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POLITECNICO DI TORINO

Dipartimento di Fisica

Invariance Principles and Extended Gravity:

Theories and Probes

Mariafelicia De Laurentis

Thesis submitted for the degree of

Doctor Philosophiae

Supervisors Coordinator

Prof. Salvatore Capozziello Prof. Fausto Rossi

Prof. Angelo Tartaglia

XXI CICLO

2006-2008

Soli Deo GloriaJ. S. Bach

I play the numbers as they are written but,

it is God who makes the physic!!!

-Sweet Nothings-John William Godward

Contents

Abstract 7

Abstract 7

List of Publications 9

Notation 13

Preface 15

1 Introduction 19

1.1 General Relativity is the theory of gravity, isn’t it? . . . . . . . . . . . . 19

1.2 A high-energy theory of gravity? . . . . . . . . . . . . . . . . . . . . . . 21

1.2.1 Searching for the unknown . . . . . . . . . . . . . . . . . . . . . 22

1.2.2 Intrinsic limits in General Relativity and Quantum Field Theory 22

1.2.3 A conceptual clash . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2.4 The vision for unification . . . . . . . . . . . . . . . . . . . . . . 24

1.3 The Cosmological and Astrophysical riddles . . . . . . . . . . . . . . . . 24

1.3.1 Cosmology in a nutshell . . . . . . . . . . . . . . . . . . . . . . . 24

1.3.2 The first need for acceleration . . . . . . . . . . . . . . . . . . . . 27

1.3.3 Cosmological and Astronomical Observations . . . . . . . . . . . 31

1.3.4 The Cosmological Constant and its problems . . . . . . . . . . . 34

1.4 Is there a way out? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.4.1 Scalar fields as matter fields in Cosmology . . . . . . . . . . . . . 39

1.4.2 The dark energy problem . . . . . . . . . . . . . . . . . . . . . . 41

1.4.3 The dark matter problem . . . . . . . . . . . . . . . . . . . . . . 45

1.4.4 Towards Quantum Gravity, but how? . . . . . . . . . . . . . . . 46

1.4.5 Status of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2 Extended Theories of Gravity 51

2.1 Theoretical motivations for Extended Theories of Gravity . . . . . . . . 51

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4 CONTENTS

2.2 What a good theory of Gravity has to do: General Relativity and itsextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.3 Structure of the Extended Theories of Gravity . . . . . . . . . . . . . . 612.3.1 Conformal transformations . . . . . . . . . . . . . . . . . . . . . 64

2.4 The Palatini Approach and the Intrinsic Conformal Structure . . . . . . 67

3 Gravity from Poincare Gauge Invariance 733.1 What can generate the Gravity? . . . . . . . . . . . . . . . . . . . . . . 733.2 Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.3 Global Poincare Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 773.4 Local Poincare Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . 783.5 Spinors, Vectors and Tetrads . . . . . . . . . . . . . . . . . . . . . . . . 813.6 Curvature, Torsion and Metric . . . . . . . . . . . . . . . . . . . . . . . 87

3.7 Field Equations for Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 90

4 Space-time deformations and conformal transformations towards ex-tended theories of gravity 974.1 Deformation and conformal transformations, how? . . . . . . . . . . . . 974.2 Generalities on space-time deformations . . . . . . . . . . . . . . . . . . 984.3 Properties of deforming matrices . . . . . . . . . . . . . . . . . . . . . . 994.4 Metric deformations as perturbations and gravitational waves . . . . . . 1024.5 Approximate Killing vectors . . . . . . . . . . . . . . . . . . . . . . . . . 1044.6 Deformations in f(R) -Theories . . . . . . . . . . . . . . . . . . . . . . . 106

5 Probing the Minkowskian limit: Gravitational waves in f(R)-Theories107

5.1 Why the gravitational waves in f(R)-Theories? . . . . . . . . . . . . . . 1075.2 Stochastic background of gravitational waves ”tuned” by f(R) gravity . 1085.3 Massive gravitational waves from f(R) theories of gravity: Potential de-

tection with LISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.4 Stochastic background of relic scalar gravitational waves from scalar-

tensor gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6 Further probe: Parametrized Post Newtonian limit 1336.1 f(R) gravity constrained by PPN parameters and stochastic background

of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.2 f(R) gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.3 f(R) viable models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.4 Constraining f(R)-models by PPN parameters . . . . . . . . . . . . . . 1466.5 Stochastic backgrounds of gravitational waves to constrain f(R)-gravity 150

7 Future perspectives and conclusions 1617.1 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.2 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

CONTENTS 5

Bibliography 187

Abstract

This thesis is devoted to the study of gravitational theories which can be seen as mod-ifications or generalisations of General Relativity. The motivation for considering suchtheories, stemming from Cosmology, High Enery Physics and Astrophysics in throrouglydiscussed (cosmological problems, dark energy and dark matter problems, the lack ofsucces so far in obtaining a successful formulation for Quantum Gravity). The basicprinciples which a gravitational theory should follow, and their geometrical interpre-tation, are analysed in a broad perspective which highlights the basic assumptions ofGeneral Relativity and suggests possible modifications which might be made. A num-ber of such modifications are presented, focusing on certain specific classes of theories:scalar-tensor theories, metric f(R) theories, Palatini f(R) theories. The caracteristicsof these theories are fully explored and attention is payed to issues of dynamical equiva-lence between them. Also, cosmological phenomenology within the realm of each of thetheories is discussed and it is shown that they can potentially address the well-knowncosmological problems. A number of viability cirteria are presented: cosmolgical tests,Solar System tests....etc...

Finally, future perspectives in the field of modified gravity are discussed and thepossibility for going beyond a trial and error approach to modified gravity is explored.

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8 ABSTRACT

List of Publications

The research presented in this thesis was mainly conducted in Politecnico di Torino andin Universit degli Studi di Napoli ”Federico II” between January 2006 and December2008. This thesis is the result of the authors own work, as well as the outcome ofscientific collaborations stated below, except where explicit reference is made to theresults of others.

The content of this thesis is based on the following research papers pubblished inrefereed Journals or refereed conference proceedings:

Refereed papers

1. S. Capozziello, M. De Laurentis, M. Francaviglia: Stochastic background ofGravitational waves as a Benchmark for Extended Theories of Gravity, Specialissue ” Problem in Modern Cosmology” Ed. Lavronov, 2009.

2. S. Capozziello, M. De Laurentis, F. Garufi, L. Milano:Relativistic orbits withgravitomagnetic effects accepted for Physica Scripta 2009.

3. S. Capozziello, M. De Laurentis, S. Nojiri, S.D. Odintsov: f(R) gravity con-strained by PPN parameters and stochastic background of gravitational waves ,accepted for General Relativity and Gravitation 2009.

4. S. Capozziello, M. De Laurentis: Gravity from Local Poincare Gauge Invariance,accettato su International Journal of Geometric Methods in Modern Physics toappear in vol. 6, N 1 February (2009).

5. S. Capozziello, M. De Laurentis, C. Corda: Massive gravitational waves fromf(R) theories of gravity potential detecton with LISA, Physics Letter B 699, 255-259 (2008).

6. S.Capozziello, M. De Laurentis: Gravitational waves from stellar encountersAstroparticle Physics Volume 30,p. 105-112 (2008) .

7. S. Capozziello, M. De Laurentis, M. Francaviglia: Higher-order gravity and thecosmological background of gravitational waves, Astroparticle Physics Volume 29,Issue 2, p. 125-129 (2008).

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10 LIST OF PUBLICATION

8. S. Capozziello, M. De Laurentis, F. De Paolis, G. Ingrosso, A. Nucita: Gravi-tational waves from hyperbolic encounters, Modern Physics Letters A, Volume 23,Issue 02, pp. 99-107 (2008).

9. S.Capozziello,C. Corda, M. De Laurentis: Stochastic background of relic scalargravitational waves from scalar-tensor gravity ,Modern Physics Letters A, Volume22, Issue 35,(2007).

10. S. Capozziello, C. Corda, M. De Laurentis: Stochastic background of gravita-tional waves ”tuned” by f(R) gravity, Modern Physics Letters A vol.22 n15 pp.1097-1104 (2007).

Proceeding

1. S. Capozziello, C.Corda,M. De Laurentis: Tuning the stochastic backgroundof gravitational waves with theory and observations Proc. of Societ Italiana diRelativit Generale e Fisica della Gravitazione XVII Congresso SIGRAV GeneralRelativity and Gravitational Physics Torino, 4-7 Settembre 2006.

2. C. Corda, M. De Laurentis:Gravitational waves from the R−1 high-order theoryof gravity, To appear in Proc. of the 10th ICATPP International Conference onAdvanced Technology and Particle Physics (Villa Olmo Como, 8-12 october 2007)published on World Scientific.

3. A. Tartaglia, A. Nagar, N. Radicella, M. De Laurentis and 26 coauthors: Sum-mary of session B3: Analytic approximations, perturbation methods and theirapplications, Classical and Quantum Gravity, Volume 25, Issue 11, pp. 114020(2008).

4. S. Capozziello, C.Corda, M. De Laurentis: Tuning the stochastic background ofgravitational waves with theory and observations, To appear in the proceedings of4th Italian-Sino Workshop on Relativistic Astrophysics, Pescara, Italy, 20-30 Jul2007. Published in AIP Conf.Proc.966:257-263, (2008).

5. S. Capozziello, M. De Laurentis, L. Izzo: Stochastic Background of gravitationalwaves tuned by f(R) gravityProc. of 3 Sueckelberg Workshop on Relativistic FieldTheories Pescara, Italy, 19-21 Jul 2008, Published in AIP Conf.Proc.(2009).

6. S. Capozziello,M. De Laurentis, L. Izzo: Detection of Cosmological StochasticBackground of Gravitational waves in f(R) gravity with FASTICA Proc. of 3Sueckelberg Workshop on Relativistic Field Theories Pescara, Italy, 19-21 Jul 2008,Published in AIP Conf.Proc.(2009).

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Submitted papers

1. S. Capozziello, M. De Laurentis, M. Francaviglia, S. Mercadante: From DarkEnergy and Dark Matter to Dark Metric, submitted to Foundations of Physics2009.

2. S. Capozziello, M. De Laurentis: Deformations in f(R) -theories, submitted toPhysics Letters B (2008).

3. S. Capozziello, M. De Laurentis, N. Radicella: Solutions to LQC with f(R)submitted to Physics Letters B (2008)

4. S. Bellucci, S. Capozziello, M. De Laurentis, V. Faraoni:Position and frequencyshifts induced by massive modes of the gravitational wave background in alternativegravity submitted to Physical Review D.

12 LIST OF PUBLICATION

Notation

13

14 NOTATION

Preface

The terms “modified gravity” and “alternative theory of gravity” have become standardterminology for theories proposed for describing the gravitational interaction which differfrom the most conventional one, General Relativity. Modified or alternative theories ofgravity have a long history. The first attempts date back to the 1920s, soon after theintroduction of Einstein’s theory. Interest in this research field, which was initiallydriven by curiosity or a desire to challenge the then newly introduced General Theoryof Relativity, has subsequently varied depending on circumstances, responding to theappearance of new motivations. However, there has been more or less continuous activityin this subject over the last 85 years.

When the research presented in this thesis began, interest in modified gravity wasalready at a high point and it has continued increasing further until the present day.This recent stimulus has mostly been due to combined motivation coming from the well-known cosmological problems related to the accelerated expansion of the universe andthe feedback from High Energy Physics.

Due to the above, and even though the main scope of this thesis is to present theresearch conducted by the author during the period january 2006 December 2008, asignificant efforts has been made so that this thesis can also serve as a guide for readerswho have recentetly developed an interest in this field. To this end, special attentionhas been paid to giving a very general analysis of the foundations of gravitation theory.Also, an effort has been made to present the theories discussed thorougly, so that readersless familiar with this subject can be introduced to them before gradually moving on totheir more complicate characteristics and applications.

The outline of this thesis is as follows: In the Introduction, several open issuesrelated to gravity are discussed, including the cosmological problems related to darkmatter and dark energy, and the search for a theory of Quantum Gravity. Through thepresentation of a historical timeline of the passage from Newtonian gravity to GeneralRelativity, and a comparison with the current status of the latter in the light of theproblems just mentioned, the motivations for considering alternative theories of gravityare introduced.

Chapter 2 is devoted a survey of what is intended for Extended Theories of Gravityin the so called ”metric” and ”Palatini” approaches. In the Chapter 3 a compact,self-contained approach to gravitation, based on the local Poincare gauge invariance, is

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16 PREFACE

proposed. Starting from the general invariance principle, we discuss the global and thelocal Poincare invariance developing the spinor, vector and tetrad formalisms. Thesetools allow to construct the curvature, torsion and metric tensors by the Fock-Ivanenkocovariant derivative. The resulting Einstein-Cartan theory describes a space endowedwith non-vanishing curvature and torsion while the gravitational field equations aresimilar to the Yang-Mills equations of motion with the torsion tensor playing the roleof the Yang-Mills field strength. A definition of space-time metric deformations on ann-dimensional manifold is given in the Chapter 4. We show that such deformations canbe regarded as extended conformal transformations. In particular, their features canbe related to the perturbation theory giving a natural picture by which gravitationalwaves are described by small deformations of the metric. As further result, deformationscan be related to approximate Killing vectors (approximate symmetries) by which it ispossible to parameterize the deformed region of a given manifold. Space-time metricdeformations can be immediately recast in terms of perturbation theory allowing acompletely covariant approach to the problem of gravitational waves (GW) and then inthe Chapter 5 we show that the stochastic background of gravitational waves, producedin the early cosmological epochs, strictly depends on the assumed theory of gravity. Inparticular, the specific form of the function f(R), where R is the Ricci scalar, is relatedto the evolution and the production mechanism of gravitational waves.

Then we given a generalization of previous results on gravitational waves (GWs)from f(R) theories of gravity where the process is further generalized, showing thatevery f(R) theory can be linearized producting a third massive mode of gravitationalradiation. The potential detectability of such massive GWs with LISA is also discussedwith the auxilium of longitudinal response functions.

Afetr we provide a distinctive spectrum of relic gravitational waves. In the frameworkof scalar-tensor gravity, we discuss the scalar modes of gravitational waves and theprimordial production of this scalar component which is generated beside tensorial one.We discuss also the upper limit for such a relic scalar component with respect to theWMAP constraints. On the other hand, detecting the stochastic background by theforthcoming interferometric experiments (VIRGO, LIGO, LISA) could be a further toolto select the effective theory of gravity.

Finally in the Chapter 6 viable f(R)-gravity models are discussed toward SolarSystem tests and stochastic background of gravitational waves. The aim is to achieveexperimental bounds for the theory at local and cosmological scales in order to selectmodels capable of addressing the accelerating cosmological expansion without cosmo-logical constant but evading the weak field constraints. Beside large scale structure andgalactic dynamics, these bounds can be considered complementary in order to selectself-consistent theories of gravity working at the infrared limit.

A number of people have contributed in this thesis in various ways. First andforemost, I would thank my PhD advisors, Salvatore Capozziello an Angelo Tartaglia,for their constant support during the course of this work. It is difficult for me to imaginehaving better advisors than Salvatore Capozziello and Angelo Tartaglia, to whom I am

17

truly grateful, not only for their guidance but also for standing by me in all my choicesand for the impressive amount of patience they have exhibited during the course of thisyears. Special thanks to Capozziello for his untiring correction of my spelling, grammarand (ab)use of the English language.

I cannot thank enough Mauro Francaviglia and Silvio Mercadante,not only for theirhard work on our common projects, but also for numerous hours of conversation anddebate moslty, but definitely not exclusively, on scientific issues. It has really been apleasure for me to collaborate with them. I am also very grateful to ......referee, my thesisexaminers, for undertaking the task of reviewing this manuscript and for their invaluablesuggestions. During the course of this research I have benefited from systematic oroccasional but always stimulating discussions with a number of people, besides thosealready mentioned. Aware of the fact that I am running the risk of forgetting a few —and apologising in advance for that — I will attempt to name them here and expressmy gratitude:

Shin’ichi Nojiri, Sergei Odintsov, Fernando Atrio Barandela,Valerio Faraoni, J. MichaelAlim, Cosimo Stornaiolo, Roberto Cianci, Stefano Vignolo, Francesco de Paolis, G. In-grosso, Achille Nucita, Fabio Garufi, Leopoldo Milano and Virgo Laboratory, GiampieroEsposito, Christian Corda, Luca Izzo, Giovanni Covone, Piero Quarati, Ninfa Radicella,Monica Capone, Matteo Ruggiero, Alessandro Nagar and Guido Rizzi.

The Politecnico has provided an ideal environment for conducting my research overthe last three years. I would like to thank all of my colleagues for contributing to that.Special thanks to my office mate Marco Zamparo, Katarzyna Szymanska and SimoneMusso for all the fun we had while sharing a room in Departiment.

Last, but definitely not least, I would like to thank my family for their love andtheir help in finding my way through life. If it was not for their continuous and untiringsupport it would have been impossible for me to start, let alone finish, this PhD.

18 PREFACE

Chapter 1

Introduction

1.1 General Relativity is the theory of gravity, isn’t it?

It is remarkable that gravity is probably the fundamental interaction which still remainsthe most enigmatic, even though it is so related with phenomena experienced in everydaylife and is the one most easily conceived of without any sophisticated knowledge. Asa matter of fact, the gravitational interaction was the first one to be put under themicroscope of experimental investigation, obviously due to exactly the simplicity ofconstructing a suitable experimental apparatus.

Galileo Galilei was the first to introduce pendulums and inclined planes to the studyof terrestrial gravity at the end of the 16th century. It seems that gravity played animportant role in the development of Galileo’s ideas about the necessity of experiment inthe study of science, which had a great impact on modern scientific thinking. However, itwas not until 1665, when Sir Isaac Newton introduced the now renowned “inverse-squaregravitational force law”, that terrestrial gravity was actually united with celestial gravityin a single theory. Newton’s theory made correct predictions for a variety of phenomenaat different scales, including both terrestrial experiments and planetary motion.

Obviously, Newton’s contribution to gravity — quite apart from his enormous con-tribution to physics overall — is not restricted to the expression of the inverse squarelaw. Much attention should be paid to the conceptual basis of his gravitational theory,which incorporates two key ideas:

1. The idea of absolute space, i.e. the view of space as a fixed, unaffected structure;a rigid arena in which physical phenomena take place.

2. The idea of what was later called the Weak Equivalence Principle which, expressedin the language of Newtonian theory, states that the inertial and the gravitationalmass coincide.

