamically favored attemperature T = TNS ≈ 80 keV. However, the Universe expands rather fast, sosome neutrons could in principle be not burned out. Let us compare the rate ofneutron burning with the expansion rate at t = tNS .

The cross section of deuterium production can be roughly estimated as thegeometric cross section,

(σv)p(n,γ)D ∼ α

m2π

1137

1(200 MeV)2

= 2 · 10−18 cm3

s,

where mπ is the pion mass determining the typical spatial range of nuclear interac-tions, r ∼ m−1

π , while the fine structure constant α accounts for suppression relatedto photon emission. Note that this estimate does not depend on the velocity ofcolliding particles, i.e., on temperature. In fact, the temperature dependence exists,and the corresponding corrections change the cross section by a factor of 1.5–2 atT ∼ TNS . Furthermore, since deuterium is a loosely bound nucleus, there is anadditional factor ωγ/pD, where ωγ ∼ ΔD is the photon energy and pD is the typicalcenter-of-mass momentum of neutron and proton in deuterium. The latter can befound from the virial theorem,

p2D

MD ΔD, (8.29)

where we assumed for the estimate that the interaction potential between protonand neutron is inversely proportional to the distance. The final estimate is

(σv)p(n,γ)D ≈ 6 · 10−20 cm3

s.


Neutron burning occurs in collisions of neutrons with protons leading to deu-terium production. Its rate per neutron is given by

Γp(n,γ)D = np · (σv)p(n,γ)D = ηB · 2ζ(3)π2

T 3 · (σv)p(n,γ)D

= 0.6 s−1, for ηB = 6.2 · 10−10, T = TNS = 80 keV,

where we expressed the proton number density through baryon-to-photon ratio ηB

and photon number density at T = TNS . Since this rate is much higher than thecosmological expansion rate, Γp(n,γ)D � H(TNS ), neutrons indeed burn out, andpractically all of them combine into deuterium.3

Problem 8.4. Estimate the temperature and age of the Universe at the time whenneutron burning terminates. What would be the residual neutron abundance if otherreactions were negligible?

8.3.2 Deuterium burning

Deuterium is the material from which tritium and helium-3 are produced. The crosssections of the reactions

D + D → 3He + n and D + D → T + p

could be estimated as geometric cross sections, but we have to take into account theCoulomb barrier: both colliding nuclei carry positive electric charge, so they repeleach other. This repulsion dominates at long distances, r � 1/mπ, and inhibits thereactions. The form of the potential is schematically shown in Fig. 8.2.

Because of the Coulomb barrier, nuclear reactions occur due to quantum tun-neling. To estimate the corresponding suppression factor for nuclei of charges Z1

Fig. 8.2 Sketch of the potential between colliding nuclei.

3That would not be the case for ηB < 10−11.


and Z2, masses M1 and M2 and velocities v1 and v2, let us work in the center-of-mass frame. In this frame, the incident kinetic energy is Ekin = Mv2/2, whereM = M1M2/(M1 +M2) is the reduced mass and v = v1−v2 is the relative velocity.

The tunneling amplitude is exponentially suppressed. For s-wave scattering wehave

A ∝ exp[−∫ r0

0

√2M(V (r) − Ekin)dr

],

where the turning point r0 is determined by the relation

Ekin(r = r0) = Ekin(r = ∞) − V (r0) =12Mv2 − αZ1Z2

r0= 0;

in writing the exponent we assumed that r0 � 1/mπ. Thus, the exponent is

−√

2παZ1Z2

∫ r0

0

√1r− 1

r0dr = −παZ1Z2

v.

As a result, the cross section is suppressed as σ ∝ exp (−2παZ1Z2/v), and includingpre-exponential factor we have

σv = σ0 · 2παZ1Z2

v· e−2παZ1Z2/v, (8.30)

where σ0 is the geometric cross section in the absence of the Coulomb suppression.The expression (8.30) is to be averaged with the Maxwell–Boltzmann distri-

bution,

〈σv〉 = σ0 · 2παZ1Z2 ·∫∞0

exp(−Mv2

2T − 2παZ1Z2v

)vdv∫∞

0 exp(−Mv2

2T

)v2dv

. (8.31)

The normalization integral in the denominator is straightforwardly calculated; it isequal to

√π/2(T/M)3/2. We evaluate the integral in the numerator by the saddle

point method and obtain∫ ∞

0

exp(−Mv2

2T− 2παZ1Z2

v

)vdv

≈ v0

√2π

MT + 4παZ1Z2

v30

exp(−Mv2

0

2T− 2παZ1Z2

v0

),

where the saddle point v0 is determined by

Mv0

T=

2παZ1Z2

v20

.

As a result, (8.31) becomes

〈σv〉 ≈ σ0 · 2√3· (2παZ1Z2)

4/3 ·(

M

T

)2/3

· exp

[−3

2(2παZ1Z2)

2/3

(M

T

)1/3].


We now introduce convenient quantities: dimensionless reduced mass of incidentnuclei A ≡ M/mp and temperature in units of billion Kelvin, T9 ≡ T/(109 K) =T/(86 keV). In these notations, the final result is

〈σv〉 = 9.3 · σ0 · (Z1Z2)4/3

A2/3T−2/39 · e−4.26·(Z1Z2)2/3A1/3T

−1/39 . (8.32)

We note that this estimate for 〈σv〉 assumes that σ0 is independent of momenta ofcolliding nuclei at energies in the interval Ekin ∼ 10 − 100 keV. This is often notthe case and such a dependence leads to more cumbersome expressions instead of(8.32). In particular, the pre-exponential factor often has different dependence ontemperature as compared to (8.32). Furthermore, sometimes the expression (8.32)is not relevant at all, since the cross section is dominated by intermediate resonancestates. Finally, the production of new nuclei may not occur in s-wave scattering;in these cases the non-zero angular momentum gives important contribution to theeffective potential determining the tunneling exponent. This yields contributions tothe cross section that have different temperature dependence in the exponent ascompared to (8.32). We omit these “details” here, and will use the correct expres-sions in appropriate places. We discuss the calculations of burning rates in moredetail in the end of Sec. 8.3.4.

Coming back to deuterium burning reactions D(D, p)T and D(D, n)3He, let usroughly estimate σ0 by making use of the typical range of nuclear force,

σ0 ∼ m−2π ∼ 10−26 cm2 ∼ 3 · 10−16 cm3

s.

As a result we obtain for these reactions (A = Z1 = Z2 = 1 for DD initial state)

〈σv〉DD 3 · 10−15 cm3

s· T−2/3

9 · e−4.26·T−1/39 . (8.33)

Deuterium burning terminates when

ΓDD = nD(T ) · 〈σv〉DD(T ) ∼ H(T ). (8.34)

This occurs roughly at temperature TNS , namely, at about T ′NS 0.8TNS ≈

65 keV [69, 71], i.e., T9 ≈ 0.75. We use this condition to determine deuterium abun-dance at deuterium freeze-out. We obtain

nD =H(T ′

NS )〈σv〉DD(T ′

NS )≈ 1014 cm−3. (8.35)

We note that this abundance depends on the parameter ηB weakly, since ηB entersnD through TNS , and TNS logarithmically depends on ηB. By comparing nD withproton number density, we obtain the relative abundance of deuterium after BBNepoch,

nD

np=

10.75ηB

· nD

nγ(T ′NS )

= 3 · 10−5, at ηB = 6.2 · 10−10. (8.36)

This ratio have not changed since then.


It is important that due to weak dependence of nD on ηB, the deuterium-to-proton ratio (8.36) is inversely proportional to ηB. Therefore, the experimentaldetermination of primordial deuterium abundance gives good measurement of thetotal baryon density.

Let us now consider one more reaction involving deuterium, D +p −→ γ + 3He.Its cross section is much smaller than the cross section of the reactions we discussedabove. The reason is that photon emission leads to the electromagnetic suppression.Hence, σ0 ∼ 10−21 cm3/s and

〈σv〉D(p,γ)3He = 8 · 10−21 cm3

s· T−2/3

9 · e−3.7·T−1/39 ,

where we made use of the general formula (8.32) with A = 2/3, Z1 = Z2 = 1. Therate of deuterium burning via this channel is proportional to proton abundance,

Γ = np · 〈σv〉D(p,γ)3He.

For ηB = 6.2 · 10−10 and T T ′NS this rate is well below the expansion rate. This

reaction would be important at large ηB: the larger is the number of baryons4 themore deuterium is burned out, and more primordial 3He is produced.

8.3.3 ∗Primordial 3He and 3H

Helium-3 and tritium produced in collisions of deuterium nuclei, then burn intohelium-4. The simple estimate (8.32) does not work for helium-3 burning reaction3He + D → p + 4He (see discussion in Sec. 8.3.2 and in the end of Sec. 8.3.4).Instead, the reaction rate in the energy range of interest is well described by theformula

〈σv〉3He(D,p)4He = 10−15 cm3

s· T−1/2

9 e−1.8T−19 . (8.37)

For T9 0.75 (T = 65keV) this rate exceeds the deuterium burning rate (8.33);helium-3 burns even after deuterium freeze-out. Helium-3 freezes out when thedeuterium abundance becomes so low that the rate of the 3He burning becomes ofthe order of the expansion rate. This occurs at time t3He such that

〈σv〉3He(D,p)4He · nD · H−1 ∼ 1, t = t3He. (8.38)

At that time there is still some helium-3 in the Universe, since the reaction D+D →3He + n continues with small probability. Number density of helium-3 produced inthe Hubble time at t t3He is estimated as

n3He ∼ 〈σv〉D(D,n)3He · n2D · H−1, t = t3He. (8.39)

4Our emphasis in this section is on BBN determination of ηB , so we keep this parameter free.

We note, however, that irrespectively of BBN, this parameter is determined with good precision

from CMB observations.


This expression gives the estimate of the residual abundance of helium-3, since atlater times, t � t3He, deuterium density gets diluted due to cosmological expansionand helium-3 is no longer produced. We estimate the ratio of the relic abundancesof helium-3 and deuterium from (8.38) and (8.39) and obtain

n3He

nD 〈σv〉D(D,n)3He

〈σv〉3He(D,p)4He, (8.40)

where the right hand side should be evaluated at time t3He. Temperature at thattime is determined by (8.38); it is quite close to T ′

NS = 65keV. Indeed, the left handside of (8.38) strongly depends on temperature since nD ∝ T−3 and H−1 ∝ T−2,while the rates of reactions D + D → 3He + p and 3He + D → 4He + p differ byone order of magnitude only. By comparing (8.38) with the relation (8.34) valid atT = T ′

NS , we see that 3He burning terminates at T3He 0.6 T ′NS . Making use of

(8.33), (8.36) and (8.37) we obtain the estimate for the helium-3 abundance afterBBN,

n3He

np 0.9 · 10−5.

Tritium burning, T + D → 4He + n, is studied in a similar way. The tritiumproduction cross section is practically the same as for 3He, while tritium burningrate is

〈σv〉T (D,n)4He = 10−15 cm3

s· T−2/3

9 e−0.5T−19 . (8.41)

It is worth noting that this rate depends on temperature weakly at T9 ∼ 1. Therate is higher than that of helium-3, so tritium burns longer than helium-3 and itsfinal abundance is smaller. The estimate for the latter proceeds in the same way asfor 3He (see (8.40)) and gives

nT

np 2 · 10−7, at ηB = 6.2 · 10−10. (8.42)

Both ratios, n3He/np and nT /np, are, crudely speaking, inversely proportional toηB. More accurate calculation [72] gives n3He/np ∝ η−0.6

B .Let us make one point here. We have claimed that practically all neutrons end

up in 4He, and this is true. There is a non-trivial reason for that, however. Namely,burning of helium-3 and tritium occurs faster than their production from deuterium.Were this not the case, deuterium would burn out first, and reactions 3He + D →4He + p, T + D→ 4He + n would terminate at the stage when neutrons are bound inhelium-3 and tritium, rather than in 4He. On the other hand, freeze-out abundancesof 3He and T are sizeable, since their burning rates are not vastly higher thandeuterium burning rate at T T ′

NS . A certain diversity of light elements in cosmicmedium after BBN is the result of rather accidental coincidences between low energythermonuclear cross sections.


8.3.4 ∗Production and burning of the heaviest elements in

primordial plasma

As an example of reactions involving the heaviest elements of primordial plasma, letus consider production and burning of 7Li in reactions T(α, γ)7Li and 7Li(p, α)4He,respectively.

Production reaction is reasonably well described by the formula (8.32), in whichσ0 ∝ m−2

π · α (the factor of α is due to photon emission). Numerically,

〈σv〉T (α,γ)7Li ∼ 10−18 cm3

s· T−2/3

9 e−8.0T−1/39 . (8.43)

The rate of tritium burning via this channel is

〈σv〉T (α,γ)7Li · nα 1.5 · 10−4 s−1, T9 = 0.75.

This rate is small as compared to the Hubble parameter. The burning reaction isalso described by formula (8.32), and we obtain

〈σv〉7Li(p,α)4He ∼ 10−15 cm3

s· T−2/3

9 e−8.5T−1/39 .

Here the parameter σ0 is determined by strong interactions only, so the rate is muchhigher than the production rate (8.43). Numerically, the burning rate of lithium-7is

〈σv〉7Li(p,α)4He · np 0.7 s−1, at T9 = 0.75, ηB = 6.2 · 10−10,

which is higher than the Hubble parameter. Hence, burning of 7Li terminates ratherlate, when the number density of protons gets diluted substantially due to thecosmological expansion. At that time, the ratio of lithium-7 and tritium abundancesfreezes out at the level (cf. (8.39))

n7Li

nT 〈σv〉T (α,γ)7Li

〈σv〉7Li(p,α)4He· nα

np∼ 2 · 10−5.

Making use of (8.42) we find that the abundance of primordial lithium-7 is verysmall.

The production of 7Be is also important. This isotope is unstable, so itsabundance is not directly measurable. Beryllium-7 transforms into lithium-7 incosmic plasma either via electron capture 7Be(e−, νe)7Li, or via the reaction7Be(n, p)7Li. Thus, lithium-7 is produced either directly, in tritium-α fusion, orthrough beryllium-7. The existence of the two production mechanisms gives rise tonon-monotonic dependence of the primordial lithium-7 abundance on ηB.

Let us describe in some details the calculation of burning rates 〈σv〉 (see Ref. [72]for further details). Averaging over energies proceeds with the Maxwell–Boltzmann distri-bution,

〈σv〉 =2

T·

r2

πMT·

Z ∞

0

σ(E) · E · e−E/T dE, (8.44)


where v is the relative velocity of colliding nuclei, M is reduced mass, E is kinetic energyin the center-of-mass frame, σ(E) is the cross section of the reaction of interest; processes2 → 2 are by far dominant.

If one of the particles is a neutron, Coulomb barrier is absent. Assuming the s-wavereaction, i.e., σ ∼ v−1, the neutron reaction cross section far from resonance energies isnatural to write as follows,

σ(E) ≡ R(E)

v(E)=

rM

2ER(E).

The function R(E) depends on E weakly in the interesting energy range E � 1MeV, so

that it can be approximated by a few terms of the Taylor series in velocity, i.e.,√

E,

R(E) =

n=mXn=0

R(n)(0)

n!En/2,

where R(n)(0) are determined by fitting experimental data at low energies. Then theintegral (8.44) can be evaluated analytically, and the neutron burning rate in the channelof interest is given by

〈σv〉(T ) =n=mXn=0

R(n)(0)

n!

Γ`

n+32

´Γ

`32

´ · T n/2.

If the reaction has resonance character (like the reaction 7Be(n, p)7Li which has resonancesat ER � 0.32 MeV and ER � 2.7 MeV), then in the case of isolated resonance the crosssection in the resonance region has the Breit–Wigner form,

σ(E) =π

2ME

(2J + 1) (1 + δij)

(2Ji + 1) (2Jj + 1)

Γin(E)Γout(E)

(E − ER)2 + (ΓR/2)2,

where ER and ΓR are energy and width of the resonance, Ji, Jj and J are total angularmomenta of the initial nuclei and the resonance, and Γin(E) and Γout(E) are partial decaywidths of the resonance state into initial and final states, respectively. The functions Γin(E)and Γout(E) are also determined from experiment. In the narrow resonance case, Γ ER,the integral (8.44) is well approximated by

〈σv〉(T ) �„

2π

MT

«3/2(2J + 1) (1 + δij)

(2Ji + 1) (2Jj + 1)

ΓinΓout

ΓR· e−ER/T . (8.45)

If both incident nuclei are charged, the important phenomenon is the exponentialCoulomb suppression. As we have seen, the exponent is determined by the Sommerfeldparameter

ζ = αZiZj

rM

2E≡ 1

2π

rEg

E,

where Eg is known as the Gamow energy. The cross section is conveniently represented as

σ(E) =S(E)

E· e−2πζ .

Assuming that S(E) is a polynomial in E, the integral (8.44) can be evaluated by thesaddle point method, which gives

〈σv〉(T ) =2σ0

T

r2

MTπe−3

“Eg4T

”1/3

· S0(E0), (8.46)


where

E0 = Eg ·„

T

2Eg

«2/3

, σ0 =2Eg√

3

„T

2Eg

«5/6

,

are the saddle point value and width, while S0(E0) is a polynomial in (σ0/E0)2 ∝

(T/Eg)1/3. The latter function can be determined from experiment. Note that the saddle

point parameter σ0/E0 = (T/Eg)1/6 is fairly large, so obtaining good accuracy requires

employing high order polynomials. In practice one often makes use of other semi-analyticalapproximations to compute the rates 〈σv〉. As an example, reactions T(D, n)4He and3He(D,p)4He proceed through resonances. Even though these resonances are wide, theresonance contributions can be approximated by (8.45). The latter approximation is quitegood numerically, while the resonance contributions turn out to be dominant in the tem-perature range of interest. We have used this approximation in the text; this is why thetemperature dependence of the rates given in (8.37), (8.41) is different from the dependencethat would follow from (8.46).

Thus, the calculation of the rates 〈σv〉 involves, in an important way, experimentaldata on nuclear reactions. In some cases, including the reactions p(n, γ)D, D(p, γ)3He, thedata are scarce, which leads to uncertainties in predictions of element abundances. Thefirst reaction, p(n, γ)D, is, however, well understood theoretically, and this knowledge isoften used in real calculations.

We mention in the end that effects due to excitations of nuclei and Debye screening ofnuclei by free electrons are not very relevant for BBN.

8.4 Comparison of Theory with Observations

BBN theory is well-developed. Numerical analysis gives precise predictions for lightelement abundances in primordial plasma. These predictions are tested by mea-suring the chemical composition of matter in those places in the present Universewhere the composition is thought to be primordial despite evolution.

The evolution effects are very strong in most places in the Universe. Primordialmatter is processed in stellar thermonuclear reactions occurring in the recent Uni-verse, at z ∼ 0− 10. Some of the primordial nuclei transform into heavier elements,while others are destroyed by hard γ-quanta emitted in the star formation pro-cesses. These γ-quanta destroy also heavier elements already produced, so thatmore light elements appear. All these processes lead to considerable changes in thelight element abundances as compared to the primordial ones.

In some regions of the Universe, however, local abundances of some elementsare thought to remain unchanged. These are regions with low star formation rate:very distant (high redshift) regions where star formation has not happened yetand/or low-metallicity regions. The latter can be found by analyzing absorptionspectra.

Deuterium is special among light nuclei: it has very small binding energy andhence is not produced in stellar nucleosynthesis; it predominantly gets destroyed. Nosubstantial sources of deuterium are known, so any measurement of local deuteriumabundance sets a lower bound on its primordial abundance. Recently, deuterium

8.4. Comparison of Theory with Observations 161

abundance has been determined by spectroscopy of high-z, low-metallicity cloudswhich absorb light of distant quasars.

Primordial helium-4 abundance is measured by spectroscopy of low-metallicityclouds of ionized hydrogen in dwarf galaxies. Production of helium-4 in stars isaccompanied by production of heavier elements, metals in astrophysics terminology,so absence of the latter in clouds suggests that helium-4 is mostly of BBN originthere.

Lithium-7 abundance is determined by spectroscopy of low-metallicity old starsin globular clusters of our Galaxy.

No regions in the Universe have been found so far, where 3He abundance couldbe measured and where 3He would be mostly of primordial origin. Primordial abun-dance of this element is not as sensitive to the parameter ηB as that of deuterium,and measurements of 3He tell more about evolution of stars and Galaxy than aboutBBN.

Spectroscopic measurements of relative local element abundances are quiteprecise by themselves. The major uncertainties are systematic and, roughlyspeaking, have to do with limited confidence on the primordial origin of theseabundances. The predictions of BBN theory together with observational data [1]are shown in Fig. 8.3. These results are in good agreement with each other andwith the value ηB = 6.2 · 10−10 obtained from CMB data. We note here that it ispremature to talk about discrepancy between the abundance of lithium-7 and otherdata, since systematic uncertainties are still pretty high.

Still, the data is precise enough to make an impact on particle physics. From theviewpoint of the Standard Model, the importance of BBN boils down to the deter-mination of a single parameter, baryon asymmetry ηB. In the context of extensionsof the Standard Model, the good agreement between BBN theory and data leads toseveral consequences.

First, as we have already discussed, BBN imposes limits on the density of newrelativistic particles at T ∼ 1MeV, see (8.28).

Second, there should be no decays or annihilations of new heavy particles withemission of numerous hard photons at BBN epoch and somewhat later. If the maindecay channel of a heavy particle X is

X → γY,

where Y is a particle with mY � mX , then the energy of produced photon isEγ ≈ mX/2. It is the emission of these photons that is dangerous for BBN. Thereare two effects. The first one is the destruction of light elements by the hard photons.Particularly sensitive to this destruction is deuterium. The second effect has to dowith the disintegration of helium-4. Even though this nucleus is tightly bound, hardphotons may well break it up into lighter nuclei. Since helium-4 is very abundant,this would lead to overproduction of lighter elements, notably, 3He. The corre-sponding bounds [73] in the space of parameters τX and ζX , where τX is X-particle


Fig. 8.3 Predictions of BBN theory for primordial abundances of 4He, D, 3He, 7Li and the obser-

vational data at 2σ confidence level [1]: statistical errors (solid lines) and statistical and systematic

errors together (dashed lines). Theoretical uncertainties are shown by thickness of the lines. Ver-

tical strip “CMB” is the CMB result for ηB . On vertical axis are: Yp =n4He·m4He

nH ·mH+n4He·m4He,

mass fraction of 4He (the same as X4He in the text); nD/nH , n3He/nH and n3Li/nH , relative

abundances of other elements. Subscript p in notations refers to primordial abundances.

lifetime and

ζX = mX

nX

nγ=

ΩX

ΩBmp ηB ,

are shown in Fig. 8.4.Examples of models with X- and Y -particles include some supersymmetric

extensions of the Standard Model, with the decaying X-particle being neutralinoand stable Y -particle being gravitino (see Sec. 9.7.1). Similar bounds [74] exist onmodels with long-lived particles decaying into hadrons.

8.4. Comparison of Theory with Observations 163

Fig. 8.4 Model-independent BBN bounds on models with long-lived particles decaying into high

energy photons [73]. The dark and light shaded regions are mostly excluded by would-be over-

abundance of 3He and underabundance of D, respectively.

Yet another source of bounds on extensions of the Standard Model is theobservation that there should be (almost) no entropy production at BBN epoch.Otherwise the nucleosynthesis temperature would be different, and the abundanceof helium-4 would be inconsistent with observations.

Chapter 9

Dark Matter

As we have discussed already, large contribution to the total energy density inthe present Universe (about 20%) comes from dark matter which consists, mostlikely, of new massive particles absent in the Standard Model of particle physics.These particles must be non-relativistic and stable (or almost stable). They shouldnot have sizeable long-ranged interactions between themselves1 and should havepractically no interactions with photons. The latter requirement comes from theobservations showing that dark matter halos are much larger than baryon parts ofgalaxies, meaning that dark matter experiences very little, if any, photon cooling(see Fig. 1.6 in Chapter 1).

In this Chapter we consider several mechanisms of the dark matter generationin the Universe and some extensions of the Standard Model which contain darkmatter particle candidates. Let us make an important comment right away: noneof these mechanisms explains approximate (valid within a factor of 5) equality

ρB,0 ∼ ρDM ,0, (9.1)

where ρB,0 and ρDM ,0 are energy (mass) densities of baryons and dark matter in thepresent Universe. This approximate equality was also valid at earlier stages, sincethe times dark matter and baryon asymmetry were generated. Several suggestionshave been made in literature on how mechanisms of baryon asymmetry and darkmatter generation may be related and lead to (9.1), but none of them appearscompelling. The approximate equality (9.1) may be accidental indeed.

9.1 Cold, Hot and Warm Dark Matter

Let us make, for the sake of concreteness, a rather natural assumption that darkmatter particles X were in kinetic equilibrium with conventional matter in the early

1Long-ranged interactions between dark matter particles would lead, among other things, to

formation of spherical halos, while observationally most halos of clusters of galaxies are ellipsoidal.

At the same time, we mention that considerable elastic cross section of dark matter particles would

be helpful in explaining the distribution of dark matter in galactic centers: numerical simulations

of non-interacting dark matter predict cuspy profiles, with strong increase of dark matter density

towards the center. These cusps apparently are not observed.

165

166 Dark Matter

Universe.2 At some moment of time these particles get out of equilibrium and sincethen they propagate freely. If the corresponding decoupling temperature Td is muchsmaller than the mass of the dark matter particle MX, these particles decouple beingnon-relativistic. In this case dark matter is cold. In the opposite case, Td � MX,there are two possibilities, MX � 1 eV and MX � 1 eV. The former corresponds tohot dark matter: its particles remain relativistic at matter-radiation equality (recallthat equality occurs at Teq ∼ 1 eV, see Sec. 4.4); this is the case, e.g., for neutrino,as we have seen in Chapter 7. In the latter case dark matter is called warm: it isnon-relativistic by equality epoch. We see in the accompanying book that densityperturbations grow differently at radiation domination and matter domination, andthat this growth strongly depends on whether dark matter is relativistic or not atequality. This is the reason for distinguishing hot and warm dark matter.

One effect specific to hot and warm dark matter is as follows. Let dark matterhave primordial density perturbations and its particles be free and relativistic inthe temperature interval Td � T � MX. At that time dark matter particles escapepotential wells and fill in underdense regions of sizes up to current horizon size. Dueto this free streaming, dark matter density perturbations of these sizes get washedout. Hence, hot and warm dark matter have low amplitudes of density perturbationsat relatively short scales.

Free streaming terminates at T ∼ MX. The horizon size at that time, stretchedby a factor (1 + z) = T/T0, is the present maximum size of suppressed density per-turbations. In the warm dark matter case the equality T ∼ MX occurs at radiationdomination, so the horizon size at that time is

lH ∼ M∗Pl

T 2∼ M∗

Pl

M2X

.

The corresponding present size is

lX,0 = lH

T

T0∼ M∗

Pl

T0MX

. (9.2)

Thus, models with warm dark matter predict the suppression of density perturba-tions of the present size l0 < lX,0. For MX ∼ 1 keV we take g∗(T ∼ MX) = 3.36(see (4.22)), so that M∗

Pl = MPl/(1.66√

g∗) = 4 · 1018 GeV. Then Eq. (9.2) gives

MX ∼ 1 keV: lX,0 ∼ 3 · 1023 cm = 0.1 Mpc,

while

MX ∼ 1 eV: lX,0 ∼ 100 Mpc. (9.3)

Hence, models with hot dark matter predict the suppression of density perturbationsof present sizes up to 100Mpc. We refine these estimates in the accompanying book.

In hot dark matter models, largest structures — superclusters of galaxies — getformed first, and then they fragment into smaller structures, clusters of galaxies.

2This assumption, in fact, might not hold, as we will see in Sec. 9.4.2.

9.1. Cold, Hot and Warm Dark Matter 167

Galaxies are the latest objects in these models. This sequence of events is in strongdisagreement with observations.

Probably the best option is cold dark matter. Hot dark matter particles (e.g.,neutrinos) should make a small contribution into the total dark matter density.

Spatial size of order 0.1Mpc is typical for perturbations that developed intosmall structures like dwarf galaxies.3 The studies of structures of this and somewhatlarger sizes gives the lower bound on the mass of dark matter particle,

MX � 1 keV. (9.4)

We discuss this bound in some details in the accompanying book, and here we pointout that warm dark matter is still a viable possibility. We emphasize that this boundapplies to dark matter which was in kinetic equilibrium with the usual matter atsome early epoch. For non-thermal momentum distribution the estimate (9.4) getsmodified by a factor of order 〈|p|〉/〈|p|〉eq, where 〈|p|〉 and 〈|p|〉eq are actual averagemomentum and thermal one, respectively.

There exist model independent bounds on the masses of dark matter particles, whichapply equally well to dark matter that had never been in kinetic equilibrium with theusual matter. Of course, these bounds are very weak. They come from the fact that darkmatter particles must be confined in galaxies. For bosons the latter requirement impliesthat their de Broglie wavelength λ = 2π/(MXvX) must be smaller than the dwarf galaxysize, 1 kpc. Making use of the fact that velocities in galaxies are vX ∼ 0.5 · 10−3, we find

MX � 3 · 10−22 eV.

The bound is much stronger for fermions, due to Pauli principle. Assuming Maxwell dis-tribution of dark matter fermions in galactic halo (this in fact is a reasonable assumption),we find for their phase space density

f(p, x) =ρX(x)

MX

· 1

(√

2πMXvX)3· e−

p2

2M2X

v2X ,

where ρX(x)/MX and v2X are the number density and velocity dispersion of dark matter

particles in a halo. The maximum of the phase space density as function of momentumoccurs at p = 0 where f(p,x) is given by

fmax(p,x) =ρX(x)

M4X

· 1

(2π)3/2v3X

.

This maximum value cannot exceed the maximum value allowed by the Pauli principle(see (5.4)),

ff =gX

(2π)3.

Taking gX = 2, vX ∼ 0.5 · 10−3 and ρ(x) ∼ 0.5 GeV/cm3 (typical mass density in a halo)we obtain the bound

MX � 25 eV.

3Mass density in galaxies exceeds the average mass density by a factor 105–106. This means that

matter in a galaxy clumped from a region whose size exceeds the size of the galaxy itself by a

factor of 50–100. This leads to the estimate given in the text.

168 Dark Matter

Stronger bound is obtained from the existence of dwarf galaxies. There, the mass densityreaches ∼ 15 GeV/cm3, which gives the bound

MX � 750 eV.

We discuss this and similar bounds in the accompanying book.We note that there is also rather formal upper bound on the mass of dark matter

particles, of order of a thousand solar masses (see, e.g., [75, 76]),

MX � 103M� ∼ 1061 GeV.

This bound comes from the stability of stellar clusters in the Galaxy, which would bedestroyed by gravitational fields of dark matter “particles” moving nearby.

9.2 Freeze-Out of Heavy Relic

Let us turn to one of the most attractive scenarios of the dark matter generation.We will discuss concrete examples in the following Sections, and now we calculatethe residual abundance of heavy relic particles in general form.

To this end, let us consider the following situation. Let there exist stable heavyparticles X which are in thermal (including chemical) equilibrium with cosmicplasma at sufficiently high temperatures. Let their interactions with the rest ofthe plasma be strong enough, so that they remain in equilibrium at temperaturessomewhat below MX. This assumption should, of course, be justified by the cal-culation of freeze-out temperature. Stability of X-particle suggests that X-particlecan be created together with its antiparticle only. Let us ignore possible complica-tions and assume that this is indeed the case. Finally, let us assume that there isno asymmetry between X-particles and their antiparticles X, i.e.,

nX − nX = 0. (9.5)

This assumption is very important: the results of this Section are not valid for theUniverse asymmetric with respect to X-particles. Another possibility leading to thesame results is that X is a truly neutral particle, i.e., X coincides with X , whileX-particles are produced and annihilate in pairs. In that case the condition (9.5) issatisfied automatically. We note here that the latter situation is inherent in super-symmetric extensions of the Standard Model. Our task is to calculate the presentabundance ΩX of X- and X-particles.

The number densities in thermal equilibrium at temperature T < MX are

nX = nX = gX

(MXT

2π

)3/2

e−MX/T . (9.6)

Here we made use of the relation (9.5) to set the chemical potential of X-particlesequal to zero. Under above assumptions, the number densities decrease due to anni-hilation processes only,

XX → light particles.

9.2. Freeze-Out of Heavy Relic 169

When the rate of annihilation is higher than the cosmological expansion rate, theabundance of X is given by the equilibrium formula (9.6). At some moment oftime the number density of X-particles becomes so small that annihilation termi-nates. The inverse process of X-X pair-production switches off at the same time: bydetailed balance, the rates of production and annihilation are the same just beforethese rates become smaller than the expansion rate. At that time the abundance ofX-particles freezes out: after that, the number of particles is constant in comovingvolume. So, to find the residual X-particle abundance we have to calculate freeze-outtemperature in the first place.

Let us consider an X-particle just created in plasma. On average, the time beforeit annihilates with some X-particle is

τ =1

nX

1〈σannv〉 , (9.7)

where σann is the annihilation cross section, and v is relative velocity of X and X.The annihilation switches off when the lifetime τ becomes of the order of the Hubbletime,

1nX

1〈σannv〉 = H−1(Tf ), (9.8)

where Tf is the freeze-out temperature in question.XX-annihilation often occurs in s-wave. In that case the velocity dependence

of the non-relativistic annihilation cross section is given by

σann =σ0

v, (9.9)

where σ0 is a constant which does not depend on velocity and which is determinedby interactions responsible for annihilation.

Let us briefly remind the reader of the way the law (9.9) emerges (detailed analysiscan be found, e.g., in the book [68]). It applies not only to the annihilation process butalso to any inelastic s-wave reaction. The main property which is assumed to hold is thatthe relevant interaction is not long ranged, i.e., it occurs inside a region of a certain sizea. As usual, let us consider the flux of non-relativistic particles X incident on particle Xat rest. Then the reaction probability per unit time is

P = Ca3|ψ(a)|2,where ψ(a) is the wave function of X-particles in the collision region near X-particle, andthe constant C is determined by the details of the interaction. It is important here thatparticles annihilate in s-wave state, so that there is no centrifugal barrier. To obtain thecross section, one divides the probability P by the flux of X-particles

j =i

2m(ψ∇ψ∗ − ψ∗∇ψ).

Away from the interaction region, the wave function is the plane wave of momentum pdescribing motion along the third axis,

ψ = eipz ,

170 Dark Matter

then the flux is equal to velocity v. Since the interaction is short ranged, the modulusof the wave function in the reaction region is |ψ(a)| = const with velocity-independentconstant of order 1. The result for the cross section is

σ =P

|j| = const · Ca3

v,

which is precisely (9.9). Note that the annihilation of heavy particles occurs with largeenergy release, ΔE ∼ MX , so the size of the interaction region is indeed small

a � 1

MX

.

It is worth noting that in the case of electrically charged particles of opposite charges,the wave function in the reaction region may be considerably different from the asymptoticone (see the book [68] for details). This occurs at kinetic energies smaller than the bindingenergy of X-X atom, E < α2MX . This effect will be irrelevant in examples below.

Making use of (9.9), and inserting the equilibrium number density (9.6) intoEq. (9.8) we obtain the equation for the freeze-out temperature Tf ,

1gXσ0

(2π

MXTf

)3/2

eMXTf = H−1(Tf ) ≡ M∗

Pl

T 2f

, (9.10)

where we assume that freeze-out occurs at radiation domination. We will alwaysassume that X-particle mass is much smaller than M∗

Pl. Then the right hand sideof Eq. (9.10) contains large factor, so the freeze-out temperature is considerablysmaller than MX. This justifies one of our assumptions.

Taking logarithm of both sides of Eq. (9.10) we obtain an equation of theform (6.18):

x = log(Axα),

where

x =MX

Tf, A =

gX

(2π)3/2σ0M

∗PlMX, α =

12. (9.11)

Let us assume that the very crude estimate of the constant σ0 is σ0 ∼ M−2X .

This estimate is sufficient to see that the condition for logarithmic approximation,log A � 1, is valid for masses MX � M∗

Pl. Then we obtain with logarithmic accuracy

Tf =MX

log(

gXMXM∗P l

σ0

(2π)3/2

) . (9.12)

Clearly, the freeze-out temperature Tf depends weakly (logarithmically) on theannihilation cross section. This temperature is smaller than MX by a factor[

loggXMXM∗

Plσ0

(2π)3/2

]−1

.

This quantifies the above claim that Tf � MX .

9.2. Freeze-Out of Heavy Relic 171

The number density of X-particles at freeze-out nX(tf ) is obtained by makinguse of 4 Eq. (9.8),

nX(tf ) =T 2

f

M∗Plσ0

. (9.13)

After freeze-out, nX changes solely due to the cosmological expansion, so its presentvalue is

nX(t0) =(

a(tf )a(t0)

)3

nX(tf ). (9.14)

We now use entropy conservation in comoving volume (see Eq. (5.30)) andwrite (9.14) as

nX(t0) =(

s0

s(tf )

)nX(tf ), (9.15)

where s(tf ) and s0 are entropy density at freeze-out and today. The present valueof the entropy density is (see (5.33))

s0 =2π2

45

(2T 3

γ + 6 · 78T 3

ν

)= 2.8 · 103 cm−3. (9.16)

Hence, the present number density of X-particles is given by

nX(t0) =s0T

2f

s(tf )M∗P lσ0

= 3.8s0

Tfσ0MPl

√g∗(tf )

, (9.17)

where we use the expressions (see Eqs. (3.32) and (5.28))

M∗P l =

MP l

1.66g1/2∗

, s(tf ) = g∗(tf ) · 2π2

45T 3

f .

Finally, making use of (9.12), we obtain the relative mass density of X- andX-particles,

ΩX = 2MXnX(t0)

ρc= 7.6

s0 log(

gXM∗PlMXσ0

(2π)3/2

)ρcσ0MPl

√g∗(tf )

. (9.18)

With known values of s0 and ρc we find numerically

ΩX = 3 · 10−10

(GeV−2

σ0

)1√

g∗(tf )log(

gXM∗PlMXσ0

(2π)3/2

)· 12h2

. (9.19)

Clearly, the strongest dependence here is on the parameter σ0. The dependence onthe mass MX is logarithmic only, while the effective number of degrees of freedomg∗ does not dramatically change during the cosmological evolution at freeze-outepoch.

Formula (9.19) is the main result of this Section. We refine it in Sec. 9.6.1.

4Inserting the expression (9.12) for temperature into the formula (9.6) for the number density

would not be a good idea, since temperature enters the exponent in (9.6), while the formula (9.12)

is valid with logarithmic accuracy only.

172 Dark Matter

To end this Section, we notice that we are mostly interested in chemical(in)equilibrium in cosmic medium. It is sometimes important to understand whetheror not the medium is in kinetic equilibrium, i.e., whether or not X-particles haveequilibrium distribution over momenta. Kinetic equilibrium is due to scattering ofX-particles off other particles, so the mean free time does not depend on the abun-dance of X-particles and hence is short compared to the mean free time with respectto annihilation, Eq. (9.7). This means that kinetic equilibrium lasts longer, i.e.,decoupling of X-particles from kinetic equilibrium occurs at temperature Td � Tf .As an example, if X-particles participate in weak interactions, then their elasticcross section off, say, electrons, is estimated at E � 100GeV on dimensionalgrounds as

σel ∼ G2F E2. (9.20)

The mean free time of X-particles is of order τel ∼ (ne · σel · v)−1, where ne iselectron number density and v is the relative velocity of electrons and X-particles,v 1 at T � 1MeV. To estimate the decoupling temperature we equate τel to theHubble time H−1(T ) and obtain (note that this calculation parallels that given inSec. 7.1) Td ∼ 1MeV. This estimate is rather crude, but it shows that the kineticequilibrium for X-particles gets violated very late.

Problem 9.1. Refine the estimate for Td assuming that the elastic cross sectionfor electrons and X-particles is given by (9.20). Hint: Make use of considerationsgiven in the beginning of Sec. 6.3.

Let us now consider concrete applications of the result (9.19).

9.3 Weakly Interacting Massive Particles, WIMPs

Let us discuss the possibility that new stable heavy particles make dark matter.Then the formula (9.19) is used to estimate their parameters (annihilation crosssection in the first place). The dark matter density today is

ΩDM ≈ 0.2 − 0.3,

where we write the conservative range; in fact, as we repeatedly pointed out, ΩDM

is known with better precision. To estimate the annihilation cross section we setσ0 ∼ 1/M2

X in the argument of logarithm in (9.19), on dimensional grounds. SettingMX = 100GeV and g∗ = 100 for the estimate, we obtain the numerical value of thelogarithmic factor in (9.19),

loggXM∗

PlMXσ0

(2π)3/2∼ log

gXM∗Pl

(2π)3/2MX∼ 30. (9.21)

Since logarithm is a slowly varying function, this estimate is valid for wide range ofmasses MX and cross sections σ0. The uncertainty in the parameter

√g∗(tf ) is also

9.3. Weakly Interacting Massive Particles, WIMPs 173

moderate: at T � 100GeV and T ∼ 100MeV we have, respectively,√

g∗(T ) ∼ 10,and

√g∗(T ) ∼ 3 (see Appendix B). Thus, formula (9.19) gives the following estimate

for the annihilation cross section of cold dark matter candidates,

σ0 ∼ 3 · 10−10 · 30 GeV−2

(3 − 10) · (0.2 − 0.3)= (0.3−1.5)·10−8 GeV−2 = (1−6)·10−36 cm2. (9.22)

Notably, this value is comparable to weak interaction cross sections at energies oforder 100GeV, namely σW ∼ α2

W /M2W ∼ 10−7 GeV−2.

The result (9.22) has several important consequences. One is that it gives acosmological lower bound on the annihilation cross section of hypothetical heavystable particles that may be predicted by extensions of the Standard Model. Ifthe cross section is below the value (9.22), mass density of these particles is unac-ceptably high. The main assumption behind this bound is that X-particles were inthermal equilibrium at some early epoch. We note here that there is a bound onthe annihilation cross section

σ0 � 4π

M2X

. (9.23)

In perturbative regime, this bound comes from the fact that the annihilation isdescribed by diagrams like that shown in Fig. 9.1. For slow X-particles, the virtualparticle Y has energy E = 2EX = 2MX in the center-of-mass frame. Its propa-gator gives the factor 1/M2

X in the cross section.5 Furthermore, there is additionalsuppression due to small coupling constant.

The bound (9.23) may actually be violated in strongly coupled theories. Thisoccurs when X-particle is a bound state of elementary particles, and the size ofthis bound state is large compared to the Compton wavelength. An example here isgiven by proton. It is difficult, however, to imagine that the bound (9.23) is violatedby many orders of magnitude.

With this reservation, one makes use of (9.22) and (9.23) to obtain the cosmo-logical upper bound on the mass of stable particles,

MX � 100 TeV. (9.24)

Fig. 9.1 Annihilation of stable hypothetical X-particles into Standard Model particles f1 and f2.

5If MY > MX , then the propagator of Y -particles is suppressed by M−2Y , which makes the

bound (9.23) even stronger.

174 Dark Matter

This bound is valid, if their interactions are sufficiently strong and the maximumtemperature in the Universe exceeded MX/30, see (9.12) and (9.21).

Problem 9.2. Let us extend the Standard Model by adding new real scalar field X

which interacts with the Higgs field H only. Let us add the following term to theStandard Model Lagrangian,

ΔL =12∂μX∂μX − κ

2H†HX2 − m2

2X2.

The discrete symmetry (X → −X) ensures the stability of the scalar particle X, soit is a dark matter candidate.

Let the vacuum expectation value of the field X be equal to zero. Assuming thatthe Higgs boson mass mh belongs to phenomenologically acceptable range 115 GeV <

mh � 300GeV, find the range of parameters (m, κ) in which X-particles make allof dark matter.

Let us mention that there is recent activity in discussing the possibility that alldark matter or its substantial part consists of much heavier particles. This scenariois realistic provided that these particles were never in thermal equilibrium andwere created in small number in the early Universe. We will briefly discuss possiblemechanisms of superheavy particle creation in Sec. 9.7.2.

By far more interesting is the possibility that X-particles are dark matter par-ticles. These dark matter candidates are called weakly interacting massive particles,WIMPs. The estimate (9.22) suggests the energy scale of X-particle interactions,σ−1/20 ∼ 10TeV. In fact, this scale is somewhat lower, since the annihilation cross

section is suppressed by coupling constant in realistic models, which we denote asαX . Assuming that the energy scale of X-particle interactions does not much exceedtheir mass, we get

σ0 ∼ α2X

M2X

. (9.25)

As an example, for αX ∼ 1/30 (W -boson coupling in the Standard Model) we obtainfrom (9.12) and (9.22) the following estimates,

Tf MX/20, (9.26)

MX ∼ 200 − 600 GeV. (9.27)

The latter estimate shows that there is a real chance to discover dark matter par-ticles at the next generation of colliders. We note, though, that unlike the estimate(9.22) for the annihilation cross section, our estimate (9.27) for the mass is verycrude. Indeed, the dark matter particle may well be lighter than given in (9.27),and the latter formula should rather be viewed as the estimate for masses of heavierparticles mediating the X-particle annihilation. Accurate calculations in concretemodels show that the mass of the dark matter particle may be considerably smallerthan 100GeV, which makes collider searches even more promising. We also note

9.3. Weakly Interacting Massive Particles, WIMPs 175

that our estimate is particularly interesting from the viewpoint of supersymmetricextensions of the Standard Model which often predict the existence of a stableparticle of mass in 100GeV range, see Sec. 9.4.2.

Thus, there are good reasons to expect that dark matter particles will be foundin reasonably near future. Experimental determination of their properties, togetherwith the precise calculation of their abundance and comparison with observationaldata will then provide a handle on the Universe at temperatures of order of tensGeV and age of 10−9 s. This is to be compared with the present knowledge: theUniverse has been probed at BBN epoch, T 1 MeV, t 1 s, see Chapter 8.

Search for dark matter particles is underway, but no conclusive evidence hasbeen obtained so far. Direct search for relic WIMPs is carried out in experimentstrying to detect energy deposition in a detector caused by elastic scattering of aWIMP off a nucleus. Dark matter, like the usual matter, is more dense in galaxies;it is expected that its mass density near the Earth is similar to that of usual matter,

ρlocalDM 0.3

GeVcm3

.

The velocity of dark matter particles is about vX ∼ 0.5 · 10−3 (orbital velocityaround the Galactic center at the position of the Sun). Hence, the energy depositionin collision with a nucleus of mass MA is estimated as

ΔE ∼ 12 MA

(2vXMAMX

MA + MX

)2

= 2MXv2X

MA/MX

(1 + MA/MX)2

= 2MAv2X

1(1 + MA/MX)2

.

The second of these formulas shows that the best choice of the target nucleus forgiven MX is MA = MX , while the third implies that for given detector material, theenergy deposition grows as MX increases. Numerically,

ΔE 50 ·(

min(MA, MX)100 GeV

)2 100 GeVMA

keV.

We see that the energy deposition is of order of tens keV, which is very small.Nevertheless, there are methods to detect it. The event rate is

ν vXnX · NA · σAX ,

where σAX is the elastic X-nucleus cross section, vX is the relative velocity, nX =ρlocalDM /MX is the local number density of dark matter particles and NA is the number

of nuclei in the detector. As an example, for elastic cross section6 σAX ∼ 10−38 cm2

and X-particle mass MX = 100GeV, the number of events in a detector of mass

6This cross section is not the sum of cross sections for individual nucleons in nucleus A: there

is coherence effect for low energy scattering of dark matter particle off nuclei which enhances the

cross section by a factor of order A.

176 Dark Matter

10 kg filled with nuclei of atomic number A = 100 is of order

ν ∼ 10−3 ·(

0.3GeVcm3

· 1100 GeV

)·(

6 · 1023 GeV · 104

100 GeV

)· 10−38 cm2 ∼ 5 · 10−8 s−1,

(9.28)which is about 1 event per year. The absence of a signal gives bounds in the spaceof model-independent parameters — dark matter mass MX and dark-matter-on-nucleon elastic cross section σNX — shown7 in Fig. 9.2.

Besides the direct search, there are ways to search for dark matter particlesindirectly. These include search for products of WIMP annihilation at present time.In this case too, no compelling dark matter signal has been observed so far.

One of the promising indirect ways is to search for monoenergetic photons whichare emitted in two-body annihilation processes XX → γγ, XX → Zγ. In this casethe signal should be enhanced in the direction towards dense regions in and outsidethe Galaxy, such as Galactic center, Magellanic Clouds, Andromeda galaxy, etc.Another way is to detect antiparticles created in the annihilation: positrons andantiprotons.

Heavy dark matter particles can also concentrate in the centers of the Earthand the Sun. This is possible if the interactions of dark matter particles with usual

]2WIMP Mass [GeV/c10 210 310

]2 [

cmSIσ

WIM

P-n

ucle

on

-4410

-4310

-4210

-4110Roszkowski 2007 (95%)CDMS, Soudan (All)ZEPLIN III, 2009XENON 10, 2008

Fig. 9.2 Parameter space for dark matter particles, (MX , σNX) (courtesy of S. V. Demidov).

Regions above the curves are excluded at 90% confidence level. Shaded regions show the parameter

regions expected in supersymmetric extensions of the Standard Model.

7The elastic cross section may strongly depend on the spin of a nucleus. This property is accounted

for in the experimental search. There exist bounds [77] on spin-dependent elastic cross sections

analogous to bounds shown in Fig. 9.2.

9.4. ∗Other Applications of the Results of Section 9.2 177

matter is strong enough, so that these particles occasionally loose energy whenpassing through the Earth or Sun, and eventually get caught by them. As a resultof the high concentration, the dark matter annihilation may be strongly enhanced.The annihilation often leads to production of high energy neutrinos, an examplebeing XX → νν. These neutrinos propagate away without colliding with matter,and can be detected by neutrino detectors. Search for high energy neutrinos fromthe centers of the Earth and the Sun is being performed at underground, underwaterand under-ice detectors, “neutrino telescopes”.

Problem 9.3. Let us assume that there exist stable, electrically charged particlesX± much heavier than proton. Let us recall that baryons in the early Universeare predominantly either protons or α-particles (4He nuclei). Mass fraction of α-particles is 25%.

(1) Find the binding energy of an “atom” consisting of X−-particle and proton.The same for X-α atom.

(2) Assuming that the number density of X−-particles is small compared to thatof baryons, find out in which state they predominantly exist today: bound statewith proton, bound state with α-particle or free state.

(3) Assuming that X±-particles were in thermal equilibrium in the early Universe,and that the asymmetry between X− and X+ equals ηX , find the present massdensity of relic X-particles as function of their mass MX and the asymmetryηX.

(4) Place the bounds on the asymmetry ηX making use of the fact that searchesfor anomalous heavy isotopes (“wild hydrogen” and “wild helium”) set upperbounds on the present mass density of these isotopes at the level [78, 79] ΩX <

10−10 ·(10 TeV /MX) at MX < 10TeV (see also [1, 79] for other mass intervals).

9.4 ∗Other Applications of the Results of Section 9.2

Before considering concrete particle physics models with dark matter candidates,let us give two examples of the use of the results of Sec. 9.2.

9.4.1 Residual baryon density in baryon-symmetric Universe

Let us find baryon (proton and neutron) number density after baryon-antibaryonannihilation in a toy Universe having no baryon asymmetry. In such a Universe,densities of baryons and antibaryons are equal to each other. We recall the baryon-antibaryon annihilation cross section at low energies8

σ0 ≈ 100 GeV−2. (9.29)

8Note that the cross section (9.29) exceeds by an order of magnitude the bound (9.23). The

reason is that baryons are bound states of quarks and gluons.

178 Dark Matter

Using this value in (9.12), we find for freeze-out temperature

Tf ≈ 20 MeV, (9.30)

where we calculated M∗Pl with g∗ = 43/4 (photons, electrons, positrons and three

neutrino species). Formula (9.19) gives the mass density of baryons in baryon-symmetric Universe,

ΩB ≈ 5 · 10−11.

This is about nine orders of magnitude smaller than the baryon density in realUniverse. Substantial amount of baryons in our Universe exists entirely due to thebaryon asymmetry.

9.4.2 Heavy neutrino

As another example, let us obtain a cosmological bound on the mass of hypotheticalheavy neutrino of the fourth generation. In the absence of mixing with other neu-trinos, such a neutrino does not experience charged current interactions with usualleptons, but participates in neutral current interactions. Let us assume that theheavy neutrino mass is not very large, mν � MZ,W . The analysis in Sec. 7 appliesto light neutrino only, for which Tf � mν . We consider here the opposite case inwhich neutrino annihilates being non-relativistic. In this case the annihilation crosssection is estimated as

σ0 ∼ G2F m2

ν ; (9.31)

it does not depend on temperature. Using this cross section in (9.19) we obtain theestimate for the present mass density of heavy neutrinos,

Ων ≈ 0.18(

6 GeVmν

)2

, (9.32)

where we set mν = 6 GeV in the argument of logarithm and used the followingvalue for the effective number of degrees of freedom,

g∗ = 2 · (1 + 8) +78· (2 · 4 + 2 · 3 + 3 · 4 · 3) = 61

34,

(photons, e±, μ±, three neutrino species, gluons and quarks9 u, d and s).We see that the existence of heavy stable neutrino of mass mν � 6GeV would be

inconsistent with cosmology [80]. For long time this had been the strongest boundon the mass of heavy neutrino. Present bound comes from the collider data on theZ-boson decay into undetected particles and reads mν > MZ/2.

Problem 9.4. Check that neutrino of mass mν = 6GeV indeed freezes out beingnon-relativistic. Find the values of the neutrino mass at which this property no

9For mν > 20 GeV, freeze-out temperature exceeds 1GeV, and g∗ receives contributions from

τ -lepton and heavy quarks. This modifies the estimate (9.32) only slightly.

9.6. ∗Stable Particles in Supersymmetric Models 179

longer holds. Show, nevertheless, that cosmologically excluded is the entire massrange of stable neutrino

20 eV � mν � 6 GeV,

where the lower value has been obtained in Chapter 7, formula (7.10).

Problem 9.5. Estimate elastic cross section of neutrino of mass mν > MZ/2 offnuclei and obtain bounds on its mass from data shown in Fig. 9.2.

9.5 Dark Matter Candidates in Particle Physics

We discuss in the following Sections some popular models of particle physics con-taining dark matter particle candidates.10 There are no such candidates in theStandard Model, so the cosmological data on dark matter require that the StandardModel must be extended. This fact is very important for particle physics.

Among the candidates we discuss, the natural one is neutralino. It is automati-cally stable in many viable supersymmetric extensions of the Standard Model andits present abundance is automatically in the right ballpark. Other candidates areless natural in the sence that their present mass density may a priori vary withinlarge margin, and the correct value is obtained by adjusting the parameters of amodel.

From the viewpoint of direct experimental searches, all candidates can be clas-sified by using two parameters only, the particle mass MX and its elastic scatteringcross section11 off nuclei σint. These parameters are shown in Fig. 9.3 for mostpopular candidates [81]. It is clear that the candidates have very diverse properties.Search for candidates from different regions in this plot is performed, as a rule, bymaking use of different detection techniques. In some cases, in particular, for veryheavy and/or very weakly interacting particles (wimpzilla and gravitino, respec-tively), no realistic detection methods are known.

The zoo of proposed dark matter candidates is huge. In what follows we do notintend to discuss even a sizeable fraction of inhabitants of this zoo. We leave asidenumerous candidates which we think are rather exotic. These include primordialblack holes, strongly interacting dark matter, axino (superpartner of axion), mirrormatter and many others.

9.6 ∗Stable Particles in Supersymmetric Models

Supersymmetry (SUSY) is the symmetry between bosons and fermions. The sim-plest (so called N = 1) SUSY theories in (3 + 1) dimensions contain both particles

10One of the candidates, massive sterile neutrino, is considered in Sec. 7.3.11In the case of axion and axino, the relevant parameter is the cross section of conversion on

nucleon into other particles, since this cross section exceeds considerably the elastic cross section.

180 Dark Matter

Fig. 9.3 Parameters for popular dark matter candidates [81]: their masses MX and elastic scat-

tering cross sections (conversion cross sections for axion and axino) off nucleon σint; 1 pb =

10−36 cm2.

and their superpartners, particles of opposite statistics whose spin differs by 1/2from that of particles. Superpartners participate in the same interactions as par-ticles. In other words, particles and their superpartners have the same quantumnumbers with respect to gauge group of the theory, while coupling constants ofother interactions (e.g., Yukawa) are related in a well-defined way. Each vectorparticle (e.g., gluon) has spin-1/2 superpartner (gluino), while each fermion (e.g.,quark) has scalar superpartner (squark). More precisely, the number of spin degreesof freedom must be the same, so there are two scalar particles (two states) per eachquark (two spin states).

Let us illustrate these properties using the supersymmetric generalization of QuantumElectrodynamics (QED). In fact, this was the first model illustrating the theoreticallydiscovered supersymmetry [82]. QED itself is a theory of massive Dirac fermion ψ (electron)interacting with Abelian gauge field Aμ (photon). The QED Lagrangian is

LQED = −1

4FμνF μν + iψγμ(∂μ + ieAμ)ψ − mψψ.

SUSY QED also contains four scalar degrees of freedom (corresponding to four spin statesof electron and positron), which are described by two complex scalar fields φ+ and φ−.These carry charges +1 and −1 of the gauge group U(1). There is an obvious contribution


to the Lagrangian,

L1 = Dμφ∗+Dμφ+ − m2

+φ∗+φ+ + Dμφ∗

−Dμφ− − m2−φ∗

−φ−,

Dμφ± = (∂μ ∓ ieAμ)φ±,

where m+ and m− are masses of scalars. In the case of unbroken supersymmetry, thesemasses are equal to the electron mass, m+ = m− = m (this is not the case if super-symmetry is spontaneously broken, see below). There is also photino, a superpartner ofphoton. This is electrically neutral Majorana fermion (two spin degrees of freedom), whichcan be described by left spinor λL. Its free Lagrangian is

L2 = iλLγμ∂μλL.

Besides the gauge interactions between the gauge field Aμ and charged fields, there are“supersymmetry completions”, the gauge-invariant Yukawa interaction between photino,electron and scalars,

L3 = i√

2eλLψ · φ− − i√

2eψλL · φ+ + h.c.,

and self-interaction in the scalar sector,

L4 = −e2

2(φ∗

+φ+ − φ∗−φ−)2.

The coupling constants of these additional interactions are related to the gauge couplingin a unique way. Unambiguous relations between various couplings is a general propertyof SUSY Lagrangians.

N = 1 supersymmetric extensions of other (3 + 1)-dimensional field theories havesimilar properties. In particular, supersymmetry completion of the usual Yukawa inter-action is interaction between scalars, and vice versa.

If supersymmetry is unbroken, masses of particles are equal to masses of theirsuperpartners. This makes unbroken supersymmetry phenomenologically unac-ceptable: no superpartners of the Standard Model particles have been discovered sofar. Hence, SUSY extensions of the Standard Model assume that supersymmetryis spontaneously broken. In that case masses of superpartners may be arbitrary.The negative results of numerous searches place lower bounds on these masses [1].These bounds are roughly at the level MS � 100GeV. On the other hand, thereare hints from the theory12 that the superpartner mass range is 30GeV–3TeV. Inthis regard, search for superpartners is one of important directions in high energyphysics, with high expectations raised by the proton-proton collider LHC at CERNof the total center-of-mass energy 14TeV.

SUSY extensions of the Standard Model give rise, generally speaking, to pro-cesses with baryon and/or lepton number violation. No processes of this sort havebeen observed so far, and there are severe bounds on their rates (the strongest onecomes from non-observation of proton decay). To get rid of these unwanted pro-cesses, one usually imposes a symmetry called R-parity which forbids baryon andlepton number violation at low energies.

12The most important hint is the partial cancellation of quadratic corrections to the Higgs boson

mass, which provides the solution, in a technical sense, of the gauge hierarchy problem (Why

MW � MPl?). Another hint is the gauge coupling unification.

182 Dark Matter

R-parity is a discrete symmetry, each particle can be either even or odd underit (positive and negative R-parity, respectively); R-parity of a state with severalparticles is a product of R-parities of all these particles. All Standard Model par-ticles are R-even, while their superpartners are R-odd. Once all interactions con-serve R-parity, superpartners are produced and disappear in pairs; in general, apair consists of different superpartners. R-parity conservation thus implies that thelightest superpartner — LSP — is stable. This is the new particle that serves asthe dark matter candidate. Since electrically charged particles of mass 30GeV –10TeV cannot be dark matter (see Problem 9.3), the potential candidates in SUSYtheories are neutralino, sneutrino and gravitino.13 Let us discuss them in turn.

9.6.1 Neutralino

A popular dark matter candidate is neutralino. In fact, most direct searches fordark matter are oriented towards WIMPs, while neutralino belongs precisely tothis category. There are three reasons for the popularity. The first is that neutralinois predicted by SUSY extensions of the Standard Model and the lightest neutralinois a stable LSP in a wide range of parameters of the theory. Second, like for anyWIMP, neutralino mass density in the Universe is more or less automatically inthe right ballpark. Third, neutralino participates in weak interactions, so its elasticscattering cross section, albeit small, may be sufficient for its direct or indirectdetection in dark matter searches (see Fig. 9.3).

Neutralino is a term used for electrically neutral Majorana fermion which is alinear combination of superpartners of Z-boson, photon and neutral Higgs bosons.There are four such combinations in the minimal SUSY extensions of the StandardModel. The point is that these theories necessarily contain two Higgs doublets, sothere are two neutral higgsinos, superpartners of the Higgs bosons. The other twoneutral fermions, photino and zino, are collectively called neutral gaugino.

Like their Standard Model partners, neutralinos participate in gauge interac-tions. Provided that the lightest neutralino is LSP, its present abundance is crudelyestimated along the lines of Sec. 9.2. We set gX = 2, g∗(tf ) 100, and use theestimate (9.25) for the annihilation cross section. Then the formula (9.19) gives

Tf MN

20, (9.33)

ΩN = 3 · 10−4 10−3

α2W

(MN

100 GeV

)2

log(

1012 · 100 GeVMN

)(9.34)

≈ 0.8 · 10−2 ·(

MN

100 GeV

)2

, (9.35)

13We restrict ourselves to minimal extensions of the Standard Model and leave aside other dark

matter candidates like axino, singlino, etc.


where αW ≈ 1/30 is the gauge coupling of SU(2)W interaction. For neutralino inthe interesting mass range

100 GeV < MN < 3 TeV

our preliminary estimate is

0.01 � ΩN � 10,

which is indeed in the right ballpark.We are going to refine the estimate (9.35) in this Section. We will make a sim-

plifying assumption that neutralino is the only new particle present in the cosmicplasma at relevant temperatures. This is the case if the masses of other superpartnersare sufficiently larger than the neutralino mass. In this case the only processes ofinterest to us are neutralino pair creation and annihilation. We note, however, thatour assumption is not innocent. If other superpartners have masses close to that ofneutralino, they also exist in the plasma at the epoch of neutralino freeze-out, andfreeze-out occurs in a more complicated way: neutralino are created and annihilatein pairs with other superpartners, with possible resonant enhancement of some ofthese processes; also, neutralinos are created in decays of superpartners, etc. Wewill see that in some versions of the theory the latter phenomena are importantand, in fact, they help to obtain the right neutralino abundance.

Still, let us proceed under our simplifying assumption. Since the only relevantprocesses are neutralino pair creation and annihilation, we write the Boltzmannequation for neutralino number density nN in the following form (cf. Sec. 5.4),

dnN

dt+ 3HnN = −〈σann

NN · v〉 · (n2N − neq 2

N ), (9.36)

where neqN is the neutralino number density in thermal equilibrium and 〈σann

NN · v〉is the product of the annihilation cross section and relative neutralino velocityaveraged with equilibrium distribution functions.

Let us describe a simple way of obtaining Eq. (9.36). The annihilation probabilityof a given neutralino per unit time is

Γann = 〈σannNN · v〉 · nN .

Therefore, the rate of neutralino annihilation in comoving volume is[d(nNa3)

dt

]ann

= −Γann · nNa3 = −〈σannNN · v〉 · n2

Na3. (9.37)

We make use of the fact that neutralino is in kinetic equilibrium with plasma (see theend of Sec. 9.2; note that our situation is similar to that described in Sec. 5.4), so itsdistribution function has thermal form. This is why 〈σann

NN ·v〉 is the thermal average.Now, in thermal equilibrium, i.e., for nN = neq

N , this rate should coincide with thecreation rate of neutralinos in collisions of the usual particles. As the latter are inthermal equilibrium, the actual creation rate is the same as the equilibrium one,[

d(nNa3)dt

]creation

= +〈σannNN

· v〉 · neq 2N

· a3. (9.38)

184 Dark Matter

The sum of (9.37) and (9.38) gives the total rate at which the number of neutralinoschanges in comoving volume, hence Eq. (9.36).

In general, the balance of particles is described by the system of Boltzmann equations.In our simple case of one type of relevant processes (pair annihilation and its inverse),more accurate treatment is as follows. Let us first consider pair annihilation in Minkowskibackground. Let p1, p2 be 3-momenta of incoming particles. The number of particles inthe interval of momenta (p,p + dp) in the volume element dx is

dN = n(t,x)F (t,p)dxdp,

where n(t,x) is the number density, and the distribution function F (t,p) is normalized ateach moment of time as Z

F (t,p)dp = 1. (9.39)

Let us consider a particle of momentum p1. In unit time it annihilates with

σ · v · n(t,x)F (t,p2)dp2 (9.40)

particles whose momenta are in the interval (p2,p2+dp2). Here v is the relative velocity ofthe two annihilating particles and σ = σ(p1,p2) is the annihilation cross section. This givesthe annihilation rate in volume dx for particles of momenta in the intervals (p1,p1 +dp1)and (p2,p2 + dp2),

1

2dN(p1, x) · σ · v · n(t,x)F (t,p2)dp2

=1

2n(t, x)F (t,p1)dxdp1 · σ · v · n(t,x)F (t,p2)dp2, (9.41)

where the factor 1/2 accounts for identical particles (otherwise the contribution of one andthe same state (p1,p2) and (p2,p1) would be double-counted).

Since each annihilation reduces the number of particles by 2, the rate of decrease ofparticles due to annihilation equals twice the expression (9.41),

n2(t,x)F (t,p1)dxdp1 · σ · v · F (t,p2)dp2. (9.42)

There is also the inverse process of pair creation in collisions of other particles. The latterare in thermal equilibrium, and repeating the argument before Eq. (9.38), we write forpair creation rate

neq 2(t,x)F eq(t,p1)dxdp1 · σ · v · F eq(t,p2)dp2,

where neq , F eq are equilibrium quantities. Finally, we integrate over momenta p2 to obtainthe equation for the number density in phase space volume dpdx,»

∂(n(t,x)F (t,p1))

∂t

–dp1dx

= −j Z

[n2(t,x)F (t,p1)F (t,p2)

− neq 2(t,x)F eq(t,p1)F eq(t,p2)] · v · σ dp2

ffdp1dx. (9.43)

The right hand side here is the collision integral.14

14When obtaining (9.43) we considered neutralino as classical particle, i.e., neglected the fact that

they obey Fermi statistics. This approximation is indeed valid in the situation we are interested

in.


This general equation is simplified in the case of interest, when the number density ishomogeneous in space,

n(t,x) = n(t), neq(t,x) = neq(t),

while the distribution over momenta corresponds to kinetic equilibrium and is time-independent in Minkowski space,

F (t,p) = F eq(p).

Then one integrates Eq. (9.43) over momentum p1 and obtains

∂n(t)

∂t= −〈σv〉(n2 − neq 2). (9.44)

Here we introduced the notation

〈σv〉 =

Zdp1dp2F eq(p1)F eq(p2) · v · σ,

which is the thermal average of the relative velocity and annihilation cross section (recallthe normalization (9.39)). Note that the right hand side of (9.44) is such that the systemrelaxes to thermal equilibrium for any sign of (n − neq).

The Boltzmann equation (9.44) should be modified when applied to matter inexpanding Universe. This modification takes into account the increase of the volume, andis given in (9.36), see also Sec. 5.4.

Let us find an approximate solution to the Boltzmann equation (9.36), specifyingto non-relativistic particles. To this end, we write this equation as the equation fordark-matter-to-entropy ratio. We introduce the variables

ΔN ≡ nN

s, Δeq

N ≡ neqN

s,

where

s =2π2

45g∗T 3 (9.45)

is entropy density at temperature T , and g∗ is the number of relativistic degreesof freedom. At neutralino freeze-out these are degrees of freedom of the StandardModel, so for realistic neutralino masses MN � 50 GeV we have (see Fig. B.4)70 � g∗ ≤ 106.75.

We now use the entropy conservation in comoving volume, Eq. (5.30), andobtain that

dnN

dt= s · dΔN

dt− 3HnN .

Equation (9.36) gives then

dΔN

dt= −〈σv〉 · s · (Δ2

N − Δeq 2N ). (9.46)

It is now convenient to change the variable from t to

x =T

MN

,

186 Dark Matter

which is small for non-relativistic particles: indeed, the kinetic energy of theseparticles is

〈Ekin〉 =MN〈v2

N〉2

=32T =

32xMN . (9.47)

We neglect possible weak dependence of g∗ on temperature at the freeze-out epochand use (3.34) to obtain

dΔN

dx=

〈σv〉Hx

· s · (Δ2N − Δeq 2

N ),

Now, we recall the expressions (9.45) and (3.31) for s = s(T ) and H = H(T ),respectively, and write the latter equation in a simple form,

dΔN

dx= 〈σv〉 ·

√πg∗

3√

5· MN · MPl · (Δ2

N − Δeq 2N ). (9.48)

We consider here the case of non-relativistic particles, so we expand the quantity〈σv〉 in a series in x = T/MN , i.e., in the relative velocity squared,15 see (9.47),〈v2〉 = 2〈v2

N〉,

〈σv〉 ≡ a0 + a1 · 〈v2〉 + · · · ≈ a0 + 6a1 · x, (9.49)

where we keep the two terms only (we will see that the first term is often suppressed).Inserting this into (9.48), we finally obtain

dΔN

dx= (a0 + 6a1 · x) ·

√πg∗

3√

5· MN · MPl · (Δ2

N − Δeq 2N ). (9.50)

This equation can be integrated numerically. Instead of doing that, let us analyze itssolution at the qualitative level. At freeze-out temperature Tf the quantity ΔN(Tf )is approximately equal to its equilibrium value Δeq

N (Tf ), while at T � Tf theequilibrium value Δeq

N (T ) is exponentially suppressed with respect to ΔN(T ). Hence,we neglect Δeq 2

N in the right hand side of (9.50) at low temperature, then integrateEq. (9.50) and in the end use the fact that ΔN(T = 0) � Δeq

N (Tf ). We find

Δ−1N (T = 0) = (a0xf + 3a1x

2f ) ·

√πg∗

3√

5· MN · MPl. (9.51)

This gives for the present neutralino mass density

ΩNh2 ≈ 0.9 · 10−10 1xf

√g∗

GeV−2

a0 + 3a1xf, (9.52)

15We make use of the fact that v2 = (v1 − v2)2 = v12 − 2v1 · v2 + v2

2, where the cross term is

zero on average.


The freeze-out temperature, and hence xf , is obtained by solving Eq. (9.10) withσ0 = 〈σv〉(xf ) = a0 + 6a1xf . The equation for xf is16

x−1f = log

[3 · √10

8π3

gN√g∗xf

· MN · MPl · (a0 + 6a1xf )

]. (9.53)

Formulas (9.52), (9.53) refine our estimate (9.19).Parameters a0 and a1 depend on the dominant neutralino annihilation channels,

and hence on neutralino mass. Let us discuss various neutralino mass ranges sepa-rately. We first consider light neutralino, MN � MZ .

We begin with very light neutralino, MN � MZ . The region in parameter spacewhere this option is viable is very restricted. Indeed, the kinematically alloweddecay Z → NN should not give too large contribution into well measured invisibleZ-decay width. Hence, the lightest neutralino in this case must be predominantlybino (superpartner of the gauge boson of the weak hypercharge group U(1)Y ), withsmall admixture ξ of higgsino and the other neutral gaugino. The dominant anni-hilation channel for very light neutralino is the s-channel annihilation into twob-quarks through exchange of virtual parity-odd Higgs boson A, see Fig. 9.4. Itscross section is estimated as follows,

〈σv〉 ∼ ξ2y2b

α2M2N

m4A

,

where mA is the mass of parity-odd boson A and yb is the Yukawa coupling ofb-quarks with the boson A.

Sufficiently large cross section is obtained for large yb ∼ 1. This is indeed thecase for large hierarchy between the vacuum expectation values of the two Higgsfields. The bounds on the masses of the boson A and lightest neutralino are about100GeV [1] and 5GeV [83], respectively. For large yb we obtain

〈σv〉 = a0 ∼(

ξ2

0.1

)·(

MN

10 GeV

)2

·(

100 GeVmA

)4

· 10−9 GeV−2.

We estimate the freeze-out temperature from (9.53) and obtain xf ∼ 1/20. Theneutralino mass density today is then found from (9.52). Numerically

ΩN ∼ 0.1 ·(

0.1ξ2

)·(

10 GeVMN

)2

·( mA

100 GeV

)4

.

Fig. 9.4 Light neutralino annihilation.

16Formulas (9.51) and (9.53) are still valid in the leading logarithm approximation, while exact

result for ΔN (T = 0) can be obtained by solving Eq. (9.50). We will not need it in what follows.

188 Dark Matter

We see that very light neutralino of mass MN 10GeV can indeed be dark matterparticle, provided the other parameters are close to the boundary of the allowedregion (in particular, ξ2 0.1).

Rather typical situation in concrete models is that the heavy17 Higgs bosonshave large masses, MH,A � MZ . In that case very light neutralinos would beoverproduced. So, let us consider the case of heavy Higgs bosons and neutralino ofintermediate mass MZ/2 < MN � MZ . The annihilation cross section in this case isoften suppressed as compared to the estimate leading to (9.34). Indeed, neutralinoin our case predominantly annihilates into the Standard Model fermions,

NN → ff. (9.54)

These processes occur via the s-channel Z-boson exchange, and also possibly viat-channel exchange of superpartners of the fermions, see Fig. 9.5. Since neutralinois non-relativistic and its mass is smaller than that of the Z-boson, momenta ofvirtual particles in these diagrams are small compared to their masses, so theycan be neglected. This means that the annihilation process (9.54) is described byeffective four-fermion interaction

L =∑

f

Nγμγ5N · fγμ(af + bfγ5)f, (9.55)

where the parameters af and bf have dimension GeV−2. These parameters aremodel-dependent (in the case of Z-boson exchange they depend on the mixingmatrix in the neutralino sector). An order of magnitude estimate for the Z-bosonexchange contribution is

af , bf ∼ GF = 1 · 10−5 GeV−2. (9.56)

Superpartners of fermions are typically heavier than Z-boson, so the t-channelexchange diagrams give smaller contributions.

The interaction (9.55) gives for the non-relativistic annihilation cross section(recall that neutralino is Majorana fermion) to the leading orders in v2 and m2

f

σv ≈ 12π

∑f

{b2f · m2

f +13[(a2

f + b2f )M2

N

] · v2

}, (9.57)

Fig. 9.5 Neutralino annihilation into the Standard Model fermions.

17The lightest Higgs boson has mass below 130–150 GeV, but it does not contribute significantly

into neutralino annihilation.


where v is the relative velocity of the two incident neutralinos. This means that theparameters a0 and a1 defined in (9.49) are equal to

a0 =12π

∑f

b2f · m2

f , a1 =16π

∑f

(a2f + b2

f )M2N .

The s-wave annihilation is suppressed by the fermion mass: we see that a0 → 0 asmf → 0. This is due to the fact that in this limit, current-current interaction (9.55)creates a pair of left-handed Standard Model fermions in a state of total angularmomentum J = 1. Non-relativistic spin-1/2 Majorana neutralinos should be in thestate of angular momentum 1 (p-wave) to produce such a pair: the state of angularmomentum 0 (s-wave) and total spin 1 is forbidden by the Pauli principle.

Making use of the above estimates for the parameters a0 and a1 we findfrom (9.52) that

ΩN ∼ 0.2 for MN 60 GeV.

So, relatively light neutralinos are reasonable dark matter candidates.Let us come back to the possibility of lighter neutralino, MN < MZ/2. Its

interaction with Z-boson must be quite weak, as we have already discussed. Oneoption still remains, though. If some superpartners of the Standard Model fermionsare light, MS � MZ , the t-channel exchange diagram of Fig. 9.5 dominates, andmay lead to acceptably high neutralino annihilation cross section. Indeed, if wetake MS ∼ MZ and improve the estimate (9.56) by accounting for the t-channelenhancement of the coefficients af , bf , we find from (9.52) that neutralino of massMN 30GeV can have the right dark matter mass density.

The problem with these scenarios, however, is that in concrete models, the neu-tralino mass is related to masses of other superpartners. The latter are stronglyconstrained by experiment, so in many models neutralino cannot be light.

Thus, let us turn to heavier neutralino. In that case, momenta in internal linesof the annihilations diagrams (e.g., diagrams shown in Fig. 9.5) are not small,Q2 = 4M2

N , so the cross sections are generically suppressed by a factor of orderM4

Z/4M4N

as compared to the case of light neutralino. With account of p-wavesuppression of annihilation into fermions this means that the annihilation crosssection is generically too small, i.e., heavy neutralinos are typically overproducedin the Universe.

Heavy neutralinos, MN � MZ , still remain dark matter candidates in a certain(typically narrow) ranges of parameters of concrete models. The annihilation maybe enhanced, e.g., by the presence of other superpartners in the cosmic plasmaat neutralino freeze-out. This occurs when these superpartners are approximatelydegenerate in mass with neutralino. Whether this is a natural solution to the neu-tralino overproduction problem, is another issue.

Problem 9.6. Consider minimal SUSY extension of the Standard Model in whichall superpartners, except for sleptons and photino, are very heavy. Let the masses

190 Dark Matter

of sleptons and photino be of order 100GeV, and photino is LSP. Neglecting sleptonmixing, find the relic photino abundance. Assuming that one, two or three sleptonsare light and have one and the same mass, find the range of slepton and photinomasses in which photino is dark matter. Hint: The Lagrangian of photino interactionwith leptons and sleptons is given in the beginning of Sec. 9.6.