Asking whether Newton’s theory, or any other physical theory for that matter, isright or wrong, would be ill-posed to begin with, since any consistent theory is apparently

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20 1. INTRODUCTION

“right”. A more appropriate way to pose the question would be to ask how suitable isthis theory for describing the physical world or, even better, how large a portion of thephysical world is sufficiently described by this theory. Also, one could ask how uniquethe specific theory is for the description of the relevant phenomena. It was obvious inthe first 20 years after the introduction of Newtonian gravity that it did manage toexplain all of the aspects of gravity known at that time. However, all of the questionsabove were posed sooner or later.

In 1855, Urbain Le Verrier observed a 35 arc-second excess precession of Mercury’sorbit and later on, in 1882, Simon Newcomb measured this precession more accuratelyto be 43 arc-seconds. This experimental fact was not predicted by Newton’s theory.It should be noted that Le Verrier initially tried to explain the precession within thecontext of Newtonian gravity, attributing it to the existence of another, yet unobserved,planet whose orbit lies within that of Mercury. He was apparently influenced by thefact that examining the distortion of the planetary orbit of Uranus in 1846 had led him,and, independently, John Couch Adams, to the discovery of Neptune and the accurateprediction of its position and momenta. However, this innermost planet was never found.

On the other hand, in 1893 Ernst Mach stated what was later called by AlbertEinstein “Mach’s principle”. This is the first constructive attack on Newton’s idea ofabsolute space after the 17th century debate between Gottfried Wilhelm Leibniz andSamuel Clarke (Clarke was acting as Newton’s spokesman) on the same subject, knownas the Leibniz–Clarke Correspondence. Mach’s idea can be considered as rather vaguein its initial formulation and it was essentially brought into mainstream physics later onby Einstein along the following lines:

” ...inertia originates in a kind of interaction between bodies...”.

This is obviously in contradiction with Newton’s ideas, according to which inertia wasalways relative to the absolute frame of space. There exists also a later, probably clearerinterpretation of Mach’s Principle, which, however, also differs in substance. This wasgiven by Dicke:

” The gravitational constant should be a function of the mass distribution in the uni-verse”.

This is different from Newton’s idea of the gravitational constant as being universal andunchanging. Now Newton’s basic axioms were being reconsidered.

But it was not until 1905, when Albert Einstein completed Special Relativity, thatNewtonian gravity would have to face a serious challenge. Einstein’s new theory, whichmanaged to explain a series of phenomena related to non-gravitational physics, appeared

1.2. A HIGH-ENERGY THEORY OF GRAVITY? 21

to be incompatible with Newtonian gravity. Relative motion and all the linked conceptshad gone well beyond the ideas of Galileo and Newton and it seemed that Special Rel-ativity should somehow be generalised to include non-inertial frames. In 1907, Einsteinintroduced the equivalence between gravitation and inertia and successfully used it topredict the gravitational redshift. Finally, in 1915, he completed the theory of GeneralRelativity, a generalisation of Special Relativity which included gravity. Remarkably,the theory matched perfectly the experimental result for the precession of Mercury’sorbit, as well as other experimental findings like the Lense-Thirring gravitomagneticprecession (1918) and the gravitational deflection of light by the Sun, as measured in1919 during a Solar eclipse by Arthur Eddington.

General Relativity overthrew Newtonian gravity and continues to be up to now anextremely successful and well-accepted theory for gravitational phenomena. As men-tioned before, and as often happens with physical theories, Newtonian gravity did notlose its appeal to scientists. It was realised, of course, that it is of limited validitycompared to General Relativity, but it is still sufficient for most applications relatedto gravity. What is more, at a certain limit of gravitational field strength and veloc-ities, General Relativity inevitably reduces to Newtonian gravity. Newton’s equationsfor gravity might have been generalised and some of the axioms of his theory may havebeen abandoned, like the notion of an absolute frame, but some of the cornerstones of histheory still exist in the foundations of General Relativity, the most prominent examplebeing the Equivalence Principle, in a more suitable formulation of course.

This brief chronological review, besides its historical interest, is outlined here also fora practical reason. General Relativity is bound to face the same questions as were facedby Newtonian gravity and many would agree that it is actually facing them now. Inthe forthcoming sections, experimental facts and theoretical problems will be presentedwhich justify that this is indeed the case. Remarkably, there exists a striking similarityto the problems which Newtonian gravity faced, i.e. difficulty in explaining particularobservations, incompatibility with other well established theories and lack of uniqueness.This is the reason behind the question mark in the title of this section.

1.2 A high-energy theory of gravity?

Many will agree that modern physics is based on two great pillars: General Relativityand Quantum Field Theory. Each of these two theories has been very successful inits own arena of physical phenomena: General Relativity in describing gravitating sys-tems and non-inertial frames from a classical viewpoint or on large enough scales, andQuantum Field Theory in revealing the mysteries of high energy or small scale regimeswhere a classical description breaks down. However, Quantum Field Theory assumesthat spacetime is flat and even its extensions, such as Quantum Field Theory in curvedspace time, consider spacetime as a rigid arena inhabited by quantum fields. GeneralRelativity, on the other hand, does not take into account the quantum nature of mat-

22 1. INTRODUCTION

ter. Therefore, it comes naturally to ask what happens if a strong gravitational fieldis present at small, essentially quantum, scales? How do quantum fields behave in thepresence of gravity? To what extent are these amazing theories compatible?

Let us try to pose the problem more rigorously. Firstly, what needs to be clarified isthat there is no precise proof that gravity should have some quantum representation athigh energies or small scales, or even that it will retain its nature as an interaction. Thegravitational interaction is so weak compared with other interactions that the charac-teristic scale under which one would expect to experience non-classical effects relevantto gravity, the Planck scale, is 10−33 cm. Such a scale is not of course accessible byany current experiment and it is doubtful whether it will ever be accessible to futureexperiments either1. However, there are a number of reasons for which one would preferto fit together General Relativity and Quantum Field Theory [1, 2]. Let us list some ofthe most prominent ones here and leave the discussion about how to address them forthe next section.

1.2.1 Searching for the unknown

Curiosity is probably the motivation leading scientific research. From this perspectiveit would be at least unusual if the gravity research community was so easily willing toabandon any attempt to describe the regime where both quantum and gravitationaleffects are important. The fact that the Planck scale seems currently experimentallyinaccessible does not, in any way, imply that it is physically irrelevant. On the contrary,one can easily name some very important open issues of contemporary physics that arerelated to the Planck scale.

A particular example is the Big Bang scenario in which the universe inevitably goesthough an era in which its dimensions are smaller than the Planck scale (Planck era).On the other hand, spacetime in General Relativity is a continuum and so in principleall scales are relevant. From this perspective, in order to derive conclusions about thenature of spacetime one has to answer the question of what happens on very small scales.

1.2.2 Intrinsic limits in General Relativity and Quantum Field Theory

The predictions of a theory can place limits on the extent of its ability to describethe physical world. General Relativity is believed by some to be no exception to thisrule. Surprisingly, this problem is related to one of the most standard processes in agravitational theory: gravitational collapse. Studying gravitational collapse is not easysince generating solutions to Einstein’s field equations can be a tedious procedure. Weonly have a few exact solutions to hand and numerical or approximate solutions are oftenthe only resort. However, fortunately, this does not prevent one from making generalarguments about the ultimate fate of a collapsing object.

1This does not imply, of course, that imprints of Quantum Gravity phenomenology cannot be foundin lower energy experiments.

1.2. A HIGH-ENERGY THEORY OF GRAVITY? 23

This was made possible after the proof of the Penrose–Hawking singularity theorems[3, 4]. These theorems state that a generic spacetime cannot remain regular beyond a fi-nite proper time, since gravitational collapse (or time reversal of cosmological expansion)will inevitably lead to spacetime singularities. In a strict interpretation, the presenceof a singularity is inferred by geodesic incompleteness, i.e. the inability of an observertravelling along a geodesic to extend this geodesic for an infinite time as measured byhis clock. In practical terms this can be loosely interpreted to mean that an observerfree-falling in a gravitational field will “hit” a singularity in a finite time and Einstein’sequation cannot then predict what happens next. Such singularities seem to be presentin the centre of black holes. In the Big Bang scenario, the universe itself emerges out ofsuch a singularity.

Wheeler has compared the problem of gravitational collapse in General Relativitywith the collapse of the classical Rutherford atom due to radiation [5]. This raises hopesthat principles of quantum mechanics may resolve the problem of singularities in GeneralRelativity, as happened for the Rutherford model. In a more general perspective, itis reasonable to hope that quantization can help to overcome these intrinsic limits ofGeneral Relativity.

On the other hand, it is not only General Relativity that has an intrinsic limit. Quan-tum Field Theory presents some disturbing ultraviolet divergences. Such divergences,caused by the fact that integrals corresponding to the Feynman diagrams diverge due tovery high energy contributions — hence the name ultraviolet — are discretely removedby a process called renormalization. These divergences are attributed to the perturba-tive nature of the quantization process and the renormalization procedure is somehowunappealing and probably not so fundamental, since it appears to cure them in a waythat can easily be considered as non-rigorous from a mathematical viewpoint. A non-perturbative approach is believed to be free of such divergences and there is hope thatQuantum Gravity may allow that (for early results see [6, 7, 8, 9, 10]).

1.2.3 A conceptual clash

Every theory is based on a series of conceptual assumption and General Relativity andQuantum Field Theory are no exceptions. On the other hand, for two theories to work ina complementary way to each other and fit well together, one would expect an agreementbetween their conceptual bases. This is not necessarily the case here.

There are two main points of tension between General Relativity and Quantum FieldTheory. The first has to do with the concept of time: Time is given and not dynamical inQuantum Field Theory and this is closely related to the fact that spacetime is consideredas a fixed arena where phenomena take place, much like Newtonian mechanics. On theother hand, General Relativity considers spacetime as being dynamical, with time alonenot being such a relevant concept. It is more of a theory describing relations betweendifferent events in spacetime than a theory that describes evolution over some runningparameter. One could go further and seek for the connection between what is mentioned

24 1. INTRODUCTION

here and the differences between gauge invariance as a symmetry of Quantum FieldTheory and diffeomorphism invariance as a symmetry of General Relativity.

The second conceptual issue has to do with Heisenberg’s uncertainty principle inQuantum Theory which is absent in General Relativity as a classical theory. It isinteresting to note that General Relativity, a theory in which background independenceis a key concept, actually introduces spacetime as an exact and fully detailed record ofthe past, the present and the future. Everything would be fixed for a super-observerthat could look at this 4-dimensional space from a fifth dimension. On the other hand,Quantum Field Theory, a background dependent theory, manages to include a degreeof uncertainty for the position of any event in spacetime.

Having a precise mathematical structure for a physical theory is always important,but getting answers to conceptual issues is always the main motivation for studyingphysics in the first place. Trying to attain a quantum theory of gravity could lead tosuch answers.

1.2.4 The vision for unification

Apart from strictly scientific reasons for trying to make a match between Quantum FieldTheory and General Relativity, there is also a long-standing intellectual desire, maybeof a philosophical nature or stemming from physical intuition, to bring the fundamentalinteractions to a unification. This was the vision of Einstein himself in his late years. Hisperspective was that a geometric description might be the solution. Nowdays most of thescientists active in this field would disagree with this approach to unification and there ismuch debate about whether the geometric interpretation or a field theory interpretationof General Relativity is actually preferable — Steven Weinberg for example even claimedin [11] that “no-one” takes a geometric viewpoint of gravity “seriously”. However, veryfew would argue that such a unification should not be one of the major goals of modernphysics. An elegant theory leading to a much deeper understanding of both gravity andthe quantum world could be the reward for achieving this.

1.3 The Cosmological and Astrophysical riddles

1.3.1 Cosmology in a nutshell

Taking things in chronological order, we started by discussing the possible shortcomingsof General Relativity on very small scales, as those were the first to appear in theliterature. However, if there is one scale for which gravity is by far of the utmostimportance, this is surely the cosmic scale. Given the fact that other interactions areshort-range and that at cosmological scales we expect matter characteristics related tothem to have “averaged out” — for example we do not expect that the universe hasan overall charge — gravity should be the force which rules cosmic evolution. Let us

1.3. THE COSMOLOGICAL AND ASTROPHYSICAL RIDDLES 25

see briefly how this comes about by considering Einstein’s equations combined with ourmore obvious assumptions about the main characteristics of the observable universe.

Even though matter is not equally distributed through space and by simple browsingthrough the sky one can observe distinct structures such as stars and galaxies, if attentionis focused on larger scales the universe appears as if it was made by patching togethermultiple copies of the same pattern, i.e. a suitably large elementary volume aroundthe Earth and another elementary volume of the same size elsewhere will have littledifference. This suitable scale is actually ≈ 108 light years, slightly larger than thetypical size of a cluster of galaxies. In Cosmology one wants to deal with scales largerthan that and to describe the universe as a whole. Therefore, as far as Cosmology isconcerned the universe can be very well described as homogeneous and isotropic.

To make the above statement useful from a quantitative point of view, we have toturn it into an idealized assumption about the matter and geometry of the Universe.Note that the universe is assumed to be spatially homogeneous and isotropic at eachinstant of cosmic time. In more rigorous terms, we are talking about homogeneity oneach one of a set of 3-dimensional space-like hypersurfaces. For the matter, we assumea perfect fluid description and these spacelike hypersurfaces are defined in terms of afamily of fundamental observers who are comoving with this perfect fluid and who cansynchronise their comoving clocks so as to measure the universal cosmic time. Thematter content of the universe is then just described by two parameters, a uniformdensity ρ and a uniform pressure p, as if the matter in stars and atoms is scatteredthrough space. For the geometry we idealize the curvature of space to be everywherethe same.

Let us proceed by imposing these assumption on the equation describing gravityand very briefly review the derivation of the equations governing the dynamics of theuniverse, namely the Friedmann equations. We refer the reader to standard textbooksfor a more detailed discussion of the precise geometric definitions of homogeneity andisotropy and their implications for the form of the metric (e.g. [11]). Additionally, forwhat comes next, the reader is assumed to be acquainted with the basics of GeneralRelativity, some of which will also be reviewed in the next chapter.

Einstein’s equation has the following form

Gµν = 8π GTµν , (1.1)

where

Gµν ≡ Rµν −1

2Rgµν (1.2)

is the Einstein tensor and Rµν and R are the Ricci tensor and Ricci scalar of the metricgµν . G is the gravitational constant and Tµν is the matter stress-energy tensor. Underthe assumptions of homogeneity and isotropy, the metric can take the form

ds2 = −dt2 + a2(t)

[dr2

1− kr2 + r2dθ2 + r2 sin2(θ)dφ2

], (1.3)

26 1. INTRODUCTION

known as the Friedmann-Lemaıtre-Robertson-Walker metric (FLRW). k = −1, 0, 1 ac-cording to whether the universe is hyperspherical (“closed”), spatially flat, or hyperbolic(“open”) and a(t) is called the scale factor2. Inserting this metric into eq. (1.1) and tak-ing into account that for a perfect fluid

T µν = (ρ+ p)uµuν + p gµν , (1.4)

where uµ denotes the four-velocity of an observer comoving with the fluid and ρ and pare the energy density and the pressure of the fluid, one gets the following equations

(a

a

)2

=8π Gρ

3− k

a2, (1.5)

a

a= −4π G

3(ρ+ 3p) , (1.6)

where an overdot denotes differentiation with respect to coordinate time t.

Eqs. (1.5) and (1.6) are called the Friedmann equations. By imposing homogeneityand isotropy as characteristics of the universe that remain unchanged with time on suit-ably large scales we have implicitly restricted any evolution to affect only one remainingcharacteristic: its size. This is the reason why the Friedmann equations are equationsfor the scale factor, a(t), which is a measure of the evolution of the size of any lengthscale in the universe. Eq. (1.5), being an equation in a, tells us about the velocity ofthe expansion or contraction, whereas eq. (1.6), which involves a, tells us about the ac-celeration of the expansion or the contraction. According to the Big Bang scenario, theuniverse starts expanding with some initial velocity. Setting aside the contribution ofthe k-term for the moment, eq. (1.5) implies that the universe will continue to expandas long as there is matter in it. Let us also take into consideration the contributionof the k-term, which measures the spatial curvature and in which k takes the values−1, 0, 1. If k = 0 the spatial part of the metric (1.3) reduces to a flat metric expressedin spherical coordinates. Therefore, the universe is spatially flat and eq. (1.5) impliesthat it has to become infinite, with ρ approaching zero, in order for the expansion tohalt. On the other hand, if k = 1 the expansion can halt at a finite density at which thematter contribution is balanced by the k-term. Therefore, at a finite time the universewill stop expanding and will re-collapse. Finally for k = −1 one can see that even ifmatter is completely dissolved, the k-term will continue to “pump” the expansion whichmeans that the latter can never halt and the universe will expand forever.

2The traditional cosmological language of “closed”/“flat”/“open” is inaccurate and quite misleadingand, therefore, should be avoided. Even if one ignores the possibility of nonstandard topologies, thek = 0 spatially flat 3-manifold is, in any sensible use of the word, “open”. If one allows nonstandardtopologies (by modding out by a suitable symmetry group) then there are, in any sensible use of theword, “closed” k = 0 spatially flat 3-manifolds (tori), and also “closed” k = −1 hyperbolic 3-manifolds.Finally the distinction between flat and spatially flat is important, and obscuring this distinction isdangerous.


Let us now focus on eq. (1.6) which, as already mentioned, governs the accelerationof the expansion. Notice that k does not appear in this equation, i.e. the accelerationdoes not depend on the characteristics of the spatial curvature. Eq. (1.6) reveals whatwould be expected by simple intuition: that gravity is always an attractive force. Letus see this in detail. The Newtonian analogue of eq. (1.6) would be

a

a= −4πG

3ρ, (1.7)

where ρ denotes the matter density. Due to the minus sign on the right hand side andthe positivity of the density, this equation implies that the expansion will always beslowed by gravity.

The presence of the pressure term in eq. (1.6) is simply due to the fact that in GeneralRelativity, it is not simply matter that gravitates but actually energy and therefore thepressure should be included. For what could be called ordinary matter (e.g. radiation,dust, perfect fluids, etc.) the pressure can be expected to be positive, as with the density.More precisely, one could ask that the matter satisfies the four energy conditions [263]:

1. Null Energy Condition: ρ+ p ≥ 0,

2. Weak Energy Condition: ρ ≥ 0, ρ+ p ≥ 0,

3. Strong Energy Condition: ρ+ p ≥ 0, ρ+ 3p ≥ 0,

4. Dominant Energy Condition: ρ ≥ |p|.

We give these conditions here in terms of the components of the stress-energy tensor ofa perfect fluid but they can be found in a more generic form in [263]. Therefore, oncepositivity of the pressure or the validity of the Strong Energy Condition is assumed,gravity remains always an attractive force also in General Relativity 3.

To sum up, even without attempting to solve the Friedmann equations, we havealready arrived at a well-established conclusion: Once we assume, according to the BigBang scenario, that the universe is expanding, then, according to General Relativityand with ordinary matter considerations, this expansion should always be decelerated.Is this what actually happens though?

1.3.2 The first need for acceleration

We derived the Friedmann equations using two assumptions: homogeneity and isotropyof the universe. Both assumptions seem very reasonable considering how the universeappears to be today. However, there are always the questions of why does the universeappear to be this way and how did it arrive at its present form through its evolution.

3When quantum effects are taken into account, one or more of the energy conditions can be violated,even though a suitably averaged version may still be satisfied. However, there are even classical fieldsthat can violate the energy conditions, as we will see latter on.

28 1. INTRODUCTION

More importantly though, one has to consider whether the description of the universeby the Big Bang model and the Friedmann equations is self-consistent and agrees notonly with a rough picture of the universe but also with the more precise current pictureof it.

Let us put the problem in more rigorous terms. First of all one needs to clarify whatis meant by “universe”. Given that the speed of light (and consequently of any signalcarrying information) is finite and adopting the Big Bang scenario, not every regionof spacetime is accessible to us. The age of the universe sets an upper limit for thelargest distance from which a point in space may have received information. This iswhat is called a “particle horizon” and its size changes with time. What we refer toas the universe is the part of the universe causally connected to us — the part insideour particle horizon. What happens outside this region is inaccessible to us but moreimportantly it does not affect us, at least not directly. However, it is possible to havetwo regions that are both accessible and causally connected to us, or to some otherobserver, but are not causally connected with each other. They just have to be insideour particle horizon without being inside each other’s particle horizons. It is intuitivethat regions that are causally connected can be homogeneous — they have had the timeto interact. However, homogeneity of regions which are not causally connected wouldhave to be attributed to some initial homogeneity of the universe since local interactionscannot be effective for producing this.

The picture of the universe that we observe is indeed homogeneous and isotropicon scales larger than we would expect based on our calculation regarding its age andcausality. This problem was first posed in the late 1960s and has been known as thehorizon problem [11, 13]. One could look to solve it by assuming that the universeis perhaps much older and this is why in the past the horizon problem has also beenreformulated in the form of a question: how did the universe grow to be so old? However,this would require the age of the universe to differ by orders of magnitude from the valueestimated by observations. So the homogeneity of the universe, at least at first sightand as long as we believe in the cosmological model at hand, appears to be built intothe initial conditions.

Another problem, which is similar and appeared at the same time, is the flatnessproblem. To pose it rigorously let us return to the Friedmann equations and morespecifically to eq. (1.5). The Hubble parameter H is defined as H = a/a. We can use itto define what is called the critical density

ρc =3H2

8π G, (1.8)

which is the density which would make the 3-geometry flat. Finally, we can use the


critical density in order to create the dimensionless fractions

Ω =ρ

ρc, (1.9)

Ωk = − k

a2H2. (1.10)

It is easy to verify from eq. (1.5) that

Ω + Ωk = 1. (1.11)

As dimensionless quantities, Ω and Ωk are measurable, and by the 1970s it was alreadyknown that the current value of Ω appears to be very close to 1 (see for example [14]).Extrapolating into the past reveals that Ω would have had to be even closer to 1, makingthe contribution of Ωk, and consequently of the k-term in eq. (1.5), exponentially small.

The name “flatness problem” can be slightly misleading and therefore it needs tobe clarified that the value of k obviously remains unaffected by the evolution. To avoidmisconceptions it is therefore better to formulate the flatness problems in terms of Ωitself. The fact that Ω seems to be taking a value so close to the critical one at earlytimes is not a consequence of the evolution and once more, as happened with the horizonproblem, it appears as a strange coincidence which can only be attributed to some finetuning of the initial conditions.

But is it reasonable to assume that the universe started in such a homogeneous state,even at scales that where not causally connected, or that its density was dramaticallyclose to its critical value without any apparent reason? Even if the universe started withextremely small inhomogeneities it would still not present such a homogeneous picturecurrently. Even if shortcomings like the horizon and flatness problems do not constitutelogical inconsistencies of the standard cosmological model but rather indicate that thepresent state of the universe depends critically on some initial state, this is definitely afeature that many consider undesirable.

So, by the 1970s Cosmology was facing new challenges. Early attempts to addressthese problems involved implementing a recurring or oscillatory behaviour for the uni-verse and therefore were departing from the standard ideas of cosmological evolution[15, 16, 17]. This problem also triggered Charles W. Misner to propose the “Mixmas-ter Universe” (Bianchi type IX metric), in which a chaotic behaviour was supposed toultimately lead to statistical homogeneity and isotropy [18]. However, all of these ideashave proved to be non-viable descriptions of the observed universe.

A possible solution came in the early 1980s when Alan Guth proposed that a periodof exponential expansion could be the answer [19]. The main idea is quite simple:an exponential increase of the scale factor a(t) implies that the Hubble parameter Hremains constant. On the other hand, one can define the Hubble radius c/H(t) which,roughly speaking, is a measure of the radius of the observable universe at a certaintime t. Then, when a(t) increases exponentially, the Hubble radius remains constant,whereas any physical length scale increases exponentially in size. This implies that in a

30 1. INTRODUCTION

short period of time, any lengthscale which could, for example, be the distance betweentwo initially causally connected observers, can become larger than the Hubble radius.So, if the universe passed through a phase of very rapid expansion, then the part ofit that we can observe today may have been significantly smaller at early times thanwhat one would naively calculate using the Friedmann equations. If this period lastedlong enough, then the observed universe could have been small enough to be causallyconnected at the very early stage of its evolution. This rapid expansion would also driveΩk to zero and consequently Ω to 1 today, due to the very large value that the scale factora(t) would currently have, compared to its initial value. Additionally, such a procedureis very efficient in smoothing out inhomogeneities, since the physical wavelength of aperturbation can rapidly grow to be larger than the Hubble radius. Thus, both of theproblems mentioned above seem to be effectively addressed.

Guth was not the only person who proposed the idea of an accelerated phase andsome will argue he was not even the first. Contemporaneously with him, Alexei Starobin-ski had proposed that an exponential expansion could be triggered by quantum correc-tions to gravity and provide a mechanism to replace the initial singularity [20]. Thereare also earlier proposals whose spirit is very similar to that of Guth, such as those byDemosthenes Kazanas [21], Katsuhiko Sato [22] and Robert Brout et al. [23]. However,Guth’s name is the one most related with these idea since he was the first to providea coherent and complete picture on how an exponential expansion could address thecosmological problems mentioned above. This period of accelerated expansion is knownas inflation, a terminology borrowed from economics due to the apparent similarity be-tween the growth of the scale factor in Cosmology and the growth of prices during aninflationary period. To be more precise, one defines as inflation any period in the cosmicevolution for which

a > 0. (1.12)

However, a more detailed discussion reveals that an exponential expansion, or atleast quasi-exponential since what is really needed is that the physical scales increasemuch more rapidly than the Hubble radius increases, is not something trivial to achieve.As discussed in the previous section, it does not appear to be easy to trigger such an erain the evolution of the universe, since accelerated expansion seems impossible accordingto eq. (1.6), as long as both the density and the pressure remain positive. In otherwords, satisfying eq. (1.12) requires

(ρ+ 3p) < 0⇒ ρ < −3p, (1.13)

and assuming that the energy density cannot be negative, inflation can only be achievedif the overall pressure of the ideal fluid which we are using to describe the universebecomes negative. In more technical terms, eq. (1.13) implies the violation of the StrongEnergy Condition [263].

It does not seem possible for any kind of baryonic matter to satisfy eq. (1.13), whichdirectly implies that a period of accelerated expansion in the universe evolution can


only be achieved within the framework of General Relativity if some new form of matterfield with special characteristics is introduced. Before presenting any scenario of thissort though, let us resort to observations to convince ourselves about whether such acosmological era is indeed necessary.

1.3.3 Cosmological and Astronomical Observations

In reviewing the early theoretical shortcomings of the Big Bang evolutionary model ofthe universe we have seen indications for an inflationary era. The best way to confirmthose indications is probably to resort to the observational data at hand for having averification. Fortunately, there are currently very powerful and precise observations thatallow us to look back to very early times.

A typical example of such observations is the Cosmic Microwave Background Ra-diation (CMBR). In the early universe, baryons, photons and electrons formed a hotplasma, in which the mean free path of a photon was very short due to constant in-teractions of the photons with the plasma through Thomson scattering. However, dueto the expansion of the universe and the subsequent decrease of temperature, it subse-quently became energetically favourable for electrons to combine with protons to formhydrogen atoms (recombination). This allowed photons to travel freely through space.This decoupling of photons from matter is believed to have taken place at a redshift ofz ∼ 1088, when the age of the universe was about 380, 000 years old or approximately13.7 billion years ago. The photons which left the last scattering surface at that time,then travelled freely through space and have continued cooling since then. In 1965 Pen-zias and Wilson noticed that a Dicke radiometer which they were intending to use forradio astronomy observations and satellite communication experiments had an excess3.5K antenna temperature which they could not account for. They had, in fact, detectedthe CMBR, which actually had already been theoretically predicted in 1948 by GeorgeGamow. The measurement of the CMBR, apart from giving Penzias and Wilson a Nobelprize publication [24], was also to become the number one verification of the Big Bangmodel.

Later measurements showed that the CMBR has a black body spectrum correspond-ing to approximately 2.7 K and verifies the high degree of isotropy of the universe. How-ever, it was soon realized that attention should be focused not on the overall isotropy,but on the small anisotropies present in the CMBR, which reveal density fluctuations[25, 26]. This triggered a numbered of experiments, such as COBE, Toco, BOOMERanGand MAXIMA [27, 28, 29, 30, 31, 32]. The most recent one is the Wilkinson MicrowaveAnisotropy Probe (WMAP) [33] and there are also new experiments planned for thenear future, such as the Planck mission [34].

The density fluctuations indicated by the small anisotropies in the temperature ofCMBR are believed to act as seeds for gravitational collapse, leading to gravitationallybound objects which constitute the large scale matter structures currently present inthe universe [35]. This allows us to build a coherent scenario about how these structures

32 1. INTRODUCTION

were formed and to explain the current small scale inhomogeneities and anisotropies.Besides the CMBR, which gives information about the initial anisotropies, one canresort to galaxy surveys for complementary information. Current surveys determiningthe distribution of galaxies include the 2 degree Field Galaxy Redshift Survey (2dFGRS) [36] and the ongoing Sloan Digital Sky Survey (SDSS) [37]. There are also othermethods used to measure the density variations such as gravitational lensing [38] andX-ray measurements [39].

Besides the CMBR and Large Scale Structure surveys, another class of observationsthat appears to be of special interest in Cosmology are those of type Ia supernovae. Theseexploding stellar objects are believed to be approximately standard candles, i.e. astro-nomical objects with known luminosity and absolute magnitude. Therefore, they can beused to reveal distances, leading to the possibility of forming a redshift-distance relationand thereby measuring the expansion of the universe at different redshifts. For thispurpose, there are a number of supernova surveys [40, 41, 42].

But let us return to how we can use the outcome of the experimental measurementsmentioned above in order to infer whether a period of accelerated expansion has oc-curred. The most recent CMBR dataset is that of the Three-Year WMAP Observations[43] and results are derived using combined WMAP data and data from supernova andgalaxy surveys in many cases. To begin with, let us focus on the value of Ωk. TheWMAP data (combined with Supernova Legacy Survey data [41]) indicates that

Ωk = −0.015+0.020−0.016, (1.14)

i.e. that Ω is very close to unity and the universe appears to be spatially flat, while thepower spectrum of the CMBR appears to be consistent with gaussianity and adiabatic-ity [44, 45]. Both of these facts are in perfect agreement with the predictions of theinflationary paradigm.

In fact, even though the theoretical issues mentioned in the previous paragraph(i.e. the horizon and the flatness problem) were the motivations for introducing theinflationary paradigm, it is the possibility of relating large scale structure formation withinitial quantum fluctuations that appears today as the major advantage of inflation [46].Even if one would choose to dismiss, or find another way to address, problems relatedto the initial conditions, it is very difficult to construct any other theory which couldsuccessfully explain the presence of over-densities with values suitable for leading to thepresent picture of our universe at smaller scales [35]. Therefore, even though it mightbe premature to claim that the inflationary paradigm has been experimentally verified,it seems that the evidence for there having been a period of accelerated expansion ofthe universe in the past is very compelling.

However, observational data hold more surprises. Even though Ω is measured to bevery close to unity, the contribution of matter to it, Ωm, is only of the order of 24%.Therefore, there seems to be some unknown form of energy in the universe, often calleddark energy. What is more, observations indicate that, if one tries to model dark energy


as a perfect fluid with equation of state p = wρ then

wde = −1.06+0.13−0.08, (1.15)

so that dark energy appears to satisfy eq. (1.13). Since it is the dominant energycomponent today, this implies that the universe should be undergoing an acceleratedexpansion currently as well. This is also what was found earlier using supernova surveys[40].

As is well known, between the two periods of acceleration (inflation and the currentera) the other conventional eras of evolutionary Cosmology should take place. Thismeans that inflation should be followed by Big Bang Nucleosynthesis (BBN), referringto the production of nuclei other than hydrogen. There are very strict bounds onthe abundances of primordial light elements, such as deuterium, helium and lithium,coming from observations [47] which do not seem to allow significant deviations fromthe standard cosmological model [48]. This implies that BBN almost certainly tookplace during an era of radiation domination, i.e. a period in which radiation was themost important contribution to the energy density. On the other hand, the formationof matter structures requires that the radiation dominated era is followed by a matterdominated era. The transition, from radiation domination to matter domination, comesnaturally since the matter energy density is inversely proportional to the volume and,therefore, proportional to a−3, whereas the radiation energy density is proportional toa−4 and so it decreases faster than the matter energy density as the universe expands.

To sum up, our current picture of the evolution of the universe as inferred fromobservations comprises a pre-inflationary (probably quantum gravitational) era followedby an inflationary era, a radiation dominated era, a matter dominated era and then asecond era of accelerated expansion which is currently taking place. Such an evolutiondeparts seriously from the one expected if one just takes into account General Relativityand conventional matter and therefore appears to be quite unorthodox.

But puzzling observations do not seem to stop here. As mentioned before, Ωm

accounts for approximately 24% of the energy density of the universe. However, one alsohas to ask how much of this 24% is actually ordinary baryonic matter. Observationsindicate that the contribution of baryons to that, Ωb, is of the order of Ωb ∼ 0.04 leavingsome 20% of the total energy content of the universe and some 83% of the mattercontent to be accounted for by some unknown unobserved form of matter, called darkmatter. Differently from dark energy, dark matter has the gravitational characteristicsof ordinary matter (hence the name) and does not violate the Strong Energy Condition.However, it is not directly observed since it appears to interact very weakly if at all.

The first indications for the existence of dark matter did not come from Cosmology.Historically, it was Fritz Zwicky who first posed the “missing mass” question for theComa cluster of galaxies [49, 50] in 1933. After applying the virial theorem in order tocompute the mass of the cluster needed to account for the motion of the galaxies nearto its edges, he compared this with the mass obtained from galaxy counts and the total

34 1. INTRODUCTION

brightness of the cluster. The virial mass turned out to be larger by a factor of almost400.

Later, in 1959, Kahn and Waljter were the first to propose the presence of darkmatter in individual galaxies [51]. However, it was in the 1970s that the first compellingevidence for the existence of dark matter came about: the rotation curves of galaxies,i.e. the velocity curves of stars as functions of the radius, did not appear to have theexpected shapes. The velocities, instead of decreasing at large distances as expected fromKeplerian dynamics and the fact that most of the visible mass of a galaxy is locatednear to its centre, appeared to be flat [52, 53, 54]. As long as Keplerian dynamics areconsidered correct, this implies that there should be more matter than just the luminousmatter, and this additional matter should have a different distribution within the galaxy(dark matter halo).

Much work has been done in the last 35 years to analyse the problem of dark matter inastrophysical environments (for recent reviews see [55, 56, 57]) and there are also recentfindings, such as the observations related to the Bullet Cluster, that deserve a specialmention4. The main conclusion that can be drawn is that some form of dark matteris present in galaxies and clusters of galaxies. What is more, taking also into accountthe fact that dark matter appears to greatly dominate over ordinary baryonic matterat cosmological scales, it is not surprising that current models of structure formationconsider it as a main ingredient (e.g. [59]).

1.3.4 The Cosmological Constant and its problems

We have just seen some of the main characteristics of the universe as inferred fromobservations. Let us now set aside for the moment the discussion of the earlier epochsof the universe and inflation and concentrate on the characteristic of the universe asit appears today: it is probably spatially flat (Ωk ∼ 0), expanding in a acceleratedmanner as confirmed both from supernova surveys and WMAP, and its matter energycomposition consists of approximately 76% dark energy, 20% dark matter and only 4%ordinary baryonic matter. One has to admit that this picture is not only surprising butmaybe even embarrassing, since it is not at all close to what one would have expectedbased on the standard cosmological model and what is more it reveals that almost 96%of the energy content of the universe has a composition which is unknown to us.

In any case, let us see which is the simplest model that agrees with the observationaldata. To begin with, we need to find a simple explanation for the accelerated expansion.The first physicist to consider a universe which exhibits an accelerated expansion wasprobably Willem de Sitter [60]. A de Sitter space is the maximally symmetric, simply-connected, Lorentzian manifold with constant positive curvature. It may be regardedas the Lorentzian analogue of an n-sphere in n dimensions. However, the de Sitter

4Weak lensing observations of the Bullet cluster (1E0657-558), which is actually a unique clustermerger, appear to provide direct evidence for the existence of dark matter [58].


spacetime is not a solution of the Einstein equations, unless one adds a cosmologicalconstant Λ to them, i.e. adds on the left hand side of eq. (1.1) the term Λgµν .

Such a term was not included initially by Einstein, even though this is technicallypossible since, according to the reasoning which he gave for arriving at the gravitationalfield equations, the left hand side has to be a second rank tensor constructed fromthe Ricci tensor and the metric, which is divergence free. Clearly, the presence of acosmological constant does not contradict these requirements. In fact, Einstein wasthe first to introduce the cosmological constant, thinking that it would allow him toderive a solution of the field equations describing a static universe [61]. The idea ofa static universe was then rapidly abandoned however when Hubble discovered thatthe universe is expanding and Einstein appears to have changed his mind about thecosmological constant: Gamow quotes in his autobiography, My World Line (1970):“Much later, when I was discussing cosmological problems with Einstein, he remarkedthat the introduction of the cosmological term was the biggest blunder of his life” andPais quotes a 1923 letter of Einstein to Weyl with his reaction to the discovery ofthe expansion of the universe: “If thereis no quasi-static world, then away with thecosmological term!” [62].