Let us describe a consistent and phenomenologically viable example with rather fewnew parameters, called mSUGRA (minimal Supergravity). The model assumes that super-symmetry is broken in some “hidden” sector, and this breaking is transferred to theStandard Model fields and their superpartners by gravitational interactions (for a reviewsee, e.g., Ref. [84]). Then at certain energy scale M all scalars of the visible sector, irre-spective of their quantum numbers, obtain one and the same SUSY breaking mass m0,while all gaugino (superpartners of gauge bosons) obtain one and the same SUSY breakingmass M1/2. Also, trilinear interactions between the Higgs doublets and other scalars aregenerated; these are assumed to be proportional to the Yukawa couplings with one and thesame proportionality coefficient A. These mass terms and trilinear couplings break super-symmetry. At the scale M this breaking is therefore universal. From viewpoint of phe-nomenology, the advantage of this universality is that dangerous flavor changing neutralcurrent interactions are suppressed.

As we mentioned above, there are two Higgs doublets; one of them, HU , interacts withup quarks while the other, HD interacts with down quarks and charged leptons. They bothobtain vacuum expectation values, so that there is one more parameter,

tan β =〈HU〉〈HD〉 .

Finally, there is yet another, discrete parameter, which is called sign of μ for technicalreasons.18

In the simplest, and yet realistic models, the mass scale M is set equal to the GrandUnification scale MGUT ∼ 1016 GeV [86]. At this scale all three gauge couplings of theStandard Model are equal (modulo normalization factor for weak hypercharge). We discussGrand Unified Theories in Sec. 11.2.2, and here we only notice that this scale is very high.

Quantum corrections modify all mass parameters {m0i, AUij, A

Dij, Mi} and coupling con-

stants {αi, YU

ij , Y Dij } as the energy scale Q changes from M to MZ . This is described by

renormalization group. At the scale MZ supersymmetry breaking is no longer universal,but all new (with respect to the Standard Model) parameters are expressed through M ,m0, M1/2, A, tan β and signμ.

The renormalization group evolution in mSUGRA model [85] is shown in Fig. 9.6.Due to large color SU(3) gauge coupling, masses of squarks and gluinos run faster thanother masses. The lightest among gauginos turns out to be bino. The situation is morecomplicated in the scalar sector, where the gauge and Yukawa interactions compete. As aresult, the lightest slepton is a superpartner of τ .

A nice feature of mSUGRA model is that the electroweak symmetry breaking occursautomatically, due to quantum corrections. As Q decreases, one of the Higgs massessquared becomes negative, which means spontaneous symmetry breaking. This is shownin Fig. 9.6, where the mass of the Higgs field is given by the combination

pm2

0 + μ2. Sincethe renormalization group running is logarithmic in Q, electroweak symmetry breakingoccurs many orders of magnitude below MGUT for small μ and SUSY breaking terms,which explains the hierarchy MW MGUT .

18In fact, the parameter μ, the supersymmetric mass of the Higgs doublets, may be complex. This

property serves as a potential source of CP-violation in the Higgs sector. This is quite interesting

from the viewpoint of the baryon asymmetry generation, see Chapter 11.


Fig. 9.6 An example of the renormalization group running of the mass parameters in mSUGRA

model (plot by T. Falk from [85]). The universal parameters at the scale MGUT ≈ 1016 GeV are:

M1/2 = 250GeV, m0 = 100 GeV, A = 0, tan β = 3 and signμ = −1.

Given the tight experimental constraints on the parameters of mSUGRA, there are onlythree narrow regions in the parameter space where relic neutralino has right abundanceto serve as dark matter [85], see Fig. 9.7 and Fig. 13.7 on color pages. The reason is thatthe region in parameter space where neutralino is light is experimentally excluded. So, thelight neutralino scenarios we discussed above are impossible. On the other hand, heavyneutralinos tend to be overproduced in the early Universe. The cosmologically favoredregions in Figs. 9.7 and 13.7 are special in the sense that the neutralino annihilation crosssection is considerably enhanced in one or another way. This way of obtaining the correctdark matter abundance does not appear particularly plausible. The situation becomesparticularly contrived for very large masses [85], see Fig. 9.8 and Fig. 13.8 on color pages.

One mechanism of enhancement of the neutralino annihilation cross section existsfor large tanβ only (lower plots in Fig. 9.7). It occurs when the masses of scalar andpseudoscalar Higgs bosons (which are automatically almost equal in the relevant parameterregion) are close to the mass of a pair of neutralinos 2MN ≈ mH ≈ mA. Then the decaychannel

NN → A∗, H∗ → Standard Model particles,

has resonance enhancement. Of couse this occurs due to tuning of parameters.Two other mechanisms exist in models with co-LSP, where the next-to-lightest super-

partner, NLSP, is almost degenerate in mass with neutralino LSP. In one case (strip inFig. 9.7 along the cosmologically disfavored region with charged LSP) NLSP is chargedslepton (mostly right stau), while in the other case (narrow strip in the left part of Fig. 9.8along the region without electroweak symmetry breaking) it is the lightest linear combi-nation of charged higgsinos and winos.

192 Dark Matter

100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

mh = 114 GeV

m0

(GeV

)

m1/2 (GeV)

tan β = 10 , μ > 0

mχ± = 104 GeV

100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

mh = 114 GeV

m0

(GeV

)

m1/2 (GeV)

tan β = 10 , μ > 0

mχ± = 104 GeV

100 1000 2000 30000

1000

1500

100 1000 2000 30000

1000

1500

mh = 114 GeV

m0

(GeV

)

m1/2 (GeV)

tan β = 50 , μ > 0

100 1000 2000 30000

1000

1500

100 1000 2000 30000

1000

1500

mh = 114 GeV

m0

(GeV

)

m1/2 (GeV)

tan β = 50 , μ > 0

Fig. 9.7 Experimentally forbidden and cosmologically favored regions [85] in the plane (M1/2, m0)

in mSUGRA model with tan β = 10 and tan β = 50 (A is chosen to be zero at the scale MGUT ≈1016 GeV). See Fig. 13.7 on color pages for details.

Due to approximate degeneracy in masses, the freeze-out epoch is one and the samefor reactions

LSP + LSP → Standard Model particles,

NLSP + LSP → Standard Model particles,

NLSP + NLSP → Standard Model particles.

Also, depending on the type of NLSP, new annihilation processes may occur in s-wave,without the suppression by small factor v2 inherent in Eq. (9.57). The extra channels areindeed important in a narrow region of the parameter space where

mNLSP − mLSP

mLSP≡ Δm

mLSP� 0.1. (9.58)


100 200 300 400 500 600 700 800 900 10000

1000

2000

3000

4000

5000

100 200 300 400 500 600 700 800 900 10000

1000

2000

3000

4000

5000

mh = 114 GeV

m0

(GeV

)

m1/2 (GeV)

tan β = 10 , μ > 0

100 1000 2000 25000

1000

1500

100 1000 2000 25000

1000

1500

m0

(GeV

)m1/2 (GeV)

μ > 0

Fig. 9.8 Left: the same as in upper part of Fig. 9.7 but for larger range of masses,see Fig. 13.8 for color version [85]. The dark upper left region is excluded by therequirement of electroweak symmetry breaking. Right: cosmologically favored regions(0.094 < ΩNh2 < 0.129) for various values of tan β = 5, 10, . . . , 55; lower strips corre-spond to smaller tan β [87].

The latter estimate follows from the fact that NLSP density is suppressed with respect toLSP by the Boltzmann factor

e−Δm

Tf = e−20· Δm

mLSP·„

mLSP /20Tf

«,

This factor is less than 15%, unless the inequality (9.58) holds.

To conclude this Section we mention that in some models neutralino is longlived but not absolutely stable. The stable particles — dark matter candidates —may then be produced in neutralino decays. This helps to solve the overproductionproblem, as the number of dark matter particles is the same as the freeze-out numberof neutralinos, while the mass of the dark matter particles is smaller. We discussthis possibility in the context of gravitino dark matter, Sec. 9.6.3, since gravitino isthe most interesting and realistic partner of neutralino in the pair LSP–NLSP.

9.6.2 Sneutrino

Sneutrino is a superpartner of neutrino. SUSY models contain three generationsof sneutrinos described by complex, electrically neutral scalar fields. If the lightestsneutrino is LSP, one might wonder whether it can be a dark matter candidate.Indeed, weak interactions of sneutrinos are the same as those of neutrinos, so theorder of magnitude estimate of the present sneutrino mass density would coincidewith (9.19) (unlike in the case of neutralino, there is no p-wave suppression of theannihilation cross section). However, sneutrino as dark matter particle is excluded

194 Dark Matter

by direct searches for dark matter. Indeed, due to weak interactions, the cross sectionof elastic scattering of sneutrino off nuclei is two to three orders of magnitude largerthan experimental bounds shown in Fig. 9.2. We note in this regard, that sneutrinois LSP in a much narrower class of models as compared to neutralino.

A lot less problematic are right sneutrinos which exist in SUSY theories withright neutrino masses in 100GeV range. The right sneutrinos interact with theStandard Model particles very weakly, so their elastic cross sections are well belowthe bounds shown in Fig. 9.2.

9.6.3 Gravitino

Consistent supersymmetric theories that include gravity are theories with localsupersymmetry — supergravities. Supersymmetry is local in the usual gauge theorysense: the action is invariant under supersymmetry transformations depending onspace-time point. In these theories graviton has its superpartner, gravitino Gμ. Ifsupersymmetry were unbroken, gravitino would be a massless spin-3/2 particle with2 polarization states, the same number as for graviton.

Supersymmetry, however, must be broken. In theories with spontaneouslybroken global supersymmetry (no supergravity), the Goldstone theorem impliesthe existence of massless particles whose properties are dictated by the propertiesof broken symmetry generators. In the case of supersymmetry, the generators arefermionic (Grassmannian) and carry spin 1/2 (they generate the transformationboson↔ fermion and change spin by 1/2), so the massless particle is a Majoranafermion ψ, goldstino. Like any Nambu–Goldstone field, goldstino interacts withother particles through its coupling to the relevant current, which is supercurrentin the case at hand,

L =1F

∂μψ · JSUSYμ . (9.59)

This is a direct analog of the Goldberger–Treiman formula which describes interac-tions of pions, the (pseudo-)Nambu–Goldstone bosons of broken chiral symmetry.The parameter F in (9.59) has dimension of mass squared and is determined by thevacuum expectation value that breaks supersymmetry. Hence,

√F is of the order of

the supersymmetry breaking scale (in complete analogy to chiral model where thepion decay constant fπ is of order of the chiral symmetry breaking scale).

Once supersymmetry is local, the super-Higgs mechanism is at work. The gold-stino becomes the longitudinal component of gravitino,

Gμ → Gμ + i√

4πMPl

F∂μψ, (9.60)

and gravitino acquires mass m3/2 proportional to F . The relation is

m3/2 =

√8π

3F

MPl. (9.61)


Phenomenologically, the value of√

F may be anywhere in the interval

1 TeV �√

F � MPl,

where the lower bound corresponds to the superpartner mass scale. Hence the widerange of possible gravitino masses,

2 · 10−4 eV � m3/2 � MPl.

The formulas (9.59) and (9.60) where ψ is the longitudinal component of gravi-tino, imply that gravitino interactions with matter have in principle rather simpleform. Indeed, the interactions of “original” gravitino components are suppressed bythe Planck scale. On the other hand, vast majority of SUSY theories have relativelylow SUSY breaking scale,

√F � MPl, so massive gravitino interacts with matter

mostly through its longitudinal component, the would-be goldstino. This interactionhas the form (9.59). To estimate the interaction strength, let us integrate the actionby parts and obtain

L = − 1F

ψ · ∂μJSUSYμ .

At low energies, the divergence of the supercurrent is due to explicit breaking ofsupersymmetry in the effective action. This breaking manifests itself in the massdifferences between particles and superpartners. Hence, the effective gravitino inter-actions with other particles are19

m2S

Ffψf ,

mS

FλγμγνψFμν , (9.62)

where f and λ are sfermion and gaugino, respectively, and mS are masses of cor-responding superpartners. The interesting and phenomenologically viable range ofparameters is

100 GeV � mS � 10 TeV.

Hence, the effective coupling (9.62) is small, and often very small (cf. Fig. 9.3). Thisis the reason for peculiar phenomenology and cosmology of gravitino. Yet there isone property of gravitino in common with other superpartners. It is clear from (9.59)and (9.60) that in theories with conserved R-parity, gravitino is odd under R-parity.Hence, it is created and annihilate in pair with itself or with other superpartner.Furthermore, gravitino is LSP provided that its mass is roughly in the range

2 · 10−4 eV � m3/2 � 100 GeV,

19One might question (9.62) as it does not appear to have smooth limit as F → 0 in which

supersymmetry is restored. This is not the case: superpartner mass parameters m2S are themselves

proportional to F . The ratio (9.62) is in fact determined by the couplings of those interactions in

the full theory which provide masses to superpartners after SUSY breaking.

196 Dark Matter

This range corresponds to fairly low SUSY breaking scale, see (9.61), namely√

F � 1010 GeV.

In theories of this sort gravitino is stable and may be a dark matter candidate. Wenote, though, that detection of gravitino dark matter is very unlikely in foreseeablefuture because of very weak gravitino interactions with matter.

A class of realistic SUSY models with gravitino LSP makes use of the gauge mech-anism of mediation of SUSY breaking to the SUSY Standard Model (for reviews see, e.g.,Refs. [88, 89]). SUSY is assumed to be broken in the secluded sector, while the super-partners of the Standard Model particles obtain masses due to the usual gauge interac-tions of the Standard Model. The interaction with the secluded sector is due to messengers,heavy fields that are charged under the Standard Model gauge group and at the same timecouple to the secluded sector. As a result of the latter interactions, the masses of scalarmessengers q get split

M2q = M2

„1 ± Λ

M

«, Λ < M,

where M is the mass of fermionic messengers, and the parameter Λ2 is proportional toSUSY breaking vacuum expectation value F .

Messengers induce one loop corrections to the gaugino masses,

Mλ(M) ∼ α

4πΛ, (9.63)

where α is the relevant gauge coupling of the Standard Model. The fields from the scalarsector of SUSY Standard Model obtain contributions to their masses at two loops,

m2f (M) ∼ (

α

4π)2Λ2. (9.64)

The proportionality coefficients in (9.63) and (9.64) are in general different for differentfields.

Hence, the non-supersymmetric masses of scalar and fermion superpartners at the scaleM depend on their quantum numbers only, and are of the same order,

Mλ ∼ mf ∼ α

4πΛ.

Flavor-independence of the gauge interactions ensures the absence of new flavor-changingparameters at the scale M . Gravitational interactions between the secluded sector andSUSY Standard Model fields lead, generally speaking, to the appearance of these param-eters. In particular, the gravitational effects generate flavor-nondiagonal masses for squarksand sleptons, whose estimate is

Δmfab∼ F/MPl ∼ m3/2.

These masses are phenomenologically acceptable (they do not lead to contradictions withdata on rare decays of leptons and mesons), provided that

Δmfab

mf

� 10−2 − 10−3, mf ∼ 500 GeV.

This gives the upper bound on the gravitino mass in gauge mediation models,

m3/2 � (10−2 − 10−3) · mf ∼ 1GeV,


as well as on the SUSY breaking scale,√

F � 109 GeV.

On the other hand, the lower bounds on m3/2 and√

F come from the experimental boundson the superpartner masses; roughly speaking,

Mλ, mf � 100 GeV.

Making use of this estimate, we find from (9.63) and (9.64) that√

F � Λ � 30 TeV,

m3/2 � 1 eV.

While gravitino is LSP in the gauge mediation models, NLSP is either neutralino or rightτ -slepton. The latter case occurs more often.

We note that gravitino production mechanisms in this class of models include processeswith messengers and with particles from the secluded sector. These interactions may changethe estimates given in the text.

Let us discuss cosmology with stable gravitino. Because of very weak gravitinointeractions, processes involving two gravitinos (including creation and annihilationof two gravitinos) are negligible. Let us begin with the possibility that gravitino wasin thermal equilibrium at high temperatures.

Gravitino G is created and destroyed in reactions

X1 + G ↔ X2 + X3 (9.65)

where Xi, i = 1, 2, 3 are other particles in the cosmic plasma. These reactionsproceed via s-, t- and u-channel exchange of virtual particles Y , see Fig. 9.9. Eachdiagram involves the coupling between the particles X1, X2, Y (e.g., gauge coupling)and the gravitino coupling (9.62). Hence, the cross section is estimated as

σG αm2

S

F 2, where α =

g2

4π.

Gravitino freeze-out temperature is determined, as usual, from the relation∑X1

σG · nX1 · vG H(Tf ) =T 2

f

M∗Pl

,

where nX is the equilibrium number density of particles in the initial state of thereaction (9.65) and vG is the gravitino velocity. Let us assume that freeze-out occurs

Fig. 9.9 Processes 2 → 2 with gravitino annihilation or production; Y, Xi, i = 1, 2, 3 are the

Standard Model particles and their superpartners. In models with R-parity at least one of the

particles Xi must be a superpartner.

198 Dark Matter

when gravitinos are still relativistic. We take nX ∼ T 3, vG = 1 and estimate thenumber of channels as g∗(Tf ). This gives the following estimate

Tf ∼ m3/2 · απ√30g∗(Tf )

· F

m2S

, (9.66)

where one of the factors F has been expressed through the gravitino mass accordingto (9.61) and we have used the definition (3.32) of M∗

Pl.Since F � m2

S , gravitino freeze-out indeed occurs when gravitino is rela-tivistic.20 Hence, the estimate of the present gravitino abundance is made in thesame way as the estimate for neutrino, see Sec. 7.2. The expression for gravitino is

n3/2,0 =34·(g3/2

2

)· 4311

· 1g∗(Tf )

· nγ,0, (9.67)

where we used the expression (5.33) for entropy at T � 1 MeV. Here g3/2 is thenumber of active degrees of freedom of gravitino; since the interaction of the trans-verse degrees of freedom is suppressed by negative power of MPl, we should setg3/2 = 2.

Problem 9.7. Derive the formula (9.67).

Problem 9.8. Find freeze-out temperature of transverse degrees of freedom ofgravitino.

The result (9.67) shows that the present gravitino mass density is

Ω3/2 =m3/2 · n3/2

ρc= 0.2

m3/2

200 eV

(g3/2

2

)·(

210g∗(Tf )

).

12h2

(9.68)

This means that stable gravitinos of masses m3/2 � 200 eV are cosmologically for-bidden, if these gravitinos were in thermal equilibrium at high temperatures. Thistranslates into the bound on the SUSY breaking scale,

√F < 2 · 103 TeV.

Gravitino of mass m3/2 200 eV would make the whole of dark matter. However,this mass is below the lower bound (9.4) dictated by the analysis of structures.Thus, if gravitinos were in thermal equilibrium in the early Universe, they cannotserve as dark matter candidates.

Problem 9.9. Show that stable gravitinos of mass m3/2 < 200 eV that were inthermal equilibrium at high temperatures are safe from the viewpoint of primordialnucleosynthesis.

20For m3/2 � 10 keV, the estimate (9.66) gives the freeze-out temperature below the scale of

superpartner masses, Tf � mS . In this case the processes (9.65) terminate, roughly speaking, at

Tf ∼ mS , since at least one of the particles X1, X2, X3 must be a superpartner. Freeze-out still

occurs in the regime of relativistic gravitino, so the argument that follows remains valid.


Gravitino as dark matter is still an option if the freeze-out temperature (9.66)was never reached in the Universe. One of the possible scenarios is as follows. Let usassume that the thermal history of the Universe began at some temperature Tmax.Let us also assume that gravitinos were not produced before the hot stage, so thatthere are no gravitinos at T = Tmax, while the Standard Model particles and theirsuperpartners are at thermal equilibrium. The gravitino production then occurs inthe conventional hot Big Bang regime.

In fact, this situation is quite realistic in inflationary theory. According to this theory,energy density before the hot stage was due to the inflaton field, and the hot plasma wasproduced as a result of the decay of this field (see details in the accompanying book).Naturally, this picture implies that the maximum temperature is well below the Planckmass. This temperature is actually model dependent and varies a lot in concrete infla-tionary models. Gravitino interactions with inflaton may be very weak, so gravitinos werenot produced prior to the hot stage in this picture.

At the hot stage, gravitinos are predominantly produced in two-body decays ofsuperpartners,

Xi → G + Xi, i = 1, . . . (9.69)

and in two-particle scattering processes,

Xi + Xj → Xk + G, i, j, k = 1, . . . (9.70)

With these two production channels, the Boltzmann equation for the gravitinonumber density is

dn3/2

dt+ 3Hn3/2 =

∑i

ΓXi· γ−1

i · nXi+∑ij

〈σij〉 · nXinXj , (9.71)

where ΓXiis the width of the decay (9.69), γi is the Lorentz-factor of Xi-particles

that gives rise to the decrease of their decay rate, nXi is the number density ofparticles Xi and 〈σij〉 is the thermal average of the total cross section of the gravitinoproduction in the processes (9.70). We neglected here the inverse processes Xk+G →Xi +Xj and Xi + G → Xi; this is the valid approximation for gravitino abundancewell below its equilibrium value. In terms of gravitino-entropy ratio Δ3/2 = n3/2/s

and temperature, the Boltzmann equation reads (we follow the logic as in Sec. 9.6.1)

dΔ3/2

d log T=∑

i

ΓXiγ−1

i

H· nXi

s+∑ij

〈σij〉 · ninj

sH(9.72)

We now make use of the formula (9.62) for the gravitino coupling to estimate thewidth and cross section,

ΓXi

M5Xi

16πF 2=

M5Xi

6m23/2M

2Pl

, (9.73)

〈σij〉 = const · αm2X

F 2= const · α8π

3· m2

X

m23/2M

2Pl

. (9.74)

200 Dark Matter

Note that at T � MX the gravitino production in scattering is dominated byits interaction with the gauge particles, the second term in (9.62), while its inter-action with matter particles (the first term in (9.62)) yields the cross section addi-tionally suppressed by M2

X/T 2. Note also that neither the width (9.73) nor the cross

section (9.74) depends on temperature. The factor 1/(16π) in (9.73) is due to phasespace, while constant in (9.74) is determined by the number of possible final states.

The two terms on the right hand side of Eq. (9.72) behave in a very differentway as functions of temperature. At high temperatures one has nXi

∼ nγ ∝ T 3

and γ−1i ∝ T−1. Hence, the first and the second terms are proportional to T−3 and

T , respectively. We see that the gravitino production in decays is not particularlysensitive to Tmax, while the number of gravitinos produced in scattering increaseslinearly with Tmax.

Let us first discuss gravitino production in decays of superpartners. There aretwo types of processes relevant here. One is the decays of all superpartners at hightemperatures when they are relativistic or almost relativistic. The other is possibledelayed decay of next-to-lightest superpartner, NLSP. Let us consider them in turn.

The contribution of the superpartner X into the first term in (9.72) behaves asT−3 at T � MX and exp(−MX/T ) at T � MX (assuming that X is in thermalequilibrium; the opposite situation may be relevant for NLSP and will be discussedlater). Hence, the gravitino production occurs predominantly at T ∼ MX , and weobtain for this contribution

Δ3/2 ∼ ΓX

g∗H(T = MX)∼ M3

X

g3/2∗ m2

3/2MPl

. (9.75)

We see that the largest contribution comes from the heaviest superpartners (pro-vided that Tmax � MX) and that the abundance is larger for lighter gravitino. Thelatter property is due to the fact that lighter gravitinos correspond to lower SUSYbreaking scale, see (9.61), and hence stronger gravitino interaction with matter. Wefind from (9.75) the contribution to the present gravitino mass density,

Ω3/2 ∼ s0

ρc

M3X

g3/2∗ m3/2MPl

Plugging in numbers, MX ∼ (a few)TeV, g∗ 200, we find that the first decaymechanism is capable of producing gravitino dark matter for m3/2 � 1 MeV. Inter-estingly, the right amount of gravitino warm dark matter, m3/2 ∼ 1 − 10keV (seeSec. 9.1), is obtained for fairly light superpartners, MX � 300GeV, see Ref. [90] fordetails.

Let us turn to the second decay mechanism, delayed decays of NLSP. Neglectingfor the moment its interactions with gravitino, we apply the analysis of Sec. 9.6.1.The NLSP freeze-out temperature is

Tf,NLSP ∼ MNLSP

20.


NLPS do not decay by that time provided that

ΓNLSP < H(Tf,NLSP) =T 2

f,NLSP

M∗Pl

.

Applying the estimate (9.73) to NLSP we see that this condition holds for gravitinoof mass m3/2 � 10 keV. Let us concentrate on this case. After NLSP decays, thenumber of gravitinos in comoving volume is the same as freeze-out number of NLSP.This immediately gives

ρ0,3/2 =m3/2

MNLSP

ρ0,NLSP ,

where ρ0,NLSP is a would-be mass density of stable NLSP. The latter has been esti-mated in Sec. 9.6.1, where we have seen that the mass density is often higher thanthe critical density,

ρ0,NLSP ∼ (10 − 1000)ρc.

So, the second decay mechanism of gravitino production leads to the right gravitinodark matter density for relatively heavy gravitino,

m3/2 ∼ (0.1 − 10)GeV ·(

100 GeVMNLSP

).

This scenario is somewhat problematic, however, since according to (9.73), theNLSP lifetime is large,

τNLSP ≡ Γ−1NLSP ∼ 5 · 104 s ·

( m3/2

1 GeV

)2

·(

100 GeVMNLSP

)5

.

For τNLSP � 100 s, NLSP decays occur at or after BBN epoch. The decay productsin the end contain hard photons, electrons and other particles which may destroylight nuclei in primordial plasma, hence spoiling the predictions of BBN theory, seethe end of Chapter 8. This problem does not occur if gravitino is sufficiently light(m3/2 � 100MeV) and/or NLSP is sufficiently heavy (MNLSP � 300GeV).

We note that in the latter scenario gravitinos are born relativistic. Hence, theyeffectively serve as warm dark matter. Their distribution in momenta is far fromthermal; this, generally speaking, is relevant for structure formation.

Problem 9.10. Find the distribution over momenta of gravitinos produced in NLSPdecays after NLSP freeze-out.

Problem 9.11. Let gravitino of mass 100MeV be dark matter particle, and themass and lifetime of NLSP be 200GeV and 10 s. Estimate the present spatial sizeof density perturbations suppressed as compared to the CDM case (see Sec. 9.1).

As an example of BBN bounds, let us consider mSUGRA model of Sec. 9.6.1 andadd light gravitino to it. Let gravitino be LSP. Let us treat the gravitino mass as a freeparameter (which, strictly speaking, is not the case in simple models of gravity mediation).

202 Dark Matter

100 1000 20000

100

200

300

400

500

100 1000 20000

100

200

300

400

500

m0

(GeV

)

M1/2 (GeV)

m3/2 = 100 GeV , tan β = 10 , μ > 0

100 1000 20000

1000

2000

100 1000 20000

1000

2000

m0

(GeV

)

1/2 (GeV)

m3/2 = 100 GeV , tan β = 57 , μ > 0

M

2000

m0

(GeV

)

1/2 (GeV)

m3/2 = 100 GeV , tan β = 57 , μ > 0

Fig. 9.10 Bounds [91] on the plane (M1/2, m0) in mSUGRA model with m3/2 = 100 GeVand two values of tanβ. See details in Fig. 13.9 on color pages.

In this model, NLSP is either neutralino or superpartner of right tau-lepton. The boundson the parameters of this model [91] are shown in Fig. 9.10 and Fig. 13.9 on color pages.

Let us now turn to gravitino production in scattering, the second term in (9.72).It is important for Tmax � MS . At T ∼ Tmax matter particles are thermalized, andtheir number densities are all of order of the photon number density. We parame-terize the sum in the right hand side of (9.72) as∑

ij

〈σij〉 · ninj = 〈σtot〉n2γ ,

with 〈σtot〉 estimated from (9.74). The largest contribution to 〈σtot〉 comes fromcolored particles; numerically const ∼ 100 in (9.74). Using this estimate for 〈σtot〉,we obtain the scattering contribution to gravitino-to-entropy ratio as the secondterm in the right hand side of Eq. (9.72) at T = Tmax. This gives

Ω3/2 =m3/2Tmax

ρc· nγ,0 · g∗(T0)

g∗(Tmax)· 〈σtot〉 · nγ(Tmax)

TmaxH(Tmax).

Numerically,

Ω3/2 ∼(

200 keVm3/2

)·(

Tmax

10 TeV

)·(

MS

1 TeV

)2

·(

15√g∗(Tmax)

)· 12h2

. (9.76)

Since there is linear dependence on Tmax, the maximum temperature in the Universemust be bounded from above.

The estimate (9.76) can be refined by integrating the Boltzmann equation (9.72)numerically. The result [92] for a SUSY model with bino NLSP and the mass


10-6 10-5 10-4 10-3 10-2 10-1 11

10

102

103

104

105

106

107

108

109

1010

m3/2 (GeV)

Tm

ax (

GeV

)

Fig. 9.11 Bounds [92] on the parameters (Tmax, m3/2). The region above the line is excluded

because of gravitino overproduction. The solid line corresponds to Ω3/2h2 � 1.

hierarchy MNLSP = 50 GeV � MS = 1 TeV is shown in Fig. 9.11. This type of cosmo-logical bounds is very restrictive, since in simple models of reheating the maximumtemperature exceeds T ∼ 108 GeV. We note that in this model, relic gravitinosmake all of dark matter for parameters just below the line shown in Fig. 9.11. Itis clear both from this Figure and from Eq. (9.76) that obtaining the right darkmatter density with gravitinos requires tuning parameters of different nature: theparameter m3/2 has to do with particle physics while Tmax characterizes the earlycosmology.

To conclude this Section, let us consider one possibility that exists in SUSYmodels with unstable but not very heavy gravitino. This case is typical for simplemSUGRA models. The gravitino lifetime is large, so these models may lead to thecosmological scenario with intermediate domination of non-relativistic gravitino.This is the first of the two scenarios considered in Sec. 5.3. For heavy gravitino, themain contribution to the decay rate comes from “original” transverse components(not goldstino), and the estimate for the width is

Γ3/2 m3

3/2

M2Pl

.

Let the maximum temperature in the Universe be so high that gravitinos were inthermal equilibrium with the rest of plasma (this assumption may be relaxed, seeSec. 5.3). Since gravitino freeze-out occurs in relativistic regime, gravitino mass

204 Dark Matter

density at T � m3/2 is given by

ρ3/2(T ) ∼ m3/2T3.

So, the intermediate matter-dominated epoch is indeed possible for long-lived grav-itino. The requirement from BBN, that (see (5.46))

Γ3/2 � T 2BBN

M∗Pl(TBBN )

gives rather strong lower bound on the gravitino mass,

m3/2 >

[T 2

NSM2Pl

M∗Pl(TNS)

]1/3

, i.e., m3/2 > 45 TeV.

Moreover, the self-consistency of the cosmological model requires either quite lightLSP (if stable), MLSP ∼ 10MeV, or even much heavier gravitino.

This bound is fairly restrictive in models with gravity mediated SUSY breaking.This bound, however, does not apply to cosmological scenarios in which gravitinomass density was never high.

9.7 ∗Other Candidates

9.7.1 Axions and other long-lived particles

Many extensions of the Standard Model contain light scalar or pseudoscalar par-ticles. In some models these new particles are so light and so weakly interactingthat their lifetime exceeds the present age of the Universe. Hence, they may serveas dark matter candidates. Depending on the context, they are called axions (seebelow), dilatons, familons, sgoldstino, etc.

Let us consider general properties of models with light scalars or pseudoscalars.These particles should interact with the usual matter very weakly, so they must beneutral with respect to the Standard Model gauge interactions. This implies thatinteractions of scalars S and pseudoscalars P with gauge fields are of the form

LSFF =CSFF

4Λ· SFμνFμν , LPFF =

CPFF

8Λ· PFμνFλρε

μνλρ, (9.77)

where Fμν is the field strength of SU(3)c, or SU(2)W , or U(1)Y gauge group. Theparameter Λ has dimension of mass and can be interpreted as the scale of newphysics related to S- and/or P -particle. This parameter has to be large, then theinteractions of S and P with gauge bosons are indeed weak at low energies. Becauseof that, the Lagrangians (9.77) contain gauge-invariant operators of the lowest pos-sible dimension; in principle, one could add to (9.77) terms like Λ−5S(Fμν)4, buttheir effects would be negligible at energies well below Λ. Dimensionless constantsCSFF , CPFF are determined by the fundamental theory valid at energies above Λ.These parameters run with the energy scale Q2 characterizing the processes con-sidered. However, the latter property will not be important for our estimates, and

9.7. ∗Other Candidates 205

we will assume that these parameters are numbers of order 1. In the standardbasis of the Standard Model gauge fields, the terms (9.77) describe interactionsof (pseudo)scalars with pairs of photons, gluons, as well as with Zγ-, ZZ- andW+W−-pairs.

Interactions with fermions can also be written on symmetry grounds. Since S

and P are singlets under SU(3)c × SU(2)W × U(1)Y , no combinations like Sff

or P fγ5f are gauge invariant, so they cannot appear in the Lagrangian (hereafterf denotes the Standard Model fermions). Gauge invariant operators of the lowestdimension have the form Hff , where H is the Higgs field. Hence, the interactionswith fermions are

LSHff =YSHff

Λ· SH f f, LPHff =

YPHff

Λ· PH fγ5f.

It often happens that the couplings YSHff and YSPff are of the order of the StandardModel Yukawa couplings, so upon electroweak symmetry breaking the low energyLagrangians have the following structure,

LSff =CSff mf

Λ· Sff, LPff =

CPff mf

Λ· P fγ5f, (9.78)

where we assume that dimensionless couplings CSff and CPff are also21 of order 1.Making use of (9.77) and (9.78) we estimate the partial widths of decays of P and

S into the Standard Model particles (of course, we are talking about kinematicallyallowed decays)

ΓP (S)→AA ∼m3

P (S)

64πΛ2, ΓP (S)→ff ∼ m2

fmP (S)

8πΛ2, (9.79)

where A denotes vector bosons, and we omitted threshold factors. By requiringthat the lifetime of new particles exceeds the present age of the Universe, τS(P ) =Γ−1

S(P ) > H−10 we find a bound on the mass of the dark matter candidates,

mP (S) < (16πΛ2H0)1/3. (9.80)

Assuming that the new physics scale is below the Planck scale, Λ < MPl, we obtainan (almost) model-independent bound,

mP (S) < 100 MeV. (9.81)

Hence, the kinematically allowed decays are P (S) → γγ, P (S) → νν and P (S) →e+e−. It follows from (9.79) that the two-photon decay mode dominates, unless themass of the new particles is close to that of electron.

Let us now consider generation of relic (pseudo)scalars in the early Universe.There are several generation mechanisms; two of them are fairly generic for the classof models we discuss. These are generation in decays of condensates and thermalgeneration (we will consider yet another mechanism later, in the model with axion).

21In some cases the sets of constants {Csff }, {CPff } have hierarchical structure. To simplify the

discussion, we do not consider these cases in what follows.

206 Dark Matter

Let us briefly discuss the two mechanisms in turn. We will denote the light particleby S for definiteness.

Let some scalar field φ be in condensate in the early Universe; we recall herethat the condensate is the homogeneous scalar field that oscillates at relatively latetimes, when mφ > H . In other words, the condensate is a collection of φ-particlesat rest, see Sec. 4.8.1. Let both particles, φ and S, interact with matter so weaklythat they never get into thermal equilibrium, and let the interaction between φ andS have the form μφS2/2, where μ is the coupling constant. Then the width of thedecay φ → SS is estimated as

Γφ→SS ∼ μ2

16πmφ. (9.82)

If the widths of other decay channels do not exceed the value (9.82), the decay ofφ-condensate occurs at temperature Tφ determined by22

Γφ→SS ∼ H(Tφ) =T 2

φ

M∗Pl

.

Let the energy density of φ-condensate at that time be equal to ρφ, so that thenumber density of decaying φ-particles is nφ ∼ ρφ/mφ, see Sec. 4.8.1. Immediatelyafter the epoch of φ-particles decays, the number density of S-particles is of orderερφ/mφ, where ε is the fraction of the condensate that decayed into S-particles.After S-particles become non-relativistic, their mass density is of order

ρS ∼ ερφ · mST 3

mφT 3φ

,

where we omitted the dependence on g∗ for simplicity. In this way we estimate themass fraction of S-particles today,

ΩS =ρS

ρc∼ mST 3

0

ρc· ερφ

mφT 3φ

∼ 0.2 ·( mS

1 eV

)· ερφ

mφT 3φ

. (9.83)

With appropriate choice of parameters, the right value ΩS 0.2 can be obtainedindeed. We note that the last factor on the right hand side of (9.83) must be small.

Let us now consider production of light (pseudo)scalars in collisions of theStandard Model particles in primordial plasma. Since the interactions (9.77) and(9.78) are very weak, and since they involve three particles, the dominant processesof S-particle creation and annihilation at temperatures T � Λ are23

S + X1 ↔ X2 + X3. (9.84)

These are analogous to the processes (9.70) which we discussed in the gravitinocase; the relevant diagrams are basically the same as in Fig. 9.9 (with S substituted

22We assume for definiteness that the energy density of φ-condensate is small compared to that

of hot matter.23At T > mh there are also processes S +h ↔ ff and crossing processes. The order-of-magnitude

estimates below apply to these processes as well.


for G) Making use of (9.77) and (9.78) we obtain the estimate for the cross sectionin the relevant case mS < T < Λ,

σS ∼ α

Λ2,

where α = g2/(4π), and g is the Standard Model coupling entering the vertexY X2X3 (see Fig. 9.9). We now equate the Hubble time to the mean free time of theStandard Model particles with respect to the emission of S-particle,

τS ∼ 1σS(P )nγ(T )

∼ 1H(T )

∼ M∗Pl

T 2,

and obtain the freeze-out temperature of S-particles,

TS ∼ α−1Λ · ΛM∗

Pl

. (9.85)

If the temperature in the Universe had ever exceeded TS(P ), the S(P )-particles werein thermal equilibrium. In that case their properties are the same as the propertiesof thermally produced gravitino. The estimate (9.68) applies, and one concludesthat the mass of the dark matter S-particle must be unacceptably small, mS ∼200 − 400 eV.

Problem 9.12. Neglecting nucleosynthesis constraints, estimate the fractional massdensity ΩS in models where S-particles freeze out being non-relativistic, i.e., maypretend to make cold dark matter.

If the maximum temperature in the Universe was lower than TS, the numberdensity of thermally produced S-particles is smaller than in the above case. Themass mS may then be in the acceptable range of 1 keV and larger. The correspondinganalysis is similar to that for gravitino, and we do not make it here.

Let us now turn to a concrete class of models with Peccei–Quinn symmetryand axions. This symmetry provides a solution to the strong CP -problem, and theexistence of axions is an inevitable consequence of the construction.

Let us briefly discuss the strong CP -problem [93–95] and its solution in models withaxion. The starting point is that one can extend the Standard Model Lagrangian (seeAppendix B) by adding the following term,

ΔL0 =αs

8π· θ0 · Ga

μνGμν a, (9.86)

where αs = g2s/(4π) is the SU(3)c gauge coupling, Ga

μν is the gluon field strength, Gμν a =12εμνλρGa

λρ is the dual tensor and θ0 is an arbitrary dimensionless parameter (the factorαs/(8π) is introduced for later convenience). The interaction term (9.86) is invariant undergauge symmetries of the Standard Model, but it violates P and CP . The term (9.86) canbe written as the divergence of the vector (we consider the theory in Minkowski space fordefiniteness)

ΔL0 =αs

4π· θ0 · ∂μKμ,

where

Kμ = εμνλρ ·„

Gaν∂λGa

ρ +1

3fabcGa

νGbλGc

ρ

«.

208 Dark Matter

and Gaμ is the gluon vector potential. Hence, the term (9.86) does not contribute to the

classical field equations, and its contribution to the action is reduced to the surface integral.For any perturbative gauge field configurations (small perturbations about Ga

μ = 0) thiscontribution is equal to zero. However, this is not the case for configurations of instantontype in 4-dimensional space of Euclidean topology. This means that CP is violated in QCDat non-perturbative level.

Furthermore, quantum effects due to quarks give rise to the anomalous term in theLagrangian24 which has the same form as (9.86) with proportionality coefficient determined

by the phase of the quark mass matrix Mq. The latter enters the Lagrangian as

Lm = qLMqqR + h.c..

By chiral rotation of quark fields one makes quark masses real (i.e., physical), but thatrotation induces a new term in the Lagrangian,

ΔLm =αs

8π· Arg(DetMq) · Ga

μνGμν a. (9.87)

There is no reason to think that Arg(DetMq) = 0. Neither there is a reason to think thatthe “tree-level” term (9.86) and anomalous contribution (9.87) cancel each other. Indeed,the former term is there even in the absence of quarks, while the latter comes from theYukawa sector, as the quark masses are due to their Yukawa interactions with the Higgsfield.

Thus, the Standard Model Lagrangian should contain the term

ΔLθ = ΔL0 + ΔLm =αs

8π(θ0 + Arg(DetMq))G

aμνGμν a ≡ αs

8π· θ · Ga

μνGμν a . (9.88)

This term violates CP , and off-hand the parameter θ is of order 1.The term (9.88) has non-trivial phenomenological consequences. One is that it

generates electric dipole moment (EDM) of neutron25 dn, which is estimated as [96, 97]

dn ∼ θ · 10−16 · e · cm. (9.89)

Neutron EDM has not been found experimentally, and the searches place strong bound [1]

dn � 3 · 10−26 · e · cm. (9.90)

This leads to the bound on the parameter θ,

|θ| < 0.3 · 10−9 .

The problem to explain such small value of θ is precisely the strong CP -problem.A solution to this problem apparently does not exist within the Standard Model.26

The solution is offered by models with axion. These models make use of the followingobservation. If at the classical level the quark Lagrangian is invariant under axial symmetryU(1)A such that

qL → eiβqL, qR → e−iβqR, (9.91)

24We have in mind the anomaly in the axial current corresponding to U(1)A transformations,

∂μJμA ∝ αs

2πGa

μνGμν a.25Neutron EDM corresponds to the interaction of neutron spin S with electric field E described

by the Hamiltonian H = dnES/|S|.26If at least one of the light quarks (e.g., u-quark) were massless, there would be no effects due to

the θ-term. At the classical level, there would exist a global U(1)A-symmetry of γ5 phase rotations

of u-quark, uL → eiβuL, uR → e−iβuR. This classical symmetry could be employed to set θ equal

to zero. However, experimental data imply that all quarks are actually massive, so this solution of

the strong CP -problem is most likely ruled out.


then the θ-term would be rotated away by applying this transformation. This global sym-metry is called Peccei–Quinn symmetry [98] U(1)PQ. Peccei–Quinn (PQ) symmetry isexplicitly broken in the Standard Model by Yukawa couplings of quarks to the Higgs field(see Sec. B.1),

LY = Y dQLHDR + Y uQLiτ2H∗UR. (9.92)

Indeed, the first term in (9.92) would be invariant under transformations (9.91) supple-mented with the phase rotation H → eiβH , while the second term would be invariant ifthe phase of the Higgs field was rotated in the opposite way, H → e−iβH .

One can, however, extend the Standard Model in such a way that the PQ symmetry isexact at the classical level. Quark masses are not invariant under the PQ transformations(9.91), so PQ symmetry is spontaneously broken. At the classical level, this leads to theexistence of massless Nambu–Goldstone field a(x), axion. As for any Nambu–Goldstonefield, its properties are determined by its transformation law under the PQ-symmetry

a(x) → a(x) + β · fPQ, (9.93)

where β is the same parameter as in (9.91) and fPQ is a constant of dimension of mass,the energy scale of U(1)PQ symmetry breaking. The mass terms in the low energy quarkLagrangian must be symmetric under the transformations (9.91), (9.93), so the quark andaxion fields enter the Lagrangian in the combination

Lm = qRmqe−2i a

fP Q qL + h.c. (9.94)

Making use of (9.87) we find that at the quantum level the low energy Lagrangian containsthe term

La = Cgαs

8π· a

fPQGa

μνGμν a, (9.95)

where the constant Cg is of order 1; it is determined by PQ charges of quarks.27 Clearly,PQ symmetry (9.91), (9.93) is explicitly broken by quantum effects of QCD, and axion ispseudo-Nambu–Goldstone boson.

Hence, θ-parameter multiplying the operator GaμνGμν a obtains a shift depending on

space-time point and proportional to the axion field,

θ → θ(x) = θ + Cga(x)

fPQ. (9.96)

Strong interactions would conserve CP provided the axion vacuum expectation value issuch that 〈θ〉 = 0. The QCD effects indeed do the job. They generate non-vanishing quarkcondensate 〈qq〉 ∼ Λ3

QCD at the QCD energy scale ΛQCD ∼ 200 MeV. This condensate

breaks chiral symmetry and in turn generates the axion effective potential28

Va ∼ −1

2θ2 mumd

mu + md〈qq〉 + O(θ4) ≈ 1

8θ2 · m2

πf2π + O(θ4), (9.97)

where mπ = 135 MeV and fπ = 93 MeV are pion mass and decay constant. The potentialhas the minimum at 〈θ〉 = 0, so the strong CP -problem finds an elegant solution. It followsfrom (9.96) and (9.97) that axion has a mass

ma ≈ Cgmπfπ

2fPQ, (9.98)

i.e., it is indeed pseudo-Nambu–Goldstone boson.

27Generally speaking, different types of quarks may have different PQ charges.28Since (9.91) is the symmetry under phase rotations, the axion potential must be periodic in θ

with period 2π, see (9.93). The simplest periodic generalization of the expression (9.97) is Va =

m2af2

PQ · (1 − cos θ). This point will not be important for us in what follows.

210 Dark Matter

The simplest way to implement PQ mechanism is to add the second Higgs doublet tothe Standard Model and choose the Yukawa interaction as

Y dQLH1DR + Y uQLiτ2H∗2 UR. (9.99)

The two Higgs fields transform under U(1)PQ-transformation (9.91) as follows,

H1 → e2iβH1, H2 → e−2iβH2,

This ensures U(1)PQ-invariance of the Lagrangian (9.99), and hence the absence of theθ-term.

Unless other new fields are added, spontaneous breaking of PQ symmetry in this sim-plest case occurs due to the Higgs expectation values. Then the classically massless field(would-be Nambu–Goldstone boson) is the relative phase of the two Higgs fields H1 andH2. To obtain the low energy Lagrangian let us write

H1 = e2iβ(x)

„0v1√

2

«, H2 = e−2iβ(x)

„0v2√

2

«, (9.100)

where v1 and v2 are the Higgs expectation values. They both contribute to the W - andZ-boson masses, hence q

v21 + v2

2 ≡ v = 247 GeV.

The kinetic term for the field β(x) comes from the kinetic terms of the Higgs doublets,

Lkin,H = ∂μH†1∂μH1 + ∂μH†

2∂μH2.

Inserting (9.100) into this expression, we find

Lkin,β =f2

PQ

2∂μβ∂μβ,

where

fPQ = 2q

v21 + v2

2 = 2v. (9.101)

The axion field is related to β(x) by

a(x) = fPQ · β(x);

with this definition the field a(x) has the standard (“canonical”) kinetic term. The axionis rather heavy in this model: we find from (9.98) that

ma ∼ 15 keV.

The interaction of this axion with quarks, gluons and photons (see below) is rather strong.So, this particle, called Weinberg–Wilczek axion [99, 100], is experimentally ruled out.

This problem is solved in models with “invisible axion”, in particular, in Dine–Fischler–Srednicki–Zhitnitsky [101, 102] (DFSZ) model and Kim–Shifman–Vainshtein–Zakharov [103, 104] (KSVZ) model. The idea is to make the scale of PQ symmetry breakingindependent of the electroweak symmetry breaking scale. In DFSZ model this occurs inthe following way. One adds into the model with the Lagrangian (9.99) complex scalarfield S which is singlet under the Standard Model gauge group. One also adds interactionsinvolving PQ invariants

S†S, H†1H2 · S2.


The field S transforms under U(1)PQ as

S → e2iβS. (9.102)

The axion field is now a linear combination of the phases of fields H1, H2 and S. Byrepeating the calculation leading to (9.101) we find

fPQ = 2q

v21 + v2

2 + v2s , (9.103)

where vs is the vacuum expectation value of the field S. The latter can be large, so it isclear from (9.103) that the mass of axion is small and, most importantly, its couplings tothe Standard Model fields are weak: these couplings are inversely proportional to fPQ ∼ vs.Note that DFSZ axion interacts with both quarks and leptons.

The KSVZ mechanism makes use of additional quark fields ΨR and ΨL which aretriplets under SU(3)c and singlets under SU(2)W × U(1)Y . Only these quarks transformnon-trivially under U(1)PQ while the usual quarks have zero PQ charge. One also intro-duces a complex scalar field S which is a singlet under the Standard Model gauge group.One writes PQ-invariant Yukawa interaction of the new fields,

L = yΨSΨRΨL + h.c.,

so that S transforms under U(1)PQ according to (9.102). PQ symmetry is spontaneously

broken by the vacuum expectation value 〈S〉 = vs/√

2. The axion here is the phase of thefield S, therefore

fPQ = 2vs. (9.104)

The KSVZ model does not contain explicit interaction of axion with usual quarks andleptons.

Thus, axion is a light particle whose interactions with the Standard Model fieldsare very weak. Its mass is related to PQ symmetry breaking scale fPQ by (9.98). Theproperty that its interactions are weak relates to the fact that it is pseudo-Nambu–Goldstone boson of a global symmetry spontaneously broken at high energy scalefPQ � MW . Like for any Nambu–Goldstone field, the interactions of axion aredescribed by the generalized Goldberger–Treiman formula

La =1

fPQ· ∂μa · Jμ

PQ. (9.105)

Here

JμPQ =

∑f

e(PQ)f · fγμγ5f. (9.106)

The contributions of fermions to the current JμPQ are proportional to their PQ

charges e(PQ)f ; these charges are model-dependent. Besides the interaction (9.105),

there are also interactions of axions with gluons, see (9.95), and photons,

Lag = Cgαs

8π· a

fPQ· Ga

μνGμν a, Laγ = Cγα

8π· a

fPQ· Fμν F μν , (9.107)

where the dimensionless constants Cg and Cγ are also model-dependent and, gen-erally speaking, are of order 1. In accordance with (9.94), the action (9.105) can be

212 Dark Matter

integrated by parts and we obtain instead

La = − 1fPQ

· a · ∂μJμPQ = − a

fPQ·∑

f

2e(PQ)f mf · fγ5f, (9.108)

plus anomalous interactions (9.107). The interaction terms (9.107) and (9.108)indeed have the form (9.77), (9.78) (with P (x) = a(x)), i.e., models with axionsbelong to the class of models with light, weakly interacting pseudoscalars. Theaxion mass, however, is not a free parameter: we find from (9.98) that

ma ≈ mπ · fπ

2fPQ≈ 0.6 eV ·

(107 GeV

fPQ

). (9.109)

The main decay channel of the light axion is decay into two photons. The lifetimeτa is found from (9.79) by setting Λ = 2πfPQ/α and using (9.109),

τa =1

Γa→γγ=

64π3m2πf2

π

α2m5a

4 · 1024 s ·(

eVma

)5

.

By requiring that this lifetime exceeds the age of the Universe, τa > t0 ≈ 14 billionyears, we find the bound on the mass of axion as dark matter candidate,

ma < 25 eV. (9.110)

There are astrophysical bounds on the strength of axion interactions f−1PQ and hence

on the axion mass. Axions in theories with fPQ � 109 GeV, which are heavier than10−2 eV would be intensely produced in stars and supernovae explosions. This wouldlead to contradictions with observations. So, we are left with very light axions,ma � 10−2 eV.

As far as dark matter is concerned, thermal production of axions is irrelevant.Indeed, the estimate of the mass density of thermally produced axions basicallycoincides with that of gravitino (9.68), which gives way too low Ωa.

It may seem that axion cannot serve as dark matter candidate. This is not thecase. There are at least two mechanisms of axion production in the early Universethat can provide not only right axion abundance but also small initial velocities ofaxions. The latter property makes axion a cold dark matter candidate, despite itsvery small mass. One mechanism has to do with decays of global strings [105] —topological defects that exist in theories with spontaneously broken global U(1) sym-metry (U(1)PQ in our case; for a discussion of this mechanism see. e.g., Ref. [106]).Another mechanism employs axion condensate [107–109], homogeneous axion fieldthat oscillates in time after the QCD epoch. Let us consider the second mechanismin some details.

As we have seen in (9.97), the axion potential is proportional to the quarkcondensate 〈qq〉. This condensate breaks chiral symmetry. The chiral symmetry isin fact restored at high temperatures.29 Hence, one expects that the axion potentialis negligibly small at T � ΛQCD. This is indeed the case: the effective potential for

29We have here an analogy to phase transitions considered in Chapter 10.


the field θ = θ + a/fPQ vanishes at high temperatures, and this field can take anyvalue,

θi ∈ [0 , 2π),

where we recall that the field θ is a phase. There is no reason to think that theinitial value θi is zero. As the temperature decreases, the axion mass starts toget generated, and the field θ, remaining homogeneous, starts to roll down fromθi towards its value θ = 0 at the minimum of the potential. This homogeneousevolution is described by the Lagrangian

L =f2

PQ

2·(

dθ

dt

)2

− m2a(T )2

f2PQθ2,

where ma(T ) is a function of temperature, so that

ma(T ) 0 at T � ΛQCD,

ma(T ) ma at T � ΛQCD.

Hereafter ma denotes the zero-temperature axion mass.The evolution of a scalar field in expanding Universe is studied in Sec. 4.8.1. It

follows from that study that axion field practically does not evolve when ma(T ) �H(T ) and at the time when ma(T ) ∼ H(T ) it starts to oscillate. Let us estimatethe present energy density of axion field in this picture, without using the concreteform of the function m(T ) for the time being.

The oscillations start at the time tosc when

ma(tosc) ∼ H(tosc). (9.111)

At this time, the energy density of the axion field is estimated as

ρa(tosc) ∼ m2a(tosc)f2

PQθ2i .

According to the discussion in the end of Sec. 4.8.1, the oscillating axion field is thesame thing as a collection of axions at rest. Their number density at the beginningof oscillations is estimated as

na(tosc) ∼ ρa(tosc)ma(tosc)

∼ ma(tosc)f2PQθ2

i ∼ H(tosc)f2PQθ2

i .

This number density, as any number density of non-relativistic particles, thendecreases as a−3 (we will explicitly see this in the end of this Section)

Axion-to-entropy ratio at time tosc is

na

s∼ H(tosc)f2

PQ

2π2

45 g∗T 3osc

· θ2i f2

PQ√g∗ToscMPl

· θ2i ,

where we use the usual relation H = 1.66√

g∗T 2/MPl. The axion-to-entropy ratioremains constant after the beginning of oscillations, so the present mass density ofaxions is

ρa,0 =na

smas0 maf2

PQ√g∗ToscMPl

s0 · θ2i . (9.112)

214 Dark Matter

In fact, it is a decreasing function of ma. Indeed fPQ is inversely proportional toma, see (9.98); at the same time, axion obtains its mass near the epoch of QCDtransition, i.e., at T ∼ ΛQCD, so Tosc depends on ma rather weakly.

To obtain preliminary estimate, let us set Tosc ∼ ΛQCD 200 MeV and makeuse of (9.98) with Cg ∼ 1. We find

Ωa ≡ ρa,0

ρc(

10−6 eVma

)θ2

i . (9.113)

The natural assumption about the initial phase is θi ∼ π/2. Hence, axion of mass10−5 − 10−6 eV is a good dark matter candidate.30 This is cold dark matter: wehave seen in Sec. 4.8.1 that effective pressure of the oscillating field is zero. This isof course consistent with the fact that oscillating field corresponds to axions at rest.

Let us note here that search for relic axions with masses ma ∼ 10−5 − 10−6 eVis a difficult but not hopeless experimental problem [110].

To refine the above estimate, let us make use of the explicit formula for theaxion mass at T > ΛQCD. It reads [111]

ma(T ) 0.1 · ma(0) ·(

ΛQCD

T

)3.7

, T > ΛQCD. (9.114)

We then find from (9.111) that the temperature at which axion field oscillationsstart is

Tosc ∼ 200 MeV ·( ma

10−9 eV

)0.2

·(

ΛQCD

200 MeV

)0.7

. (9.115)

Note that the assumption Tosc > ΛQCD is justified for ma > 10−9 eV. Inserting theestimate (9.115) into (9.112) we get

Ωa 0.2 · θ2i ·(

4 · 10−6 eVma

)1.2

· 12h2

.

We see that our earlier estimate (9.113) is quite reasonable, and the dependence onthe axion mass is close to inverse proportionality. This is related to strong depen-dence on temperature in (9.114).

To end this Section, let us check explicitly that the homogeneous oscillating field ofvariable mass has the property that

na(t) =ρa(t)

ma(t)

decays as a−3. Let us still use the notation θ for this field. Let us write the field equationin expanding Universe,

d2θ

dt2+ 3H(T )

dθ

dt+ m2

a(T )θ = 0. (9.116)

30We note that axion of lower mass ma < 10−6 eV may also serve as dark matter particle, if for

some reason the initial phase θi is much smaller than π/2. We note also that the mechanism of

axion production in decays of topological defects is capable of producing right axion abundance

for ma > 10−5 eV as well.


Let us multiply this equation by dθ/dt and find

1

2

d

dt

„dθ

dt

«2

+ 3H ·„

dθ

dt

«2

+m2

a(t)

2

d

dtθ2 = 0. (9.117)

We solve this equation approximately, making use of the fact that for ma(T ) � H(T ), thefollowing equality for averages over the oscillation period holds,*„

dθ

dt

«2+

= m2a(t)〈θ2〉.

So, we obtain the equation

d〈θ2〉dt

+

„3H +

1

ma(t)

dma(t)

dt

«〈θ2〉 = 0.

It gives

m(t)〈θ2〉(t) =const

a3.

The left hand side here coincides with na, in view of the fact that

ρa = const ·*„

dθ

dt

«2+

= const · m2a(t)〈θ2〉

(the constant here equals f2PQ for the field θ, and equals to 1 for canonically normalized

field).

9.7.2 Superheavy relic particles

Less natural dark matter candidates are particles of very large mass (we will callthem X-particles)

MX � 100 TeV.

We recall that the assumption that these particles were in thermal equilibriumat T � MX would lead to overproduction of these particles, see (9.24). Hence,one has to assume that thermal (more precisely, chemical) equilibrium had beennever reached. The superheavy non-thermal relic particles are sometimes called“wimpzillas”, see, e.g., Ref. [112].

Let us briefly discuss several production mechanisms of superheavy dark matter.We begin with production in collisions of light particles in hot plasma. This mech-anism works if the maximum temperature in the Universe Tmax is smaller, but notmuch smaller than MX. Generalizing the analysis of Sec. 9.2, we find that the rightabundance, ΩX ≈ 0.2 is obtained when there is a very definite relation betweenTmax and MX,

MX

Tmax= 25 +

12· log(M2

X〈σ〉), (9.118)

where σ is the production cross section of X-particles in the plasma. The sup-pression of the relic mass density, as compared to the equilibrium case, is due to

216 Dark Matter

the Boltzmann factor here. Note that the right mass density of X-particles can beobtained at the expense of fine-tuning between the particle physics parameter MX

and cosmology-related parameter Tmax.

Problem 9.13. Derive the relation (9.118).

Heavy particles can be created before the hot stage, at the very process of for-mation of hot plasma. Such a process, reheating, occurs, in particular, in successfulmodels with inflationary stage. We discuss reheating in the accompanying book.Here we only mention that production of particles at reheating may be efficientup to masses MX ∼ 1016 GeV, even though the maximum temperature of nearlyequilibrium plasma after reheating is several orders of magnitude smaller.

Even more exotic possibility is that inflation ends by vacuum phase transition.It occurs via spontaneous creation of bubbles of new vacuum, expansion of thesebubbles and collisions of bubble walls. The collision of two walls may be viewedlocally as a collision of particles of mass m and Lorentz-factor γ. The scale of m

is the energy scale of the phase transition; the same scale determines the finaltemperature, Tmax � m. On the other hand, the Lorentz-factor γ may be large,and one expects that particles of mass up to MX ∼ γm can be produced in wallcollisions. So, this mechanism is also capable of producing heavy relic of very largemass.

Let us also note that time-dependent gravitational field produces particles aswell. This mechanism can work both at inflationary stage and afterwards. Themost efficient production occurs at the time when MX ∼ H . That can happen atthe radiation-dominated epoch as well. In the latter case, the number density ofX-particles produced when MX ∼ H is given at later times by [113]

ρX 5 · 10−4 · MX ·(

MX

t

)3/2

.

The present mass fraction of the heavy relic produced in this way is then

ΩX ∼(

MX

109 GeV

)5/2

.

We see that this mechanism is successful for MX ∼ 109 GeV and that it overproducesheavy particles of larger masses. Detection of superheavy dark matter with MX �109 GeV would probably mean that the Hubble parameter at the hot stage wasnever as large as MX .

Chapter 10

Phase Transitions inthe Early Universe

As we mentioned in Chapter 1, there are no direct experimental indications yet thattemperatures above a few MeV existed in the Universe. Nevertheless, it is naturalto assume that the Universe in the remote past had much higher temperatures.1

In this regard, the study of properties of cosmic plasma at high temperatures is ofconsiderable interest.

As in any evolving hot system, there could occur phase transitions in the earlyUniverse associated with the rearrangement of the ground state. At temperaturesabove 200MeV quarks and gluons do not form bound states — hadrons — andthe medium is in the phase of quark-gluon matter. At these temperatures, thereis no quark condensate, i.e., the phase of unbroken chiral symmetry is realized.If temperature in the Universe ever exceeded 200MeV, then at some moment oftime there was the phase transition from quark-gluon plasma to hadronic matter2

comprised of colorless particles: pions, kaons, nucleons and other hadrons. Moreover,at the same or almost the same time there must have occurred the chiral transition,responsible for the formation of the quark condensate.

It is quite likely that there was an era of even higher temperatures, T �MEW ∼ 100GeV. Oversimplifying the situation, we can say that at these temper-atures electroweak symmetry was unbroken, and the Higgs expectation value waszero. When the temperature dropped, the electroweak phase transition [30–33] mayhave occurred,3 which resulted in the non-zero Higgs expectation value and spon-taneous breaking of electroweak symmetry SU(2)W × U(1)Y down to the electro-magnetic U(1)em.

1For instance, in Chapter 9 we noted that a simple and efficient (and therefore plausible) mech-

anism of non-baryonic dark matter generation works at temperatures of tens of GeV or above.

Many mechanisms generating the baryon asymmetry of the Universe (though not all) require

even higher temperatures, from 100 GeV up to 1015 GeV (see Chapter 11), depending on specific

mechanism.2In fact, lattice data suggest that instead of phase transition there was smooth crossover.3The subtlety here is that in the Standard Model and its many extensions, “phases” of “broken”

and “unbroken” symmetry are not really distinguishable, and the electroweak phase transition

may in general be absent. We discuss this issue in Sec. 10.3.

217

218 Phase Transitions in the Early Universe

Depending on the maximum temperature at the hot stage of the cosmologicalexpansion, and on physics at ultra-short distances and at ultra-high energies,phase transitions could occur at even higher temperatures. Namely, if the Uni-verse had temperature of about 1016 GeV (which is very strong and hardly realisticassumption), and physics at these energies is described by Grand Unified Theory,then there was the Grand Unified phase transition at temperature comparable toMGUT ∼ 1016 GeV. It is not excluded that phase transitions occurred also at inter-mediate temperatures, MEW � T � MGUT .

The study of phase transitions in the Universe is not only of academic interest;it also sheds light on some of the mysteries of cosmology. Among them are theproblems of the baryon asymmetry and dark matter. Phase transitions are alsoresponsible for possible formation of topological defects in the early Universe andplay an important role in some inflationary models.

In this Chapter we recall the general classification of phase transitions andintroduce methods for describing phase transitions in the early Universe. We discusspredominantly theories with the Higgs mechanism and are interested in phase transi-tions leading to spontaneous breaking of the corresponding symmetry. An importantexample here is the Standard Model and electroweak transition; it is this examplethat we have in mind in what follows. As usual in field theory, the applicability ofanalytical methods is limited to theories with small couplings (this is indeed thecase not only in the electroweak sector of the Standard Model but also in quantumchromodynamics at high temperatures, T � 1GeV), but we will see that this is notenough: detailed analytical description of phase transitions is possible only whenthe vacuum value of the mass of the Higgs boson is sufficiently small. Nevertheless,analytical methods often enable one to make qualitatively correct conclusions aboutthe type of the phase transition and study the most important cosmological impli-cations. The results of this Chapter are used in the Chapters on baryogenesis andtopological defects.

In theories where couplings are not small, analytical study of phase transitionsfrom “first principles” is usually impossible; the most reliable source of informationhere are numerical methods based on lattice field theory. An important example isthe confinement–deconfinement transition and the transition with chiral symmetrybreaking in QCD. They occur at temperatures T ∼ 200MeV, when the QCD gaugecoupling αs(T ) is large. We will not present any detailed study of QCD transitionsin this book, although there is no doubt that they actually occurred in the earlyUniverse (assuming that temperatures T � 200MeV were indeed realized). Thereason is that these transitions apparently left no traces in the present Universe(with the exception of rather exotic proposals such as the formation of quark nuggetswith large number of strange quarks [114]; we also mention in this context axions ascandidates for dark matter: one of the mechanisms of their generation is based on thevery fact that the chiral phase transition occurred in the early Universe, while theresults are practically insensitive to the dynamics of this transition, see Sec. 9.7.1).

10.1. Order of Phase Transition 219

10.1 Order of Phase Transition

Phase transitions occur because of mismatch between the properties of the groundstates of the theory at zero and non-zero temperatures. As we show below, in the-ories with the Higgs mechanism this is caused by non-trivial temperature-dependentterms in the effective potential. At finite temperature, the equilibrium state ofthe medium corresponds to the minimum of the Grand thermodynamic potential(Landau potential). As we discussed in Chapter 5, chemical potentials are negligiblysmall in the early Universe at interesting temperatures T � 1GeV, and the Grandpotential reduces to the (Helmholtz) free energy F . Hence, we consider below thefree energy of primordial plasma. To find the expectation value of the Higgs field〈φ〉T at temperature T , one considers a system in which the average value of theHiggs field is fixed and equal to φ everywhere in space, but otherwise there isthermal equilibrium. The free energy of such a system depends, of course, on thechosen value φ, as well as on temperature. Because of spatial homogeneity, the freeenergy is proportional to the spatial volume Ω,

F = ΩVeff (T, φ). (10.1)

The free energy density of medium at temperature T with the average Higgs fielduniform and equal to φ is called the effective potential Veff (T, φ). In thermal equi-librium, the free energy is at minimum with respect to all macroscopic parameters,including the average Higgs field. Therefore, 〈φ〉T is the absolute minimum of theeffective potential Veff (T, φ) at given temperature (we will often omit the argumentT in Veff ).

At zero temperature, the free energy reduces to the energy of the system, and theeffective potential coincides with the scalar potential V (φ) entering the field theoryaction.4 At finite temperature, Veff (T, φ) does not coincide with V (φ). As a result,symmetry broken at zero temperature may be restored at high temperatures. Thisis a general statement valid for any system. In particular, such a phenomenon occursin the Standard Model of particle physics.5 At zero temperature, the ground state isnot invariant under SU(2)W × U(1)Y gauge group: the symmetry is spontaneouslybroken down to the gauge group U(1)em due to non-zero vacuum expectation valueof the Higgs doublet (see Fig. 10.1(a))

v = 〈φ〉 ≈ 247 GeV.

At finite temperature, the effective Higgs potential of the Standard Model getsadditional contributions. Because of these contributions the Higgs expectation valueis zero at high temperatures T � v, i.e., the symmetry is restored (see Fig. 10.1(b)).

4In fact, even at zero temperature the effective potential does not coincide with the scalar

potential entering the classical action. This is due to quantum corrections. In weakly coupled

theories, quantum corrections to the effective potential are often small.5We again emphasize that we oversimplify the situation, see the discussion in Sec. 10.3.


(a) (b)

Fig. 10.1 The effective Higgs potential at zero (a) and high (b) temperatures.

As the Universe cools down, the transition from 〈φ〉T = 0 to 〈φ〉T �= 0 occurs at acertain temperature Tc, the temperature of the phase transition. Depending on theparameters of the theory, the transition can be quite long or nearly instantaneous,occur at once throughout the entire system, or proceed in its individual parts.

Two types of phase transitions are most common; these are phase transitionsof the 1st and 2nd order. From the standpoint of the general formalism, 1st orderphase transition is accompanied by a jump in heat capacity; in field theory thiscorresponds to a jump in the expectation value 〈φ〉T as a function of temper-ature, see Fig. 10.2(a). On the contrary, 2nd order phase transition is characterizedby continuous behavior of the heat capacity and the expectation value 〈φ〉T , seeFig. 10.2(b). This difference is illustrated in Fig. 10.3 where the families of effectivepotentials Veff (φ, T ) as functions of φ at various temperatures T are shown. The leftpanel of Fig. 10.3 shows the 1st order phase transition, culminating in an abrupt

(a) (b)

Fig. 10.2 The expectation value 〈φ〉T as a function of temperature for the systems with 1st order

(a) and 2nd order (b) phase transition.


Fig. 10.3 Shapes of the effective potential Veff (φ) at various temperatures: upper darker curves

correspond to higher temperatures. Left and right panels describe systems with 1st and 2nd order

phase transition, respectively. Black circles show the expectation value 〈φ〉T .

change of 〈φ〉T . The right part of Fig. 10.3 corresponds to the 2nd order phasetransition: the expectation value 〈φ〉T is a smooth function of temperature.

The famous example of the 1st order phase transition is boiling of liquid.Examples of the 2nd order phase transition are transitions in ferromagnets, order-disorder transitions in alloys of metals, transitions into superconducting and super-fluid states.

The notion of different phases and respective phase transition is particularlywell-defined in the cases where the phases differ by symmetry and/or there is aparameter (called the order parameter) equal to zero in one phase and differentfrom zero in the other. The above examples of the 2nd order phase transitionsbelong to this category (the order parameter in ferromagnet is spontaneous mag-netization, in superconductor it is the density of the Cooper pair condensate, etc.).Another example is the chiral phase transition of QCD with massless quarks, theorder parameter here is quark condensate. If the system is such that there is noorder parameter, then phase transitions are also possible, but their existence orabsence may depend on internal or external parameters. A well-known example isthe water-vapor transition, which is of the 1st order at low pressure, and is not aphase transition at all at high pressure. In the latter case, the substance properties(e.g., density) change with temperature continuously, albeit rather quickly, so thesystem exhibits a phenomenon called smooth crossover, rather than phase tran-sition proper. The same situation occurs in the electroweak sector of the StandardModel of particle physics: if gauge and Yukawa couplings are fixed, then at small


vacuum mass of the Higgs boson the 1st order phase transition occurs at temper-ature T ∼ 100GeV (we will see that in Sec. 10.2), while at large vacuum mass thereis smooth crossover instead [115].

The way the transition proceeds is quite different for the 1st and 2nd orderphase transitions. We are interested in the case where the rate at which temperaturechanges in time is low compared to the typical rate of particle interactions in themedium; this is the case for the early Universe. For the 2nd order phase transition,the medium properties (for example, the expectation value 〈φ〉T ) change slowlyand homogeneously over entire space; at every moment of time the medium is ina state close to thermal equilibrium. The same applies to smooth crossover. Thesituation is different in the case of the 1st order phase transition. Before the phasetransition, the expectation value 〈φ〉T equals to zero, but as soon as the minimumof the effective potential at φ = 〈φ〉T �= 0 becomes deeper than the minimum atφ = 0, the ground state with 〈φ〉T �= 0 becomes thermodynamically favorable, seeFig. 10.4. The transition from the state φ = 0 to the state φ = 〈φ〉T cannot occurhomogeneously over entire space: the field value φ in such a process would evolvehomogeneously from φ = 0 to φ = 〈φ〉T , and free energy (10.1) in infinite volumewould be infinitely large in the intermediate states as compared to its initial valueat φ = 0. The transition proceeds via spontaneous nucleation of bubbles of thenew phase, their subsequent expansion and mergers, see Fig. 10.5. Nucleation of abubble with φ = 〈φ〉T �= 0 in the medium with φ = 0 is local in space and mayoccur due to thermal fluctuations.6 Bubbles expand, their walls collide, the newphase percolates, and after this “boiling” the system eventually returns to spatiallyhomogeneous state of thermal equilibrium, but with φ = 〈φ〉T �= 0; the released freeenergy converts into heat.

This boiling is a highly inequilibrium process. We have already noted that themost important stages of the evolution of the hot Universe are those when the cosmic

Fig. 10.4 Shape of the effective potential of the system undergoing the 1st order phase transition.

6Sometimes the dominant process is quantum tunneling.


Fig. 10.5 Spontaneous nucleation and subsequent growth of bubbles of the new phase at the 1st

order phase transition.

plasma is out of thermal equilibrium. Therefore, the 1st order phase transitions areof particular interest for cosmology.

Let us estimate the nucleation probability of a bubble of the new phase attemperature T . Let V− = Veff (T, φ = 0) and V+ = Veff (T, φ = 〈φ〉T ) be free energydensities of the old and new phase, respectively, V+ < V−, see Fig. 10.4. The freeenergy of a bubble of size R, relative to the free energy of medium with φ = 0without a bubble, contains the volume and surface terms. The former is due to thefact that the free energy density inside the bubble is smaller than the free energydensity of the surrounding medium; it is negative and equal to

43πR3 (V+ − V−) .

The surface term exists because the field φ near the surface is inhomogeneous anddiffers from both zero and 〈φ〉T ; the contributors here are effective potential Veff (φ)and the gradient term in the free energy, the latter being now a functional F [φ(x)].The surface term is proportional to the bubble area and equal to 4πR2 · μ, whereμ is the free energy per unit area (surface tension). Thus, the free energy of thebubble, relative to the free energy of the old phase, is (see Fig. 10.6)

F (R) = 4πR2μ − 4π

3R3 · ΔV, (10.2)

where

ΔV = V− − V+ > 0

is the difference between the free energy densities of the old and new phases (latentheat of the phase transition). From (10.2) we see that at sufficiently small sizes


the free energy of the bubble decreases with decrease of R; this means that spon-taneously nucleated small bubble collapses due to surface tension force, and thesystem returns to the initial homogeneous state with φ = 0. On the contrary, forsufficiently large R the free energy decreases with increase of R, i.e., the bubbleexpands and the system converts to the new phase. The minimum size at which thebubble begins to expand is determined by the equation

∂F

∂R= 0.

Hence, it is equal to

Rc =2μ

ΔV. (10.3)

The bubble of this size is called the critical bubble; its free energy is positive andequal to

Fc = 4πR2cμ − 4π

3R3

c · ΔV =16π

3μ3

ΔV. (10.4)

Importantly, both the size of the critical bubble and its free energy are larger forsmaller ΔV .

Spontaneous nucleation of bubbles of the new phase in hot medium occurs viathermal fluctuations, i.e., thermal jumps on top of the barrier shown in Fig. 10.6.The probability of such a jump per unit time per unit spatial volume is mainlydetermined by the Boltzmann factor e−Fc/T :

Γ = AT 4e−FcT (10.5)

(the Arrhenius formula), where the factor T 4 is introduced on dimensional grounds,and pre-exponential factor A does not depend very strongly on temperature andother parameters. The formula (10.5) is valid for Fc � T , i.e., when the proba-bility of the bubble nucleation is small. This formula together with Eq. (10.4) implyimmediately that for finite cooling rate the medium remains in supercooled state

Fig. 10.6 Free energy of a bubble of the new phase as a function of its radius.


with φ = 0 for some time, even though the new phase is already thermodynami-cally favorable. This occurs when ΔV = V− − V+ is still so small that the bubblenucleation rate is smaller than the cooling rate. In the cosmological context, thenucleation of bubbles begins when the probability of the bubble nucleation perHubble volume per Hubble time is of order 1, i.e.,

AT 4e−FcT ∼ H4(T ) =

(T 2

M∗Pl

)4

. (10.6)

In specific models, this equation determines the extent to which the cosmic plasma issupercooled before the phase transition, and the latent heat ΔV released as a resultof the phase transition. These properties are model-dependent. However, one canmake a general statement concerning the picture of the first order phase transition inthe Universe (if it happened): the phase transition begins when handful of bubbleshave nucleated in the entire Hubble volume. Their size at the time of nucleation isdetermined by microscopic physics,7 and it is much smaller than the Hubble scaleH−1(T ), while the distance between their centers is comparable to the Hubble size.Bubbles have time to expand by many orders of magnitude before they begin topercolate, and only a small number of new bubbles are produced during that time.

For instance, at T ∼ 100GeV (the electroweak scale) the Hubble size is of order

H−1 =M∗

Pl

T 2∼ 1 cm.

The bubble size at the time of nucleation is, roughly speaking, of order T−1, i.e.,Rc ∼ 10−16 cm (in fact, it is one to two orders of magnitude larger). Thus, thephase transition in the Universe occurs through the formation of several bubblesof subnuclear size in a cubic centimeter of cosmic plasma, their expansion up tomacroscopic size and merger as a result of collisions of their walls.

Note that theoretically there is a possibility that the phase transition does not completeat all in the expanding Universe, despite the fact that the effective potential has the formshown in Fig. 10.4. The zero-temperature scalar potential may have a local minimum atφ = 0, the false vacuum. It has positive energy density V−, so that even in the absence ofparticles, the Universe filled with false vacuum expands at the Hubble rate

H− =

r8π

3GV−.

If the rate of bubble nucleation per Hubble time per Hubble volume is small, Γ H4−,

the bubble walls do not collide, because the centers of neighboring bubbles move fromeach other with velocities exceeding the speed of light (in other words, nucleated bubblesare outside the event horizons of each other; for definition of event horizon see Sec. 3.2.3).Regions of false vacuum grow faster than regions of the new phase, and the phase transitiondoes not complete.

We also note another theoretical possibility. Namely, because of the gravitational inter-actions, the false vacuum decay may not occur since bubbles of the true vacuum do notnucleate at all [116]. This takes place when the energy density of the true vacuum is neg-ative, and the gravitational effects are strong. The study of this effect is beyond the scopeof this book.