In any case, once the cosmological term is included in the Einstein equations, deSitter space becomes a solution. Actually, the de Sitter metric can be brought into theform of the FLRW metric in eq. (1.3) with the scale factor and the Hubble parametergiven by5.

a(t) = eH t, (1.16)

H2 =8π G

3Λ. (1.17)

This is sometimes referred to as the de Sitter universe and it can be seen that it isexpanding exponentially.

The de Sitter solution is a vacuum solution. However, if we allow the cosmologicalterm to be present in the field equations, the Friedmann equations (1.5) and (1.6) willbe modified so as to include the de Sitter spacetime as a solution:

(a

a

)2

=8π Gρ+ Λ

3− k

a2, (1.18)

a

a=

Λ

3− 4π G

3(ρ+ 3p) . (1.19)

From eq. (1.19) one infers that the universe can now enter a phase of accelerated expan-sion once the cosmological constant term dominates over the matter term on the righthand side. This is bound to happen since the value of the cosmological constant stays

5Note that de Sitter space is an example of a manifold that can be sliced in 3 ways — k = +1, k = 0,k = −1 — with each coordinate patch covering a different portions of spacetime. We are referring herejust to the k = 0 slicing for simplicity.

36 1. INTRODUCTION

unchanged during the evolution, whereas the matter density decreases like a3. In otherwords, the universe is bound to approach a de Sitter space asymptotically in time.

On the other hand Ω in eq. (1.11) can now be split in two different contributions,ΩΛ = Λ/(3H2) and Ωm, so that eq. (1.11) takes the form

Ωm + ΩΛ + Ωk = 1. (1.20)

In this sense, the observations presented previously can be interpreted to mean thatΩΛ ∼ 0.72 and the cosmological constant can account for the mysterious dark energyresponsible for the current accelerated expansion. One should not fail to notice thatΩm does not only refer to baryons. As mentioned before, it also includes dark matter,which is actually the dominant contribution. Currently, dark matter is mostly treated asbeing cold and not baryonic, since these characteristics appear to be in good accordancewith the data. This implies that, apart from the gravitational interaction, it does nothave other interactions — or at least that it interacts extremely weakly — and can beregarded as collisionless dust, with an effective equation of state p = 0 (we will returnto the distinction between cold and hot dark matter shortly).

We have sketched our way to what is referred to as the Λ Cold Dark Matter orΛCDM model. This is a phenomenological model which is sometimes also called theconcordance model of Big Bang Cosmology, since it is more of an empirical fit to the data.It is the simplest model that can fit the cosmic microwave background observations aswell as large scale structure observations and supernova observations of the acceleratingexpansion of the universe with a remarkable agreement (see for instance [43]). As aphenomenological model, however, it gives no insight about the nature of dark matter,or the reason for the presence of the cosmological constant, neither does it justify thevalue of the latter.

While it seems easy to convince someone that an answer is indeed required to thequestion “what exactly is dark matter and why is it almost 9 times more abundant thanordinary matter”, the presence of the cosmological constant in the field equations mightnot be so disturbing for some. Therefore, let us for the moment put aside the darkmatter problem — we will return to it shortly — and consider how natural it is to tryto explain the dark energy problem by a cosmological constant (see [63, 64, 65, 66] forreviews).

It has already been mentioned that there is absolutely no reason to discard the pres-ence of a cosmological constant in the field equations from a gravitational and mathe-matical perspective. Nonetheless, it is also reasonable to assume that there should bea theoretical motivation for including it — after all there are numerous modificationsthat could be made to the left hand side of the gravitational field equation and still leadto a consistent theory from a mathematical perspective and we are not aware of anyother theory that includes more than one fundamental constant. On the other hand, itis easy to see that the cosmological term can be moved to the right hand side of thefield equations with the opposite sign and be regarded as some sort of matter term.It can then be put into the form of a stress-energy tensor T µ

ν = diag(Λ,−Λ,−Λ,−Λ),


i.e. resembling a perfect fluid with equation of state p = −ρ or w = −1. Notice thevery good agreement with the value of wde inferred from observations (eq. (1.15)), whichexplains the success of the ΛCDM model.

Once the cosmological constant term is considered to be a matter term, a naturalexplanation for it seems to arise: the cosmological constant can represent the vacuumenergy associated with the matter fields. One should not be surprised that empty spacehas a non-zero energy density once, apart from General Relativity, field theory is alsotaken into consideration. Actually, Local Lorentz Invariance implies that the expectationvalue of the stress energy tensor in vacuum is

〈Tµν〉 = −〈ρ〉gµν , (1.21)

and 〈ρ〉 is generically non-zero. To demonstrate this, we can take the simple example ofa scalar field [67]. Its energy density will be

ρφ =1

2φ2 +

1

2(∇spφ)2 + V (φ), (1.22)

where ∇sp denotes the spatial gradient and V (φ) is the potential. The energy densitywill become constant for any constant value φ = φ0 and there is no reason to believethat for φ = φ0, V (φ0) should be zero. One could in general assume that there is someprinciple or symmetry that dictates it, but nothing like this has been found up to now.So in general one should expect that matter fields have a non-vanishing vacuum energy,i.e. that 〈ρ〉 is non-zero.

Within this perspective, effectively there should be a cosmological constant in thefield equations, given by

Λ = 8π G〈ρ〉. (1.23)

One could, therefore, think to use the Standard Model of particle physics in order toestimate its value. Unfortunately, however, 〈ρ〉 actually diverges due to the contributionof very high-frequency modes. No reliable exact calculation can be made but it is easyto make a rough estimate once a cutoff is considered (see for instance [11, 67]). Takingthe cutoff to be the Planck scale (MPlank = 1018 GeV), which is a typical scale at whichthe validity of classical gravity is becoming questionable, the outcome is

ρΛ ∼ (1027 eV)4. (1.24)

On the other hand, observations indicate that

ρΛ ∼ (10−3 eV)4. (1.25)

Obviously the discrepancy between these two estimates is very large for being attributedto any rough approximation. There is a difference of 120-orders-of-magnitude, which islarge enough to be considered embarrassing. One could validly claim that we should notbe comparing energy densities but mass scales by considering a mass scale for the vacuum

38 1. INTRODUCTION

implicitly defined through ρΛ = M4Λ. However, this will not really make a difference,

since a 30-orders-of-magnitude discrepancy in mass scale hardly makes a good estimate.This constitutes the so-called cosmological constant problem.

Unfortunately, this is not the only problem related to the cosmological constant. Theother known problem goes under the name of the coincidence problem. It is apparentfrom the data that ΩΛ ∼ 0.72 and Ωm ∼ 0.28 have comparable values today. However,as the universe expands their fractional contributions change rapidly since

ΩΛ

Ωm=ρΛ

ρm∝ a3. (1.26)

Since Λ is a constant, ρΛ should once have been negligible compared to the energydensities of both matter and radiation and, as dictated by eq. (1.26), it will come todominate completely at some point in the late time universe. However, the striking factis that the period of transition between matter domination and cosmological constantdomination is very short compared to cosmological time scales6. The puzzle is, there-fore, why we live precisely in this very special era [67]. Obviously, the transition frommatter domination to cosmological constant domination, or, alternatively stated, fromdeceleration to acceleration, would happen eventually. The question is, why now?

To sum up, including a cosmological constant in the field equations appears as aneasy way to address issues like the late time accelerated expansion but unfortunatelyit comes with a price: the cosmological constant and coincidence problems. We willreturn to this discussion from this point later on but for the moment let us close thepresent section with an overall comment about the ΛCDM model. Its value shoulddefinitely not be underestimated. In spite of any potential problems that it may have,it is still a remarkable fit to observational data while at the same time being elegantlysimple. One should always bear in mind how useful a simple empirical fit to the datamay be. On the other hand, the ΛCDM model should also not be over-estimated. Beinga phenomenological model, with poor theoretical motivation at the moment, one shouldnot necessarily expect to discover in it some fundamental secrets of nature.

1.4 Is there a way out?

In the previous sections, some of the most prominent problems of contemporary physicswere presented. As one would expect, since these questions were initially posed, manyattempts to address one or more of them have been pursued. These problems may beviewed as being unrelated to each other, or grouped in different categories at will. Forinstance, one could follow a broad research field grouping, much like the one attempted inthe previous section, dividing them into problems related with Cosmology and problemsrelated with high energy physics, or group them according to whether they refer to

6Note that in the presence of a positive cosmological constant there is an infinite future in which Λis dominating.

1.4. IS THERE A WAY OUT? 39

unexplained observations or theoretical shortcomings. In any case there is one commondenominator in all of these problems. They are all somehow related to gravity.

The way in which one chooses to group or divide these problems proposes a naturalpath to follow for their solution. In this section let us very briefly review some of themost well-known and conventional solutions proposed in the literature, which mainlyassume that all or at least most of these issues are unrelated. Then we can proceed toargue why and how the appearance of so many yet to be explained puzzles related togravity and General Relativity may imply that there is something wrong with our currentunderstanding of the gravitational interaction even at a classical level, resembling thehistorically recorded transition from Newtonian gravity to General Relativity describedin section 1.1. With that we will conclude this introductory chapter.

1.4.1 Scalar fields as matter fields in Cosmology

We have already discussed the need for an inflationary period in the early universe.However, we have not yet attempted to trace the cause of such an accelerated expansion.Since the presence of a cosmological constant could in principle account for that, one istempted to explore this possibility, as in the case of late time acceleration. Unfortunately,this simple solution is bound not to work for a very simple reason: once the cosmologicalconstant dominates over matter there is no way for matter to dominate again. Inflationhas to end at some point, as already mentioned, so that Big Bang Nucleosynthesis andstructure formation can take place. Our presence in the universe is all the evidenceone needs for that. Therefore, one is forced to seek other, dynamical solutions to thisproblem.

As long as one is convinced that gravity is well described by General Relativity, theonly option left is to assume that it is a matter field that is responsible for inflation.However, this matter field should have a rather unusual property: its effective equationof state should satisfy eq. (1.13), i.e. it should have a negative pressure and actuallyviolate the Strong Energy Condition. Fortunately, matter fields with this property doexist. A typical simple example is a scalar field φ.

A scalar field minimally coupled to gravity, satisfies the Klein–Gordon equation

∇2φ+ V ′(φ) = 0, (1.27)

where ∇µ denotes the covariant derivative, ∇2 ≡ ∇µ∇µ, V (φ) is the potential and theprime denotes partial differentiation with respect to the argument. Assuming that thescalar field is homogeneous and therefore φ ≡ φ(t) we can write its energy density andpressure as

ρφ =1

2φ2 + V (φ), (1.28)

pφ =1

2φ2 − V (φ), (1.29)

40 1. INTRODUCTION

while, in a FLRW spacetime, eq. (1.27) takes the following form:

φ+ 3Hφ+ V ′(φ) = 0. (1.30)

It is now apparent that if φ2 < V (φ) then the pressure is indeed negative. In factwφ = pφ/ρφ approaches −1 when φ2 ≪ V (φ).

In general a scalar field that leads to inflation is referred to as the inflaton. Sincewe invoked such a field instead of a cosmological constant, claiming that in this way wecan successfully end inflation, let us see how this is achieved. Assuming that the scalardominates over both matter and radiation and neglecting for the moment the spatialcurvature term for simplicity, eq. (1.5) takes the form

H2 ≈ 8π G

3

(1

2φ2 + V (φ)

). (1.31)

If, together with the condition φ2 < V (φ), we require that φ is negligible in eq. (1.30)then eqs. (1.31) and (1.30) reduce to

H2 ≈ 8π G

3V (φ), (1.32)

3Hφ ≈ −V ′(φ). (1.33)

This constitutes the slow-roll approximation since the potential terms are dominant withrespect to the kinetic terms, causing the scalar to roll slowly from one value to another.To be more rigorous, one can define two slow-roll parameters

ε(φ) = 4π G

(V ′

V

)2

, (1.34)

η(φ) = 8π GV ′′

V, (1.35)

for which the conditions ε(φ)≪ 1 and η(φ)≪ 1 are necessary in order for the slow-rollapproximation to hold [68, 69]. Note that these are not sufficient conditions since theyonly restrict the form of the potential. One also has to make sure that eq. (1.33) issatisfied. In any case, what we want to focus on at this point is that one can startwith a scalar that initially satisfies the slow-roll conditions but, after some period, φcan be driven to such a value so as to violate them. A typical example is that ofV (φ) = m2φ2/2, where these conditions are satisfied as long as φ2 > 16π G but, as φapproaches the minimum of the potential, a point will be reached where φ2 > 16π Gwill cease to hold. Once the slow-roll conditions are violated, inflation can be naturallydriven to an end since φ2 can begin to dominate again in eq. (1.29).

However, just ending inflation is not enough. After such an era the universe wouldbe a cold and empty place unable to evolve dynamically to anything close to the picturewhich we observe today. A viable model for inflation should include a mechanism that


will allow the universe to return to the standard Big Bang scenario. This mechanismis called reheating and consists mainly of three processes: a period of non-inflationaryscalar field dynamics, after the slow-roll approximation has ceased to be valid, thecreation and decay of inflaton particles and the thermalization of the products of thisdecay [35]. Reheating is an extensive and intricate subject and analyzing it goes beyondthe scope of this introduction. We refer the reader to [70, 71, 72, 73, 74, 75, 76] for moreinformation.

On the same grounds, we will refrain here from mentioning specific models for in-flation and from discussing subtleties with using inflation in order to address problemsof initial conditions such as those stated in paragraph 1.3.2. We refer the reader to theliterature for further reading [75, 76, 77, 340, 79].

Before closing this paragraph, it should be mentioned that scalar fields can be usedto account for the late-time accelerated expansion of the universe in the same way asthe inflaton is used in inflationary models. Since, however, this subject overlaps withthe subject of dark energy, we will discuss it in the next sub-section which is dedicatedto the dark energy problem.

1.4.2 The dark energy problem

We have already seen that there seems to be compelling observational evidence that theuniverse is currently undergoing an accelerated expansion and we have also discussedthe problems that arise if a cosmological constant is considered to be responsible for thisacceleration within the framework of the ΛCDM model. Based on that, one can classifythe attempts to address the problem of finding a mechanism that will account for thelate-time accelerated expansion in two categories: those that try to find direct solutionsto the cosmological constant and the coincidence problems and consequently attemptto provide an appealing theoretical explanation for the presence and the value of thecosmological constant, and those that abandon the idea of the cosmological constantaltogether and attempt to find alternative ways to explain the acceleration.

Let us state two of the main approaches followed to solve the cosmological constantproblem directly:

The first approach resorts to High Energy Physics. The general idea is simple andcan be summed up in the question: Are we counting properly? This refers to thequite naive calculation mentioned previously, according to which the energy densityof the cosmological constant as calculated theoretically should be 10120 times largerthan its observed value. Even though the question is simple and reasonable, givinga precise answer to it is actually very complicated since, as mentioned already, littleis known about how to make an exact calculation of the vacuum energy of quantumfields. There are indications coming from contemporary particle physics theories, suchas supersymmetry (SUSY), which imply that one can be led to different values for theenergy density of vacuum from the one mentioned before (eq. (1.24)). For instance,since no superpartners of known particles have been discovered in accelerators, one can

42 1. INTRODUCTION

assume that supersymmetry was broken at some scale of the order of 103GeV or higher.If this is the case, one would expect that

ρΛ ∼M4SUSY ≥ (1012eV)4. (1.36)

This calculation gives an estimate for the energy density of the vacuum which is 60orders of magnitude smaller than the one presented previously in eq. (1.24). However,the value estimated here is still 60 orders of magnitude larger than the one inferred fromobservations (eq. (1.25)). Other estimates with or without a reference to supersymmetryor based on string theory or loop quantum gravity exist. One example is the approachof Ref. [80] where an attempt is made to use our knowledge from condensed mattersystems in order to explain the value of the cosmological constant. We will not, however,list further examples here but refer the reader to [63, 65] and references therein formore details. In any case, the general flavour is that it is very difficult to avoid thecosmological constant problem by following such approaches without making some finetuning within the fundamental theory used to perform the calculation for the energydensity of vacuum. Also, such approaches mostly fail to address the second problemrelated to the cosmological constant: the coincidence problem.

he second direct approach for solving problems related to the cosmological constanthas a long history and was given the name “anthropic principle” by Brandon Carter[81, 82, 83]. Unfortunately, the anthropic principle leaves a lot of room for differentformulations or even misinterpretations. Following [63] we can identify at least threeversions, starting from a very mild one, that probably no one really disagrees with butis not very useful for answering questions, stating essentially that our mere existencecan potentially serve as an experimental tool. The second version on the other hand is arather strong one, stating that the laws of nature are by themselves incomplete and be-come complete only if one adds the requirement that conditions should allow intelligentlife to arise, for only in the presence of intelligent life does science become meaningful.It is apparent that such a formulation places intelligent life or science at the centre ofattention as far as the universe is concerned. From this perspective one cannot help butnotice that the anthropic principle becomes reminiscent of the Ptolemaic model. Addi-tionally, to quote Weinberg: “...although science is clearly impossible without scientists,it is not clear that the universe is impossible without science”. The third and mostmoderate version of the anthropic principle, known as the “weak anthropic principle”states essentially that observers will only observe conditions which allow for observers.This version is the one mostly discussed from a scientific perspective and even thoughit might seem tautological, it acquires a meaning if one invokes probability theory.

To be more concrete, as opposed to the second stronger formulation, the weak an-thropic principle does not assume some sort of conspiracy of nature aimed at creatingintelligent life. It merely states that, since the existence of intelligent observes requirescertain conditions, it is not possible for them in practice to observe any other condi-tions, something that introduces a bias in any probabilistic analysis. This, of course,requires one extra assumption: that parts of the universe, either in space or time, might


indeed be in alternative conditions. Unfortunately we cannot conclude at this pointwhether this last statement is true. Assuming that it is, one could put constrains onthe value of the cosmological constant by requiring that it should be small enough forgalaxies to form as in [84] and arrive at the conclusion that the currently observed valueof the cosmological constant is by no means unlikely. Some modern theories do allowsuch alternative states of the universe to co-exist (multiverse), and for this reason it hasrecently been argued that the anthropic principle could even be placed on firm groundby using the ideas of string theory for the “anthropic or string landscape”, consistingof a large number of different false vacua [85]. However, admitting that there are limitson our ability to unambiguously and directly explain the observable universe inevitablycomes with a disappointment. It is for this reason that many physicists would refrainfrom using the anthropic principle or at least they would consider it only as a last resort,when all other possibilities have failed.

Let us now proceed to the indirect ways of solving problems related with the cosmo-logical constant. As already mentioned, the main approach of this kind is to dismiss thecosmological constant completely and assume that there is some form of dynamical darkenergy. In this sense, dark energy and vacuum energy are not related and therefore thecosmological constant problem ceases to exist, at least in the strict formulation givenabove. However, this comes with a cost: as mentioned previously, observational dataseem to be in very good agreement with having a cosmological constant, therefore im-plying that any form of dynamical dark energy should be able to mimic a cosmologicalconstant very precisely at present times. This is not something easy to achieve. In orderto be clearer and also to have the possibility to discuss how well dynamical forms ofdark energy can address the cosmological constant and coincidence problems, let us usean example.

Given the discussion presented earlier about inflation, it should be clear by nowthat if a matter field is to account for accelerated expansion, it should have a specialcharacteristic: negative pressure or more precisely p ≤ −ρ/3. Once again, as in theinflationary paradigm, the obvious candidate is a scalar field. When such a field is usedto represent dark energy it is usually called quintessence [86, 87, 88, 89, 90, 91, 92, 93, 94].Quintessence is one of the simplest and probably the most common alternative to thecosmological constant.

If the scalar field is taken to be spatially homogeneous, its equation of motion inan FLRW spacetime will be given by eq. (1.30) and its energy density and pressurewill be given by eqs. (1.28) and (1.29) respectively, just like the inflaton. As dictatedby observations through eq. (1.15), a viable candidate for dark energy should have aneffective equation of state with w very close to minus one. In the previous section itwas mentioned that this can be achieved for a scalar field if the condition φ2 ≪ V (φ)holds. This should not be confused with the slow-roll condition for inflation, whichjust requires that φ2 < V (φ) and also places a constraint for φ. However, there is asimilarity in the spirit of the two conditions, namely that in both cases the scalar fieldis required, roughly speaking, to be slowly-varying. It is worth mentioning that the

44 1. INTRODUCTION

condition φ2 ≪ V (φ) effectively restricts the form of the potential V .Let us see how well quintessence can address the cosmological constant problem.

One has to bear in mind that the value given in eq. (1.25) for the energy density ofthe cosmological constant now becomes the current value of the energy density of thescalar ρφ. Since we have asked that the potential terms should be very dominant withrespect to the kinetic terms, this value for the energy density effectively constrains thecurrent value of the potential. What is more, the equation of motion for the scalar field,eq. (1.30) is that of a damped oscillator, 3Hφ being the friction term. This impliesthat, for φ to be rolling slowly enough so that φ2 ≪ V (φ) could be satisfied, thenH ∼

√V ′′(φ). Consequently, this means that the current value of V ′′(φ) should be

that of the observed cosmological constant or, taking also into account that√V ′′(φ)

represents the effective mass of the scalar mφ, that

mφ ∼ 10−33 eV. (1.37)

Such a small value for the mass of the scalar field raises doubts about whetherquintessence really solves the cosmological constant problem or actually just transfers itfrom the domain of Cosmology to the domain of particle physics. The reason for this isthat the scalar fields usually present in quantum field theory have masses many ordersof magnitude larger than that given in eq. (1.37) and, hence, this poses a naturalnessquestion (see [65] for more details). For instance, one of the well-known problems inparticle physics, the hierarchy problem, concerns explaining why the Higgs field appearsto have a mass of 1011 eV which is much smaller that the grand unification/Planckscale, 1025-1028 eV. As commented in [67], one can then imagine how hard it couldbe to explain the existence of a field with a mass equal to 10−33 eV. In all fairness toquintessence, however, it should be stated that the current value of the energy densityof dark energy (or vacuum, depending on the approach) is an observational fact, and soit does not seem possible to completely dismiss this number in some way. All that isleft to do, therefore, is to put the cosmological constant problem on new grounds thatwill hopefully be more suitable for explaining it.

One should not forget, however, also the coincidence problem. There are attemptsto address it within the context of quintessence mainly based on what is referred to astracker models [95, 96, 97, 98, 99, 100, 101]. These are specific models of quintessencewhose special characteristic is that the energy density of the scalar parallels that ofmatter or radiation for a part of the evolution which is significant enough so as toremove the coincidence problem. What is interesting is that these models do not ingeneral require specific initial conditions, which means that the coincidence problem isnot just turned into an initial conditions fine-tuning problem. Of course, the dependenceof such approaches on the parameters of the potential remains inevitable.

It is also worth mentioning that φ should give rise to some force, which judgingfrom its mass should be long-range, if the scalar couples to ordinary matter. From aparticle physics point of view, one could expect that this is indeed the case, even ifthose interactions would have to be seriously suppressed by powers of the Planck scale


[102, 103]. However, current limits based on experiments concerning a fifth-force or timedependence of coupling constants, appear to be several orders of magnitude lower thanthis expectation [102, 103]. This implies that, if quintessence really exists, then thereshould be a mechanism — probably a symmetry — that suppresses these couplings.

Yet another possibility for addressing the cosmological constant problems, or moreprecisely for dismissing them, comes when one adopts the approach that the acceleratedexpansion as inferred by observations is not due to some new physics but is actuallydue to a misinterpretation or an abuse of the underlying model being used. The BigBang model is based on certain assumptions, such as homogeneity and isotropy, andapparently all calculations made rely on these assumptions. Even though at present onecannot claim that there is compelling evidence for this, it could be, for example, thatthe role of inhomogeneities is underestimated in the standard cosmological model anda more detailed model may provide a natural solution to the problem of dark energy,even by changing our interpretation of current observations (for instance see [104] andreferences therein).

1.4.3 The dark matter problem

As we have already seen, the presence of dark matter is indirectly inferred from ob-servations through its gravitational interaction. Therefore, if one accepts that GeneralRelativity describes gravity correctly, then an explanation for the nature of dark matteras some form of matter yet to be observed in the universe or in the laboratory shouldbe given. Note that dark matter is used here generically to mean matter that does notemit light. So, to begin with, its nature could be either baryonic and non-baryonic. Thecandidates for baryonic dark matter are mostly quite conventional astrophysical objectssuch as brown dwarfs, massive black holes and cold diffuse gas. However, there is preciseevidence from observations that only a small fraction of dark matter can be baryonic(see for example [43] and [105, 106] for reviews). Therefore, the real puzzle regards thenature of non-baryonic dark matter.

One can separate the candidates into two major categories: hot dark matter, i.e. non-baryonic particles which move (ultra-)relativistically, and cold dark matter i.e. non-baryonic particles which move non-relativistically. The most prominent candidate forhot dark matter is the the neutrino. However, studies of the cosmic microwave back-ground, numerical simulations and other astrophysical observations indicate that darkmatter has clumped to form some structures on rather small scales and therefore it can-not consist mainly of particles with high velocities, since this clumping would then havebeen suppressed (see for example [107, 108] and references in [106]). For this reason,and because of its simplicity, cold dark matter currently gives the favoured picture.

There are many cold dark matter candidates and so we will refrain from listing themall or discussing their properties in detail here and refer the reader to the literature[106]. The most commonly considered ones are the axion and a number of weaklyinteracted massive particles (WIMPs) naturally predicted in supersymmetry theories,

46 1. INTRODUCTION

such as the neutralino, the sneutrino, the gravitino, the axino etc. There are a numberof experiments aiming for direct and indirect detection of dark matter and some ofthem, such as the DAMA/NaI experiment [109], even claim to have already achievedthat (see [110] for a full list of dark matter detection experiments and [105] for a reviewof experimental searches for dark matter). Great hope is also being placed on theLarge Hadron Collider (LHC) [111], which is due to start operating shortly, to constrainthe parameter space of particles arising from supersymmetric theories. Finally, theimprovement of cosmological and astrophysical observations obviously plays a crucialrole. Let us close by saying that the general flavour or expectation seems to be thatone of the proposed candidates will soon be detected and that the relevant dark matterscenario will be verified. Of course expectations are not always fulfilled and it is best tobe prepared for surprises.

1.4.4 Towards Quantum Gravity, but how?

In Section 1.2 we discussed some of the more prominent motivations for seeking a highenergy theory of gravity which would allow a matching between General Relativity andQuantum Field Theory. These triggered research in this direction at a very early stageand already in the 1950s serious efforts were being made towards what is referred toas Quantum Gravity. Early attempts followed the conventional approach of trying toquantize the gravitational field in ways similar to the quantization of Electromagnetism,which had resulted in Quantum Electrodynamics (QED). This led to influential papersabout the canonical formulation of General Relativity [112, 113]. However, it was soonrealized that the obvious quantization techniques could not work, since General Relativ-ity is not renormalizable as is the case with Quantum Electrodynamics [114]. In simpleterms, this means that if one attempts to treat gravity as another particle field and toassign a gravity particle to it (graviton) then the sum of the interactions of the gravi-ton diverges. This would not be a problem if these divergences were few enough to beremovable via the technique called renormalization and this is indeed what happens inQuantum Electrodynamics, as also mentioned in Section 1.2. Unfortunately, this is notthe case for General Relativity and renormalization cannot lead to sensible and finiteresults.

It was later shown that a renormalizable gravitation theory — although not a unitaryone — could be constructed, but only at the price of admitting corrections to GeneralRelativity [114, 188]. Views on renormalization have changed since then and moremodern ideas have been introduced such as the concept of effective field theories. Theseare approximate theories with the following characteristic: according to the length-scale,they take into account only the relevant degrees of freedom. Degrees of freedom which areonly relevant to shorter length-scales and higher energies and are, therefore, responsiblefor divergences, are ignored. A systematic way to integrate out short-distance degreesof freedom is given by the renormalization group (see [116] for an introduction to theseconcepts).


In any case, quantizing gravity has proved to be a more difficult task than initiallyexpected and quantum corrections seem to appear, introducing deviations away fromGeneral Relativity [117, 118, 119]. Contemporary research is mainly focused on twodirections: String Theory and Loop Quantum Gravity. Analysing the basis of eitherof these two approaches would go beyond the scope of this introduction and so wewill only make a short mention of them. We refer the reader to [120, 121, 122] and[123, 124, 125, 126, 127] for text books and topical reviews in String Theory and LoopQuantum Gravity respectively.

String Theory attempts to explain fundamental physics and unify all interactionsunder the key assumption that the building blocks are not point particles but one dimen-sional objects called strings. There are five different versions of String Theory, namelyType I, Type IIA, Type IIB and two types of Heterotic String Theory. M-Theory isa proposed theory under development that attempts to unify all of the above types.A simplified version of the idea behind String Theory would be that its fundamentalconstituents, strings, vibrate at resonant frequencies. Different strings have differentresonances and this is what determines their nature and results in the discriminationbetween different forces.

Loop Quantum Gravity follows a more direct approach to the quantization of gravity.It is close to the picture of canonical quantization and relies on a non-perturbativemethod called loop quantization. One of its main disadvantages is that it is not yetclear whether it can become a theory that can include the description of matter as wellor whether it is just a quantum theory of gravitation.

It is worth mentioning that a common problem with these two approaches is that,at the moment, they do not make any experimentally testable predictions which aredifferent from those already know from the standard model of particle physics. Asfar as gravity is concerned, String Theory appears to introduce deviations from GeneralRelativity (see for example [128, 129, 130]), whereas, the classical limit of Loop Quantumgravity is still under investigation.

1.4.5 Status of Gravity

In this introductory chapter, an attempt has been made to pose clearly a series ofopen questions related, in one way or the other, to gravity and to discuss some of themost common approaches currently being pursued for their solution. This brings usto the main question motivating the research presented in this thesis: could all or atleast some of the problems mentioned earlier be somehow related and is the fact thatGeneral Relativity is now facing so many challenges indicative of a need for some newgravitational physics, even at a classical level?

Let us be more analytic. In Section 1.1 we presented a brief chronological reviewof some landmarks in the passage from Newtonian Gravity to General Relativity. Onecould find striking similarities with what has happened in the last decades with GeneralRelativity itself. For instance, the cosmological and astrophysical observations which

48 1. INTRODUCTION

are interpreted as indicating the existence of dark matter and/or dark energy couldbe compared with Le Verrier’s observation of the excess precession of Mercury’s orbit.Remarkably, the first attempt to explain this phenomenon, was exactly the suggestionthat an extra unseen — and therefore dark, in a way — planet orbited the Sun insideMercury’s orbit. The basic motivation behind this attempt, much like the contemporaryproposals for matter fields to describe dark matter and dark energy, was to solve theproblem within the context of an otherwise successful theory, instead of questioningthe theory itself. Another example one could give, is the theoretical problems facedby Newtonian gravity once Special Relativity was established. The desire for a unifieddescription of coordinate frames, inertial or not, and the need for a gravitational theorythat is in good accordance with the conceptual basis of Special Relativity (e.g. Lorentzinvariance) does not seem to be very far from the current desire for a unified descriptionof forces and the need to resolve the conceptual clash between General Relativity andQuantum Field Theory.

The idea of looking for an alternative theory to describe the gravitational interactionis obviously not new. We already mentioned previously that attempts to unify gravitywith quantum theory have included such considerations in the form of making quantumcorrections to the gravitational field equations (or to the action, from a field theoryperspective). Such corrections became effective at small scales or high energies. Addi-tionally, many attempts have been made to modify General Relativity on both smalland large scales, in order to address specific problems, such as those discussed earlier.Since we will refer to such modification extensively in the forthcoming chapters we willrefrain from listing them here to avoid repetition. At present we will confine ourselvesto giving two very early examples of such attempts which were not triggered so muchby a theoretical or observational need for a new theory, but by another important issuein our opinion: the desire to test the uniqueness of General Relativity as the only viablegravitational theory and the need to verify its conceptual basis.

Sir Arthur Stanley Eddington, the very man who performed the deflection of lightexperiment during the Solar eclipse of 1919 which was one of the early experimentalverifications of General Relativity, was one of first people to question whether Einstein’stheory was the unique theory that could describe gravity [131]. Eddington tried todevelop alternative theories sharing the same conceptual basis with General Relativ-ity, most probably for the sake of theoretical completeness, since at the time there wasno apparent reason coming from observations. Robert Dicke was also one of the pio-neers in exploring the conceptual basis of General Relativity and questioning Einstein’sequivalence principle. He reformulated Mach’s principle and together with Carl Bransdeveloped an alternative theory, known as Brans–Dicke theory [199, 133]. Part of thevalue of Dicke’s work lies on the fact that it helped people to understand that we donot know as much as we thought about the basic assumptions of General Relativity, asubject that we will discuss shortly.

Even though the idea of an alternative theory for gravitation is not new, a newperspective about it has emerged quite recently. The quantum corrections predicted in


the 1960s were expected to appear only at small scales. On the other hand, Eddington’smodification or Brans–Dicke theory were initially pursued as a conceptual alternativeof General Relativity and had phenomenological effects on large scales as well. Now,due to both the shortcomings of Quantum Gravity and the puzzling cosmological andastrophysical observations, these ideas have stopped being considered unrelated. It seemworthwhile to consider the possibility of developing a gravitation theory that will be inagreement with observations and at the same time will be closer to the theories thatemerge as a classical limit of our current approaches to Quantum Gravity, especiallysince it has been understood that quantum corrections might have an effect on largescale phenomenology as well.

Unfortunately, constructing a viable alternative to General Relativity with the abovecharacteristics is far from being an easy task since there are numerous theoretical andobservational restrictions. Two main paths have been followed towards achieving thisgoal: proposing phenomenological models tailored to fit observations, with the hopethat they will soon gain some theoretical motivation from high energy physics and cur-rent Quantum Gravity candidates, and developing ideas for Quantum Gravity, with thehope that they will eventually give the answer in the form of an effective gravitationaltheory through their classical limit which will account for unexplained observations. Inthis thesis a different approach will be followed in an attempt to combine and comple-ment these two. At least according to the author’s opinion, we seem to be still at tooearly a stage in the development of our ideas about Quantum Gravity to be able togive precise answers about the type and form of the expected quantum corrections toGeneral Relativity. Current observations still leave scope for a wide range of differentphenomenological models and so it seems a good idea to attempt exploring the limitsof classical gravity by combining theory and observations. In a sense, this approach liessomewhere in the middle between the more conventional approaches mentioned before.Instead of starting from something known in order to extrapolate to the unknown, weattempt here to jump directly into the unknown, hoping that we will find an answer.

To this end, we will examine theories of gravity, trying to determine how far onecan go from General Relativity. These theories have been chosen in such a way as topresent a resemblance with the low energy effective actions of contemporary candidatesfor Quantum Gravity in a quest to study the phenomenology of the induced corrections.Their choice has also been motivated by a desire to fit recent unexplained observations.However, it should be stressed that both of these criteria have been used in a loosemanner, since the main scope of this study is to explore the limits of alternative theoriesof gravity and hopefully shed some light on the strength and validity of the severalassumptions underlying General Relativity. The main motivation comes from the fearthat we may not know as much as we think or as much as needed to be known beforemaking the key steps pursued in the last 50 years in gravitational physics; and from thehope that a better understanding of classical gravity might have a lot to offer in thisdirection.

As a conclusion to this introduction it is worth saying the following: it is probably too

50 1. INTRODUCTION

early to conclude whether it is General Relativity that needs to be modified or replacedby some other gravitational theory or whether other solutions to the problems presentedin this chapter, such as those mentioned earlier, will eventually give the required answers.However, in scientific research, pursuing more than one possible solution to a problemhas always been the wisest and most rewarding choice; not only because there is analready explored alternative when one of the proposed solutions fails, but also due to thefact that trial and error is one of the most efficient ways to get a deeper understandingof a physical theory. Exploring alternative theories of gravity, although having somedisadvantages such as complexity, also presents a serious advantage: it is bound to befruitful even if it leads to the conclusion that General Relativity is the only correcttheory for gravitation, as it will have helped us both to understand General Relativitybetter and to secure our faith in it.

Chapter 2

Extended Theories of Gravity

2.1 Theoretical motivations for Extended Theories of Grav-

ity

Due to the problems of Standard Cosmological Model and to the problems of the solutionfound to solve them, and, first of all, to the lack of a denitive quantum gravity theory,alternative theories have been considered in order to attempt, at least, a semi-classicalscheme where Gen- eral Relativity and its positive results could be recovered. One of themost fruitful approaches has been that of Extended Theories of Gravity (ETG) whichhave become a sort of paradigm in the study of gravitational interaction. They arebased on corrections and enlargements of the Einstein theory. The paradigm consists,essentially, in adding higher-order curvature invariants and minimally or non-minimallycoupled scalar fields into dynamics which come out from the effective action of quantumgravity [404].

Other motivations to modify GR come from the issue of a full recovering of theMach principle which leads to assume a varying gravitational coupling. The principlestates that the local inertial frame is determined by some average of the motion ofdistant astronomical objects [135]. This fact implies that the gravitational coupling canbe scale-dependent and related to some scalar field. As a consequence, the concept of“inertia” and the Equivalence Principle have to be revised. For example, the Brans-Dicke theory [136] is a serious attempt to define an alternative theory to the Einsteingravity: it takes into account a variable Newton gravitational coupling, whose dynamicsis governed by a scalar field non-minimally coupled to the geometry. In such a way,Mach’s principle is better implemented [136, 137, 138].