7Modulo a factor of logM∗

P lT

, as is clear from (10.3), (10.4) and (10.6).


Let us discuss in general terms how to calculate the surface tension of the bubblewall. Let us neglect the curvature of the wall (i.e., consider the bubble of large size R)and take the difference between the free energies of the old and the new phase ΔV

small. In this case, the configuration of the field φW (r) inside the wall is the minimumof the free energy F [φ(r)] considered as a functional of the inhomogeneous field φ(r).At the one side of the wall, r � R (i.e., inside the bubble), the field tends to φ = φ+,while at the other side, r � R, it approaches φ = 0. If R is sufficiently large, thenthe coordinate (r − R) inside the bubble can be formally extended to −∞, and wewrite the boundary conditions as

φW (x) → 〈φ〉T as (r − R) → −∞, (10.7)

φW (x) → 0 as (r − R) → +∞. (10.8)

We assume that the temperature corrections to the gradient term in the energyfunctional are small (this assumption is indeed valid in weakly coupled theories).Then the free energy (relative to the free energy of the old phase) can be writtenas a functional of the field φ(r),

F [φ] =∫ ∞

0

4πr2dr

[12

(dφ

dr

)2

+ Veff (φ) − V−

]. (10.9)

The wall thickness is small compared to R for large bubble, and a slowly varyingfactor 4πr2 can be treated as a constant inside the wall. Thus,

F [φ] = 4πR2

∫ +∞

−∞dr

[12

(dφ

dr

)2

+ Veff (φ) − V−

], (10.10)

where

r = r − R,

and we formally extended the integration over this variable to −∞ (cf. (10.7)).The field configuration obeys the Euler–Lagrange equation for the extremum of thefunctional (10.10),

d2φ

dr2=

∂Veff (φ)∂φ

. (10.11)

This equation is formally identical to the equation of one-dimensional classicalmechanics of a particle in the potential

U(φ) = −Veff (φ),

where r plays the role of time. We can now neglect ΔV and set U(〈φ〉T ) = U(0).Then the potential U(φ) has two equally high maxima, see Fig. 10.7(a). The solutionφW (r) to Eq. (10.11) describes the roll down of a particle from right hump, inaccordance with (10.7), and subsequent roll up to left hump. The whole processoccurs in infinite time, see (10.8). Using the analogy with the classical particle, it


Fig. 10.7 (a) The potential of the analogous classical mechanics problem; (b) The field configu-

ration for the bubble of the new phase.

is straightforward to find the solution in quadratures to Eq. (10.11) with boundaryconditions (10.7), (10.8),∫ φW

φ1

dφ√2 (Veff − V−)

= − (R − r) , (10.12)

where the limit of integration is chosen in such a way that at r = R the field φ(r)takes an intermediate value φ1 between φ = 0 and φ = 〈φ〉T . The configurationφW (r) is shown in Fig. 10.7(b). Note that in the one-dimensional scalar field theorywith degenerate minima of the scalar potential, this solution is called “kink”. Inview of (10.12), the free energy (10.10) of the wall reads

FW = 4πR2μ,

where

μ =∫ 〈φ〉T

0

√2 [Veff (φ) − V−]dφ. (10.13)

Note that the surface tension μ is finite in the limit ΔV → 0.The expression (10.2) for the free energy of the bubble, as well as the analysis

of the field behavior near the wall are valid when the wall thickness is small com-pared to the bubble size R, i.e., thin-wall approximation is applicable. Accordingto (10.3), it really works if the difference of free energies ΔV is a small parameter.Otherwise, the configuration of the critical bubble should be obtained by findingthe extremum (saddle point configuration) of the free energy functional (10.9) withthe only boundary condition φ(r → ∞) = 0. Details can be found in book [117].

Problem 10.1. Check the formulas (10.12), (10.13).

Problem 10.2. Let the effective potential be

Veff (φ) =λ

4φ2 (φ − v)2 − εφ2,

where λ, v and ε are positive parameters. Find the conditions at which the thin-wallapproximation is valid. Find the surface tension and wall thickness in the thin-wall


Fig. 10.8 Shape of scalar potential with two non-degenerate minima.

approximation, as well as the size of the critical bubble Rc; estimate the probability ofnucleation of a bubble of the new phase inside the phase with φ = 0 in the thin-wallapproximation at temperature T .

Let us make a brief comment on the false vacuum decay at zero temperature. We havein mind scalar field models in which the scalar potential (at zero temperature) has a localminimum (e.g., at φ = 0), i.e., it has the form shown in Fig. 10.8. The state in which thefield expectation value is spatially homogeneous and equal to zero is metastable; it is thefalse vacuum. False vacuum decay also occurs via spontaneous formation of bubbles ofthe new phase, but in contrast to the medium at finite temperature, the bubble emerges notdue to thermal fluctuations, but as a result of tunneling process [118–120]. The descriptionof tunneling in the semiclassical approximation is given, for example, in book [117]. Inweakly coupled theories, the probability of the bubble nucleation is exponentially small,

Γ ∝ e−const

α ,

where α is small coupling constant. Finally, in a certain range of temperatures the bubbleformation is dominated by a combination of thermal fluctuation and tunneling.

10.2 Effective Potential in One-Loop Approximation

In accordance with (10.1), the effective potential is the free energy density of theplasma in a state where the average Higgs field takes one and the same value φ

everywhere in space. The free energy F of the system is related to its energy E andentropy S by the thermodynamical relation F = E−TS, so that for the free energydensity we have

f = ρ − Ts,

10.2. Effective Potential in One-Loop Approximation 229

where, as usual, ρ and s are energy density and entropy density, respectively. Weknow from Sec. 5.2 that entropy density is expressed in terms of energy density andpressure,

s =ρ + p

T,

hence the free energy density is8

f = −p.

So, to calculate the effective potential one needs to find the pressure of the systemunder the constraint that the average Higgs field is equal to φ in the whole space.

Let us consider theories with small couplings at temperatures of interest to us.For example, at temperature T ∼ 100GeV, typical for electroweak interactions, thecosmic plasma is composed of quarks, leptons, W±- and Z-bosons, Higgs bosons,as well as photons and gluons, whose couplings are small. In this Section we neglectthe interactions between particles in the cosmic plasma, i.e., we consider the freeenergy of the ideal gas of elementary particles. The free energy of the gas dependsnon-trivially on the average value φ of the Higgs field, since the latter determinesparticle masses, and, consequently, particle contributions to pressure. For reasonsgiven in Appendix D, the ideal gas approximation is called one-loop approximationin this context.

In this approximation, the pressure is a sum of the contributions of the homo-geneous field φ itself and of each type of particles and antiparticles,

f = Veff (T, φ) = V (φ) +∑

i

fi, (10.14)

where V (φ) is scalar potential entering the scalar field action9

Sφ =∫

[∂μφ∂μφ − V (φ)] d4x. (10.15)

The first term in (10.14) arises from the fact that the energy-momentum tensor fortime-independent and spatially homogeneous scalar field is

Tμν(φ) = gμν · V (φ),

i.e., the homogeneous scalar field gives a contribution to the pressure equal to p(φ) =T11 = T22 = T33 = −V (φ). Of course, this is a reformulation of the fact that thefree energy in vacuo coincides with the energy and its density equals V (φ) for

8The fact that the medium chooses the phase with the lowest free energy has a simple physical

interpretation: in this phase pressure is at maximum, and sufficiently large region of this phase,

spontaneously created inside the phase of lower pressure, will expand, “pushing away” the phase

of lower p.9In the Standard Model of particle physics, the Higgs field is a complex doublet with kinetic term

in Lagrangian given in (B.8). This explains the choice of the coefficient in front of the kinetic term

in (10.15). The relation between the field φ considered here and the Standard Model Higgs boson

is φ(x) =v+h(x)√

2.


homogeneous scalar field. In what follows, we assume that the scalar potential isgiven by the standard expression

V (φ) = λ

(φ2 − v2

2

)2

,

where v is the vacuum expectation value at zero temperature, λ is the Higgs self-coupling, and λ � 1.

The second term in (10.14) is the medium contribution in the ideal gasapproximation,

fi = −pi[T, mi(φ)],

where pi[T, mi(φ)] is the contribution to pressure coming from particles and antipar-ticles of i-th type, whose mass equals mi and depends on φ. According to Sec. 5.1,we have

fi = −pi = − gi

6π2

∫ ∞

0

k4dk√k2 + m2

i

1

e

√k2+m2

iT ∓ 1

, (10.16)

where gi is the number of spin states, upper (lower) sign corresponds to bosons(fermions). Contributions of heavy particles with mi � T are exponentially small,therefore the interesting case is mi � T .

Integral (10.16) cannot be evaluated analytically. We analyze it for the particularcase of high temperature, T � m, and make use of the expansion in m/T (sub-script i will be omitted wherever possible). This approach is called high-temperatureexpansion. In dimensionless variables

x = k/T and zi = mi/T,

the expression (10.16) takes the form

fi = − gi

6π2T 4 · I(zi)∓, I(z)∓ =

∫ ∞

0

x4dx√x2 + z2

1e√

x2+z2 ∓ 1. (10.17)

We are interested in the behavior of these integrals at small z.To the zeroth order in z, the contributions fi correspond to pressures of free

gases of massless particles (see Sec. 5.1); they do not depend on φ, and will beomitted. The integrand in Eq. (10.17) is a function of z2, so one might expect thatI(z) is a series in z2. The first term in this series is

I(z) = z2

(dI

dz2

)z2=0

= −z2

2

(∫ ∞

0

xdx

ex ∓ 1+∫ ∞

0

x2exdx

(ex ∓ 1)2

). (10.18)

The integrals here are finite, so the first non-trivial term in the high-temperatureexpansion is indeed quadratic in z. Performing the integration with the use of for-mulas given in the end of Sec. 5.1 (in the process, it is convenient to integrate thesecond term in (10.18) by parts), we obtain in this order

Veff (φ) = λ

(φ2 − v2

2

)2

+T 2

24

[ ∑bosons

gim2i (φ) +

12

∑fermions

gim2i (φ)

]. (10.19)


This expression immediately leads to the important conclusion that in theories likethe Standard Model, the symmetry is unbroken at high temperatures, although itis broken at T = 0.

Indeed, in these models particles acquire masses due to the scalar fieldcondensate,

mi(φ) = hiφ, (10.20)

where hi are coupling constants. The only exception is the Higgs boson itself; itscontribution to (10.19) is small, and we will neglect it. In the Standard Model, forquarks and charged leptons denoted by f , and for W±- and Z-bosons the relation(10.20) takes a particular form

mf (φ) = yfφ, MW (φ) =g√2φ, MZ(φ) =

√g2 + g′2√

2φ, (10.21)

where yf are Yukawa couplings, and g and g′ are gauge couplings (see Appendix Bfor notations and details). The vacuum value is φ = v/

√2, and we return to the

zero-temperature formulas for the particle masses (see Appendix B), namely,

mf =yf√2v, MW =

g

2v, MZ =

√g2 + g′2

2v. (10.22)

Let us first pretend that the effective potential contains the terms (10.19) only.Then the behavior of the effective potential near φ = 0 is

Veff (φ) =(−λv2 +

α

24T 2)

φ2 + λφ4 (10.23)

(φ-independent terms are omitted), where

α =∑

bosons

gih2i +

12

∑fermions

gih2i (10.24)

is a positive quantity. At low temperatures, the expression (10.23) has a minimumat φ �= 0 (symmetry is broken), while at high temperatures the only minimum isφ = 0, corresponding to the restored symmetry. Minimum at φ = 0 disappears andturns into maximum when the first term in (10.23) flips sign, which happens attemperature (the notation will be clarified later)

Tc2 = 2v

(6λ

α

)1/2

. (10.25)

In what follows, when discussing weakly coupled theories in general terms, weassume that the relevant couplings hi are small, h � 1, and the following order-of-magnitude relation holds,

λ ∼ h2.

The latter ensures that the Higgs boson mass mh ∼ √λv is of the same order as

masses of all other particles contributing noticeably to the effective potential. Withthis prescription, the estimate for the critical temperature reads

Tc2 ∼ v.

This follows from (10.24) in the case of not too large number of particle species.


In the Standard Model, the main contributions to α come from the heaviestparticles, W±- and Z-bosons and t-quark. Equations (10.20), (10.21) and (10.22)yield, in terms of zero-temperature masses,

α =2v2

(6M2

W + 3M2Z + 6m2

t

).

Here we used the fact that W+- and W−-bosons together have six polarizations,Z-boson has three polarizations, and t-quark with its antiparticle have four; also,t-quark has three color states. Recalling that the Higgs boson mass is

mh =√

2λv,

we arrive at the following one-loop expression for the critical temperature Tc2 inthe Standard Model,

Tc2 =(

6m2h

6M2W + 3M2

Z + 6m2t

)1/2

· v = 121 ·( mh

100 GeV

)GeV (10.26)

(we remind that MW = 80.4GeV, MZ = 91.2GeV, mt ≈ 175GeV, v = 247GeV,see Appendix B).

If the high-temperature expansion of the integrals (10.16) were really a series inz2 ≡ m2(φ)/T 2, one would conclude that we are dealing with the 2nd order phasetransition: corrections of the fourth and higher orders in φ are small comparedto terms written in (10.23) (see below), and the position of the minimum of theexpression (10.23) smoothly moves from φ = 0 towards large φ as temperaturedecreases from Tc2 to zero. In other words, the behavior of the expression (10.23)corresponds to the right plot in Fig. 10.3. However, integrals (10.16) are not analyticin z2, and the one-loop effective potential actually corresponds to the 1st order phasetransition. The lack of analyticity can be seen from the behavior of contributionsto the integrals (10.17) coming from the low-momentum region k � T , i.e., x � 1(infrared region). For small z and x, the expansion of the exponential terms in theintegrand gives

I(IR)− =

∫ Λ

0

x4dx

x2 + z2, bosons

(10.27)

I(IR)+ =

12

∫ Λ

0

x4dx√x2 + z2

, fermions,

where Λ � 1 is a fictitious parameter separating the infrared region. Formalexpansion of the integrands in these formulas in z2 would result in the order z4

contributions of the form

z4

∫ Λ

0

dx

x2, bosons

z4

∫ Λ

0

dx

x, fermions.


The first of them linearly diverges at the lower limit of integration, while the secondone diverges logarithmically. We can therefore expect that besides the above termsof order z2, bosons and fermions give contributions of order z3, and z4 log z, respec-tively. Contributions of the latter type give the terms in the effective potential ofthe form

m4i (φ) log

φ

T= h4

i φ4 log

φ

T. (10.28)

Most of them are of little interest, because at λ � h4i (which holds for not too

small Higgs boson mass), they are small compared to λφ4 coming from the scalarpotential V (φ) (only contributions due to t-quark are important in determiningcertain parameters of the phase transition). On the contrary, terms of order z3

are very important: it is due to these terms that the transition (in the one-loopapproximation) is of the 1st order.

To calculate the term of order z3 in the boson integral I−, we divide this integralinto two parts using the fictitious parameter Λ,

I− =∫ ∞

Λ

x4dx√x2 + z2

1e√

x2+z2 − 1+ I

(IR)− .

The first term is analytic in z2, while for the second term the approximation (10.27)is sufficient, i.e.,

I(IR)− =

∫ Λ

0

(x2 − z2

)dx + z4

∫ Λ

0

dx

x2 + z2.

The first term here is again analytic in z2, while the second term gives the contri-bution of order z3 we are after,

I(IR)− −→ π

2z3 + O

(z4

Λ

).

As a result, the effective potential in the one-loop approximation reads

Veff (φ) = λ

(φ2 − v2

2

)2

+T 2

24

( ∑bosons

gim2i (φ) +

12

∑fermions

gim2i (φ)

)

− T

12π

∑bosons

gim3i (φ) + O

(m4

i (φ) logmi(φ)

T

).

In models where particle masses are related to the Higgs expectation value by(10.20), this expression is rewritten as

Veff (φ) =α

24(T 2 − T 2

c2

)φ2 − γTφ3 + λφ4, (10.29)

where the parameter γ is positive and equal to

γ =1

12π

∑bosons

gi|hi|3 =√

26π

∑bosons

gi

(mi

v

)3

. (10.30)

Here we use the notations introduced in (10.24) and (10.25).


Problem 10.3. Calculate the terms of order φ4 log φT in the high-temperature

expansion of the effective potential, using the relation (10.20). Show that at λ ∼ h2i

(the Higgs boson mass is comparable to those of other particles) and hi � 1 (thecouplings are small), these terms are small compared to those written in (10.29) inthe entire interesting range of φ, namely, 0 < φ � v.

The behavior of effective potential (10.29) corresponds to the left plot inFig. 10.3, i.e., to the 1st order phase transition. Extrema of the effective potentialare determined by the equation

∂Veff

∂φ=

α

12(T 2 − T 2

c2

)φ − 3γTφ2 + 4λφ3 = 0. (10.31)

At temperature Tc0 such that

9γ2T 2c0 =

4αλ

3(T 2

c0 − T 2c2

),

the effective potential acquires two extrema at φ �= 0: minimum and maximum.This temperature exceeds Tc2 only slightly: for λ ∼ h2 we have γ ∼ h3, αλ ∼ h4,and thus

T 2c0 − T 2

c2

T 2c2

=27γ2

4αλ∼ h2. (10.32)

Due to the small difference of T and Tc2 in the interesting temperature range, onecan replace T by Tc2 in the second term in (10.31). It is important that the secondminimum of the effective potential (if φ = 0 is treated as the first minimum) appearsat nonzero φ = Φc(Tc0),

Φc0 = Φc(Tc0) =3γ

8λTc0. (10.33)

As temperature decreases, the second minimum becomes deeper, and the values ofthe effective potential at this minimum and at the minimum φ = 0 become equal attemperature Tc1, such that both Eq. (10.31) and equation10 Veff = 0 are satisfied.The solution to the latter system of equations gives the first critical temperatureTc1,

T 2c1 − T 2

c2

T 2c2

=6γ2

αλ,

and the position of the second minimum at this temperature,

Φc1 = Φc(Tc1) =γ

2λTc1. (10.34)

In view of (10.30), this value is much smaller than the critical temperature,Φc(Tc1) ∼ hTc1 in the weak coupling limit h � 1. Once temperature decreasesdown to the second critical temperature Tc2, the minimum of the effective potential

10Recall that we dropped φ-independent terms in the effective potential, i.e., we set

Veff (φ = 0) = 0.


at φ = 0 disappears and turns into maximum. At that instant the second minimumis at

Φc2 = Φc(Tc2) =3γ

4λTc2. (10.35)

Thus, the evolution of the one-loop effective potential shown in the left plot inFig. 10.3 takes place in weakly coupled theories in a narrow temperature intervalnear the critical temperature (10.25), Tc2 ≤ T ≤ Tc0, where Tc0 is defined by (10.32).Immediately after the phase transition, the expectation value of the Higgs field ismuch smaller than its vacuum value,

Φc ∼ γ

λTc2 ∼ hTc2 ∼ hv � v.

Note that the high-temperature expansion is justified in weakly coupled theories,since

mi(Φ) = hiΦ ∼ h2Tc2 � Tc2.

Another point concerns the latent heat of the phase transition. At T = Tc2, the valueof the effective potential at the minimum (10.35), relative to its value at φ = 0, is

Veff (Tc2, Φc(Tc2)) = − 27256

γ4

λ3T 4

c2.

This value (with opposite sign) gives the maximum latent heat of the transition,which is roughly equal to

−Veff ∼ h6T 4c2 � T 4

c2. (10.36)

Thus, the energy released during the phase transition is small compared to theenergy of particles in the plasma, whose density is of order T 4

c . The cosmic plasmais heated by the phase transition only slightly.

Problem 10.4. Using the expression (10.29) for the effective potential and theresults of Sec. 10.1, find the wall profile of the critical bubble and the surface tensionμ at temperature close to Tc1. Find the temperature at which the bubble nucleationrate becomes of the order of the Hubble parameter, i.e., the Universe “boils”. Whatis the ratio of the bubble size and the Hubble length? Find the ratio of the transitionlatent heat released at this moment to the energy density of particles in the plasma,

thus refining the estimate (10.36). Give numerical estimates for the Standard ModelHiggs boson of mass 40GeV (experimentally forbidden) and 120GeV.

High-temperature expansion of the integrals (10.16) obviously does work at smallvalues of φ. Therefore, the result that within the one-loop approximation the phasetransition is of the 1st order, is justified from this point of view. Of course, the useof the high-temperature expansion is not necessary for one-loop calculations. Inte-grals (10.16) are easily computed numerically, and thus the exact one-loop effectivepotential is straightforwardly obtained. The corresponding graphs for the StandardModel are shown in Figs. 10.9 and 10.10. It is seen that at φ � T , results for


50 100 150 200 250 300

5

10

15

20

20 40 60 80 100 120 140

0.2

0.4

0.6

20 40 60 80 100 120 140

0.05

0.1

0.15

0.2

10 20 30 40 50 60

-0.0005

0.0005

0.001

0.0015

Fig. 10.9 One-loop effective potentials at different temperatures, obtained numerically (solid

lines) and analytically within the high-temperature expansion (dashed lines) for the Higgs boson

mass mh = 50GeV. Note the difference in scales of the axes for different temperatures.

50 100 150 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10 20 30 40 50

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

0.00014

10 20 30 40 50

0.00002

0.00004

0.00006

5 10 15 20

-1.10-6-5.10-7

5.10-71.10-6

1.5.10-62.10-6

50 100 150 200

-0.01

-0.005

0.005

0.01

0.015

0.02

50 100 150 200

-0.05

-0.025

0.025

0.05

0.075

Fig. 10.10 The same as in Fig. 10.9, but for the Higgs boson mass mh = 150GeV.

the effective potential obtained by using the high-temperature expansion are inreasonable agreement with the exact one-loop calculation. At the same time, theresults for a number of characteristics of the phase transition agree only qualita-tively, within a factor of 3 or so (see Fig. 10.11). For analytical estimates of thesecharacteristics, the omitted higher order terms in m/T are important. Indeed, theratios Φc(Tc1)/Tc1 and Φc(Tc2)/Tc2 are inversely proportional to the coefficient infront of φ4 in the effective potential (see formulas (10.34), (10.35)), and this coeffi-cient gains a significant contribution when the omitted higher-order terms in (10.28)are taken into account.


25 50 75 100 125 150

0.5

1

1.5

2

2.5

3

3.5

25 50 75 100 125 150

1

2

3

4

Fig. 10.11 Comparison of numerical results (black lines) with the analytical results obtained

using the high-temperature expansion (gray line) for the one-loop values of (a) Φc(Tc1)/Tc1 and

(b) Φc(Tc2)/Tc2 at various Higgs boson masses mh .

Problem 10.5. Making use of the high-temperature expansion, calculate the valuesof Φc(Tc1)/Tc1 and Φc(Tc2)/Tc2 taking into account the contributions of the form(10.28). Check that the results are in better agreement with exact numerical results.

Much more serious issue is the applicability of the one-loop approximation itself,and, more generally, the applicability of the perturbation theory at finite temper-atures. These issues are discussed in the next Section. An important conclusionis that in the Standard Model, the one-loop approximation does not describe thetransition correctly, given the experimental constraint mh > 114GeV.

Considering the Standard Model and its extensions, one should bear in mindanother circumstance. The Higgs field φ itself is not gauge invariant in the StandardModel. Gauge-invariant, physical quantities are products like φ†φ (more pre-cisely, H†H , see Appendix B), but these are invariant under all symmetries ofthe Lagrangian, and therefore cannot serve as order parameters. In other words,“phases” of “broken” and “unbroken” symmetry most likely11 are not really distin-guishable [122, 123]. If so, smooth crossover may occur at high temperatures insteadof phase transition. The picture of smooth crossover is indeed observed in latticestudies of the Standard Model at high temperatures for mh > 80GeV [115].

Though in the Standard Model the electroweak phase transition is certainly notof the 1st order, in some extensions with additional bosons coupled to the SM Higgsboson this is not true. In particular, the phase transition can be of the 1st orderfor extended Higgs sector, where several scalar fields acquire vacuum expectationvalues. The realistic example is the two-doublet Higgs model where one Higgs fieldgives masses to the up quarks, while another does to the down quarks and chargedleptons. In this case the discussion of this Section can be straightforwardly modified.New bosons contribute to the effective potential, most importantly to its cubic part.

11The reservation here has to do with statement in literature on the existence of “hidden” sym-

metry and nonlocal order parameter in the Standard Model [121].


The parameter analogous to γ in (10.29) is larger than in the Standard Model, andthe electroweak phase transition can be of the 1st order. Thus, discussion of thisSection is directly applicable in this situation. We consider a concrete example inSec. 11.5.

10.3 Infrared Problem

In this Section we show that the finite-temperature perturbation theory is not alwaysapplicable, even if couplings are small [111, 124]. Let us discuss this issue at thequalitative level first. The physical reason for failure of the perturbation theory isthat the distribution function of bosons fB = [exp(ω/T )−1]−1 is large at low particleenergies ω. Owing to that, the interaction between the bosons at low energies isenhanced in the medium. Indeed, for small momenta and masses of particles, p � T ,m � T , the bosonic distribution function has the form

fB(p) =T

ωp.

In quantum field theory, this means that number densities of low-momentumparticles,

〈〈a†pap′〉〉 = fB(p)δ(p − p′),

are large in comparison with the commutator

〈〈[ap, a†p′ ]〉〉 = δ(p − p′),

provided that the particles are light, m(T ) � T . Therefore, the infrared part of alight boson field Φ is classical field. The linearized expression for it is

Φ(x) =1

(2π)3/2

∫d3p√2ωp

(e−ipxap + e−ipxa†

p

),

where ap, a†p can be treated as c-numbers at low momenta. Hence, omitting

numerical factors, we obtain for the field fluctuation

〈〈Φ2(x)〉〉 =∫

d3p

ωpfB(p).

We are interested in the contribution from the infrared (IR) region,

〈〈Φ2(x)〉〉IR =∫

p2dp

ωp

T

ωp= T

∫dp

p

p3

ω2p

.

Thus, the amplitude of the field fluctuations with momenta of order p is estimated as

〈〈Φ2(x)〉〉p = Tp3

ω2p

.

The field is in the linear regime, if the quadratic (free) contribution to the freeenergy is larger than the contribution due to interactions. At p2 � m2, the free

10.3. Infrared Problem 239

contribution can be estimated as (∇Φ)2 = p2Φ2. Following the convention adoptedin this Chapter, we choose the self-interaction of the field Φ as h2Φ4, where h isa small coupling. Comparison of the interaction and free contributions to the freeenergy gives

h2Φ4

(∇Φ)2∼ h2T 2p6

ω4p

(p2T

p3

ω2p

)−1

∼ h2Tp

ω2p

. (10.37)

This implies for the massless field

h2Φ4

(∇Φ)2∼ h2T

p.

The field is in the linear regime if this ratio is small, which is true only at p � h2T .In the infrared region, p � h2T , on the contrary, the regime is strongly non-linear,and the perturbation theory is inapplicable. For massive field, the ratio (10.37) issmall for all momenta, only if mΦ � h2T . Otherwise, the infrared region is at strongcoupling. Note that this is the strong coupling regime in the classical field theoryat finite temperature.

In gauge theories, this argument applies to spatial components of non-Abeliangauge field, whose interaction contains, in particular, the commutator term of thetype g2A4. Thus, instead of h2 we have g2, and the condition of applicability of theperturbation theory has the form

MW (T ) � g2T.

For the component A0 the argument is inapplicable because of the Debye massmD ∼ gT � g2T .

Thus, the perturbative calculation of the effective potential is justified whenMW (T ) � g2T , i.e.,

φ � gT. (10.38)

We see that the effective potential cannot be perturbatively calculated near φ = 0.Moreover, the description of the first order phase transition presented in the previousSection is valid only when the position of the second minimum of the effectivepotential given by Eqs. (10.33), (10.34) or (10.35), satisfies Φc � gTc. In terms ofcouplings, this restriction has the form

λ � γ

g.

Given that γ ∼ g3 (see (10.30)), and omitting numerical factors (they actuallywork towards stronger restriction), we get from this λ � g2, i.e., in terms of thezero-temperature mass

m2h � M2

W . (10.39)

It is difficult to refine this estimate. In other words, it is impossible to findanalytically up to what values of mh the phase transition is of the 1st order. Still,


the restriction (10.39) suggests that the one-loop results of Sec. 10.2 most likelyhave nothing to do with reality in the Standard Model, taking into account theexperimental constraint mh > 114GeV.

Since the difficulty is due to interactions of low-momentum particles, it is calledinfrared problem. The main role here is played by the self-interactions of gaugebosons inherent in any non-Abelian gauge theory.

Let us see at the formal level that the applicability of the perturbation theory forcalculating the effective potential is indeed limited to the values of the Higgs field obeying(10.38). We are interested in contributions to the effective potential associated with self-interactions of gauge bosons. These contributions, as in Sec. 10.2, depend on the field φthrough the masses of vector bosons (10.21),

M(φ) ∼ gφ,

where we omitted indices labeling W - and Z-bosons, neglected the difference between MW

and MZ and skipped factors of order 1.As shown in Appendix D, the effective potential is given by the functional integral

e−βV (φ) =

ZDAμe−S(β)[A], (10.40)

where we neglected all but gauge fields. Here β = T−1, the functional S(β)[A] is theEuclidean action in the Euclidean time interval 0 ≤ τ ≤ β,

S(β)[A] =

Z β

0

dτ

Zd3x

»1

4F b

μνF bμν +

M2(φ)

2Ab

μAbμ

–, (10.41)

summation over indices is performed with the Euclidean metric. The integration in (10.40)is over fields Ab

μ(x, τ ) periodic in τ with period β.

Because of the periodicity, the field Abμ(x, τ ) can be represented as a discrete sum

Aμ(x, τ ) =1√β

aμ(x) +X

n=±1,...

1√β

a(n)μ (x)eiωnτ , (10.42)

where

ωn =2πn

β≡ 2πnT

are the Matsubara frequencies, we omitted the group index and extracted explicitly theterms in (10.42) with zero Matsubara frequency. Upon substituting expansions (10.42) into(10.41), we arrive at the action of 3-dimensional Euclidean theory with an infinite set of

fields aμ(x), a(n)μ (x). We are interested in the infrared region, more precisely, in the region

of spatial momenta

|p| gT.

At these momenta, only light 3-dimensional fields are relevant, whose masses are much

smaller than gT . Note first, that a0, a(n)0 are not light fields: they acquire the Debye mass

mD ∼ gT (see Sec. D.5). Therefore, the fields a0, a(n)0 can be ignored. Now, the fields a

(n)i

with n �= 0 are also heavy: the term F a0iF

a0i in the original Lagrangian leads to the term in

the action Xn=±1,±2,...

Zd3xω2

na(n)i a

(−n)i ,

i.e., the mass term in the 3-dimensional theory with large masses |ωn|. As a result, theonly light fields are ai(x), i.e., homogeneous in Euclidean time components of spatial

10.3. Infrared Problem 241

vector potentials Ai. Substituting Ai(x) = β−1/2ai(x) into the action (10.41), we obtainthe effective 3-dimensional action describing the infrared properties of the theory at finitetemperature,

Seff =

Zd3x

„1

4fb

ijfbij +

1

2M2(φ)ab

iabi

«, (10.43)

where

f bij = ∂ia

bj − ∂ja

bi + g

√Tfbcdac

iadj (10.44)

and f bcd are structure constants of the non-Abelian gauge group (in the case of gauge

group SU(2) these are fbcd = εbcd). The factor T 1/2 = β−1/2 emerges in (10.44) due to thenormalization in (10.42), chosen in such a way that quadratic part of the 3-dimensionalaction (10.43) has canonical form.

As a digression, we make an observation regarding fermions. Within the Euclideanapproach all 3-dimensional fermions are heavy: they are antiperiodic in β, and theirMatsubara frequencies ωn′ = 2πTn′, n′ = ± 1

2 ,± 32 , . . . are all of order T . Therefore,

fermionic fields are insignificant for the infrared properties of the theory. This also followsfrom the consideration given in the beginning of this Section: the distribution function offermions fF = [exp(ω/T ) + 1]−1 does not diverge as ω → 0.

Coming back to the action (10.43), we note that it is the action of 3-dimensional vectorfields with mass M(φ) and dimensionful coupling

g(3) = g√

T . (10.45)

The ratio of dimensionful quantities [g(3)]2/M(φ) is the effective coupling constant. Indeed,in the framework of perturbation theory, the effective potential (more precisely, −βV (φ))is given by the sum of one-particle irreducible diagrams without external lines (seeAppendix D) of the type shown in Fig. 10.12. For our purpose, these are diagrams in3-dimensional theory with the action (10.43). Diagrams with n loops give contributions

to βVeff proportional to [g(3)]2(n−1). On dimensional grounds, these contributions are oforder

βV(n)eff ∼ [g(3)]2(n−1)

[M(φ)]n−4= [M(φ)]3

„[g(3)]2

M(φ)

«n

.

Fig. 10.12 An example of a diagram contributing to the effective potential.


Thus, the expansion parameter in the perturbation theory is [g(3)]2/M(φ), i.e., theperturbation theory is applicable at12 M(φ) � g2T . This confirms the result of the quali-tative analysis given in the beginning of this Section.

To conclude this Section, we make a few remarks. First, we note that at momenta andenergies small compared to g2T , the theory is effectively reduced to a 3-dimensional gaugetheory (in general, with scalar fields) with gauge coupling (10.45). Such a theory exhibitsquite non-trivial nonperturbative properties. These have been explored in a number oflattice studies.

Second, for the fields Ai(x) independent of Euclidean time τ , the partition function(10.40) is

e−βF =

ZDAi(x)e−βH[Ai],

where

H[Ai] =

Zd3x

„1

4F b

ijFbij + · · ·

«,

and dots denote terms involving possible scalar fields (here we have preserved the four-dimensional normalization of the vector field). The expression on the right side is thepartition function of classical four-dimensional gauge field theory (in general, with scalarfields) with the Hamiltonian H[Ai]. Thus, our analysis leads to understanding [125, 126]that the behavior of quantum theory at high temperatures at length and time scalescomparable to or in excess of (g2T )−1, basically coincides with the behavior of classicalfield theory.13 This fact is useful for studying static, and especially dynamical (related toevolution in time) properties of the theory at high temperatures.

12In the case of fields a0 and a(n)i the role of M(φ) is played by the Debye mass mD ∼ gT

and ωn ∼ T , respectively, so that these fields are indeed unimportant in the infrared region for

sufficiently small g.13There is a subtlety associated with the need for ultraviolet cutoff in the classical field theory:

for the Rayleigh–Jeans distribution, the energy density diverges in the ultraviolet region.

Chapter 11

Generation of Baryon Asymmetry

As we discussed in Secs. 1.3 and 5.2, there are baryons and no antibaryons in thepresent Universe.1 The present baryon abundance is characterized by the baryon-to-photon ratio ηB = nB,0/nγ,0 6.2 · 10−10. Another way to quantify the baryonasymmetry is to use the baryon-to-entropy ratio (see Sec. 5.2)

ΔB =nB − nB

s 0.88 · 10−10. (11.1)

This ratio stays constant at the hot stage of the cosmological evolution providedthere are no processes that violate baryon number or produce large entropy. One ofthe problems for cosmology is to explain the baryon asymmetry [34, 35]. Accordingto our discussion in Secs. 1.5.5 and 5.2, the initial state is natural to assumebaryon-symmetric.

Let us stress right away that there is no unambiguous answer to the questionof the origin of the baryon asymmetry. It could be produced either at hot stage orat earlier post-inflationary reheating epoch. We discuss reheating in the accompa-nying book, and here we consider several mechanisms that could work at hot stage.Before discussing concrete mechanisms, let us study general conditions necessaryfor baryogenesis.

Besides the baryon asymmetry, there can be, and most likely there is, leptonasymmetry. If it is not large, its value does not seem to be possible to measure. Leptonasymmetry resides today in the excess of neutrinos over antineutrinos, or vice versa(electron number density equals that of protons by electric neutrality), while measuringrelic neutrino abundance will be impossible in foreseeable future.

11.1 Necessary Conditions for Baryogenesis

Baryon asymmetry generation at a certain cosmological epoch is possible only ifthree conditions are met. These are dubbed Sakharov conditions. Namely, three

1Modulo exotica like possible existence of compact objects made of antimatter. The latter are

predicted in some models [127] and do not contradict observational data [128].

243

244 Generation of Baryon Asymmetry

properties must hold simultaneously:

(1) Baryon number non-conservation (with qualification, see below)(2) C- and CP -violation.(3) Thermal inequilibrium.

The fact that the first condition is necessary to produce asymmetry is self-evident.Now, if C- or CP -invariance is exact, processes with quarks and leptons occur inthe same way as processes with their antiparticles, and no asymmetry is generated.Hence the second condition.

The latter result follows, at formal level, from the evolution law of the density matrix,

ρ(t) = e−iH(t−ti)ρ(ti)eiH(t−ti), (11.2)

where H is the Hamiltonian of the system, ti is initial moment of time. C- or CP -invariancemeans that the corresponding unitary operator UC or UCP commutes with the Hamil-tonian. Say, for CP -invariance

UCP HU−1CP = H.

This gives

UCP ρ(t)U−1CP = ρ(t),

provided that the initial state is symmetric, i.e., UCP ρ(ti)U−1CP = ρ(ti). Together with the

fact that the baryon number operator is CP -odd, UCP BU−1CP = −B, the latter property

gives

〈B(t)〉 = Tr[Bρ(t)] = 0.

The medium remains baryon symmetric.

Finally, the third condition is also fairly obvious. In thermal equilibrium withrespect to baryon number violating reactions, the system is in a state with zerobaryon chemical potential, i.e., zero baryon number density. Baryon asymmetrytends to get washed out, rather than generated, as the system approaches thermalequilibrium.

The latter conclusion needs qualification. It would be literally valid if baryonnumber were the only relevant quantum number. We will see that lepton numbersare also important in the Standard Model of particle physics: baryon number itselfis not conserved at temperatures above T ∼ 100 GeV, while its linear combinationswith lepton numbers are. In particular, conserved quantum number is (B−L) whereL = Le + Lμ + Lτ is the total lepton number. In thermal equilibrium at non-zero(B − L) both baryon and lepton numbers do not vanish,

B = C · (B − L), L = (C − 1) · (B − L), (11.3)

where the constant C is of order 1 (but somewhat less than 1). This property is usedin leptogenesis mechanisms: they produce lepton asymmetry (and hence (B − L))at very high temperature by processes beyond the Standard Model, and then thisasymmetry is partially reprocessed into baryon asymmetry by electroweak physics.

11.1. Necessary Conditions for Baryogenesis 245

Each of the three Sakharov conditions, generally speaking, is associated withone or another small parameter. Since all three should work at the same time, thenet baryon asymmetry is naturally small in most models. We will see how theseconditions are fulfilled in concrete mechanisms.

To end this Section, let us see that irrespective of the baryogenesis, the veryfact that baryon asymmetry exists implies that there is no relic antimatter in theUniverse.

To estimate the relic abundance of antibaryons, let us write the equation forantibaryon number density nB. Baryon number changes in comoving volume dueto annihilations with baryons and due to the inverse process of baryon-antibaryonpair creation. The annihilation contributes

d(nBa3)ann

dt= −Γann · nBa3,

where

Γann = σann · v · nB (11.4)

is the annihilation probability for an antibaryon per unit time. Here σann is theannihilation cross section (we disregard for the sake of argument the fact that anni-hilation cross sections of neutrons and protons are different); in non-relativistic limitσann = σ0/v (see Sec. 9.3), and

σ0 ∼ 1 Fm2 = 10−26 cm2 ∼ 25 GeV−2. (11.5)

nB = ηB · nγ in (11.4) is the baryon number density.In thermal equilibrium the number of baryons does not change (modulo cos-

mological expansion), so pair production and annihilation of antibaryons occur atthe same rate — the same number of events per unit volume per unit time. Thisobservation gives for pair production rate

d(nBa3)prod

dt= σ0 · neq

Bneq

Ba3,

where neqB and neq

Bare equilibrium baryon and antibaryon number densities. Hence,

the Boltzmann equation for the antibaryon number density is

d(nBa3)dt

= −σ0 · (nBnBa3 − neqB neq

B a3). (11.6)

Problem 11.1. Obtain Eq. (11.6) by making use of Eq. (5.56). Hint: Recall thatΓann = τ−1

ann in Eq. (11.4) is the annihilation rate per antibaryon (i.e., τann is thelifetime of antibaryon in the medium). Make use of Eq. (5.48) and the fact thatbaryon number density is much greater than that of antibaryons.

For large annihilation rate Γann the antibaryon density is close to equilibriumone. This situation occurs, as we will see, at high enough temperatures. Number of


baryons freezes out when the production and annihilation rates become too low tokeep the system in equilibrium. The freeze out condition is∣∣∣∣d(neq

Ba3)

dt

∣∣∣∣ ∼ σ0 · neqB

neqB

a3. (11.7)

It remains to find the equilibrium antibaryon density as function of temperature.Since chemical potentials of particles and antiparticles differ only by sign in thermalequilibrium, we write at T � mp (we neglect the difference between proton andneutron masses and do not write spin factors)

neqB =

(mpT

2π

)3/2

e−mp−μB

T ,

neqB

=(

mpT

2π

)3/2

e−mp+μB

T .

(11.8)

Note that at T � mp the baryon density neqB is not exponentially small only for

μB = mp + O(T ). Using nB = ηB · nγ , we find from (11.8)

neqB

=1

nB

(mpT

2π

)3

e−2mp

T ∼ m3p

ηB

· e− 2mpT .

The right hand side rapidly changes with temperature, so∣∣∣∣d(neqB a3)dt

∣∣∣∣ =∣∣∣∣a3 dneq

B

dt

∣∣∣∣ .We now use |T /T | = H(T ) = T 2/M∗

Pl and obtain from (11.9) that∣∣∣∣a3 dneqB

dt

∣∣∣∣ ∼ a3 mp

TH · neq

B.

The freeze out relation (11.7) takes the form

mp

TH ≡ mp

T

T 2

M∗Pl

∼ Γann ∼ σ0ηBT 3.

In this way we obtain freeze out temperature,

T ∼(

mp

M∗Pl · ηB · σ0

)1/2

∼ 10 keV. (11.9)

The antibaryon abundance is fantastically small at this temperature: we findfrom (11.9)

nB ∼ 10−105, (11.10)

no matter in which units. The visible Universe contains no single relic antibaryon.

Clearly, one naturally questions the validity of statistical physics methods, which weused in our calulation, for the system with so small number of antibaryons. Still, there isno doubt that all antibaryons annihilate away in baryon-asymmetric Universe.

Problem 11.2. Solve the Boltzmann equation (11.6) in quadratures. Compute theintegral by saddle point method and find the temperature that gives the largest

11.2. Baryon and Lepton Number Violation in Particle Interactions 247

contribution into the present value of nB, thus refining the estimate (11.9). Findthe present value of nB. Is it consistent with (11.10)?

Problem 11.3. Find the relic positron abundance at the present epoch.

11.2 Baryon and Lepton Number Violation in Particle Interactions

In this Section we discuss two mechanisms of baryon number violation. One indeedworks in Nature, since it exists already in the Standard Model [129]. Another isinherent in Grand Unified Theories, GUTs (see, e.g., Ref. [130]). Even though thereis no direct experimental evidence for Grand Unification, this hypothesis is veryplausible. Finally, we will see in what follows that lepton number violation is alsorelevant for baryogenesis; we discuss one of the mechanisms of this violation at theend of this Section.

11.2.1 Electroweak mechanism

Baryon and lepton number violation are non-perturbative in the Standard Model;they are not visible in Feynman diagrams. We will only give here sketch of whatis going on; interested reader may find details in the book [117] or in appropriatereviews.

The phenomenon we are going to discuss is called ’t Hooft effect [93]; it isdue to gauge interactions corresponding to the subgroup SU(2)W of the StandardModel group SU(3)c × SU(2)W × UY (see Appendix B). These interactions involveleft quarks and leptons. The classical Lagrangian is invariant under common phaserotations of all quark fields; this symmetry would give rise to baryon number con-servation. It is also invariant under phase rotations of leptons of each generationseparately (we neglect neutrino masses and mixing at this point); these would corre-spond to the conservation of three lepton numbers Ln, n = e, μ, τ , see Appendix B.However, the corresponding currents jB

μ , jLnμ are anomalous at the quantum level,

∂μjB

μ = 3g2

16π2V μν aV a

μν , (11.11)

∂μjLnμ =

g2

16π2V μν aV a

μν , n = 1, 2, 3, (11.12)

where V aμν = ∂μV a

ν − ∂νV aμ + gεabcV b

μ V cν is the field strength of the SU(2)W gauge

field, V aμν = 1

2εμνλρV λρa is the dual tensor, g is SU(2)W gauge coupling. The reasonfor this anomaly is that left and right fermions interact with the field V a

μ in differentways2 (in fact, right fermions do not interact with V a

μ at all).

2In QCD, left and right quarks interact with gluon field in the same way, and baryon number is

conserved in strong interactions.


Equations (11.11) and (11.12) show that baryon and lepton numbers are notconserved,3 if there emerge non-zero gauge fields in vacuo or in medium,

ΔB = B(tf ) − B(ti) =∫ tf

ti

dt

∫d3x∂μjB

μ = 3∫ tf

ti

d4xg2

16π2V μν aV a

μν , (11.13)

where ti and tf denote initial and final time. Similar relation is valid for each of thelepton numbers. Baryon number violation requires strong fields, V a

μν ∝ 1g , making

the integral (11.13) different from zero (this integral takes integer values only).Energy of very strong fields is proportional to 1

g2 . Thus, we conclude that baryonand lepton number violation occur when the system overcomes energy barrier, seeFig. 11.1. The estimate for its height is

Esph ∼ MW

g2,

where the factor MW is inserted on dimensional grounds.4 The calculation of thebarrier height (sphaleron energy) gives [131]

Esph =2MW

αW

B

(mh

MW

),

Fig. 11.1 Schematic plot of static energy as functional of classical gauge and Higgs fields. The hor-

izontal line corresponds to (infinite-dimensional) space of all field configurations {V, φ}. Absolute

minima labeled by an integer n = 0,±1,±2, . . . are pure gauge configurations of zero energy, in

which the fields V(x), φ(x) have different topological properties (topologically distinct vacua).

The integral in (11.13) is equal to 1 for the fields that evolve from the vacuum with topological

number n to the vacuum (n+1) as time runs from ti to tf . This integral is zero for fields evolving

near one vacuum. The “maximum” with energy Esph is in fact a saddle point: energy decreases

along one direction in the configuration space and increases along all other directions.

3We do not discuss the actual physics leading to non-conservation; see in this regard the book

[117] and references therein.4The reason for notation Esph is as follows. The height of energy barrier is equal to energy

of saddle point configuration, which extremizes the static energy. This configuration is called

sphaleron, from Greek σφαλερoν, unreliable, ready to fall.


where

αW ≡ g2

4π,

the function B(mh/MW ) takes values in the interval

B = 1.56,mh

MW

� 1; B = 2.72,mh

MW

� 1,

and we neglect weak dependence on sin θW . Thus, the barrier height in the StandardModel is 7.5−13 TeV.

At zero temperature and zero fermion density the transmission through theenergy barrier is only possible via quantum tunneling. This tunneling is describedby instanton [132]. The tunneling probability is exponentially small,

Γ ∝ e−4π

αW .

Since αW ∼ 1/30, the suppression factor is extremely small, Γ ∝ 10−165. Baryonnumber violating processes practically do not occur in usual circumstances.

Another situation occurs at finite temperatures [129]. In that case, the barriermay be overcome via thermal jumps on its top (better to say, on the saddle point).At relatively low temperatures a naive estimate for the suppression factor in therate is given by the Boltzmann formula for configuration of energy Esph,

Γsph ∝ e−Esph

T .

This estimate, however, is incorrect at interesting temperatures when the sup-pression factor is not extremely small. The probability of the realization of a givenconfiguration is determined by its free energy, rather than energy. In our case themain effect is that the Higgs expectation value, and hence MW , depends on tem-perature, see Chapter 10. So, the better estimate is

Γsph = C · T 4e−Fsph(T )

T , (11.14)

where

Fsph(T ) =2MW (T )

αW

B

(mh

MW

), (11.15)

and one can use zero-temperature masses in the argument of the function B (in fact,the latter statement is not quite correct, but we will not need further refinement, inview of weak dependence of B on its argument). Γsph is the over-barrier transitionrate per unit time per unit volume, so we have written the factor T 4 in (11.14) ondimensional grounds. The pre-exponential factor C is dimensionless; it depends onthe Higgs expectation value and couplings. The most important in (11.14) is theexponential factor, so we will set

C ∼ 1

in our estimates.


Thus, at low enough temperatures the rate of electroweak baryon number vio-lation is given by (11.14). This formula does not work, however, at high tempera-tures, when Fsph � T and the exponential suppression is absent. The latter situationoccurs in unbroken phase,5 when 〈φ〉 = 0 and hence MW (T ) = 0. In that situationone makes use of the estimate (modulo logarithm)

Γsph = κ′α5

W T 4, (11.16)

where κ′ is a numerical coefficient. This estimate follows, to a certain extent,

from the results of Sec. 10.3. We have seen there that at high temperatures andMW (T ) = 0 the theory possesses the non-perturbative parameter g2

3 = g2T . Thesphaleron transition rate is mostly determined by this parameter; using it, we wouldobtain dimensional estimate Γsph ∼ g4

3 ∼ (αW T )4. The additional factor αW is dueto specific plasma effects [133]. The coefficient κ

′ has been found by lattice simula-tions and it turned out to be rather large [134],

κ′ 25.

Note that if one substitutes the non-perturbative scale g23 into (11.15) instead of

MW (T ), the sphaleron free energy becomes of the order of temperature, so theBoltzmann suppression is indeed absent.

Let us make use of the estimates (11.14) and (11.16) to find the range of tem-peratures in which electroweak baryon number violating processes are in thermalequilibrium in the early Universe. This is the case when a particle participates inat least one of these processes per Hubble time, i.e., Γsph � nH , where n ∼ T 3 isthe number density of particles of a given type. Thus, the condition for equilibriumwith respect to the sphaleron processes is

Γsph

T 3� H(T ) =

T 2

M∗Pl

(11.17)

At high temperatures, we use (11.16) and find

T � Tsph ∼ 1012 GeV. (11.18)

At relatively low temperatures the condition (11.17) gives

MW (T )T

� αW

2B(mh/MW )log

M∗Pl

T.

Making use of the crude estimate T ∼ 100 GeV in the argument of logarithm, wefind numerically

MW (T )T

� 0.66B(mh/MW )

∼ 0.24 − 0.43, (11.19)

depending on zero temperature Higgs boson mass. Thus, the sphaleron transitionsswitch off after electroweak transition only, i.e., at T � 100 GeV. We conclude that

5Our terminology here is standard, though not quite appropriate, see discussion in Sec. 10.


electroweak baryon number violation is in thermal equilibrium in wide temperatureinterval extending from 1012 GeV to about 100 GeV.

Let us now discuss selection rules for electroweak baryon number violating pro-cesses. They follow from the relations (11.11) and (11.12) and have the form

ΔB = 3ΔLe = 3ΔLμ = 3ΔLτ .

In other words, there are three conserved combinations of baryon and leptonnumbers, which can be chosen as follows,

(B − L), (Le − Lμ), (Le − Lτ ). (11.20)

At temperatures 100 GeV � T � 1012 GeV these numbers may be non-vanishing,while baryon and lepton numbers themselves are adjusted in such a way that theGrand canonical potential is at its minimum.

Let us find baryon and lepton number densities at temperatures above the electroweaktransition but below (11.18) at given (B − L) number density, assuming that the threelepton number densities are equal to each other. Let us make the calculation for νf

fermionic generations and νs Higgs doublets. Quantum numbers of all particles are givenin Appendix B. Above the electroweak transition, it is convenient to work in terms ofthe Higgs doublets with components h+ and h0, their antiparticles and gauge bosons oftwo polarizations. To calculate particle number densities, we introduce chemical potentialsto all conserved quantum numbers. Relevant quantum numbers are (B − L) and weakhypercharge Y (see Problem 5.2). Then particles of type I have chemical potential

μI = μ(BI − LI) + μY

YI

2,

while for antiparticles μI = −μI . Here BI , LI and YI are baryon number, lepton numberand weak hypercharge of particle I , μ and μY are chemical potentials to (B −L) and Y/2.As an example, for left electron and neutrino we write

μν = μeL = −μ − 1

2μY ,

while for charged and neutral scalars (components of one of the Higgs doublets) we have

μh+ = μh0 =1

2μY ,

etc. We make use of the result of Problem 5.2 and write for asymmetry of fermions of typeF of all generations

nF − nF = ΔnF =1

2νf · μF

T 2

3,

while the asymmetry of scalars of type H is

nH − nH = ΔnH = νs · μH · T 2

3

(we assume here that asymmetries are small, so that μI T ). This gives

Δnh+ + Δnh0 = νs · μY · T 2

3,

Δnν + ΔnlL = νf

„−1

2μY − μ

«· T 2

3,


ΔnlR =1

2νf (−μY − μ) · T 2

3,

ΔnuL + ΔndL = νf

„1

6μY +

1

3μ

«· 3 · T 2

3,

ΔnuR =1

2νf

„2

3μY +

1

3μ

«· 3 · T 2

3,

ΔndR =1

2νf

„−1

3μY +

1

3μ

«· 3 · T 2

3,

(11.21)

where the factor 3 in the last three formulas accounts for quark colors. Gauge bosons carryzero baryon number, lepton number and weak hypercharge, so there is no asymmetry inthem.

The system is neutral with respect to all gauge charges, including weak hypercharge(this is the analog of electric neutrality of the usual plasma), thereforeX

I

YI · ΔnI = 0.

Making use of (11.21) we find

νf

»5

3μY +

4

3μ

–+

1

2νs · μY = 0. (11.22)

This explains why we introduced the chemical potential μY : had we set μY = 0 in thebeginning, the medium would not be neutral with respect to weak hypercharge at μ �= 0.

We now eliminate one of the chemical potentials, say, μ, by making use of Eq. (11.22)and express all asymmetries (11.21) in terms of the only remaining chemical potential. Inthis way we obtain for baryon number density, which we denote simply by B,

B ≡ 1

3(ΔnuL + ΔndL + ΔnuR + ΔndR) = −T 2

3

„1

2νf +

1

4νs

«μY , (11.23)

while the lepton asymmetry is

L =T 2

3

„7

8νf +

9

16νs

«μY .

Thus, the value of (B − L) is related to μY by

B − L = −T 2

3

„11

8νf +

13

16νs

«μY .

Let us again use (11.23) and obtain finally

B =8νf + 4νs

22νf + 13νs· (B − L). (11.24)

This gives the constant C entering (11.3). There are three generations of fermions whichare effectively massless at electroweak temperature, so in the theory with one Higgs doubletwe find

C =8νf + 4νs

22νf + 13νs(νf = 3, νs = 1) =

28

79. (11.25)

We stress that this value, as well as the whole analysis, is valid above the electroweaktransition temperature only; below this temperature, more precisely, at φ(T ) ∼ T , where


φ(T ) is the Higgs expectation value, the parameter C is a function of the ratio φ(T )/Tand is somewhat (but not dramatically) different from (11.25), see, e.g., [135].

Problem 11.4. Considering temperatures above electroweak transition, introduce, besidesμ and μY , the chemical potential μ3 to the diagonal (third) component T 3 of weak isospin.Find asymmetries in all particles including vector bosons and show that the requirement ofneutrality with respect to the weak isospin is equivalent to μ3 = 0 at any μ and μY . Note:This result is not valid below electroweak transition temperature.

Problem 11.5. For temperatures above electroweak transition, consider general situationin which the densities of all conserved quantum numbers (11.20) are non-zero. Show thatthe baryon asymmetry is still given by (11.24).

11.2.2 Baryon number violation in Grand Unified Theories

Another mechanism of baryon and lepton number violation exists in Grand UnifiedTheories (GUTs). These contain new superheavy particles, vectors and scalars (andalso fermions in supersymmetric theories) whose interactions with the StandardModel particles violate baryon and lepton numbers already at the level of pertur-bation theory. As an example, there is a vertex shown in Fig. 11.2(a). This vertexdescribes the interaction of vector boson V with two quarks (unlike the gauge ver-tices of the Standard Model which contain both quark and antiquark). The existenceof this vertex does not yet mean that baryon number is violated: were this vertex theonly one, we could assign baryon number 2/3 to the boson V , and baryon numberwould still be conserved. However, there is also a vertex shown in Fig. 11.2(b)(we will explain shortly why it involves antilepton and not lepton). Therefore,baryon number is indeed violated: V -boson exchange leads to the process shownin Fig. 11.3.

Interactions of Fig. 11.3 give rise to proton decay, see Fig. 11.4. There are verystrong experimental bounds on the proton lifetime: depending on the decay mode

τp > 1032 − 1033 yrs. (11.26)

(b)(a)

Fig. 11.2 Vector boson interaction with quarks q, antiquarks q and antileptons l.


Fig. 11.3 V -boson exchange leading to baryon number violation.

Fig. 11.4 Proton decay diagram.

This leads to strong bounds on baryon number violating interactions. Thediagram 11.4 implies the following estimate for the proton width,

Γp ≡ 1τp

∼ α2V

M4V

m5p,

where αV = g2V

4π , gV is the coupling constant in each vertex; the factor M−2V in

amplitude, and hence M−4V in the width, comes from the V -boson propagator, and

the factor m5p is introduced on dimensional grounds. Together with (11.26) this gives

the bound

MV � 1016 GeV. (11.27)

Thus, we are dealing with interactions that could be of interest for cosmology onlyif the temperatures in the Universe were as high as the GUT energy scale

MGUT ∼ 1016 GeV. (11.28)

If the theory does not involve new fermions beyond the Standard Model ones,then the baryon number violating interactions are indeed given only by diagramsof the types shown in Fig. 11.2, with the diagram 11.2(b) containing preciselyantilepton. This is true both for new vector boson V and possible new scalar bosonS. This follows from the invariance under the Standard Model gauge group; wewill see that in the end of this Section. This result is important, since all theseinteractions conserve (B − L) if we assign (B − L) = 2

3 to vectors V and scalarsS. Hence, baryon asymmetry cannot be due to these interactions only: they wouldnot generate non-vanishing (B − L), and electroweak processes studied in the pre-vious Section would wash the baryon asymmetry out. We note that similar situationoccurs in supersymmetric models as well.


Fig. 11.5 (B − L)-violating vertex.

If there are new fermions in the theory, (B −L) may be violated simultaneouslywith B. A simple example is given by a theory with extra fermion ΛL which isneutral with respect to the Standard Model gauge interactions and whose baryonand lepton numbers are zero. Then the theory may contain both vertices shown inFig. 11.2 and vertex of Fig. 11.5. Final states in Fig. 11.2 have (B − L) = 2

3 , whilethe final state in Fig. 11.5 has (B − L) = − 1

3 , so (B − L) is not conserved indeed.Another possibility is that ΛL is lepton, i.e., its lepton number is L = 1.

The reason for the existence of interactions shown in Fig. 11.2 in GUTs is as follows.Grand Unification of strong and electroweak interactions assumes that at ultra-highenergies, all these gauge interactions are one and the same force. In other words, theStandard Model gauge group SU(3)c ×SU(2)W ×UY is a subgroup of simple gauge groupG, and known fermions combine (possibly together with new fermions) into representationsof G. This immediately implies that a multiplet containing the lepton doublets (as wellas possibly different multiplet containing right charged lepton) must also contain fermionsthat transform non-trivially under SU(3)c, i.e., carry color. If one does not introduce manynew fermions, the color partners of leptons are identified with quarks (or antiquarks).Now, since quarks and leptons are in one gauge multiplet, there must be a gauge bosoninteracting as shown in Fig. 11.2(b). A multiplet with colored SU(2)W -doublets may alsocontain colored SU(2)W -singlets. In other words, a multiplet under group G with leftquark doublets may contain also SU(2)W -singlet colored fermions, which should also beleft. These are left anti-quarks. So, there may exist multiplets of the full gauge group Gthat contain both quarks and antiquarks. This leads to interaction shown in Fig. 11.2(a).

The simplest, but probably unrealistic example is given by the theory with gaugegroup G = SU(5) [136], whose algebra includes the Standard Model gauge algebras in thefollowing way,

SU(3)c :

„SU(3) 03×2

02×3 02×2

«

SU(2)W :

„03×3 03×2

02×3 SU(2)

« (11.29)

U(1)Y : Y =

r5

3T 24, T 24 =

1

2√

15· diag(2, 2, 2,−3,−3) (11.30)

(subscripts in (11.29) denote dimensions of matrices; 0m×n is matrix with zero entries).New fermions are not needed to complete SU(5) multiplets, while all Standard Modelfermions of one generation are comfortably placed in representations 5 (antifundamental


representation of SU(5)) and 10 (antisymmetric representation of SU(5)):

5 = (dcL1, d

cL2, d

cL3, e

−L , νe),

10 =

0BBBBBB@

0 uc 3L −uc 2

L u1L d1

L

−uc 3L 0 uc 1

L u2L d2

L

uc 2L −uc 1

L 0 u3L d3

L

−u1L −u2

L −u3L 0 e+

L

−d1L −d2

L −d3L −e+

L 0

1CCCCCCA

,

where superscript refers to color and ucL and dc

L denote left antiquark fields (antitripletsunder SU(3)c and singlets under SU(2)W ).

Problem 11.6. Show that all fermions have correct quantum numbers with respect tothe Standard Model gauge group SU(3)c × SU(2)W × UY embedded in SU(5) according to(11.29), (11.30).

Gauge bosons, as always, belong to adjoint representation of SU(5) (24-plet). Besidesthe Standard Model gauge bosons corresponding to the algebras (11.29), (11.30), there are12 extra gauge bosons whose fields are embedded in SU(5) algebra as follows,0

BBBBBB@

0 0 0 V 1μ U1

μ

0 0 0 V 2μ U2

μ

0 0 0 V 3μ U3

μ

V 1∗μ V 2∗

μ V 3∗μ 0 0

U1∗μ U2∗

μ U3∗μ 0 0

1CCCCCCA

.

Here every field Uaμ , V a

μ is complex and describes two vector bosons. Their interactionswith quarks and leptons have precisely the form shown in Fig. 11.2.

Problem 11.7. Write all terms in the Lagrangian describing the interactions of Vμ-and Uμ-bosons with quarks and leptons. Hint: To simplify notations, notice that (Vμ, Uμ)together form SU(2)W -doublet and that they are SU(3)c-triplets.

The energy scale (11.28) emerges in GUTs in a natural way [137]. As we pointed outin Sec. 9.4.2, couplings logarithmically depend on energy (more precisely, on momentumtransfer Q). The gauge couplings of the groups SU(3)c, SU(2)W and U(1)Y are verydifferent at low energies, but as the energy increases, they get closer, as schematicallyshown in Fig. 11.6. In supersymmetric extension of the Standard Model (but not in theStandard Model itself), all three couplings become equal to αGUT ≈ 1/25 at Q = MGUT ≈1016 GeV [86]. This is precisely what should happen in a theory with a single gaugeinteraction above energy MGUT with single coupling. Gauge coupling unification is a verystrong argument both for Grand Unification and for the supersymmetric extension of theStandard Model at relatively low energies.

Let us continue our general discussion and find the constraints that the StandardModel gauge symmetries impose on the structure of baryon number violating interactions.Let us assume for the time being that there are no new fermions. It is convenient in thisSection to treat all fermions as left, i.e., instead of right quark and lepton fields UR, DR

and ER consider left fields denoted by UcL, Dc

L and EcL and describing left antiquarks and

antileptons. These are singlets under SU(2)W , antitriplets (UcL, Dc

L) and singlets (EcL)

under color SU(3)c; their weak hypercharges are opposite in sign to hypercharges of therespective particles, i.e,

YUcL

= −4

3, YDc

L=

2

3, YEc

L= 2.


Fig. 11.6 Sketch of the evolution of the Standard Model gauge couplings with momentum transfer

in the supersymmetric extension of the Standard Model. Here α3 ≡ αs ≡ g2s

4π, α2 ≡ αW ≡ g2

4π,

where gs and g are gauge couplings of SU(3)c and SU(2)W ; α1 = 53

g′2

4π, where g′ is gauge coupling

of U(1)Y ; the factor 53

is due to the fact that the normalization of the generator U(1)Y is different

from the standard normalization of the generators of GUT gauge group (as an example, trace of

the generator T 24 squared in SU(5) equals the standard value 1/2, while trace of Y 2 equals 5/6,

see (11.30); hence, gauge couplings of SU(5) and U(1)Y are related by gSU(5) =q

53g′).

Let us also recall the weak hypercharges of left doublets,

YQL =1

3, YLL = −1.

In terms of left fields, the only Lorentz-invariant renormalizable interaction with vectorfield has the Lorentz structure

ψLiγμVμψLj ∝ V ψiψj , (11.31)

while Yukawa interaction with scalar field S can only have the form (compare withMajorana mass term for neutrino, Appendix C)

ψcLSψL ∝ Sψiψj or ψLSψc

L ∝ Sψiψj , (11.32)

where ψi, ψj are various Standard Model fermion fields. We stress that (11.31) containsboth ψ and ψ, while (11.32) contains only ψ or only ψ.

Interactions (11.31), (11.32) must be invariant under all gauge symmetries of theStandard Model. Let us begin with color SU(3)c. If there exists vertex of Fig. 11.2(a), thenV is either color antitriplet or sextet (since SU(3)c representations obey 3 × 3 = 6 + 3).SU(3)c forbids other vertices in the sextet case, so this case is not interesting. For anti-triplet V , there are two more vertices allowed by SU(3)c: this is the vertex of Fig. 11.2(b)and similar vertex with lepton instead of antilepton. The same holds true for a scalarS. We see that interactions with (anti)triplet bosons exhaust all possible baryon numberviolating interactions, unless new fermions are introduced.

Further constraints come from the requirement of invariance under SU(2)W and weakhypercharge. Let us begin with vector particles V . Since Uc

L and DcL are antiquark fields,

there are two combinations of the fields that can enter the interactions of the type (11.31)and give rise to the vertex of Fig. 11.2(a):

UcLQ

„2,

5

3

«(11.33)

DcLQ

„2,−1

3

«(11.34)


(we show in parenthesis the representations of SU(2)W — doublet 2 in our case — andweak hypercharge of each operator). The combination of quark and lepton or antileptoninteracting with V must have the same Lorentz structure (11.31) and the same quantumnumbers. The doublet under SU(2)W combinations are

QEcL

„2,

5

3

«, LUc

L

„2,

1

3

«, LDc

L

„2,

5

3

«, (11.35)

(recall that doublet representation of SU(2) coincides with its conjugate). We see thatthe only possibility is that the vector boson V is a doublet under SU(2)W , has weakhypercharge 5

3 and has the following interactions:

V †UcLQ + V †QEc

L + V †LDcL + h.c.,

where we do not write Lorentz structure and couplings. The corresponding vertices areshown in terms of usual quarks and leptons in Fig. 11.7.

Let us now turn to scalars. Now we have to consider interactions with Lorentz structure(11.32). The combinations leading to the vertex of type of Fig. 11.2(a) are

QLQL

„3,

1

3

«(11.36)

QLQL

„1,

1

3

«, Uc

LUcL

„1,

4

3

«, Uc

LDcL

„1,

1

3

«, Dc

LDcL

„1,−2

3

«. (11.37)

There is one triplet combination of the type (11.32) involving antiquark and giving rise tothe vertex of the type of Fig. 11.2(b),

QL

„3,

1

3

«, (11.38)

and three singlet combinations,

QL

„1,

1

3

«, Uc

LEcL

„1,

1

3

«, Dc

LEcL

„1,

4

3

«. (11.39)

The former matches (11.36) and the latter should match one of the combinations in (11.37).In this way we come to interactions listed in Table 11.1. The structure of vertices for allthese interactions is shown in Fig. 11.8. Thus, we again see that symmetries of the StandardModel ensure (B − L) conservation unless there are new fermions.

Let us now include into the theory a new fermion ΛL. Let it be a singlet under theStandard Model gauge interactions. Color anti-triplet combinations that could interactwith vectors and lead to vertices shown in Fig 11.5 are

QLΛL

„2,−1

3

«, ΛLUc

L

„1,−4

3

«, ΛLDc

L

„1,

2

3

«

Fig. 11.7 Vector boson interactions that violate baryon number. V 4/3 and V 1/3 denote upper

and lower components of the weak doublet, 4/3 and 1/3 are electric charges. Quarks and leptons

are assumed to belong to the first generation; in fact, each vertex may contain arbitrary linear

combination of quarks and leptons of all generations with quantum numbers shown here.


Table 11.1 Baryon number violating interactions in terms of the usual quark

and lepton fields.

Quantum numbers with respect to Interacts with

SU(3)c × SU(2)W × U(1)Y

vector V 3, 2, 53

URQL, QLER, LDR

scalar S1 3, 3, 13

QLQL, QLL

scalar S2 3, 1, 13

QLQL, URDR, QLL, URER

scalar S3 3, 1, 43

URUR, DRER

(a) (b)

Fig. 11.8 Interactions with scalars also conserve (B − L).

The first of them has quantum numbers (11.34), so that color antitriplet vector boson withthese quantum numbers can have vertices shown both in Fig. 11.2(a) and in Fig. 11.5. Thescalar combinations of quarks and new fermions are

QLΛL

„2,−1

3

«, Uc

LΛL

„1,−4

3

«, Dc

LΛL

„1,

2

3

«.

The second and third of them match one of the combinations in (11.37), so there are twotypes of scalars that have vertices shown both in Fig. 11.8(a) and in Fig. 11.5.

The neutral fermion ΛL may have Majorana mass, see Appendix C. If it is stable, itis cosmologically allowed and for appropriate values of parameters it can be dark mattercandidate, provided it pair annihilates via, say, exchange of a new neutral scalar Σ, seeFig. 11.9, and the annihilation cross section is sufficiently large.6 On the other hand, if

Fig. 11.9 Annihilation of two ΛL.

6The existence of these processes means that the lepton number of ΛL is naturally set equal

to zero.


(a) (b)

Fig. 11.10 Lepton number violation in interactions with the Higgs boson.

baryon asymmetry is generated in the processes shown in Figs. 11.2 and 11.5, then thereshould be practically no interactions with lepton and Higgs doublets like

hΛH†ΛLL + h.c. (11.40)

These would give rise to the vertex of Fig. 11.10 that violates lepton number and (B −L),and hence in combination with electroweak processes would wash out baryon asymmetry.

Problem 11.8. Assuming that masses of ΛL- and Σ-particles are small compared tomasses of V - and S-bosons participating in baryon number violating processes, and that atenergies below these masses ΛL-particles participate only in interactions shown in Fig. 11.9,find cosmological bounds on masses mΛ, mΣ and Yukawa couplings. Consider the casesmΛ � mΣ and mΛ mΣ separately. At what values of these parameters is the fermionΛL a dark matter candidate?

Problem 11.9. Let the fermion ΛL have Majorana mass mΛ � mh, where mh is themass of the Standard Model Higgs boson.

(1) At what values of the coupling hΛ in (11.40) is the interaction of Fig. 11.10 irrelevantfor cosmology? Assume that the annihilation processes of Fig. 11.9 lead to negligiblenumber of Λ-particles at low temperatures.

(2) The interactions (11.40) contribute to masses of the conventional neutrinos via see-saw mechanism (see Appendix C). Using the previous result, find bounds on thesecontributions.

Problem 11.10. Let the interactions of Fig. 11.9 be absent, while interactions ofFig. 11.10 be non-negligible. Let the baryon asymmetry be generated at T � mΛ (say,in decays of super-heavy scalar bosons S → qq and S → qΛL, see Figs. 11.5 and 11.8).Assuming that fermions ΛL have large Majorana mass mΛ � mh, find bounds on mΛ

and hΛ from the following requirements: (a) baryon asymmetry must be preserved untilthe present epoch; (b) fermions ΛL must have decayed long before BBN epoch. Estimatein this case the contribution of interactions (11.40) to Majorana masses of conventionalneutrinos.

Let us now turn to another case mentioned above, namely, that the neutral fermionΛL has lepton number +1, while lepton number is conserved below MGUT modulo elec-troweak effects, and new lepton has short enough lifetime. The minimal possibility, fromthe viewpoint of the extension of the Standard Model, is that ΛL is the left componentof a Dirac fermion Λ. Then the Standard Model Lagrangian can be extended by adding


terms (we do not write kinetic term for Λ)

MΛΛΛ + hΛΛH†L + h.c.,

If the mass of Λ is large compared to the mass of the Higgs boson, MΛ > mh, then themain decay channels of the lepton Λ are Λ → h0ν, Λ → h±l∓, while for MΛ < mh itsdecay proceeds via the Higgs boson exchange. Heavy leptons of this type are cosmologicallyacceptable.

Problem 11.11. Are there massless neutrinos in the latter extension of the StandardModel?

11.2.3 Violation of lepton numbers and Majorana masses

of neutrino

As we discuss in Appendix C, one of the plausible ways to explain small neutrinomasses is the see-saw mechanism [138–141]. One adds new fields Nα

L , heavy sterileneutrinos, which we treat as left fermions. These fields are neutral with respectto the gauge interactions of the Standard Model, hence the name “sterile”. Thesuperscript α labels the species of these new fields; in what follows we assume,although this is not completely necessary, that α = 1, 2, 3, i.e., there are three newfields, according to the number of Standard Model generations. The fields Nα

L haveMajorana masses and interact with the Standard Model fields, so additional termsin the Lagrangian are

L =Mα

2N c

LαNLα + (yαβN cLαH†Lβ + h.c.), (11.41)

where Lα are left leptonic doublets of the Standard Model, H is related to theStandard Model Higgs field H by (see Appendix B) Hi = εijH

∗j (i, j are the indices

of the doublet representation of SU(2)W ), summation over α, β is assumed. TheYukawa couplings yαβ are in general complex while the masses Mα are real. Thesecond term in (11.41) is the only interaction of NLα consistent with the StandardModel gauge symmetries, unless other new fields are introduced.

Neglecting the second term in (11.41) (this is an excellent approximation inthe present context) one observes that the fields Nα

L describe three fermions ofmasses Mα. We will assume that Mα � v; this range of masses is indeed preferredfrom the viewpoint of baryogenesis. The Yukawa interaction in (11.41) leads tothe instability of N -particles. We will be interested in high temperature situation,when the Higgs expectation value is zero (see Chapter 10). Then the doublet H

describes one electrically neutral and one electrically charged particle plus theirantiparticles. The Yukawa interaction explicitly written in (11.41) gives rise to thedecay (Fig. 11.11(a)),

Nα → hlβ , (11.42)

where lβ is charged lepton or neutrino of generation β, while h is one of the Higgsparticles. The Hermitean conjugate interaction term gives rise to the decay into


(a) (b)

Fig. 11.11 Decays of heavy neutrino.

antilepton in the final state7 (Fig. 11.11(b)),

Nα → hlβ . (11.43)

Clearly, the existence of both of these processes means that lepton numbers arenot conserved, whatever lepton numbers are assigned to Nα. The same conclusionfollows from the existence of a process (Fig. 11.12)

hlα → hlβ ,

as well as double-β decay. It is useful for what follows to estimate the decay width,

ΓNα ∼ y2

8πMα. (11.44)

More precisely, the total decay width is given by (assuming Mα � v)

ΓNα =∑

β

|yαβ |28π

Mα.

Let us stress that in the tree level approximation, partial widths of the decays (11.42)and (11.43) are equal to each other,

Γ(Nα → hlβ) = Γ(Nα → hlβ). (11.45)

Fig. 11.12 Lepton-Higgs scattering with lepton number violation.

7The scalar particles in (11.42) and (11.43) are actually different: say, in the case of charged

lepton h = h+ and h = h− in (11.42) and (11.43), respectively. This will be unimportant for what

follows, and we will use somewhat vague notation h.

11.3. Asymmetry Generation in Particle Decays 263

This is not the case once loop corrections are included. This is very important fromthe viewpoint of the baryon asymmetry generation, see discussion in Sec. 11.4.

Problem 11.12. Consider a theory of one fermion field NL (the subscript L willbe omitted in this and next problem) whose Lagrangian is

L = iNγμ∂μN +M

2N cN.

Obtain the field equation and its positive-energy solutions for |p| � M and |p| � M,

where p is spatial momentum. Wave functions of which particles do they describe?

Problem 11.13. Quantize the model of the previous problem. Upon adding theYukawa term (yN cψϕ + h.c.), where ψ and ϕ are massless left fermion and scalar,respectively, find the decay widths of N → ψϕ and N → ψϕ∗ at the tree level, thusconfirming (11.44) and (11.45).

11.3 Asymmetry Generation in Particle Decays

Theories extending the Standard Model sometimes provide a simple mechanismof the baryon asymmetry generation due to particle decays. As we discussed inSecs. 11.2.2 and 11.2.3, these theories may contain new particles whose decaysviolate baryon and/or lepton numbers. In fact, what is important is (B − L)-violation: these decays occur before the electroweak transition, so the electroweakprocesses discussed in Sec. 11.2.1 make the baryon asymmetry equal8 to the gen-erated (B − L) asymmetry up to a numerical factor of order 1.