Besides, every unification scheme as Superstrings, Supergravity or Grand UnifiedTheories, takes into account effective actions where non-minimal couplings to the geom-etry or higher-order terms in the curvature invariants are present. Such contributionsare due to one-loop or higher-loop corrections in the high-curvature regimes near the full(not yet available) quantum gravity regime [404]. Specifically, this scheme was adopted

51

52 2. EXTENDED THEORIES OF GRAVITY

in order to deal with the quantization on curved spacetimes and the result was that theinteractions among quantum scalar fields and background geometry or the gravitationalself-interactions yield corrective terms in the Hilbert-Einstein Lagrangian [139]. More-over, it has been realized that such corrective terms are inescapable in order to obtainthe effective action of quantum gravity at scales closed to the Planck one [140]. Allthese approaches are not the “full quantum gravity” but are needed as working schemestoward it.

In summary, higher-order terms in curvature invariants (such asR2, RµνRµν , RµναβRµναβ ,RR, or RkR) or non-minimally coupled terms between scalar fields and geometry(such as φ2R) have to be added to the effective Lagrangian of gravitational field whenquantum corrections are considered. For instance, one can notice that such terms occurin the effective Lagrangian of strings or in Kaluza-Klein theories, when the mechanismof dimensional reduction is used [141].

On the other hand, from a conceptual viewpoint, there are no a priori reason torestrict the gravitational Lagrangian to a linear function of the Ricci scalar R, minimallycoupled with matter [142]. Furthermore, the idea that there are no “exact” laws ofphysics could be taken into serious account: in such a case, the effective Lagrangiansof physical interactions are “stochastic” functions. This feature means that the localgauge invariances (i.e. conservation laws) are well approximated only in the low energylimit and the fundamental physical constants can vary [381].

Besides fundamental physics motivations, all these theories have acquired a hugeinterest in cosmology due to the fact that they “naturally” exhibit inflationary behaviorsable to overcome the shortcomings of Cosmological Standard Model (based on GR). Therelated cosmological models seem realistic and capable of matching with the CMBRobservations [144, 145, 146]. Furthermore, it is possible to show that, via conformaltransformations, the higher-order and non-minimally coupled terms always correspondto the Einstein gravity plus one or more than one minimally coupled scalar fields [147,148, 149, 150, 151].

More precisely, higher-order terms appear always as contributions of order two inthe field equations. For example, a term like R2 gives fourth order equations [152],R R gives sixth order equations [151, 153], R2R gives eighth order equations [154]and so on. By a conformal transformation, any 2nd-order derivative term correspondsto a scalar field1: for example, fourth-order gravity gives Einstein plus one scalar field,sixth-order gravity gives Einstein plus two scalar fields and so on [151, 155].

Furthermore, it is possible to show that the f(R)-gravity is equivalent not only to ascalar-tensor one but also to the Einstein theory plus an ideal fluid [156]. This featureresults very interesting if we want to obtain multiple inflationary events since an earlystage could select “very” large-scale structures (clusters of galaxies today), while a latestage could select “small” large-scale structures (galaxies today) [153]. The philosophy

1The dynamics of such scalar fields is usually given by the corresponding Klein-Gordon Equation,which is second order.

2.1. THEORETICAL MOTIVATIONS FOR EXTENDED THEORIES OF GRAV ITY 53

is that each inflationary era is related to the dynamics of a scalar field. Finally, theseextended schemes could naturally solve the problem of “graceful exit” bypassing theshortcomings of former inflationary models [146, 157].

In addition to the revision of Standard Cosmology at early epochs (leading to the In-flation), a new approach is necessary also at late epochs. ETGs could play a fundamentalrole also in this context. In fact, the increasing bulk of data that have been accumulatedin the last few years have paved the way to the emergence of a new cosmological modelusually referred to as the Concordance Model.

The Hubble diagram of Type Ia Supernovae (hereafter SNeIa), measured by boththe Supernova Cosmology Project [158] and the High - z Team [159] up to redshift z ∼ 1,has been the first evidence that the Universe is undergoing a phase of accelerated expan-sion. On the other hand, balloon born experiments, such as BOOMERanG [160] andMAXIMA [161], determined the location of the first and second peak in the anisotropyspectrum of the cosmic microwave background radiation (CMBR) strongly pointing outthat the geometry of the Universe is spatially flat. If combined with constraints com-ing from galaxy clusters on the matter density parameter ΩM , these data indicate thatthe Universe is dominated by a non-clustered fluid with negative pressure, genericallydubbed dark energy, which is able to drive the accelerated expansion. This picture hasbeen further strengthened by the recent precise measurements of the CMBR spectrum,due to the WMAP experiment [162, 163, 164], and by the extension of the SNeIa Hubblediagram to redshifts higher than 1 [165]. After these observational evidences, an over-whelming flood of papers has appeared: they present a great variety of models trying toexplain this phenomenon. In any case, the simplest explanation is claiming for the wellknown cosmological constant Λ [166]. Although it is the best fit to most of the availableastrophysical data [162], the ΛCDM model fails in explaining why the inferred value of Λis so tiny (120 orders of magnitude lower!) if compared with the typical vacuum energyvalues predicted by particle physics and why its energy density is today comparable tothe matter density (the so called coincidence problem).

As a tentative solution, many authors have replaced the cosmological constant witha scalar field rolling down its potential and giving rise to the model now referred to asquintessence [167, 168]. Even if successful in fitting the data, the quintessence approachto dark energy is still plagued by the coincidence problem since the dark energy andmatter densities evolve differently and reach comparable values for a very limited portionof the Universe evolution coinciding at present era. To be more precise, the quintessencedark energy is tracking matter and evolves in the same way for a long time. But then, atlate time, somehow it has to change its behavior into no longer tracking the dark matterbut starting to dominate as a cosmological constant. This is the coincidence problem ofquintessence.

Moreover, it is not clear where this scalar field originates from, thus leaving a greatuncertainty on the choice of the scalar field potential. The subtle and elusive natureof dark energy has led many authors to look for completely different scenarios able togive a quintessential behavior without the need of exotic components. To this aim, it is


worth stressing that the acceleration of the Universe only claims for a negative pressuredominant component, but does not tell anything about the nature and the number ofcosmic fluids filling the Universe.

This consideration suggests that it could be possible to explain the accelerated expan-sion by introducing a single cosmic fluid with an equation of state causing it to act likedark matter at high densities and dark energy at low densities. An attractive feature ofthese models, usually referred to as Unified Dark Energy (UDE) or Unified Dark Matter(UDM) models, is that such an approach naturally solves, al least phenomenologically,the coincidence problem. Some interesting examples are the generalized Chaplygin gas[169], the tachyon field [170] and the condensate cosmology [171]. A different class ofUDE models has been proposed [172] where a single fluid is considered: its energy den-sity scales with the redshift in such a way that the radiation dominated era, the matterera and the accelerating phase can be naturally achieved. It is worth noticing that suchclass of models are extremely versatile since they can be interpreted both in the frame-work of UDE models and as a two-fluid scenario with dark matter and scalar field darkenergy. The main ingredient of the approach is that a generalized equation of state canbe always obtained and observational data can be fitted.

Actually, there is still a different way to face the problem of cosmic acceleration. Asstressed in [173], it is possible that the observed acceleration is not the manifestation ofanother ingredient in the cosmic pie, but rather the first signal of a breakdown of ourunderstanding of the laws of gravitation (in the infra-red limit).

From this point of view, it is thus tempting to modify the Friedmann equationsto see whether it is possible to fit the astrophysical data with models comprising onlythe standard matter. Interesting examples of this kind are the Cardassian expansion[174] and the DGP gravity [175]. Moving in this same framework, it is possible to findalternative schemes where a quintessential behavior is obtained by taking into accounteffective models coming from fundamental physics giving rise to generalized or higher-order gravity actions [176] (for a comprehensive review see [177]).

For instance, a cosmological constant term may be recovered as a consequence of anon - vanishing torsion field thus leading to a model which is consistent with both SNeIaHubble diagram and Sunyaev - Zel’dovich data coming from clusters of galaxies [178].SNeIa data could also be efficiently fitted including higher-order curvature invariants inthe gravity Lagrangian [179, 181, 182, 183]. It is worth noticing that these alternativemodels provide naturally a cosmological component with negative pressure whose originis related to the geometry of the Universe thus overcoming the problems linked to thephysical significance of the scalar field.

It is evident, from this short overview, the high number of cosmological models whichare viable candidates to explain the observed accelerated expansion. This abundanceof models is, from one hand, the signal of the fact that we have a limited numberof cosmological tests to discriminate among rival theories, and, from the other hand,that a urgent degeneracy problem has to be faced. To this aim, it is useful to remarkthat both the SNeIa Hubble diagram and the angular size - redshift relation of compact

2.1. THEORETICAL MOTIVATIONS FOR EXTENDED THEORIES OF GRAV ITY 55

radio sources [184] are distance based methods to probe cosmological models so thensystematic errors and biases could be iterated. From this point of view, it is interestingto search for tests based on time-dependent observables.

For example, one can take into account the lookback time to distant objects since thisquantity can discriminate among different cosmological models. The lookback time isobservationally estimated as the difference between the present day age of the Universeand the age of a given object at redshift z. Such an estimate is possible if the objectis a galaxy observed in more than one photometric band since its color is determinedby its age as a consequence of stellar evolution. It is thus possible to get an estimateof the galaxy age by measuring its magnitude in different bands and then using stellarevolutionary codes to choose the model that reproduces the observed colors at best.

Coming to the weak-field-limit approximation, which essentially means consideringSolar System scales, ETGs are expected to reproduce GR which, in any case, is firmlytested only in this limit [185]. This fact is matter of debate since several relativistictheories do not reproduce exactly the Einstein results in the Newtonian approximationbut, in some sense, generalize them. As it was firstly noticed by Stelle [188], a R2-theorygives rise to Yukawa-like corrections in the Newtonian potential. Such a feature couldhave interesting physical consequences. For example, some authors claim to explainthe flat rotation curves of galaxies by using such terms [189]. Others [190] have shownthat a conformal theory of gravity is nothing else but a fourth-order theory containingsuch terms in the Newtonian limit. Besides, indications of an apparent, anomalous,long-range acceleration revealed from the data analysis of Pioneer 10/11, Galileo, andUlysses spacecrafts could be framed in a general theoretical scheme by taking correctionsto the Newtonian potential into account [191, 371].

In general, any relativistic theory of gravitation yields corrections to the Newtonpotential (see for example [193]) which, in the post-Newtonian (PPN) formalism, couldbe a test for the same theory [185]. Furthermore the newborn gravitational lensingastronomy [194] is giving rise to additional tests of gravity over small, large, and verylarge scales which soon will provide direct measurements for the variation of the Newtoncoupling [195], the potential of galaxies, clusters of galaxies and several other featuresof self-gravitating systems.

Such data will be, very likely, capable of confirming or ruling out the physical con-sistency of GR or of any ETG. In summary, the general features of ETGs are that theEinstein field equations result to be modified in two senses: i) geometry can be non-minimally coupled to some scalar field, and/or ii) higher than second order derivativeterms in the metric come out. In the former case, we generically deal with scalar-tensortheories of gravity; in the latter, we deal with higher-order theories. However combina-tions of non-minimally coupled and higher-order terms can emerge as contributions ineffective Lagrangians. In this case, we deal with higher-order-scalar-tensor theories ofgravity.

Considering a mathematical viewpoint, the problem of reducing more general theo-ries to Einstein standard form has been extensively treated; one can see that, through


a “Legendre” transformation on the metric, higher-order theories, under suitable reg-ularity conditions on the Lagrangian, take the form of the Einstein one in which ascalar field (or more than one) is the source of the gravitational field (see for example[142, 197, 196, 198]); on the other side, as discussed above, it has been studied themathematical equivalence between models with variable gravitational coupling with theEinstein standard gravity through suitable conformal transformations (see [199, 200]).

In any case, the debate on the physical meaning of conformal transformations is far tobe solved [see [347] and references therein for a comprehensive review]. Several authorsclaim for a true physical difference between Jordan frame (higher-order theories and/orvariable gravitational coupling) since there are experimental and observational evidenceswhich point out that the Jordan frame could be suitable to better match solutions withdata. Others state that the true physical frame is the Einstein one according to theenergy theorems [198]. However, the discussion is open and no definitive statement hasbeen formulated up to now.

The problem should be faced from a more general viewpoint and the Palatini ap-proach to gravity could be useful to this goal. The Palatini approach in gravitationaltheories was firstly introduced and analyzed by Einstein himself [203]. It was, how-ever, called the Palatini approach as a consequence of an historical misunderstanding[204, 205].

The fundamental idea of the Palatini formalism is to consider the (usually torsion-less) connection Γ, entering the definition of the Ricci tensor, to be independent of themetric g defined on the spacetimeM. The Palatini formulation for the standard Hilbert-Einstein theory results to be equivalent to the purely metric theory: this follows from thefact that the field equations for the connection Γ, firstly considered to be independent ofthe metric, give the Levi-Civita connection of the metric g. As a consequence, there isno reason to impose the Palatini variational principle in the standard Hilbert-Einsteintheory instead of the metric variational principle.

However, the situation completely changes if we consider the ETGs, depending onfunctions of curvature invariants, as f(R), or non-minimally coupled to some scalarfield. In these cases, the Palatini and the metric variational principle provide differentfield equations and the theories thus derived differ [198, 206]. The relevance of Pala-tini approach, in this framework, has been recently proven in relation to cosmologicalapplications [176, 177, 207, 208, 209].

It has also been studied the crucial problem of the Newtonian potential in alternativetheories of Gravity and its relations with the conformal factor [211]. From a physicalviewpoint, considering the metric g and the connection Γ as independent fields means todecouple the metric structure of spacetime and its geodesic structure (being, in general,the connection Γ not the Levi-Civita connection of g). The chronological structure ofspacetime is governed by g while the trajectories of particles, moving in the spacetime,are governed by Γ.

This decoupling enriches the geometric structure of spacetime and generalizes thepurely metric formalism. This metric-affine structure of spacetime is naturally trans-

2.2. WHAT A GOOD THEORY OF GRAVITY HAS TO DO: GENERAL RELATIVI TY AND ITSEXTENSIONS 57

lated, by means of the same (Palatini) field equations, into a bi-metric structure ofspacetime. Beside the physical metric g, another metric h is involved. This new metricis related, in the case of f(R)-gravity, to the connection. As a matter of fact, the con-nection Γ results to be the Levi-Civita connection of h and thus provides the geodesicstructure of spacetime.

If we consider the case of non-minimally coupled interaction in the gravitationalLagrangian (scalar-tensor theories), the new metric h is related to the non-minimal cou-pling. The new metric h can be thus related to a different geometric and physical aspectof the gravitational theory. Thanks to the Palatini formalism, the non-minimal couplingand the scalar field, entering the evolution of the gravitational fields, are separated fromthe metric structure of spacetime. The situation mixes when we consider the case ofhigher-order-scalar-tensor theories. Due to these features, the Palatini approach couldgreatly contribute to clarify the physical meaning of conformal transformation [210].

2.2 What a good theory of Gravity has to do: General

Relativity and its extensions

From a phenomenological point of view, there are some minimal requirements that anyrelativistic theory of gravity has to match. First of all, it has to explain the astrophysicalobservations (e.g. the orbits of planets, the potential of self-gravitating structures).

This means that it has to reproduce the Newtonian dynamics in the weak-energylimit. Besides, it has to pass the classical Solar System tests which are all experimentallywell founded [185].

As second step, it should reproduce galactic dynamics considering the observedbaryonic constituents (e.g. luminous components as stars, sub-luminous componentsas planets, dust and gas), radiation and Newtonian potential which is, by assumption,extrapolated to galactic scales.

Thirdly, it should address the problem of large scale structure (e.g. clusteringof galaxies) and finally cosmological dynamics, which means to reproduce, in a self-consistent way, the cosmological parameters as the expansion rate, the Hubble constant,the density parameter and so on. Observations and experiments, essentially, probe thestandard baryonic matter, the radiation and an attractive overall interaction, acting atall scales and depending on distance: the gravity.

The simplest theory which try to satisfies the above requirements was formulated byAlbert Einstein in the years 1915-1916 [212] and it is known as the Theory of GeneralRelativity. It is firstly based on the assumption that space and time have to be entangledinto a single spacetime structure, which, in the limit of no gravitational forces, has toreproduce the Minkowski spacetime structure. Einstein profitted also of ideas earlierput forward by Riemann, who stated that the Universe should be a curved manifoldand that its curvature should be established on the basis of astronomical observations[213].


In other words, the distribution of matter has to influence point by point the localcurvature of the spacetime structure. The theory, eventually formulated by Einstein in1915, was strongly based on three assumptions that the Physics of Gravitation has tosatisfy.

The ”Principle of Relativity”, that amounts to require all frames to be good framesfor Physics, so that no preferred inertial frame should be chosen a priori (if any exist).

The ”Principle of Equivalence”, that amounts to require inertial effects to be locallyindistinguishable from gravitational effects (in a sense, the equivalence between theinertial and the gravitational mass).

The ”Principle of General Covariance”, that requires field equations to be ”generallycovariant” (today, we would better say to be invariant under the action of the group ofall spacetime diffeomorphisms) [214].

And - on the top of these three principles - the requirement that causality has to bepreserved (the ”Principle of Causality”, i.e. that each point of spacetime should admita universally valid notion of past, present and future).

Let us also recall that the older Newtonian theory of spacetime and gravitation -that Einstein wanted to reproduce at least in the limit of small gravitational forces(what is called today the ”post-Newtonian approximation”) - required space and timeto be absolute entities, particles moving in a preferred inertial frame following curvedtrajectories, the curvature of which (i.e., the acceleration) had to be determined as afunction of the sources (i.e., the ”forces”).

On these bases, Einstein was led to postulate that the gravitational forces have tobe expressed by the curvature of a metric tensor field ds2 = gµνdx

µdxν on a four-dimensional spacetime manifold, having the same signature of Minkowski metric, i.e.,the so-called ”Lorentzian signature”, herewith assumed to be (+,−,−,−). He alsopostulated that spacetime is curved in itself and that its curvature is locally determinedby the distribution of the sources, i.e. - being spacetime a continuum - by the four-dimensional generalization of what in Continuum Mechanics is called the ”matter stress-energy tensor”, i.e. a rank-two (symmetric) tensor Tm

µν .