As we discussed in Sec. 11.2.2, scalar bosons of GUTs, S (or vector bosons V ),may have decay channels

(1) : S → qq

(2) : S → qΛ(11.46)

where q denotes conventional quarks and Λ is a new fermion neutral with respect toSU(3)c × SU(2)W × U(1)Y . To simplify formulas below we assume (though this isnot necessary) that the decay S → ql has negligible partial width. Lepton numberof Λ is either 0 or +1; we will take LΛ = 0 for definiteness. The values of (B − L)of the final states of the first and second type are, respectively,

(B − L)(1) =23

and (B − L)(2) = −13. (11.47)

The boson S is color antitriplet, so there exists its antiparticle, triplet S. Its decaychannels are

(1) : S → qq,

(2) : S → qΛ(11.48)

8The possibility that the baryon asymmetry is generated in the electroweak processes themselves

is studied in Sec. 11.5.


and (B − L) of final states are given by

(B − L)(1) = −23

and (B − L)(2) =13. (11.49)

Suppose that at temperatures exceeding the S-boson mass mS, bosons S and S

are in thermal equilibrium. Then their number densities, modulo color and spinfactors, are the same as those of other particles. As the Universe cools down, thermalequilibrium breaks down: S- and S-bosons decay, while the inverse processes of theirproduction do not occur. These decays produce (B −L)-asymmetry, provided thatthe probabilities of decays (1) and (2) are not the same as probabilities of (1) and(2), respectively. Let us denote partial widths of decays (1), (2), (1) and (2) by Γ(1),Γ(2), Γ(1) and Γ(2). (B −L)-asymmetry produced in decays of one S-boson and oneS-boson is

δ =1

Γtot

[(23Γ(1) − 1

3Γ(2)

)−(

23Γ(1) −

13Γ(2)

)],

where Γtot is the total S-boson width. The latter equals the total S boson widthdue to CPT -theorem,

Γtot = Γ(1) + Γ(2) = Γ(1) + Γ(2)

(we assume for simplicity that there are no other channels). Making use of the latterequality, we obtain for “microscopic asymmetry”

δ =Γ(1) − Γ(1)

Γtot. (11.50)

The difference between partial widths of the decays (1) and (1) is possible only ifC and CP are violated; this is the way the second condition of Sec. 11.1 shows up.

Partial widths of a particle and antiparticle coincide at the tree level, e.g. (seeFig. 11.8(a)),

Γtree(S → qq) = Γtree(S → qq). (11.51)

Indeed, modulo one and the same kinematic factor, the tree level probabilities areequal to the modulus squared of the coupling g(1) in the interaction vertex, which isthe same for particle and antiparticle. However, the equality (11.51) does not hold,generally speaking, at one loop level (cf. Sec. B.4). To obtain non-zero microscopicasymmetry (11.50), it is sufficient to have other bosons S′ with the same quantumnumbers as S, with the couplings to all these bosons being complex. Note thatcomplex couplings mean CP -violation in models we consider here. At the one looplevel, the amplitude of decay S → qq is given then by the sum of the diagramsshown in Fig. 11.13 (there are other diagrams, but considering these is sufficient forthe sake of argument).

Modulo common kinematical factor, partial width Γ(1) is, at the one loop level,

Γ(1) = const · |g(1) + Dg(2)g∗(2′)g(1′)|,


Fig. 11.13 Diagrams for S → qq decay at the one loop level. Bosons S′ have the same quantum

numbers as S. Shown are couplings in interaction vertices.

where D is the one loop Feynman integral for the diagram of Fig. 11.13, and sum-mation over all S′-bosons is assumed. The analogous amplitude for antiparticlecontains complex conjugate couplings, therefore

Γ(1) = const · |g∗(1) + Dg∗(2)g(2′)g∗(1′)|.

As a result, the microscopic asymmetry is9

δ = −2Im(D) ·Im(g(1)g

∗(2)g(2′)g

∗(1′))

|g(1)|2 + |g(2)|2 . (11.52)

Crudely speaking, this is proportional to coupling constant squared, while Im(D)contains small loop factor. Assuming that the phases of couplings g(i) and g(i′) areneither small nor correlated, we obtain crude estimate,

δ ∼ g2

4πf

(mS

mS′

), (11.53)

where the function f of mass ratio mS/mS′ is of order 1 at mS � mS′ and decaysas mS/mS′ decreases (fermion masses have been neglected).

Problem 11.14. Find the asymmetry δ in the above case assuming that S andS′ are scalars. Find its dependence on the ratio of masses of S- and S′- bosonsassuming mS � mS′ (but not necessarily mS � mS′).

Let us now turn to cosmological generation of (B−L)-asymmetry. The simplestcase is when S- and S-particles are in thermal equilibrium at T � mS, while atT � mS their decays are the dominant processes. Both of these properties are non-trivial. The first implies that the temperature in the Universe was indeed as high asT � mS, and, furthermore, that production and annihilation of SS-pairs is fast atthese temperatures. The second is valid only if pair creation and annihilation switchoff at T � mS, and if production of S and S in processes inverse to (11.46) and(11.48) (inverse decays) is negligible at those temperatures. The latter requirement

9Note that in a theory with one S-boson one would get δ = 0 at the one loop level. Indeed, in

that case one should set g(1′) = g(1) and g(2′) = g(2) in (11.52).


is indeed satisfied when the decay width of S-particles is small compared to theexpansion rate at T ∼ mS,

Γtot � H(T ∼ mS) =m2

S

M∗Pl

. (11.54)

If all these conditions are fulfilled, then at T � mS the number density of S-particlesand their antiparticles is the same as of all other species, nS ∼ T 3, i.e.,

nS

s∼ 1

g∗,

where s is the entropy density. The decays of S and S at later times produce theasymmetry

ΔB−L ∼ δnS

s∼ δ

g∗. (11.55)

Clearly, the asymmetry is quite large in this simplest case: the required valueΔB−L ∼ 10−10 is obtained for δ ∼ 10−8 (assuming that the number of degreesof freedom g∗ is similar to its value gSM

∗ ∼ 100 at T ∼ mS).The requirement (11.54) implies, in fact, that the S-particle mass is very large.

Indeed, we have an estimate

Γtot ∼ g2

4πmS. (11.56)

Then the requirement (11.54) gives

mS � g2

4πM∗

Pl, (11.57)

As an example, for g2/4π ∼ 10−2 one obtains mS � 1016 GeV. The possibility thatthe Universe had such a large temperature is rather problematic (see accompanyingbook). For smaller couplings the requirement (11.57) is easier to fulfill, but themass of S-particles must be large in this case too. Indeed, without fine-tuning ofparameters the estimate for the microscopic asymmetry is given by (11.53), i.e.,δ � g2/4π. Making use of (11.55) one obtains the value ΔB−L ∼ 10−10 for g2/4π �10−8, i.e.,

mS � 1010 GeV.

The general statement is that the asymmetry generation in particle decays can occuronly for large masses and hence at high temperatures.

We note that the departure from thermal equilibrium (the third necessary con-dition of Sec. 11.1) is achieved rather trivially in the scenario just described: atT � mS decay processes occur while inverse decays do not.

Let us now consider the case in which the inequality (11.54) is not valid. Let usintroduce the parameter

K =Γtot

H(T = mS)=

ΓtotM∗Pl

m2S

(11.58)


and let us take it large,

K � 1.

The dependence on this parameter is not particularly strong, so in this case thegeneration of the required (B −L)-asymmetry is also possible. Again, the mass mS

must be quite large. Unlike in the previous case, we now need not assume that S andS were in thermal equilibrium at T � mS; furthermore, the maximum temperaturein the Universe may be even somewhat smaller than mS.

We will see in the end of this Section that at K � 1 the estimate for (B − L)-asymmetry is

ΔB−L = const · δ

g∗K log K, (11.59)

where constant is of order 1. Similar to the small width case (11.54) this estimateimplies that S-particle mass is large. If all relevant couplings are of the same order,then the microscopic asymmetry is again estimated by (11.53), i.e., δ � g2

4π , whilethe estimate for the width is given by (11.54). Making use of the definition (11.58)we find, modulo logarithm,

mS � ΔB−L · g∗M∗Pl,

i.e., we again obtain

mS � 1010 GeV.

This bound is weaker in models where couplings differ by orders of magnitude (e.g.,couplings g(1′) and g(2′) in Fig. 11.13 are considerably greater than g(1) and g(2)),but in any case the mass of S-particles must be large.

To summarize, heavy particle decays provide efficient mechanism of generationof (B −L) asymmetry and hence baryon asymmetry. This mechanism may work in(B − L)-violating GUTs. It may work also in supersymmetric GUTs, and in thatcase S-particles may be not only bosons but also fermions. Some (but not fatal)problem with GUTs is that their mass scale is extremely high (see (11.26)), so themaximum temperature in the Universe could be lower than this scale (this is indeedthe case in most inflationary models).

Let us obtain the estimate (11.59). Let us assume for the time being that at T � mS

the dominant processes are decays and inverse decays of S and S. One might think thatinverse decays of the type

qq → S (11.60)

should be very rare, since the center-of-mass energy of the two incoming particles mustbe adjusted to the mass of S-particle with precision of order of its width Γtot. Still, theprobability of inverse decays is not small in thermal equilibrium, as the rates of directand inverse processes are equal. Hence, for quarks and leptons in thermal equilibrium thenumber of inverse decays per unit time in comoving volume is

d(na3)inv.

dt= Γtot · neq

S · a3, (11.61)

where neqS is the equilibrium number density of S-particles.


Problem 11.15. Let us neglect cosmological expansion. Show by explicit calculation thatthe rate of the inverse processes (11.60) is„

dn

dt

«qq→S

= Γ(S → qq) · neqS ,

where Γ(S → qq) is the partial width of the decay S → qq. Simplifications: Consider lowtemperatures T mS; do not account for color of quarks and S-particles.

S-particle decays give the following contribution to the evolution of their number incomoving volume,

d(na3)dec

dt= −Γtot · nS · a3. (11.62)

By adding (11.61) and (11.62) we obtain the Boltzmann equation for the number density,

d(nSa3)

dt= −Γtot · (nS · a3 − neq

S · a3). (11.63)

Let us now consider (B−L) density in comoving volume nB−La3. Without loss of generalitywe assign (B − L) = 0 to S- and S-particles. Evolution of nB−L is due to several effects.First, it is created in decays of S and S at rate

δ · Γtot · nS · a3

Second, even if (B − L) is zero in the medium, CP -violation in inverse decays give thecontribution

−δ · Γtot · neqS · a3, (11.64)

so that (B − L) is not produced in thermal equilibrium.10 Finally, if there is (B − L) inthe medium, it gets washed out due to production of S-particles. Indeed, if the mediumcontains more quarks than antiquarks, then there are more inverse decays qq → S thanthe conjugate processes qq → S (recall that S-particles have B − L = 0). Evolution of(B −L) due to the latter mechanism is proportional to the inverse decay rate (11.61) andexcess of fermions with positive (B − L), i.e., the contribution to the rate is

−cΓtot · neqS · a3 · nB−L

nq,

where nq is equilibrium quark density introduced on dimensional grounds, and theparameter c ∼ 1 accounts for the number of decay channels. As a result we obtain theequation

d(nB−L · a3)

dt= δ · Γtot · (nS · a3 − neq

S · a3) − cΓtot · neqS · a3 · nB−L

nq.

The first and second terms in the right hand side describe the generation and wash-out of(B − L), respectively. It is convenient to make use of Eq. (11.63) and write

d(nB−La3)

dt= −δ · d(nSa3)

dt− cΓtot · neq

S · a3 · nB−L

nq. (11.65)

The system of equations (11.63) and (11.65) determines number density of S-particles and(B − L) asymmetry at all times if they are known at initial time ti.

10We simplify the situation slightly: the rate (11.64) includes the contribution of resonant scat-

tering of fermions, e.g., qq → S → qΛ. Still, Eq. (11.64) is correct, since it follows from the

requirement that (B − L) is not generated in thermal equilibrium when nS = neqS .


It is convenient to introduce the variables

NS =nS

T 3and NB−L =

nB−L

T 3, (11.66)

so that nSa3 = const · NS, nB−La3 = const · NB−L with one and the same constant. Also,let us use, instead of time, the variable

z =mS

T

and make use of the relation (3.34)

− T

T= H(T ) = H(T = mS) · T 2

m2S

.

Recalling that nq ∝ T 3 we obtain the following form of Eqs. (11.63) and (11.65)

dNS

dz= −Kz(NS − Neq

S ), (11.67)

dNB−L

dz= −δ · dNS

dz− Kz · Neq

S · NB−L, (11.68)

where the parameter K is defined in (11.58) and

K = cT 3

nq· K ∼ K.

We are interested in the case K � 1, when the relevant temperatures are low, T mS ,i.e., z � 1. The equilibrium number density of S-particles in this regime is (spin and colorfactors are omitted)

neqS =

„mST

2π

«3/2

e−msT ,

i.e., modulo irrelevant constant c, we have

NeqS = cz3/2e−z. (11.69)

Hence, we have explicit expressions for all quantities entering (11.67), (11.68).Let us begin with Eq. (11.67). Its solution is

NS(z) =

Z z

zi

e−K2 (z2−z

′2)Kz′NeqS (z′)dz′ + e−

K2 (z2−z2

i )NS(zi),

where zi ≡ mST is the initial value of the variable z, at which the initial value of the

S-particle relative number density is NS(zi). This solution shows that for K � 1 the initialvalue of the number density gets irrelevant soon (the second term rapidly tends to zero).Now, at K � 1 the integral is saturated at z′ near z (if Neq

S (z) is not very small), therefore

NS(z) = NeqS + O

„1

K

«. (11.70)

Both properties are quite obvious: large K means high rates of decays and inverse decays,so S-particles are close to thermal equilibrium.

Let us turn to Eq. (11.68). Neglecting corrections of order K−1 and making use of(11.70), we write it in the following form,

dNB−L

dz= −δ · dNeq

S

dz− Kz · Neq

S · NB−L. (11.71)

Departure from thermal equilibrium that leads to the asymmetry generation is now rathersubtle: even though the density of S-particles is close to equilibrium at each moment oftime, it changes in time (and hence in z), so the equilibrium is incomplete.


The solution to Eq. (11.71) is

NB−L(z) = −δ ·Z z

zi

e−I(z′,z) dNeqS

dz′ dz′ + e−I(zi,z)NB−L(zi), (11.72)

where

I(z, z′) =

Z z′

z

NeqS (z′′)Kz′′dz′′.

We see that initial anisotropy is irrelevant at late times, just like the initial S-particledensity, provided that the initial temperature is of order of mS or only slightly less thanmS (so that Neq

S (zi) is not very small).

Problem 11.16. Let the maximum temperature in the Universe be Ti, and K � 1.Estimate the maximum value of Ti such that most of the initial (B − L)-asymmetry ispreserved up to the present epoch.

Assuming that the initial temperature is sufficiently high, we neglect the second termin (11.72) and write for the resulting asymmetry

NB−L = −δ ·Z ∞

0

e−I(z) dNeqS

dzdz, (11.73)

where

I(z) ≡ I(z,∞) =

Z ∞

z

NeqS (z′)Kz′dz′.

The integral in (11.73) is saturated at rather large z (i.e., at temperatures well below mS):the asymmetry produced at z ∼ 1 (i.e., T ∼ mS) due to the first term in (11.71) is washedout in the course of evolution. We find from (11.69) that at large z

dNeqS

dz= −cz3/2e−z = −Neq

S (z). (11.74)

Hence, the integrand in (11.73) is the product of two exponential factors: the decayingfactor e−z from (11.74) (the decreasing density of S-particles makes the generation of

(B − L) slower) and increasing factor e−I(z) (wash-out becomes less efficient). Such anintegral is determined by the behavior of the exponent

f(z) = I(z) + z.

This function has a minimum at z = z∗ such that

NeqS (z∗)Kz∗ = 1. (11.75)

We recall (11.74) and obtain11 (see Sec. 6.1 regarding equations of the latter type)

z∗ = log K + O(log log K) = log K + O(log log K). (11.76)

Both the integral I(z∗) and its first and second derivatives are of order one at the saddlepoint, so

f(z∗) = z∗ + O(1),

and

d2f

dz2= O(1).

11The approximation z∗ = log K works poorly at moderate K but this is unimportant for our

purposes.

11.4. Baryon Asymmetry and Neutrino Masses: Leptogenesis 271

The latter property means that the integral (11.73) is saturated in the region around z∗whose size is

Δz ∼ 1 z∗.

We obtain in the saddle point approximation

NB−L = δ · const · NeqS (z∗),

and finally, making use of (11.75), we find

NB−L = const · δ

Kz∗= const · δ

K log K, (11.77)

with constant of order 1. This is precisely the quoted result (11.59).Let us now discuss the result obtained and calculation performed. First, the temper-

ature of the asymmetry generation, T∗, is determined by (11.76), i.e.,

T∗ =mS

log K. (11.78)

Wash-out effect is not very strong at this temperature (I(z∗) ∼ 1), while the density ofS-particles and the number of their decays are still sizeable,

NeqS � dNeq

S

dz∼ 1

K log K.

It is these properties that make the resulting asymmetry suppressed rather mildly atlarge K. Second, the result (11.77) is valid provided that the maximum temperature inthe Universe exceeded T∗ which is somewhat lower than mS. Third, the asymmetry isgenerated in the temperature interval which is small compared to temperature itself,

ΔT

T∗=

Δz

z∗=

1

log K.

Finally, let us point out that the equation we actually used, Eq. (11.71), does not containinformation on the processes that keep the number density of S-particles in thermal equi-librium. Instead of decays and inverse decays (or together with them) these may be pro-cesses of pair creation and annihilation of S and S. It is this situation that happens forcolored S and S which we considered in Sec. 11.2.2. Still, the result (11.59) remains validfor K � 1. What we have not included into our analysis is non-resonant scattering ofthe type qq → qΛ occurring via S-boson exchange. The latter process tends to wash out(B − L) and its effect is sometimes important, see Sec. 11.4.

11.4 Baryon Asymmetry and Neutrino Masses: Leptogenesis

As we discussed in Sec. 11.2.3, violation of lepton number, and hence (B − L)may be related to non-zero neutrino masses. Hence, it is appealing to explain thebaryon asymmetry within the same approach that deals with neutrino masses. Thelepton asymmetry may indeed be generated in decays of N -particles considered inSec. 11.2.3. This asymmetry is then partially reprocessed into baryon asymmetryby electroweak processes of Sec. 11.2.1. This scenario [142] is known as leptogenesis.

The mechanism of the lepton asymmetry generation is basically the same asin Sec. 11.3. The only difference is that S-particles are now replaced by Majoranafermions Nα. Without fine tuning of parameters, the lepton asymmetry is produced


in decays of the lightest of these particles: asymmetry generated in decays of heavierparticles is washed out by the processes involving the lightest ones. Let N1 be thelightest of N -particles. Then the relevant decays are

N1 → lh (11.79)

and

N1 → lh (11.80)

As we discussed in Sec. 11.3, the microscopic asymmetry is generated if theirpartial widths are different. This occurs at one loop, if couplings in the Lagrangian(11.41) are complex, and hence CP is violated. The relevant diagrams are shown inFig. 11.14. Thus, one loop partial width of decay N1 → lh is given by

Γ(N1 → lh) = const ·∑α

∣∣∣∣∣∣y1α +∑β,γ

D

(M1

Mγ

)· y∗

1βyγαyγβ

∣∣∣∣∣∣2

, (11.81)

where Mγ is mass of particle Nγ (we neglect masses of the Standard Model particles:we will see that leptogenesis requires Mα � 100 GeV), D(M1/Mγ) is the sum ofloop integrals shown in Fig. 11.14.

The partial width of N1 → lh decay is obtained from (11.81) by replacing theYukawa couplings by their complex conjugates, yαβ → y∗

αβ. We denote

Im D

(M1

Mγ

)=

18π

f

(M1

Mγ

),

and find the microscopic lepton asymmetry,

δ ≡ Γ(N1 → lh) − Γ(N1 → lh)Γtot

=18π

∑γ=2,3

f

(M1

Mγ

)· Im(

∑α y1αy∗

γα)2∑α |y1α|2 . (11.82)

We used here the fact that Im(y1αy∗γα) = 0 for γ = 1, so the diagrams with N1 in

the loop do not contribute. We will concentrate on the mass hierarchy M1 � M2,3,when

f

(M1

Mγ

)= −3

2M1

Mγ. (11.83)

Fig. 11.14 Amplitude of the decay N1 → lαh is the sum of tree-level and one loop diagrams.

Summation over generation indices β and γ is assumed in the latter. Yukawa couplings entering

the vertices are shown explicitly.


So, the microscopic asymmetry becomes (the overall sign is irrelevant for us here)

δ =3M1

16π

1∑α |y1α|2

∑αβγ

Im[y1αy1β

(y∗

γα

1Mγ

y∗γβ

)]. (11.84)

Problem 11.17. Show by calculating Feynman diagrams that (11.83) is indeedvalid for Mγ � M1.

Let us make the following comment regarding the expressions (11.82) and(11.84). They contain combinations of the Yukawa couplings which are differentfrom the combinations entering neutrino mass matrix (see (C.67)),

mαβ = −v2

2

∑γ

yγα1

Mγyγβ. (11.85)

As an example, the unitary transformation of the form

y → yU (11.86)

changes the neutrino mass matrix mαβ but leaves intact the expressions (11.82) and(11.83). This is not surprising: the transformation (11.86) corresponds to the changeof basis for lepton fields lα, while the asymmetry is insensitive to this basis (weneglect masses of the Standard Model leptons, so all choices of the basis are equiv-alent). This shows that, generally speaking, the asymmetry δ is not directly relatedto the parameters of PMNS matrix, and the measurement of neutrino oscillationparameters does not tell whether there is asymmetry in N -particle decays. Still,the very fact that oscillations exist suggests that the matrix of Yukawa couplingsyαβ has non-trivial structure. Additional hint towards asymmetry in N -particledecays would be given by the observation of CP -violation in neutrino oscillations:it would show that elements of neutrino mass matrix, and hence Yukawa couplings,are complex at least in the gauge basis.

We will get back to the expression (11.84) for the microscopic asymmetry, andnow we turn to the cosmological lepton asymmetry generation in decays of particlesN1. The analysis repeats the treatment in Sec. 11.3 word by word, so we simply usethe results of that Section. The generation is most efficient for

Γtot(M1) � H(T = M1), (11.87)

but one has to assume in that case that particles N1 are produced in cosmic plasmaat T � M1 by interactions other than Yukawa interactions (11.41). We recall that

Γtot =M1

8π

∑α

|y1α|2,

and that H = T 2/M∗Pl, and find that the relation (11.87) can be written as

m1 � 4π

M∗Pl

· v2 ∼ 10−3 eV, (11.88)


where

m1 =∑

α

|y1α|22M1

· v2 (11.89)

is the sum of absolute values of the N1-particle contribution into neutrino massmatrix (see Section (C.4)). We see that the case under study requires strong hier-archy of Yukawa couplings:12 if all yαβ were of the same order, then the contributionsof the lightest particle N1 into mass matrix (11.85) would be the largest, and allneutrinos would have masses below 10−3 eV, in contradiction to experiment (see thediscussion in Appendix C before Eq. (C.55)). Still, let us continue with this case andobtain the estimate for the mass M1, assuming for definiteness the direct neutrinomass hierarchy (C.57) with small mass of the lightest mass eigenstate. Once theinequality (11.87) holds, the estimate for the lepton (and hence baryon) asymmetryis (see (11.55))

ΔL ∼ δ

g∗.

Now, the formula (11.84) can be written as

δ = − 3M1

8πv2

1∑α |y1α|2

∑αβ

Im

(y1αy1β

∑γ=2,3

m(γ)∗αβ

), (11.90)

where

m(γ)αβ = yαγ

v2

2Mγyγβ (11.91)

is the contribution of Nγ-particle into neutrino mass matrix. Making use of (C.57)we find

δ � 3M1

8πv2matm.

The asymmetry ΔL ∼ 10−10 is obtained for δ ∼ 10−8 (assuming that g∗ ∼ 100 likein the Standard Model), so

M1 � 109 GeV.

Without fine tuning of parameters this gives the minimum mass scale of N -particlesin the leptogenesis scenario; the maximum temperature in the Universe must exceed109 GeV.

Probably the most natural possibility is that there is direct neutrino masshierarchy (C.57) related to inverse mass hierarchy of Nα, so that m3 ∝ M−1

1 ,m2 ∝ M−1

2 , m1 ∝ M−13 with M1 � M2 � M3. In that case the right hand side of

(11.89) is estimated as

m1 ∼ matm 0.05 eV, (11.92)

12This possibility is not particularly natural, but there is no reason to discard it; recall that there

is strong hierarchy of the Yukawa couplings of quarks and leptons in the Standard Model.


so that the inequality (11.88) and hence (11.87) do not hold. Then the generationof lepton asymmetry is suppressed. Let us use the estimate (11.59) and write

ΔL const · δ

g∗K log K,

where the constant is of order 1, and

K =Γtot

H(T ∼ M1)=

m1M∗Pl

4πv2. (11.93)

Using the estimate (11.92) we see that the suppression factor is of order

K ∼ 100,

so the correct value ΔL ∼ 10−10 is obtained for larger microscopic asymmetry

δ � 10−5. (11.94)

On the other hand, for γ = 2, 3 the right hand side of (11.91) is estimated asm

(γ)αβ ∼ msol, so (11.90) and (11.94) give

M1 � 1012 GeV.

We see that even in the least favorable case the required mass scale is not unac-ceptably large. We note that in the case we study here, one does not need to invokeextra mechanisms of N -particle production: they are efficiently produced in inversedecays. What is required is that the maximum temperature in the Universe exceeds(see (11.78))

T � M1

log K.

Thus, leptogenesis scenario indeed works in a wide range of parameters. Inter-estingly, the range of neutrino masses suggested by the oscillation experiments isnot very far from 4πv2

M∗P l

∼ 10−3 eV, so that the suppression factor (11.93) is notvery large in any case. In other words, neutrino masses m � 1 eV are such thatthere is crude (within two or three orders of magnitude) equality between the cos-mological expansion rate at leptogenesis and the width of the lightest N -particle,Γtot ∼ (1 − 1000) · H(T ∼ M1). This coincidence is a rather strong argument infavor of leptogenesis.

If neutrino masses are much larger than matm 0.05 eV, they must be degen-erate. In that case one obtains interesting bounds from the non-resonant scatteringshown in Fig. 11.15,

lh → lh, (11.95)

and crossing processes. These tend to wash out the lepton asymmetry. The newcontribution to the Boltzmann equation for lepton number density, as compared toSec. 11.3, is

d(nL · a3)dt

∝ Γlh · (nL · a3),


Fig. 11.15 Lepton scattering off scalars that washes out the lepton asymmetry.

where Γlh is the rate of processes of the type (11.95). In terms of the variablez = M1/T this contribution reads

dNL

dz∝ − M∗

Pl

M1TΓlh(z) · NL,

where NL = nL

T 3 . Wash-out of the lepton number is important after the time atwhich the asymmetry is generated, i.e., at z � z∗ = log K (see Sec. 11.3). Sincez∗ is typically large, the N -particles are non-relativistic at z > z∗ and the crosssections of the processes (11.95) are estimated as

σlh = const ·∑αβγ

∣∣∣∣yγαyγβ

Mγ

∣∣∣∣2

(recall that N -particles are fermions, their propagator at low momenta is 1Mγ

).Making use of the expression (C.67) for neutrino mass matrix, the cross section iscast into the form

σlh = const · Tr(mm†)v4

= const · 1v4

∑m2

ν . (11.96)

It follows already from dimensional argument that

Γlh = const · σlh · T 3.

As a result, we find that at z > z∗, when decays and inverse decays are switchedoff, lepton asymmetry obeys

dNL

dz= −const · M∗

PlT2

M1

∑m2

ν

v4· NL = −const · M∗

PlM1

z2

∑m2

ν

v4· NL, (11.97)

where the constant is of order 1. We see that the scattering processes suppress theasymmetry by a factor

exp(−∫ ∞

z∗dz

constz2

· M∗PlM1 ·

∑m2

ν

v4

)

= exp(−const

z∗· M∗

PlM1 ·∑

m2ν

v4

). (11.98)

Let us require that this factor is not very small. This gives∑m2

ν � v4z∗M∗

PlM1.

11.5. Electroweak Baryogenesis 277

Even for relatively small M1 ∼ 109 GeV when z∗ ∼ 1 (the case (11.88)) the resultingbound is quite strong (we take into account neutrino degeneracy)

mν � 1 eV. (11.99)

In the case M1 � 1012 GeV (and z∗ ∼ 10) this bound is even stronger,

mν � 0.1 eV.

Accurate calculations of the wash-out effect in scattering give [143]

mν < 0.12 eV (11.100)

for most values of the parameters of the model. This bound does not contradictexisting experimental and cosmological bounds and again suggests that the neu-trino masses are just right for leptogenesis. On the other hand, if double-beta decayexperiments would measure neutrino mass above the bound (11.100), the leptoge-nesis scenario13 will be much less attractive.

Problem 11.18. Discarding experimental bounds on neutrino masses and neutrinooscillation data, and assuming that all Yukawa couplings are of the same order, showthat successful leptogenesis is possible only for

mν � 1 eV.

11.5 Electroweak Baryogenesis

It is natural to ask whether electroweak baryon number violation discussed inSec. 11.2.1 can by itself generate baryon asymmetry at temperature of order100 GeV. Such a possibility — electroweak baryogenesis — is of particular interestbecause the relevant energy range of 100 GeV – 1 TeV is accessible to collider exper-iments. Examining this energy range will either rule out or give strong argumentsin favor of electroweak baryogenesis in our Universe.

Both the requirement of strong enough CP -violation and the thermal inequi-librium condition are quite non-trivial in this scenario. CP -violation in the CKMmatrix of the Standard Model is insufficient for baryogenesis; strong deviation fromthermal equilibrium is also not inherent in the Standard Model. However, someextensions of the Standard Model have both extra sources of CP -violation and pos-sibility of enhancement of thermal inequilibrium at the electroweak transition, sothe electroweak baryogenesis still remains an option.

13We talk here about leptogenesis at the hot Big Bang stage (“thermal leptogenesis”). The analysis

in the text does not apply to leptogenesis at reheating which we do not consider here.


11.5.1 Departure from thermal equilibrium

The Universe expansion at temperatures of order 100 GeV is pretty slow. TheHubble time is

tU ∼ H−1 =M∗

Pl

T 2∼ 1014 GeV−1 ∼ 10−10 s.

This is much longer than the time scale of electroweak interactions between particlesin the medium,

tint ∼ 1αW T

∼ 1 GeV−1 ∼ 10−24 s.

Hence, one of the necessary conditions of Sec. 11.1, strong deviation from thermalequilibrium, is very non-trivial. It appears that the only way this condition canbe satisfied is that the electroweak phase transition is first order. As we discussedin Sec. 10.1, 1st order phase transition occurs via spontaneous creation of bubblesof the new phase, their subsequent growth and percolation. This boiling of matterin the Universe is a strongly inequilibrium phenomenon. We will see that baryonasymmetry may indeed be generated in this process.

Even the requirement that the electroweak phase transition is 1st order is insuf-ficient. Medium after the phase transition quickly equilibrates, and the asymmetrygenerated in the course of transition may be washed out. This does not happen onlyif the electroweak baryon number violation rate is smaller than the expansion rateafter the transition. As we will now see, the latter requirement is not fulfilled in theStandard Model (in fact, there is no phase transition in the Standard Model at all),but can be satisfied in some of its extensions.

Electroweak baryon number violation is switched off in the broken phase afterthe phase transition, if the inequality inverse to (11.19) holds,

MW (T )T

� 0.66B(mh/MW )

.

Recalling that MW (T ) = gφ(T )/√

2, αW = g2/4π ≈ 1/30 and B ≈ 2, we find thatthe latter inequality gives14 √

2Φc

Tc� 1, (11.101)

where Tc and Φc are the phase transition temperature and the Higgs expectationvalue just after the transition. The requirement (11.101) strongly restricts thephysics at the electroweak scale. To see this, we make use of the one-loop results ofSec. 10.2. In the one loop approximation we have

Φc

Tc= c · γ

λ,

where λ is the Higgs self-coupling, the parameter γ is defined by (10.30), and theconstant c ranges from 1/2 to 3/4 depending on how delayed the phase transition

14We recall that the normalization of the field φ adopted in Sec. 10 is such that 〈φ〉 = v/√

2.


is (see Eqs. (10.34), (10.35)). Making use of the relation between the Higgs bosonmass and self-coupling, mh =

√2λv, we obtain from (11.101) a bound on the Higgs

boson mass,

m2h < c · 2

3π· 1v·∑

bosons

gim3i , (11.102)

where the realistic value is c = 1/2. Within the Standard Model with one Higgsdoublet, the right hand side obtains contributions from W - and Z-bosons of masses80.4 GeV and 91.2 GeV and numbers of spin states g = 6 and g = 3, respectively(the contribution of the Higgs boson itself may be neglected for estimates). Recallingthat v = 247 GeV, we obtain from (11.102) that the bound in the Standard Modelis [144]

mh < 50 GeV.

This is not the case in Nature: the experimental bound on the mass of the StandardModel Higgs boson is

mexph > 114 GeV (11.103)

(see Appendix B). Thus, electroweak baryogenesis is impossible within the Standardmodel.

In fact, a less restrictive requirement of the 1st order phase transition is notsatisfied in the Standard Model as well. We pointed out in Sec. 10.3 that for allowedvalues of the Higgs boson mass, the electroweak transition is smooth crossover. Thecosmic medium always remains close to thermal equilibrium, and baryon asymmetryis not generated at all. However, this problem may be overcome in some extensionsof the Standard Model, as we explained in Sec. 10.2.

To obtain a simple example, let us add an extra color triplet scalar field,15 whichis a singlet under SU(2)W × U(1)Y . Let us choose the Lagrangian for this field as

Lχ = Dμχ†Dμχ − λχH†H · χ†χ, (11.104)

where H is the Higgs doublet. It is important that this Lagrangian does not containexplicit mass term,16 and the χ-boson obtains its mass in the Higgs vacuum with〈H†H〉 ∝ v2. The χ-boson gives its contribution into the right hand side of (11.102),so the bound (11.102) does not contradict the experimental limit (11.103) formχ > 140 GeV. We notice that both high temperature expansion and one-loopapproximation used in the derivation of (11.102) work poorly in this range of param-eters, but the result is qualitatively correct. Similar mechanism exists in supersym-metric extension of the Standard Model [145]. The analog of χ-boson there is the

15Color is useful for having larger number of degrees of freedom of χ-particles; in our case gχ = 6.

We note that taken at face value, the model has a problem of stability of χ-bosons; this problem

is easily solved by adding interactions leading to χ-boson decays into conventional particles.16If Lχ would contain mass term, the formula (10.30), as well as other formulas of Sec. 10.2 would

not be valid, since we assumed in our derivation that particle masses are related to the Higgs

expectation value by (10.20).


scalar superpartner of t-quarks, and it is again necessary that its explicit (“soft”)mass be small. The latter implies that the mass of t-quark superpartner is close tothe mass of t-quark itself. Similar situations are inherent in other extensions withscalar singlets and/or additional scalar doublets.

In fact, when discussing the model (11.104) we somewhat simplified the analysis. The

point is that the field χ obtains “thermal” mass meff(T ) ∝ pλχT due to interactions with

the Higgs doublet. This mass is due to thermal effects, and hence it depends on temper-ature. It is analogous to the second term in (10.19), but now in the effective potential of thefield χ. Color triplet χ-boson gets contribution to its thermal mass also from interactionswith gluons. Therefore, one has to make sure that the thermal mass is small compared tothe Higgs-induced mass after the phase transition.

Problem 11.19. Does the latter property hold in the model (11.104)?

Thus, electroweak baryogenesis requires the existence of new bosons that areabundant in the medium after the electroweak transition and hence make sub-stantial contribution into the Higgs effective potential. Clearly, the mass of thesenew particles must be at most a few hundred GeV. These particles, if they indeedexist, will be discovered at colliders.

11.5.2 ∗Thick wall baryogenesis

As we discussed in Sec. 10.1, bubbles of new phase spontaneously created at thefirst order phase transition expand to macroscopic sizes before their walls collide.Hence, most of the matter in the Universe interacts with the walls, while the regionswhere the walls collide have small volume (the ratio of the corresponding volumesis d/R, where d is wall thickness and R is the bubble size at percolation). Thus,when calculating the baryon asymmetry one neglects processes occurring in wallcollisions; the baryon asymmetry is generated in interactions of matter with bubblewalls. Importantly, the walls rapidly move through the medium; their velocity isdetermined by “friction” produced by the medium and in most models is in therange of 0.1 – 0.01 of the speed of light.

In this and next Sections we describe mechanisms of baryon asymmetry gener-ation in interactions of cosmic plasma particles with moving bubble wall. Our studywill be rather sketchy, since we will not take into account a number of effects thatare of various degree of importance for this quite complicated dynamical process.For details see, e.g., Ref. [146].

A fairly simple, though not completely realistic mechanism of electroweak baryo-genesis works in so called adiabatic regime. Let us assume that the thickness of abubble wall is much larger than the mean free path of particles in cosmic plasma.Let us also assume that the wall moves through the plasma sufficiently slowly. Thenat any given time and everywhere in space (including the region inside the wall),the medium is in local thermal equilibrium with respect to fast processes of elasticscattering, pair creation and annihilation, etc. On the other hand, the electroweak


baryon number violating processes are not so fast even in the unbroken phase, ascan be seen from (11.16). So, it is natural that the latter processes may be out ofthermal equilibrium. This may lead to the baryon asymmetry generation.

We are going to illustrate this mechanism making use of a simplified model withthe gauge group SU(2)L, two Higgs doublets H1 and H2 and one pair of fermions17

QL (doublet) and qR (singlet), whose Yukawa interactions are similar to those ofthe Standard Model,

Lint = h1QLH1qR + h.c. (11.105)

In what follows we take the coupling h1 real (by redefining the Higgs doublet one canalways set the Yukawa coupling with H2 equal to zero and h1 real). The theory withtwo or more Higgs doublets contains extra, with respect to the Standard Model,source of CP -violation in the Higgs sector. Both in broken phase and in the wallregion, the fields H1 and H2 have expectation values

〈H1,2〉 =

(0

φ1,2

), (11.106)

where φ1,2 are complex functions of the coordinate normal to the wall. The commonphase of φ1 and φ2 can be set equal to zero by the gauge choice, since the gaugetransformation with gauge function exp(iα τ3

2 ) gives

φ1 → ei α2 φ1, φ2 → ei α

2 φ2. (11.107)

Therefore, one can set

φ1 = eiθρ1, φ2 = e−iθρ2, (11.108)

where ρ1 and ρ2 are real. The phase θ is physical.In the presence of the background fields (11.106) the quadratic fermion

Lagrangian, besides the kinetic term, contains the term

Lf = h1qLρ1eiθqR + h.c.,

where qL is the lower component of the doublet QL.If there is CP -violation in the Higgs sector, then both ρ1 and θ vary across the

wall. Since the wall moves, both ρ1 and θ depend on time at a given point of space.Therefore, the fermion Lagrangian depends on time,

Lf = h1qLρ1(t)eiθ(t)qR + h.c.. (11.109)

Time-dependent phase θ(t) leads to CP -violation in the fermion sector and in theend to the baryon asymmetry, once QL and qR are identified with quarks.

17SU(2)L gauge theory with one left doublet has global anomaly. This is unimportant for our

purposes. In realistic extensions of the Standard Model the major effect comes from the Yukawa

interactions of t-quark, and our simplified model is thus adequate.


The simplest way to analyse this mechanism is to perform time-dependent phaserotation of the field18 qR

qR → e−iθ(t)qR. (11.110)

This phase rotation makes the Yukawa coupling θ-independent, but induces extraterm in the fermion Lagrangian, as the kinetic term gives

iqRγμ∂μqR → iqRγμ∂μqR + qRγ0qRθ.

The latter term modifies the Hamiltonian as follows,

H → H − θNR, (11.111)

where

NR =∫

qRγ0qRd3x

is the right quark number operator.We are assuming that transitions from right to left quark (e.g., qR → QL+H) are

fast processes as compared to the rate of time-variation of θ, while the anomalousbaryon number processes are not. If we neglect the latter, the baryon number densityvanishes. In the case of the Hamiltonian (11.111) this means that there is chemicalpotential μB to baryon number, which is the only conserved quantum number inour model. Hence, the calculation of the free energy proceeds by replacing theHamiltonian (11.111) with

H − θNR − μB(NR + NL),

where (NR +NL) = B is the baryon number. Hence, the effective chemical potentialfor right quarks equals (μB + θ), while it is equal to μB for left quarks. Using theresult of Problem 5.2, we find the expression for the baryon number,

ΔB = ΔR + ΔL =T 2

6

[(μB + θ

)+ 2μB

],

where we accounted for two left quark species, hence the factor 2 in the last term.We now require ΔB = 0 and obtain

μB = −13θ.

Let us now turn on the electroweak processes that violate baryon number. Sincethere is non-vanishing chemical potential μB, they generate baryon number by par-tially washing out μB. Let us make use of Eqs. (5.50) and (5.51) and write

dnB

dt= −ΔF · ΔB

TΓsph,

18The phase rotation of QL is an anomalous symmetry, so if we performed that phase rotation

instead, there would appear θ-dependent term in the effective Lagrangian. This would be a dif-

ferent, though equivalent approach. We are not going to use it.


where ΔF and ΔB are the changes of free energy and baryon number at onesphaleron process, and Γsph is the rate of the sphaleron processes. For one leftquark doublet ΔB = 1 and ΔF = μB · ΔB = μB. Thus,

dnB

dt= −μB

TΓsph =

13

θ

TΓsph. (11.112)

Here we use the assumption that the sphaleron processes are slow, and hence weneglect their effect on the evolution of μB. In this way we obtain baryon numberdensity generated by the wall passing through a given region of space,

nB =1

3T

∫θΓsph(t)dt. (11.113)

Here Γsph depends on time, since the Higgs expectation values at a given point inspace depend on time. As a reasonable approximation we take Γsph given by (11.16),

Γsph = κ′α5

W T 4, κ′ ∼ 25

as long as√|φ1|2 + |φ2|2 < T , and Γsph = 0 after that (see Eq. (11.101); the W -

boson mass in the two-Higgs doublet model is determined by v =√|φ1|2 + |φ2|2).

We recall the formula for entropy density s = (2π2/45)g∗T 3 and obtain the estimate

nB

s κ

′ α5W

g∗Δθ,

where Δθ is the change in the phase from the unbroken phase to the point wheresphaleron transitions switch off. For g∗ ∼ 100 and αW 1/30 we find

nB

s 10−8 · Δθ, (11.114)

which is quite acceptable for generating the required asymmetry (11.1). Thisestimate is indeed valid in realistic extensions of the Standard Model modulo thenumerical factor in (11.113). The dominant effect there often comes from t-quark.

The necessary conditions of asymmetry generation are fulfilled here in thefollowing way:

(1) Baryon number is violated because Γsph is non-zero;(2) The source of CP -violation is the time-dependent phase θ;(3) Departure from thermal equilibrium occurs due to the time-dependence of the

phase θ and low rate of electroweak baryon number violating processes.

To end this Section, let us see that the time-dependence of the relative phase of theHiggs fields is indeed possible. Let us consider the Higgs potential of the form

V (H1, H2) = V1(H†1H1) + V2(H

†2H2)

+ λ+[Re(H†2H1) − v1v2 cos 2ξ]2 + λ−[Im(H†

2H1) − v1v2 sin 2ξ]2,

where the minima of the functions V1 and V2 are at |φ1| = v1 and |φ2| = v2, respectively,and λ±, ξ are dimensionless parameters. For λ± > 0 the scalar potential has its minimum at

φ1 = eiξv1, φ2 = e−iξv2,


which corresponds to the vacuum phase

θvac = ξ.

Let the phase transition occur in such a way that both Higgs fields develop the expectationvalues. φ1 and φ2 are small at the beginning of the transition, so the relevant part of thepotential is quadratic in the fields,

Veff = V1,eff (|φ1|2) + V2,eff (|φ2|2) − v1v2Re(φ∗2φ1 · λe2iζ), (11.115)

where

λe2iζ = λ+ cos 2ξ + iλ− sin 2ξ,

i.e.,

tan 2ζ =λ−λ+

tan 2ξ.

The minimum of the potential (11.115) with respect to the relative phase is at

θi = ζ.

This phase gives the direction in the space of the Higgs fields along which the fields rollin the beginning of the phase transition, i.e., in the wall region near the unbroken phase.Hence, the phase θ changes across the wall from θi = ζ (front, near the unbroken phase)to θvac = ξ (back, near the broken phase), as required.

11.5.3 ∗Thin wall case

The case when the wall thickness is small compared to the mean free path of particlesis more realistic, but at the same time more complicated. We will discuss here themost important physical processes leading to the thin wall baryogenesis, leavingaside the detailed analysis.

We are going to use the same model as in previous Section, but with differentvalues of parameters. As before, we assume that CP -violation occurs due to thephase of the scalar field that varies in space. We will work in the wall rest framefor the time being, so instead of (11.109) we have now the following Lagrangian forfermions interacting with the wall,

Lf = h1qLρ(z)eiθ(z)qR + h.c., (11.116)

where the coordinate z is normal to the wall. The function ρ(z) changes from zero(as z → −∞, unbroken phase, we consider the wall moving from right to left) to Φc

(as z → +∞, broken phase); according to (11.101) we have

Φc � T. (11.117)

Let us consider fermion with small Yukawa coupling,19 h1 � 1. Then its effectivemass in the broken phase is also small,

mf = h1Φc � T.

19This is not the case for t-quark, which requires special treatment.


We will make our order-of-magnitude estimates for the wall thickness of the orderof inverse temperature,

lw ∼ T−1. (11.118)

We will also assume (and this is indeed the case) that the wall velocity is small,

vw � 1.

Finally, without loss of generality we set the phase θ(z) equal to zero in the brokenphase, θ(z) = 0 as z → ∞.

Some fraction of fermions moving towards the wall from the unbroken phase getsreflected by the wall. Since the phase θ(z) depends on z, the reflection coefficientsRL and RL for the left fermion and its antiparticle are different. Particles whosemomenta pz well exceed the inverse wall thickness do not experience the reflection(they are well described in the WKB approximation), while particles with lowermomenta do. Still, the reflection coefficient for the latter is small, R � 1: theheight of the energy barrier is equal to mf , which is small compared to pz fortypical momenta pz ∼ l−1

w . On the other hand, particles with pz < mf do notpenetrate through the wall at all. Hence, the important range of momenta is

mf < pz < l−1w . (11.119)

In this range, we can use perturbation theory in h1, so that the reflection amplitudeis of order h1, and reflection probability (amplitude squared) is of order h2

1. Theasymmetry of reflection is thus roughly estimated as

RL − RL ∼ h21θCP , (11.120)

where θCP is determined by the variation of the phase θ in the wall region.It is useful to note at this stage that left fermion after having been reflected

becomes right fermion, see below.If the wall is at rest, the system is in thermal equilibrium, and CP -asymmetric

flux of left and right particles vanishes. This is because the asymmetries of fluxesof reflected and transmitted particles cancel each other. This is no longer the casefor moving wall: the flux of right particles from the wall to the symmetric phaseis different from the flux of their antiparticles (or vice versa). The asymmetry isproportional to the wall velocity. To estimate this asymmetry, we note that momentapx, py of the reflected particles are arbitrary, while pz is in the range (11.119).Hence, we have the estimate for the asymmetry in the flux of right particles andtheir antiparticles

JR ∼ vwT 2l−1w

[RL − RL

]pz∼l−1

w, (11.121)

where we take into account that right particles are left before reflection. Here T 2l−1w

is the number density of particles of relevant momenta, and the factor vw accountsfor the fact that the asymmetry vanishes in the static limit vw → 0. The flux of leftreflected particles is

JL = −JR. (11.122)


Hence, there is an excess of right particles and deficit of left ones (if JR > 0) inunbroken phase in front of the wall. The opposite situation takes place behind thewall in broken phase. The moving wall separates particles of different types.

Once the inequality (11.101) holds, the asymmetry behind the wall does not leadto baryon and lepton number violation. In front of the wall it does: even though theoverall fluxes of baryon and lepton numbers are zero according to (11.122), thereis deficit of left fermions there. These are precisely the particles that participate inelectroweak sphaleron processes. Hence, the latter processes tend to wash out thedeficit of left fermions, thus partially equilibrating the system and generating thenet baryon asymmetry.

The asymmetric fermion reflection changes the medium locally, at distance l

from the wall (we will soon estimate the value of l). At a given point of space, thetime interval from the appearance of the asymmetry to the moment when the wallpasses through that point equals t = l/vw. The baryon number violating processesoccur precisely during this time interval. To estimate l we notice that in time t, thereflected particle experiences t/tf collisions and, according to the Brownian law,moves away to distance

l = lf

√t

tf, (11.123)

where tf and lf are mean free time and mean free path of the particle; lf = tf inour relativistic case. Together with t = l/vw, Eq. (11.123) gives

l =lfvw

, t =tfv2

w

. (11.124)

The excess of left particles generated in the region of size l by collisions with thewall in time t (per unit wall area) is of order

NL = JL · t,so the excess number density is

nindL = JL · t

l=

JL

vw. (11.125)

The sphaleron processes are typically so slow that they wash out small fraction ofthe excess (11.125). Then (see (11.113))

dnB

dt∼ −μL

TΓsph,

where for relativistic particles μL ∼ nL/T 2. This gives

nB ∼ nindL

T 3Γsph · t,

where the time interval during which the sphaleron processes work is given by(11.124). Combining formulas (11.121), (11.122), (11.124), (11.125) and recallingthat the entropy density is s ∼ g∗T 3, we obtain the baryon asymmetry

ΔB ≡ nB

s∼ 1

v2wg∗

tflw

Γsph

T 4

[RL − RL

]pz∼l−1

w. (11.126)


Now, mean free time of left leptons is of order

tf ∼ (α2W T )−1.

The mean free time of quarks is shorter due to collisions caused by strong inter-actions, so the most important20 is τ -lepton: indeed, according to (11.120), theasymmetry is larger for larger h1 = m

v . We use Γsph = κ′α5

W T 4, lw ∼ T−1 as wellas the estimate (11.120) in (11.126) and find for τ -lepton

ΔB ∼ κ′α3

W

v2wg∗

h2τθCP ,

where h2τ = m2

τ

v2 ∼ 10−4. For vw ∼ 3 · 10−2 (which is quite realistic) and κ′ 25 we

obtain numerically

ΔB ∼ 10−6θCP . (11.127)

This is sufficient for the generation of the realistic asymmetry.In our analysis, we have not taken into account a number of important effects

such as dynamical masses of particles, conservation of (B−L) and weak hypercharge,Debye screening of gauge charges in the plasma, etc. Still, the order of magnitudeestimate (11.127) is correct, so the mechanism we described is indeed efficient.

Problem 11.20. At what wall velocities is the approximation of slow sphaleronprocesses, made in the text, valid?

Problem 11.21. At what wall velocities are the transitions of left τ-lepton intoright one, τL + Z → τR + h, irrelevant?

Let us refine the estimate (11.120). To do that, we need to solve the Dirac equation forfermion that interacts with the wall according to (11.116). Proceeding to work in the wallrest frame, we make the Lorentz transformation along the wall after which the fermionmotion is normal to the wall. In our reference frame we have ∂1ψ = ∂2ψ = 0, so the term(11.116) leads to the following form of the Dirac equation,

i∂0qR + iσ3∂zqR + m∗(z)qL = 0, (11.128)

i∂0qL − iσ3∂zqL + m(z)qR = 0, (11.129)

where m(z) = h1ρeiθ. The wave function of left fermion incident on the wall from the left(i.e., from unbroken phase where m(z) = 0), is

q(in)L = e−iωt+ipz ·

„0

1

«, z → −∞, (11.130)

where p = ω > 0. Indeed, this is precisely the solution to Eq. (11.129) with m = 0. To cal-culate the reflection coefficient of left fermion, we have to find the solution to Eqs. (11.128),(11.129), which contains the incoming wave (11.130), outgoing wave at z → −∞ that moves

20Our analysis does not apply to t-quark, in particular because left t-quark quickly becomes right

due to the Yukawa interaction. Hence, t-quark does not play major role in the mechanism under

study.


to the left, and only outgoing wave at z → +∞. We see from Eqs. (11.128), (11.129) thatthe solution has the structure

qL = e−iωtψL(z) ·„

0

1

«, qR = e−iωtψR(z) ·

„0

1

«.

The complex functions ψL and ψR obey

−i∂zψR + ωψR + m∗(z)ψL = 0, (11.131)

i∂zψL + ωψL + m(z)ψR = 0. (11.132)

The reflected wave behaves at z → −∞ as ψ ∝ e−ipz. We see that the reflected wave isright, since Eq. (11.132) with m = 0 does not admit solutions of this form. This generalresult is actually a consequence of angular momentum conservation. Hence, the reflectedwave is

ψR = Ae−ipz, z → −∞, (11.133)

where A is the reflection amplitude, and p = ω. To find the amplitude A we make use ofperturbation theory in m(z). At zeroth order we have ψL = e−ipz , ψR = 0. At the firstorder, ψR is determined from Eq. (11.131), which gives

−i∂zψR + ωψR = −m∗(z)eipz.

The general solution of the latter equation is

ψR = e−ipz

»−i

Z z

z0

m∗(z′)e2ipz′dz′ + c

–, (11.134)

where z0 and c are arbitrary constants. We choose the constant c in such a way thatthere is no left-moving wave beyond the wall, i.e., at z → +∞. We choose z0 to be in theregion behind the wall. There we have m∗ = mf (without loss of generality, h1 is real andθ(z → +∞) = 0), and the solution at z → +∞ is

ψR = e−ipz

»−mf

2p(e2ipz − e2ipz0 ) + c

–.

Hence, the requirement of the absence of incoming wave at z → +∞ gives

c = −mf

2pe2ipz0 .

The solution (11.134) indeed has the form (11.133), and we find

A = −i

Z −∞

z0

m∗(z)e2ipzdz − mf

2pe2ipz0 , (11.135)

where we have to take the limit z0 → +∞. The reflection amplitude A for antiparticle hasthe same form with m∗(z) substituted for m(z).

Problem 11.22. Prove the last statement.

The reflection coefficient for left fermion RL is equal to the modulus of the amplitude(11.135) squared. We are interested in the asymmetry

RL − RL = |A|2 − ˛A

˛2. (11.136)

To proceed further, we write the amplitude (11.135) in the following form,

A = i

Z +∞

−∞[m∗(z) − mfΘ(z)] e2ipzdz − mf

2p,


where Θ(z) is the usual step function, and we have taken the limit z0 → ∞. This is thefinal analytical result for the amplitude. To estimate the asymmetry (11.136), let us assumethat the function Re[m(z)] − mfΘ(z) is symmetric in z. Then

A = −M

2p+

Z +∞

−∞Im[m(z)]e2ipzdz, (11.137)

where

M = mf + 2

Z +∞

−∞{Re[m(z)] − mfΘ(z)} sin(2pz)pdz.

We note that there is an order of magnitude relation

M ∼ mf . (11.138)

From (11.137) we obtain the asymmetry

RL − RL = −2M

p

Z +∞

−∞Im[m(z)] cos 2pzdz.

Finally, we introduce

θCP (p) =1

lwmf

Z +∞

−∞Im[m(z)] cos 2pzdz, (11.139)

and obtain

RL − RL = −2Mmf lwp

θCP (p).

Because of the oscillatory factor cos 2pz, the integral in (11.139) rapidly tends to zero atlarge momenta, p � lw, while at p � l−1

w it is determined by the phase θ(z) of the functionm(z). Making use of (11.138) we obtain the estimate

RL − RL ∼ 2m2f lw

pzθCP , pz � l−1

w , (11.140)

where we recalled that p is the momentum normal to the wall. This estimate is independentof px, py, as the Lorentz boost along the wall does not change pz . We stress that (11.139)contains fermion mass in the broken phase, mf = h1Φc. The estimate (11.120) is obtainedfrom (11.140) at pz ∼ l−1

w by assuming that the wall thickness is of order Φ−1c (i.e., by

assuming the validity of (11.117) and (11.118)).The asymmetry in reflection of right particles is

RR − RR = −(RL − RL). (11.141)

Problem 11.23. Prove the last statement in the general case by making use of the prop-erties of Eqs. (11.128) and (11.129).

To end this Section we notice that both electroweak mechanisms we have dis-cussed employ extra sources of CP -violation as compared to the Standard Model.This extra CP -violation is in fact interesting from the viewpoint of precision particlephysics experiments.

One class of experiments sensitive to new CP -violating phases is the search forelectric dipole moments (EDMs) of electron and neutron, de and dn. EDM d is


the parameter of the Hamiltonian describing the interaction of spin S with electricfield E,

H = −d · E · S|S| .

The corresponding relativistic Lagrangian for fermion ψ is

L = −di

2ψγμγνγ5ψFμν . (11.142)

EDM violates P and T (see Sec. B.3) and therefore CP . EDM of composite par-ticle, neutron, is of the order of EDMs of quarks in it, dn ∼ du, dd. The presentexperimental limits are

de < 1.4 · 10−27 · e · cm = 0.7 · 10−13 · e · GeV−1, (11.143)

dn < 3 · 10−26 · e · cm = 1.5 · 10−12 · e · GeV−1, (11.144)

where e is the electron charge. In the two-doublet model (11.105), the additional CP -violation in the Higgs sector induces quark EDM via one-loop diagram of Fig. 11.16,where H denotes all Higgs fields, and summation over these fields is assumed. Theestimate for this diagram is

du ∼ θCPe

(4π)2mqiUuqiU

∗qiuYuYqi

m2h

, (11.145)

where mh is the Higgs boson mass scale, θCP is the CP -violating phase, U is thequark mixing matrix (so that the Yukawa matrix is Y = Ydiag ·U), (4π)−2 is the loopfactor, and the quark mass mqi comes about due to the fact that the Lagrangian(11.142) violates helicity.

The dominant contribution into (11.145) comes from virtual t-quark, so forUut ∼ 1 we obtain

dn ∼ du ∼ θCP · 1.6 · 10−11 ·(

1 TeV

mh

)2

e · GeV−1. (11.146)

By comparing (11.146) to (11.144) we see that the phase θCP must be fairly small,θCP � 10−3 for mh ∼ 100 GeV, provided that mixing in Higgs-quark interactionsis of order 1. Similar bounds on CP -violating phases exist in other models. These

Fig. 11.16 EDM of u-quark at one loop level.

11.6. ∗Affleck–Dine Mechanism 291

bounds are completely independent of the baryon asymmetry. On the other hand,electroweak baryogenesis requires sizeable CP -violating phases. Thus, most modelswith successful electroweak baryogenesis predict EDMs comparable to the limits(11.143), (11.144). As an example, the result (11.114) for the asymmetry in themodel of the previous Section suggests that θCP ∼ 10−2. Making use of the estimate(11.146) we obtain the prediction

dn ∼ 10−13 ·(

1 TeV

mh

)2

· e · GeV−1.

This is close to the experimental limit for realistic Higgs mass, say mh ∼ 200 GeV.The general conclusion is that the new mechanisms of CP -violation required forelectroweak baryogenesis will most probably be indirectly discovered in the EDMexperiments of the next generation.

11.6 ∗Affleck–Dine Mechanism

11.6.1 Scalar fields carrying baryon number

Some extensions of the Standard Model contain not only quarks, but also scalar par-ticles that carry baryon number. Scalars carrying lepton numbers are also possible.These particles have not been found experimentally, so they must be quite heavy;21

roughly speaking, their masses must exceed a few hundred GeV. Yet the dynamicsof the corresponding scalar fields in expanding Universe may lead to baryon asym-metry generation; the class of baryogenesis scenarios of this sort is generically calledthe Affleck–Dine mechanism [147]. Certainly, the necessary conditions of Sec. 11.1must be satisfied in this case as well. In particular, baryon number should not beexact symmetry, now of the Lagrangian of these scalar fields; CP must be violatedas well.

To construct a prototype model, let us extend the Standard Model by addingcomplex scalar field φ carrying baryon number, Bφ �= 0, and a fermion ψ with zerobaryon number. The kinetic and mass terms in the action for ψ have the standardform, while the action for the scalar field is chosen as

Sφ =∫

d4x√−g [gμν∂μφ∗∂νφ − V (φ)] , (11.147)

where

V (φ) = m2φ∗φ +λ

2(φ∗φ)2 − λ′

4(φ4 + φ∗4). (11.148)

21Assuming that their interactions with Standard Model particles are not particularly weak.


The parameters λ and λ′ are real and positive,22 and we assume that λ′ � λ.Finally, there is a Yukawa term involving φ, ψ and a spinor combination q of quarkfields,

Lint = hqψφ + h.c., (11.149)

where h is the Yukawa coupling. In fact, q may be (and often is) a compositeoperator, a product of the Standard Model fields; it may or may not carry color.The only important property is that the operator q carries baryon number Bq suchthat

Bφ = Bq,

and the Yukawa term (11.149) conserves baryon number. We do not discuss herethe transformation properties of the fields φ and ψ under the Standard Model gaugegroup SU(3)c × SU(2)W × U(1)Y ; the relevant representations can be straightfor-wardly found.

Problem 11.24. Find the representations of φ and ψ under SU(3)c × SU(2)W ×U(1)Y and the composite operator q such that the full action is gauge invariant.

Were the last term in (11.148) absent, the model would be invariant under globalphase rotations23

φ → eiαBφφ, q → eiαBφq, ψ → ψ.

The corresponding conserved quantum number is nothing but baryon number, andthe baryon number density is

nB = iBφ(∂tφ∗ · φ − φ∗ · ∂tφ) + nB,q, (11.150)

where nB,q = 13 (nq − nq) is the baryon number density of quarks. For small but

finite constant λ′ in (11.148) baryon number is conserved only approximately.

Problem 11.25. Obtain the expression (11.150) from the Noether theorem.

The situation similar to what has been just described occurs in supersymmetric exten-sions of the Standard Model, see Sec. 9.6. The composite field φ is a combination of squarks,sleptons and Higgs fields, while the field ψ is a combination of gauginos. There is indeed

the interaction of the form (11.150), and up to numerical factor, the coupling h coincideswith the gauge coupling gs of the color group SU(3)c. The interaction term λ′(φ4 + h.c.)is forbidden by gauge invariance under SU(3)c, but higher order interactions violatingbaryon number and (B −L) are allowed (e.g., λ′φ6). (B −L)-violation is important, sincethe generation of (B − L) asymmetry is necessary and sufficient for the generation ofthe present baryon asymmetry, once one employs mechanisms operating at temperaturesabove 100 GeV.

22The third term in (11.148) is, generically, 14(λ′φ4 + h.c.) with complex λ′. However, by redefining

the field φ it can be cast into the form (11.148). Sign of this term in (11.148) is chosen for future

convenience.23Modulo electroweak anomaly which is irrelevant here.


A generic feature of SUSY extensions of the Standard Model is the existence of flatdirections (moduli) in the field space, along which the scalar potential is small up tolarge values of the scalar fields. In terms of the potential (11.148) in which φ is the fieldparameterizing flat direction, this means that the mass m is low while the constants λ andλ′ are very small. Fairly realistic values are

m ∼ 1 TeV, λ, λ′ ∼ m2

M2Pl

∼ 10−32. (11.151)

In this case the fourth order terms in the potential (11.148) start to dominate over themass term only at φ ∼ MPl.

11.6.2 Asymmetry generation

Let us continue with the model described above. Let the field φ at initial time (rightafter the end of inflation in inflationary theory) be spatially homogeneous and takesome complex value φi = rieiθi/

√2. The analysis of further evolution is simple in

the case

m2|φi|2 � λ|φi|4. (11.152)

Let us discuss this case in some details; the opposite case is the subject ofProblem 11.27. We know from Sec. 4.8.1 that the field stays practically constant forsome time during which the slow roll conditions are satisfied. Once these conditionsget violated, the field, remaining homogeneous, evolves towards the minimum of thepotential, φ = 0, and in a few Hubble times comes to the vicinity of this minimum.Near the minimum, real and imaginary parts of the field evolve independently (thepotential is quadratic), and each of them evolves according to (4.55), i.e.,

Reφ ≡ φR =CR

a3/2(t)cos(mt + βR),

Im φ ≡ φI =CI

a3/2(t)cos(mt + βI).

(11.153)

Note that for time-independent scale factor a the trajectory (11.153) in the fieldspace is an ellipse in complex plane (for βI �= βR). In reality, the scale factor a(t)grows, so the ellipse turns into elliptical spiral, see Fig. 11.17. The presence of thebaryon number violating term in (11.148) (the term with λ′) is important: were itnot for this term, the phases βR and βI would be equal (see below), and the ellipsewould degenerate into interval.

The field (11.153) carries baryon number density

nB = iBφ(∂tφ∗ · φ − φ∗∂tφ) = 2Bφ(φR∂tφI − φI∂tφR) (11.154)

which is equal to

nB = 2BφmCICR

a3(t)sin(βR − βI). (11.155)


Fig. 11.17 Trajectory φ(t) in complex plane.

In the absence of interactions with quarks, baryon number density decreases asnB ∝ a−3, so baryon number is conserved in comoving volume (baryon numberviolation due to the last term in (11.148) is negligible at small φ). Interactionwith quarks (11.149) transmits this baryon number into quarks, and the Universebecomes baryon asymmetric in the end.

As we noticed in Sec. 4.8.1, the field (11.153), in quantum language, describes acoherent state of φ-bosons and their antiparticles at rest. The φ-boson and its antiparticlecarry baryon number Bφ and (−Bφ), respectively. Non-vanishing baryon number (11.155)means that the numbers of φ-bosons and their antiparticles are different. The decay of theoscillating field φ into quarks and fermions ψ may, with reservations, be viewed as decays ofφ-particles and their antiparticles. In this language, the generation of asymmetry betweenquarks and antiquarks in the decay of the field (11.153) is completely obvious.

Let us now turn to the estimate of the generated asymmetry. It is convenient torewrite the action (11.147) in terms of variables r and θ, defined by

φ =r√2eiθ.

The action for the homogeneous field in expanding Universe is

Sρ,θ =∫

dta3(t)(

12r2 +

12r2θ2 − V (r, θ)

),

where

V (r, θ) =m2

2r2 +

λ

8r4 − λ′

8r4 cos 4θ. (11.156)

This gives the equation for θ,

1a3

∂

∂t(a3r2θ) = −∂V

∂θ. (11.157)

The expression for the baryon number density (11.154) has the following form interms of the variables r and θ,

nB = Bφr2θ.


Hence, Eq. (11.157) reads

1a3

∂

∂t(a3nB) = −Bφ

∂V

∂θ. (11.158)

If the potential V were independent of θ, this would be the baryon number conser-vation law for comoving volume. For the θ-dependent potential Eq. (11.158) is theequation describing the generation of baryon number.

Let us assume that the initial phase θi is non-zero. This means that the initialstate is CP -asymmetric; it is in this place where one of the necessary conditionsfor asymmetry generation shows up. This assumption is important, as the evolutionfor θi = 0 proceeds along Imφ = 0, and baryon number density (11.154) is alwayszero. For non-vanishing θi, the baryon number density at later time t is

a3(t)nB(t) = −Bφ

∫ t

ti

∂V

∂θa3(t′)dt′. (11.159)

The integral in (11.159) is saturated in a few Hubble times24 after the field startsrolling down towards φ = 0: before that, ∂V/∂θ is constant but a3(t) is small; afterthat ∂V/∂θ ∝ r4 is small (proportional to a−6, see (11.153)). This immediatelygives the estimate

nB(t) ∼ a3(tr)a3(t)

· ∂V

∂θ(tr) · 1

H(tr),

where tr is the time at which the field starts to roll down. We recall that a(t) ∝ T−1

and obtain the estimate for asymmetry,

ΔB ≡ nB

s∼ 1

g∗T 3r H(Tr)

· ∂V

∂θ(tr). (11.160)

Note that we made an implicit assumption that the field rolls down its potentialat radiation domination. We will proceed under this assumption, and also underthe assumption that the field φ never dominates the energy density. We will discusslater on under what conditions these assumptions are valid.

In the case of the potential (11.157) we write the expression (11.160) as follows,

ΔB(t) ∼ λ′

g∗r4i

Hr(M∗PlHr)3/2

sin 4θi, (11.161)

where we used the standard relation H = T 2/M∗Pl. We made use of the fact here

that before starting to roll down, the field stays almost constant and equal to itsinitial value.

Let us now recall that the field starts to roll at the time when the slow rollcondition gets violated. In the case (11.152) under study, this occurs when

m2 ∼ H2r (11.162)

24It is at this point that the assumption (11.152) simplifies the analysis.


Making use of this relation, we write the estimate (11.161) as

ΔB ∼ λ′g−1/4∗

( ri

m

)5/2(

ri

MPl

)3/2

sin 4θi. (11.163)

Clearly, the resulting asymmetry strongly depends both on parameters of the model(mass m and coupling λ′) and on the initial amplitude ri and phase θi of the scalarfield φ. Hence, the value of ΔB appears to be the result of a choice of initial data.

To find out under what conditions the expression for the asymmetry (11.163)gives the required value ΔB ∼ 10−10, we have to specify the initial data ri and θi.One possible (but not at all unique) choice is to assume that the initial phase is notsmall, sin 4θi ∼ 1, while the initial amplitude is of the order of the Planck scale,ri ∼ MPl. Then one finds from (11.163) and m � MPl that the required asymmetryis obtained only for very weak baryon number violation, λ′ � 10−10. Otherwise theasymmetry is too large. This result illustrates rather general property of the Affleck–Dine mechanism: it often (though not always) generates too large asymmetry.

At ri ∼ MPl, the assumption that the slow roll regime terminates when theUniverse is already at the hot stage of its evolution is valid for quite small m andvery small λ only. Let Tmax be the maximum temperature in the Universe. Thenthe relation (11.162) indeed holds at Ti < Tmax only if

m <T 2

max

M∗Pl

.

For Tmax ∼ 1015 GeV (which is the order of magnitude of the maximum possiblereheating temperature in realistic inflationary models, see accompanying book) andri ∼ MPl we have

m < 1012 GeV. (11.164)

Furthermore, our assumption that m2|φ|2 � λ|φ|4 holds only for λ � 10−14, whilethe required baryon asymmetry is obtained from (11.163) at λ′ � 10−24. Theseestimates illustrate another property of the Affleck–Dine mechanism: it needs almostflat scalar potential V (φ) for the field φ. As we pointed out, almost flat potentialsare rather natural in supersymmetric theories, see (11.151).

Let us discuss one more property of the Affleck–Dine mechanism, still assumingthat ri is roughly of order MPl. It is clear from (11.162) that at the beginning ofoscillations, the energy of the field φ is roughly comparable to the energy of hotmatter,

ρφ(tr)ρrad(tr)

∼ m2r2i

H2M2Pl

∼ r2i

M2Pl

.

The energy density of the oscillating field (11.153) decreases in the same way asthat of non-relativistic matter, ρφ ∝ a−3(t), see (4.61), and the decay of oscillationsoccurs at H ∼ Γ, see Sec. 5.3, where Γ is the decay width of the φ-boson. For Γ � m

(which is consistent with the fact that φ has tiny self-interaction) this implies that


the energy of the oscillating field starts to dominate in the Universe soon afterthe oscillations begin. The existence of the intermediate matter-dominated stage isanother characteristic property of the Affleck–Dine mechanism.

Problem 11.26. If there is indeed the intermediate matter-dominated stage, theestimate (11.163) is not valid. Estimate ΔB in that case, still assuming that slow rollconditions (see Eq. (4.49) and Problem 4.11) get violated at radiation domination.

Problem 11.27. Consider the case

λ|φi|4 � m2|φ2i |.

(a) Show that the amplitude of oscillations in the case of quartic potential, V = λφ4,

decreases as a−1 (as opposed to a−3/2 for quadratic potential). This means that∂V/∂θ in the integrand in (11.159) decreases as a−4(t′) until the time whenλr4 ∼ m2r2, and only afterwards it decreases as a−6.

(b) Making use of this observation and assuming that the energy of the field φ neverdominates in the Universe, show that the resulting asymmetry is of order

ΔB ∼ λ′g−1/4∗

ri

mλ3/4

(ri

MPl

)3/2

sin 4θi

(c) Show that for ri roughly of order MPl the required properties of the theory (smallm, very small λ′, etc.) are qualitatively the same as in the case considered inthe text.

Let us now discuss a version of the Affleck–Dine mechanism (cf. Ref. [148]), inwhich the initial value ri is determined dynamically, while the flatness requirementsfor the scalar potential are less severe. Let us assume that besides the terms writtenin (11.147), there is one more term in the action,

SRφ = −∫

d4x√−gcR |φ|2, (11.165)

where c is a positive dimensionless constant which we choose somewhat larger (butnot much larger) than 1. For the FLRW Universe one has R = −12H2, so this extraterm effectively changes the mass term in the potential,

m2 |φ|2 → (m2 − 12cH2) |φ|2. (11.166)

At early times, when H � m, the potential has maximum at φ = 0, and in thelimit λ′ → 0 there is a valley of minima on a circle in complex plane,

|φ|2 ≡ r2

2=

12cH2

λ. (11.167)

Slow roll conditions are not satisfied in this case for radial motion (this is the reasonfor the choice c > 1), so r(t) is approximately given by (11.167). At small but finiteλ′ the valley (11.167) is slightly inclined, the potential depends on the phase θ, butthis phase is almost flat direction. The evolution along this direction may occur inthe slow roll regime.


Problem 11.28. Show that for H � m and λ′ � λ the evolution of the phase θ

occurs in the slow roll regime.

Hence, for λ′ � λ the phase θ = θi remains constant until the time when

Hi ∼ m. (11.168)

At about this time the effective mass term (11.166) changes sign. The field rollsdown towards φ = 0 starting from

ri ∼ m√λ

, θ = θi. (11.169)

At this time and later, the case (11.152) is a reasonable approximation, so the result(11.161) holds. Making use of (11.168) and (11.169) we obtain the asymmetry

ΔB ∼ λ′

g∗λ2

(m

M∗Pl

)3/2

sin 4θi.

Clearly, the required value is obtained in a wide range of parameters. We note,though, that this version also requires rather high reheating temperature: sinceHi < H(Tmax) = T 2

max/M∗Pl and at the same time Eq. (11.168) holds, we have an

estimate

Tmax > (mM∗Pl)

1/2, (11.170)

which is typically quite large.

Problem 11.29. Consider this mechanism for c � 1 where c is the parameter in(11.165). Can this mechanism work at the hot stage and yield the observed baryonasymmetry for Tmax � 109 GeV?

Thus, the Affleck–Dine mechanism can successfully work at the hot stage, albeitin a rather restricted range of parameters. The scalar potential must be quite flat,and the maximum temperature in the Universe quite high. An interesting possibilityhere is that the Universe is temporarily dominated by the oscillating scalar field,and hence experiences intermediate matter dominated stage.

If the maximum temperature in the Universe is not very high, and the inequality(11.170) is not valid, then the field φ rolls down to the minimum of its potentialbefore the hot stage. This is a fairly realistic situation: it happens for Tmax �1010 GeV and the mass of φ of order 1 TeV or higher. In that case the Affleck–Dinemechanism may work at the latest stage of inflation or at reheating. The corre-sponding estimates are different from those given above, but the general conclusionon the possibility of the baryon asymmetry generation remains valid.

11.7 Concluding Remarks

The baryon asymmetry generation mechanisms considered in this Chapter by nomeans exhaust the possibilities proposed in literature. Among other things, the

11.7. Concluding Remarks 299

asymmetry generation may occur at the latest stage of inflation or at reheating,rather than at the hot stage. Unfortunately, many mechanisms (like those studied inSecs. 11.3 and 11.4) make use of physics beyond the energy scale accessible at futurecolliders. Hence, direct experimental evidence in favor of one of these mechanismswill be hard if not impossible to obtain. The exception is the electroweak mechanismthat will be supported or falsified by collider experiments in near future. As tothe Affleck–Dine mechanism, it may be supported by the discovery of so calledbaryon isocurvature perturbations in the spectrum of density perturbations in theUniverse.25 Search for isocurvature perturbations is one of the goals of the CMBstudies. We consider this aspect in the accompanying book.

25Generation of baryon isocurvature perturbations is a possible, but not necessary consequence of

the Affleck–Dine mechanism, so this mechanism cannot be ruled out by the cosmological data.

Chapter 12

Topological Defects and Solitonsin the Universe

In this Chapter we consider cosmological aspects of field theory models which admitsoliton or soliton-like solutions. These solutions are specific (sometimes macro-scopic) field configurations whose stability is ensured either by non-trivial topologyof the space of vacua (topological defects) or by the existence of conserved globalquantum numbers (non-topological solitons, Q-balls). Of interest are both particle-like solitons (monopoles, Q-balls) and extended objects (cosmic strings, domainwalls); we will jointly call them defects in what follows. Generally speaking, theremay be also unstable solutions whose lifetime is comparable to the age of the Uni-verse. We will not discuss the latter, since their cosmological consequences are quitesimilar to those of stable solutions.

We point out right away that no undisputable evidence for defects of any sort inour Universe has been found so far. Nevertheless, they are inherent in many exten-sions of the Standard Model, so their study is of considerable interest. We will seethat there are numerous effects occurring in models with defects, including pecu-liarities of the expansion history, new mechanisms of structure formation, lensingof distant sources, features in CMB angular anisotropy, new processes generatingthe baryon asymmetry, etc. The energy scale of the defect formation is considerablyhigher than the electroweak scale. Experimental discovery of defects would thusgive evidence that the temperature in the Universe reached that energy scale.1 Fur-thermore, this discovery would show the existence of physics beyond the StandardModel and give a hint on what this new physics is. This would open up a windowto energy scales well exceeding the reach of particle colliders.

A general reason for the existence of topological defects in field theoretic models is thenon-trivial topology of the space of vacua in these models. This means that the groundstate — vacuum — is not unique, and, furthermore, some homotopy group πN of themanifold of vacua M is non-trivial,

πN (M) �= 0. (12.1)

In other words, there exist non-trivial mappings from N-dimensional sphere SN to thevacuum manifold M. In most cases the defect configuration corresponds to a non-trivial

1In fact, there exist mechanisms of defect production that do not require so high temperature.

But in any case, the energy density at the time of defect production must be very high.

301

302 Topological Defects and Solitons in the Universe

mapping from spatial asymptotics (sphere SN in the general case) to vacuum manifold(since the fields tend to vacuum values at spatial infinity, otherwise energy would beinfinite). In space-time of dimension d + 1 the property (12.1) suggests the existence oftopological defects of space-time dimension d − N . Stability of these defects is due tothe fact that destruction of a defect would require rearrangement of the fields at spatialinfinity; that would cost infinite energy. Non-trivial configurations thus have topologicalcharges which are conserved. In the case of 4-dimensional space-time, three types of defectsare possible, whose dimensions are 2 + 1 (walls), 1 + 1 (strings) and 0 + 1 (particle-likedefects, e.g., monopoles). There may exist hybrid defects, including combinations of defectsof different dimensionality.

We note that the relevant symmetry of a theory may be local (gauge) or global. Accord-ingly, the defects are called local or global. In the former case the energy is localized. On thecontrary, in the global defect case, the gradient energy falls of with distance so slowly thatthe energy integral diverges. In the physically relevant situation with numerous defects thismeans that one cannot neglect the interaction between them. These interactions renderthe energy density finite.

12.1 Production of Topological Defects in the Early Universe

Rather general property of field theory models with topological defects is that thedefects exist only in the phase with spontaneously broken symmetry, whereas theyare absent in the unbroken phase. We will encounter this situation in Secs. 12.2–12.5.As we discussed in Chapter 10, symmetry is usually restored at high temperatures,and the broken phase appears in the Universe at lower temperatures as a result ofphase transition. Thus, the existence of topological defects is possible only after thisphase transition, i.e., at T < Tc, where Tc is the phase transition temperature.2

Defects may be produced after the phase transition thermally, in collisions ofparticles. This mechanism is often inefficient. If defects equilibrate, their density issmall. Indeed, considering for definiteness particle-like defects like monopoles (seeSec. 12.2), we quote the result that their mass MTD is typically much larger than thecritical temperature, so the equilibrium abundance is suppressed by the Boltzmannfactor

n(eq)TD ∝ e−

MT DTc . (12.2)

Strings and domain walls are suppressed even stronger.There exists, however, another way of producing topological defects, which is

called Kibble mechanism [149]. We will consider this mechanism in detail in Sec. 12.2where we study magnetic monopoles, but we will see that it is quite general. Thismechanism works at the phase transition epoch. The phase transition proceedsindependently in regions of the Hubble size which are causally disconnected in thehot Big Bang theory. Inside a Hubble region the field configurations are correlated,

2In fact, the production of topological defects is possible at both thermal phase transitions

occurring at the hot Big Bang stage and non-thermal phase transitions that may take place,

e.g., at post-inflationary reheating.

12.2. ∗’t Hooft–Polyakov Monopoles 303

so the system ends up in one and the same vacuum. On the other hand, vacua indifferent Hubble regions may belong to different points of the vacuum manifold.Hence, the topology of the field configuration at the spatial scale exceeding theHubble scale may coincide with the defect topology. The relaxation processes afterthe phase transition do not change the topology, so the result is the formation of adefect. This is precisely the Kibble mechanism. It is clear from the above reasoningthat this mechanism is quite universal; it works for defects of different types anddepends rather weakly on MTD/Tc.

More precisely, the number density of defects produced by the Kibble mechanismis determined, right after the phase transition, by the correlation length lcor, whichdefinitely does not exceed the Hubble size,3

lH(tc) � H−1(tc).

So, the number of defects is at least of order 1 per the Hubble volume. Makinguse of the relation between the Hubble parameter and temperature at radiationdomination, Eq. (3.31), we have

nTD(tc) � lH(tc)−3 =T 6

c

M∗ 3Pl

. (12.3)

This estimate is general for point-like topological defects, modulo a numerical factor(the latter may be quite different from unity, though). For extended defects (cosmicstrings, domain walls), the above argument applies to the distance lD(tc) betweenthem, i.e., one has lD(tc) � lH(tc).

Further evolution is specific for defects of different types. As a rule, this evolutionis out of thermal equilibrium: the equilibrium number density of defects falls offexponentially, while the actual density exhibits power law behavior. The evolutionof various defects will be considered in the following sections, and here we mentionone property. Right after the phase transition, defects do not affect the expansionrate, since their energy density is smaller than that of hot matter by a factor T α

c /MαPl

where the positive exponent α depends on the type of defects (it follows from (12.3)that α = 3 for point-like defects). However, at later times the energy density ofdefects may become significant and may even dominate the expansion.

12.2 ∗’t Hooft–Polyakov Monopoles

12.2.1 Magnetic monopoles in gauge theories

The simplest model admitting monopole (and antimonopole) solution [150, 151] isthe Georgi–Glashow model. This is the gauge model with gauge group SU(2) and

3In the case of the first order phase transition, lcor is the typical size of bubbles at percolation.

For the second order phase transition, lcor is the size of a region in which the energy needed to

unwind the defect is of the order of temperature.


triplet of scalar fields φa, a = 1, 2, 3 which transform under adjoint representation.The Lagrangian is

L = −14F a

μνF aμν +12DμφaDμφa − λ

4(φaφa − v2)2, (12.4)

where

F aμν = ∂μAa

ν − ∂νAaμ + gεabcAb

μAcν ,

Dμφa = ∂μφa + gεabcAbμφc.

The last term in (12.4) is self-interaction of the scalar fields leading to spontaneoussymmetry breaking. The vacuum energy density is determined by minimizing thescalar potential, so that

〈φaφa〉 = v2, (12.5)

This equation determines the vacuum manifold; in our case it is a 2-sphere S2vac.

Each point in this manifold is invariant only under the subgroup U(1) of the gaugegroup SU(2). Hence, the symmetry breaking pattern is SU(2) → U(1). We identifythe unbroken U(1) as the gauge group of electromagnetism in this toy model.

By choosing the vacuum as

〈φa〉 = δa3v, (12.6)

we find that the field A3μ remains massless (gauge field of unbroken U(1), photon),

while the fields

W±μ =

1√2(A1

μ ± iA2μ) and h = φ3 − v

obtain masses

mV = gv and mh =√

2λv,

respectively.

Problem 12.1. By writing the quadratic action for small perturbations aboutthe vacuum (12.6), find the spectrum of the theory, thus confirming the abovestatements.

Let us study static field configurations of the form

Aa0 = 0, Aa

i = Aai (x), φa = φa(x),

and require that the energy is finite. The energy functional is

E =∫

d3x[14F a

ijFaij +

12Diφ

aDiφa +

λ

4(φaφa − v2)2

]. (12.7)

Hence, at spatial infinity (r → ∞, where r2 ≡ x2) the fields tend to vacuum values,φaφa = v2, Aa

μ = 0, up to gauge transformation. Furthermore, the convergence ofthe energy integral requires sufficiently fast decay at infinity of the gauge-covariantquantities F a

ij , Diφa and (φaφa − v2).


The length of the vector (φ1, φ2, φ3) is equal to v at spatial infinity. Hence, anyfinite energy configuration defines the mapping from a sphere S2

∞ at spatial infinityto vacuum manifold S2

vac. Since

π2(S2) = Z �= 0,

this mapping may be topologically non-trivial, and the configuration topologicallystable.

The simplest non-trivial mapping is given by the hedgehog configuration of thescalar field, see Fig. 12.1,

φa(r → ∞) = vna, na ≡ xa

r. (12.8)

The energy (12.7) is finite provided that the asymptotics of the vector field is

Aai (x) =

1gr

εaijnj . (12.9)

Thus, we write the following Ansatz,

φa = vna · (1 − f(r)),

Aai =

1gr

εaijnj(1 − a(r)).(12.10)

This Ansatz is invariant under spatial rotations supplemented by rotations in theinternal space, i.e., by global SU(2)-transformations. The functions f(r) and a(r)must obey the requirements of the absence of singularity at the origin and finitenessof energy. This gives

f(r → ∞) = a(r → ∞) = 0,

[1 − f(r → 0)] ∝ r, [1 − a(r → 0)] ∝ r2.

Fig. 12.1 Asymptotic behavior of the Higgs field for the monopole configuration. The direction

of the Higgs field in internal space (shown by arrows) is the same as the direction of the radius

vector in the physical space.


The actual configuration of the soliton is obtained by solving the field equations interms of f(r) and a(r).

Problem 12.2. Show that the Ansatz (12.10) is consistent with the field equations.Show that at g ∼ √

λ, the functions f(r) and a(r) decay exponentially at spatialinfinity.

The soliton just described is a magnetic monopole. Indeed, according to (12.9),the electric components of the field strength vanish, while the magnetic componentsare

Bai = −1

2εijkF a

jk =1

gr2nina. (12.11)

Their direction in the internal space coincides with the direction of the Higgs field,so this field is actually the magnetic field of the unbroken electromagnetic U(1)gauge group. The gauge-invariant field strength

Bi = Bai

φa

v=

1g

ni

r2(12.12)

equals the magnetic field of magnetic monopole of magnetic charge gm = 1/g. Themassive vector and Higgs fields decay exponentially away from the monopole center,and far from the center the field coincides with that of the Dirac monopole. Thesoliton we discuss is known as the ’t Hooft–Polyakov monopole.

It is useful to note that there is also an antimonopole solution. It has the differentbehavior of the Higgs field,

φa = −vna(1 − f(r)),

whereas the field Aai is the same as for monopole. The magnetic field (Ba

i φa/v) hasthe sign opposite to that in (12.12).

For mV ∼ mh, the monopole mass can be estimated on dimensional grounds.One rewrites the integral (12.7) by changing the variables

x = (gv)−1ξ, Aaμ = vAa

μ, φa = vϕa.

Then the energy (12.7) becomes

E =mV

g2

∫d3ξ

[14Fa

ijFaij +

12Diϕ

aDiϕa +

m2h

8m2V

(ϕaϕa − 1)2]

,

where Diϕa = ∂iϕ

a + εabcAbiϕ

c, etc. The monopole configuration minimizes theenergy functional, and for mV ∼ mh the integrand does not contain large or smallparameters. Hence, the monopole energy (mass) is estimated as

mM 4πmV

g2=

4πv

g,

where the factor 4π corresponds to angular integration. Thus, the monopole massexceeds the energy scale v of symmetry breaking SU(2) → U(1).


The existence of magnetic monopoles is a general property of Grand UnifiedTheories, GUTs. In GUT context, monopoles have masses of order mM ∼ 1017 GeV.They are produced at the GUT phase transition which occurs at Tc ∼ 1016 GeV. Ofcourse, this is true under the assumption that so high temperatures indeed existedin the Universe.

Let us describe the necessary (and very often sufficient) condition for the existenceof magnetic monopoles in a general gauge theory with the Higgs mechanism. Let theLagrangian be invariant under gauge group G and the ground state be invariant undersubgroup H of the group G. In other words, the symmetry breaking pattern is G → H .The manifold M of vacua is then the coset space G/H (here we assume that G actson M transitively, i.e., all vacua are related to each other by symmetry transformations;this assumption is not, in fact, completely necessary). Stable monopole configurations arepossible if the vacuum manifold M contains non-contractible 2-spheres, i.e., if its secondhomotopy group is non-trivial,

π2(G/H) �= 0. (12.13)

If the gauge group G is simple or semi-simple (gauge groups are always compact), and Hincludes one factor U(1), then

π2(G/H) = π1(H) = Z,

so monopoles exist. In the case of the Standard Model, the gauge group G = SU(3) ×SU(2) × U(1) is not semi-simple, the unbroken subgroup is H = SU(3) × U(1) andπ2(G/H) = 0; thus, there are no monopoles. The situation is entirely different in GUTs.There, the gauge group is, as a rule, simple (sometimes semi-simple), and the symmetrygets broken at high energies down to SU(3)c × SU(2)W ×U(1)Y , and at lower energies toH = SU(3)c × U(1)em (an example is the SU(5)-theory, discussed in Sec. 11.2.2). Hence,π2(G/H) = Z, and monopoles always exist.

12.2.2 Kibble mechanism

Let us make use of the Georgi–Glashow model and the ’t Hooft–Polyakov monopoleto discuss the Kibble mechanism. The Higgs expectation value vanishes at high tem-peratures, the system is in symmetric phase and monopoles do not exist. The Higgsexpectation value becomes non-zero as a result of the phase transition. Just afterthe phase transition, the directions of the Higgs field are uncorrelated at distancesexceeding the correlation length lcor. As a result, there are regions in the Universewith the Higgs field shown in Fig. 12.2(a), as well as regions where the Higgs fieldconfigurations show the patterns of Figs. 12.2(b) and 12.2(c). The configurationof Fig. 12.2(a) is topologically trivial, and in the course of evolution it relaxes toa state without a monopole. On the other hand, the configuration of Fig. 12.2(b)has hedgehog topology, cf. Fig. 12.1; it relaxes to a monopole. The configuration ofFig. 12.2(c) becomes an antimonopole in the end. The probabilities of all three typesof configurations are roughly the same (and precisely the same for configurations ofFigs. 12.2(b) and 12.2(c)), so the number densities of monopoles and antimonopolesjust after the phase transition are estimated as

nM = nM ∼ l−3cor.


(b) (c)(a)

Fig. 12.2 Possible Higgs field configurations just after the phase transition. Circle radii are of

order lcor.

The Higgs condensate is definitely formed in an independent way at separationexceeding the cosmological horizon, so that lcor � H−1(Tc), as we noticed inSec. 12.1. Hence, the abundance of monopoles produced by the Kibble mechanismis indeed estimated as in (12.3).

12.2.3 Residual abundance: the monopole problem

Monopoles are non-relativistic at temperatures below the phase transition in whichthey are produced. If the monopole-antimonopole annihilation were negligible, theirnumber density would decrease nM(t) ∝ a−3(t) and the monopole-entropy ratiowould remain constant. Hence, the mass density of monopoles today would beequal to

ρM,0 = mMnM,0 = mM

nM,Tc

s(Tc)·s0 ∼ √

g∗·mM

T 6c

M3Pl

g∗,0T30

T 3c

∼ 1012 mM

1016 GeV

(Tc

1016 GeV

)3√g∗102

GeVcm−3, (12.14)

where we made use of the expression (5.28) for entropy density and estimated themonopole number density according to (12.3). The scale 1016 GeV is typical forGUTs, so we see that the mass density of monopoles would exceed the criticaldensity ρc ∼ 10−5 GeVcm−3 by 17 orders of magnitude. As we will see momentarily,monopole-antimonopole annihilation does not change the result qualitatively: thereremain too many monopoles. This is the essence of the monopole problem [152, 153],as we should either suppose that the temperature in the Universe was never as highas Tc ∼ 1016 GeV, or give up the idea of Grand Unification.

Let us now take into account monopole-antimonopole annihilation. First, letus check that monopoles are in kinetic equilibrium with plasma. A non-relativisticmonopole in plasma experiences effective friction force due to the electromagneticinteractions with plasma particles. This force is estimated as f ∼ n(T ) · σ · Δp,where n(T ) is the number density of charged particles in plasma, σ ∼ αg2

m/T 2

is the interaction cross section and Δp ∼ TvM is the momentum that monopole


of velocity vM looses in each interaction. We now write the second Newton lawneglecting the cosmological expansion,

mM

dvM

dt= −κT 2vM ,

where κ ∼ αg2mg∗ ∼ g∗. We see that the monopole velocity changes significantly

in the time interval tM ∼ mM/(κT 2). This time is always smaller than the Hubbletime M∗

Pl/T 2, so the monopoles are indeed in kinetic equilibrium with plasma. Theyhave thermal velocity of order vM ∼ vT =

√T/mM , and mean free path is

lM ∼ vT · tM =1

κT

√mM

T.

The mean free path grows as temperature decreases.The efficient monopole-antimonopole annihilation occurs at high temperatures

when the mean free path is short. The monopole and antimonopole attract eachother, while interactions with plasma damp their velocities. As a result, they formmonopolonium, the monopole-antimonopole bound state. Monopolonium then anni-hilates into conventional particles. Unlike direct annihilation, which has very smallcross section, this two-stage process reduces the monopole abundance significantly.

The bound state can be formed when the electromagnetic interaction energyof monopole and antimonopole exceeds temperature, i.e., at r � r0 = g2

m/T . Thisdetermines the cross section of monopolonium production, which subsequently leadsto annihilation,

σann ∼ r20 ≡ g4

m/T 2.

This estimate is valid at high temperatures, when lM is small compared to r0,otherwise monopolonium is not formed, and annihilation practically does not occur.

The monopole abundance at high temperatures is such that the monopole meanfree time with respect to annihilation is of order of the Hubble time,

σannnMvM ∼ T 2

M∗Pl

.

We recall that monopoles have thermal velocities and obtain

nM

s· g∗g4

m

M∗Pl√

mMT∼ 1, (12.15)

where we used s ∼ g∗T 3. We see that annihilation dilutes nM/s as the Universeexpands. This regime, however, terminates at lM ∼ r0, and monopole abundancefreezes out. The latter relation gives freeze-out temperature,

Tf ∼ mM

g4mκ2

.

The monopole abundance at freeze-out is then found from (12.15),

nM

s∼ 1

κg6m

mM

M∗Plg∗

. (12.16)


This ratio stays constant until today; using g2m ∼ α−1 ∼ 100, κ ∼ g∗ ∼ 100 we get

the present mass density

ρM,0 = mM ·nM

s· s0 ∼ 107·

( mM

1016 GeV

)2

GeVcm−3.

This is much smaller than the estimate (12.14), but still way too large for GUTmonopoles with mM ∼ 1017 GeV.

In retrospect, monopole problem was a strong reason for not extrapolating thehot Big Bang theory up to temperature 1016 GeV. This was (and still is) one of thearguments in favor of inflation.

Problem 12.3. Show that at T � Tf the monopole-antimonopole annihilationpractically does not change the ratio (12.16). Hint: Assume that the effect of plasmaon annihilation process is negligible. Take into account Coulomb enhancement ofthe annihilation cross section.

Problem 12.4. Find the temperature in an unrealistic Universe where monopolesof mass mM ∼ 1017 GeV were produced at GUT phase transition at T ∼ 1016 GeV,

at the time when the Hubble parameter takes the same value as in our presentUniverse.

12.3 ∗Cosmic Strings

12.3.1 String solutions

The minimal model admitting cosmic strings — dimension-1 topological defects —is the Abelian Higgs model.4 Its Lagrangian is

L = Dμφ∗Dμφ − 14FμνFμν − λ

(φ∗φ − v2

2

)2

,

Dμφ = ∂μφ − ieAμφ, Fμν = ∂μAν − ∂νAμ,

(12.17)

where φ is complex scalar field, Aμ is the gauge field of U(1) gauge group. TheLagrangian (12.17) is invariant under U(1) gauge transformations

φ → φeiα(x), φ∗ → φ∗e−iα(x), Aμ → Aμ +1e∂μα(x).

The U(1) symmetry is spontaneously broken in the theory (12.17): the vacua areobtained by minimizing the scalar potential and obey

〈φ∗φ〉 =v2

2(12.18)

Neither of this vacua is invariant under U(1); the symmetry G = U(1) is brokencompletely (G → H , H = I). According to (12.18) the vacuum manifold is a circle

4Models with global Abelian symmetry group may admit global strings.

12.3. ∗Cosmic Strings 311

S1 (this is in accordance with the relation G/H = U(1), as U(1) is itself a circle),

〈φ〉 =v√2eiα, α ∈ [0, 2π).

The spectrum is obtained, as usual, by writing the Lagrangian (12.17) as Taylorseries about the vacuum values of the fields φvac = 〈φ〉, Avac

μ = 0 and using thegauge Imφ = 0, i.e., φ = v/

√2 + h with real h. One finds that all fields become

massive as a result of spontaneous symmetry breaking; the model contains realvector and scalar fieds with masses

MV = ev and Mφ =√

2λv,

respectively.In the hot Universe at T � v symmetry is restored, and both real and imag-

inary components of the complex field φ take random values at different points inspace. The random phase α(x), defined formally as arctan[Reφ/Imφ], is distributedhomogeneously. At critical temperature Tc, the Higgs field acquires non-zero value,and the values of the phase α(x) get fixed. Generally speaking, they are different atdifferent points. Due to the gradient term in the energy, the configuration becomesmore homogeneous in the course of evolution. The phases α(x) may become equalover the horizon scale in the end, but this is not necessarily the case: the initialstate may be such that there is a non-trivial winding around some contour C. Single-valuedness of the field requires that the phase obeys

Δα =∮C

dα

dθdθ = 2πN, (12.19)

where θ is the asimuthal angle in physical space and N is an integer. Configurationswith N �= 0 become strings in the end, and the mechanism of their production whichwe just described is nothing but the Kibble mechanism. Cosmic string may againbe illustrated by Fig. 12.1, where the plane of the plot is now the plane orthogonalto the string, and arrows show the directions of vector (Reφ, Imφ) in internal space.With this clarification, the Kibble mechanism is again illustrated by Fig. 12.2; itleads to the production of about one piece of string of length lcor per volume l3cor.

Continuity of the field φ guarantees that strings must be either closed or infinite.The latter are open strings stretching across the entire horizon.5 The contituity of φ

requires also that φ vanishes along a line surrounded by the contour of integration Cin (12.19). This means that the string configuration has large energy density there,E ∼ λv4.

At large distance from the center, the string configuration minimizes the scalarpotential (12.18), but has non-trivial angular dependence of φ and Aμ. In the caseof infinite straight string stretching along z-axis, the asymptotics in the plane (x, y)are (modulo irrelevant constant phase)

φ → v√2eiNθ, Aμ → N

e∂μθ, (12.20)

5It is hard to “get rid” of strings, if they exist in the Universe, for this reason.


The winding number N is a topological invariant. The behavior of the field Aμ in(12.20) is such that asymptotically, i.e., at x2 +y2 → ∞, there are no physical fields,

Dμφ → 0, Fμν → 0,

and string energy per unit length is finite. It follows from (12.20) that there ismagnetic field B = ∇× A inside the string core. Its flux is given by∫

Bds =∮

Aθdθ =2π

eN.

We see that it is quantized in units of 2π/e.These defects are called Abrikosov vortices in condensed matter theory; these

are magnetic flux tubes in superconductors. In particle theory, these objects areknown as Abrikosov–Nielsen–Olesen strings [154, 155], or simply cosmic strings.

In gauge-Higgs theories of general type, the existence of cosmic strings is possibleprovided the vacuum manifold M is not simply connected, i.e., it contains non-contractibleloops. The necessary (and often sufficient) condition for the existence of cosmic strings isthus non-trivial first homotopy group; in notations used in (12.13),

π1(G/H) �= 0. (12.21)

Rotationally symmetric string Ansatz is

φ =v√2[1 − f(ρ)]eiNθ, Ai = −N

eρ

εijxj

ρ[1 − a(ρ)], i, j = 1, 2, (12.22)

where ρ =√

(x1)2 + (x2)2 is the radial coordinate in the plane (x, y). Here, therotation of the plane can be undone by the phase rotation of φ. Large-ρ asymptoticsshould coincide with (12.20), which means

f(ρ → ∞) → 0, a(ρ → ∞) → 0.

The fields must be non-singular at the origin, so we have the following behavior,

f(ρ → 0) → 1, a(ρ → 0) → 1.

Analytical solutions of the field equations are unknown, but the solutions arestraightforward to find numerically by making use of the Ansatz (12.22).

Let us estimate the string energy per unit length, i.e., the string tension. Inanalogy to the estimate of the monopole mass, we write

μ =dE

dz=∫

d2x

[Diφ

∗Diφ +14F 2

ij + λ

(φ∗φ − v2

2

)2]

= v2

∫d2ξ

[Diϕ

∗Diϕ +14F2

ij +λ

e2

(ϕ∗ϕ − 1

2

)2]

, (12.23)


where the change of variables is φ = vϕ, x = (ev)−1ξ, Ai = vAi. This gives theestimate for the tension,

μ ∼ πv2, (12.24)

for N ∼ 1 and λ ∼ e2. We also see that the string thickness, i.e., the radius of theregion where energy density is considerable, is estimated as ξ ∼ 1, i.e., ρ ∼ (ev)−1.The tension can be calculated exactly in the special case Mφ = MV with the result

μ = πv2. (12.25)

This supports the estimate (12.24).

Problem 12.5. Express the string tension in kg/cm for v ∼ 1016 GeV. Comparethe mass of the Earth with the mass of a string encircling the Earth along equator.

The fact that N is a conserved topological number does not tell what is theminimum energy configuration for given N at |N | > 1: this can be one string ofwinding number N or N strings of unit winding number. It turns out that the resultdepends on the parameters of the theory. For MV < Mφ strings with |N | > 1 decayinto |N | = 1 strings; conversely, for MV > Mφ energy decreases if strings merge(these two cases correspond to type II and type I superconductors in condensedmatter theory). The production of strings with |N | > 1 is suppressed in the earlyUniverse; the phase transition creates predominantly strings with |N | = 1.

To understand how cosmic strings affect the space-time geometry, let us findenergy-momentum tensor of a string. The general formula for the theory (12.17) is

Tμν = −Lgμν + Dμφ∗Dνφ − FμλFνρgλρ.

This gives for the configuration (12.22) the following expression,

Tμν = diag(1, 0, 0,−1) · L. (12.26)

Note that the configuration (12.22), and, consequently, tensor (12.26) are invariantunder Lorentz boosts along the string. Clearly, this implies that motion of the entireconfiguration along the string direction is unphysical.

In what follows we will be interested in length scales much greater than the stringthickness, l � (ev)−1. For this purpose, the energy-momentum tensor (12.26) canbe set equal to

Tμν = μ · diag(1, 0, 0,−1)δ(x)δ(y), (12.27)

where μ is string tension estimated according to (12.24). The formula (12.27) repre-sents the approximation of infinitely thin string; this approximation will be sufficientfor our purposes.

Problem 12.6. Obtain the structure of the tensor (12.27) from the arguments basedon dimensions, localization of energy, rotational symmetry, energy-momentum con-servation and invariance under boosts along the string direction.


The only non-vanishing components of the energy-momentum tensor (12.27) arethe energy density T00 and pressure along the string T33. They are equal up to sign,so we are dealing with a relativistic object which cannot be studied in the Newtonianapproximation. Still, one expects that the space-time metric is nearly flat far awayfrom the string, so that one can use linearized equation for the Newtonian potentialΦ. In background Minkowski space the latter is defined by g00 = 1 + 2Φ and obeysthe equation (see (A.115) and (A.117))

ΔΦ = 8πG

(T00 − 1

2ημνTμν

)= 4πG(T00 + T11 + T22 + T33). (12.28)

The Newtonian potential is zero for the source (12.27) since the right hand sideof (12.28) vanishes. Hence, gravitational field is absent away from the string core;straight strings neither attract nor repel each other or surrounding matter! We willdiscuss the geometry of space in the presence of a string in Sec. 12.3.3.

We discuss here straight strings. For strings with ripples, the energy-momentum tensorcan also be treated as localized along a line, provided that one considers distances largerthan the length scale of ripples. However, in that case the energy density T00 and pressurealong the string T33 no longer coincide up to sign; the effective energy-momentum tensorobeys |T00| > |T33|. The energy density T00 is somewhat larger than tension of straightstring, T00 > μ, since a distant observer sees “more string” per unit length. For similarreason T33 < μ. It turns out that energy density and pressure are related by T00 ·|T33| = μ2.Since |T00| �= |T33| for strings with ripples, the right hand side of Eq. (12.28) does notvanish: strings with ripples generate static gravitational field.

As it evolves, cosmic string sweeps a (1+1)-dimensional manifold, world surface.The action for thin cosmic string equals the area of the world surface (just like theaction for a particle is the length of its world line):

S = −μ

∫ √−γd2ξ. (12.29)

Here ξ0 and ξ1 are timelike and spacelike coordinates on world surface, Xμ = Xμ(ξ)are space-time coordinates of a point on world surface, and γ = det(γαβ), where

γαβ = ∂αXμ(ξ)∂βXν(ξ)gμν

is induced metric on world surface. This action is known as Nambu–Goto action; itdescribes the dynamics of cosmic strings everywhere except for their crossing pointswhere finiteness of string width is important.

The Nambu–Goto action can be obtained as the leading approximation to the action ofmoving curved cosmic string in the Abelian Higgs model (12.17). The idea of the derivationis to find approximate expression for the string configuration, insert it into the action(12.17) and integrate over coordinates transverse to the string.

Let ξ0 and ξ1 be timelike and spacelike coordinates on the manifold Xμ(ξ) where theHiggs field φ takes zero value. This manifold is identified with the string world surface.Assuming that the string curvature is small compared to its thickness, one approximatesthe string configuration by Eq. (12.22) generalized to non-zero string velocity. The twovectors parallel to the world surface at a given point Xμ(ξ) are ∂Xμ/∂ξβ , and one can

choose two tangential spacelike vectors e(α)μ , α = 1, 2, such that they are orthonormal,


e(α)μ e(β)

ν gμν = −δαβ , and orthogonal to the world surface, e(α)μ ∂Xμ/∂ξβ = 0. The coordi-

nates xμ of each point near the world surface can now be written as

xμ ≡ xμ(ζ) = Xμ(ξ) +2X

α=1

e(α)μ(ξ)ηα, ζ = (ξ0, ξ1, η1, η2),

where we introduced two new coordinates ηα, α = 1, 2 which in the case of straight stringparameterize points on the plane orthogonal to the string. The Jacobian of the coordinatetransformation from xμ to ζμ is

√−g · det

„∂x

∂ζ

«=

s−det

„gμν

∂xμ

∂ζλ

∂xν

∂ζρ

«≈ √−γ + · · · , (12.30)

where we write only the terms which are not suppressed by the radius of the string cur-vature.

In terms of new coordinates, the string configuration is

φ[x(ζ)] = φ(s)(η1, η2) Aμ[x(ζ)] =

α=2Xα=1

e(α)μA(s)α (η1, η2),

where (s) denotes the configuration of straight string in the plane (η1, η2). This is the sameconfiguration as in (12.22) with η1, η2 substituted for x1, x2. To the leading order in thestring curvature we have

Dμφ∗Dμφ ≈ Dμφ(s)∗Dμφ(s), F 2μν ≈ F (s)2

μν , V (φ) ≈ V (φ(s)). (12.31)

The Nambu–Goto action is now obtained by substituting the Jacobian (12.30) and expres-sions (12.31), (12.17) into the action of the Abelian Higgs model and integrating over thetransverse coordinates ηα.

Problem 12.7. Show that there are no corrections to the Nambu–Goto action (12.29)which are linear in the ratio of the string thickness to its radius of curvature.

12.3.2 Gas of cosmic strings

For a distant observer, a closed cosmic string of radius R looks like a particle of massMs = 2πRμ. If dissipation of string energies (say, by gravitational wave emission)is negligible, their effect on expansion rate is the same as that of relativistic ornon-relativistic matter, depending on their velocities. In fact, the dissipation isimportant; we consider the realistic situation in Sec. 12.3.4.

The properties of the gas of infinite strings are less obvious. In what followswe consider non-interacting strings: indeed, their interaction mediated by the fieldsAμ and φ weakens exponentially with separation,6 while gravitational interaction isweak. To find equation of state for gas of cosmic strings, consider first the config-uration of N straight strings at rest, which are parallel to z-axis and separated bydistance L well exceeding the string thickness. Energy-momentum tensor of such aconfiguration is

T (0)μν (x, y) = μ · diag(1, 0, 0,−1)·

N∑i

δ(x − xi)δ(y − yi).

6We consider here Abrikosov–Nielsen–Olesen strings; the situation for global string is different.


In the limit of large number of strings the average tensor becomes

〈T (0)μν 〉 =

∫T

(0)μν (x, y)dxdy∫

dxdy=

μ

L2· diag(1, 0, 0,−1). (12.32)

This expression is valid for static configuration. Let now the strings move along thex-axis at velocity u. Energy-momentum tensor is obtained from (12.32) by Lorentzboost,7

〈Tμν〉(ux) =μ

L2

⎛⎜⎜⎝

γ2 γ2u 0 0γ2u γ2u2 0 00 0 0 00 0 0 −1

⎞⎟⎟⎠ , γ =

1√1 − u2

.

We are going to average over the boost directions, so we omit terms linear in u.The tensor averaged over boosts along x and y is (recall that the motion along thez-axis is unphysical for strings stretched along z-axis)

〈Tμν〉(uxy) =μ

L2· diag

(γ2,

γ2u2

2,γ2u2

2,−1

).

We now repeat the procedure for string configurations parallel to x- and y-axesand average over all three directions. The resulting energy-momentum of the gas ofinfinite cosmic strings moving with velocity u in random directions is

T gS

μν ≡ 〈Tμν〉(u) =μ

L2· diag

(γ2,

u2γ2 − 13

,u2γ2 − 1

3,u2γ2 − 1

3

). (12.33)

As expected, in the limit u → 1 the equation of state of the string gas coincideswith that of radiation, p = ρ/3. More relevant for the Universe is the equation ofstate for slow strings

p = −13ρ. (12.34)

According to the results of Sec. 3.2.4 this gives

ρ, p ∝ a−2(t). (12.35)

The physical reason for this behavior is that expansion affects distances betweenstrings in two transverse directions, rather than in three directions as in the case ofparticles.

Problem 12.8. Neglecting string collisions and dissipation of their energy (whichis not in fact a good approximation), estimate the number of strings in the visibleUniverse, if they were produced in the course of phase transition at Tc ∼ 100GeV.The same for Tc ∼ 1016 GeV. Hint: Assume that strings were produced by the Kibblemechanism, so that just after the phase transition there was one piece of string ofHubble length per Hubble volume.

7This transformation accounts not only for the change of the energy-momentum of each string

but also Lorentz-contraction of the distance between the strings.


The dependence (12.35) shows that strings, if exist, could become importantat later stages of the cosmological expansion. Their energy density behaves in thesame way as the contribution of spatial curvature to the Friedmann equation, see(4.2), and decreases slower than that of radiation or non-relativistic matter.

Problem 12.9. Under conditions of Problem 12.8 and assuming, within the AbelianHiggs model (12.17), that strings were produced in the course of phase transitionat Tc ∼ v, obtain the bound on the energy scale v by requiring that the string gasmakes small contribution to the present energy density.

If the late time expansion of the Universe were dominated by cosmic string gas,the scale factor would grow as

a(t) ∝ t.

This shows that string domination at present epoch is inconsistent with observa-tions: the cosmological expansion accelerates. In fact, CMB and SNe Ia data ruleout string domination at a high confidence level (see (4.44) and Fig. 4.7); the cos-mological data require dark energy instead. It then follows from (12.35) that stringsnever dominated and, if dark energy density is constant or almost constant, theywill never dominate in the future.

12.3.3 Deficit angle

Let us now consider other effects of infinite or very long cosmic strings. We beginwith studying the spatial geometry in the presence of an infinite string. Even thoughgravitational potential due to cosmic string vanishes away of its core, the effecton geometry is non-trivial. To see this, we consider small perturbation hμν aboutMinkowski metric. Let us choose harmonic gauge,

∂μhμν − 1

2∂νhμ

μ = 0. (12.36)

Then linearized Einstein equations are (see Sec. A.9)

�hμν = −16πG

(Tμν − 1

2ημνT λ

λ

), (12.37)

where � = ∂λ∂λ is D’Alembertian in Minkowski space-time. In the case of cosmicstring, the right hand side of (12.37) is given by (12.27), so its non-zero componentsare (11) and (22). Thus, the non-zero components of perturbations are h11 and h22,and they both obey

(∂2x + ∂2

y)h11(22) = 16πGμδ(x)δ(y).

We conclude that in the harmonic gauge

hμν = 4G log(

x2 + y2

ρ20

)· diag(0, 1, 1, 0), (12.38)


where ρ0 is a parameter of dimension of length, determined by the string thickness.With perturbation included, the metric is

ds2 = dt2 − dz2 −[1 − 4Gμ log

(ρ2

ρ20

)]· (dρ2 + ρ2dθ2),

where we switched to cylindrical coordinates. The latter expression is valid forGμ � 1 and ρ � ρ0, but 4Gμ log

(ρ2

ρ20

)� 1; in this case the perturbation (12.38) is

small while the details of the string core are irrelevant. It is convenient to performcoordinate transformation to the radial coordinate ρ, such that

dρ2 =(

1 − 4Gμ log(

ρ2

ρ20

))dρ2. (12.39)

To the leading order in Gμ this gives

ρ = ρ ·(

1 − 4Gμ logρ

ρ0+ 4Gμ

),

and

ρ2 ·[1 − 4Gμ log

(ρ2

ρ20

)]= ρ2 · (1 − 4Gμ)2. (12.40)

It follows from (12.39) and (12.40) that the metric of static straight string stretchingalong z-axis can be written as [156] (we omit tilde over ρ from now on)

ds2 = dt2 − dz2 − dρ2 − (1 − 4Gμ)2ρ2dθ2. (12.41)

Problem 12.10. Find asymptotically flat form of metric for string with rippleswhose energy-momentum tensor is Tμν = diag(μ′, 0, 0,−μ′′)δ(x)δ(y), μ′μ′′ = μ2,

μ′ > μ′′, to the leading order in Gμ′, Gμ′′. Show that space-time is curved outsidethe string core.

Metric (12.41) has conical singularity. Circles ρ = const have length smaller than2πρ. Metric (12.41) becomes Minkowskian upon the coordinate transformation

θ → (1 − 4Gμ)θ.

However, the polar angle θ now takes values in a narrower interval

0 ≤ θ < 2π(1 − 4Gμ).

In this regard, one introduces the notion of deficit angle, whose value is

Δθ = 8πGμ . (12.42)

Thus, space-time is locally flat but the geometry on (x, y)-plane is conical.8

We note that we obtained the result (12.41) by making use of the linearizedEinstein equations. However, the result that metric of straight string is flat awayfrom its core and has deficit angle is in fact exact.

8We obtained the expression (12.41) for 4Gμ log“

ρ2

ρ20

”� 1. It is clear, however, that (12.41) is a

solution to the Einstein equations for larger ρ as well: the metric is locally flat and its Ricci tensor

vanishes, Rμν = 0.


Fig. 12.3 Light propagation from a point A′ = A′′ behind the string S to an observer O.

The presence of deficit angle leads to a number of interesting physical phe-nomena. One is double image of an object behind the string. To illustrate thispictorially, let us use the coordinates in which metric is Minkowskian. Consider anobserver at distance d from straight cosmic string: this is the point O in Fig. 12.3,while the string is at point S; the figure shows tangential plane to the string. Thelines SA′ and SA′′ must be identified, and the shaded region cut out; the deficitangle is Δθ. Light rays coming to the observer at equal angles α′ = α′′ to thedirection OS towards the string, actually emanate from one and the same point,since the points A′ and A′′ are identified. Thus, the observer will see two imagesof the object placed at A′ = A′′. The angular distance between the two images,Δα = α′ + α′′, is proportional to deficit angle. For small Δθ one has

Δθ =l + d

lΔα, (12.43)

where l and d are the distances from the string to the source and observer, respec-tively. While the distance to the source is measurable (say, by determining redshift),the distance d to the string is not. Thus, the measurement of the angular distancecan only give the lower bound on deficit angle,

Δθ ≥ Δα.

According to (12.42), deficit angle is proportional to the string tension, so a mea-surement of Δα would place a bound on the energy scale v of the theory,

v � MPl

2

√Δα

2π.

To see this lensing effect in the coordinate frame (12.41) one would cut outthe shaded region in Fig. 12.3 and glue together its boundaries SA′ and SA′′. Theresulting surface would be a cone with apex at the string position S. Light cantravel from the source A′ = A′′ to the observer O along two straight lines, whichpass left and right of the apex. Hence the two images of the source.

Problem 12.11. Making use of the geodesic equation in metric (12.41) give alter-native derivation of the lensing effect of cosmic string.


We neglected the cosmological expansion in the above analysis. In general, however,the relationship between Δθ and Δα depends on cosmological evolution. As an example,the analog of the relation (12.43) in flat matter dominated Universe is

Δα = Δθ ·„

1 − 1 − (1 + zS)−1/2

1 − (1 + zA)−1/2

«, (12.44)

where zS and zA are redshifts of string and source, respectively.

Problem 12.12. Derive the relation (12.44). Find similar relation for the Universe dom-inated by cosmological constant. Generalize that formula, as well as (12.44) to stringinclined to the plane of Fig. 12.3 at angle δ.

We have considered point-like source for simplicity; in that case the observationalproof that two images are due to single source would be the identity of the twospectra. The situation with extended source is more complicated, but in that casetoo, there exists a region behind the string such that sources in it produce twoidentical images.

Problem 12.13. Find the shapes of images of a spherical object behind the stringin a general case.

Another phenomenon due to cosmic string is specific distortion of CMBanisotropy. If a string and an observer are at rest with respect to CMB, then lensingby the string leads to repetitions in the pattern of anisotropy. The source is nowthe surface of last scattering, and the distance to it much exceeds the distance tothe string. Therefore, the angular distance between identical temperature spots isequal to deficit angle, Δθ = Δα.

If the string moves in the direction transverse to the line of sight, there is a neweffect of systematic shift between frequencies (and hence temperature) of photonspassing the string on different sides [160]. This effect is illustrated in Fig. 12.4,where the coordinate frame and notations are the same as in Fig. 12.3. Let usagain consider two images A′ and A′′ of one and the same source. Let the string S

move with velocity u⊥ in direction normal to the line of sight. Then all points onthe line A′SA′′ move with the same velocity. The source A′ has velocity u⊥ whoseprojection on the line OA′ points in the same direction as the photon momentum.

Fig. 12.4 Light propagation from source A′ = A′′ behind the string S moving with velocity u⊥.


For the source A′′ this projection is opposite to photon momentum. Hence, thelongitudinal Doppler effect leads to shifts of photon frequencies

OA′: Δω = u⊥Δθ

2γ, OA′′: Δω = −u⊥

Δθ

2γ,

where γ = 1/√

1 − u2⊥ and we have set Δα ≈ Δθ, having in mind CMB photons.

We see that photons crossing the string trajectory in front and behind the string getredshifted and blueshifted, respectively. This leads to additional CMB anisotropy.For small Δθ and realistic u⊥ the effect is small,

δT

T= γu⊥Δθ = 1.7 · 10−6γ

u⊥10−1

μ

(1016 GeV)2.

Yet the analysis of CMB angular spectrum together with data on large scalestructure rule out string networks [161] with μ � (0.6 · 1016 GeV)2.

It is known from observations that CMB temperature fluctuations are randomand Gaussian. The small jump of temperature across the string projection ontocelestial sphere is a non-Gaussian feature. Non-Gaussianity of this sort has notbeen observed, which also leads to a bound on μ [162]. This bound is of the sameorder as other bounds, μ � (0.7 · 1016 GeV)2 for an infinite straight string movingwith velocity u⊥ = 1/

√2.

Yet another effect due to cosmic string, moving now in dust-like medium, isthe formation of overdense region (sheet-like wake) behind the string [163]. This isillustrated in Fig. 12.5, where the coordinate frame is the same as in Fig. 12.3. Thestring is perpendicular to the plane of the figure. Let there be two dust particlesA1 and A2 in this plane. Let these particles be at rest and at distance r/2 fromthe string trajectory OS. The string moves with velocity u. After the string passesbetween the particles, the latter move towards each other and meet behind thestring (the meeting point is A′

1 = A′2 in Fig. 12.5; recall that the lines SA′

1 and SA′2

are identified).Let us calculate the velocities of the dust particles in direction towards each

other. For particle A1 this direction is orthogonal to the line SA′1. In the string rest

frame this particle moves along the line A1A′1 with velocity u. Hence, its velocity

Fig. 12.5 Motion of a string S in dust-like medium.


at point A′1 in the direction to particle A2 is

vy = u · sin Δθ

2= 4πGμu, (12.45)

where we set Gμ � 1. Taking, as an example, u ∼ 0.1, we find numerically

|Δvy| = 0.8 · 10−6 · μ

(1016 GeV)2.

We see that this velocity may be fairly high.

12.3.4 Strings in the Universe

Relatively large velocities of particles in wakes of moving cosmic strings lead toformation of overdense regions there. These overdensities may in principle serve asseeds for formation of structures — galaxies, clusters of galaxies, etc. The structuresformed in this way would stretch along the string trajectories and hence they wouldbe 2-dimensional. We note here that galaxy distribution indeed shows that there are2-dimensional and also one-dimensional structures, walls and filaments. However,detailed analysis reveals that only small fraction of matter gets into the wakes [164],so that observed voids (galaxy-poor regions) do not form.

Another structure formation mechanism in models with cosmic strings isaccretion of non-relativistic matter onto string loops [165, 166]. The resulting densityperturbation spectrum is almost scale invariant, in accordance with observations.However, the string mechanism of structure formation predicts CMB anisotropyangular spectrum in gross contradiction to observations [167], see Fig. 12.6. Hence,string mechanism of structure formation cannot be dominant. The bounds on thestring contribution into CMB anisotropy, and hence into structure formation, areat the level of 10%.

We note in this regard that the standard mechanism of structure formation,which assumes that the density perturbations are of primordial nature, also leads toformation of one- and two-dimensional structures. This has been shown by numericalsimulations; the general reason for this phenomenon is complex dynamics of grav-itating systems in non-linear regime. We also note that sizeable, though not dom-inant, string contribution to density perturbations is predicted by some inflationarymodels (e.g., subclass of hybrid inflation).

Many properties of production and evolution of cosmic strings cannot be foundanalytically, so one relies upon numerical simulations. These show, in particular,that just after the phase transition the mass fraction of infinite strings is four timeslarger than that of closed strings. The initial velocities of long strings are small, butthey increase in time and on average become rather large, u ≈ 0.15. Numerical sim-ulations have shown also that the energy density of cosmic strings decreases in time,so that it never dominates the cosmological expansion. Hence, unlike monopoles and


101 102 1030

1000

2000

3000

4000

5000

6000

multipole moment: l

l(l+

1) C

l / 2

π [μ

K2 ]

WMAPAbelian Higgs stringsSemilocal stringsTextures

Fig. 12.6 Results of measurements of CMB anisotropy (WMAP) and predictions of CMB

anisotropy spectrum in ΛCDM model with primordial density perturbations (solid line fitting

the data) and in the model with density perturbations generated by topological defects [167]:

lower solid line and dash-dotted line for two variants of cosmic strings, dashed line for models

with textures (see Sec. 12.5). The absence of peaks in the latter spectra is due to the fact that

perturbations of a given wavelength are not built in before the Hot Big Bang epoch, but rather

are generated during fairly long period of time.

domain walls, models with strings produced in phase transitions are still cosmolog-ically allowed.

There are two main processes determining the evolution of cosmic strings in theUniverse. One is string intersections and self-intersections, which lead to productionof loops out of long strings. The second is gravitational wave emission by relativelyshort strings, which leads to disappearance of small loops. Let us first discuss thelatter process. The power of gravitational wave emission is determined by the thirdtime derivative of the quadrupole moment [53] Q of a string loop,

Pgw ∼ 1M2

Pl

(d3Q

dt3

)2

.

The quadrupole moment of a loop of radius R is estimated as Q ∼ μR3. The stringequation of motion which follows from the action (12.29) shows that loops rotateand oscillate at velocities of order 1, i.e., dR/dt ∼ 1. This gives

Pgw = Cgwμ2

M2Pl

,

where the constant Cgw is obtained by numerical simulations and turns out to berather large, Cgw 102 (see e.g. [157]). Thus, loop looses its energy at the timescale

tgw ∼ μR

Pgw=

M2PlR

Cgwμ. (12.46)


As a result, the size of the loop becomes of order of its thickness, R ∼ v−1, and thestring decays into high energy particles.9

Let us now turn to string intersections. Numerical analysis shows that inter-section almost always leads to reconnection [158], so self-intersections give rise todecays into shorter loops. Somewhat unexpected result of numerical simulations ofcosmic string dynamics in the Universe is that the evolution of the cosmic stringenergy density soon after the phase transition coincides with the evolution of thetotal energy density, ρ ∝ t−2 [159]. At any given moment of time each Hubblevolume contains a dozen long strings stretching behind the horizon, many longclosed strings and numerous small loops. This behavior is due to efficient productionof closed strings and their subsequent decay via gravitational wave emission.

To understand this picture at qualitative level, let us estimate the energy densityof closed loops at radiation domination. The number density of loops of size R, whereR � H−1, decreases due to the cosmological expansion,

nl(R, t) ∝ a(t)−3 ∼ t−3/2.

In the scale-invariant regime we obtain on dimensional grounds

nl(R, t) ∼ 1

(Rt)3/2∼( μ

Et

)3/2

,

where E ∼ Rμ is the mass of a loop. The energy density is then

ρl(t) ∼∫ Emax

Emin

EdEdnl(R, t)

dE=(μ

t

)3/2∫ Emax

Emin

dE

E3/2. (12.47)

This integral is saturated at lower limit, hence

ρl(t) ∼(μ

t

)3/2 1√Emin

, (12.48)

where Emin is the minimum possible loop energy. We see that the energy densityis dominated by the lightest but numerous loops.

The minimum size of loops is determined by the requirement that the lifetime(12.46) exceeds the Hubble time, so we get

Rmin(t) ∼ Cgwμ

M2Pl

t, Emin(t) ∼ Cgwμ2

M2Pl

t.

We finally obtain

ρl(t) √

μMPl√Cgw

1t2

,

i.e., the energy density of strings indeed follows that of radiation, ρrad ∼ M2Pl/t2.

The relative contribution of strings remains small, ρl/ρrad ∼ √μ/M2

Pl. Still, it

9We note that this is one of the possible mechanisms of generation of ultra-high energy cosmic

rays. For μ = (1016 GeV)2, this mechanism works with string loops of present size of order R ∼1Mpc: it follows from (12.46) that the lifetime of these strings is of the order of the present age

of the Universe.


is much larger than the relative contribution of infinite strings, which is of orderμ/M2

Pl. This shows that it is small loops rather than long strings that may affectthe spectrum of density perturbations.

The behavior nl(R, t) ∝ (Rt)−3/2 can be seen from the following argument. Let nl(R, t)be the number density of loops of size R at time t. Loops of sizes in interval R and R+dRhave energy density

ρl(R, t)dR = μRnl(R, t)dR.

Energy density of loops decreases due to the cosmological expansion, and at the same timemore loops are produced by (self)intersections of long strings. Scale-invariant behaviormeans that the loop production is determined by a scale-invariant function which wedenote by f(R/lH), where lH is the horizon size, the length scale inherent in the Universe.This function parameterizes the energy loss of long strings due to production of loops ofsizes between R and R + dR as follows,

μdR

lH· f(R/lH).

Hence, equation for balance of loops is

dρl(R, t)

dt+ 3H(t)ρl(R, t) =

μ

l4Hf(R/lH),

where H(t) = 1/(2t) and lH = 2t. The solution to this equation is

ρl(R, t) =1

16

μ

(Rt)3/2

Z ∞

R/t

pξf(ξ/2)dξ.

For short loops, R/t → 0, this indeed gives the evolution law (12.47).All the discussion above assumed that the sizes of loops produced in intersections are

much larger than the minimum size. Numerical simulations [168] show that the loop sizesare indeed comparable to the horizon size; at both radiation and matter domination thetypical size of a newborn loop is R(t) = αt, α � 0.1. We note that earlier simulationswith worse resolution could not determine the value of α but suggested that it was small,α 1. The interpretation was that the typical size of a newborn loop was determinedby the size of ripples of long strings rather than the horizon size. Were that the case, thenewborn loops would decay in Hubble time, and would be irrelevant from the viewpointof density perturbations. More sizeable would be density perturbations created by a dozenof moving infinite strings.

Finally, let us estimate the energy of gravitational waves produced by cosmicstrings. With account of redshift of their frequencies, the energy density in gravitywaves ρgw obeys the equation

ρgw + 4Hρgw = −dρl

dt= 2

√μMPl√Cgw

1t3

.

Its solution at radiation domination is

ρgw =1t2

∫ t

t1

(−dρl

dt′

)t′2dt′ 2

√μMPl√Cgw

1t2

logt

t1, (12.49)

where t1 denotes the time when the first gravity waves are emitted, i.e., the timewhen the cosmic strings begin to decay. The energy density in the Universe atradiation domination is (see Sec. 3.2)

ρ =3M2

Pl

32πt2,


hence the relative contribution of gravity waves is given by

ρgw

ρ 64π

3MPl

√μ√

Cgw

logt

t1.

We see that this contribution grows slowly, logarithmically, but it can be quite large.Decaying strings may be intense source of relic gravity waves.

The effect of gravity wave production enables one to place strong bounds onmodels with cosmic strings. In particular, pulsar timing measurements give thebound [169]

μ � (2 · 1015 GeV)2.

We note that additional sources of gravity waves are singularities on strings, kinksand cusps. These singularities emerge both as a result of the evolution of closedstrings and due to (self-)intersections, see, e.g., Ref. [170]. The singularities havevelocities close to the speed of light and emit intense pulses of gravity waves. Thesingularities may also be sources of particles of large masses and/or super-highenergies. The latter aspect is of interest for cosmic ray physics.

The possibility of production of strings in the Universe is considered also in the contextof superstring theory, the candidate theory unifying gravity with other forces. In super-string theory, there exist fundamental strings and other string-like objects. In simple casestheir existence in the Universe is ruled out: their tension is in the range μ ∼ (1−10−2) ·M2

Pl,which is inconsistent with CMB data. Furthermore, production of very heavy strings isproblematic from the viewpoint of inflationary theory. Finally, in simple versions of stringtheory long strings are unstable at cosmological time scale.

Nevertheless, there are versions of string theory in which strings have interesting andphenomenologically acceptable properties, see e.g. [171] and references therein. Besidesfundamental strings (F -strings), another type of objects, D-strings, is of importance. Thisis a subclass of so called Dp-branes (where p refers to spatial dimensionality of the object).

Properties of these types of strings are quite different from those of cosmic strings.Production mechanisms of F - and D-strings are also different from the Kibble mechanism.Unlike cosmic strings, they have small probability to reconnect when intersecting. Fur-thermore, intersection of F - and D-strings may lead to formation of hybrid FD-stringsconnecting F - and D-strings. The production of loops at reconnections, and hence energyloss, is thus strongly suppressed. In some string theories, F -, D- and FD-strings formstable networks of cosmological scale. Tensions of these strings need not be extremelylarge, so such networks are not ruled out.

12.4 ∗Domain Walls

Let us consider the simplest field theory model admitting domain wall solution.This is the model of one real self-interacting scalar field φ with the Lagrangian

L =12∂μφ∂μφ − λ

4(φ2 − v2)2. (12.50)

12.4. ∗Domain Walls 327

This Lagrangian is invariant under change of sign of the field, φ → −φ (symmetrygroup Z2). As the result of spontaneous symmetry breaking, the field φ acquiresvacuum expectation value

〈φ〉 = +v or 〈φ〉 = −v,

i.e., the space of vacua is disconnected and consists of two points. Phase transitionin the Universe leads to domains of two types: one with positive field10 〈φ〉 = +v,and another with negative value 〈φ〉 = −v. The field configurations interpolating inspace between different vacua φ = +v and φ = −v are known as domain walls.

The general necessary condition for the existence of domain walls is, in notations usedin (12.13),

π0(G/H) �= 0.

This condition means that the vacuum manifold M = G/H consists of several disconnectedcomponents.

In the simplest case of static infinite domain wall stretching in flat space alongthe plane z = 0, the profile depends only on z and is the solution to the fieldequation

d2φ

dz2− λ(φ2 − v2)φ = 0, (12.51)

with boundary conditions φ(z → ±∞) = ±v (kink) or φ(z → ±∞) = ∓v (antikink).These configurations are given by

kink: φ(z) = v tanhz

Δ, Δ2 =

2λv2

(12.52)

antikink: φ(z) = −v tanhz

Δ. (12.53)

Problem 12.14. Show that kink is indeed the solution to Eq. (12.51).

The parameter Δ has the meaning of the domain wall thickness. To see this, let uscalculate the energy-momentum tensor of the domain wall. The general expressionfor the energy-momentum tensor is

T scμν = ∂μφ∂νφ − Lημν ,

and for the solution (12.52) we obtain

T DW

μν =λv4

2cosh4 z4Δ

diag(1,−1,−1, 0). (12.54)

We see that the energy-momentum tensor does not depend on x and y and is non-zero only in the region −Δ � z � Δ. Thus, Δ is indeed the wall thickness. Wenote in passing that domain walls produced in the Universe are not flat, generally

10The expectation value of φ at finite temperature does not coincide with v, but this is unimportant

for what follows.


speaking. Nevertheless, the kink approximation works well unless domain walls arestrongly curved.

The integrals

η ≡∫

T DW

00 dz =2√

2λ

3v3 and

∫T DW

11 dz =∫

T DW

22 dz = −η

give the surface energy density of the wall and pressure along the wall. These areequal up to sign.11 We note that T DW

33 = 0, which means that there is no tensionacross the wall.

Large spatial components of the energy-momentum tensor imply that domainwalls are relativistic objects, which may be expected to have non-trivial propertiesonce gravity is turned on. To illustrate this, let us make use of the linearized equation(12.28) for the Newtonian potential. With the domain wall energy-momentum tensor(12.54) we obtain

ΔΦ = −4πGT DW

00 .

This would give gravitational field of opposite sign as compared to point-like object.Hence, if a domain wall could be at rest it would antigravitate: non-relativisticparticles would be repelled by domain wall.

In fact, the notion of a static domain wall, as well as usage of the linearized Ein-stein equations are not valid. Hence, the picture we just presented is oversimplified.We have given it only to illustrate that gravitating domain walls have non-trivialproperties.

The Kibble mechanism works for domain walls too. In general, after the phasetransition the Universe consists of domains of different vacua divided by domainwalls, which in turn contain closed domains of the opposite vacuum, etc. Domainwalls make either open or closed non-intersecting surfaces.12

In vast majority of particle physics models, the existence of even a single“infinite” (horizon size) domain wall in the present Universe is ruled out. Asimple way to see this is to estimate the energy of a domain wall of horizon size,MDW ∼ ηH−2

0 , and compare it to the total energy inside the present horizon,ρcH

−30 . The requirement that the wall energy is smaller than the total energy,

ηH−20 � ρcH

−30 ,

gives the bound on the surface energy density,

η � ρcH−10 ∼ (10 MeV)3. (12.55)

Barring the possibility of extremely small couplings, this means that the energy scaleof physics responsible for domain wall must be less than 10MeV, which is very small

11The latter property is no longer valid for curved domain walls.12We consider the model (12.50) whose space of vacua consists of two points only. There can be

more complicated situations (e.g., there are models with symmetry Zn, n > 2). In those cases,

there are more types of domains, and hence more types of walls; these walls may intersect.

12.5. ∗Textures 329

from the viewpoint of extensions of the Standard Model. Reversing the argument,extensions of the Standard Model with discrete symmetries face the “domain wallproblem” [172]: there must be a mechanism that forbids domain wall production inthe early Universe, or a mechanism that destroys them in the course of evolution.This requirement is certainly far from trivial.

Problem 12.15. Find equation of state for gas of non-interacting domain wallsmoving at non-relativistic velocities. Hint: Make use of approach similar to that ofSec. 12.3.2.

To end this Section we note that there are other cosmological effects of domainwalls besides their effect on the cosmological expansion. In particular, domain wallis a source of gravity, so it affects CMB photons. The corresponding bounds on thetension of a domain wall, if it existed in the present Universe, are even strongerthan (12.55).

12.5 ∗Textures

Textures are unstable topologically non-trivial field configurations [149, 173]. Theseare also produced in the course of phase transitions, and the role of topology is thattheir production occurs via the Kibble mechanism. Being unstable, textures do notsurvive up to the present epoch, and their presence in the early Universe can bedetected only indirectly. One possibility is that they contribute to the generation ofdensity perturbations that develop into structures in the Universe [174]. We note,though, that no effects due to textures have been observed so far.

Like stable topological defects, textures may exist in theories with spontaneouslybroken global or gauge symmetry. An example of the former type is given by amodel with global symmetry O(4) broken down to O(3). The model contains fourreal scalar fields ϕa, a = 1, . . . , 4, and the Lagrangian is

L =12∂μϕa∂μϕa − λ

4(ϕaϕa − v2)2. (12.56)

Symmetry is broken at zero temperature, and the vacuum can be chosen as

〈ϕa〉 = vδa4 . (12.57)

The fields ϕ1, ϕ2, ϕ3 are massless in this vacuum; these are Nambu–Goldstonebosons13 corresponding to global symmetry breaking O(4) → O(3). The field h

defined by h = ϕ4 − v acquires the mass mh which we assume to be large enough.

Problem 12.16. By calculating the quadratic action for perturbations aboutvacuum (12.57) show that the spectrum of the model is indeed as described above.

13The existence of massless Nambu–Goldstone bosons is a very general property of models with

spontaneously broken global symmetries. It may lead to phenomenological problems; we do not

discuss this aspect here.


The scalar potential of the model (12.56) vanishes for any fields obeying

ϕaϕa = v2. (12.58)

This equation determines the vacuum manifold; in our case this manifold is a3-sphere S3

vac. The energy density of field configurations of the sizes greater thanm−1

h is not very large provided the relation (12.58) is valid at every point in space.These configurations have finite total energy as long as the gradients of ϕa vanishas r → ∞, i.e., ϕa(x) tends to a constant independent of angles and r. Without lossof generality we set

ϕa(r → ∞) = vδa4 .

From the topological viewpoint this means that our 3-dimensional space effectivelyhas topology of sphere S3

space, since all points at spatial infinity are mapped into onepoint in S3

vac. The field configurations ϕa(x) just described define, therefore, map-pings from 3-sphere S3

space to 3-sphere S3vac, which may have non-trivial topology.

The topological number is the number of times the sphere S3vac is wrapped by the

mapping S3space → S3

vac; this number, called degree of mapping, is integer.The simplest topologically non-trivial configuration is

ϕ = v

⎛⎜⎜⎝

cosφ sin θ sin χ

sin φ sin θ sinχ

cos θ sin χ

cosχ

⎞⎟⎟⎠ , (12.59)

where φ and θ are spherical angles in our space, while the function χ(r) obeys

χ(r = 0) = 0, χ(r → ∞) = π. (12.60)

This configuration has unit topological number. This is precisely a texture ofminimum topological number.

Problem 12.17. Show that the configuration (12.59) has topological number 1.

The configuration (12.59) is unstable. Its energy is entirely due to the gradientterm,

E =12

∫d3x∇ϕa · ∇ϕa . (12.61)

Under rescaling x → x′ = αx, energy (12.61) scales as E → E′ = αE. Hence, thisconfiguration is unstable against shrinking;14 once formed, texture shrinks. Whenits size becomes of order m−1

h , the property (12.58) need not hold any longer, andthe texture unwinds and disappears. Its energy gets transmitted into energy ofelementary excitations, Nambu–Goldstone bosons.

14This illustrates the fact that topology alone is not sufficient to ensure the existence of non-trivial

stable configurations.

12.5. ∗Textures 331

We note in passing that in static 3-dimensional space whose geometry is S3, configu-rations similar to textures may be stable. The solution still has the form (12.59), but nowχ is the third coordinate on S3

space. Metric in coordinates χ, θ, φ is

dΩ2 = a2[dχ2 + sin2 χ(dθ2 + sin2 θdφ2)]. (12.62)

The solution (12.59) is non-trivial on entire S3space, and vector (12.59) coincides with the

normal to S3space, if the latter is embedded into fictitious 4-dimensional Euclidean space.

Problem 12.18. Show that the field configuration (12.59) is a solution to the field equa-tions in the space with metric (12.62).

Let us give heuristic argument in favor of stability of the solution (12.59) in space withmetric (12.62). Consider the family of field configurations of the following form

ϕ = v

0B@

cos φ sin θ sin(χ/α)sin φ sin θ sin(χ/α)

cos θ sin(χ/α)cos(χ/α)

1CA, ϕ = v

0B@

0001

1CA

for 0 < χ ≤ πα, for πα < χ ≤ π.

(12.63)

They have topological number 1 and describe shrinking texture as α changes from 1 to 0.The energy of the configuration (12.63) is

E = 4π·a · v2·Z πα

0

»1

α2+ 2

sin2(χ/α)

sin2 χ

–sin2 χdχ

= 2π2· a · v2

»1

2α+ α − sin 2πα

4πα2

–. (12.64)

Since both α = 0 and α = 1 are minima of energy, both limiting configurations (i.e., theoriginal configuration with α = 1, and shrunk configuration with α = 0) appear classicallystable; there is potential barrier between the configuration (12.59) and shrunk texture, atleast along the path in configuration space defined by (12.63).

Textures are created in the course of phase transition by the Kibble mech-anism. As the Universe expands, textures shrink and disappear by emitting Nambu–Goldstone bosons. The energy density in textures decreases, while the spatial regionswhere the field takes one and the same value expand and eventually become of thehorizon size. The latter process is fast, so almost immediately after the phase tran-sition, there is about 1 texture of size H−1(Tc), per Hubble volume H−3(Tc).

As the Universe expands, the process continues, and there remains about onetexture of the horizon size per horizon volume. Since the energy density in textureis determined by the gradient energy, we have an estimate

δρ ∼ (∇ϕ)2 ∼ v2H2(t) ∼ v2t−2.

The background energy density has the same dependence on time at both radiationand matter domination, so textures give perturbations of energy density at horizon


crossing independent of the spatial scale of the horizon,

δρ

ρ= const

v2

M2Pl

.

Importantly, these perturbations exist in wide range of spatial scales, starting fromH−1(Tc). Hence, to zeroth approximation density perturbations have scale-invariantspectrum [174].

Approximate scale invariance is precisely the property of the measured spectrumof density perturbations. However, the texture mechanism of the generation ofdensity perturbations would lead to the CMB anisotropy spectrum inconsistentwith observations, see Fig. 12.6. Thus, textures as the main source of density per-turbations are ruled out [167].

Problem 12.19. Estimate the number density and energy density of Nambu–Goldstone bosons produced in the Universe within the model (12.59). Which valuesof v are ruled out by BBN? What is the fraction of Nambu–Goldstone bosons in theenergy of relativistic matter today for v = 1016 GeV?

12.6 ∗Hybrid Topological Defects

Some models admit structures consisting of topological defects of different dimen-sionality. This may be the case if spontaneous symmetry breaking occurs in twoor more steps. Each step leads to vacuum rearrangement, possibly new topologicalproperties of vacuum manifold and hence new types of defects. From topology view-point, a chain of phase transitions

GT1→ H1

T2→ H2 → · · · , T1 > T2 > · · · ,

may give rise to vacuum manifolds with non-trivial homotopy groups

πN1(G/H1) �= 0, πN2(G/H2) �= 0, · · ·Examples of hybrid structures are fleece (strings with ends on domain walls), neck-laces (strings with magnetic monopoles on them), etc. As an example, necklaces areproduced in two subsequent phase transitions:

step 1 : G → G′ × U(1)

step 2 : G′ × U(1) → H × ZN

The first step leads to monopole production while at the second step thesemonopoles get connected by strings. The necklace case corresponds to N = 2,so that each monopole gets connected to two others.

The evolution of hybrid defects is, generally speaking, different from that oftopological defects constituting the hybrids. Their detailed discussion is beyond thescope of this book; we only point out here that among hybrids, the most interestingare necklaces.

12.7. ∗Non-topological Solitons: Q-balls 333

12.7 ∗Non-topological Solitons: Q-balls

Besides topological solitons discussed in previous Sections, there exist other localizedfield configurations, whose stability is unrelated to topology. In particle physics,particularly popular are Q-balls which are stable due to the existence of conservedglobal charge (hence the name) and absence of massless particles carrying thischarge.

12.7.1 Two-field model

A simple model admitting non-topological solitons [175] contains real scalar field χ

and complex scalar field φ. The Lagrangian is

L = ∂μφ∗∂μφ +12∂μχ∂μχ − V (χ) − h2χ2|φ|2, (12.65)

where the potential V (χ) has absolute minimum at

χ = v �= 0, (12.66)

so that

V (χ = v) = 0, V (χ = 0) = V0 > 0.

The mass of the field φ in vacuum (12.66) is

mφ = hv.

We assume that the field δχ ≡ χ − v is also massive.The Lagrangian (12.65) is invariant under global U(1) symmetry

φ → eiαφ, φ∗ → e−iαφ∗, χ → χ. (12.67)

Note that vacuum (12.66) with φ = 0 is invariant under this symmetry. Due tothese properties, there exists conserved charge

Q = i

∫d3x(φφ∗ − φ∗φ). (12.68)

Let us find the state of minimum energy at a given value of Q. One of the statesof charge Q is a collection of φ-particles at rest in vacuum (12.66); their numberequals Q. The energy of this state is equal to Q · mφ. The competing Q-ball statehas the following structure. Inside a region of yet to be determined size r0, the fieldχ is zero, while away from this region this field takes vacuum value (12.66); χ(r)smoothly changes from zero to v near the boundary, see Fig. 12.7. All Q particlesφ are sitting at the lowest energy level inside the region of vanishing χ. Their massis zero there, and the momentum is of order 1/r0. Hence, the energy of each of theφ-particles is b/r0, where b is of order 1, and their total energy is bQ/r0. The totalQ-ball energy is thus

E =43πr3

0 ·V0 + 4πr20 ·σ + b

Q

r0, (12.69)


Fig. 12.7 Q-ball configuration: profile of the field χ(r) and wave function of φ-particle φ0(r).

where the first term is the energy of the field χ inside the Q-ball (recall that V0 =V (χ = 0)), and the second one is energy in the transition region near r0, σ beingthe tension of the Q-ball boundary. We will find (see (12.71)) that the Q-ball sizeis large for large Q, so the boundary term can be neglected, and the Q-ball energyis given by

E(r0) =43πr3

0 · V0 + bQ

r0. (12.70)

We now minimize this expression with respect to r0 and find the Q-ball radius atgiven Q,

r0(Q) =(

bQ

4πV0

)1/4

, (12.71)

so the Q-ball energy (mass) is

E[r0(Q)] ≡ MQ = const · V 1/40 Q3/4, (12.72)

with constant of order 1. We see that Q-ball energy grows with Q slower thanthe energy mφQ of free φ-particles in vacuum (12.66). This means that Q-ball isindeed the lowest energy state at sufficiently large Q. It is stable against decay intoφ-particles at

MQ < mφQ, (12.73)

i.e., at Q > Qc where the critical charge is of order

Qc ∼ V0

m4φ

. (12.74)

Note that for the potential V (χ) = λ · (χ2 − v2)2 the estimate (12.74) gives

Qc ∼ λ

h4,


so that the critical charge is large for λ ∼ h2 � 1, i.e., in the weak coupling regimewithout fine-tuning of parameters. If the estimate (12.74) gives formally Qc � 1 (say,for λ � h4), then the above calculation is inadequate for the critical Q-ball, andaccurate analysis is required for finding Qc. We note also that the above calculationis made under two assumptions. First, we assumed that φ-particles are the lightestparticles carrying global U(1) charge, otherwise the estimate (12.74) would containthe mass of the lightest charged particle instead of mφ. Second, it is important thatφ-particles are bosons, so they can all occupy the lowest energy level inside theQ-ball.

Problem 12.20. Let the particles φ charged under U(1) be fermions. Are Q-ballssimilar to those studied in the text stable in weakly coupled theories with mφ ∼ mχ,

where mχ is χ-particle mass in vacuum (12.66)? Hint: Take for definiteness V (χ) =λ · (χ2 − v2)2, λ � 1.

Let us give, for later convenience, an alternative but equivalent description ofQ-balls entirely within classical field theory. Such a description is adequate at largeQ. In classical field theory, one considers a classical configuration of the fields χ

and φ, instead of studying the state of quantum φ-particles. For time-independentχ and time-dependent φ the energy is

E =∫

d3x(

φ∗ · φ + ∇φ∗ · ∇φ + h2χ2|φ|2 +12∇χ · ∇χ + V (χ)

).

To construct the Q-ball, one finds the minimum of this functional at given value ofclassical charge (12.68). As the Ansatz for χ(x) one again chooses the configurationof Fig. 12.7, while for φ(x, t) one writes

φ(x, t) = Aeiωtf(x), (12.75)

where A is yet unknown amplitude and f(x) is normalized by∫|f(x)|2d3x = 1. (12.76)

Making use of the latter condition, one finds the energy and charge

E = ω2A2 + A2 · Ef +43πr3

0V0, (12.77)

Q = 2ωA2, (12.78)

where we again neglected the surface term in energy of field χ and introduced thenotation

Ef =∫

d3x(|∇f |2 + h2χ2(r)|f |2).

One chooses the function f(x) in such a way that it minimizes Ef under normal-ization condition (12.76). Thus, f(x) obeys

−Δf + h2χ2(r)f = λ2f. (12.79)


where λ is the Lagrange multiplier. Equation (12.79) coincides with the Schrodingerequation in potential h2χ2(r); its lowest eigenvalue λ determines Ef :

Ef = λ2.

Thus, f(x) coincides with the wave function of the ground state of φ-particle inQ-ball (f0(r) in Fig. 12.7), and

λ = Eφ =b

r0,

where b is the same coefficient as in (12.69).Hence, the energy functional (12.77) has the form

E = ω2A2 + A2 · b2

r20

+43πr3

0V0. (12.80)

It remains to find the minimum of this functional with respect to remainingunknowns ω, A and r0 at given value of charge (12.78). One obtains from (12.78)that ω = Q/(2A2). Then minimization of (12.80) with respect to A gives

A2 =Qr0

2b,

and the frequency is ω = b/r0, as one should expect. The energy as function ofthe only remaining parameter r0 is given precisely by Eq. (12.70). Further analysiscoincides with that given above, so we see that classical field theory approach isindeed equivalent to quantum mechanical one.

Let us discuss simple mechanism of the cosmological production of Q-balls asdark matter candidates. Assume that the field χ has zero expectation value at hightemperatures, and at some critical temperature Tc it develops non-zero expectationvalue χc due to first order phase transition. Let us assume further that the φ-particlemass is large in the new phase15

mφ(Tc) = hχc � Tc. (12.81)

We assume finally that by the phase transition, the Universe is Q-asymmetric, withthe asymmetry characterized by

ηφ =nφ − nφ

nφ + nφ

∼ nφ − nφ

nγ(12.82)

(we assume that φ is the lightest particle carrying the global U(1) charge).As we discussed in Sec. 10.1, 1st order phase transition occurs via nucleation

of a few bubbles of new phase per Hubble volume; these bubbles then expandand eventually percolate. As a result, there remain islands of the old phase χ = 0.These islands are precursors of Q-balls, and, indeed, entire Q-charge of the Universeconcentrates in these islands. The reason is that because of (12.81), φ-particles

15The latter assumption is in fact non-trivial. According to (10.30), (10.35) it implies, in particular,

that self-coupling of φ is small, λ � h4.


cannot penetrate from regions with χ = 0 to regions with χ = χc. Hence, theislands of old phase carry large charges Q. To make crude estimate, let us assumethat there remains of order 1 island of old phase per Hubble volume,16 i.e., thenumber density of Q-balls just below temperature Tc is estimated as

nQ(Tc) ∼ H3(Tc) =T 6

c

M∗ 3Pl

.

Every Q-ball collects the charge from the entire Hubble volume, so the Q-ballcharges are of order

Q ∼(

ηφ · nγ · 1H3

)T=Tc

∼ ηφ ·(

M∗Pl

Tc

)3

. (12.83)

The ratio of Q-ball number to entropy remains constant during subsequent evo-lution, so we obtain for the present number density of Q-balls

nQ,0 =nQ(Tc)s(Tc)

· s0 ∼ nQ(Tc)g∗(Tc)T 3

c

· s0 ∼ √g∗

T 3c

M3Pl

· s0. (12.84)

Making use of (12.72) and (12.83) we now estimate the present mass density ofQ-balls,

ρQ,0 = MQ · nQ,0 ∼ V1/40 η

3/4φ g

−5/4∗

(Tc

MPl

)3/4

· s0. (12.85)

Finally, for our crude estimate we set V1/40 and Tc equal to the energy scale of the

model, v, recall that s0 ∼ 103 cm−3 and choose g∗ ∼ 100. This gives

ρQ,0 ∼ 3 · 10−9 · η3/4φ

( v

1 TeV

)7/4 GeVcm3

.

We see that Q-balls are dark matter candidates provided that the energy scale v ishigh: even for ηφ ∼ 1 the required mass density ρQ,0 ∼ 0.2ρc ∼ 10−6 GeV · cm−3

is obtained for v somewhat larger than 10TeV. For parameters in this range thepresent number density of Q-balls calculated according to (12.84) is

nQ,0 (2 · 1013 cm)−3,

i.e., the average distance between Q-balls is of the order of the size of Earth orbit.The Q-ball masses for these parameters are

MQ =ρQ,0

nQ,0∼ 104 tons.

According to (12.83) and (12.71), the charge and size of such a Q-ball are

Q ∼ 1040, r0 ∼ 0.05 nm.

16This is not quite correct, in particular because the velocity of bubble wall is somewhat smaller

than speed of light.


For larger values of energy scale v (and, correspondingly, Tc), Q-balls are producedearlier, their present number density is larger, while the masses of Q-balls — darkmatter candidates — are smaller.

Problem 12.21. Find the value of v for which the present average number densityof Q-balls is nQ,0 = (108 cm)−3, so that they are dark matter candidates if theirmasses are MQ = ρCDM/nQ,0 ∼ 10−6 g. Estimate the charge asymmetry needed forproducing these dark matter Q-balls.

12.7.2 Models with flat directions

Somewhat different class of Q-balls [176] exists in models with sufficiently flat scalarpotentials. The simplest of these models contains single complex scalar field and hasthe Lagrangian

L = ∂μφ∗∂μφ − V (φ∗φ), (12.86)

where the potential V (φ∗φ) has absolute minimum at the origin; its other propertieswill be specified later on. This Lagrangian is invariant under the U(1) symmetry ofglobal phase rotations (12.67), and vacuum φ = 0 does not break this symmetry. Inwhat follows we set V (0) = 0, so that vacuum has zero energy.

Field equation in the model (12.86) is

∂μ∂μφ +∂V

∂φ∗ = 0, (12.87)

and the total energy is given by the integral

E =∫

[|∂0φ|2 + |∂iφ|2 + V (φ∗φ)]d3x. (12.88)

The mass of φ-particle in vacuum φ = 0 is

m =

√∂2V

∂φ∂φ∗ (0). (12.89)

In analogy with (12.75), let us try to find a Q-ball solution to Eq. (12.87). Thissolution should carry global charge (12.68), so one expects that it oscillates (ininternal space) and is spherically symmetric,

φ = eiωtf(r), φ∗ = e−iωtf(r), (12.90)

By substituting this Ansatz into Eq. (12.87) we find

d2f

dr2= −2

r

df

dr− d

df

(12ω2f2 − 1

2V (f)

). (12.91)

Formally, this equation coincides with the equation for classical particle of unit masswith coordinate f , moving with friction in the potential

Veff(f) =12ω2f2 − 1

2V (f),


while the radial coordinate plays the role of time. We find the asymptotics of f(r)at large r by requiring that energy (12.86) is finite. The energy functional for theconfiguration (12.90) has the form

E = 4π

∫[ω2f2 + (∂rf)2 + V (f)]r2dr. (12.92)

Its finiteness requires that

f(r → ∞) → 0. (12.93)

This means that the solution we search for is localized, as should be the case forQ-ball.