Once a metric gµν is given, its curvature is expressed by the Riemann (curvature)tensor

Rαβµν = Γα

βν,µ − Γαβµ,ν + Γσ

βνΓασµ − Γσ

βµΓασν (2.1)

where the comas are partial derivatives. Its contraction

Rαµαν = Rµν , (2.2)

is the ”Ricci tensor” and the scalar

R = Rµµ = gµνRµν (2.3)

is called the ”scalar curvature” of gµν . Einstein was led to postulate the following


equations for the dynamics of gravitational forces

Rµν =κ

2Tm

µν (2.4)

where κ = 8πG, with c = 1 is a coupling constant. These equations turned out to bephysically and mathematically unsatisfactory.

As Hilbert pointed out [214], they were not of a variational origin, i.e. there wasno Lagrangian able to reproduce them exactly (this is slightly wrong, but this remarkis unessential here). Einstein replied that he knew that the equations were physicallyunsatisfactory, since they were contrasting with the continuity equation of any reasonablekind of matter. Assuming that matter is given as a perfect fluid, that is

Tmµν = (p + ρ)uµuν − pgµν (2.5)

where uµuν is a comoving observer, p is the pressure and ρ the density of the fluid,then the continuity equation requires Tm

µν to be covariantly constant, i.e. to satisfy theconservation law

∇µTmµν = 0 , (2.6)

where ∇µ denotes the covariant derivative with respect to the metric.In fact, it is not true that ∇µRµν vanishes (unless R = 0). Einstein and Hilbert

reached independently the conclusion that the wrong field equations (2.4) had to bereplaced by the correct ones

Gµν = κTmµν (2.7)

where

Gµν = Rµν −1

2gµνR (2.8)

that is currently called the ”Einstein tensor” of gµν . These equations are both variationaland satisfy the conservation laws (2.6) since the following relation holds

∇µGµν = 0 , (2.9)

as a byproduct of the so-called ”Bianchi identities” that the curvature tensor of gµν hasto satisfy [11].

The Lagrangian that allows to obtain the field equations (2.7) is the sum of a ”matterLagrangian” Lm, the variational derivative of which is exactly Tm

µν , i.e.

Tmµν =

δLm

δgµν(2.10)

and of a ”gravitational Lagrangian”, currently called the Hilbert-Einstein Lagrangian

LHE = gµνRµν√−g = R

√−g , (2.11)

where√−g denotes the square root of the value of the determinant of the metric gµν .


The choice of Hilbert and Einstein was completely arbitrary (as it became clear afew years later), but it was certainly the simplest one both from the mathematical andthe physical viewpoint. As it was later clarified by Levi-Civita in 1919, curvature isnot a ”purely metric notion” but, rather, a notion related to the ”linear connection” towhich ”parallel transport” and ”covariant derivation” refer [215].

In a sense, this is the precursor idea of what in the sequel would be called a ”gaugetheoretical framework” [216], after the pioneering work by Cartan in 1925 [217]. But atthe time of Einstein, only metric concepts were at hands and his solution was the onlyviable.

It was later clarified that the three principles of relativity, equivalence and covariance,together with causality, just require that the spacetime structure has to be determinedby either one or both of two fields, a Lorentzian metric g and a linear connection Γ,assumed to be torsionless for the sake of simplicity.

The metric g fixes the causal structure of spacetime (the light cones) as well as itsmetric relations (clocks and rods); the connection Γ fixes the free-fall, i.e. the locallyinertial observers. They have, of course, to satisfy a number of compatibility relationswhich amount to require that photons follow null geodesics of Γ, so that Γ and g canbe independent, a priori, but constrained, a posteriori, by some physical restrictions.These, however, do not impose that Γ has necessarily to be the Levi-Civita connectionof g [218].

This justifies - at least on a purely theoretical basis - the fact that one can envisagethe so-called ”alternative theories of gravitation”, that we prefer to call ”Extended The-ories of Gravitation” since their starting points are exactly those considered by Einsteinand Hilbert: theories in which gravitation is described by either a metric (the so-called”purely metric theories”), or by a linear connection (the so-called ”purely affine the-ories”) or by both fields (the so-called ”metric-affine theories”, also known as ”firstorder formalism theories”). In these theories, the Lagrangian is a scalar density of thecurvature invariants constructed out of both g and Γ.

The choice (2.11) is by no means unique and it turns out that the Hilbert-EinsteinLagrangian is in fact the only choice that produces an invariant that is linear in secondderivatives of the metric (or first derivatives of the connection). A Lagrangian that,unfortunately, is rather singular from the Hamiltonian viewpoint, in much than same wayas Lagrangians, linear in canonical momenta, are rather singular in Classical Mechanics(see e.g. [219]).

A number of attempts to generalize GR (and unify it to Electromagnetism) alongthese lines were followed by Einstein himself and many others (Eddington, Weyl, Schrodinger,just to quote the main contributors; see, e.g., [220]) but they were eventually given upin the fifties of XX Century, mainly because of a number of difficulties related to thedefinitely more complicated structure of a non-linear theory (where by ”non-linear” wemean here a theory that is based on non-linear invariants of the curvature tensor), andalso because of the new understanding of Physics that is currently based on four fun-damental forces and requires the more general ”gauge framework” to be adopted (see


[221]).

Still a number of sporadic investigations about ”alternative theories” continued evenafter 1960 (see [185] and refs. quoted therein for a short history). The search of acoherent quantum theory of gravitation or the belief that gravity has to be consideredas a sort of low-energy limit of string theories (see, e.g., [222]) - something that we arenot willing to enter here in detail - has more or less recently revitalized the idea thatthere is no reason to follow the simple prescription of Einstein and Hilbert and to assumethat gravity should be classically governed by a Lagrangian linear in the curvature.

Further curvature invariants or non-linear functions of them should be also consid-ered, especially in view of the fact that they have to be included in both the semi-classicalexpansion of a quantum Lagrangian or in the low-energy limit of a string Lagrangian.

Moreover, it is clear from the recent astrophysical observations and from the currentcosmological hypotheses that Einstein equations are no longer a good test for gravitationat Solar System, galactic, extra-galactic and cosmic scale, unless one does not admitthat the matter side of Eqs.(2.7) contains some kind of exotic matter-energy which isthe ”dark matter” and ”dark energy” side of the Universe.

The idea which we propose here is much simpler. Instead of changing the matterside of Einstein Equations (2.7) in order to fit the ”missing matter-energy” contentof the currently observed Universe (up to the 95% of the total amount!), by addingany sort of inexplicable and strangely behaving matter and energy, we claim that it issimpler and more convenient to change the gravitational side of the equations, admittingcorrections coming from non-linearities in the Lagrangian. However, this is nothing elsebut a matter of taste and, since it is possible, such an approach should be explored. Ofcourse, provided that the Lagrangian can be conveniently tuned up (i.e., chosen in a hugefamily of allowed Lagrangians) on the basis of its best fit with all possible observationaltests, at all scales (solar, galactic, extragalactic and cosmic).

Something that - in spite of some commonly accepted but disguised opinion - canand should be done before rejecting a priori a non-linear theory of gravitation (basedon a non-singular Lagrangian) and insisting that the Universe has to be necessarily de-scribed by a rather singular gravitational Lagrangian (one that does not allow a coherentperturbation theory from a good Hamiltonian viewpoint) accompanied by matter thatdoes not follow the behavior that standard baryonic matter, probed in our laboratories,usually satisfies.


2.3 Structure of the Extended Theories of Gravity

With the above considerations in mind, let us start with a general class of higher-order-scalar-tensor theories in four dimensions 2 given by the action

A =

∫d4x√−g

[F (R,R,2R, ..kR,φ)− ε

2gµνφ;µφ;ν + Lm

], (2.12)

where F is an unspecified function of curvature invariants and of a scalar field φ. Theterm Lm, as above, is the minimally coupled ordinary matter contribution. We shall usephysical units 8πG = c = ~ = 1; ε is a constant which specifies the theory. Actually itsvalues can be ε = ±1, 0 fixing the nature and the dynamics of the scalar field which canbe a standard scalar field, a phantom field or a field without dynamics (see [223, 224]for details).

In the metric approach, the field equations are obtained by varying (4.22) withrespect to gµν . We get

Gµν =1

G

[T µν +

1

2gµν(F − GR) + (gµλgνσ − gµνgλσ)G;λσ

+1

2

k∑

i=1

i∑

j=1

(gµνgλσ + gµλgνσ)(j−i);σ

(i−j ∂F

∂iR

)

;λ

−gµνgλσ

((j−1R);σi−j ∂F

∂iR

)

;λ

], (2.13)

where Gµν is the above Einstein tensor and

G ≡n∑

j=0

j

(∂F

∂jR

). (2.14)

The differential Eqs.(4.23) are of order (2k + 4). The stress-energy tensor is due to thekinetic part of the scalar field and to the ordinary matter:

Tµν = Tmµν +

ε

2[φ;µφ;ν −

1

2φα

; φ;α] . (2.15)

The (eventual) contribution of a potential V (φ) is contained in the definition of F .From now on, we shall indicate by a capital F a Lagrangian density containing also thecontribution of a potential V (φ) and by F (φ), f(R), or f(R,R) a function of suchfields without potential.

By varying with respect to the scalar field φ, we obtain the Klein-Gordon equation

εφ = −∂F∂φ

. (2.16)

2For the aims of this review, we do not need more complicated invariants like RµνRµν , RµναβRµναβ ,CµναβCµναβ which are also possible.

2.3. STRUCTURE OF THE EXTENDED THEORIES OF GRAVITY 63

Several approaches can be used to deal with such equations. For example, as we said, bya conformal transformation, it is possible to reduce an ETG to a (multi) scalar-tensortheory of gravity [193, 149, 150, 151, ?].

The simplest extension of GR is achieved assuming

F = f(R) , ε = 0 , (2.17)

in the action (4.22); f(R) is an arbitrary (analytic) function of the Ricci curvaturescalar R. We are considering here the simplest case of fourth-order gravity but we couldconstruct such kind of theories also using other invariants in Rµν or Rα

βµν . The standardHilbert-Einstein action is, of course, recovered for f(R) = R. Varying with respect togαβ , we get the field equations

f ′(R)Rαβ −1

2f(R)gαβ = f ′(R);

µν

(gαµgβν − gαβgµν) , (2.18)

which are fourth-order equations due to the term f ′(R);µν ; the prime indicates thederivative with respect to R. Eq.(2.18) is also the equation for Tµν = 0 when the matterterm is absent.

By a suitable manipulation, the above equation can be rewritten as:

Gαβ =1

f ′(R)

1

2gαβ

[f(R)−Rf ′(R)

]+ f ′(R);αβ − gαβf ′(R)

, (2.19)

where the gravitational contribution due to higher-order terms can be simply reinter-preted as a stress-energy tensor contribution. This means that additional and higher-order terms in the gravitational action act, in principle, as a stress-energy tensor, relatedto the form of f(R). Considering also the standard perfect-fluid matter contribution,we have

Gαβ =1

f ′(R)

1

2gαβ

[f(R)−Rf ′(R)

]+ f ′(R);αβ − gαβf ′(R)

+Tm

αβ

f ′(R)= T curv

αβ +Tm

αβ

f ′(R),

(2.20)where T curv

αβ is an effective stress-energy tensor constructed by the extra curvature terms.In the case of GR, T curv

αβ identically vanishes while the standard, minimal coupling isrecovered for the matter contribution. The peculiar behavior of f(R) = R is due tothe particular form of the Lagrangian itself which, even though it is a second orderLagrangian, can be non-covariantly rewritten as the sum of a first order Lagrangianplus a pure divergence term. The Hilbert-Einstein Lagrangian can be in fact recast asfollows:

LHE = LHE√−g =

[pαβ(Γρ

ασΓσρβ − Γρ

ρσΓσαβ) +∇σ(pαβuσ

αβ)]

(2.21)

where:

pαβ =√−ggαβ =

∂L∂Rαβ

(2.22)


Γ is the Levi-Civita connection of g and uσαβ is a quantity constructed out with the

variation of Γ [11]. Since uσαβ is not a tensor, the above expression is not covariant;

however a standard procedure has been studied to recast covariance in the first ordertheories [225]. This clearly shows that the field equations should consequently be secondorder and the Hilbert-Einstein Lagrangian is thus degenerate.

From the action (4.22), it is possible to obtain another interesting case by choosing

F = F (φ)R − V (φ) , ε = −1 . (2.23)

In this case, we get

A =

∫d4x√−g

[F (φ)R +

1

2gµνφ;µφ;ν − V (φ)

](2.24)

V (φ) and F (φ) are generic functions describing respectively the potential and the cou-pling of a scalar field φ. The Brans-Dicke theory of gravity is a particular case of theaction (2.24) for V (φ)=0 [226]. The variation with respect to gµν gives the second-orderfield equations

F (φ)Gµν = F (φ)

[Rµν −

1

2Rgµν

]= −1

2T φ

µν − gµνgF (φ) + F (φ);µν , (2.25)

here g is the d’Alembert operator with respect to the metric g The energy-momentumtensor relative to the scalar field is

T φµν = φ;µφ;ν −

1

2gµνφ;αφ

α; + gµνV (φ) (2.26)

The variation with respect to φ provides the Klein - Gordon equation, i.e. the fieldequation for the scalar field:

gφ−RFφ(φ) + Vφ(φ) = 0 (2.27)

where Fφ = dF (φ)/dφ, Vφ = dV (φ)/dφ. This last equation is equivalent to the Bianchicontracted identity [227]. Standard fluid matter can be treated as above.

2.3.1 Conformal transformations

Let us now introduce conformal transformations to show that any higher-order or scalar-tensor theory, in absence of ordinary matter, e.g. a perfect fluid, is conformally equiv-alent to an Einstein theory plus minimally coupled scalar fields. If standard matter ispresent, conformal transformations allow to transfer non-minimal coupling to the mattercomponent [198]. The conformal transformation on the metric gµν is

gµν = e2ωgµν (2.28)

2.3. STRUCTURE OF THE EXTENDED THEORIES OF GRAVITY 65

in which e2ω is the conformal factor. Under this transformation, the Lagrangian in(2.24) becomes

√−g(FR+

1

2gµνφ;µφ;ν − V

)=√−ge−2ω

(FR− 6Fgω+

−6Fω;αωα; +

1

2gµνφ;µφ;ν − e−2ωV

) (2.29)

in which R and g are the Ricci scalar and the d’Alembert operator relative to themetric g. Requiring the theory in the metric gµν to appear as a standard Einsteintheory [201], the conformal factor has to be related to F , that is

e2ω = −2F. (2.30)

where F must be negative in order to restore physical coupling. Using this relation andintroducing a new scalar field φ and a new potential V , defined respectively by

φ;α =

√3Fφ

2 − F2F 2

φ;α, V (φ(φ)) =V (φ)

4F 2(φ), (2.31)

we see that the Lagrangian (2.29) becomes

√−g(FR+

1

2gµνφ;µφ;ν − V

)=√−g(−1

2R+

1

2φ;αφ

α; − V

)

which is the usual Hilbert-Einstein Lagrangian plus the standard Lagrangian relativeto the scalar field φ. Therefore, every non-minimally coupled scalar-tensor theory, inabsence of ordinary matter, e.g. perfect fluid, is conformally equivalent to an Einsteintheory, being the conformal transformation and the potential suitably defined by (2.30)and (2.31). The converse is also true: for a given F (φ), such that 3Fφ

2 − F > 0, wecan transform a standard Einstein theory into a non-minimally coupled scalar-tensortheory. This means that, in principle, if we are able to solve the field equations in theframework of the Einstein theory in presence of a scalar field with a given potential,we should be able to get the solutions for the scalar-tensor theories, assigned by thecoupling F (φ), via the conformal transformation (2.30) with the constraints given by(2.31). Following the standard terminology, the “Einstein frame” is the framework ofthe Einstein theory with the minimal coupling and the “Jordan frame” is the frameworkof the non-minimally coupled theory [198].In the context of alternative theories of gravity, as previously discussed, the gravitationalcontribution to the stress-energy tensor of the theory can be reinterpreted by means ofa conformal transformation as the stress-energy tensor of a suitable scalar field and thenas “matter” like terms. Performing the conformal transformation (2.28) in the fieldequations (2.19), we get:

Gαβ =1

f ′(R)

1

2gαβ

[f(R)−Rf ′(R)

]+ f ′(R);αβ − gαβf ′(R)

+ (2.32)


+2

(ω;α;β + gαβω − ω;αω;β +

1

2gαβω;γω

;γ

).

We can then choose the conformal factor to be

ω =1

2ln |f ′(R)| , (2.33)

which has now to be substituted into (2.20). Rescaling ω in such a way that

kφ = ω , (2.34)

and k =√

1/6, we obtain the Lagrangian equivalence

√−gf(R) =√−g(−1

2R+

1

2φ;αφ

α; − V

)(2.35)

and the Einstein equations in standard form

Gαβ = φ;αφ;β −1

2gαβφ;γφ

;γ + gαβV (φ) , (2.36)

with the potential

V (φ) =e−4kφ

2

[P(φ)−N

(e2kφ

)e2kφ

]=

1

2

f(R)−Rf ′(R)

f ′(R)2. (2.37)

Here N is the inverse function of P ′(φ) and P(φ) =∫

exp(2kφ)dN . However, theproblem is completely solved if P ′(φ) can be analytically inverted. In summary, a fourth-order theory is conformally equivalent to the standard second-order Einstein theory plusa scalar field (see also [142, 196]).This procedure can be extended to more general theories. If the theory is assumed tobe higher than fourth order, we may have Lagrangian densities of the form [204, 151],

L = L(R,R, ...kR) . (2.38)

Every operator introduces two further terms of derivation into the field equations.For example a theory like

L = RR , (2.39)

is a sixth-order theory and the above approach can be pursued by considering a conformalfactor of the form

ω =1

2ln

∣∣∣∣∂L∂R

+ ∂L∂R

∣∣∣∣ . (2.40)

In general, increasing two orders of derivation in the field equations (i.e. for every termR), corresponds to adding a scalar field in the conformally transformed frame [151].A sixth-order theory can be reduced to an Einstein theory with two minimally coupled

2.4. THE PALATINI APPROACH AND THE INTRINSIC CONFORMAL STRU CTURE 67

scalar fields; a 2n-order theory can be, in principle, reduced to an Einstein theory plus(n−1)-scalar fields. On the other hand, these considerations can be directly generalizedto higher-order-scalar-tensor theories in any number of dimensions as shown in [148].

As concluding remarks, we can say that conformal transformations work at threelevels: i) on the Lagrangian of the given theory; ii) on the field equations; iii) on the so-lutions. The table below summarizes the situation for fourth-order gravity (FOG), non-minimally coupled scalar-tensor theories (NMC) and standard Hilbert-Einstein (HE)theory. Clearly, direct and inverse transformations correlate all the steps of the tablebut no absolute criterion, at this point of the discussion, is able to select which is the“physical” framework since, at least from a mathematical point of view, all the framesare equivalent [198]. This point is up to now unsolved even if wide discussions arepresent in literature [347].