The behavior near r = 0 is found by requiring that “friction force” in Eq. (12.91)is finite. This gives [

df

dr(r → 0)

]∝ r1+ε, ε ≥ 0. (12.94)

The Q-ball profile f(r) corresponds to the “particle” that starts to move at zerovelocity at initial “time” r = 0 from some point

f0 = f(r = 0) (12.95)

and rolls down along the potential Veff(f) in such a way that the point f = 0 isreached in infinite “time”, f(r → ∞) → 0.

Clearly, the existence of such a solution implies that, first, the effective potentialVeff(f) has a minimum at f = 0, and second, the particle has positive energy at thestart of its motion, since it experiences friction force and its energy is zero in theend. Hence, the “initial” value of the field must be such that

V (f0)f20

≡ ω20 ≤ ω2. (12.96)

The region r � r0 where f(r) is considerably different from zero, 0 < f(r) � f0, isthe inner region of Q-ball, and r0 is the Q-ball size.

The charge of the Q-ball is given by

Q = 8πω

∫ ∞

0

f2(r)r2dr =8π

3ωf2

0 r30 , (12.97)

where the second equality is obtained by assuming that

f(r) ≈ f0 · θ(r0 − r).

This is a good approximation in the thin wall regime, when the size Δr of the regionwhere f(r) changes is small compared to the size of Q-ball itself, Δr � r0. We notethat in the general case, the second equality in (12.97) can be considered as thedefinition of Q-ball size r0.


Assuming the thin wall regime, we estimate the Q-ball energy (12.92) as

E ≈ 4π

3r30 [ω2f2

0 + V (f0)], (12.98)

where we neglected the contribution of the gradient term (∇f)2. The latter comesfrom the Q-ball surface region, and is therefore small for large Q. The Q-ball size isobtained by minimizing the energy (12.98) with respect to r0 at given value of thecharge (12.97). To this end, we use (12.97) to express the frequency ω through r0

and insert the result into (12.98). This gives

E ≈ 4π

3r30V (f0) +

3Q2

16πr30f

20

.

The minimum of this expression with respect to r0 is at

4π

3r30 =

Q

2f0

1√V (f0)

.

The energy of the Q-ball is

E =Q

f0

√V (f0),

while the frequency ω reaches the critical value ω0 given by (12.96). Finally,the energy of stable stationary solution must be at minimum with respect tothe remaining parameter, the field value in the center f0. This is only possibleif the function V (f)/f2 has a minimum at finite f = f0 �= 0,

minf

[V (f)f2

]=

V (f0)f20

�= 0, f0 �= 0, f0 �= ∞. (12.99)

It is this value that the field takes in the soliton center.The properties that the potential V (f) has global minimum at f = 0 and at

the same time that the function V (f)/f 2 has minimum away from zero are quitenon-trivial, especially if one recalls that V (f) is actually a function of f2 = φ∗φ (thelatter follows from invariance under global U(1)). As an example, the conventionalrenormalizable potential V (φ) = m2φ∗φ + λ(φ∗φ)2 does not have these properties.The existence of Q-balls needs more exotic scalar potentials which are sufficientlyflat at least somewhere in the field space. We will briefly discuss in what modelsthis is indeed the case, and here we proceed by assuming that the potential has therequired properties.

The Q-ball stability condition is still given by (12.73), which we write as

Q

f0

√V (f0) < Q · m. (12.100)

Making use of (12.89) we find that the latter condition is yet another non-trivialrequirement imposed on the scalar potential,

2V (f0)f20

<d2V (0)

df2. (12.101)


Provided the potential has properties (12.99), (12.101) and has global minimum atthe origin, there exist stable Q-balls of the type we have described.

We note that we have performed the analysis for large Q-balls, as we neglectedthe surface contribution to the energy. Detailed analysis shows that some modelsadmit rather small Q-balls whose charges are not exceedingly larger than 1.

Problem 12.22. Find approximate Q-ball solution (assuming that Q is large) ina model with gauge group U(1). Show that the ratio E/Q increases with Q becauseof the Coulomb contribution to the energy. Show that at large charge the Q-ball isunstable against emitting charged particles from its surface. We note, though, thatfor sufficiently small gauge coupling, there is a range of Q-ball charges in whichQ-balls are stable in models with gauge U(1) symmetry.

Let us now consider the case when the function V (f)/f2 does not have aminimum at finite f . If V (f) increases with f slower than f2, then the minimumof V (f)/f2 is at f → ∞ and the value at minimum is zero,

minf

[V (f)f2

]=[V (f)f2

]f→∞

= 0. (12.102)

This is possible only in models with very flat scalar potentials. In this case ourprevious analysis does not go through. However, in the case of very flat potential,

f → ∞ : V (f) ∝ fα, α < 2,

the scalar potential can be neglected in Eq. (12.91) and there exists approximatesolution obeying all above conditions (12.93)–(12.95),

f(r) = f0sin ωr

ωr· θ(r0 − r). (12.103)

This soluton is continuous across the Q-ball boundary r = r0 provided that thesoliton size is related to the oscillation frequency,

r0 =π

ω. (12.104)

This is in contrast with the case (12.99). The form of solution (12.103) shows thatthe thin wall approximation is not valid, as f(r) varies in the entire region r <

r0. Nevertheless, the solution is explicitly known, so we can proceed further. Byexpressing the frequency in terms of the size, we find the charge of the Q-ball,

Q = 4f20 r2

0 . (12.105)

and its energy (12.92)

E = 4πf20 r0 +

4π

3r30bV (f0), (12.106)


where the parameter b is of order 1; it is determined by the shape of the scalarpotential. We now have to minimize the energy with respect to parameters f0 andr0 at given value of the charge (12.105). By expressing r0 through f0 from (12.105)we find energy as function of f0,

E(f0) = 2πf0Q1/2 +

π

6bQ3/2

f30

V (f0). (12.107)

Its minimum occurs at

b

12d

df0

(V (f0)

f30

)=

1Q

. (12.108)

The left hand side decreases as function of f0 for potentials under discussion, so thesolution f0(Q) to Eq. (12.108) increases with Q. In particular, for V (f) ∼ fα wehave from (12.108) that

f0(Q) ∝ Q1

4−α ,

and the energy (12.107) behaves as

E(Q) ∝ Q6−α

2(4−α) .

For α < 2 the exponent here is smaller than 1, so the Q-ball energy is smaller thanenergy of Q particles at least for large Q; the Q-ball is stable. We note that in thelimit α → 0 (very flat potential at large fields), energy behavior is

E ∝ Q3/4. (12.109)

This is analogous to (12.72).Q-balls exist in some realistic extensions of the Standard Model, including super-

symmetric ones [177–180]. Our analysis goes through in these models, except forone important point. The analogs of our φ-particles are usually not the lightestparticles carrying Q-charge, so the bound (12.100) gets modified. It is the mass ofthe lightest charged particle that enters the right hand side of the bound (12.100)in realistic models.

The charge Q in realistic theories may be baryon number, lepton number or acombination thereof. Flatness of the scalar potential is a natural property of super-symmetric theories, which often have flat directions at the tree level. Quantum cor-rections lift the flat directions, but the potential remains nearly flat, since quantumcorrections give weak dependence on the field, V ∝ log |φ|.

In SUSY theories, Q may be baryon number, and then the field φ is a combi-nation of squark fields. The parameter m in the generalized formula (12.100) is theproton mass in that case. If Q is lepton number, then the field φ is a combinationof sleptons, and m in (12.100) is neutrino mass. In models with potentials of thetype (12.102), Q-balls are stable on cosmological time scale for large values of theircharge only.


Problem 12.23. Estimate the charges of stable B-ball and L-ball (Q being baryonand lepton number, respectively) in models with potentials of the type (12.102). Fornumerical estimate consider the following potential at large f,

V (f) (1 TeV)4 · (f/1 TeV)α,

and study the cases α = 1 and α → 0.

Problem 12.24. Let us consider a model with Q-balls in which scalar particles areunstable against decay into massless fermions. Q-balls in this model are unstable,but may have long lifetime, since because of Pauli blocking, fermions evaporate fromthe Q-ball surface rather than from its interior. Assuming that the only parameterdetermining the evaporation rate per unit area is the frequency ω, find the charge forwhich the Q-ball lifetime exceeds the age of the Universe. Make numerical estimatesfor the same potentials as in the previous problem.

Properties of large Q-balls are affected by gravity. At very large Q these objectsare gravitationally unstable and collapse into black holes.

Problem 12.25. Estimate the critical charge above which Q-balls collapse intoblack holes in models of the types (12.99) and (12.102). Take all dimensionful param-eters of order 1 TeV for numerical estimates. Compare with the results of previoustwo problems.

The dominant mechanism of Q-ball production is the decay of flat directionsof the scalar potential, moduli fields [179, 181]. In the limit r0 → ∞ our Q-balls degenerate into homogeneous scalar condensate filling the entire space. It canbe formed in much the same way as the condensate of the Affleck–Dine field, seeSec. 11.6, but now this condensate carries certain density of U(1) charge. As we willsee below, the condensate is actually unstable; it decays by producing very differentcharge densities in different places in space. The regions with high charge densityeventually evolve into Q-balls, while in regions with low charge density the chargeis carried by free particles. Detailed analysis shows that this mechanism indeedefficiently produces Q-balls, so that they carry fairly large fraction of the chargeinitially contained in the condensate.

Let us consider this mechanism in some detail. We have seen in Sec. 11.6 thatthe asymmetry (here we are talking about asymmetry with respect to global U(1)charge Q) is produced in narrow time interval near the epoch at which the slow rollconditions get violated,17

V ′(φi)φi

∼ H2(tr), (12.110)

where φi is the initial value of the field, and subscript r refers to the end of slowroll regime, cf. Sec. 11.6. Let small explicit breaking of U(1) result in Q-asymmetry

ηQ =nQ

s,

17We use here the general relation (4.49) rather than (4.50) valid for power-law potentials.


where nQ is the Q-charge density. Unlike in the previous Section, the Q-chargedensity is due to the field φ itself, as this field evolves as shown in Fig. 11.17.

Let us first show that the homogeneous scalar condensate carrying Q-charge isunstable against production of inhomogeneities. To this end, we neglect the cosmo-logical expansion (we will discuss this point later on) and write the scalar field as

φ(x) = f(x) · eiα(x).

The field equations for f and α are

α − Δα +2f

αf − 2f

∇α · ∇f = 0 (12.111)

f − Δf − fα2 + fΔα + V ′(f) = 0 (12.112)

Let us begin with the situation in which the scalar field rotates along a circle ininternal space,

f = const, α = ωt, ω = const, (12.113)

and it follows from (12.112) that

ω2 =V ′

f. (12.114)

Note that at t > tr, when slow roll conditions are violated, one has ω > H . In theend, this justifies our approximation of static Universe.

Let us consider linear perturbations about the homogeneous condensate(12.113). Making use of translational invariance we write

δf(x) = δf · eiλt+ipx,

δα(x) = δα · eiλt+ipx.(12.115)

Equations (12.111) and (12.112) give

(2iλfω) · δf + f2 · (λ2 − p2) · δα = 0,

(λ2 − p2 − V ′′(f) + ω2) · δf − (2iλfω) · δα = 0.

We combine these equations to obtain the equation for eigenvalues at each p2,

λ4 − λ2 · (2p2 + V ′′ + 3ω2) + p2 · (p2 + V ′′ − ω2) = 0. (12.116)

It is straightforward to see that all its solutions λ2 are real. For

p2 + V ′′ − ω2 < 0, (12.117)

one of the roots is negative, λ2 < 0. This corresponds to imaginary λ and henceinstability of the condensate: some modes (12.115) increase exponentially in time.Importantly, these modes are not homogeneous in space: all values of λ are real atp = 0.


Making use of (12.111) we find that the condition (12.117) can be satisfied ifthe potential is sufficiently flat,

V ′′ <V ′

f. (12.118)

The latter inequality is valid for power-law potential, V ∝ fα, at α < 2, i.e., whenthe model admits Q-balls. The strongest instability (the largest −λ2) occurs for

|p| ∼ ω, (12.119)

and the exponent is |λ| ∼ ω. This is the final justification of our static Universeapproximation.

The development of instability in non-linear regime is difficult, if not impos-sible, to study analytically. Still, the very fact that the condition for instability(12.118) coincides with the condition for the existence of Q-balls suggests that theprocess ends up by Q-ball formation. Numerical simulations support this conclusion[182, 183]. The estimate (12.119) shows that Q-charge is collected into Q-ball fromthe region of volume

|p|−3 ∼ ω−3.

Both evolution of the condensate and the development of instability occur rightafter the end of the slow roll regime, so a crude estimate is

|p| ∼ |λ| ∼ ω ∼ H(tr),

where we made use of (12.110) and (12.114). Thus, of order 1 Q-ball is producedper Hubble volume at time tr.

Further estimates are quite similar to our estimates in the end of Sec. 12.7.1.Let us consider for definiteness almost flat potential at large f ,

V (f) = v4

(f

v

)α

, α � 1,

where v is a parameter of dimension of mass. In this case one has the estimate(12.109), i.e., the Q-ball mass is of order

MQ ∼ vQ3/4.

The results of Sec. 12.7.1 are directly translated to our model, and we obtain (see(12.85))

ρQ,0 = vη3/4Q g

−5/4∗

(Tr

MPl

)3/4

s0.

The difference, though, is that the temperature Tr is determined by (12.110), sothat in the limit of small α we have

Tr ∼ v

√MPl

fi.


This introduces additional uncertainty into the estimate of the present Q-ball massdensity related to the unknown initial value of the field φ. If we set fi ∼ MPl as inSec. 11.6, then the results of Sec. 12.7.1 are literally valid. With appropriate choiceof parameters, Q-balls of our model serve as dark matter candidates.

If Q-charge is baryon number, then Q-balls may play a role in the generationof the baryon asymmetry [180]. As an example, B-balls may carry baryon numberthrough the epoch when electroweak baryon and lepton number violating interac-tions are at work and tend to wash out the asymmetry.

Of particular interest are B-balls which are unstable at fairly late cosmologicalepoch. Their decays transmit their baryon number into quarks. This is basicallya version of the Affleck–Dine mechanism. Most intriguing is the situation whenQ-balls are unstable and decay into baryons and new stable particles, dark mattercandidates. Such a mechanism may possibly explain approximate equality betweenthe present energy densities of baryons and dark matter. As an example, the decayof squark condensate produces three neutralino LSP per baryon. If most of baryonnumber initially was in Q-balls, one obtains a simple relation between mass densitiesof baryons and LSP dark matter,

ρB

ρCDM∼ mp

3mLSP. (12.120)

For realistic LSP masses, mLSP ∼ 10–100GeV, this is only two or three orders ofmagnitude smaller than the real value. We note in this regard that the neutralinodensity may get diluted by their annihilation near Q-ball surface.

To end this Section we note that the class of non-topological solitons is notexhausted by Q-balls. This class includes also quark nuggets, cosmic neutrino balls,soliton stars and other hypothetical objects.

Chapter 13

Color Pages

Fig. 13.1 Spatial distribution of galaxies and quasars from the analysis of early observational

data of SDSS [4]. Green dots mark all galaxies (within a given solid angle) with brightness

(apparent magnitude) exceeding a certain value. Red dots show galaxies of a special type (Large

Red Galaxies), which are very luminous and form fairly homogeneous population; in the comoving

frame their spectrum is shifted towards the red wave-band as compared to ordinary galaxies.

Turquoise and blue points show the positions of ordinary quasars. The value of parameter h is

about 0.7 (see Sec. 1.2.2).

347

348 Color Pages

Fig. 13.2 WMAP data [9]: angular anisotropy of CMB temperature, i.e., variation of the tem-

perature of photons coming from different directions in the sky (shown by color). The average

temperature and dipole component are subtracted. The observed variation of temperature is at

the level of δT ∼ 100 μK, i.e., δT/T0 ∼ 10−4 − 10−5.

Fig. 13.3 CMB temperature anisotropy measured by various instruments [8]. The theoretical

curve is the best fit of the ΛCDM model (see Chapter 4) to the WMAP data; this fit is in

agreement with other experiments as well.

Color Pages 349

Fig. 13.4 Cluster of galaxies CL0024 + 1654 [22]. Blue color in the left panel illustrates dark

matter distribution; elongated blue objects in the right panel are multiple images of a galaxy

behind the cluster.

Fig. 13.5 Observation of “Bullet cluster” 1E0657-558, two colliding clusters of galaxies [24]. Lines

show gravitational equipotential surfaces, the bright regions in the right panel are regions of hot

baryon gas.

350 Color Pages

0.0 0.1 0.2 0.3 0.4 0.5-1.5

-1.0

-0.5

0.0

m

w

SNe

BAO

CMB

Fig. 13.6 Regions in the plane of parameters (ΩM , w) allowed (for the flat Universe) by observa-

tions of CMB anisotropy, by large scale structures (BAO) and by SNe Ia data [41]. The intersection

region corresponds to the combined analysis of all these data. Regions of smaller and larger size

correspond to 68.3%, 95.4% and 99.7% confidence level, respectively.

Color Pages 351

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500

600

700

800

0

100

200

300

400

500

600

700

800

mh = 114 GeV

m0

(GeV

)m

0 (G

eV)

m0

(GeV

)m

0 (G

eV)

m1/2

(GeV)

m1/2

(GeV) m1/2

(GeV)

m1/2

(GeV)

tan β = 10 , μ > 0

mχ± = 104 GeV

mh = 114 GeV

tan β = 10 , μ > 0

mχ± = 104 GeV

100 1000 2000 30000

1000

1500

mh = 114 GeV

tan β = 50 , μ > 0

100 1000 2000 30000

1000

1500

0

mh = 114 GeV

tan β = 50 , μ > 0

Fig. 13.7 Experimentally forbidden and cosmologically favored regions [85] in the plane

(M1/2, m0) in mSUGRA model with tan β = 10 and tan β = 50 (A is chosen to be zero at

the scale MGUT ≈ 1016 GeV). Forbidden regions in each plot: the region left of the thick blue

dashed line is excluded by the bound on light chargino mass mχ± > 104 GeV, the region in the

lower left corner left of blue dash-dotted line is excluded by the bound on the light slepton mass

me > 99GeV, the region left of red dash-dotted line is excluded by the bound on the lightest

Higgs boson mass, the green region in the left part is excluded by the measurement of b → sγ

decay width, the light pink strip is excluded by the measurement of the muon anomalous magnetic

moment. Brown region on the right is cosmologically unfavored, since it corresponds to charged

slepton LSP (mostly stau). Neutralino dark matter regions are shown in light blue; the neutralino

mass density range in the left and right plots are 0.1 < ΩNh2 < 0.3 (conservative estimate) and

0.094 < ΩNh2 < 0.129, respectively. In the bulk of experimentally allowed region, neutralinos are

overproduced.

352 Color Pages

100 200 300 400 500 600 700 800 900 10000

1000

2000

3000

4000

5000

mh = 114 GeV

m0

(GeV

)

m0

(GeV

)

m 1/2 (GeV) m 1/2

(GeV)

tan β = 10 , μ > 0

100 1000 2000 25000

1000

1500

μ > 0

Fig. 13.8 Left: The same as in upper part of Fig. 13.7 but for larger range of masses. The

pink upper left region is excluded by the requirement of electroweak symmetry breaking. Right:

Cosmologically favored regions (0.094 < ΩN h2 < 0.129) for various values of tan β = 5, 10, . . . , 55;

lower strips correspond to smaller tan β [85].

100 1000 2000 100 1000 20000

100

200

300

400

500

m0(

GeV

)

m0(

GeV

)

M 1/2 (GeV) M 1/2 (GeV)

m 3/2 = 100 GeV, tan β = 10 , μ > 0

0

1000

20002000m 3/2 = 100 GeV, tan β = 57 , μ > 0m 3/2 = 100 GeV, tan β = 57 , μ > 0

Fig. 13.9 Bounds [91] on the plane (M1/2, m0) in mSUGRA model with m3/2 = 100GeV and

two values of tan β. The region where gravitino is LSP is to the right of solid black line. The

green shaded region is excluded from b → sγ decays. The region to the right of the solid red line

is excluded by BBN. The blue dash-dotted line divides the regions with neutralino NLSP (upper

parts of figures) and the lightest slepton (lower parts). In the blue shaded region, NLSP, if stable,

would have the right dark matter mass density (0.094 ≤ Ωh2 ≤ 0.129). The present gravitino mass

density coincides with the dark matter mass density at pink dashed lines. The gravitino mass

density is smaller than observed ΩDM below this line (left panel) and between these lines (right

panel). Thus, there are cosmologically allowed regions in this model.

Color Pages 353

Fig. 13.10 Allowed region of parameter space for oscillations νe ↔ ν obtained from solar neutrino

experiments and KamLAND [210].

Appendix A

Elements of General Relativity

A.1 Tensors in Curved Space-Time

In this Appendix we introduce the basic concepts of General Relativity. Our pre-sentation does not pretend to be mathematically rigorous and comprehensive; itsmain purpose is to introduce the notions used in the main text and to gather in asingle place a number of useful relations and formulas. More systematic treatment ofGeneral Relativity (GR) is given in the textbooks [53, 184, 185]. Readers, interestedin more mathematically rigorous description of differential geometry, are invited toconsult the book [186]. Conventions and notations used in this book are collectedin Sec. A.11.

The main object of study in GR is curved four-dimensional space (manifold) Mdescribing space-time. For better understanding of abstract mathematical notionsintroduced below, it is sometimes helpful to think of this space as embedded intoa flat enveloping space (ambient manifold) of higher dimension. We emphasize,however, that our space-time is embedded nowhere,1 and we never rely on the possi-bility of such an embedding. It is worth noting here that all definitions and facts con-tained in this Appendix are straightforwardly generalized to (pseudo-)Riemannianspaces of arbitrary dimension.

The interval (square of the invariant distance) ds2 between two nearby pointsin space-time is2

ds2 = gμν(x)dxμdxν , (A.1)

where indices μ, ν take values 0, 1, 2, 3, and metric gμν(x) can be considered as a4 × 4 symmetric matrix. Thus, the metric is determined by ten independent func-tions of the coordinates gμν(x), μ ≤ ν. Hereafter we assume that the metric has thesignature (+,−,−,−), i.e., that the matrix gμν(x) has one positive and three neg-ative eigenvalues at each point x. Vectors dxμ of positive, zero and negative values ofds2 correspond to time-like, light-like (null) and space-like directions, respectively.

1We are not discussing here models with extra spatial dimensions.2In what follows, unless otherwise stated, summation over repeated indices is implied.

355

356 Elements of General Relativity

The basic principle of GR is that all choices of the local coordinate frame areequivalent to each other. It is therefore natural to consider the functions (fields) onthe manifold M, which transform in a certain way under coordinate transformation,

xμ → x′μ(xμ). (A.2)

The simplest example of such an object is scalar field φ(x), whose defining propertyis that it transforms as

φ′(x′) = φ(x).

Hereafter all quantities with prime belong to the new coordinate frame, while quan-tities without prime refer to the original one. The above relation shows that thevalue of the field at a given point of the manifold does not change under coordinatetransformation. Another important example of an object, “well”-transforming undercoordinate transformations, is a contravariant vector: a set of four functions Aμ(x),transforming in the same way as small increments of coordinates dxμ, i.e.,

A′ν(x′) =∂x′ν

∂xμAμ(x). (A.3)

The covariant vector is a set of four variables Aμ(x) which transform in the sameway as the derivatives ∂

∂xμ , i.e.,

A′ν(x′) =

∂xμ

∂x′ν Aμ(x). (A.4)

Using the transformation laws (A.3) and (A.4), it is easy to obtain transformationlaw of the contraction AμBμ,

A′μ(x′)B′μ(x′) =

∂x′μ

∂xνAν(x)

∂xλ

∂x′μ Bλ(x)

=∂xλ

∂xνAν(x)Bλ(x)

= Aν(x)Bν(x). (A.5)

We see that this contraction transforms as a scalar, i.e., its value at each point doesnot depend on the choice of coordinates.

From the point of view of geometry, contravariant vector Aμ(x) can be thought ofas a tangent vector to the surface M, if the latter is embedded into some envelopingspace. For example, a derivative of a scalar function φ(x) along the direction definedby the tangent vector Aμ(x) has the form

∂Aφ(x) = Aμ(x)∂μφ(x). (A.6)

Invariance of contraction (A.5) under coordinate transformations shows thatcovariant vectors Bμ(x) can be regarded as linear functionals that map, via con-traction, the tangent space to real numbers.

Similarly, we can define tensor with an arbitrary number of upper and lowerindices. Such an object transforms in the same way as the product of an appropriate

A.1. Tensors in Curved Space-Time 357

number of covariant and contravariant vectors. For example, tensor Bμνλ transforms

under coordinate transformations as follows,

B′μνλ(x′) =

∂x′μ

∂xσ

∂xτ

∂x′ν∂xρ

∂x′λ Bστρ(x)

Directly generalizing the above result for contraction of covariant and contravariantvectors, it is straightforward to prove that by contracting an upper and a lowerindices in tensor of arbitrary rank we again obtain tensor.

From the fact that the interval ds2 defines the distance between two points, whichis independent of the choice of coordinate frame, it follows that metric gμν(x) is acovariant tensor of the second rank, i.e., it transforms as

g′μν(x′) =∂xλ

∂x′μ∂xρ

∂x′ν gλρ(x). (A.7)

Problem A.1. Prove the transformation law (A.7).

Another important example of second rank tensor is the Kronecker δ-symbol δνμ

defined in an arbitrary coordinate frame as unit diagonal matrix,

δνμ = diag(1, 1, 1, 1).

Let us check that this definition of δνμ is consistent with the tensor transformation

law. If a tensor is equal to δνμ in an original coordinate frame, then in a new coor-

dinate frame it is equal to

∂x′μ

∂xλ

∂xρ

∂x′ν δλρ =

∂x′μ

∂xλ

∂xλ

∂x′ν .

The right hand side here is again the Kronecker symbol δμν , so that δμ

ν is a tensorindeed. Starting from the metric tensor gμν and the Kronecker tensor δμ

ν one candefine a new contravariant second rank symmetric tensor gμν by the followingequality,

gμνgνλ = δμλ . (A.8)

In other words, the matrix gμν is inverse to the matrix gμν .

Problem A.2. Prove that gμν is indeed a tensor.

Making convolutions with tensors gμν and gμν , we can define the operations ofraising and lowering of indices. For example,

Aν = gνμAμ, Bμν = gμλgνρBλρ.

If Aμ and Bλρ are tensors, then Aν and Bμν are tensors too.Another important object needed to construct the action functional in GR is

the determinant of the metric tensor,

g ≡ det gμν .


In order to determine how g transforms under coordinate transformations, we writethe transformation law (A.7) in the matrix form:

g′(x′) = J g(x)JT . (A.9)

Hats indicate that all objects in Eq. (A.9) are matrices of size 4× 4. The symbol J

denotes the Jacobian matrix corresponding to the change of coordinates (A.2),

Jμν =

∂xμ

∂x′ν ,

while JT is the transposed matrix. Equality (A.9) implies the following transfor-mation law for g,

g′(x′) = J2g(x), (A.10)

where J is the Jacobian determinant of the coordinate transformation (A.2),

J ≡ det(

∂xμ

∂x′ν

).

It follows from the transformation law (A.10) that the product√−gd4x

defines the invariant 4-volume element. Since the matrix gμν has three negativeand one positive eigenvalues, the determinant g is negative. Therefore, the quantity√−g is real.

In Minkowski space, in addition to the Kronecker delta, there is one more tensorwhich is invariant under Lorentz transformations. It is the Levi-Civita symbol εμνλρ.We remind that εμνλρ is totally antisymmetric in its indices and, consequently, it isuniquely defined by the condition

ε0123 = 1.

However, the Levi-Civita symbol is not invariant under arbitrary coordinate trans-formations. Indeed, if a tensor is equal to εμνλρ in a given coordinate frame, thenin another frame it is equal to

ε′μνλρ =∂x′μ

∂xα

∂x′ν

∂xβ

∂x′λ

∂xγ

∂x′ρ

∂xδεαβγδ = J−1εμνλρ. (A.11)

The transformation law (A.11) shows that the natural generalization of the Levi-Civita symbol to the case of arbitrary curvilinear coordinates and to curved spaceis the Levi-Civita tensor3

Eμνλρ =1√−g

εμνλρ.

3More accurately, Eμνλρ is pseudotensor because it transforms with “wrong” sign under the

coordinate transformations which change the spatial orientation (i.e., transformations with J < 0).

A.2. Covariant Derivative 359

This tensor is completely antisymmetric in all its indices and reduces to εμνλρ whenmetric gμν coincides with the metric of Minkowski space in Cartesian coordinates,ημν = diag(1,−1,−1,−1).

A.2 Covariant Derivative

In order to build the action invariant under arbitrary coordinate transformations,we have to define the covariant differentiation operator ∇μ that converts tensorsinto tensors. For scalar field, it is natural to demand that this operation coincideswith the usual differentiation,

∇μφ(x) ≡ ∂μφ(x). (A.12)

It follows from the definition (A.4), that the derivative ∇μφ is covariant vector.One cannot define the covariant derivative of vector field Aμ(x) in the same way.

In order to differentiate vector field, one must learn how to subtract tangent vectorsbelonging to different points of the space M. Therefore, we have to define the ruleof parallel transport of vectors from one point to another.

Consider the parallel transport of contravariant vector Aμ from the point withcoordinates xμ to the point with coordinates

xμ = xμ + dxμ

(see Fig. A.1). Imposing the natural requirement of linearity (the sum of two vectorsupon parallel transport becomes the sum of images), we observe that to the leadingorder in the increments of coordinates dxμ, the image Aμ of the vector Aμ has thefollowing general form

Aμ(x) = Aμ(x) − Γμνλ(x)Aν(x)dxλ. (A.13)

The quantities Γμνλ entering here are called connection coefficients.4 To determine

the transformation law of the connection coefficients under arbitrary coordinatetransformations, we perform coordinate transformation in both parts of equality

Fig. A.1 Parallel transport of a vector.

4In the Riemannian geometry, connection coefficients are also called Christoffel symbols, see

below.


(A.13) and make use of the fact that the quantities Aμ, Aμ and dxμ transformaccording to the law (A.3). The left hand side of (A.13) transforms into

A′μ(x′) =∂x′μ(x)

∂xνAν(x) =

(∂x′μ(x)

∂xν+

∂2x′μ(x)∂xν∂xλ

dxλ

)Aν(x)

=∂x′μ(x)

∂xνAν(x) +

∂2x′μ(x)∂xν∂xλ

dxλAν(x), (A.14)

where we work to the linear order in dxμ. Upon coordinate transformation, theright-hand side of (A.13) takes the following form,

A′μ(x′) − Γ′μνλ(x′)A′ν(x′)dx′λ =

∂x′μ

∂xνAν(x) − Γ′μ

νλ(x′)∂x′ν

∂xρAρ(x)

∂x′λ

∂xσdxσ.

(A.15)

Equating the results of the transformations (A.14) and (A.15), convoluting bothsides of the resulting equality with ∂xμ

∂x′ν , and comparing the result with the originalrule of parallel transport (A.13), we obtain the following transformation law for theconnection coefficients,

Γ′μνλ(x′) =

∂xρ

∂x′ν∂xσ

∂x′λ∂x′μ

∂xξΓξ

ρσ +∂x′μ

∂xρ

∂2xρ

∂x′ν∂x′λ . (A.16)

The second term on the right hand side of (A.16) shows that connection is nottensor.

To define the covariant derivative of vector field we transport the vector Aμ(x)to the point x = x + dx, subtract the resulting vector from the value of the vectorfield at the point x and write

Aμ(x) − Aμ(x) = ∇νAμ · dxν .

Using the parallel transport rule (A.13), we arrive at the following definition of thecovariant derivative of vector field Aμ(x),

∇νAμ(x) = ∂νAμ(x) + ΓμλνAλ(x). (A.17)

The transformation law of the connection coefficients (A.16) guarantees that ∇νAμ

is a second rank tensor with one covariant and one contravariant indices.The parallel transport rule of covariant vector Bμ follows from the fact that the

contraction AμBμ is a scalar transported in a trivial way,

(AμBμ)(x) = (AμBμ)(x). (A.18)

The relation (A.18) and parallel transport law of contravariant vector (A.13) givethe following parallel transport rule for covariant vector Bμ,

Bμ(x) = Bμ(x) + ΓνμλBν(x)dxλ. (A.19)

Consequently, the covariant derivative has the form

∇νBμ(x) = ∂νBμ(x) − ΓλμνBλ(x). (A.20)


Now that we have defined the covariant derivatives of scalar and vectors of bothtypes, it is not difficult to generalize these definitions to tensors of arbitrary rank.This is done using the Leibniz rule,

∇μ(AB) = (∇μA)B + A∇μB,

where A and B are two arbitrary tensors whose indices are not explicitly written.For example, the covariant derivative of third rank tensor with one upper and twolower indices reads

∇μBνλτ = ∂μBν

λτ + ΓνρμBρ

λτ − ΓρλμBν

ρτ − ΓρτμBν

λρ.

In principle, one could consider manifolds with arbitrary set of connection coef-ficients transforming according to the law (A.16). However, GR is based on the(pseudo-)Riemannian geometry,5 in which additional conditions are imposed on theconnection coefficients Γμ

νλ. The first of these conditions is that the operation of par-allel transport (or, equivalently, the operation of covariant differentiation) commuteswith the operation of raising and lowering of indices. This means, in particular, that

gμν∇λAν = ∇λ(gμνAν)

for an arbitrary vector Aν . The Leibniz rule implies that this is possible only if themetric tensor gμν is covariantly constant,

∇μgνλ = 0. (A.21)

More explicitly, this condition reads

∂μgνλ = Γρνμgρλ + Γρ

λμgνρ.

Connections satisfying the condition (A.21) are called metric connections (sincethey are consistent with metric). The second condition imposed on the connectioncoefficients is the requirement that they are symmetric in lower indices,

Cλμν ≡ Γλ

μν − Γλνμ = 0. (A.22)

The transformation law of the connection (A.16) implies that Cλμν is a tensor (called

torsion tensor in the general case). Hence, the validity of (A.22) does not dependon the choice of coordinate frame. Manifold equipped with metric and torsionlessmetric connection is precisely what is called Riemannian manifold, and in this casethe connection coefficients are called Christoffel symbols.

Equations (A.21) and (A.22) enable us to unambiguously express the Christoffelsymbols in terms of the metric tensor,

Γμνλ =

12gμρ(∂νgρλ + ∂λgρν − ∂ρgνλ). (A.23)

We always assume in what follows that the equality (A.23) holds.6

5Pseudo-Riemannian geometry differs from the Riemannian one by the signature of metric, which

for the Riemannian geometry is Euclidean. We often do not pay attention to this terminological

subtlety.6In geometries more general than Riemannian, objects given by (A.23) are also called Christoffel

symbols. Other terms used for them are metric connection and Riemannian connection.


Problem A.3. Consider a 2-dimensional surface Σ embedded in 3-dimensionalEuclidean space R3. The space R3 induces metric in Σ : if yi (i = 1, 2) are coordi-nates on the surface Σ, then the square of the distance between nearby points on Σcan be written as

ds2 = gijdyidyj ,

where the metric gij(y) is uniquely determined by the requirement that ds be thedistance in R3. There is an obvious definition of tangent plane at each point onthe surface Σ; contravariant vectors, as discussed above, are vectors belonging tothe tangent plane. Their components Ai(y) in the chosen coordinate frame on Σcan, for instance, be defined by the relation

∂Aφ(y) = Ai(y)∂φ

∂yi,

where φ(y) is a function on the surface Σ, and ∂Aφ is its derivative along thedirection determined by the vector �A. Parallel transport of tangent vector alongthe surface Σ is naturally defined in the following manner (see Fig. A.2): first wetransport the vector �A from point y to point y as a vector in R3 (yielding the vector�A|| in Fig. A.2), and then we project it onto the tangent plane at point y. Let thesurface Σ be (locally) defined by the equations

xα = fα(y1, y2), α = 1, 2, 3,

where xα are coordinates in R3.

(1) Calculate the components of the metric gij(y).(2) Calculate the Christoffel symbols Γk

ij(y) on the surface Σ, which are associatedwith the above operation of parallel transport.

(3) Show that the properties (A.21) and (A.22) are satisfied, i.e., the geometry onthe surface Σ is Riemannian.

(4) Propose a generalization of the parallel transport of a vector, such that thetorsion tensor (A.22) is non-zero. Demonstrate this property by explicit cal-culation of the connection coefficients. Does the relation (A.21) remain valid?

Problem A.4. Derive the formula (A.23) from Eqs. (A.21) and (A.22).

Fig. A.2 Parallel transport of a tangent vector.


Problem A.5. Check the following properties of the Christoffel symbols andcovariant derivative:

Γμνμ = ∂ν ln

√−g, (A.24)

gμνΓλμν = − 1√−g

∂μ(√−ggλμ), (A.25)

∇μAμ = 1√−g∂μ(

√−gAμ), (A.26)

for antisymmetric tensor Aμν :

∇μAμν =1√−g

∂μ(√−gAμν), (A.27)

for scalar φ :

∇μ∇μφ =1√−g

∂μ(√−ggμν∂νφ), (A.28)

where ∇μφ ≡ gμν∇νφ.

The property (A.26) leads to the following generalization of the Gauss formula,∫(∇νAν)

√−gd4x =∫

∂ν(√−gAν)d4x =

∫ √−gAνdΣν ,

where dΣν is the element of surface which bounds the integration region. Togetherwith the Leibniz rule for the covariant derivatives, this formula allows for integrationby parts of invariant integrals. For example∫

Aμ∇νBμν√−gd4x = −∫

(∇νAμ)Bμν√−gd4x + surface terms.

To conclude this Section, we note the following fact. By chosing a suitable coor-dinate frame, we can locally, at a given point, set all Christoffel symbols equal tozero; this is fully consistent with the equivalence principle because this enables usto switch off locally the gravitational field.7 In this coordinate frame all covariantderivatives coincide with usual ones, and all first derivatives of the metric tensorvanish (by virtue of (A.21)). The transformation to such a frame at a chosen point,which we place at the origin, is

xμ → x′μ = xμ +12Γμ

νλ(0)xνxλ, (A.29)

where Γμνλ(0) are the values of the Christoffel symbols in coordinates x at the origin.

Using the relation (A.16), it is straightforward to see that all Christoffel symbolsindeed vanish at the origin in the new frame. Note that the key role here is playedby the symmetry of the Christoffel symbols in the lower indices, formula (A.22).

Since the transformation (A.29) is identity at the origin, there is still freedom inthe metric tensor. This freedom can be used to reduce the metric tensor at the originto the Minkowski tensor. To this end, one simply chooses xμ = Jμ

ν x′ν , where Jμν does

7In fact, the stronger statement is valid: it is possible to make all Christoffel symbols equal to

zero along any predetermined world line.


not depend on coordinates. In matrix notations we then have the relation (A.9).The matrix gμν can be cast into diagonal form by orthogonal transformation, andthen converted to the Minkowski tensor by rescaling of coordinates. The resultingcoordinate frame has thus the properties

gμν(0) = ημν , Γμνλ(0) = 0,

It is called the locally-Lorentz frame.

A.3 Riemann Tensor

It is seen from formula (A.23) that the Christoffel symbols differ from zero if metricnon-trivially depends on the coordinates xμ. It should be understood, however, thatthe deviation of Γλ

μν from zero does not imply that the space is not flat. Since thequantities Γλ

μν do not form a tensor, they can be identically equal to zero in onecoordinate frame and differ from zero in another frame.

Problem A.6. Find the Christoffel symbols in polar coordinates on 2-dimensionalplane and in spherical coordinates in 3-dimensional Euclidean space.

The quantity which characterizes the geometry of manifold, rather than the choiceof coordinate frame, is the Riemann tensor (curvature tensor) Rμ

νλρ. The Riemanntensor determines how the commutator of covariant derivatives acts on tensors. Forexample, for an arbitrary contravariant vector Aλ we have

∇μ∇νAλ −∇ν∇μAλ = AσRλσμν . (A.30)

Problem A.7. Check that the equality (A.30) indeed defines the tensor Rμνλρ. In

particular, check that all terms with derivatives of Aλ, which could appear in theleft hand side of this equality, cancel out.

Explicit expression for the Riemann tensor is

Rμνλρ = ∂λΓμ

νρ − ∂ρΓμνλ + Γμ

σλΓσνρ − Γμ

σρΓσνλ. (A.31)

In order to better understand the geometric meaning of the Riemann tensor, let usconsider parallel transport of vector Aλ from the point x with coordinates xμ tothe point x with coordinates

xμ = xμ + dyμ + dzμ,

where directions of vectors dyμ and dzμ do not coincide (see Fig. A.3). This paralleltransport can be done in two different ways: (i) one first moves the vector Aλ alongthe path 1 to point y with coordinates

xμ(y)

= xμ + dyμ,

and then proceeds along the path 2 to point x; (ii) one does the opposite, namely,first performs parallel transport along the path 3 to the point z with coordinates

xμ(z) = xμ + dzμ,

A.3. Riemann Tensor 365

Fig. A.3 Parallel transport of a vector along different trajectories (12) and (34).

and then moves along the path 4 to point x. Of course, in flat space the result ofparallel transport does not depend on the choice of path. In the case of curved spacethis is, generally speaking, incorrect. Using the parallel transport rule (A.13), onecan see immediately that the result of the transport does not depend on the pathto the linear order in increments of coordinates. However, in the quadratic order weobtain

Aλ(12) − Aλ(34) = AσRλσμνdzμdyν , (A.32)

where Aλ(12) and Aλ(34) are images of the vector Aλ under parallel transport alongpaths (12) and (34), respectively. Thus, the tensor Rμ

νλρ determines the dependenceof the parallel transport on the path along which it is done. Consequently, theRiemann tensor is indeed a non-trivial characteristic of the curvature.

Problem A.8. Obtain equality (A.32). In particular, check that the second orderterms in dxμ, omitted in the parallel transport law (A.13), do not contribute to thedifference (Aλ(12) − Aλ(34)) to the quadratic order.

Problem A.9. Using the Christoffel symbols found in Problem A.6, check byexplicit calculation that all components of the Riemann tensor are zero in polarcoordinates on plane and in spherical coordinates in 3-dimensional Euclidean space.

The above analysis is valid, with minimal changes, for covariant vector Aμ. Theanalog of Eq. (A.30) in that case is

∇μ∇νAλ −∇ν∇μAλ = −AσRσλμν . (A.33)

The action of the commutator of covariant derivatives

[∇μ,∇ν ] ≡ ∇μ∇ν −∇ν∇μ

on tensor of arbitrary rank follows from the fact that the operator [∇μ,∇ν ] obeysthe Leibniz rule. For example,

[∇μ,∇ν ] Aρλ = Rρ

σμνAσλ − Rσ

λμνAρσ. (A.34)


Let us list some important properties of the Riemann tensor.

(1) Tensor

Rμνλρ ≡ gμσRσνλρ

is antisymmetric in the first and second pair of indices.(2) Tensor Rμνλρ is symmetric under permutations of pairs of indices (μν) ↔ (λρ).(3) For any three indices, the sum of three components of the tensor Rμνλρ with

cyclic permutation of these indices is zero. For instance,

Rρμνλ + Rρλμν + Rρνλμ = 0. (A.35)

(4) The Bianchi identity holds:

∇ρRλσμν + ∇νRλ

σρμ + ∇μRλσνρ = 0. (A.36)

Problem A.10. Using the explicit expression (A.31) for the Riemann tensor, provethe properties (1) and (2).

Problem A.11. Using properties (1), (2), (3) determine the number of independentcomponents of the Riemann tensor at each point in a space of dimension D = 2, 3, 4.

Proving the properties (3) and (4) by using the explicit formula (A.31) would be toocumbersome. Instead, it is convenient to make use of the definition (A.30). Namely,let us write the following equality (Jacobi identity), valid for arbitrary operators,

[A, [B, C]] + [C, [A, B]] + [B, [C, A]] = 0.

Problem A.12. Prove the Jacobi identity.

Let us choose the operators A, B, C as covariant derivatives. Let us then act by theJacobi identity on arbitrary scalar φ first,

[∇ρ, [∇μ,∇ν ]]φ + [∇μ, [∇ν ,∇ρ]]φ + [∇ν , [∇ρ,∇μ]]φ = 0. (A.37)

Let us write for the first term

[∇ρ, [∇μ,∇ν ]]φ = ∇ρ[∇μ,∇ν ]φ − [∇μ,∇ν ]∇ρφ = −[∇μ,∇ν ]∇ρφ = ∂σφRσρμν ,

and similarly for the other terms. Here we first used the fact that [∇μ,∇ν ]φ =∂μ∂νφ − Γλ

μν∂λφ is symmetric in μ, ν, and then utilized (A.33). In this way weobtain from the identity (A.37) that

(Rσρμν + Rσ

μνρ + Rσνρμ)∂σφ = 0,

which implies (A.35) by virtue of the arbitrariness of ∂σφ.Let us now act by the Jacobi identity on arbitrary vector Aλ and get

[∇ρ, [∇μ,∇ν ]]Aλ + [∇μ, [∇ν ,∇ρ]]Aλ + [∇ν , [∇ρ,∇μ]]Aλ = 0. (A.38)

Then, using the definition (A.30), we have

[∇ρ, [∇μ,∇ν ]]Aλ = ∇ρ(RλσμνAσ) − [∇μ,∇ν ](∇ρA

λ). (A.39)

A.3. Riemann Tensor 367

Evaluating the right hand side of Eq. (A.39) and using the Leibniz rule, we find

[∇ρ, [∇μ,∇ν ]]Aλ = ∇ρRλσμνAσ + Rλ

σμν∇ρAσ − Rλ

σμν∇ρAσ + Rσ

ρμν∇σAλ

= ∇ρRλσμνAσ + Rσ

ρμν∇σAλ. (A.40)

Upon substituting the expression (A.40) and similar expressions for the other twodouble commutators to the Jacobi identity (A.38) we obtain, taking into accountthe property (3) of the Riemann tensor,

(∇ρRλσμν + ∇νRλ

σρμ + ∇μRλσνρ)A

σ = 0.

Consequently, the Bianchi identity is indeed satisfied.By contracting indices of the Riemann tensor Rμνλρ one can construct a new

tensor with smaller number of indices, still characterizing the curvature of space. Theabove symmetry properties of the Riemann tensor guarantee that the contractionof any two of its indices results in either zero or the following symmetric tensor ofthe second rank,

Rμν ≡ Rλμλν ,

called the Ricci tensor. In many places of this book we need the explicit form ofthis tensor,


λν + ΓλρλΓρ

μν − ΓλρμΓρ

νλ. (A.41)

By contracting the indices of the Ricci tensor we obtain the scalar curvature,

R ≡ gμνRμν .

Problem A.13. Find the metric, Christoffel symbols, Riemann and Ricci tensorsand scalar curvature for 2-dimensional sphere S2.

Problem A.14. Show that for an arbitrary 2-dimensional surface, the quantity√gR is total derivative and hence the integral of the scalar curvature over the

invariant volume,

14π

∫d2x

√gR, (A.42)

does not depend on the choice of metric on the surface (Gauss–Bonnet theorem).Thus, in two dimensions this integral is a characteristics of topology. Note that thescalar curvature is twice the Gaussian curvature in two dimensions. Integral (A.42)is in one-to-one correspondence with the degree of the Gauss mapping and coincideswith the Euler characteristic of 2-dimensional surface. Find the value of this integralfor sphere and torus.


A.4 Gravitational Field Equations

We now have at our disposal all ingredients needed for constructing General Rela-tivity. In GR, the metric tensor is dynamical field (“gravitational field”), and theequations of GR arise as extremum conditions for the action functional. As we havealready mentioned, one of the basic principles of GR is that all local coordinateframes are equivalent. This means that the equations for the gravitational field gμν

should be written in terms of covariant quantities and their form should not dependon the choice of coordinates. To fulfill this requirement, the gravitational field actionSgr must be a scalar, i.e., it must be written as an integral of a scalar LagrangianLgr over the invariant 4-volume,

Sgr =∫

d4x√−gLgr .

The simplest possibility is to take the Lagrangian equal to a constant (−Λ) inde-pendent of metric,

SΛ = −Λ∫

d4x√−g. (A.43)

This term can indeed enter the action for the gravitational field and play animportant role in cosmology. Since the action is dimensionless, the parameter Λhas dimension (mass)4. This parameter is called the cosmological constant or, forreasons that are explained in Chapter 3, vacuum energy density. However, theaction (A.43) cannot be the complete gravitational field action. Indeed, SΛ doesnot contain derivatives of the metric gμν , and, consequently, upon its variation onewould obtain purely algebraic equation that would not enable one to interpret gμν

as truly dynamical field.Another scalar quantity at our disposal is the scalar curvature R, or, in general,

an arbitrary function f(R). To understand which choice of the function f(R) as theLagrangian is most natural, we recall that the commonly used field equations are ofthe first or second order in derivatives. For the field equations to be second order atmost, it is usually required that the Lagrangian does not contain second or higherderivatives. Indeed, consider field theory with the action of the form

S =∫

d4xL(φ, ∂φ, ∂2φ, . . .). (A.44)

Here, the symbol φ denotes all fields of theory, and we omitted possible tensorindices. The variation of (A.44) under small variation of the fields, φ → φ + δφ,reads

δS =∫

d4x

(∂L∂φ

δφ +∂L

∂(∂φ)∂δφ +

∂L∂(∂2φ)

∂2δφ + · · ·)

.

Assuming, as usual, that the variation of the fields δφ vanishes at infinity, andintegrating by parts, we arrive at the field equation of the form

∂L∂φ

− ∂∂L

∂(∂φ)+ ∂2 ∂L

∂(∂2φ)+ · · · = 0, (A.45)

A.4. Gravitational Field Equations 369

which, generally speaking, contains the field derivatives of higher than secondorder.

The Riemann tensor (A.31), and, consequently, the scalar curvature R, containthe first derivatives of the Christoffel symbols Γλ

μν . The latter, in turn, containthe first derivatives of the metric tensor gμν . Therefore, if the Lagrangian densityLgr depends non-trivially on the scalar curvature, the action necessarily involvesthe second derivatives. By the above argument, one could conclude that it wouldbe impossible to write covariant action for the gravitational field, which wouldlead to second order equations. Note, however, that if the Lagrangian density L inEq. (A.45) depends on second derivatives only via the terms of the form

f(φ)∂2φ (A.46)

and does not contain higher derivatives, then the equations of motion do not involvehigher derivatives. In fact, one can integrate the term (A.46) in the Lagrangianby parts and come to the Lagrangian which contains first derivatives only. It isstraightforward to check that the action

SEH = − 116πG

∫d4x

√−gR (A.47)

depends on the second derivatives exactly in this way.

Problem A.15. Using integration by parts, find the action equivalent to the action(A.47), which does not contain second derivatives. Is the Lagrangian in this actiona scalar? Is the action itself a scalar?

This action is called the Einstein–Hilbert action. As we will see later, the constantG here, which has the dimension M−2, is Newton’s gravity constant.

The full gravitational field action in GR is the sum of the terms (A.43) and(A.47),

Sgr = SΛ + SEH . (A.48)

In order to obtain the gravitational field equations one has to calculate the variationof the action δSgr under variation of the metric

gμν → gμν + δgμν .

Let us start with the first, simpler term SΛ. To vary SΛ, we make use of the followingformula of linear algebra,

det(M + δM) = det (M)[1 + Tr(M−1δM) + o(δM)

], (A.49)

where M is an arbitrary non-degenerate matrix.

Problem A.16. Prove the formula (A.49).


Applying the relation (A.49) to the determinant of the metric tensor, we obtain

δg = ggμνδgμν . (A.50)

Using this result, we arrive at the following expression for the variation of SΛ,

δSΛ = −Λ∫

d4xδ(√−g) = −Λ

2

∫d4x

√−ggμνδgμν . (A.51)

Let us now calculate the variation of the Einstein–Hilbert action SEH . It can bewritten as a sum of the following three terms,

δSEH = δS1 + δS2 + δS3,

where

δS1 = − 116πG

∫d4xR δ(

√−g),

δS2 = − 116πG

∫d4x

√−gRμνδgμν

and

δS3 = − 116πG

∫d4x

√−ggμνδRμν . (A.52)

Taking into account the relation (A.51), we immediately obtain the explicitexpression for δS1:

δS1 = − 132πG

∫d4x

√−gRgμνδgμν . (A.53)

In order to compute δS2, we note that varying Eq. (A.8) we get

gρλδgμρ = −gμρδgρλ.

Contracting both sides of this equality with matrix gλν we obtain

δgμν = −gμρδgρλgλν . (A.54)

Consequently,

δS2 =1

16πG

∫d4x

√−gRμνδgμν . (A.55)

It remains to find the variation δS3, which at first glance looks the most complicated.To compute δS3, we note that the transformation rules of the Christoffel symbols(A.16) imply that the variation δΓμ

νλ is a tensor. Then, using (A.31), we arrive atthe following expression for the variation of the Riemann tensor:

δRμνλρ = ∂λδΓμ

νρ − ∂ρδΓμνλ + δΓμ

σλΓσνρ + Γμ

σλδΓσνρ − δΓμ

σρΓσνλ − Γμ

σρδΓσνλ.

By direct calculation one can check that

δRμνλρ = ∇λ(δΓμ

νρ) −∇ρ(δΓμνλ), (A.56)

A.5. Conformally Related Metrics 371

where the covariant derivatives are taken with the unperturbed metric. From theequality (A.56) we get the following expression for the variation of the Ricci tensor

δRμν = ∇λ(δΓλμν) −∇ν(δΓλ

μλ) . (A.57)

Substituting Eq. (A.57) into Eq. (A.52), we obtain

δS3 = − 116πG

∫d4x

√−ggμν[∇λ(δΓλ

μν) −∇ν(δΓλμλ)]

= − 116πG

∫d4x

√−g∇λ(gμνδΓλμν − gμλδΓσ

μσ), (A.58)

where we put the tensor gμν inside the covariant derivatives in the second equalityand renamed the summation indices ν and λ in the second term. Finally, makinguse of the property (A.26), we rewrite δS3 as an integral of the total derivative,

δS3 = − 116πG

∫d4x∂λ(gμνδΓλ

μν − gμλδΓσμσ).

Consequently, δS3 does not contribute to the field equations. Assembling togetherthe variations (A.53) and (A.55), we derive the variation of the Einstein–Hilbertaction,

δSEH =1

16πG

∫d4x

√−g

(Rμν − 1

2gμνR

)δgμν (A.59)

This formula and Eq. (A.51) give the following gravitational field equations (Einsteinequations in vacuo):

Rμν − 12gμνR = 8πGΛgμν . (A.60)

We see that the Einstein equations are indeed second order in derivatives. TheEinstein equations are often written in the form

Gμν = 8πGΛgμν ,

where

Gμν ≡ Rμν − 12gμνR

is the Einstein tensor.

A.5 Conformally Related Metrics

It is useful for some applications to have relations between the Ricci tensors andscalar curvatures of metrics, conformally related to each other. Suppose that thereare two metrics gμν and gμν , such that

gμν(x) = e2ϕ(x)gμν(x), (A.61)

where ϕ(x) is some scalar function of coordinates. Our purpose is to expressRμν and R — the Ricci tensor and scalar curvature constructed from the metric


gμν — in terms of Rμν and R obtained from the metric gμν . To this end, we first findthe relationship between the Christoffel symbols. The direct substitution of (A.61)in (A.23) yields

Γμνλ = Γμ

νλ + δμλ∂νϕ + δμ

ν ∂λϕ − gνλgμρ∂ρϕ.

As a result of substituting this expression in Eq. (A.41) and straightforward (buttedious) calculation we obtain

Rμν = Rμν − 2∇μ∇νϕ − gμνgλρ∇λ∇ρϕ + 2∂μϕ∂νϕ − 2gμνgλρ∂λϕ∂ρϕ, (A.62)

where the covariant derivatives are evaluated with the metric gμν . Hence, for thescalar curvature R = gμνRμν we have

R = e−2ϕ(R − 6gμν∇μ∇νϕ − 6gμν∂μϕ∂νϕ), (A.63)

while for the Einstein tensor we get

Gμν ≡ Rμν − 12gμνR

= Gμν − 2∇μ∇νϕ + 2∂μϕ∂νϕ + gμν(2∇λ∇λϕ + ∂λϕ∂λϕ), (A.64)

where indices in the right hand side are raised and lowered by the metric gμν .Finally, for the integral entering the gravitational field action, the relationship hasthe form ∫

R√

−gd4x =∫

e2ϕR√−gd4x + 6

∫e2ϕgμν∂μϕ∂νϕ

√−gd4x.

The latter relation is obtained with the use of (A.63) and integration by parts.As an example of application of these formulas, we consider “non-linear gravi-

tational theories” with actions of the form

S = −∫

d4x√−gf(R), (A.65)

where f(R) is an arbitrary function of the scalar curvature R. We are going toprove that in vacuo, these theories are dynamically equivalent to conventionalgravity (General Relativity described by the Einstein–Hilbert Lagrangian) plus self-interacting scalar field.8

We first find the field equations obtained by varying the action (A.65). We writethe variation again as the sum of three terms,

δS = δS1 + δS2 + δS3,

δS1 = −∫

d4x f(R) δ(√−g),

δS2 = −∫

d4x√−gf ′(R)Rμνδgμν ,

δS3 = −∫

d4x√−ggμνf ′(R)δRμν ,

8For the sake of convenience, in Eq. (A.65) and to the end of this Section we adopt the system

of units where 16πG = 1.

A.5. Conformally Related Metrics 373

where f ′(R) = ∂f(R)/∂R. The variations δS1 and δS2 are simple generalizations ofthe analogous expressions for the Einstein–Hilbert action (see (A.53) and (A.55)),

δS1 = −12

∫d4x

√−gf(R)gμνδgμν , (A.66)

δS2 =∫

d4x√−gf ′(R)Rμνδgμν . (A.67)

To calculate δS3 we use (A.57), where we substitute the variation of the Christoffelsymbols

δΓλμν =

12gλρ(∇μδgνρ + ∇νδgμρ −∇ρδgμν).

As a result we obtain from (A.57) that

δRμν =12(−∇λ∇λδgμν + ∇λ∇μδgλν + ∇λ∇νδgλμ −∇ν∇μδgλ

λ). (A.68)

Then the variation δS3 reads

δS3 = −∫

d4x√−gδgμν(∇μ∇ν − gμν∇λ∇λ)f ′(R), (A.69)

where we integrated by parts twice. In the case of the Einstein–Hilbert Lagrangianf ′ = 1, so the expression (A.69) vanishes identically.

Finally, equating to zero the variation δS we obtain the field equations for thetheory with the action (A.65):

12f(R)gμν − f ′(R)Rμν + (∇μ∇ν − gμν∇λ∇λ)f ′(R) = 0. (A.70)

Note that these equations contain fourth order derivatives.It is convenient to introduce the new variables gμν by making a conformal

transformation:

gμν = ψ−1gμν , ψ = f ′(R). (A.71)

We will assume that ψ > 0, then the new metric gμν has the same signature as themetric gμν . The relationship between the Ricci tensor and scalar curvature of thesetwo metrics is given by Eqs. (A.62), (A.63) with ϕ = −1

2 ln ψ. We have, therefore,

Rμν = Rμν + ψ−1∇μ∇νψ +1

2ψgμν∇λ∇λψ

− 12ψ2

(∇νψ∇μψ + 2gμν∇λψ∇λψ), (A.72)

R = ψR + 3∇μ∇μψ − 92ψ−1∇λψ∇λψ, (A.73)

where symbols with tilde are calculated, and indices are raised and lowered withthe metric gμν .

Let R0(ψ) be a solution to the equation

f ′ [R0(ψ)] − ψ = 0


(the analysis below can be extended to the case of multiple solutions), i.e., R0 isthe inverse function to f ′. It follows from the second of Eq. (A.71) that

R = R0(ψ). (A.74)

In terms of new variables, Eq. (A.70) takes the form

Rμν − 12Rgμν

= ψ−2

{12[f(R0(ψ)) − ψR0(ψ)]gμν +

32∇μψ∇νψ − 3

4gμν∇λψ∇λψ

}.

(A.75)

We still have to account for Eq. (A.74). Its left hand side is given by (A.73) whereR is obtained by contracting (A.75) with gμν . In this way we get the equation

ψ∇λ∇λψ − ∇λψ∇λψ +13{ψR0(ψ) − 2f [R0(ψ)]} = 0. (A.76)

So, instead of fourth order equation (A.70), we obtain an extended system of secondorder equations (A.75), (A.76) in terms of the new variables gμν and ψ . This systemcoincides with the equations of conventional gravity interacting with the scalar fieldψ. The action whose variation with respect to gμν and ψ gives Eqs. (A.75) and(A.76) is

S = −∫

d4x√

−gR

+∫

d4x√

−g

{32

gμν∇μψ∇νψ

ψ2+

R0(ψ)ψ

− f [R0(ψ)]ψ2

}. (A.77)

Problem A.17. Obtain Eqs. (A.75) and (A.76) by varying the action (A.77).

The kinetic term in the action (A.77) can be cast into the canonical form by thereplacement ψ = e

√23 φ. Finally, we arrive at the action

S =∫

d4x√−g{−R + gμν∇μφ∇νφ

+ e−√

23 φR0

(e√

23φ)− e−2

√23 φf

[R0

(e√

23 φ)]}

.

It describes self-interacting scalar field φ in the framework of GR. As we have seen,this theory is dynamically equivalent to the theory of “non-linear” gravity with theaction (A.65).

Problem A.18. Consider scalar-tensor theory of gravity with the action

S =∫

d4x√−g

(−R +

12ω(ϕ)∂μϕ∂νϕ − V (ϕ)

),

where ϕ is scalar field. Under what conditions on functions ω(ϕ) and V (ϕ) thistheory is equivalent to f(R)-gravity?

A.6. Interaction of Matter with Gravitational Field. Energy-Momentum Tensor 375

Note that as a result of the conformal transformation, matter fields begin tointeract with the dilaton field φ. This results in various physical effects. In particular,for homogeneous solution φ = φ(t) the “cosmic time” (time in the FLRW metric)differs from the “atomic time” (time determining the evolution and interaction ofmatter fields). In this sense, in the presence of matter fields f(R)-gravity is notequivalent to GR with scalar field.

A.6 Interaction of Matter with Gravitational Field.Energy-Momentum Tensor

Equations (A.60) describe the dynamics of the gravitational field in vacuo. However,it is of primary interest to study gravity in the presence of matter fields which serveas sources of gravitational field. To study this general situation, we add to the action(A.48) a new term,

Sm =∫

d4x√−gLm, (A.78)

that describes matter and its coupling to gravity. Here the Lagrangian density Lm

is a scalar function of the gravitational field gμν and matter fields. We collectivelydenote the latter by ψ:

Lm = Lm(ψ, gμν).

Once we add the term (A.78), the Einstein equation (A.60) get modified as follows,

Rμν − 12gμνR = 8πG(Λgμν + Tμν). (A.79)

Here we turned to tensors with lower indices. The symmetric tensor Tμν is definedby the following relation,

δSm =12

∫d4x

√−gTμνδgμν . (A.80)

The latter relation can be rewritten with account of (A.54),

δSm = −12

∫d4x

√−gT μνδgμν .

Equation (A.79) (with upper indices) is obtained then by using (A.59).To understand the physical meaning of the tensor Tμν , we calculate it for two

simple theories: scalar field theory and theory of electromagnetic field. The covariantaction describing the real scalar field interacting with gravity reads

Ssc =∫

d4x√−gLsc =

∫d4x

√−g

(12gμν∂μφ∂νφ − V (φ)

), (A.81)

where the scalar potential V (φ) can be an arbitrary function of the field φ. Generallyspeaking, one can add to the action (A.81) yet another term which vanishes in flatspace,

Sξ = ξ

∫d4x

√−gRU(φ), (A.82)


where U(φ) is an arbitrary function. We restrict ourselves to the choice ξ = 0. Inthis case, the interaction of the scalar field with gravity is called minimal.

Using the definition (A.80), we obtain the following expression for the tensorTμν of the scalar field,

T scμν = ∂μφ∂νφ − gμνLsc. (A.83)

Problem A.19. Find the tensor Tμν for free massless scalar field non-minimallycoupled to gravity, ξ �= 0, by choosing

U(φ) = φ2,

and V (φ) = 0. At what value of the parameter ξ does the trace gμνTμν vanish onthe equations of motion? For what V (φ) does this property remain valid?

We now find the explicit form of the tensor Tμν for the electromagnetic field.The action for the vector field Aμ coupled to gravity has the form

Sem = −14

∫d4x

√−gFμνFλρgμλgνρ, (A.84)

where Fμν is the conventional field strength tensor,

Fμν = ∂μAν − ∂νAμ. (A.85)

At first glance, the ordinary derivatives in the definition (A.85) should be replacedby covariant derivatives in curved space. But it is straightforward to check that forsymmetric connection, the terms with the connection cancel out due to antisym-metrization in μ and ν, so that ∇μAν −∇νAμ = ∂μAν −∂νAμ, and Fμν is a tensor.Using the action (A.84), we obtain the following expression for the tensor Tμν ofelectromagnetic field,

T emμν = −FμλFνρg

λρ +14gμνFλρF

λρ. (A.86)

Now we note that in Minkowski space, the tensor Tμν coincides with the energy-momentum tensor for both scalar and electromagnetic field. For scalar field, T sc

μν isexactly equal to the Noether energy-momentum tensor, while for the electromag-netic field, T em

μν differs on the equations of motion from the Noether tensor by thedivergence of an antisymmetric tensor.

Problem A.20. Prove these statements.

In particular, the (00)-components of these tensors in Minkowski space are

T sc00 =

12(∂0φ)2 +

12(∂iφ)2 + V (φ)

and

T em00 =

12F 2

0i +14F 2

ij ≡ 12(E2 + H2).

These are indeed the energy densities of scalar and electromagnetic fields.


Generally, the tensor Tμν defined by Eq. (A.80) is called metric energy-momentum tensor. We emphasize that it is always symmetric. Below we prove thatin Minkowski space and on equations of motion, it is always equal to the Noetherenergy-momentum tensor modulo the divergence of an antisymmetric tensor.

The energy-momentum tensor is conserved in flat space,

∂μT μν = 0. (A.87)

This leads to the conservation of energy and momentum. It is natural to assume thatthe generalization of the conservation law (A.87) to curved space is the covariantconservation equation

∇μT μν = 0. (A.88)

To see that this is indeed the case, we take the divergence of both sides of theEinstein equations (A.79),

∇μ

(Rμν − 1

2gμνR

)= 8πG∇μTμν . (A.89)

Let us prove that the left hand side of Eq. (A.89) is identically zero. To this end,we first contract the indices λ and μ in the Bianchi identity (A.36). As a result, weobtain the following identity,

∇ρRσν −∇νRσρ + ∇λRλσνρ = 0.

Then we contract it with gσρ and arrive at

0 = ∇ρRρν −∇νR + ∇λRλν = 2∇μ

(Rμν − 1

2gμνR

).

Thus we obtained the identity

∇μ

(Rμν − 1

2gμνR

)= 0,

which implies that the covariant conservation law of energy-momentum tensor(A.88) is a necessary condition for the consistency of the Einstein equations.

On the other hand, the energy-momentum tensor is entirely determined by theaction of matter fields. Therefore, to check the consistency of the whole systemof field equations, one needs to prove that the conservation law (A.88) is a conse-quence of matter field equations. The proof makes use of the invariance of the actionunder coordinate transformations. We begin with evaluating the first variation ofthe metric gμν under small coordinate transformation

x′μ = xμ + ξμ. (A.90)

Substituting Eq. (A.90) to the general formula (A.7), we get

g′μν(x′) = (δμλ + ∂λξμ)(δν

ρ + ∂ρξν)gλρ(x)

= gμν(x) + gνλ∂λξμ + gμλ∂λξν , (A.91)


where we neglected terms of the second order in ξμ. Expanding the left hand sideof Eq. (A.91) we write

g′μν(x′) = g′μν(x) + ∂λg′μν(x)ξλ + O(ξ2) = g′μν(x) + ∂λgμν(x)ξλ + O(ξ2).

In this way we obtain the following relationship between the functions gμν(x) andg′μν(x) taken at points with the same coordinates in the old and new coordinateframes,

g′μν(x) = gμν(x) − ∂λgμν(x)ξλ + gνλ∂λξμ + gμλ∂λξν . (A.92)

Problem A.21. Check by explicit calculation that the relation (A.92) can be writtenin the following covariant form:

g′μν = gμν + ∇μξν + ∇νξμ. (A.93)

Now, the invariance of the matter action under coordinate transformations meansthat the variation of this action is equal to zero when simultaneously metric gμν(x)is transformed as in (A.93) and matter fields are transformed accordingly, ψ′(x) =ψ(x) + δψξ(x). The form of δψξ is dictated by the transformation properties ofmatter fields under the coordinate transformation (A.90). For instance, for scalarfield φ

δφξ = −ξμ∂μφ.

So, in the general case we have

δξSm =12

∫d4x

√−gTμν(∇μξν + ∇νξμ) +∫

d4x√−g

δLm

δψδψξ = 0. (A.94)

The identity (A.94) is valid regardless of the field equations. Now we assume inaddition that the matter field equations are satisfied. Then the second term on theleft hand side of Eq. (A.94) vanishes. Therefore, we proved that the matter fieldequations guarantee the validity of equality∫

d4x√−gTμν(∇μξν + ∇νξμ) = 0.

Since the vector ξμ is arbitrary and Tμν is symmetric, we arrive, upon integrationby parts, at the covariant conservation law (A.88), as required.

We now use the identity (A.94) to prove that in flat space, the metric energy-momentum tensor Tμν coincides on the equations of motion with the Noether tensorτμν modulo total derivative. In flat space, the identity (A.94) takes the form∫

d4xTμν∂μξν +∫

d4xδLm

δψδψξ = 0, (A.95)

where we again used the symmetry of the tensor Tμν . In Minkowski space, the actionis invariant under the variations of matter fields δψξ corresponding to shifts (A.90)with constant ξμ. Consequently, the second term in (A.95) can be written as∫

d4xδLm

δψδψξ = −

∫d4xτμν∂μξν , (A.96)


where τμν coincides on the equations of motion with the conserved Noether energy-momentum tensor.

Problem A.22. Modifying the proof of the Noether theorem, show the validity ofthe relation (A.96), where τμν is equal on the equations of motion to the Noetherenergy-momentum tensor.

Integrating by parts Eq. (A.95), we see that

∂μ(Tμν − τμν) = 0.

This is the identity valid irrespective of the equations of motion. The identity canonly hold if the difference (Tμν − τμν) is the divergence of an antisymmetric tensor,

Tμν − τμν = ∂λAμνλ, where Aμνλ = −Aλνμ, (A.97)

as promised.The total derivative of the form (A.97) does not contribute to the total energy-

momentum 4-vector

P ν ≡∫

d3xT 0ν.

Indeed, because of the antisymmetry property of Aμνλ, its contribution is of theform ∫

d3x∂iA0νi .

This integral is zero for fields vanishing at spatial infinity. Hence, metric and Noetherenergy-momentum tensors are equivalent in flat space in the sense that they giveequal values of energy and momentum.

Problem A.23. Consider tensor of the form

Θμν = (ημν∂2 − ∂μ∂ν)f,

where f is an arbitrary function. Clearly, this tensor is identically conserved in flatspace. Express it in the form Θμν = ∂λAμνλ, where Aμνλ = −Aλνμ.

Problem A.24. Check explicitly that the metric energy-momentum tensor of scalarfield coupled non-minimally to gravity, see (A.81) and (A.82), differs in flat spacefrom the Noether energy-momentum tensor by the total derivative of the type (A.97)for arbitrary U(φ) and ξ.

To conclude our discussion of energy-momentum tensor in GR, we make the fol-lowing observation. In flat space, the differential conservation law (A.87) implies theexistence of four conserved quantities, the components of the energy-momentum4-vector. However, Eq. (A.88), in general, does not imply the existence of fourintegrals of motion in curved space, which could be interpreted as the energy andmomentum of the system. In this regard, the concepts of energy and momentum,generally speaking, are not defined in GR. For spatially localized gravitating systemsone can define the energy and momentum by making use of the asymptotics of thegravitational field far away from the system, but in general such a construction is


impossible. In particular, talking about the total mass of the Universe does notmake sense.

A.7 Particle Motion in Gravitational Field

As a digression from the discussion of the properties of the Einstein equations, letus study the motion of point particles in external gravitational field. The action forpoint particle in GR has the same form as in special relativity,

Sp = −m

∫ds. (A.98)

In GR, the interval along the world line of a particle involves the space-time metric,

ds =√

dxμdxνgμν =

√dxμ

dτ

dxν

dτgμνdτ,

where we introduced the affine parameter τ along the world line, and gμν =gμν [x(τ)]. In terms of the affine parameter, the action (A.98) is

Sp = −m

∫ √xμxνgμν(x)dτ, (A.99)

where dot denotes differentiation with respect to τ . Equation of motion obtainedby varying the action (A.99) is

− d

dτ

(gμν xν

√xαxα

)+

12

xλxν∂μgνλ√xαxα

= 0. (A.100)

Problem A.25. Derive Eq. (A.100).

The affine parameter τ can be chosen in such a way that the 4-velocity vector

uμ =dxμ

dτ(A.101)

has unit length at each point of the world line,

gμνuμuν = 1. (A.102)

This choice identifies the affine parameter with the proper time of the particle, sinceEq. (A.102) is equivalent to

ds = dτ.

With this choice of the world line parameterization, equation of motion (A.100)becomes

− d

ds(gμνuν) +

12∂μgνλuλuν = 0. (A.103)

Contracting this equation with gμρ, we obtain

−duρ

ds− gμρ

(dgμν

ds− 1

2∂μgνλuλ

)uν = 0. (A.104)

A.7. Particle Motion in Gravitational Field 381

Since gμν = gμν [x(s)], the definition of the 4-velocity givesdgμν

ds= ∂λgμνuλ.

Substituting this expression into Eq. (A.104) and using the formula (A.23) for theChristoffel symbols, we finally arrive at the following form of the equation of motion,

duν

ds+ Γν

μλuμuλ = 0. (A.105)

Multiplying this equation by ds and noticing that

dxλ = uλds

along the particle world line, we rewrite Eq. (A.105) as

duν + Γνμλuμdxλ = 0.

We recall now the parallel transport law of contravariant vectors (A.13) and observethat the geometric meaning of Eq. (A.105) is that under parallel transport alongthe world line the normalized tangential vectors uμ[x(τ)] transform into each other.Curves which satisfy this property are geodesics (shortest paths), and Eq. (A.105)is the geodesic equation.

The action (A.98) does not make sense for massless particles, m = 0. To findthe world lines of these particles (e.g., rays of light) one directly uses the geodesicequation

duν

dτ+ Γν

μλuμuλ = 0, (A.106)

where τ is now an arbitrary parameter along the world line, and 4-velocity uμ isstill defined by Eq. (A.101). In the massless case, the geodesic must be light-like(null),

ds2 = 0,

or, in differential form,

gμνxμxν ≡ gμνuμuν = 0. (A.107)

Problem A.26. Show that Eq. (A.106) is consistent with the requirement (A.107).

Problem A.27. Check that the equation of motion of massive point particle as wellas the equation for light-like geodesic can be obtained from the action

Sη = −12

∫dτ η

[η−2xμxνgμν(x) + m2

], (A.108)

where η(τ) is a new auxiliary dynamical variable (“dynamical” here means that oneof the equations of motion is obtained by varying the action with respect to η). Thisnew variable transforms under the change of parameterization as

η′ [τ ′(τ)] = η(τ)[∂τ ′(τ)

∂τ

]−1

.

Note that η2(τ) can be considered as internal metric on the world line, while theaction (A.108) can be viewed as the action of four fields xμ(τ) in one-dimensionalspace with dynamical metric.


A.8 Newtonian Limit in General Relativity

Let us now discuss how the main object of the Newtonian gravity, the gravitationalpotential, arises in General Relativity and how the Newtonian gravity law followsfrom GR. To this end, we study the motion of a particle in weak static gravitationalfield, i.e., in space with metric

gμν = ημν + hμν(x), (A.109)

where ημν is the Minkowski metric, and all components of hμν(x) are small,

hμν(x) � 1. (A.110)

Furthermore, we consider non-relativistic particles,

vi ≡ dxi

dt� 1.

Let us write the explicit form of various components of the geodesic equation (A.105)to the linear order in the velocity vi and gravitational field hμν . We first note thatin the linear order, the proper time of a particle ds is related to the coordinate timedt by

ds =(

1 +h00

2

)dt. (A.111)

Problem A.28. Find in the general case the relation between coordinate time andproper time of a particle moving with the coordinate 3-velocity vi = dxi

dt . Show thatin the linear order this relation indeed reduces to (A.111).

Hence, the 4-velocity components uμ are related to the metric and physical velocityvi as follows,

u0 ≡ dt

ds≈ 1 − h00

2,

ui ≡ dxi

ds≈ vi.

It is straightforward to check that in the linear order, the zeroth component ofthe geodesic equation is satisfied identically. Indeed, the first term d2t

ds2 vanishes, inview of (A.111), because the metric is static. The second term Γμ

νλuνuλ containssmall factor from the very beginning, since all Christoffel symbols vanish for theunperturbed metric ημν . Furthermore, the component Γ0

00 is equal to zero for staticmetric, hence the second term must involve the velocity ui at least once. So, allterms in the zeroth component of the geodesic equation are at least of the secondorder.

Spatial components of the geodesic equation in the linear approximation read

dvi

dt+ Γi

00 = 0,

A.8. Newtonian Limit in General Relativity 383

where we again took into account that the second term is inherently small due tothe presence of the Christoffel symbols, so that the contributions involving velocitiesui drop out. Recalling the explicit expression (A.23) for the Christoffel symbols, wearrive at the following equation describing the motion of non-relativistic particlesin weak static gravitational field,

dvi

dt= −∂iΦ, (A.112)

where we introduced new function Φ(x) defined by

g00 = 1 + 2Φ.

Equation (A.112) coincides with the equation of non-relativistic mechanicsdescribing the motion of a particle in an external potential Φ(x), so the field Φ(x)is identified, in weak field limit, with the Newtonian gravitational potential. Notethat, as follows from the above analysis, the motion of non-relativistic particles inweak static fields depends only on 00-component of metric.

To cross check the interpretation of the field Φ(x) as the Newtonian potential,let us see that the Poisson equation

ΔΦ = 4πGρ (A.113)

indeed follows from the Einstein equations in the Newtonian limit. Here Δ ≡ (∂i)2

is the Laplace operator. At the same time we will identify the constant G, enteringthe Einstein–Hilbert action, with Newton’s gravity constant.

To this end, let us find, starting from the Einstein equations (A.79), the metricproduced by a static distribution of non-relativistic matter with energy densityρ(x). Before doing that, it is convenient to rewrite the Einstein equations in theequivalent form. Taking the trace of both sides of the Einstein equations, we obtain

R = −8πG(4Λ + T ), (A.114)

where

T ≡ gμνTμν

is the trace of energy-momentum tensor. Substituting (A.114) back into the Einsteinequations, we arrive at the following equivalent equations,

Rμν = 8πG

(Tμν − 1

2gμνT − gμνΛ

). (A.115)

This form of the Einstein equations is sometimes more convenient in practical cal-culations than the original one, because, as a rule, the curvature tensor Rμν hasmore cumbersome structure than the energy-momentum tensor Tμν .

Coming back to the problem of calculating the gravitational field, we assume thatthe cosmological constant is absent, i.e., Λ = 0, and that both energy density ρ(x)and all its spatial derivatives are small. The gravitational field produced by such a


body is weak, i.e., the metric has the form (A.109). The only nonzero componentof the energy-momentum tensor for static distribution of non-relativistic matter is

T00 = ρ(x). (A.116)

Consider now the 00-component of Eq. (A.115). For weak gravitational field, weneglect the quadratic terms in the expression (A.41) for the Ricci tensor. Fur-thermore, the second term in the expression for R00 vanishes for static metric.Consequently, the left hand side of the 00-component of Eq. (A.115) takes the form

R00 = ∂λΓλ00 =

12Δg00, (A.117)

where the latter equality is again obtained for weak and static field. Substitutingthis expression and the explicit form (A.116) of the energy-momentum tensor inEq. (A.115) we obtain (at Λ = 0) Eq. (A.113). Hence, Φ is indeed the gravitationalpotential and G is Newton’s gravity constant.

A.9 Linearized Einstein Equations about Minkowski Background

Let us generalize Eq. (A.113) to the case of weak but otherwise arbitrary gravita-tional field about Minkowski background. In this case the metric has the form (cf.(A.109))

gμν(x) = ημν + hμν(x),

where |hμν(x)| � 1, and perturbations hμν(x) can depend on both time and spatialcoordinates. We make use of the Einstein equations in the form (A.115), and setΛ = 0, so that Minkowski space is the solution for Tμν = 0. In fact, the computationof the Ricci tensor to the linear order in hμν has already been performed: we can usethe formula (A.68), considering it as the expression for the deviation of the Riccitensor from its zero value in Minkowski space. Thus, we make the replacementδgμν → hμν in (A.68), replace the covariant derivatives with ordinary ones andraise and lower indices by Minkowski metric. As a result, we obtain the linearizedequation (A.115) in the following form,(−∂λ∂λhμν + ∂λ∂μhλν + ∂λ∂νhλμ − ∂μ∂νhλ

λ

)= 16πG

(Tμν − 1

2ημνT λ

λ

), (A.118)

where Tμν is assumed to be small.Equation (A.118) is invariant under gauge transformations

hμν → hμν + ∂μξν + ∂νξμ,

Tμν → Tμν ,(A.119)

where ξμ(x) are small gauge parameters. The transformation (A.119) is nothingbut the linearized transformation (A.92); since Tμν is small, it does not change,

A.10. Macroscopic Energy-Momentum Tensor 385

in the linear order, under the small coordinate transformation (A.90). It is oftenconvenient to take advantage of this gauge freedom and impose the harmonic gauge

∂μhμν − 1

2∂νhλ

λ = 0.

In this gauge, the linearized Einstein equations take particularly simple form

�hμν = −16πG

(Tμν − 1

2ημνT λ

λ

),

where � ≡ ∂λ∂λ is the D’Alembertian in Minkowski space. Clearly, this equationdescribes massless gravitational field sourced by the energy-momentum tensor.

A.10 Macroscopic Energy-Momentum Tensor

To find solutions to the Einstein equations describing the expanding Universe filledwith matter (e.g., relativistic plasma or “dust”), we need the appropriate expressionfor the energy-momentum tensor. It is sufficient for our purposes to treat matteras ideal fluid and make use of the hydrodynamic approximation to the energy-momentum tensor. To obtain its explicit expression in curved space-time, we firstconsider the case of flat space. Isotropic fluid without internal rotation has thefollowing energy-momentum tensor in its own rest frame,

T μν =

⎛⎜⎜⎝

ρ 0 0 00 p 0 00 0 p 00 0 0 p

⎞⎟⎟⎠ , (A.120)

where ρ and p are energy density and pressure. Let us first generalize this expressionto moving fluid. We note that in the rest frame, the 4-velocity vector is

uμ = (1, 0, 0, 0).

Therefore, we define a tensor object by

(p + ρ)uμuν − pημν . (A.121)

Clearly it coincides with the energy-momentum tensor (A.120) in the rest frame.Since both energy-momentum tensor and the object (A.121) transform accordingto the tensor law, they coincide in all reference frames. The simplest way to gen-eralize the expression (A.121) to curved space-time is to replace Minkowski metricημν by the general space-time metric gμν . Indeed, as we discussed above, thereexists locally Lorentz reference frame at every given point of space-time. In thisframe, the metric tensor at that point coincides with Minkowski tensor, and thematter energy-momentum tensor has the form (A.121). Performing the coordinatetransformation to arbitrary frame, we arrive at the following final expression for theenergy-momentum tensor,

T μν = (p + ρ)uμuν − pgμν . (A.122)


It is worth noting that, generally speaking, expression (A.122) is valid only in thecase of weak gravitational field. In strong field, there may appear additional termsdepending on the curvature tensor.

In general, density ρ, pressure p and 4-velocity uμ are arbitrary functions of timeand spatial coordinates, with the restrictions that

uμuμ = 1 (A.123)

and

∇μT μν = 0. (A.124)

Equality (A.123) is a direct consequence of the definition of 4-velocity, uμ = dxμ/ds,while equality (A.124) is the covariant conservation law for the energy-momentumtensor.

Problem A.29. Write various components of the conservation law (A.124)explicitly in the case of flat space. Show that in the non-relativistic limit (i.e., for|v| � 1, p � ρ) the resulting equations coincide with the hydrodynamic continuityequation and the Euler equation.

To conclude this Section, we note that in the linearized theory with Λ = 0,Eqs. (A.115) and (A.117) give the following equation for the Newtonian potentialin the presence of static source,

ΔΦ = 4πG(T00 + Tii) (A.125)

(summation over i is assumed). For the energy-momentum tensor of the form(A.120) we have

ΔΦ = 4πG(ρ + 3p). (A.126)

In this sense, the source of the gravitational field in GR is not energy but ratherthe combination (ρ+3p). In particular, an object made of hypothetical matter withρ+3p < 0 would repel rather than attract non-relativistic particles (in other words,it would “antigravitate”). The homogeneous isotropic Universe filled with such amatter would undergo accelerated expansion, see Sec. 3.2.4.

A.11 Notations and Conventions

Indices μ, ν, . . . refer to space-time and take values 0, 1, 2, 3. The summation overrepeated indices is assumed.

Indices i, j, . . . refer to space, i, j = 1, 2, 3. Spatial vectors are marked withbold-face font. The summation over repeated lower spatial indices is assumed, e.g.,aibi = ab, aiai = a2.

The signature of metric is (+,−,−,−).The Riemann tensor is defined by

[∇μ,∇ν ]Aλ = AσRλσμν .

A.11. Notations and Conventions 387

Its explicit expression is given in (A.31). The Ricci tensor is equal to


λν + ΓλρλΓρ

μν − ΓλρμΓρ

νλ.

Minkowski metric is denoted by ημν = diag(1,−1,−1,−1). The metric with smallperturbations about the spatially flat Friedmann–Lemaıtre–Robertson–Walkersolution reads

ds2 = a2(η)(ημν + hμν)dxμdxν ,

where x0 = η is conformal time. In other words

gμν = a2(η)(ημν + hμν).

Indices of hμν are raised and lowered by Minkowski metric.Our sign convention is that

e−iωt, ω > 0

is negative-frequency function.

Appendix B

Standard Model of Particle Physics

In this Appendix we introduce the main elements of the Standard Model of par-ticle physics. Of course, our presentation cannot be exhaustive; we do not considernumerous concrete phenomena in the world of elementary particles. Our purpose isto briefly describe those aspects which are used in the main text.

B.1 Field Content and Lagrangian

The Standard Model is in excellent agreement with all known to date experimentaldata (with the exception of neutrino oscillations, see Appendix C), obtained in low-energy physics, high-precision measurements and high-energy physics [1]. The basisof the Standard Model is quantum field theory, see, e.g., books [187–190].

The Standard Model describes the following particles thought to be elementaryto date:

(a) gauge bosons: photon, gluon, W±-bosons, Z-boson;(b) quarks: u, d, s, c, b and t;(c) leptons: electrically charged (electron e, muon μ and τ -lepton) and neutral

(electron neutrino νe, muon neutrino νμ and τ -neutrino ντ );(d) neutral Higgs boson h.

Particles of types “a” and “d” are bosons, particles of types “b” and “c” arefermions. Fields describing particles of type “a” are gauge fields. They are vectorsunder the Lorentz group and serve as mediators of gauge interactions. In particlephysics, fields of types “b” and “c” are often called matter fields; we will avoid thisterminology. They are spinors under the Lorentz group and participate in gauge andYukawa interactions.1 The field describing the Higgs boson is scalar; it participatesin the Yukawa interactions and self-interactions. In addition, the Higgs field playsa special role: its vacuum expectation value gives masses to all massive StandardModel particles.

1Non-Abelian gauge fields also carry charge of their own gauge group and participate in gauge

interactions.

389

390 Standard Model of Particle Physics

The Standard Model gauge group is SU(3)c × SU(2)W × U(1)Y . It describesstrong interactions (color group SU(3)c with gauge coupling gs) and electroweakinteractions (SU(2)W × U(1)Y with gauge couplings g and g′, respectively). Theelectroweak gauge group is in the Higgs phase, with the electromagnetic groupU(1)em left unbroken. Accordingly, W±- and Z-bosons are massive, and photonremains massless. The Standard Model fields form complete multiplets with respectto these gauge groups, i.e., they belong to certain representations of these groups.

Gauge fields are in the adjoint representations of their groups: there are eightgluon fields Ga

μ (a = 1, . . . , 8, according to the number of generators of SU(3)c),three gauge fields V i

μ of SU(2)W (i = 1, 2, 3 corresponding to the generators ofSU(2)W ) and one field Bμ of U(1)Y . As a result of the Higgs mechanism, threecombinations of the fields V i

μ and Bμ are massive and describe W±- and Z-bosons,

W±μ =

1√2(V 1

μ ∓ iV 2μ ), (B.1)

Zμ =1√

g2 + g′2(gV 3

μ − g′Bμ). (B.2)

The fourth combination,

Aμ =1√

g2 + g′2(g′V 3

μ + gBμ), (B.3)

remains massless and describes photon. The relation between the fields Zμ and Aμ

and original gauge fields is also written in the form

Zμ = cos θW · V 3μ − sin θW · Bμ,

Aμ = cos θW · Bμ + sin θW · V 3μ ,

where θW is the weak mixing angle,

tan θW =g′

g.

The experimental value of sin θW is2

sin θW = 0.481.

Table B.1 contains dimensions of representations of vector fields and theirAbelian charges. Note that in this Table as well as in a number of subsequentformulas, we use matrix notations

Gμ ≡8∑

a=1

Gaμ

λa

2,

Vμ ≡3∑

i=1

V iμ

τ i

2,

2Here and in what follows, if not stated otherwise, we omit important details related to radiative

corrections.

B.1. Field Content and Lagrangian 391

Table B.1 Dimensions of representations and charges of

gauge and Higgs fields; symbol 0∗ indicates that fields Bμ

and Aμ are gauge fields of groups U(1)Y and U(1)em,

respectively.

Field�Group SU(3)c SU(2)W U(1)Y U(1)em

Gμ 8 1 0 0

Vμ 1 3 0

Bμ 1 1 0∗

W±μ 1 ±1

Zμ 1 0

Aμ 1 0∗

H 1 2 1

where λa are the Gell-Mann matrices, and τ i are the Pauli matrices (λa/2 and τ i/2are generators of SU(3)c and SU(2)W , respectively).

Fermion fields “b” and “c” form three generations of quarks and leptons,

I : u, d, νe, e

II : c, s, νμ, μ

III : t, b, ντ , τ.

Particles within a single generation are discriminated by gauge interactions (theyhave different gauge quantum numbers), while the three “partner” particles of dif-ferent generations (e.g., u-, c- and t-quarks or electron, muon and τ -lepton) havethe same gauge quantum numbers but different masses and Yukawa couplings tothe Higgs boson.

One way to describe fermionic fields in (3 + 1)-dimensional Minkowski space3 isto introduce left and right two-component (Weyl) spinors χL and χR. These spinorstransform independently under the proper Lorentz group.4

Of the two-component spinors one can construct Lorentz scalars, vectors andtensors. In particular, one can show that bilinear combinations

χTL iσ2χL, χT

R iσ2χR,

are scalars, and

χTR σμχL, χT

L σμχR

3We leave without discussion the description of fermion fields in curved space-time. It is briefly

considered in the accompanying book.4More precisely, they transform according to fundamental (χL) and antifundamental (χT

R) rep-

resentations of group SL(2, C), which is the universal covering group for the proper Lorentz group

SO(3, 1). The covering is two-fold; this property is responsible, in particular, for the fact that not

fermion fields themselves, but their bilinear combinations are physical observables.


are vectors. Here

σμ = (�, σ),

σμ = (�,−σ),

where σ are the conventional Pauli matrices acting on the Lorentz indices.

Problem B.1. Check the validity of the above statements. Hint : Make use ofthe equivalence of the fundamental and antifundamental representations of SU(2);find the transformation law for spinors χL and χR under the Lorentz boosts and3-dimensional rotations.

The full Lorentz group, besides proper transformations (boosts and rotations),also contains reflection of space (parity transformation) P and inversion of timeT . Two-component spinor χL (or χR) does not transform to itself under spatialreflection. The relevant representation of the full Lorentz group is in terms of four-component Dirac spinor ψ. The Dirac spinor includes two two-component Weylspinors χL and χR,

ψ =

(χL

χR

).

The free Dirac fermions are solutions to the Dirac equation

iγμ∂μψ = mψ,

where m is fermion mass and γμ are four 4× 4 Dirac matrices obeying anticommu-tation relations

{γμ, γν} = ημν .

In the chiral (Weyl) representation, the Dirac matrices are

γμ =

(0 σμ

σμ 0

).

In this representation, the Dirac equation is written in the matrix form,(0 iσμ∂μ

iσμ∂μ 0

)(χL

χR

)= m

(χL

χR

)

Note that in the massless case, m = 0, the Dirac equation splits into two sep-arate equations for each of the components χL, χR; massless solutions χL and χR

are eigenfunctions of the helicity operator pσ|p| with eigenvalues −1 and +1, respec-

tively.5 Therefore, in the massless case the minimum option is to introduce onetwo-component spinor χL, so that the theory contains only particles of left helicity

5In both massless and massive cases, helicity is the projection of spin onto the direction of motion.

However, helicity is Lorentz invariant only for massless fermions.


and antiparticles of right helicity. This is precisely the way neutrino is described inthe Standard Model. Of course, parity is broken in such a situation.

To describe the Standard Model interactions in terms of the Dirac 4-componentspinors, it is necessary to extract the components χL and χR. This is done by usingprojection operators

P∓ =1 ∓ γ5

2, γ5 ≡ iγ0γ1γ2γ3.

In the chiral representation

γ5 =

(−1 00 1

).

In what follows we use the notations

ψL ≡ P−ψ =1 − γ5

2ψ, ψR ≡ P+ψ =

1 + γ5

2ψ. (B.4)

In the chiral representation of the Dirac matrices

ψL =

(χL

0

), ψR =

(0

χR

).

For some applications a useful observation is that χcR ≡ iσ2χ

∗R is left spinor.

Problem B.2. Prove the last statement above.

If not otherwise stated, we use 4-component spinors in what follows. The mostcommonly used Lorentz structures bilinear in fermion fields are

ψψ, scalar, ψγμψ, vector,ψγ5ψ, pseudoscalar, ψγ5γμψ, pseudovector.

(B.5)

where

ψ ≡ ψ†γ0

is the Dirac conjugate spinor.

Problem B.3. Check the validity of the Lorentz assignment (B.5). Express thesestructures in terms of the Weyl fermions.

Within the Standard Model, neutrinos have only left components, in contrast toquarks and charged leptons. With respect to strong interactions, both left and rightquarks form fundamental representations (triplets), so that from the standpoint ofstrong interactions the separation of quarks into left and right components is notrequired. On the other hand, right quarks and right charged leptons are singletsunder SU(2)W , while the left fermions form doublets

Q1 =(

u

d

)L

, Q2 =(

c

s

)L

, Q3 =(

t

b

)L

,

L1 =(

νe

e

)L

, L2 =(

νμ

μ

)L

, L3 =(

ντ

τ

)L

.

(B.6)


Similarly, the convenient notation for the right fermions is

Un = uR, cR, tR ; n = 1, 2, 3 ;

Dn = dR, sR, bR ;

En = eR, μR, τR.

(B.7)

Dimensions of the fermion representations and Abelian charges are given inTable B.2.

The scalar Higgs field H is singlet under the group of strong interactions,doublet under SU(2)W and carries U(1)Y charge +1. These properties are reflectedin Table B.1.

In terms of the fields explicitly covariant under the gauge group SU(3)c ×SU(2)W × U(1)Y the Standard Model Lagrangian reads

LSM = −12TrGμνGμν − 1

2TrVμνV μν − 1

4BμνBμν

+ iLnDμγμLn + iEnDμγμEn + iQnDμγμQn + iUnDμγμUn + iDnDμγμDn

− (Y lmnLmHEn + Y d

mnQmHDn + Y umnQmHUn + h.c.)

+DμH†DμH − λ

(H†H − v2

2

)2

. (B.8)

Here the first line contains gauge fields, whose strength tensors are

Bμν ≡ ∂μBν − ∂νBμ,

Vμν ≡ ∂μVν − ∂νVμ − ig[Vμ, Vν ],

Gμν ≡ ∂μGν − ∂νGμ − igs[Gμ, Gν ],

where the square brackets denote commutators; for instance, [Vμ, Vν ] ≡ VμVν−VνVμ.The field Bμ is real and fields Vμ, Gμ are Hermitean. In terms of real fields Ga

μ

and V iμ

TrGμνGμν =12Ga

μνGa μν , TrVμνV μν =12V i

μνV i μν ,

Table B.2 Dimensions of representations and charges of

fermions of the first generation; fermions of the second

and third generations have exactly the same quantum

numbers.

Field�Group SU(3)c SU(2)W U(1)Y U(1)em

L ≡„

νe

e

«L

1 2 −1

„0

−1

«

E ≡ eR 1 1 −2 −1

Q ≡„

u

d

«L

3 2 +1/3

„+2/3

−1/3

«

U ≡ uR 3 1 +4/3 +2/3

D ≡ dR 3 1 −2/3 −1/3


with

Gaμν = ∂μGa

ν − ∂νGaμ + gsf

abcGbμGc

ν , V iμν = ∂μV i

ν − ∂νV iμ + gεijkV j

μ V kν , (B.9)

where fabc and εijk are the structure constants of SU(3) and SU(2), respectively(εijk is completely antisymmetric symbol). The last terms in (B.9) are responsiblefor gluon self-interactions and for interactions between W±-, Z-bosons and photons.

The second line in (B.8) includes free Lagrangians of fermions and fermion cou-plings to gauge fields. Covariant derivatives entering there are uniquely determinedby gauge group representations of fermions and their U(1)Y -charges: for fermion f

in representations Ts and TW of SU(3)c and SU(2)W

Dμf ≡(

∂μ − igsTas Ga

μ − igT iW V i

μ − ig′Yf

2Bμ

)f,

where T as and T i

W are generators of SU(3)c and SU(2)W and Yf is U(1)Y -chargeof this fermion. For quarks T a

s = λa/2, while for leptons T as = 0. For left doublets

(B.6) we have T iW

= τ i/2 and for right singlets T iW

= 0. Note that summation overgenerations is assumed in the second line of (B.8). In terms of the fields used in(B.8), gauge interactions are diagonal in generations.

The third line in (B.8) describes the Yukawa interactions of fermions with theHiggs field H ; h.c. means the Hermitean conjugation and summation over repeatedindices m, n labeling generations is assumed. Yukawa coupling matrices Y l

mn, Y dmn

and Y umn are complex and not diagonal in generations. Below we briefly discuss

the consequences of this property. We emphasize that the Lagrangian (B.8) doesnot describe neutrino oscillations: the relevant Yukawa terms are absent. We con-sider in Appendix C extensions of the Standard Model capable of describing thephenomenon of neutrino oscillations.

Let us make a remark concerning the third line in (B.8). Like the completeLagrangian of the Standard Model, this line is invariant under SU(3)c ×SU(2)W ×U(1)Y . To see this explicitly, we begin with the first term in the third line. Left leptonand Higgs field are doublets under SU(2)W , while right lepton is singlet. Therefore,the first term has SU(2)W -structure (L†H)E; it is SU(2)W -singlet. According toTable B.2, the total U(1)Y -charge of the fields in this term is zero, which meansU(1)Y -invariance. The situation with the second Yukawa term is similar. The lastterm includes

Hα ≡ iτ2αβH∗ β = εαβH∗ β , (B.10)

where α = 1, 2 and εαβ is antisymmetric symbol. The field H is in the fundamentalrepresentation6 of SU(2)W . Therefore, the third Yukawa term is SU(2)W -singlet. Itis also invariant under U(1)Y , this is the reason for using H there rather than thefield H itself.

6H∗ is in anti-fundamental representation. The relation (B.10) gives isomorphism between the

anti-fundamental and fundamental representations. Such an isomorphism exists for SU(2) but not

for SU(N) with N > 2.


It is important to emphasize that the third line in (B.8) is the most general gaugeinvariant renormalizable Lagrangian combining together left and right fermions.In particular, explicit mass terms of fermions, which would also have the Lorentzstructure (fL) · fR + h.c., are forbidden by the invariance under SU(2)W × U(1)Y .

The last line in Eq. (B.8) is the Lagrangian of the Higgs field itself. Accordingto Table B.1, the covariant derivative of the Higgs field is

DμH =(

∂μ − igτ i

2V i

μ − ig′

2Bμ

)H.

The scalar potential of the theory — the last term in (B.8) — has minimum atnon-zero value of the Higgs field such that

H†H =v2

2.

Using gauge invariance, one can show that the Higgs vacuum and the perturbationsabout this vacuum, without loss of generality, can be written in the form (unitarygauge)

H(x) =

(0

v√2

+ h(x)√2

). (B.11)

Thus, there is only one physical scalar excitation about the Higgs vacuum. This isthe Higgs boson, described by the field h.

The Higgs vacuum breaks the gauge symmetry SU(2)W×U(1)Y down to U(1)em.Conversion from explicitly SU(2)W ×U(1)Y -invariant fields to physical vector fieldsin this vacuum is accomplished by the change of variables (B.1), (B.3). Indeed, uponthis change, free gradient terms of the fields W±

μ , Zμ and Aμ maintain canonicalform

− 14

[3∑

i=1

(∂μV iν − ∂νV i

μ)2 + (∂μBμ − ∂νBμ)2]

= −12(∂μW+

ν − ∂νW+μ )(∂μW− ν − ∂μW− ν) − 1

4ZμνZμν − 1

4FμνFμν ,

where

Fμν = ∂μAν − ∂νAμ, Zμν = ∂μZν − ∂νZμ. (B.12)

The vacuum expectation value in (B.11) leads to mass terms in quadraticLagrangian about the Higgs vacuum,

DμH†DμH −→ g2v2

4W+

μ W μ− +(g2 + g

′2)v2

8ZμZμ.

This is precisely what the Higgs mechanism is about. The change of variables (B.1),(B.3) is chosen in such a way that these mass terms are diagonal. Thus, the massesof W±- and Z-bosons are

MW =gv

2, MZ =

v√

g2 + g′2

2=

MW

cos θW

.


The Yukawa couplings are responsible for the fact that quarks and chargedleptons are massive as well. The masses of fermions are given by

mf =yf√

2v,

where yf are eigenvalues of the matrix of Yukawa couplings entering (B.8). Theonly fermions that remain massless are neutrinos (the latter property is in fact ashortcoming of the Standard Model).

The Lagrangian of the Standard Model in terms of physical fields is obtained bysubstituting (B.1), (B.3) and (B.11) into (B.8) and performing the transformationof fermion fields from the original basis in which gauge interactions are diagonal,to the basis where diagonal structure is possessed by the fermion mass matrix (andYukawa terms). As we discuss in Sec. B.3, the latter transformation gives rise tothe Cabibbo–Kobayashi–Maskawa (CKM) matrix Vmn describing quark mixing inweak interactions. Note that we often use the same notation for fermions in the twobases.

It is convenient to present the full Lagrangian of the Standard Model written interms of physical fields as a sum of several terms,

LSM = LQCD + Lfreelept + Lf,em + Lf,weak + LY + LV + LH + Lint

HV . (B.13)

Here

LQCD = −14Ga

μνGa μν +∑

quarks

q

(iγμ∂μ − mq − igs

λa

2Ga

μ

)q

is the Lagrangian containing quarks, gluons and their interactions with each other;summation runs over all types of quarks. This is the Lagrangian of the theory ofstrong interactions, quantum chromodynamics (QCD). The second term in (B.13)is the free Lagrangian of leptons

Lfreelept =

∑n

ln(iγμ∂μ − mln)ln +∑

n

νniγμ∂μνn.

Here n runs over generations, ln = e, μ, τ . The third and the fourth terms describeelectromagnetic and weak interactions of quarks and leptons, respectively,

Lf,em = e∑

f

qf fγμAμf,

where

e = g sin θW =gg′√

g2 + g′2(B.14)

is proton electric charge, so that eqf is electric charge of fermion f ;

Lf,weak =g

2√

2

∑n

(νnγμ(1 − γ5)W+μ ln + h.c.)

+g

2√

2

∑m,n

(umγμ(1 − γ5)W+μ Vmndn + h.c.)

+g

2 cos θW

∑f

fγμ(tf3 (1 − γ5) − 2qf sin2 θW )fZμ. (B.15)


Here the sum over f means summation over all quarks and leptons, tf3 is weakisospin equal to +1/2 for up quarks u, c, t and neutrinos and −1/2 for down quarksd, s, b and charged leptons. The first two terms in Lf,weak are couplings of leptonsand quarks to W -bosons (charged currents), the third term is coupling to Z-boson(neutral currents). Note that the emission and absorption of W -boson changes thetype (flavor) of fermion (and for quarks, generally speaking, the generation numberas well, thanks to the non-diagonal CKM matrix Vmn), while the interaction withZ-boson does not change flavor.

Problem B.4. Check that the following relation holds for all fermions,

qf =Yf

2+ tf3 .

The term LY in the Lagrangian (B.13) describes the Yukawa interactions offermions with the Higgs boson,

LY = −∑

f

yf√2f fh = −

∑f

mf

vffh,

where summation runs over all fermion species except for neutrinos. The Yukawacouplings are proportional to the fermion masses; this is of course a reflection of thefact that all fermions obtain masses due to interactions with the Higgs field.

The term LV in (B.13) contains the free Lagrangians of photons, W±- andZ-bosons and their interaction with each other,

LV = −14FμνF μν − 1

4ZμνZμν +

M2Z

2ZμZμ

− 12|W−

μν |2 + M2W |W−

μ |2 +g2

4(W−

μ W+ν − W+

μ W−ν )2

− ig

2(Fμν sin θW + Zμν cos θW )(W−

μ W+ν − W+

μ W−ν ), (B.16)

where Fμν and Zμν are defined by q. (B.12), and

W−μν ≡ (∂μ + ieAμ + ig cos θW Zμ)W−

ν − (μ ↔ ν). (B.17)

The relation between the electromagnetic coupling e and gauge couplings g, g′

is again given by (B.14). Note that the Lagrangian (B.16), as well as the fullLagrangian (B.13), is invariant under the unbroken gauge symmetry U(1)em, and,according to (B.17), W±-bosons carry electric charge ±e.

The term LH describes the Higgs sector of the Standard Model,

LH =12∂μh∂μh − 1

2m2

hh2 − λvh3 − λ

4h4,

where

mh =√

2λv

B.2. Global Symmetries 399

is the Higgs boson mass. The Higgs self-coupling λ (in other words, the Higgs bosonmass mh) is the only parameter of the Standard Model not measured experimen-tally.7 There is only a lower limit on this mass,

mh > 114.4 GeV.

Finally, the term LintHV stands for the Higgs boson coupling to the massive vector

bosons,

LintHV =

g2

2vh|W−

μ |2 +g2 + g

′2

4vhZμZμ +

g2

4h2|W−

μ |2 +g2 + g

′2

8h2ZμZμ.

To date, all particles of the Standard Model, except for the Higgs boson, havebeen observed experimentally. The values of the Standard Model parameters are8

[1]:

me = 0.511 MeV, mu = 1.5 − 3.3 MeV, md = 3.5 − 6.0 MeV,

mμ = 105.7 MeV, mc = 1.14 − 1.34 GeV, ms = 0.07 − 0.13 GeV,

mτ = 1.78 GeV, mt = 169.1 − 173.3 GeV, mb = 4.13 − 4.37 GeV,

MZ = 91.2 GeV, MW = 80.4 GeV, v = 247 GeV,

α ≡ e2

4π=

1137

, sin2 θW = 0.231, αs(MZ) = 0.118.

Uncertainties in quark masses (except for t-quark) are predominantly theoretical;they are due to the fact that quarks do not exist in free state.

B.2 Global Symmetries

In addition to gauge symmetries, there are global Abelian symmetries in theStandard Model: the Lagrangian (B.13) is invariant under simultaneous phase rota-tions of all quark fields,

q → eiβ/3q, q → e−iβ/3q (B.18)

and independently under phase rotations of lepton fields of each generation,

(νe, e) → eiβe(νe, e), (νe, e) → e−iβe(νe, e) (B.19)

(νμ, μ) → eiβμ(νμ, μ), (νμ, μ) → e−iβμ(νμ, μ) (B.20)

(ντ , τ) → eiβτ (ντ , τ), (ντ , τ ) → e−iβτ (ντ , τ ) (B.21)

7Yukawa interactions have not yet been directly observed either. However, measurements of par-

ticle masses enable one to predict the Yukawa couplings for all massive fermions in the framework

of the Standard Model. On the contrary, accurate predictions for the mass or self-coupling of the

Higgs boson do not exist. Radiative corrections due to the Higgs boson lead to weak (logarithmic)

dependence of some observables on mh, that actually makes it possible to determine the range of

the Higgs boson mass, mh � 200 GeV.8We omit the details related to the dependence of these parameters on the renormalization scale,

except for the case of αs.


Here β, βe, βμ and βτ are independent parameters of the transformations. Theconserved quantum number associated with the symmetry (B.18) is baryon number9

B =13(Nq − Nq),

where Nq and Nq are total numbers of quarks and antiquarks of all types, respec-tively. The baryonic charge of quarks by definition is equal to 1/3, and the chargeof antiquarks is (−1/3). With this assignment, the total baryonic charge of proton,which consists of two u-quarks and one d-quark, equals 1. Hence, in terms of numbersof baryons and antibaryons we have

B =∑

(NB − NB),

where the summation runs over all types of baryons. A clear manifestation thatbaryon number is conserved in Nature with high precision is proton stability: protonis the lightest particle carrying baryon number, so it must be absolutely stable ifthe baryon number conservation is exact. Proton decay has not yet been discovered,and the experimental limit on the lifetime is

τp > 1032 − 1033 years,depending on the decay mode.

Symmetries (B.19)–(B.21) are associated with three independently conservedlepton numbers (electron, muon and tau)

Le = (Ne + Nνe) − (Ne+ + Nνe), (B.22)

Lμ = (Nμ + Nνμ) − (Nμ+ + Nνμ), (B.23)

Lτ = (Nτ + Nντ ) − (Nτ+ + Nντ ), (B.24)

where Ne, Nνe , Ne+ , Nνe are numbers of electrons, electron neutrinos, positronsand electron antineutrinos, respectively, and similarly for other generations.

A manifestation of the lepton quantum number conservation is the absence ofprocesses violating lepton numbers, but otherwise allowed. An example of such aprocess is the decay

μ → eγ.

In this process, electron and muon numbers would be violated. The experimentalbound on its branching ratio is

Br(μ → eγ) < 1.2 · 10−11.

This is one of the best results showing the conservation of the muon and electronnumbers. Note that the observed neutrino oscillations show that lepton numbersare actually violated in Nature, see Appendix C.

In many models generalizing the Standard Model, baryon and/or lepton numbersare violated, which should lead to new physical phenomena, such as proton decay.The search for these phenomena is an important challenge for low-energy particle

9Baryon number is conserved in perturbation theory only. Nonperturbative effects violate baryon

and lepton numbers, but these effects are very small under normal conditions (but not in the early

Universe, see Chapter 11).

B.3. C-, P-, T-Transformations 401

physics experiments. At present, besides neutrino oscillations (see Appendix C), noexperimental evidence for violation of baryon or lepton numbers has been found.

B.3 C-, P-, T-Transformations

Let us briefly consider discrete transformations: time reversal,

T-transformation: (x0,x) T−→ (−x0,x),

spatial reflection (parity transformation),

P-transformation: (x0,x) P−→ (x0,−x)

and charge conjugation,

C-transformation: f(x) C−→ f c(x).

P- and T-transformations, together with the proper Lorentz group, form the fullLorentz group.

From the standpoint of scattering processes, time reversal means the replacementof initial state by final state and vice versa; spatial reflection implies the inversionof spatial momenta of all particles, and charge conjugation interchanges particlesand antiparticles. In quantum field theory, the CPT-theorem is valid, which statesthat physical processes must be invariant under the joint action of all three trans-formations. One of its consequences is the equality of masses and total decay widthsof particle and its antiparticle.

Lorentzian scalars, vectors and tensors10 can be even and odd underP-transformations. In the latter case they are called pseudoscalars, axial vectorsand pseudotensors. For instance, scalar (parity-even) and pseudoscalar (parity-odd)transform under P-transformation as

φ(x0,x) P−→ φ′(x0,x) = φ(x0,−x)

and

φ(x0,x) P−→ φ′(x0,x) = −φ(x0,−x),

respectively. P-transformation of vector (parity-even) reads

Vν(x0,x) P−→ V ′ν(x0,x) = δ0

νV0(x0,−x) − δiνVi(x0,−x),

while P-transformation of axial vector is

Aν(x0,x) P−→ A′ν(x0,x) = −δ0

νA0(x0,−x) + δiνAi(x0,−x),

Several spinor bilinear combinations of definite parity are given in (B.5).It is clear from the form of the Lagrangian (B.13), (B.15), that weak interac-

tions violate parity: weak bosons couple to both vector currents ψmγμψn, and axialcurrents ψmγμγ5ψn. Moreover, weak interactions violate CP. The source of CP-violation is a complex parameter in the Cabibbo–Kobayashi–Maskawa matrix. Onthe contrary, strong and electromagnetic interactions violate neither P nor C.

10We do not discuss here the properties of spinors under parity.


B.4 Quark Mixing

To understand how the Cabibbo–Kobayashi–Maskawa (CKM) matrix emerges, letus consider the transformation from the gauge basis of fermions to the mass basis.The Lagrangian (B.8) is written in terms of fields in gauge basis: all gauge inter-actions have diagonal form. As a result of spontaneous electroweak symmetrybreaking, the Yukawa terms in the Lagrangian (B.8) in the unitary gauge (B.11)give rise to fermion mass terms,

Lm = − v√2Y l

mneLmeRn − v√2Y d

mndLmdRn − v√2Y u

mnuLmuRn + h.c., (B.25)

as well as Yukawa couplings to the Higgs boson,

LY = − h√2Y l

mneLmeRn

− h√2Y d

mndLmdRn

− h√2Y u

mnuLmuRn

+ h.c. (B.26)

The gauge interactions with gluons, photons and Z-bosons are still diagonal infermions, while interactions with W±-bosons are diagonal in generations,

LW =g√2νnγμW−

μ eLn +g√2uLnγμW−

μ dLn + h.c.

The Yukawa couplings can be written in the form

Y lmn = UeL

mpYlp (UeR)−1

pn , Y dmn = UdL

mpYdp (UdR)−1

pn , Y umn = UuL

mpYup (UuR)−1

pn ,

where constants Y lp , Y d

p and Y up are real, and UeL , . . . , UuR are unitary matrices;

summation over repeated index p is assumed. It is precisely these matrices thattransform the quark fields into the mass basis,

eRm= UeR

mneRn, dRm

= UdRmndRn

, uRm= UuR

mnuRn,

eLm = UeLmneLn , dLm = UdL

mndLn , uLm = UuLmnuLn .

Indeed, in terms of fields with tilde, the terms (B.25) and (B.26) are diagonal; forexample

v√2Y l

mneLmeRn =∑

p

v√2Y l

p¯eLp eRp .

Upon this transformation, kinetic terms of the fermion fields and also gauge inter-actions with gluons, photons and Z-boson remain diagonal, while the interactionwith the W±-bosons contains mixing matrices of generations.

Within the Standard Model, mixing in the lepton sector is unphysical: it can beeliminated by redefining the neutrino fields,

νm → νm: νm = (U eL)−1mnνn.

Since all gauge interactions of neutrinos are proportional to unit matrix (in thespace of generations), this redefinition does not give rise to mixing in the neutrinosector. The reason for the absence of mixing in the lepton sector of the StandardModel is that neutrinos are massless there.

B.4. Quark Mixing 403

In the quark sector the situation is different, and the transformation to the massbasis leads to mixing between quarks of different generations in the interactionvertices with W±-bosons,

LW =g√2

¯uLmγμW−

μ VmndLn+ h.c., (B.27)

where Vij ≡ (UuL)−1ik UdL

kj is the unitary 3×3 Cabibbo–Kobayashi–Maskawa mixingmatrix.

A 3 × 3 unitary matrix of general form is specified by 9 real parameters (thedimension of the group of unitary matrices U(3)). In terms of the orthogonal sub-group SO(3) ⊂ U(3) these parameters can be divided into 3 rotation angles and6 phases. However, five of the six phases are unphysical. They can be eliminated from(B.27) by redefining the fermion fields fn → fneiβn . This phase rotation does notlead to complex coefficients in other parts of the Lagrangian (B.13), since fermionfields enter all its terms, except for (B.27), in combinations explicitly invariant underphase rotations, such as fnγμfn. One remaining phase in the CKM matrix is thesource of CP-violation in weak interactions (see below).

Let us check that there is only one CP-violating phase. To this end, we explicitlyextract phase factors in the elements of the mixing matrix,

V =

⎛⎜⎝

O11eiβ11 O12eiβ12 O13eiβ13



⎞⎟⎠ .

Here all quantities Omn and βmn, m, n = 1, 2, 3 are real, and we take for definitenessOmn ≥ 0 and 0 ≤ βmn < 2π. There are 9 phase factors in the representation above.However, they are not independent, since the unitarity of Vmn implies 3 additionalconstraints:11

O11O21ei(β11−β21) + O12O22ei(β12−β22) + O13O23ei(β13−β23) = 0,

O11O31ei(β11−β31) + O12O32ei(β12−β32) + O13O33ei(β13−β33) = 0, (B.28)

O21O31ei(β21−β31) + O22O32ei(β22−β32) + O23O33ei(β23−β33) = 0.

As a result of the phase rotation of fermion fields

dLn = e−iβ1n ˜dLn , n = 1, 2, 3, (B.29)

¯uLm = e−i(βm1−β11) ¯uLm , m = 2, 3,

five out of nine phases of the mixing matrix are set equal to zero without using(B.28) (the common phase rotation of fermion fields does not change the Lagrangianat all). The remaining 4 phases are dependent. They are expressed through single

11The general unitarity condition reads VmnV ∗pn = δmp, m, p = 1, 2, 3. Three out of these equations

(diagonal ones, m = p) do not involve phases βmn at all, while the conditions with m > p are not

independent, because they are complex conjugate to the conditions with m < p.


phase and the elements Omn by making use of the relations (B.28), which after thetransformation (B.29) take the form

O11O21 + O12O22e−iβ22 + O13O23e−iβ23 = 0,

O11O31 + O12O32e−iβ32 + O13O33e−iβ33 = 0,

O21O31 + O22O32ei(β22−β32) + O23O33ei(β23−β33) = 0.

The first condition enables one to express β22 in terms of β23 (and elements Omn),from the second condition one finds the relation between β32 and β33. Finally, uponsubstitution of the obtained expressions into the third condition, β33 is expressedthrough the only remaining phase β23. The latter determines all phase factors inthe elements of the CKM matrix.

Equivalently, one can use so-called standard parameterization [1] in which thematrix V reads

V =

⎛⎜⎝

c12c13 s12c13 s13e−iδ13

−s12c23 − c12s23s13eiδ13 c12c23 − s12s23s13eiδ13 s23c13

s12s23 − c12c23s13eiδ13 −c12s23 − s12c23s13eiδ13 c23c13

⎞⎟⎠, (B.30)

where cij = cos θij , sij = sin θij , the parameters θij are mixing angles, and δ13 isthe CP-violating phase. The values of these parameters have been determined withgood accuracy from numerous experiments:

s12 = 0.2272, s13 = 0.0040,

s23 = 0.04221, δ13 = 57◦ ± 10◦.

The absolute values of the elements of the quark mixing matrix are

|V | =

⎛⎜⎝

0.9742 0.226 0.00360.226 0.9733 0.0420.0087 0.041 0.99913

⎞⎟⎠ , (B.31)

with experimental errors of the order of the last digit. Note that CP-violating phaseof the CKM matrix is large. Nevertheless, the resulting CP-violating effects in theStandard Model are strongly suppressed due to the smallness of mixing angles,which is clearly seen in the standard parameterization (B.30). Note also that thereis strong hierarchy between the entries in (B.31): the diagonal elements are close to1, while off-diagonal elements obey V13, V31 � V23, V32 � V12, V21 � 1.

The general fact that complex entries in Vmn give rise to CP-violation, followsfrom the transformation law of fields under CP,

ψL

P−→ ηγ0ψL, ψL

P−→ η−1ψLγ0, (B.32)

ψL

C−→ CψT

L, ψL

C−→ ψT

L(C−1)T , (B.33)

B.4. Quark Mixing 405

where η is the sign factor (η2 = ±1), and C is the charge conjugation matrix suchthat ψγμψ

C−→ −ψγμψ. For the bilinear combinations of spinors which form thecharged current coupled to W -boson (see (B.27)), the above formulas yield a simpletransformation law (we no longer write tilde over fields in the mass basis)

uLmγ0VmndLn

CP−→ −dLnγ0VmnuLm , uLmγiVmndLn

CP−→ dLnγiVmnuLm .

Taking into account the transformation law of W -bosons under CP-conjugation

W±0

CP−→ −W∓0 , W±

iCP−→ W∓

i ,

we finally obtain that under CP-transformation, the interaction (B.27) converts into

LCPW =

g√2dLnγμW+

μ VmnuLm + h.c. =g√2uLmγμW−

μ V ∗mndLn + h.c. (B.34)

Comparison of (B.34) with (B.27) shows that the coupling (B.27) violates CP if theentries of mixing matrix Vmn are complex-valued.

This result reflects the general property of field theory: the initial Lagrangiantransforms under CP into the Lagrangian with complex-conjugate couplings. If thereare unremovable phases in the set of couplings, then CP is broken.

As an illustration, let us estimate the difference of partial widths of decays

t → W+b and t → W−b,

These widths would be equal if CP were exact symmetry of the Standard Model(CPT ensures that the total widths of particles and antiparticles are the same, butdoes not forbid different partial widths). At the tree level, the partial widths areidentical, and the main contribution to their difference comes from the interferenceof the tree and one-loop diagrams. For the decay t → W+b, these diagrams areshown in Fig. B.1. Similar diagrams contribute to the decay t → W−b.

If one denotes by V33Mtree the tree-level contribution to the amplitude of thedecay t → W+b, then the one-loop term is

Mtree · g2

16π2

3∑n,l=1

V3nV ∗lnVl3(Anl + iπBnl),

Fig. B.1 Tree-level and one-loop diagrams giving the main contribution to the difference of partial

widths of t → W +b and t → W−b in the Standard Model. The vertices are proportional to the

elements of the CKM matrix which are explicitly indicated.


where Anl, Bnl are real functions depending on masses of W -boson, t- and b-quarks,and also on masses of virtual quarks in the loop. In what follows we need only thefunction Bnl, which is conveniently written in the form

Bnl =∫ 1

0

dx

∫ 1−x

0

dy Cnl − 11 − m2

ul/m2

t

∫ 1

0

xdx Cn,

Cnl = θ

[(x +

m2b

m2t

y

)(1 − x − y) − m2

dn

m2t

x − m2ul

m2t

y − M2W

m2t

(1 − x − xy)

],

Cn = θ

[−x2 +

(1 +

m2dn

m2t

− M2W

m2t

)x − m2

dn

m2t

],

where θ(z) is the usual step-function. The imaginary part Bnl of the diagramsis related to the kinematics of virtual processes, rather than complex-valuedness ofcouplings: it arises from the integration over virtual momenta near mass shell. Sincethis is a kinematic effect, it is the same for decays of t and t. Hence, the one-loopdiagram for t → W−b gives

M∗tree ·

g2

16π2

3∑n,l=1

V ∗3nVlnV ∗

l3(Anl + iπBnl).

The decay widths are determined by the absolute values of the amplitudes squared,

|M(t → W +b)|2 = |Mtree|2 (B.35)

×⎧⎨⎩1 +

g2

16π2

⎛⎝V ∗

33

3∑l,n=1

V3nV ∗lnVl3(Anl + iπBnl) + h.c.

⎞⎠+ O(g4)

⎫⎬⎭ (B.36)

|M(t → W−b)|2 = |Mtree|2 (B.37)

×⎧⎨⎩1 +

g2

16π2

⎛⎝V33

3∑l,n=1

V ∗3nVlnV ∗

l3(Anl + iπBnl) + h.c.

⎞⎠+ O(g4)

⎫⎬⎭ (B.38)

In this way, we finally obtain to the leading order in g2:

ΔCP =Γ(t → W+b) − Γ(t → W−b)Γ(t → W+b) + Γ(t → W−b)

= − g2

8π

∑l,n

Im(V ∗33V3nV ∗

lnVl3)Bnl.

Numerically, the relative difference of widths is tiny,

ΔCP = 7 · 10−8.

Therefore, this particular effect is unlikely to be observed experimentally in fore-seeable future. We note in this regard that the phase in the CKM matrix leads to anumber of other CP-violating effects (notably, in processes involving neutral kaonsand B-mesons), some of which have been observed in experiments. We note also thatthe above mechanism responsible for the difference of partial widths of particles andantiparticles is quite generic. In various theories generalizing the Standard Model,this mechanism is used to generate the baryon asymmetry (see Chapter 11).

B.5. Effective Fermi Theory 407

B.5 Effective Fermi Theory

Processes at low energies, E � MW , are well described by the effective Fermi theory.The Lagrangian of this theory is obtained from (B.13) by integrating out massivevector fields.

In the Standard Model, the interaction of massive vector bosons with fermionshas the form

Lf,weak =g

2 cos θW

JNCμ Zμ +

g

2√

2(JCC

μ Wμ,− + h.c.),

where the neutral JNCμ and charged JCC

μ currents are

JNCμ =

∑f

fγμ(tf3 (1 − γ5) − 2qf sin2 θW )f (B.39)

JCCμ =

∑m

νmγμ(1 − γ5)em +∑m,n

umγμ(1 − γ5)Vmndn. (B.40)

At E � MZ there are only fermions (leptons and quarks, except for heavy t-quark)in the initial and final states. At these energies, the main contribution to the weakamplitudes comes from single exchange of virtual massive vector boson. In thelanguage of Feynman diagrams “integrating out” means shrinking the propagators ofW - and Z-bosons to a point (see Fig. B.2). As a result, the Standard Model diagramsof the type shown in Fig. B.3(a) transform into diagrams shown in Fig. B.3(b). Thelatter correspond to the Fermi theory of effective four-fermion interaction. Thus,integrating out Z-boson gives rise to the interaction

LN = −GF√2

JNCμ JNC μ,

while integrating out W±-bosons yields

LC = −GF√2

JCCμ JCC † μ.

Fig. B.2 Point-like approximation to the propagators of massive vector bosons.

(a) (b)

Fig. B.3 The relation between diagrams of the Standard Model and of the Fermi theory.


The effective coupling GF , Fermi constant, has dimension m−2; it determines thestrength of the four-fermion interaction. This coupling is related to the fundamentalparameters of the Standard Model as follows (see Fig. B.2)

GF ≡ g2

4√

2M2W

.

Its numerical value is

GF = 1.17 · 10−5 GeV−2.

Note that similar procedure of integrating out the Higgs boson also leads to effectivefour-fermion interaction of the form ψmψm · ψnψn. However, the effective couplingshere are proportional to the corresponding fermion Yukawa couplings, which aresmall compared to the weak gauge coupling.12 Therefore, the impact of this effectiveinteraction on processes at low energies can be neglected.

B.6 Peculiarities of Strong Interactions

Although mediators of strong interactions, gluons, are massless, strong interactionsare also described by effective theory at low energies (more accurately, at smallmomentum transfers Q). In contrast to the Fermi theory, the reason is that QCD isin strong coupling regime at energies of order ΛQCD 200 MeV, and even somewhathigher.

The energy-dependent (“running”) strong gauge coupling αs(Q) ≡ g2s(Q)/(4π)

increases as energy decreases, and becomes large at Q ∼ ΛQCD (at the one-looplevel it tends to infinity as Q → ΛQCD). Thus, at energy E ∼ ΛQCD, not only theperturbation series in gauge coupling blows up, but the very description in termsof quarks and gluons loses any sense.

This feature of the theory is fully consistent with the fact that quarks and gluonsare not observed in free state. They are bound inside colorless particles — hadrons(mesons and baryons) — and begin to play an independent role only in processeswith characteristic momentum transfer higher than ΛQCD. This phenomenon gotthe name confinement of quarks and gluons. It is clear that the characteristic sizeof light hadrons — regions where u-, d- and s-quarks and gluons are confined — isprecisely Λ−1

QCD.At low energies, interactions of the lightest hadrons (protons, neutrons, pions,

kaons, etc.) between themselves and with leptons and photons are well describedin the framework of the chiral perturbation theory. To describe heavier hadronsand strong processes in the intermediate energy region E � ΛQCD, various phe-nomenological approaches are used. A number of variables characterizing QCD inthe strong coupling regime can be calculated from the first principles by putting the

12An exception is the t-quark Yukawa coupling, but t-quark does not participate in low energy

processes discussed here.

B.7. The Effective Number of Degrees of Freedom in the Standard Model 409

theory on the lattice and calculating the functional integral numerically by MonteCarlo method. From the point of view of cosmology, of special interest are latticecalculations of QCD transition temperature which give TQCD 170 MeV [191].

B.7 The Effective Number of Degrees of Freedomin the Standard Model

Using the particle spectrum of the Standard Model and accounting for peculiar-ities of strong interactions, it is straightforward to estimate the effective numberof relativistic degrees of freedom g∗ as a function of temperature of the primordialplasma. The simplest way is to use step-function approximation to the temperatureevolution of g∗(T ) near particle thresholds and QCD transition. The result is shownin Fig. B.4.

At temperature T < 100 MeV, only photons, electrons and neutrinos are rela-tivistic, so at 1 MeV� T � 100 MeV the effective number of relativistic degrees offreedom is

g∗(T � 100 MeV) = 2γ +78(4e + 3 · 2ν) =

434

= 10.75.

At T � 100 MeV, additional contribution comes from muon. Above the temperatureof the QCD transition TQCD 170 MeV, the plasma contains light quarks (u, d, s)and gluons. Lattice calculations show that their interactions do not affect dramati-cally the thermodynamic quantities such as energy density or pressure. Therefore,at T ∼ TQCD the effective number of relativistic degrees of freedom changes by

Δg(QCD)∗ 8 · 2 +

78· 3 · 3 · 4 = 47.5.

The first term here comes from gluons (massless vectors in eight color states), thesecond one is due to u-, d-, s-quarks and antiquarks, each in three color states. Steps

0.001 0.01 0.1 1 10 100

20

40

60

80

100

Fig. B.4 The effective number of degrees of freedom in the primordial plasma as a function of

temperature. Only Standard Model particles are taken into account. The Higgs boson mass is set

equal to mh = 120 GeV.


at higher temperatures are associated with heavier particles (see the SM spectrumgiven in the end of Sec. B.1), as shown in Fig. B.4.

At T > 200 GeV the effective number of degrees of freedom, in the frameworkof the Standard Model with one Higgs doublet, is

g∗(T �200 GeV)=2γ + 2 · 3W + 3Z + 1h + 8(c) · 2G +78(3 · 4e + 3 · 2ν + 6 · 3(c) · 4q)

= 106.75 (B.41)

Here subscripts indicate types of particles, superscript (c) refers to the number ofcolor states, the last factors are the numbers of spin states. Note that in the abovecalculation, three polarizations of W - and Z-bosons and one degree of freedom ofthe Higgs boson are taken into account. This is appropriate for the Standard Modelin the Higgs phase. In the phase of unbroken electroweak symmetry, W - and Z-bosons are massless and have 2 polarizations each, while the Higgs field — complexdoublet — describes 4 scalar particles. The number of degrees of freedom is thesame in these two phases, so the result (B.41) is valid in any case.

Appendix C

Neutrino Oscillations

Neutrino oscillations — transitions that change neutrino flavor — is a uniqueto date direct evidence for the incompleteness of the Standard Model of particlephysics, obtained in laboratory experiments rather than from cosmology or astro-physical observations. Neutrino oscillations occur if neutrinos are massive and thereis mixing between lepton generations analogous to quark mixing considered inSec. B.4. Within the Standard Model, the Lagrangian cannot contain renormal-izable gauge-invariant terms which would lead to neutrino masses. In this sense,neutrino oscillations are phenomenon beyond the Standard Model.

Historically, the first data pointed at neutrino oscillations were obtained in mea-surements of solar and atmospheric neutrino fluxes. These discoveries have beenconfirmed afterwards by experiments with neutrinos from nuclear reactors andaccelerators.

C.1 Oscillations and Mixing

In this Section we discuss in general terms the mechanism leading to neutrino oscil-lations. Here we set aside the aspects related to the fact that neutrino has spin 1/2.These aspects are discussed in Sec. C.4.

C.1.1 Vacuum oscillations

In extensions of the Standard Model allowing for non-zero neutrino masses, therecan occur oscillations between neutrinos of different types (flavors). Namely, neu-trinos are produced in weak processes in full compliance with the Standard Model.1

However, in the basis of the Standard Model fields where all other particle massmatrices are diagonal, and lepton gauge interactions are diagonal in generations too(gauge basis), neutrinos oscillate: the Hamiltonian describing their free propagationis non-diagonal.

1In this Appendix we are interested in processes at not too high energies, for which this is certainly

the case.

411

412 Neutrino Oscillations

The situation here is quite analogous to quark mixing. In the basis where thegauge interactions of quarks are diagonal (gauge basis), the quark mass matrix isnon-diagonal. Vice versa, in the basis where quark mass matrix is diagonal (massbasis) gauge couplings are non-diagonal. In the case of quarks it is convenient towork exclusively in the mass basis. When studying neutrino oscillations, one usesboth gauge and mass basis.

We assume here that there are three types of neutrinos in Nature. Electron,muon, and τ -neutrino are the states which are created together with respectivecharged antileptons in the two-particle decays of W+-bosons (either real or virtual),W+ → e+νe, W+ → μ+νμ, W+ → τ+ντ . Since the creation and detection ofneutrino occur via weak interactions, it is the states |νe〉, |νμ〉 and |ντ 〉 that areobservable. They form the basis vectors of the gauge basis. In literature on neutrinooscillations, the gauge basis is often called “flavor basis”, and the states |νe〉, |νμ〉and |ντ 〉 are called “flavor states”. We will use the latter terminology. The basisvectors of the flavor basis |να〉, α = e, μ, τ are related to the basis vectors of themass basis |νi〉, i = 1, 2, 3 by unitary transformation traditionally written in theform

|νi〉 = Uαi|να〉 (C.1)

(summation over repeated indices is assumed). Here the unitary matrix Uαi is themixing matrix in the neutrino sector, dubbed Pontecorvo–Maki–Nakagawa–Sakata(PMNS) matrix. The states |νi〉 are eigenstates of the free Hamiltonian, i.e., theyhave definite masses mi. The inverse transformation from the mass basis to theflavor basis reads

|να〉 = (U †)iα|νi〉 ≡ U∗αi|νi〉. (C.2)

With the definition (C.1), the neutrino mass matrix in the flavor basis has simpleform,

Mαβ = 〈να|M |νβ〉 = (UM (m)U †)αβ , (C.3)

where M (m) is diagonal mass matrix in the mass basis,

M(m)ij = miδij . (C.4)

The neutrino free evolution in the rest frame is determined by eigenvalues of themass matrix,

|νj(t)〉 = e−imj t|νj(0)〉. (C.5)

Suppose that at time t = 0 there is a pure flavor state — for instance, electronneutrino |νe〉 produced in the decay of (possibly virtual) W+. Then at time t, othercomponents of the state vector in the flavor basis also become non-zero. This impliesnon-vanishing probability of detecting muon or τ -neutrino at that time.

For practical applications, one needs to calculate the transition probability να →νβ in the laboratory frame at a distance L from the source of neutrino να. Thenthe formula (C.5) for neutrino evolution in the mass basis has to be generalized,

|νj(t, L)〉 = e−i(Ejt−pjL)|νj(0)〉,

C.1. Oscillations and Mixing 413

where pj and Ej are neutrino momentum and energy. Realistically, neutrinos areultra-relativistic. Let us assume that neutrino energy is fixed,2 and write in the ultra-relativistic case pj =

√E2 − m2

j = E − m2j/2E. Omitting phase factor common to

all neutrino species, we find that the evolution of states in the mass basis in termsof the traveled distance is given by

|νj(L)〉 = e−im2

j2E L|νj(0)〉.

Note that this evolution corresponds to the effective Hamiltonian

Heff =M2

2E, (C.6)

where M is the neutrino mass matrix, which takes the forms (C.3) and (C.4) in themass and flavor basis, respectively. As follows from (C.1), the transition amplitudeof neutrino να to neutrino νβ is equal to

A(α → β) =∑

j

〈νβ |νj(L)〉〈νj(0)|να〉 =∑

j

〈νβ |νj〉e−im2

j2E L〈νj |να〉

=∑

j

Uβje−im2

j2E LU∗

αj (C.7)

This formula enables one to calculate the probability of transition between twostates of the flavor basis after traveling the distance L:

P (να → νβ) = |A(α → β)|2

= δαβ − 4∑j>i

Re[U∗αjUβjUαiU

∗βi] sin

2

(Δm2

ji

4EL

)(C.8)

+ 2∑j>i

Im[U∗αjUβjUαiU

∗βi] sin

(Δm2

ji

2EL

),

where

Δm2ji ≡ m2

j − m2i .

The expression (C.9) describes oscillations with amplitude determined by the neu-trino mixing matrix and the oscillation lengths depending on the difference of neu-trino masses squared and energy. Note that in realistic situations the oscillationpattern may be washed out if the source has large spatial size and/or averaging isperformed over a certain interval of neutrino energies.

2There is a discussion in literature of delicate issues like “are neutrino states eigenstates of the

energy operator P0 or the momentum operator P ?” The answers to questions of this sort are

important for correct description of the oscillations of not too fast neutrino, as well as for studying

the applicability limits of the oscillation picture (it is obviously not valid at large distances from

the source where neutrinos of different masses come in substantially different times). We do not

enter this discussion here.


It is clear that along with the oscillations of neutrinos there should be oscilla-tions of antineutrinos. The latter have indeed been observed experimentally. Anti-neutrino oscillations are also described within the above formalism. CPT-theorem(see Sec. B.3) gives the following relation between the probabilities of neutrino andantineutrino transitions,

P (να → νβ) = P (νβ → να). (C.9)

The transition probability of νβ → να is equal to the transition probability ofνα → νβ calculated with complex conjugate neutrino mixing matrix (see (C.9)).Hence, the relation (C.9) leads to equality

P (να → νβ ; U) = P (να → νβ ; U∗).

This equality implies that neutrino and antineutrino oscillation probabilities maybe different only if the matrix U is complex (see Eq. (C.9)). This would mean CP-violation in lepton sector. Note that non-trivial CP-phase is possible only if thenumber of neutrino species exceeds two: in the case of two types of neutrinos, themixing matrix U can be made real by redefining the fields (see below).

An important example is the case of oscillations between two types of neutrinos.In this case 2×2 unitary matrix Uαi, i, α = 1, 2 is determined by 4 real parameters.Three of the four are unphysical: one can get rid of them by phase rotations ofneutrino and charged lepton fields. After that the neutrino mixing matrix is

Uα,i =

(cos θ sin θ

− sin θ cos θ

). (C.10)

It depends on one parameter, the mixing angle θ.

Problem C.1. Prove the above statement. Hint: See discussion on CKM matrix inSec. B.4.

In the case of two-neutrino oscillations, the formula (C.9) gets simplified,

P (να → νβ) = δαβ + (−1)δαβ sin2 2θ sin2

(Δm2

4EL

). (C.11)

In other words, the probability of transition of neutrino να to another type ofneutrino νβ is equal to

P (να → νβ �=α) = sin2 2θ · sin2

(Δm2

4EL

), (C.12)

while the survival probability of neutrino να is

P (να → να) = 1 − P (να → νβ �=α) = 1 − sin2 2θ · sin2

(Δm2

4EL

). (C.13)

The mixing angle θ determines the oscillation amplitude, A = sin2 2θ. The oscil-lation length is

Losc =4πE

Δm2= (2.5 km) · E

GeVeV2

Δm2. (C.14)

At this distance the neutrino να returns to its original state, while the maxima ofthe oscillation probability are at distances Lk = Losc(1/2 + k), k = 0, 1, 2, . . . .


C.1.2 Three-neutrino oscillations in special cases

As we discuss later, there is a hierarchy between the differences of neutrino massessquared,

|Δm231| � Δm2

21. (C.15)

Hereafter we stick to the following convention concerning the numbering of the masseigenstates: the mass m3 differs significantly from m1 and m2; the masses m1 andm2 are close to each other, and

m2 > m1.

The property (C.15) suggests either normal or inverted mass hierarchy, as illustratedin Fig. C.1. Let us show that due to this property, the formulas describing theoscillations between the three types of neutrinos, are similar in two special cases tothose of two-neutrino oscillations. These cases are, in fact, of great interest, sincethe first of them often occurs for accelerator and reactor neutrinos, while the secondone is relevant for solar neutrinos.

Let us begin with the case when the energy E and the distance between theneutrino production and detection are such that

Δm221

2EL � 1. (C.16)

Then the first oscillating term in (C.9) is expressed as∑j>i

U∗αjUβjUαiU

∗βi · sin2

(Δm2

ji

4EL

)

= U∗α3Uβ3(Uα1U

∗β1 + Uα2U

∗β2) · sin2

(Δm2

31

4EL

). (C.17)

Let us now take into account the unitarity condition

(UU †)αβ ≡∑

i

UαiU∗βi = δαβ (C.18)

and write the expression (C.17) in the form

U∗α3Uβ3(δαβ − Uα3U

∗β3) · sin2

(Δm2

31

4EL

). (C.19)

Fig. C.1 Normal (a) and inverted (b) hierarchies of neutrino masses.


The latter formula is valid for all α and β and contains only |Uα3|2 and |Uβ3|2.Similar calculation shows that the last term in (C.9) is equal to zero. These resultsimply, in particular, that CP-violating effects are strongly suppressed in the regime(C.16) (they vanish in the limit Δm2

212E L → 0). The regime (C.16) is indeed realized,

as a rule, in terrestrial experiments; because of this fact, and also because of thesmallness of the angle θ13 (see below), the observation of CP-violation in neutrinooscillations is extremely difficult.

Equations (C.9) and (C.19) give for the survival probability

P (να → να) = 1 − sin2 2θeff sin2

(Δm2

31

4EL

), (C.20)

where by definition

sin2 θeff = |Uα3|2.

The formula (C.20) is analogous to the expression (C.13) valid for two-neutrinooscillations. The probability of appearance of neutrino of type β �= α is

P (να → νβ) =|Uβ3|2∑

β′ �=α |Uβ′3|2 sin2 2θeff sin2

(Δm2

31

4EL

). (C.21)

It differs from (C.12) by the first factor which accounts for admixtures of two stateswith β �= α in the mass eigenstate ν3.

We now turn to the second special case. It occurs when the neutrino productionregion is sufficiently large and/or neutrino have fairly large energy spread, so thatthe oscillations with phases proportional to Δm2

31 and Δm232 get averaged,⟨

sin2

(Δm2

31

4EL

)⟩=

12,

⟨sin(

Δm231

2EL

)⟩= 0.

The quantity of interest for applications is the survival probability P (να → να).Again using the unitarity condition (C.18), it can be represented as

P (να → να) = 1 − 2|Uα3|2(1 − |Uα3|2) − 4|Uα2|2|Uα1|2 sin2

(Δm2

21

4EL

).

We introduce the mixing angle θ′eff by the relations

cos2 θ′eff =|Uα1|2

|Uα1|2 + |Uα2|2 ≡ |Uα1|21 − |Uα3|2 , sin2 θ′eff =

|Uα2|2|Uα1|2 + |Uα2|2

As a result, we obtain finally

P (να → να) = |Uα3|4 + (1 − |Uα3|2)2[1 − sin2 2θ′eff sin2

(Δm2

21

4EL

)]. (C.22)


In the case of small admixture of neutrino να in the mass eigenstate ν3, i.e., when|Uα3|2 � 1, the latter formula also transforms into (C.13).

C.1.3 Mikheev–Smirnov–Wolfenstein effect

The formulas presented so far are valid for vacuum neutrino oscillations, and,more generally, when the influence of medium where neutrinos propagate is negli-gible. However, matter does affect neutrino properties in some situations. The cor-responding phenomenon is called Mikheev–Smirnov–Wolfenstein effect [192, 193](MSW); it is due to coherent forward neutrino scattering off electrons present inmatter.

MSW effect is accounted for by introducing an effective term in the Hamiltoniandescribing neutrino propagation. Recall that at relatively low energies, chargedcurrent interactions are described by the Fermi theory whose Lagrangian in thelepton sector is

LCC = −GF√2

νeγμ(1 − γ5)e · eγμ(1 − γ5)νe = −2

√2GF νeγ

μe · eγμνe, (C.23)

(see Sec. B.5), where in the last equality we made use of the fact that neutrinosare left fermions, so that 1

2 (1 − γ5)νe = νe, see Eq. (B.4). For matter with electronnumber density ne we have

〈〈ekγ0klel〉〉 = 〈〈e†e〉〉 = ne, (C.24)

where double brackets denote matter average and k, l are spinor indices. Assumingthat electric currents are negligible (this is certainly the case for non-relativisticmatter), we write

〈〈ekγiklel〉〉 = 0. (C.25)

Given that the operators ek and el anticommute, we obtain from (C.24) and (C.25)

〈〈ekel〉〉 = −14γ0

kl · ne

(in the representation where γ0 is symmetric). Averaging the Lagrangian (C.23)over matter, we obtain the contribution to the effective Lagrangian describing thepropagation of electron neutrino,

Leff = −2√

2GF νeγμ〈〈ee〉〉γμνe = 2

√2GF ne

14νeγ

μγ0γμνe

= −√2GF neνeγ

0νe.

Hence, we conclude that matter effect is accounted for by the following substitutionin the Dirac operator iγμ∂μ,

iγ0∂0 → iγ0∂0 −√

2GF neγ0,


i.e., the operator i∂0 is replaced by

i∂0 − V,

where

V =√

2GF ne (C.26)

is the matter contribution to the effective Hamiltonian. We emphasize that thiscontribution exists for electron neutrino only, as muons and τ -leptons are absent inmatter.

The last statement is not entirely correct. The effective four-fermion Lagrangianhas the neutral current terms. The relevant structure is (see Sec. B.5)∑

α

eγμe · ναγμνα,

where summation runs over all neutrino types. These terms lead to new contri-bution to the neutrino effective Hamiltonian which has the form analogous to (C.26).However, this contribution is now the same for all types of neutrinos. The latterproperty means that in the basis |να〉 as well as in any other basis, this contributionis proportional to the unit matrix. Hence, it does not affect neutrino oscillations,leading only to additional time-dependent overall phase in the state vector. Thereis no need to consider this contribution in what follows.

Thus, the effective Hamiltonian describing neutrino propagation in matter isdifferent from (C.6),

Heff (L) =M2

2E+ V (L). (C.27)

The effective potential operator V has the only non-zero matrix element in theflavor basis,

V (L)αβ = V (L)δeαδeβ ,

where V (L) =√

2GF ne(L) and ne(L) is the electron density at distance L fromneutrino source.

The matter effect on neutrino propagation leads to a number of important andinteresting phenomena. One of them is discussed in Sec. C.2.1. Here we make onesimple observation. Namely, even in the two-neutrino case, the oscillation probabil-ities for neutrino and antineutrino are different in matter. The physical reason isthat there are electrons in matter and no positrons. Hence, the presence of matterexplicitly violates CP. At more formal level, the CP-transformation converts theelectron density operator ne into (−ne), so the matter contribution in the effectiveHamiltonian for antineutrino differs by sign from (C.26). This fact is responsible,in particular, for additional difficulty in search for CP-violation in neutrino oscil-lations: neutrino beams produced by accelerators will pass through the Earth, andonly then will be detected, so one will have to discriminate between the “true”CP-violation (arising due to the phase in the PMNS matrix) and the matter effect.

C.2. Experimental Discoveries 419

C.2 Experimental Discoveries

C.2.1 Solar neutrinos and KamLAND

At the Earth, the major contribution to the cosmic neutrino flux comes from ther-monuclear reactions in the center of the Sun, which are the source of solar energy.We list here the main reactions leading to the neutrino emission by the Sun,

p + p → 2H + e+ + νe (C.28)

p + e + p → 2H + νe (C.29)3He + p → 4He + e+ + νe (C.30)

7Be + e− → 7Li + νe (C.31)8B → 8Be + e+ + νe (C.32)

13N → 13C + e+ + νe

15O → 15N + e+ + νe.

Only electron neutrinos are produced in these reactions. The energies of these neu-trinos range from zero to tens MeV; the energy spectrum of solar neutrinos is shownin Fig. C.2. Neutrino flux and spectrum are reliably calculated within the Standard

Fig. C.2 Solar neutrino spectrum at the Earth [194] in the absence of oscillations. Contributions

of various thermonuclear reactions are shown with estimates of accuracy of calculations. Neutrino

flux from continuum is given in units cm−2s−1MeV−1. Energy ranges accessible for various solar

neutrino detectors are indicated in the upper part.


Solar Model [194, 195] (SSM); validity of this model has been confirmed by bothmeasurements of the solar luminosity and the helioseismological data.

Low-energy neutrinos interact extremely weakly with matter, they pass throughthe Sun and Earth with virtually no absorption or scattering. Despite the largeneutrino flux, their detection is very difficult. The number of events per unit massof a detector is small, so one has to install massive detectors (tens of tons to severaltens of kilotons, depending on the type of detector), collect statistics for many years,and reduce the background by using radioactively pure materials, placing detectorsdeep underground (where the cosmic ray flux is substantially reduced), etc.

Historically, the first experiment designed to measure the solar neutrino fluxwas built in the Homestake mine [196] (USA). It lasted for almost 30 years. Solarneutrinos were captured in the reaction

37Cl + νe → 37Ar + e−. (C.33)

Handful of 37Ar atoms were chemically extracted on regular basis from the 615 tonstarget, and then their number was determined by counting the decays of radioactive37Ar. Experiments of this type are called radiochemical. They measure the inte-grated neutrino flux, weighted by the energy-dependent capture cross section. TheHomestake experiment was sensitive mainly to the boron neutrinos produced inreaction (C.32), with substantial contribution expected from reaction (C.31) andother reactions (but not (C.28)). The measured integrated flux of electron neutrinosΦCl turned out to be smaller than the calculated SSM flux ΦCl

SSM,

ΦCl

ΦClSSM

= 0.34 ± 0.05. (C.34)

This result was the first indication that on the way from the center of the Sun tothe Earth, electron neutrino transforms into neutrinos of other types, which do notparticipate in the reaction (C.33).

Boron neutrino flux in the high-energy part of the spectrum (Eνe � 5 MeV)was then measured by the Kamiokande detector [197], and later by Super-K [198](Kamioka mine, Japan3), neutrino energies were Eνe > 7.0 MeV and Eνe > 5.0 MeVat different stages of the experiments. These detectors used water as detectormaterial; target masses were about 1 kiloton and 22.5 kilotons for Kamiokandeand Super-K, respectively. Neutrino participates in the elastic scattering reaction

ν + e− → ν + e−, (C.35)

which results in the production of relativistic electron, whose Cherenkov radiationwas detected. The measured flux of solar neutrinos was again lower than the SSM

3Hereafter we indicate only the geographical position of the detector. The experiments are per-

formed by collaborations of scientists from various countries; the lists of collaboration members

can be found in the original literature.


Fig. C.3 Neutrino-electron elastic scattering via exchange of W -boson (a) and Z-boson (b).

prediction (Super-K data),

ΦS−K

ΦS−KSSM

= 0.41 ± 0.07. (C.36)

Note that the error here (and to a lesser extent, in (C.34)) is mainly due to SSMuncertainties; the flux ΦS−K itself is measured with much better precision.

Elastic scattering off electron is experienced by both electron neutrino and νμ,ντ (see details in Appendix B, in particular, the Lagrangian (B.15) and discussionin Sec. B.5). νee

− scattering occurs through the exchange of W -boson (chargedcurrents, Fig. C.3(a)) and Z-boson (neutral currents, Fig. C.3(b)). On the otherhand, only Z-boson exchange contributes to νμe and ντ e scattering (Fig. C.3(b)).The Z-boson exchange results in smaller scattering amplitude than that due toW -boson, so the effective neutrino flux detected in the elastic scattering reaction(C.35) is proportional to

Φνeeff ∝ Φνe + 0.15(Φνμ + Φντ ). (C.37)

This property is important for interpreting the results obtained with the detectorSNO, which we discuss later on.

The next solar neutrino experiments were radiochemical; they used the reaction

71Ga + νe → 71Ge + e−,

followed by chemical extraction of 71Ge atoms and counting of their radioactivedecays. These are the SAGE experiment [199] (Baksan Neutrino Observatory,Russia, 60 tons of gallium) and GALLEX/GNO [200] (Gran Sasso Laboratory,Italy, 30 tons of gallium). Unlike in other experiments, the largest contributionto the measured integrated flux comes from neutrinos produced in reaction (C.28),though sizeable contributions are due to neutrinos from the reactions (C.31), (C.32)and others. The integrated flux measured in the gallium experiments (duration ofmeasurements exceeded 10 years) is also significantly lower than the SSM prediction


(note the consistency between the two results obtained, in fact, using somewhat dif-ferent techniques),

ΦGa

ΦGaSSM

= 0.54 ± 0.06 SAGE

ΦGa

ΦGaSSM

= 0.56 ± 0.06 GALLEX/GNO

The key role of these data was that they eliminated hypothetical possibility thatthe observed deficit of boron neutrinos results from some error in SSM, i.e., it hasastrophysical nature. Indeed, in contrast to (C.32), reaction (C.28), most relevantfor gallium experiments, directly determines energy production in the Sun, so thep-p neutrino flux can be deduced in practically model-independent way from thewell-measured luminosity of the Sun (barring very exotic possibilities). Hence, aftergallium experiments, neutrino flavor transition became the only explanation of thesolar neutrino deficit.

Final argument that directly established the fact of transitions of νe to νμ andντ on their way from the center of the Sun, came from the SNO detector [201](Sudbury Neutrino Observatory, Canada). This detector used 1 thousand tons ofheavy water as the detector material.4 Neutrinos were detected in the reaction ofelastic scattering (C.35) as well as in the reactions

νe + 2H → p + p + e− (CC) (C.38)

ν + 2H → p + n + ν (NC) (C.39)

Like Kamiokande and Super-K, SNO detector was sensitive to boron neutrinos ofenergies Eν � 5 MeV. In reaction (C.35), the neutrino flux combination (C.37)was measured; the result was in agreement with Super-K (though it had largerstatistical uncertainty). On the other hand, the charged-current reaction (C.38)exists for electron neutrino only, so electron neutrino flux Φνe was obtained bymeasuring its rate. Finally, the reaction (C.39) is purely neutral-current, so its rateis determined by

ΦNC = Φνe + Φνμ + Φντ . (C.40)

Neutrino fluxes measured in reactions (C.38), (C.39) as compared to the SSMpredictions are

ΦSNOνe

Φνe, SSM= 0.30 ± 0.05, (C.41)

ΦSNONC

ΦNC, SSM= 0.87 ± 0.19. (C.42)

The result (C.42) shows that the Standard Solar Model predicts the emitted fluxof boron neutrinos correctly, while from (C.41) it follows directly that about 2/3

4At later stages 2 tons of salt were added to increase the sensitivity to neutral currents.


of them are converted from νe to νμ and ντ when traveling to the Earth. It is alsoimportant that the results (C.41) and (C.42) are consistent with (C.36), taking intoaccount (C.37).

In fact, the agreement between the experimental data is even better than it mightseem from (C.36), (C.41) and (C.42). As we have already noted, large contributionto errors in (C.36), (C.41), (C.42) is due to the uncertainty in the SSM calculation.By themselves, the experimental data have an error of less than 10%; this is theprecision they agree among themselves. We emphasize that irrespective of SSM, thethree measured combinations of fluxes, Φνe , ΦNC and Φνe

eff have two independentparameters Φνe and Φνμ + Φντ .

The fundamental result of electron neutrino oscillations has been confirmed byKamLAND experiment [202] (Kamioka mine, Japan). KamLAND detector contains1 thousand tons of liquid scintillator and detects antineutrinos produced in nuclearreactions at Japanese nuclear power plants. Distances to them range from 70 kmto 250 km and more, so that the effective baseline is about 180 km, in contrastto earlier reactor experiments with much shorter baseline. KamLAND observedthe deficit of electron antineutrinos as compared to the value calculated under no-oscillation hypothesis,

ΦKamLAND

Φno osc= 0.66 ± 0.06.

Thus, electron antineutrino of energies E 3 − 6 MeV (the range relevant forKamLAND) experiences transition into other types already at distance of about100 km.

To describe the solar neutrino data and KamLAND results at the present level ofexperimental accuracy, it is sufficient to use the two-neutrino picture of oscillationsbetween electron neutrino νe and some linear combination ν of muon neutrino andτ -neutrino. This is the second special case studied in Sec. C.1.2: the relevant masssquared difference Δm2

sol ≡ Δm221 is the smallest one, and the PMNS matrix indeed

has |Ue3|2 � 1, see Sec. C.2.3. In the two-neutrino picture, the data are describedby the following parameters of oscillations (precise ranges are given in Sec. C.3)

Δm2sol 10−4 eV2, (C.43)

θsol 35◦. (C.44)

Notably, of great importance to solar neutrinos is MSW effect. In the two-neutrino approximation, the effective Hamiltonian in the flavor basis (νe, ν) hasthe form

H(L) =Δm2

sol

4E

(− cos 2θsol sin 2θsol

sin 2θsol cos 2θsol

)+ V (L)

(1 0

0 0

), (C.45)

where θsol is the vacuum mixing angle, so that the mass eigenstates are

|ν2〉 = |νe〉 sin θsol + |ν〉 cos θsol, |ν1〉 = |νe〉 cos θsol − |ν〉 sin θsol.

Recall that in vacuum, by definition, the heavier state is |ν2〉.


The electron density is ne = 6 ·1025 cm−3 in the center of the Sun and decreaseswith the distance from the center. Hence, the estimate for the maximum value ofthe potential V is

Vmax = V (L = 0) 8 · 10−12 eV.

This implies that the relationΔm2

sol

4E∼ Vmax

holds at E ∼ 3 MeV. For neutrino of significantly lower energies (e.g., pp-neutrino)the matter effects are negligible, and one can use (C.22) with |Ue3|2 � 1. At E �3 MeV, on the contrary, the effect of solar matter is important.

As an example, consider neutrino produced in 8B decay. Its characteristic energyis 4 – 10 MeV. In this case, matter term V dominates in the center of the Sunover the neutrino mass term in the Hamiltonian. Let |νi(L)〉 be eigenvectors of thematrix (C.45) at given distance L from the solar center, and |ν2(L)〉 refers to largereigenvalue. Since V > 0, see (C.26), |ν2〉 coincides with |νe〉 in the center. Thus, thedecay of 8B produces the eigenstate |ν2〉.

Let us consider further evolution of this state, making use of the adiabaticapproximation. Recall that in this approximation, quantum system is always stuckat one and the same energy level. In our case, this means that neutrino is alwaysin the state |ν2(L)〉. So, on the solar surface it is in the state which coincides withthe state |ν2〉 in vacuum. This is still mass eigenstate, therefore it does not oscillateduring the further propagation in vacuum.5 Thus, the probability of detection ofelectron neutrino at the Earth is

P (νe → νe) = |〈νe|ν2〉|2 = sin2 θsol. (C.46)

Equation (C.44) then implies that P (νe → νe) < 0.5. We emphasize that theobserved fact that the measured flux of boron νe is smaller than half of the pre-dicted one (see (C.41)) is direct evidence for the MSW effect: in the case of vacuumoscillations, electron neutrino survival probability averaged over energies cannot beless than 50 %, once two-neutrino approximation is valid, see Eq. (C.11).

We note one feature evident from the above analysis. Consider unrealistic caseof small vacuum mixing, | sin θsol| � 1. In that case, the survival probability (C.46)is low. In the case of vacuum oscillations the situation is opposite: Eq. (C.11) givesPνe→νe = 1 − O(sin2 2θsol). Here we have an example of the Mikheev–Smirnovresonance that enhances neutrino transition in matter. The resonance picture is asfollows. In the absence of mixing, vectors |νe〉 and |ν〉 would be the eigenvectors ofthe operator (C.45) with sin 2θsol = 0. In the center of the Sun and in vacuum, |νe〉corresponds to larger and smaller eigenvalue, respectively (the assumption that thelighter neutrino is mainly |νe〉 in vacuum is important here). The evolution of levelswith L would have the form shown in Fig. C.4(a). If small mixing is switched on, the

5Note that in the adiabatic regime, neutrino oscillations never occur: neutrino is constantly in

the eigenstate |ν2〉 of the local effective Hamiltonian.


Fig. C.4 The evolution of levels of neutrino Hamiltonian with distance from the center of

the Sun. (a) No mixing, (b) Small but non-vanishing mixing.

levels no longer intersect, as we know from quantum mechanics, and the evolutionof the levels becomes as shown in Fig. C.4(b). The heavier neutrino in the center ofthe Sun almost coincides with electron neutrino, while it is almost ν in vacuum. Inthe adiabatic evolution, the transitions from level to level do not occur; this explainsthe low survival probability of electron neutrino for small but finite sin θsol.

Problem C.2. Show that for small sin θsol, the eigenvalues of the Hamiltonian(C.45) indeed evolve with L as shown in Fig. C.4(b).

Problem C.3. Consider the (unrealistic) model of the Sun, in which the density offree electrons ne(L) changes linearly with L from its value in the center, 6·1025 cm−3,

to zero at the surface (at L� = 7 ·105 km). In what region of parameters Δm2sol and

sin θsol the evolution of the neutrino state with L is adiabatic? Consider separatelythe cases of weak and strong mixing, | sin θsol| � 1 and | sin θsol| ∼ 1.

Problem C.4. Find an analog of Eq. (C.46) in the case of oscillations betweenthree types of neutrinos, assuming |Ue3|2 � 1.

To end this discussion, we note that the MSW-effect can lead to a number ofother features in neutrino experiments, such as “day-night” effect at high energies:the difference in the measured flux of solar neutrinos at night (when neutrinos passthrough the Earth) and daytime.

Both vacuum regime (E < 3 MeV) and matter-dominated regime (E > 3 MeV)are probed, in one and the same experiment, by Borexino [203]. This is 278-ton scin-tillation detector (Gran Sasso Laboratory, Italy) that makes use of ν-e scattering.It has unique possibility to observe the signal from 862 keV electron neutrinosborn in 7Be electron capture, as well as from boron neutrinos. The measured fluxin beryllium line is about 2/3 of that predicted by SSM with no oscillations; thedeficit of boron neutrinos is also confirmed. Results for both beryllium and boronneutrinos are fully consistent with the prediction of the model with the oscillation


parameters (C.43) and (C.44). For beryllium neutrino, the vacuum oscillations areresponsible for the electron neutrino disappearance, while the MSW effect is at workfor boron neutrino. Borexino found no day-night asymmetry in the signal, which forthe relevant energy range is also in agreement with oscillation parameters (C.43)and (C.44).

C.2.2 Atmospheric neutrinos, K2K and MINOS

Oscillations of muon neutrino were discovered in experiments of another class.They first showed up in measurements of atmospheric neutrino flux withKamiokande [204] and Super-K [205, 206].

Our Galaxy is filled with cosmic rays — charged particles (protons and nuclei)propagating in space. Their interaction with the Earth atmosphere gives rise tosecondary particles. As a result, large number of particles are produced, amongwhich the dominant component is the lightest hadrons, pions (with rather smalladmixture of kaons). Charged pions π± do not reach the Earth surface and decayin the atmosphere, producing muons and muon (anti)neutrinos,

π+ → μ+νμ, π− → μ−νμ. (C.47)

If the energy of the primary particle is not too high, muons, in turn, also decay,again giving rise to neutrinos:

μ+ → e+νeνμ, μ− → e−νeνμ. (C.48)

Neutrinos produced in reactions like (C.47), (C.48), are called atmospheric; neutrinoenergies relevant for oscillations range from hundreds MeV to tens GeV.

Problem C.5. At what energies of primary particle most muons reach the Earthsurface? Assume that the average multiplicity (number of particles produced in acollision) is 10 to 500 at energies of tens GeV to hundreds EeV. Hints: Muonspractically do not interact with the atmosphere; hadron (including pion) mean freepath with respect to inelastic scattering is about 10% of atmospheric depth.

The cosmic ray flux is isotropic, so the flux of atmospheric neutrinos in theabsence of oscillations should be isotropic as well.6 However, the observed flux ofmuon neutrinos and antineutrinos actually depends on the zenith angle (right panelof Fig. C.5). This means that muon neutrinos coming from above and flying just afew kilometers from the production point to the detector, have no time to oscillate;on the other hand, neutrinos coming from below pass through the entire Earth, andhave time to partly transform into other types of neutrinos. At the same time, theeffect of oscillations on the electron neutrino flux is small (left panel of Fig. C.5).

6For neutrinos with energies of several GeV and above, the isotropy gets lost: the flux has a peak

in the horizontal direction due to the fact that horizontal muons travel longer in the atmosphere

and thus have more time to decay. This phenomenon is straightforwardly accounted for.


0

100

200

300

-1 -0.5 0 0.5 1

Num

ber

of E

vent

s Sub-GeV e-likeP > 400 MeV/c

cosθ-1 -0.5 0 0.5 1

cosθ

0

100

200

300

400Sub-GeV μ-likeP > 400 MeV/c

Fig. C.5 Dependence of neutrino fluxes at energies below 1 GeV on zenith angle [206]: the left and

right panel refer to electron and muon neutrinos, respectively, P is the momentum of an outgoing

lepton. Double solid lines show the prediction in the absence of oscillations, single solid lines are

for oscillations with parameters obtained by fitting the data.

The muon neutrino deficit, together with very small excess of electron neutrinos,suggest that there is νμ − ντ oscillation. This result is confirmed by the entire setof data on atmospheric neutrinos, including the measurements of absolute fluxes ofνe, νe and νμ, νμ, neutrino fluxes at energies above 1 GeV, etc.

Muon neutrino oscillations were confirmed by the K2K experiment [207]. Herethe source of muon neutrinos are pions produced by proton beam from the accel-erator of KEK Laboratory in Japan and decaying according to (C.47). These neu-trinos are detected by Super-K. The distance from production to detection is 250 km(the distance between the KEK Laboratory and Kamioka mine), and the neutrinoenergy is 0.5 – 3 GeV. K2K experiment found disappearance of muon neutrinos:their flux at Super-K is smaller than the flux mesured by the “near” neutrinodetector located directly at the KEK Laboratory. The results of K2K experimentare in good agreement with atmospheric neutrino data.

Study of muon neutrino oscillations is the main purpose of another exper-iment[208], where neutrinos produced by the Fermilab accelerator (Batavia, USA)are detected by underground detector MINOS (Minnesota, USA). The results ofthis experiment are in agreement with the atmospheric and K2K data.

The results of experiments briefly reviewed in this Section are also well describedwithin two-neutrino oscillation picture.7 The simplest and most plausible possi-bility is the oscillations of νμ to ντ (otherwise new types of neutrino have to beintroduced). In that case, the oscillation parameters are estimated as follows (moreprecise estimates are given in Sec. C.3)

Δm2atm (2 − 3) · 10−3 eV2, (C.49)

θatm 45◦. (C.50)

7The two-neutrino oscillation approach, generally speaking, is not applicable to atmospheric

neutrinos. In this case, almost complete absence of any distortion of electron neutrino flux is

accidental: it is due to the interplay between nearly maximum mixing (C.50) and the relation

between muon and electron neutrino fluxes produced in the atmosphere, Φνe/Φνμ � 1/2.


Note that θatm = 45◦ corresponds to maximum mixing: the parameter sin2 2θatm

entering (C.11) is equal to 1. Matter effects are of little importance for acceleratorneutrinos, so in that case we are dealing with vacuum oscillations.

Problem C.6. Using the parameters of νμ – ντ system, determine the energy atwhich neutrinos oscillate substantially when crossing the Earth. Do parent muonshave enough time to decay in the atmosphere?

Problem C.7. Estimate what part of νμ disappears (transforms into ντ ) in theK2K experiment.

C.2.3 CHOOZ: limit on |Ue3|The constraint on the admixture of electron neutrino νe in the mass eigenstate ν3

follows from the results of the CHOOZ experiment in France [209]. It uses electronantineutrinos from nuclear reactor. Antineutrino energies are in the several MeVrange, and the distance from the reactor to the detector is about 1 km. The Earthmatter effects are negligible here, and, moreover, inequality (C.16) is satisfied. Thus,this is the first of the special cases discussed in Sec. C.1.2, and the survival proba-bility of electron antineutrino is (see (C.20))

P (νe → νe) = 1 − 4|Ue3|2(1 − |Ue3|2) sin2

(Δm2

31

4EL

).

The baseline corresponds to the maximum probability of neutrino oscillations atenergy 2 – 3 MeV for the mass squared difference Δm2

atm: Eqs. (C.14) and (C.49)imply that Losc/2 1 km precisely at these energies. Were |Ue3| large enough,the disappearance of electron antineutrinos and, most importantly, the distortionof their energy spectrum would have been observed. These effects were not found,and the results of the experiment yield the bound

|Ue3|2 < 0.032.

This is the only element of the PMNS matrix whose value is unknown to date.

C.3 Oscillation Parameters

Figure C.6 (see also Fig. 13.10 on color pages) shows the regions in the parameterspace8 tan2 θsol and Δm2 allowed by solar neutrino experiments and KamLAND.It is seen that all data are consistent with each other in the region (C.43), (C.44).

8Sometimes the data are parameterized by sin2 2θ rather than tan2 θ. In the case of vacuum

oscillations, the former is more natural, see eq. (C.11). As can be seen from (C.46), sin2 2θ is not

an appropriate parameter in cases where matter effects are significant. Indeed, sin 2θ does not

change under the replacement θ → (π/2 − θ), while the probability of oscillations in matter does.

C.3. Oscillation Parameters 429

Fig. C.6 Allowed region of parameter space for oscillations νe ↔ ν obtained from solar neutrino

experiments and KamLAND [210], see Fig. 13.10 for color version.

sin2(2θ)

0.6 0.7 0.8 0.9 11

|Δm

2 | (

10−3

eV

2 )

1.0

1.5

2.0

2.5

3.0

3.5

4.0

MINOS 90%

MINOS 68%

MINOS best oscillation fit

Super–K 90%

Super–K L/E 90%

K2K 90%

Fig. C.7 Allowed region of parameter space for oscillations νμ ↔ ντ [211]: lines encircle areas

allowed by Super-K, K2K and MINOS experiments.

Figure C.7 shows similar data on atmospheric neutrino, K2K and MINOS. Herethe allowed region is around (C.49), (C.50).

Let us mention here the anomaly in the data of LSND experiment [212], whichcould be explained by νμ ↔ νe oscillations with Δm2

LSND ∼ 1 eV2, sin2 2θLSND ∼10−2. Were it confirmed, at least 4 neutrino states would be necessary: the sum of allΔm2 must be zero, and this is impossible with the known values of Δm2

sol, Δm2atm.

In fact, the number of neutrino species would be even larger than 4, since the fourthneutrino with LSND parameters is inconsistent with observations of structures inthe Universe and CMB data. Indeed, according to Eq. (7.23), LSND neutrinoswould give large contribution to dark matter mass density. These neutrinos would


be hot dark matter particles. Such a possibility is inconsistent with cosmologicalobservations.

The LSND anomaly is ruled out, however, by MiniBooNE experiment [213](which has reported its own anomaly, though [214]).

To date, three neutrino mass eigenstates |νi〉, i = 1, 2, 3 are sufficient fordescribing the results of all neutrino oscillation experiments; there is no need tointroduce new neutrino beyond νe, νμ, ντ . The PMNS matrix is conveniently rep-resented as follows,

U =

⎛⎝1 0 0

0 c23 s23

0 −s23 c23

⎞⎠⎛⎝ c13 0 s13eiδ

0 1 0−s13e−iδ 0 c13

⎞⎠⎛⎝ c11 s12 0−s12 c12 0

0 0 1

⎞⎠⎛⎝eiδ1/2 0 0

0 eiδ2/2 00 0 1

⎞⎠

where sij = cos θij , cij = cos θij and θij ∈ [0, π/2], the last factor (phases δ1, δ2) isonly applicable for the Majorana neutrino mass matrix (see Sec. C.4). Except forthe latter phases, the number of parameters in the neutrino mixing matrix is thesame as in the quark mixing matrix (see Sec. B.4). The results shown in Figs. C.6and C.7 correspond to the following absolute values of the mixing matrix elements(3σ C.L.)

|Uαi| =

⎛⎝0.79 − 0.88 0.47 − 0.61 < 0.18

0.19 − 0.52 0.42 − 0.73 0.58 − 0.820.20 − 0.53 0.44 − 0.74 0.56 − 0.81

⎞⎠ .

Current constraints on the mixing angles are [1]

0.82 ≤ sin2 2θ12 ≤ 0.89, 68% C.L. (C.51)

sin2 2θ13 ≤ 0.19, 90% C.L. (C.52)

0.92 ≤ sin2 2θ23, 90% C.L. (C.53)

Note that most of the mixing matrix elements are of the same order, Uαj ∼ 0.5. This“anarchy” in the neutrino mixing matrix distinguishes it from its quark counterpart,the CKM matrix (B.31) exhibiting noticeable hierarchy between its elements.

As we already mentioned, the smallness of the mixing angle θ13 leads to sup-pression of CP-violating effects in the neutrino sector; other sources of suppressionhave been discussed above.

The results shown in Figs. C.6 and C.7 also determine the differences of massessquared. Joint analysis of existing data yields [1]

7.7 · 10−5 eV2 ≤ Δm221 ≡ Δm2

sol ≤ 8.4 · 10−5 eV2, 68% C.L.

1.9 · 10−3 eV2 ≤ |Δm232| ≡ Δm2

atm ≤ 3.0 · 10−3 eV2, 90% C.L.(C.54)

As we discussed above, these values can be obtained with two different hierarchiesin neutrino masses, see Fig. C.1.

Which of these two hierarchies exists in Nature is not yet known. At thesame time, Eq. (C.54) gives lower limits on neutrino masses: at least one of them

C.4. Dirac and Majorana Masses. Sterile Neutrinos 431

must be

m ≥ matm ≡√

Δm2atm 0.05 eV, (C.55)

while another is not less than

msol ≡√

Δm2sol 0.009 eV. (C.56)

Minimal possibility is that

m1 � msol, m2 = msol, m3 = matm, (C.57)

(normal hierarchy without degeneracy), but other options are also widely dis-cussed, including the case of rather heavy neutrinos almost degenerate in masses,m1, m2, m3 � matm.

C.4 Dirac and Majorana Masses. Sterile Neutrinos

There are two different types of fermion masses in 3 + 1 space-time dimensions:Majorana and Dirac. The two Lorentz-invariant mass terms in the Lagrangian forfermion f are

LM

f =mM

2f c

LfL + h.c., (C.58)

LD

f = mDfRfL + h.c., (C.59)

where f c is charge-conjugate fermion field.Recall that 4-component Dirac spinor can be expressed in terms of 2-component

Weyl spinors χL, ξR (see Sec. B.1 of Appendix B). In the Weyl basis of the Diracmatrices

f =(

χL

ξR

).

Hence, the mass terms (C.58) read

LM

f =mM

2χT

L iσ2χL + h.c.,

LD

f = mDξ†RχL + h.c..

The Dirac mass is only possible if the theory contains both left and right fermioncomponents, while the left component (or right component) is sufficient by itselffor the Majorana mass. Electrically charged fermions can only have Dirac masses,otherwise the charge would not be conserved.

The Standard Model contains only left components of neutrinos, so its minimalgeneralization with massive neutrino and without additional fields involves theMajorana mass term,

LM

ν =mαβ

2νc

LανLβ+ h.c., (C.60)


where we used the flavor basis for neutrino fields, and summation over flavor indicesis assumed. The mass matrix of light neutrinos mαβ is symmetric, and it can bediagonalized by transformation m = UT mdiagU , where U is a unitary matrix. Thisis precisely the PMNS matrix in the case of Majorana neutrinos.

Problem C.8. Show that the expression νcLανLβ

is symmetric in α and β. Hint:note that fermion fields anticommute.

Since the Majorana mass term mixes field with its charge-conjugate, the notionsof particle and antiparticle are not quite adequate for Majorana neutrino. Thismeans, in particular, that there is no conserved lepton number: the Majorana massviolates lepton number explicitly. Indeed, the expression (C.60) is not invariantunder phase rotations ν → eiαν, ν → e−iαν. In the case of ultra-relativisticMajorana neutrino, the eigenstates of the Hamiltonian are states with left andright helicities, and up to corrections suppressed by the ratio m/E the left-helicityand right-helicity states coincide with neutrino and antineutrino states of masslessneutrino theory, respectively.

Problem C.9. Prove the above statement. To do this, obtain the analog of the Diracequation in the case of the Majorana mass, find its solution in terms of creationand annihilation operators and compare it with the solution to the Weyl equationfor massless left fermions.

Problem C.10. Show that in the ultra-relativistic case, the off-diagonal Majoranamass (C.60) gives rise to oscillations between states of one and the same helicity(i.e., left-helicity state oscillates into left-helicity state of different flavor, and notinto right-helicity state, and vice versa).

The result of the latter problem implies that oscillations να ↔ νβ , να ↔ νβ , butnot να ↔ νβ , are possible in the case of the Majorana mass term. Here neutrino andantineutrino are understood as left- and right-helicity states, respectively. With thisconvention, all results of massless theory concerning neutrino interactions remainvalid: for example, it is antineutrino (right-handed state) that is predominantlyproduced in the neutron decay, while the admixture of left-handed state (neutrino)is suppressed by powers of mν/Eν .

Mass term (C.60) cannot be obtained from any SU(3)c × SU(2)W × U(1)Y -invariant renormalizable interaction. Giving up renormalizability, one writes downthe interaction of the form

Lint =∑α,β

ξαβ

ΛνLαH · HT Lc

β + h.c., (C.61)

where we introduced dimensionless couplings ξαβ (indices α, β = 1, 2, 3 label gener-ations); Λν is the energy scale of a theory that generalizes the Standard Model athigh energies and leads to non-renormalizable interaction (C.61) at low energies; the


field H is related to H by (B.10). The Higgs field H acquires vacuum expectationvalue (see Appendix B), and the coupling (C.61) gives rise to the mass term

v2

2Λνξαβ νc

ανβ + h.c.,

which coincides with (C.60). Note that the term (C.61) is of the lowest possibleorder in Λ−1

ν among all terms capable of giving masses to neutrinos; this is thereason we picked it up.

For ξαβ ∼ 1, the scale of new interaction must be of order Λν ∼ 1015 GeV toyield neutrino masses of order 10−2 eV. This scale is close to the Grand Unificationscale and about two orders of magnitude smaller than the putative string theoryscale.

Non-renormalizable effective interaction (C.61) can arise from renormalizableinteraction involving new heavy fields, just like the Fermi four-fermion interactionemerges upon integrating out massive vector bosons of the Standard Model. Thesmallness of neutrino masses as compared to the masses of other Standard Modelfermions requires strong hierarchy between the known Yukawa couplings and thenew couplings, y2

SM � ξαβ , and/or between the electroweak scale and the massscale of new heavy fields. In specific models, these hierarchies may have one oranother natural explanation.

One possibility here is so-called see-saw mechanism. To begin with, let us con-sider this mechanism in the case of one type of conventional neutrino ν. This neu-trino is a component of the left lepton doublet L of the Standard Model. Let there beanother left fermion NL, which is a singlet under the Standard Model gauge groupSU(3)c × SU(2)W ×U(1)Y (equivalently, one introduces right fermion Nc

L). In con-

trast to the known fields of the Standard Model, the field NL can have Majoranamass M unrelated to the vacuum expectation value of the Standard Model Higgsfield. It is remarkable that the gauge invariance of the Standard Model allows theYukawa interaction which couples NL and ν to the Standard Model Higgs field H .So, the renormalizable Lagrangian for the fields NL and L includes the terms

L =M

2N c

LNL + yNc

LH†L + h.c., (C.62)

where y is the Yukawa coupling. As a result of electroweak symmetry breaking, thefield H† obtains vacuum expectation value (v/

√2, 0), leading to the mass terms

Lm =M

2N c

LNL + yv√2N c

Lν + h.c. (C.63)

Combining left fermions NL and ν into the column

ψ =(

ν

NL

), (C.64)


we find that the mass term (C.63) can be written as

Lm =12ψcmψ + h.c.,

where the matrix m is9 (0 mD

mD M

)(C.65)

and

mD =yv√

2.

At M � mD, the eigenvalues of the mass matrix (C.65) are (modulo the irrelevantsign and corrections suppressed by mD/M)

mν =m2

D

M=

y2v2

2M, (C.66)

mN = M,

where the smaller eigenvalue (C.66) corresponds to the eigenvector(1

−mD

M

)

(again up to small corrections). It is seen from (C.64) that the main componentof this eigenvector is the ordinary neutrino ν. Thus, as a result of this mechanism,neutrino acquires Majorana mass mν , which is naturally small for M � mD. Notethat at y = 10−6−1 (the range of known Yukawa couplings of the Standard Model),the mass mν ∼ 10−2 eV is obtained at

M ∼ 103 − 1015 GeV,

i.e., the condition M � mD is satisfied indeed.It is worth noting that the result (C.66) can be obtained by integrating out

the heavy field NL. To this end, let us write the equation obtained by varying theaction with respect to NL. With the gradient term iNLγμ∂μNL in the Lagrangianand mass terms (C.63), we obtain

−i∂μNLγμ + MN cL +

yv√2νc = 0.

When the momentum and energy are small compared to M , the first term on theleft hand side is negligible, and the field NL is algebraically expressed through thefield ν,

NL = − yv√2M

ν.

9If instead of left field NL, one uses the right field NcL, then the second term in (C.63) has the

form of the Dirac mass term in which NcL serves as the right component. Hence the notation mD .


We substitute this expression back into the original Lagrangian and thereby obtainthe effective Lagrangian for the field ν. The correction to the kinetic term is negli-gible, and the main effect is the mass term

Lmν = −y2v2

4Mνcν + h.c.

We see that the Majorana neutrino mass is indeed given by (C.66) up to signomitted there. We also note that starting from the original Lagrangian (C.62) andintegrating out the heavy field NL, we obtain the effective Lagrangian of the form(C.61) with Λν = M and ξ = y2.

We now turn to the realistic case of three types of neutrinos. In this case, itis natural to introduce three fields Nα, α = 1, 2, 3 (subscript L is omitted) andgeneralize the Lagrangian (C.62) as follows,

L =12MαβNc

αNβ + yαβNcαH†Lβ + h.c.

Here Mαβ and yαβ are 3×3 matrices (generally speaking, complex) and the matrixMαβ is symmetric. One can always choose the basis of fields Nα in such a way thatthe matrix Mαβ is real and diagonal,

M = diag(M1, M2, M3).

In this basis, the field Nα describes heavy sterile neutrino of a certain mass. Theeasiest way to find the effective mass term for light neutrinos is to integrate outheavy fields Nα, as outlined above. As a result, we obtain the Majorana mass termfor light neutrinos in the form (C.60) with the mass matrix

m = −mDM−1mTD, (C.67)

where

mDαβ =yαβv√

2.

In general, the light neutrino masses and the parameters of the PMNS matrix non-trivially depend on the elements of both the diagonal matrix M and matrix ofYukawa couplings yαβ.

Let us now discuss the possibility that the known neutrinos have the Diracmasses. In this case one adds new light fields νRα, which are the right componentsof neutrinos. Then the Dirac mass term reads

LDν = mαβ νRανLβ + h.c., (C.68)

where the flavor basis is used again. These right components must be neutral(sterile) with respect to the Standard Model gauge group, otherwise they wouldgive contribution, for example, to the total width of Z-boson, whlich is measuredwith high precision and is consistent with the Standard Model prediction.

Since the Dirac mass is invariant under charge conjugation, the notion of leptonnumber makes sense in this theory: the mass term (C.68), like all other terms of the


Standard Model Lagrangian, are invariant under the transformation

να → eiξνα, να → e−iξ να.

Were the matrix mαβ diagonal, there would exist conserved lepton numbers foreach lepton flavor separately. The observed neutrino oscillations show that themass matrix mαβ is not diagonal, hence they reveal violation of individual leptonnumbers.

The mass terms (C.68) can emerge, for example, due to renormalizable Yukawainteraction (cf. (C.62))

L =∑α,β

yαβLαHνRβ + h.c. (C.69)

The Yukawa couplings yαβ have to be extremely small. In a number of theStandard Model extensions (for example, in supersymmetric theories and GUTs),the smallness of the Yukawa couplings is obtained naturally due to the presenceof an intermediate energy scale at which the effective interaction (C.69) emerges.As a result, the Yukawa couplings are suppressed by the ratio (or its power) of theintermediate and the gravitational scales. An illustration is the interaction involvinga new scalar S, which is a singlet under the Standard Model gauge group,

L =S

MPl·∑α,β

YαβLαHνRβ + h.c.,

where the dimensionless couplings Yαβ can be of order unity. If the field S acquiresnonzero vacuum expectation value v � 〈S〉 � MPl, then the effective renormal-izable interaction (C.69) appears at lower energies, with Yukawa couplings of order〈S〉/MPl � 1.

To conclude this Section, we note that one cannot exclude the possibility thatthere are both Majorana and Dirac neutrino mass terms in the Lagrangian, andthat both types of masses are important for describing neutrino properties. Thispossibility, however, does not look very natural, since the mechanisms that lead tothe two different types of mass terms, generally speaking, are different. Hence itis hard to expect that these mechanisms lead to mass parameters which are equalwithin an order of magnitude or so.

C.5 Search for Neutrino Masses

The present direct experimental limits on neutrino masses are [1]:

mνe < 2 eV, (C.70)

mνμ < 0.19 MeV, (C.71)

mντ < 18.2 MeV. (C.72)

C.5. Search for Neutrino Masses 437

These limits are valid regardless of the type of neutrino mass. In the Majoranacase, the constraint on the combination of neutrino masses relevant for neutrinolessdouble-β decay of nuclei (for details see, e.g., Ref. [1]) is stronger:

mν < 0.35 eV.

For comparison, the current limit on the sum of neutrino masses, following fromthe measurements of CMB anisotropy and studies of structures in the Universe, isat the level ∑

i

mνi < 0.2 − 1.0 eV,

depending on which cosmological parameters are fixed from other observations. Wediscuss the cosmological aspects of massive neutrinos in the accompanying book.

It is expected that the sensitivity of direct laboratory experiments to the massof the electron neutrino will soon reach 0.2 − 0.02 eV (depending on the type ofmass). Accuracy of cosmological estimates for the sum of the neutrino masses isalso improving.

Appendix D

Quantum Field Theory atFinite Temperature

In this Appendix we briefly describe the method of calculation of various quan-tities (free energy, effective potential, static Green’s functions) in quantum fieldtheory at finite temperature. We consider the most interesting for cosmology caseof zero chemical potentials, although the overall approach allows for an appropriategeneralization.

We begin with a general comment. It is sometimes useful to treat quantum fieldtheory as quantum mechanics of large but finite number of degrees of freedom.Indeed, field theory can be regularized by introducing spatial lattice of smallbut finite spacing (ultraviolet regularization) and considering the system in a3-dimensional box of finite, albeit large size (infrared regularization). For our pur-poses, time is conveniently treated as continuous variable.1 Then fields φ(x, t) arefunctions of the lattice sites2 and time, φ(x, t) → φ(xn, t), where xn are coordinatesof the lattice site, labeled by a discrete index n = (n1, n2, n3). With this regular-ization, the number of dynamical coordinates φ(xn, t) is large but finite. In this wayfield theory reduces to quantum mechanics.

We use this approach to obtain formal results.3 Namely, we develop finite tem-perature techniques in quantum mechanics, and then merely extend it to quantumfield theory.

D.1 Bosonic Fields: Euclidean Time and Periodic BoundaryConditions

Let us consider quantum-mechanical system with dynamical coordinates q =(q(1), q(2), . . . , q(N)). Here we assume that q are bosonic coordinates, as usual inquantum mechanics. Let this system be at temperature T . As is known from

1In lattice numerical simulations, time is also discretized. This would be inconvenient for us.2Gauge fields are naturally considered as living on links of the lattice, rather than on sites. This

is insignificant for us.3We are not going to discuss subtle points concerning removal of ultraviolet and infrared regu-

larizations. In brief, no new ultraviolet divergencies or infrared pathologies appear at finite tem-

perature as compared to zero temperature theory.

439

440 Quantum Field Theory at Finite Temperature

statistical physics, in thermal equilibrium the expectation values of operators atfixed time are given by

〈O〉T =Tr(e−βHO)

Tr(e−βH), (D.1)

where the operator H is the Hamiltonian of the system, the parameter β is

β =1T

,

and trace is taken over all states of the system. Free energy F is determined by

e−βF = Tr(e−βH), (D.2)

Our first goal is to find a convenient representation for the right hand side of thisequality.

Consider a system with one degree of freedom q and the Hamiltonian

H =p2

2+ V (q). (D.3)

As complete set of states in (D.3) we choose eigenstates of the operator q, i.e., wework in the coordinate representation. Then

e−βH =∫

dq〈q|e−βH |q〉. (D.4)

Here and in what follows we omit the numerical factor in front of the integral, whichleads only to overall shift of the free energy, F → F +const. We are interested in theexpectation values (D.1) in which this pre-factor cancels out, and also in differencesof free energies of different phases, so the shift cancels out as well.

Let us obtain a representation for the right hand side of (D.4) in the form offunctional integral, i.e., path integral in our case (for details see, e.g., Ref. [190]).We write

〈q|e−βH |q〉 = 〈q|∏

i

e−ΔτiH |q〉

= 〈q|(1 − Δτ1 · H)(1 − Δτ2 · H) · · · (1 − Δτn · H)|q〉,where we divided the interval of length β into n small segments of lengthsΔτ1, . . . ,Δτn; we take the limit n → ∞, Δτi → 0 in the end. We insert the unitoperator between each bracket and write∫

dq〈q|e−βH |q〉 =∫ n∏

k=0

dqk δ(q0 − qn) · 〈q0|(1 − Δτ1 · H)|q1〉

× 〈q1|(1 − Δτ2 · H)|q2〉 · · · 〈qn−1|(1 − Δτn · H)|qn〉.(D.5)

D.1. Bosonic Fields: Euclidean Time and Periodic Boundary Conditions 441

Now, we make use of the relations

〈q′|V (q)|q〉 = V (q)δ(q′ − q) =∫

dp

2πV (q)eip(q−q′),

〈q′| p2

2|q〉 =

∫dp

2π

p2

2eip(q−q′).

Omitting the numerical coefficient, we have for each factor in (D.5)

〈qk−1|(1 − Δτk · H)|qk〉 =∫

dpkeipk(qk−qk−1)e−

»p2

k2 +V (qk)

–Δτk

, (D.6)

where we again wrote

1 −(

p2k

2+ V (qk)

)Δτk = e

−»

p2k2 +V (qk)

–Δτk

.

The integral over dpk in (D.6) is Gaussian and can be calculated, as usual, byshifting pk → pk − iqk, where

qk =qk−1 − qk

Δτk.

This gives

〈qk−1|(1 − Δτk · H)|qk〉 = e−[

q2k2 +V (qk)

]Δτk .

Substituting this expression in (D.5) we obtain in the limit n → ∞, Δτi → 0 thepath integral representation for the free energy,

e−βF =∫

q(β)=q(0)

Dq e−S(β)E [q(τ)], (D.7)

where

S(β)E =

∫ β

0

dτ

[q2

2+ V (q)

], (D.8)

and q = dq/dt.Let us explain the notation introduced here. SE is the Euclidean action of the

system with the Hamiltonian (D.1). It is obtained from the original action

S =∫

dt

[12

(dq

dt

)2

− V (q)

]

by formal replacement

t = −iτ, (D.9)

and then considering τ as real. The replacement (D.9) turns S into iSE, so that

eiS → e−SE , (D.10)

Further, in accordance with (D.8) the theory is considered in a finite interval ofEuclidean time τ , whose length equals β ≡ T−1. Finally, the functional integral(D.7) is taken over paths periodic in τ with period β.


The representation (D.7) of the free energy is intuitively clear. The operatore−βH can be regarded as an evolution operator e−iHtβ in the imaginary (Euclidean)time interval tβ = −iβ. In accordance with this, the matrix element

〈qf |e−βH |qi〉is path integral over trajectories in Euclidean time, which start at q = qi and end atq = qf . It is clear from (D.4) that the relevant trajectories are periodic, qi = qf = q,without any other conditions imposed on them.

The above derivation is directly generalized to quantum mechanics of multipledegrees of freedom and, in accordance with what is said in the beginning of thisAppendix, to quantum theory of any bosonic fields. Making use of collective notationφ for all of the bosonic fields, we write the representation for the free energy in aform similar to (D.7),

e−βF =∫

Dφ(x, t) · e−S(β)E [φ(x,t)],

where the integration is performed over field configurations periodic4 in Euclideantime τ with period β, the Euclidean action has the form

S(β)E =

∫ β

0

dτ

∫d3xLE(φ, φ)

and is obtained from the original action by formally replacing τ → −iτ , iS → −SE,as in (D.9), (D.10). In other words, the Euclidean LagrangianLE in the case of gaugetheories with scalar fields is obtained from the original Lagrangian by replacing theMinkowski metric with the Euclidean metric and by changing the signs in front ofthe scalar potential and in front of the Lagrangian of gauge fields. Schematically,

LE =14F a

μνF aμν + Dμφ†Dμφ + V (φ), (D.11)

where summation over 4-dimensional indices μ, ν is performed with the Euclideanmetric.

Problem D.1. Show that the outlined procedure for obtaining the Euclidean actionindeed leads to the expression (D.11), if the original Lagrangian in Minkowski spacehas the form

L = −14ημνηλρF a

μλF aνρ + ημνDμφ†Dνφ − V (φ),

where F aμν is the gauge field strength, φ denotes collectively all scalar fields which

transform according to some (generally speaking, reducible and complex) represen-tation of the gauge group. Hint: First, impose the gauge condition Aa

0 = 0, and thenrestore gauge invariance in the Euclidean formulation.

4In the case of non-Abelian gauge theories, the configurations must be periodic up to “large”

(topologically non-trivial) gauge transformations, see ref. [111]. This subtlety will be insignificant

for us.

D.2. Fermionic Fields: Antiperiodic Boundary Conditions 443

D.2 Fermionic Fields: Antiperiodic Boundary Conditions

In the case of fermions, the functional integral representation for the free energyhas to be derived anew. We restrict ourselves to the case of the action quadratic infermionic fields, although the result will be valid for the general case as well. Moreprecisely, we consider theories where fermionic part of the Lagrangian in Minkowskispace has the form

L = iψγμ∂μψ − ψMψ, (D.12)

where M accounts for fermion mass and interaction with bosonic fields (for example,in electrodynamics, M = m− eγμAμ). There can be several fermionic fields; gener-alization to this case is straightforward. Bosonic fields are considered external andfixed for the time being.

Given that ψ ≡ ψ†γ0, we write the Lagrangian (D.12) in the form

L = iψ†∂0ψ − H, (D.13)

where

H = −iψ†γ0γi∂iψ + ψ†γ0Mψ. (D.14)

As is seen from (D.13), pψ = iψ† is the generalized momentum conjugate to thegeneralized coordinate ψ, and H is the Hamiltonian of the theory.

In contrast to bosonic fields, fermionic fields obey anticommutation relations; atequal times

{ψ(x, t), ψ(x′, t)} ={ψ†(x, t), ψ†(x′, t)

}= 0,{

ψ(x, t), ψ†(x′, t)}

= δ(x − x′).

The latter equality is equivalent to the canonical relation {ψ(x, t), pψ(x′, t)} =iδ(x − x′). If spatial lattice and finite spatial box are introduced, we come toquantum mechanics of operators obeying (in the Schrodinger representation) anti-commutation relations

{ψm, ψn} = {ψ†m, ψ†

n} = 0, {ψm, ψ†n} = δmn,

while the discretization of (D.14) leads to the Hamiltonian of the type

H = ψ†mhmnψn.

Our purpose is to find the functional integral representation for Tr(e−βH) in thistheory.

Let us consider a theory with one fermion operator ψ and its conjugate ψ†. Theysatisfy the following relations

{ψ, ψ} = {ψ†, ψ†} = 0, {ψ, ψ†} = 1.


These coincide with the anticommutation relations for the fermion creation andannihilation operators. For definiteness, we assume that ψ is the creation operator.Then the space of states of the system has two basis vectors |0〉, |1〉, such that

ψ†|0〉 = 0, ψ|0〉 = |1〉,ψ†|1〉 = |0〉, ψ|1〉 = 0.

It is convenient to consider this space of states as space of functions Ψ(ψ) of theanticommuting (Grassmannian) variable ψ, whose fundamental property is nilpo-tency,

ψ · ψ = 0. (D.15)

Let us associate the vector |0〉 with the unit function, Ψ(ψ) = 1, and the vector |1〉with the function Ψ1(ψ) = ψ. Then the linear space with two basis vectors |0〉 and|1〉 is equivalent to the space of functions of the form

Ψ(ψ) = α + βψ,

where α and β are complex numbers. In fact, all functions Ψ(ψ) are of this type.This is easy to see by writing the Taylor expansion in ψ and using (D.15). Theoperators ψ and ψ† act in this space as follows

ψΨ(ψ) = ψΨ(ψ), ψ†Ψ(ψ) =∂

∂ψΨ(ψ).

It is useful to present these formulas in an integral form. We introduce the Berezinintegral; by definition, ∫

dψ = 0,

∫dψ · ψ = 1.

This definition is sufficient for evaluating the integral of any function Ψ(ψ). It isstraightforward to check by direct substitution that the following relations hold:

Ψ(ψ) =∫

dψdψ†e−ψ†(ψ−ψ)Ψ(ψ), (D.16)

ψΨ(ψ) =∫

dψdψ†e−ψ†(ψ−ψ)ψΨ(ψ), (D.17)

ψ†Ψ(ψ) =∫

dψdψ†e−ψ†(ψ−ψ)ψ†Ψ(ψ), (D.18)

ψ†ψΨ(ψ) =∫

dψdψ†e−ψ†(ψ−ψ)ψ†ψΨ(ψ), (D.19)

where all variables and differentials ψ, ψ, ψ†, dψ, dψ† are treated as anticommuting.Now we are ready to write the functional integral for the quantity

(e−βHΨ)(ψ),

where the Hamiltonian is of the form H = cψ†ψ. We proceed in analogy to thebosonic case and write

e−βHΨ = (1 − HΔτ1) · · · (1 − HΔτn) · Ψ.

D.2. Fermionic Fields: Antiperiodic Boundary Conditions 445

Using the formulas (D.16) and (D.19) we obtain

(e−βHΨ)(ψ)

=∫ n∏

k=1

dψkdψ†ke−ψ†

1(ψ−ψ1)−H(ψ1)Δτ1 . . . e−ψ†n(ψn−1−ψn)−H(ψn)ΔτnΨ(ψn).

It is useful to note that, as in the bosonic case, ψk−1 − ψk = ψ(τk) · Δτk at smallΔτk. Hence, in the limit n → ∞, Δτi → 0 we obtain the functional integral repre-sentation,

(e−βHΨ)(ψ) =∫

DψDψ†e−S(β)E Ψ(ψi), (D.20)

where

S(β)E =

∫ β

0

[ψ† ∂ψ

∂τ+ H(ψ†ψ)

]dτ.

Note that the functional integral in (D.20) includes integration over ψi and ψ†i at

“initial time” τ = 0 (with ψi = ψ(τ = 0)), but does not include the integration overψ and ψ† at “final time” τ = β. Just as in the bosonic case, the Euclidean actionSE is obtained from the action in real time

S =∫

dt(iψ†∂tψ − H)

via the formal substitution t → −iτ , iS → −SE.It remains to find the boundary conditions leading to Tr(e−βH). Let us write

(e−βHΨ)(ψ) =∫

dψiU(ψ, ψi)Ψ(ψi), (D.21)

where

U(ψ, ψi) =∫

D′ψDψ†e−S(β)E ,

and prime means that the integration is not performed over the initial “value”ψ(τ = 0) = ψi (this integration is left for (D.21)). The general expression for thefunction of two Grassmannian variables reads

U(ψ, ψi) = u0 + u1ψ + u−1ψi + u2ψψi.

Then we obtain∫dψiU(ψ, ψi) · 1 = u−1 − u2ψ,

∫dψiU(ψ, ψi) · ψi = u0 − u1ψ.

In the operator language this means

e−βH |0〉 = u−1|0〉 − u2|1〉, e−βH |1〉 = u0|0〉 − u1|1〉.Consequently,

Tr(e−βH) = u−1 − u1 =∫

dψiU(−ψi, ψi).


So, we get finally

Tr(e−βH) =∫

ψ(β)=−ψ(0)

DψDψ†e−S(β)E ,

i.e., the integration is performed over Grassmannian trajectories with antiperiodicboundary conditions on ψ(τ) in the interval (0, β). The variable ψ†(τ) can alsobe considered antiperiodic: in the interval (0, β) any function ψ†(τ) is representedas a sum of periodic and antiperiodic functions, and the periodic part does notcontribute to S

(β)E , since it is convoluted with antiperiodic ψ(τ).

The above derivation is fully applicable to the systems of many fermionic degreesof freedom, and, consequently, to the theory of fermionic fields. Here an importantrole is played by the relations (D.17) and (D.18) which we have not used so far.Although our derivation is given for the case where bosonic fields are external, itis easy to understand that this is actually not a limitation: in a theory with bothbosonic and fermionic fields, the integral over fermions can be considered as internal(there bosonic fields are fixed), and then the integral over bosonic fields is evaluated.Thus, the free energy is given by the integral

e−βF ≡ Z =∫

DφDψ†Dψe−S(β)E , (D.22)

where S(β)E is the Euclidean action in the interval (0, β), and bosonic fields φ

(fermionic fields ψ, ψ†) satisfy periodic (antiperiodic) boundary conditions there.To conclude this Section, we note that the formalism can be generalized to

the case of non-zero chemical potential. In general, the chemical potential is intro-duced when the medium has non-zero density of conserved (at given temperature)quantum number. In the cosmological context, the baryon and lepton numbers areof the greatest interest. The corresponding operators have structure like

Q =∫

d3xψγ0ψ.

The nonzero average density n = ψγ0ψ is taken into account by the additional term(−μQ) in the effective Hamiltonian, i.e.,

Heff = H − μQ, (D.23)

where μ stands for the chemical potential. Within the formalism considered, thisleads to the following change of the Euclidean action,

S(β)E → S

(β)E − μ

∫ β

0

dτ

∫d3xψγ0ψ. (D.24)

The partition function is then again given by (D.22). In this case, the quantityF (T, μ) is called the Grand potential or Landau potential.

D.3. Perturbation Theory 447

D.3 Perturbation Theory

The approach described in Secs. D.1, D.2 is useful for calculating the free energy,the effective potential Veff (T, φ) introduced in Chapter 10, as well as the staticGreen’s functions. The latter characterize the response of the system to time-independent external probes. For example, suppose that static source J(x) is intro-duced into a theory of quantum field φ. This implies the following modification ofthe Hamiltonian

H → H −∫

J(x)φ(x)d3x ≡ HJ ,

where φ(x) is the Schrodinger field. In the presence of this source, the partitionfunction

ZJ = e−βFJ = Tr(e−βHJ ),

is represented in the form of functional integral (D.22), and its expansion in J hasthe static Green’s functions as coefficients,

G(x1, . . . ,xn) = Z−1

∫Dφe−S(β)[φ]

× 1β

∫ β

0

dτ1φ(x1, τ1) × · · · × 1β

∫ β

0

dτnφ(xn, τn) (D.25)

(the normalization by partition function Z without the source and factors β−1 areintroduced for convenience); the subscript E in the notation of the Euclidean actionhere and below is omitted.

A simple example is the field expectation value in the presence of the staticsource,

〈φ(x)〉J =Tr(e−βHJ φ(x))

Tr(e−βHJ ).

To the leading order in J it is equal to (assuming 〈φ〉J=0 = 0)

〈φ(x)〉J =∫

G(x,y)J(y)dy.

The difference between G(x,y) and the free propagator at zero temperature corre-sponds to the modification of the Coulomb or Yukawa law in the presence of themedium.

Note that the static correlation functions (D.25) do not represent all interestingclasses of Green’s functions. Computational technique for correlators at differenttimes (for instance, the Keldysh method) is quite complicated and not requiredhere.

To generalize (D.25) we consider the Euclidean Green’s functions

G(x1, τ1; . . . ;xn, τn) = Z−1

∫Dφe−S(β)[Φ]φ(x1, τ1) . . . φ(xn, τn), (D.26)


where φ denotes collectively all fields in the theory, the integration is performedover the bosonic and fermionic fields, periodic and antiperiodic in the interval [0, β],respectively. The normalization factor Z is given by a similar integral, see (D.22).

Proceeding from the representation (D.26), it is straightforward to construct thediagram technique for perturbative calculations, similar to the Feynman techniquein theories at zero temperature. As usual, we first consider free theories with sources.In the scalar and fermion cases, the expressions for the quadratic actions are

S(β)ϕ =

∫ β

0

dτ

∫d3x

[12∂μϕ∂μϕ +

m2

2ϕ2 − Jϕϕ

], (D.27)

S(β)ψ =

∫ β

0

dτ

∫d3x

[ψγμ∂μψ + mψψ − Jψψ − ψJψ

]. (D.28)

Here x0 ≡ τ , summation over indices is performed with the Euclidean metric, andthe Euclidean γ-matrices are Hermitean and obey the rule {γμ, γν} = δμν .

Problem D.2. Check that the Euclidean action of the free Dirac field with a sourcehas the form (D.28).

As the field ϕ is periodic in τ with period β, so is the source Jϕ(x, τ). And viceversa, Jψ and Jψ are antiperiodic.

The functional integral (D.22) with the quadratic action and source (D.27) isGaussian and is calculated by shifting ϕ(x, τ) → ϕ(x, τ) + ϕc(x, τ), where thefunction ϕc(x, τ) is the solution to the equation

−∂μ∂μϕc + m2ϕc = Jϕ. (D.29)

According to our prescription, ϕc has to be periodic in τ with period β. One writes

ϕc(x, τ) =∫ β

0

dτ ′∫

d3x′D(x, τ ;x′, τ ′)Jϕ(x′, τ ′),

where D is the free propagator at finite temperature. Given the periodicity of Jϕ,it is straightforward to see that Eq. (D.29) and periodic boundary conditions areobeyed if the free propagator has the form

D(x, τ ;x′, τ ′) =1

(2π)3β

∑n∈Z

∫d3p

eip(x−x′)+iωn(τ−τ ′)

p2 + ω2n + m2

,

where

ωn =2πn

β, n = 0,±1,±2, . . . (D.30)

are the Matsubara frequencies of bosonic fields. In contrast to the field theory atzero temperature, the frequencies form a discrete set.

The free propagator for the vector field is constructed similarly and is also asum over frequencies (D.30).

In the case of fermionic field, the analog of Eq. (D.29) is

γμ∂μψc + mψc = Jψ.

D.4. One-Loop Effective Potential 449

Both Jψ(x, τ) and ψc(x, τ) are antiperiodic in τ with period β. The solution is

ψc(x, τ) =∫ β

0

dτ ′∫

d3x′S(x, τ ;x′, τ ′)Jψ(x′, τ ′),

where the free propagator is given by

S(x, τ ;x′, τ ′)

=1

(2π)3β

∑n′=± 1

2 ,± 32 ,...

∫d3p

−iγ0ωn′ − iγp + m

p2 + ω2n′ + m2

eip(x−x′)+iω′n(τ−τ ′).

(D.31)

Here

ωn′ =2πn′

β, n′ = ±1

2,±3

2, . . . (D.32)

are the Matsubara frequencies for fermions. The fact that n′ runs over half-integervalues is obviously due to the antiperiodicity of fermionic fields.

Further development of the diagram technique proceeds along the same lines asin the (Euclidean) field theory at zero temperature. The expressions for interactionvertices in theories at T = 0 and T �= 0 coincide. Since integration over dτ inthe action runs from 0 to β, instead of the δ-function of energy conservation thefollowing factor appears at every vertex,

βδ(∑

ω)

, (D.33)

where∑

ω is a sum over the Matsubara frequencies of all lines (considered asincoming), and the function δ(

∑ω) equals 1 if

∑ω = 0 and zero in all other cases.

Note that turning on chemical potential, according to (D.24), leads to thereplacement ∂0 → ∂0−μ in the action (D.28). The corresponding change in the freefermion propagator (D.31) is the replacement

ωn′ → ωn′ + iμ

in the pre-exponential factor of the integrand in (D.31), while the Matsubara fre-quencies remain intact in the exponential factor exp[iωn′(τ − τ ′)].

D.4 One-Loop Effective Potential

As the first example of the use of the technique described in Secs. D.1, D.2, let usre-derive the expression (10.16) for the first temperature-dependent correction tothe effective potential. Our aim is to calculate the free energy as function of thehomogeneous background scalar field φ, neglecting interactions between particles inthe medium. Our starting point is the formula (D.22) for the free energy. In ourapproximation, the action S(β) is quadratic in quantum fields, and the background


Fig. D.1 Some diagrams contributing to the first-order correction to the effective potential.

field φ enters into it only through the particle masses. The integral (D.22) is fac-torized into a product of integrals over different fields, so the free energy has thestructure (10.14) indeed.

Note that in perturbation theory, the higher-order corrections to the effectivepotential are given by diagrams without external lines, where masses and verticesdepend on the background field φ. The simplest of these diagrams are schematicallyshown in Fig. D.1. These diagrams begin with two loops. Therefore, within thedescribed formalism, the zeroth order is naturally called one-loop approximation.

Turning back to the one-loop approximation, consider, for example, the contri-bution of the scalar field, whose action is given by formula (D.27) with Jϕ = 0. Theintegral over ϕ of the type (D.22) is Gaussian and equal to∫

Dϕe−S(β)[ϕ] =[Det(−∂μ∂μ + m2)

]−1/2,

where m2 = m2(φ), and the determinant can be understood as the product ofeigenvalues of the operator (−∂μ∂μ + m2) with boundary conditions of periodicityin τ with period β. If the system is put into spatial box of large size L, then theeigenvalues are

λn,n1,n2,n3 = p2 + ω2n + m2,

where

p =(

2πn1

L,2πn2

L,2πn3

L

), n1, n2, n3 ∈ Z, (D.34)

and ωn are Matsubara frequencies (D.30). Therefore, the contribution to the freeenergy reads

Fϕ =∑

n

∑n1,n2,n3

12β

log[p2 + ω2

n + m2

Λ2

],

where the parameter Λ makes the ratio dimensionless. This parameter leads onlyto an overall shift in the free energy, and hence it is insignificant. In the limit oflarge L ∑

n1,n2,n3

→ L3

∫d3p

(2π)3,


so that the free energy is indeed proportional to the volume, and the contributionto the effective potential is

fϕ =12β

∑n

∫d3p

(2π)3log[p2 + ω2

n + m2

Λ2

]. (D.35)

It is convenient to calculate not this contribution itself, but rather its derivativewith respect to m2 (we are not interested in the contribution independent of m2,i.e., of the field φ),

∂fϕ

∂m2=

12β

∫d3p

(2π)3∑

n

1p2 + ω2

n + m2.

To calculate the sum over all integer n, we note that it can be represented in theform ∑

n=0,±1,...

u(n) =12i

∮cot(πz)u(z)dz, (D.36)

where integration is performed along closed contour in the complex plane encirclingthe real axis counterclockwise, see Fig. D.2(a). To prove the formula (D.36), it issufficient to note that cotπz has poles at integer z = 0,±1, . . . with residues equalto π−1. In our case

u(z) =

[p2 +

(2π

βz

)2

+ m2

]−1

. (D.37)

As u(z) has singularities (poles) at the imaginary axis only, the contour of inte-gration in (D.36) can be deformed as shown in Fig. D.2(b). The integral picks upcontributions from two poles,

z = ± iβ

2π

√p2 + m2,

(a) (b)

Fig. D.2 (a) The contour of integration in Eq. (D.36); (b) Deformed contour.


and we obtain∂fϕ

∂m2=∫

d3p(2π)3

1

4√

p2 + m2coth

[β

2

√p2 + m2

].

This expression can be represented as

∂fϕ

∂m2=

∂fϕ(T = 0)∂m2

+∂f

(T )ϕ

∂m2,

where∂fϕ(T = 0)

∂m2=∫

d3p(2π)3

1

4√

p2 + m2=

∂

∂m2

[12

∫d3p

(2π)3√

p2 + m2

](D.38)

does not depend on temperature, while the temperature-dependent part is

∂f(T )ϕ

∂m2=∫

d3p(2π)3

12√

p2 + m2

1

e√

p2+m2T − 1

. (D.39)

The zero-temperature contribution (D.38) to the effective potential is simply thesum of zero-point energies of oscillators of the field ϕ,

fϕ(T = 0) =1L3

∑n1,n2,n3

12

√p2 + m2, (D.40)

where for clarity we returned back to the theory in finite spatial volume; themomentum p is given by (D.34). This contribution is not interesting for us here,although under certain relations between couplings it can lead to interesting conse-quences in theories at zero temperature.5

The contribution (D.39), relevant at finite temperatures, exactly corresponds toboson integral (10.16) with gi = 1 (we consider one real scalar field ϕ). Indeed, itis equal to

∂f(T )ϕ

∂m2=

14π2

∫ ∞

0

p2dp√p2 + m2

1

e√

p2+m2T − 1

. (D.41)

On the other hand, the derivative of the integral (10.16) with respect to m2 reads

− 16π2

∫ ∞

0

k4dk∂

∂m2

[1√

k2 + m2· 1

e√

k2+m2T − 1

]

= − 112π2

∫ ∞

0

k4dk1k

∂

∂k

[1√

k2 + m2· 1

e√

k2+m2T − 1

], (D.42)

which coincides with (D.41) after integration by parts.The calculation of the fermionic contribution to the one-loop thermal effective

potential proceeds in a similar way. For the theory with the action (D.28) andJψ = Jψ = 0, the functional integral (D.22) equals∫

DψDψe−S(β)ψ = Det [γμ∂μ + m(φ)] ,

5Zero-temperature one-loop contribution (D.40) is ultraviolet divergent. This divergence is elim-

inated by means of the usual renormalization of mass and self-coupling of the Higgs field φ.


where the eigenfunctions of the Euclidean Dirac operator must be antiperiodic inτ with period β. For fixed 3-momentum p and Matsubara frequency (D.32), thereare two doubly degenerate eigenvalues of the Dirac operator,

λ± = m ± i√

p2 + ω2n′ .

As a result, for each momentum we have the factor (λ+λ−)2 in the determinant,and instead of (D.35) we obtain

fψ = − 2β

∫d3p

(2π)3∑

n′=± 12 ,± 3

2 ,...

ln[p2 + ω2

n′ + m2

Λ2

].

We emphasize that the difference in sign as compared to (D.35) is due to the factthat we are dealing with fermions. When calculating ∂fψ/∂m2, we encounter thesum over half-integer n′, which can be written as∑

n′=± 12 ,± 3

2 ,...

u(n′) = − 12i

∮tan(πz)u(z)dz, (D.43)

where the integration contour is the same as shown in Fig. D.2(a), and u(z) isstill given by formula (D.37). Further calculation basically repeats the calculationfor the scalar field. The contribution of fermions also has zero-temperature andfinite-temperature parts. The former,

fψ(T = 0) = −2∫

d3p(2π)3

√p2 + m2,

can be interpreted as the contribution of the Dirac sea (negative energy states,ω = −

√p2 + m2, doubly degenerate at each p). The temperature-dependent term

∂f(T )ψ

∂m2=∫

d3p(2π)3

2√p2 + m2

1

e√

p2+m2T + 1

coincides with the derivative of the fermion integral (10.16), given that the totalnumber of spin states of fermion and antifermion is g = 4.

Thus, in the framework of the formalism presented in this Appendix, the dif-ference between the Bose and Fermi statistics manifests itself in the differencebetween the Matsubara frequencies (D.30) and (D.32). In Sec. 10.3 we discuss theimportance of this difference for the infrared properties of the theory at high tem-peratures.

Problem D.3. In the one-loop approximation, find the Grand potential and fermionnumber density of fermionic matter at chemical potential μ and temperature T .Consider the limiting cases T � μ � m and T � μ. Hint: Make use of the property

∂F (μ, T )∂μ

= −〈Q〉T ,μ.

This property follows from (D.23).


Fig. D.3 One-loop photon self-energy.

D.5 Debye Screening

As the second example, let us consider the one-loop contribution Πμν to the photonself-energy in quantum electrodynamics at finite temperature and zero chemicalpotential, see Fig. D.3. As usual, it modifies the photon propagator

Dμν → [D−1μν + Πμν

]−1,

where Πμν is also called photon polarization operator. Let us consider the staticpropagator, see (D.25). So, we are interested in the self-energy at zero Matsubarafrequency,

Πμν(p) = Πμν(p, ωn = 0).The sought-for contribution modifies the static Maxwell equations, which in themedium obtain the following form (in momentum space)

p2A0 + Π00A0 + Π0iAi = j0,

p2Ai − pipA + Πi0A0 + ΠikAk = ji,

where jμ(p) are time-independent charge and current densities.Before doing the calculation, we note that due to gauge invariance of electrody-

namics, which holds in the presence of matter as well, the self-energy Πμν(p, ωn)has to be transverse,

pμΠμν = 0,

where pμ = (ωn,p). Lorentz-invariance is explicitly broken by the presenceof matter, but the symmetry with respect to spatial rotations remains intact.Therefore, the general structure of the self-energy is

Π00 = Π(E),

Πi0 = −pip0

p2Π(E),

Πij =pipjp

20

p4Π(E) +

(δij − pipj

p2

)Π(M),

where “electric” and “magnetic” terms, Π(E) and Π(M), depend on p2 and p0 ≡ ωn.In the static limit p0 ≡ ωn = 0, only Π00 and transverse part of Πij are non-zero,thus the modified Maxwell equations take the following form,

(p2 + Π(E))A0 = j0, (D.44)

(p2 + Π(M))(

δik − pipk

p2

)Ai = jk. (D.45)

D.5. Debye Screening 455

We will be interested in the field behavior at large distances. Hence, in the end wetake the limit p2 → 0. The order of limits is important here: one should first setp0 ≡ ωn = 0, and then take the limit of small p2.

The interaction of fermions with electromagnetic field is introduced, as usual,by the replacement ∂μ → ∂μ− ieAμ in the action (D.28), so the diagram in Fig. D.3gives

Πμν(p, p′) = −e2

∫tr [γμS(x, y)γνS(y, x)] eip

(i)λ

xλ

e−ip(f)ρ yρ

d4xd4y,

where pμ = (ωni ,p(i)), p

′μ = (ωnf ,p(f)) are momenta of incoming and outgoingphotons, integration over x0 and y0 is performed in the interval (0, β), and thefermion propagator is given by (D.31). Extracting the δ-function of energy andmomentum conservation (the former is understood in the sense of (D.33)) andsetting ωni = ωnf = 0, p(i) = p(f) = p, we obtain

Πμν(p) =e2

(2π)3β

∫d3q

∑n′

Tr [γμ(−iq + m)γν(−i(q + p) + m)](q2 + m2)((q + p)2 + m2)

,

where q = γμqμ, q0 = 2πn′β , photon momentum is equal to pμ = (0,p), the sum

is evaluated over half-integer n′, and momentum squared is understood in theEuclidean sense. We again perform summation over the Matsubara frequencies bymaking use of (D.43). Taking the limit of small photon momentum, we arrive at

Πμν(p→0, ω=0) =2e2

(2π)3

∫d3q

∮dq0

2πitan(

β

2q0

)·2qμqν−δμν(q2

0+q2+m2)(q2

0 + q2 + m2)2(D.46)

Poles of the integrand are at q0 = ±i√

q2 + m2; they show that in fact we aredealing with the photon forward scattering off fermions and antifermions existingin the medium; this is schematically illustrated in Fig. D.4, where crosses denoteparticles in the medium. Here we encounter the situation where nominally one-loopcalculation corresponds to tree-level diagrams for scattering off the medium, whilethe interaction between the particles of the medium is not taken into account. Thissituation is analogous in some sense to that in Sec. D.4 (formally one-loop calcu-lation of the effective potential corresponds to the approximation of non-interactingparticles in the medium).

Fig. D.4 The interpretation of self-energy in terms of photon rescattering in medium.


Integrating over dq0 in (D.46) and omitting temperature-independent terms, weobtain for the 00-component

Π00(p → 0, ω = 0) ≡ Π(E) =e2

π2

∫ ∞

0

dq

ωq· ω2

q + q2

eωqT + 1

, (D.47)

where ωq =√

q2 + m2 (when performing this calculation, it is convenient to makeuse of the integration by parts analogous to that used in (D.42)). At the sametime, the spatial components Πij are equal to zero in the limit of small photonmomentum,

Π(M)(p → 0, ω = 0) = 0.

Recalling (D.44) and (D.45), we see that our results imply that electric field getsscreened in the medium, while magnetic field does not (of course, the latter propertyhas been demonstrated within one-loop approximation only). Indeed, in the coor-dinate representation, the solution to Eq. (D.44), in the case of point charge q placedat the origin, has the following form at large distances,

A0(x) = q

∫d3p

(2π)3eipx

p2 + m2D

=q

4π

e−mD |x|

|x| ,

where m2D = Π(E)(p → 0, ω = 0) is the Debye mass squared. The exponential

fall-off here precisely means screening of electric field. For the magnetic field thisphenomenon of exponential fall-off at large distances is absent.

Note that at T � m, the Debye mass is exponentially small (this is becausewe consider medium at zero chemical potential; the density of fermions is thusexponentially small at low temperature), while in the opposite limit

mD =e

πT, T � m,

i.e., the Debye screening radius rD = m−1D

decreases with temperature.At the end of this Section, we mention that the Debye screening arises also

when the medium contains charged bosons rather than fermions; the contributionof bosons to the Debye mass squared is of the same order of magnitude as that offermions of the same mass and electric charge.

Finally, the Debye screening occurs at non-zero net particle densities, too. Andthe temperature may be fairly low: in this case the Debye radius is determined by thedensity of charged particles. Physically interesting example here is electron-protonplasma.

Problem D.4. Find the Debye radius in electrically neutral electron-proton plasmaat given electron number density and temperature T in two cases: mp � T � me

and me � T � Δ, where Δ is the binding energy of electron in hydrogen atom(Δ = 13.6 eV). Hint: Perform the calculation at fixed chemical potentials of electronsand protons; in order to find their relation to the electron number density use theresults of Problem D.3.

Books and Reviews

We give here an (incomplete) list of books and reviews where various aspects of theHot Big Bang theory and related issues are discussed.

Books

Ya. B. Zeldovich and I. D. Novikov, The Structure and Evolution of the Universe(Relativistic Astrophysics, Volume 2), University of Chicago Press, 1983.

A. D. Linde, Particle Physics and Inflationary Cosmology, Harwood, Chur, 1990.E. W. Kolb and M. S. Turner, The Early Universe, Addison-Wesley, Redwood City, 1990 —

Frontiers in physics, 69.A. D. Dolgov, M. V. Sazhin and Ya. B. Zeldovich, Basics of Modern Cosmology, Ed.

Frontieres, Gif-sur-Yvette, 1991.P. J. E. Peebles, Principles of Physical Cosmology, Princeton University Press, 1993.A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects,

Cambridge University Press, 1994.J. A. Peacock, Cosmological Physics, Cambridge University Press, 1999.S. Dodelson, Modern Cosmology, Academic Press, Amsterdam, 2003.V. Mukhanov, Physical Foundations of Cosmology, Cambridge University Press, 2005.S. Weinberg, Cosmology, Oxford University Press, 2008.

General Reviews

A. D. Dolgov and Y. B. Zeldovich, Cosmology and elementary particles, Rev. Mod. Phys.53 (1981) 1.

R. H. Brandenberger, Particle physics aspects of modern cosmology, arXiv:hep-ph/9701276.

M. S. Turner and J. A. Tyson, Cosmology at the millennium, Rev. Mod. Phys. 71 (1999)S145 [arXiv:astro-ph/9901113].

W. L. Freedman and M. S. Turner, Measuring and understanding the Universe, Rev. Mod.Phys. 75 (2003) 1433 [arXiv:astro-ph/0308418].

V. Rubakov, Introduction to cosmology, PoS RTN2005 (2005) 003.

457

458 Books and Reviews

Lectures at Summer Schools on High Energy Physics

J. A. Peacock, Cosmology and particle physics, Proc. 1998 European School of High-EnergyPhysics, St. Andrews, Scotland, 23 Aug–5 Sep 1998.

M. Shaposhnikov, Cosmology and astrophysics, Proc. 2000 European School ofHigh-Energy Physics, Caramulo, Portugal, 20 Aug–2 Sep 2000.

V. A. Rubakov, Cosmology and astrophysics, Proc. 2001 European School of High-EnergyPhysics, Beatenberg, Switzerland, 2001.

I. I. Tkachev, Astroparticle physics, Proc. 2003 European School on High-Energy Physics,Tsakhkadzor, Armenia, 24 Aug–6 Sep 2003. arXiv:hep-ph/0405168.

Reviews on Topics Covered in Separate Chapters

If necessary, relevant Sections are indicated in parenthesis.

Chapter 4

S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 61 (1989) 1.V. Sahni and A. A. Starobinsky, The case for a positive cosmological Lambda-term, Int.

J. Mod. Phys. D 9 (2000) 373 [arXiv:astro-ph/9904398].S. Weinberg, The cosmological constant problems, arXiv:astro-ph/0005265.A. D. Chernin, Cosmic vacuum, Phys. Usp. 44 (2001) 1099 [Usp. Fiz. Nauk 44 (2001)

1153].T. Padmanabhan, Cosmological constant: The weight of the vacuum, Phys. Rept. 380

(2003) 235 [arXiv:hep-th/0212290].P. J. E. Peebles and B. Ratra, The cosmological constant and dark energy, Rev. Mod.

Phys. 75 (2003) 559 [arXiv:astro-ph/0207347].V. Sahni, Dark matter and dark energy, Lect. Notes Phys. 653 (2004) 141 [arXiv:astro-

ph/0403324].E. J. Copeland, M. Sami and S Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys.

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Appendix C

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W. M. Alberico and S. M. Bilenky, Neutrino oscillations, masses and mixing, Phys. Part.Nucl. 35 (2004) 297 [Fiz. Elem. Chast. Atom. Yadra 35 (2004) 545] [arXiv:hep-ph/0306239].

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Part. Sci. 56 (2006) 569 [arXiv:hep-ph/0603118].


A. Yu. Smirnov, Recent developments in neutrino phenomenology, In the Proceedings ofIPM School and Conference on Lepton and Hadron Physics (IPM-LHP06), Tehran,Iran, 15–20 May 2006, p. 0003 [arXiv:hep-ph/0702061].

Appendix D

E. V. Shuryak, Quantum chromodynamics and the theory of superdense matter, Phys.Rept. 61, 71 (1980).

D. J. Gross, R. D. Pisarski and L. G. Yaffe, QCD and instantons at finite temperature,Rev. Mod. Phys. 53 (1981) 43.

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Index

4-velocity, 32, 380

Abelian Higgs model, 310angular diameter distance, 79anomalous non-conservation of current,

247antibaryons, residual abundance, 245Arrhenius formula, 224asymmetry

(B − L), 265microscopic, 264

baryon number, 400Berezin integral, 444Bianchi identity, 366, 377Big Rip, 19binding energy

of hydrogen, 113of nucleus, 148, 149

Boltzmann equation, 109, 184bubble of new phase, 222, 278

critical, 224surface tension, 223

Cabibbo–Kobayashi–Maskawa (CKM)matrix, 397, 401

Chaplygin gas, 59chiral symmetry, 209

restoration, 212, 217Christoffel symbols, 359comoving frame, 32confinement, 408conformal time, 34conical singularity, 318

Cosmic Microwave Background (CMB), 9,127

Gaussianity, 321

polarization, 10

spectrum, 9

temperature, 9

anisotropy, 10, 320, 322, 332

anisotropy, dipole component, 10

cosmological constant problem, 18

Coulomb barrier, 153

CPT-theorem, 264, 401

CP -violation, 264, 404, 405

critical density, 61

crossover, 221

curvature scalar, 46, 367

cusp in cosmic string, 326

Debye radius, 126, 456

density perturbations, 25, 322, 331

isocurvature, 299

detailed balance, 107

deuterium bottleneck, 150

de Sitter horizon, 56

de Sitter space, 55

Dirac matrices, 392

Dirac spinor, 392

distribution function

Bose–Einstein, 92

Fermi–Dirac, 92

Maxwell–Boltzmann, 93

Planckian, 9

double-β decay, 262, 437

D-string, 326

471

472 Index

Einstein equations, 45, 371, 375Einstein tensor, 371Einstein–Hilbert action, 369electric dipole moments de, dn, 289energy-momentum tensor, 376

covariant conservation, 47, 377metric, 378Noether, 378of domain wall, 327of string, 313of string gas, 316

equilibriumchemical, 91, 111kinetic, 117thermal, 91, 111

event horizon, 51, 56

Fermi constant, 133, 144, 408field

conformal, 35messenger, 196scalar

condensate, 206, 231, 343fast roll, 81minimal, 376slow roll, 81, 293

flat direction, 293, 343fleece, 332free streaming, 166F -string, 326functional integral, 440, 444fundamental string, 326

galactic halo, dark, 16galaxies, 3

clusters, 3Gamow energy, 159gauge couplings, 390

unification, 256gauge mediation, 196gauge transformations in General

Relativity, 377, 384Gauss formula, 363Gauss–Bonnet theorem, 367geodesic

equation, 381line, 381

Georgi–Glashow model, 303global charge, 333

gluino, 180Goldberger–Treiman formula, 194goldstino, 194Goldstone theorem, 194Grand Unified Theories, 247, 253

SU(5), 255Grassmannian variable, 444gravitational lensing, 14, 319

strong, 14gravitational waves

primordial, 28production, 325

Green’s function, 447Greisen–Zatsepin–Kuzmin effect, 11

Higgs field, 389expectation value, 219, 396, 399

after phase transition, 235, 278high-temperature expansion, 230homotopy group, 301

instanton, 208, 249inverse decays, 265

Kibble mechanism, 302kink, 227

in cosmic string, 326Kronecker tensor, 357

lepton asymmetry, 137, 146lepton numbers, 244, 400Levi-Civita tensor, 358locally-Lorentz frame, 364

magneticcharge, 306flux, 312

masses of Standard Model particles, 399Matsubara frequencies, 448, 449Megaparsec, 2Mikheev–Smirnov resonance, 424moduli fields, 293, 343monopolonium, 309mSUGRA model, 190

Nambu–Goldstone field, 194, 209, 329Nambu–Goto action, 314necklace, 332neutralino, 179

Index 473

neutrinobound on chemical potential, 146bound on number of species, 151effective potential in matter, 139, 418flavor basis, 412mass basis, 412oscillation amplitude, 414oscillation length, 413, 414

Newtonian gravitational potential, 383non-linear gravity, f(R), 372number of degrees of freedom, 93

effective, 53, 95

order parameter, 221

particle horizon, 51path integral, 440Pauli principle, 189Peccei–Quinn symmetry, 209phase transition, 302

critical temperature Tc0, 234critical temperature Tc1, 234critical temperature Tc2, 231electroweak, 217, 278Grand Unified, 218, 307latent heat, 223

photino, 181, 189photometric distance, 72Planck mass, length, time, 1polarization operator of photon, 454Pontecorvo–Maki–Nakagawa–Sakata

(PMNS) matrix, 412processes, fast, slow, 105proper time, 32, 380proton decay, 253, 400

quark condensate, 209, 217quark-gluon matter, 23, 217quasars, 4

recombinationLyman-α transitions, 118, 122transitions from continuum, 118two-photon transitions, 118

reduced Planck mass M∗Pl, 53

resonance states of nuclei, 155, 159Ricci tensor, 45, 367Riemann tensor, properties, 366R-parity, 182, 195Rutherford cross section, 126

Saha equation, 113Sakharov conditions, 243see-saw mechanism, 261, 433self-energy of photon, 454slepton, 189, 292, 342Sommerfeld parameter, 159spatial curvature, bound, 62sphaleron, 248sphaleron processes

rate, 249selection rules, 251

squark, 180, 292, 342string tension, 312strong CP -problem, 207Sunyaev–Zeldovich effect, 11supercurrent, 194supergravity, 194supernovae, type Ia, 71superpartner, 180

lightest (LSP), 182, 195next-to-lightest (NLSP), 191, 200

superstring theory, 326symmetry breaking, 302, 310, 396symmetry restoration, 219

Thomson cross section, 116’t Hooft effect, 247topology of space, 9

vacuumfalse, 228manifold, 301

weak mixing angle θW , 390Weyl spinor, 391

G

INTRODUCTION TO

THE THEORY OF THE


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ISBN-13 978-981-4343-97-8(pbk)ISBN-10 981-4343-97-8(pbk)

This book is written from the viewpoint of a deep connection between cosmology and particle physics. It presents the results and ideas on both the homogeneous and isotropic Universe at the hot stage of its evolution and in later stages. The main chapters describe in a systematic and pedagogical way established facts and concepts on the early and the present Universe. The comprehensive treatment, hence, serves as a modern introduction to this rapidly developing field of science. To help in reading the chapters without having to constantly consult other texts, essential materials from General Relativity and the theory of elementary particles are collected in the appendices. Various hypotheses dealing with unsolved problems of cosmology, and often alternative to each other, are discussed at a more advanced level. These concern dark matter, dark energy, matter-antimatter asymmetry, etc. This book is accompanied by another book by the same authors, Introduction to the Theory of the Early Universe: Cosmological Perturbations and Inflationary Theory.

introduction to the theory of the early universe: hot big bang theory

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