LFOG ←→ LNMC ←→ LHE

l l lFOG Eqs. ←→ NMC Eqs. ←→ Einstein Eqs.l l l

FOG Solutions ←→ NMC Solutions ←→ Einstein Solutions

2.4 The Palatini Approach and the Intrinsic Conformal

Structure

As we said, the Palatini approach, considering g and Γ as independent fields, is “intrin-sically” bi-metric and capable of disentangling the geodesic structure from the chrono-logical structure of a given manifold. Starting from these considerations, conformaltransformations assume a fundamental role in defining the affine connection which ismerely “Levi-Civita” only for the Hilbert-Einstein theory.

In this section, we work out examples showing how conformal transformations assumea fundamental physical role in relation to the Palatini approach in ETGs [210].

Let us start from the case of fourth-order gravity where Palatini variational principleis straightforward in showing the differences with Hilbert-Einstein variational principle,involving only metric. Besides, cosmological applications of f(R) gravity have shown therelevance of Palatini formalism, giving physically interesting results with singularity -free solutions [207]. This last nice feature is not present in the standard metric approach.

An important remark is in order at this point. The Ricci scalar entering in f(R)is R ≡ R(g,Γ) = gαβRαβ(Γ) that is a generalized Ricci scalar and Rµν(Γ) is the Riccitensor of a torsion-less connection Γ, which, a priori, has no relations with the metricg of spacetime. The gravitational part of the Lagrangian is controlled by a given realanalytical function of one real variable f(R), while

√−g denotes a related scalar densityof weight 1. Field equations, deriving from the Palatini variational principle are:

f ′(R)R(µν)(Γ)− 1

2f(R)gµν = Tm

µν (2.41)


∇Γα(√−gf ′(R)gµν) = 0 (2.42)

where ∇Γ is the covariant derivative with respect to Γ. As above, we assume 8πG = 1.We shall use the standard notation denoting by R(µν) the symmetric part of Rµν , i.e.

R(µν) ≡ 12 (Rµν +Rνµ).

In order to get (2.42), one has to additionally assume that Lm is functionally in-

dependent of Γ; however it may contain metric covariant derivativesg

∇ of fields. Thismeans that the matter stress-energy tensor Tm

µν = Tmµν(g,Ψ) depends on the metric g

and some matter fields denoted here by Ψ, together with their derivatives (covariantderivatives with respect to the Levi-Civita connection of g). From (2.42) one sees that√−gf ′(R)gµν is a symmetric twice contravariant tensor density of weight 1. As previ-ously discussed in [206, 210], this naturally leads to define a new metric hµν , such thatthe following relation holds:

√−gf ′(R)gµν =√−hhµν . (2.43)

This ansatz is suitably made in order to impose Γ to be the Levi-Civita connection ofh and the only restriction is that

√−gf ′(R)gµν should be non-degenerate. In the caseof Hilbert-Einstein Lagrangian, it is f ′(R) = 1 and the statement is trivial.

The above Eq.(2.43) imposes that the two metrics h and g are conformally equivalent.The corresponding conformal factor can be easily found to be f ′(R) (in dim M = 4)and the conformal transformation results to be ruled by:

hµν = f ′(R)gµν (2.44)

Therefore, as it is well known, Eq.(2.42) implies that Γ = ΓLC(h) and R(µν)(Γ) =Rµν(h) ≡ Rµν . Field equations can be supplemented by the scalar-valued equationobtained by taking the trace of (2.41), (we define τ = trT )

f ′(R)R − 2f(R) = gαβTmαβ ≡ τm (2.45)

which controls solutions of (2.42). We shall refer to this scalar-valued equation as thestructural equation of the spacetime. In the vacuum case (and spacetimes filled withradiation, such that τm = 0) this scalar-valued equation admits constant solutions,which are different from zero only if one add a cosmological constant. This means thatthe universality of Einstein field equations holds [206], corresponding to a theory withcosmological constant [228].

In the case of interaction with matter fields, the structural equation (2.44), if ex-plicitly solvable, provides an expression of R = F (τ), where F is a generic function,and consequently both f(R) and f ′(R) can be expressed in terms of τ . The mattercontent of spacetime thus rules the bi-metric structure of spacetime and, consequently,both the geodesic and metric structures which are intrinsically different. This behaviorgeneralizes the vacuum case and corresponds to the case of a time-varying cosmologicalconstant. In other words, due to these features, conformal transformations, which allow


to pass from a metric structure to another one, acquire an intrinsic physical meaningsince “select” metric and geodesic structures which, for a given ETG, in principle, donot coincide.

Let us now try to extend the above formalism to the case of non-minimally coupledscalar-tensor theories. The effort is to understand if and how the bi-metric structure ofspacetime behaves in this cases and which could be its geometric and physical interpre-tation.

We start by considering scalar-tensor theories in the Palatini formalism, calling A1

the action functional. After, we take into account the case of decoupled non-minimalinteraction between a scalar-tensor theory and a f(R) theory, calling A2 this actionfunctional. We finally consider the case of non-minimal-coupled interaction betweenthe scalar field φ and the gravitational fields (g,Γ), calling A3 the corresponding actionfunctional. Particularly significant is, in this case, the limit of low curvature R. Thisresembles the physical relevant case of present values of curvatures of the Universe andit is important for cosmological applications.

The action (2.24) for scalar-tensor gravity can be generalized, in order to betterdevelop the Palatini approach, as:

A1 =

∫ √−g [F (φ)R +ε

2

g

∇µ φg

∇µ

φ− V (φ) + Lm(Ψ,g

∇ Ψ)]d4x . (2.46)

As above, the values of ε = ±1 selects between standard scalar field theories andquintessence (phantom) field theories. The relative “signature” can be selected by con-formal transformations. Field equations for the gravitational part of the action are,respectively for the metric g and the connection Γ:

F (φ)[R(µν) −1

2Rgµν ] = T φ

µν + Tmµν∇Γ

α(√−gF (φ)gµν) = 0 (2.47)

R(µν) is the same defined in (2.41). For matter fields we have the following field equa-tions:

εφ = −Vφ(φ) + Fφ(φ)RδLm

δΨ= 0 . (2.48)

In this case, the structural equation of spacetime implies that:

R = −τφ + τm

F (φ)(2.49)

which expresses the value of the Ricci scalar curvature in terms of the traces of thestress-energy tensors of standard matter and scalar field (we have to require F (φ) 6= 0).The bi-metric structure of spacetime is thus defined by the ansatz:

√−gF (φ)gµν =√−hhµν (2.50)

such that g and h result to be conformally related

hµν = F (φ)gµν . (2.51)


The conformal factor is exactly the interaction factor. From (2.49), it follows that inthe vacuum case τφ = 0 and τm = 0: this theory is equivalent to the standard Einsteinone without matter. On the other hand, for F (φ) = F0 we recover the Einstein theoryplus a minimally coupled scalar field: this means that the Palatini approach intrinsicallygives rise to the conformal structure (2.51) of the theory which is trivial in the Einstein,minimally coupled case. Beside fundamental physics motivations, these theories haveacquired a huge interest in Cosmology due to the fact that they naturally exhibit infla-tionary behaviors able to overcome the shortcomings of Cosmological Standard Model(based on GR). The related cosmological models seem realistic and capable of match-ing with the Cosmic Microwave Background Radiation (CMBR) observations [?] As afurther step, let us generalize the previous results considering the case of a non-minimalcoupling in the framework of f(R) theories. The action functional can be written as:

A2 =

∫ √−g [F (φ)f(R) +ε

2

g

∇µ φg

∇µ


∇ Ψ)]d4x (2.52)

where f(R) is, as usual, any analytical function of the Ricci scalar R. Field equations(in the Palatini formalism) for the gravitational part of the action are:

F (φ)[f ′(R)R(µν) −1

2f(R)gµν ] = T φ

µν + Tmµν∇Γ

α(√−gF (φ)f ′(R)gµν) = 0 . (2.53)

For scalar and matter fields we have, otherwise, the following field equations:

εφ = −Vφ(φ) +√−gFφ(φ)f(R)

δLm

δΨ= 0 (2.54)

where the non-minimal interaction term enters into the modified Klein-Gordon equa-tions. In this case the structural equation of spacetime implies that:

f ′(R)R− 2f(R) =τφ + τm

F (φ). (2.55)

We remark again that this equation, if solved, expresses the value of the Ricci scalarcurvature in terms of traces of the stress-energy tensors of standard matter and scalarfield (we have to require again that F (φ) 6= 0). The bi-metric structure of spacetime isthus defined by the ansatz:

√−gF (φ)f ′(R)gµν =√−hhµν (2.56)

such that g and h result to be conformally related by:

hµν = F (φ)f ′(R)gµν . (2.57)

Once the structural equation is solved, the conformal factor depends on the values ofthe matter fields (φ,Ψ) or, more precisely, on the traces of the stress-energy tensors


and the value of φ. From equation (2.55), it follows that in the vacuum case, i.e. bothτφ = 0 and τm = 0, the universality of Einstein field equations still holds as in the caseof minimally interacting f(R) theories [206]. The validity of this property is related tothe decoupling of the scalar field and the gravitational field.

Let us finally consider the case where the gravitational Lagrangian is a generalfunction of φ and R. The action functional can thus be written as:

A3 =

∫ √−g [K(φ,R) +ε

2

g

∇µ φg

∇µ


∇ Ψ)]d4x (2.58)

Field equations for the gravitational part of the action are:

[∂ K(φ,R)

∂R

]R(µν) −

1

2K(φ,R)gµν = T φ

µν + Tmµν∇Γ

α

(√−g[∂ K(φ,R)

∂R

]gµν

)= 0 .

(2.59)For matter fields, we have:

εφ = −Vφ(φ) +

[∂ K(φ,R)

∂φ

]δLmat

δΨ= 0 . (2.60)

The structural equation of spacetime can be expressed as:

∂K(φ,R)

∂RR− 2K(φ,R) = τφ + τm (2.61)

This equation, if solved, expresses again the form of the Ricci scalar curvature in terms oftraces of the stress-energy tensors of matter and scalar field (we have to impose regularityconditions and, for example, K(φ,R) 6= 0). The bi-metric structure of spacetime is thusdefined by the ansatz:

√−g∂K(φ,R)

∂Rgµν =

√−hhµν (2.62)

such that g and h result to be conformally related by

hµν =∂K(φ,R)

∂Rgµν (2.63)

Again, once the structural equation is solved, the conformal factor depends just on thevalues of the matter fields and (the trace of) their stress energy tensors. In other words,the evolution, the definition of the conformal factor and the bi-metric structure is ruledby the values of traces of the stress-energy tensors and by the value of the scalar fieldφ. In this case, the universality of Einstein field equations does not hold anymore ingeneral. This is evident from (2.61) where the strong coupling between R and φ avoidsthe possibility, also in the vacuum case, to achieve simple constant solutions.

We consider, furthermore, the case of small values of R, corresponding to smallcurvature spacetimes. This limit represents, as a good approximation, the present epoch


of the observed Universe under suitably regularity conditions. A Taylor expansion ofthe analytical function K(φ,R) can be performed:

K(φ,R) = K0(φ) +K1(φ)R + o(R2) (2.64)

where only the first leading term in R is considered and we have defined:

K0(φ) = K(φ,R)R=0K1(φ) =

(∂K(φ,R)

∂R

)

R=0

. (2.65)

Substituting this expression in (2.61) and (2.63) we get (neglecting higher order approx-imations in R) the structural equation and the bi-metric structure in this particularcase. From the structural equation, we get:

R =1

K1(φ)[−(τφ + τm)− 2K0(φ)] (2.66)

such that the value of the Ricci scalar is always determined, in this first order approx-imation, in terms of τφ, τm, φ. The bi-metric structure is, otherwise, simply defined bymeans of the first term of the Taylor expansion, which is

hµν = K1(φ)gµν . (2.67)

It reproduces, as expected, the scalar-tensor case (2.51). In other words, scalar-tensortheories can be recovered in a first order approximation of a general theory where gravityand non-minimal couplings are any (compare (2.66) with (2.55)). This fact agrees withthe above considerations where Lagrangians of physical interactions can be consideredas stochastic functions with local gauge invariance properties [381].

Finally we have to say that there are also bi-metric theories which cannot be confor-mally related (see for example the summary of alternative theories given in [185]) andtorsion field should be taken into account, if one wants to consider the most generalviewpoint [186, 187]. We will not take into account these general theories in this review.

After this short review of ETGs in metric and Palatini approach, we are going toface some remarkable applications to cosmology and astrophysics. In particular, wedeal with the straightforward generalization of GR, the f(R) gravity, showing that,in principle, no further ingredient, a part a generalized gravity, could be necessary toaddress issues as missing matter (dark matter) and cosmic acceleration (dark energy).However what we are going to consider here are nothing else but toy models which arenot able to fit the whole expansion history, the structure growth law and the CMBanisotropy and polarization. These issues require more detailed theories which, up tonow, are not available but what we are discussing could be a useful working paradigm assoon as refined experimental tests to probe such theories will be proposed and pursued.In particular, we will outline an independent test, based on the stochastic background ofgravitational waves, which could be extremely useful to discriminate between ETGs and


GR or among the ETGs themselves. In this latter case, the data delivered from ground-based interferometers, like VIRGO and LIGO, or the forthcoming space interferometerLISA, could be of extreme relevance in such a discrimination.

Finally, we do not take into account the well known inflationary models based onETGs (e.g. [144, 146]) since we want to show that also the last cosmological epochs,directly related to the so called Precision Cosmology, can be framed in such a new”economic” scheme.

Chapter 3

Gravity from Poincare Gauge

Invariance

3.1 What can generate the Gravity?

Following the prescriptions of General Relativity, the physical spacetime is assumed tobe a four-dimensional differential manifold (see [229] for a general discussion on gravitytheories and their prescriptions). In Special Relativity, this manifold is the Minkwoskiflat-spacetime M4 while, in General Relativity, the underlying spacetime is assumed tobe curved in order to describe the effects of gravitation.

Utiyama [230] was the first to propose that General Relativity can be seen as agauge theory based on the local Lorentz group SO(3, 1) in much the same way thatthe Yang-Mills gauge theory [231] was developed on the basis of the internal iso-spingauge group SU(2). In this formulation the Riemannian connection is the gravitationalcounterpart of the Yang-Mills gauge fields. While SU(2), in the Yang-Mills theory, is aninternal symmetry group, the Lorentz symmetry represents the local nature of spacetimerather than internal degrees of freedom. The Einstein Equivalence Principle, assertedfor General Relativity, requires that the local spacetime structure can be identified withthe Minkowski spacetime possessing Lorentz symmetry. In order to relate local Lorentzsymmetry to the external spacetime, we need to solder the local space to the externalspace. The soldering tools are the tetrad fields. Utiyama regarded the tetrads as objectsgiven a priori. Soon after, Sciama [232] recognized that spacetime should necessarilybe endowed with torsion in order to accommodate spinor fields. In other words, thegravitational interaction of spinning particles requires the modification of the Riemannspacetime of General Relativity to be a (non-Riemannian) curved spacetime with torsion.Although Sciama used the tetrad formalism for his gauge-like handling of gravitation,his theory fell shortcomings in treating tetrad fields as gauge fields. Kibble [233] madea comprehensive extension of the Utiyama gauge theory of gravitation by showing thatthe local Poincare symmetry SO(3, 1) ⋊ T (3, 1) (⋊ represents the semi-direct product)

75

76 3. GRAVITY FROM POINCARE GAUGE INVARIANCE

can generate a spacetime with torsion as well as curvature. The gauge fields introducedby Kibble include the tetrads as well as the local affine connection. There have been avariety of gauge theories of gravitation based on different local symmetries which gaverise to several interesting applications in theoretical physics [234, 235, 236, 237, 238,239, 240, 241, 242, 243].

Following the Kibble approach, it can be demonstrated how gravitation can be for-mulated starting from a pure gauge viewpoint. In particular, the aim of this chapter isto show, in details, how a theory of gravitation is a gauge theory which can be obtainedstarting from the local Poincare symmetry.

A gauge theory of gravity based on a nonlinear realization of the local conformal-affine group of symmetry transformations has been formulated [244]. The coframe fieldsand gauge connections of the theory have been obtained. The tetrads and Lorentz groupmetric have been used to induce a spacetime metric. The inhomogenously transforming(under the Lorentz group) connection coefficients gave rise to gravitational gauge po-tentials used to define covariant derivatives accommodating minimal coupling of matterand gauge fields. On the other hand, the tensor valued connection forms have beenused as auxiliary dynamical fields associated with the dilation, special conformal anddeformation (shear) degrees of freedom inherent to the bundle manifold. This allowedto define the bundle curvature of the theory. Then boundary topological invariants havebeen constructed. They served as a prototype (source free) gravitational Lagrangian.Finally the Bianchi identities, covariant field equations and gauge currents have beenobtained.

Here, starting from the Invariance Principle, we consider first the Global PoincareInvariance and then the Local Poincare Invariance. This approach lead to construct agiven theory of gravity as a gauge theory. This viewpoint, if considered in detail, canavoid many shortcomings and could be useful to formulate self-consistent schemes forquantum gravity.

3.2 Invariance Principle

As it is well-known, the field equations and conservation laws can be obtained from aleast action principle. The same principle is the basis of any gauge theory so we startfrom it to develop our considerations. Let us start from a least action principle and theNoether theorem.

Let χ(x) be a multiplet field defined at a spacetime point x and Lχ(x), ∂jχ(x); xbe the Lagrangian density of the system. The action integral of the system over a givenspacetime volume Ω is defined by

I(Ω) =

∫

Ω

Lχ(x), ∂jχ(x); x d4x. (3.1)

3.2. INVARIANCE PRINCIPLE 77

Now let us consider the infinitesimal variations of the coordinates

xi → x′i = xi + δxi, (3.2)

and the field variablesχ(x)→ χ′(x′) = χ(x) + δχ(x). (3.3)

Correspondingly, the variation of the action is given by

δI =

∫

Ω′

L′(x′) d4x′ −∫

Ω

L(x) d4x =

∫

Ω

[L′(x′)||∂jx

′j || − L(x)]d4x. (3.4)

Since the Jacobian for the infinitesimal variation of coordinates becomes

||∂jx′j|| = 1 + ∂j(δx

j), (3.5)

the variation of the action takes the form,

δI =

∫

Ω

[δL(x) + L(x) ∂j(δx

j)]d4x (3.6)

whereδL(x) = L′(x′)− L(x). (3.7)

For any function Φ(x) of x, it is convenient to define the fixed point variation δ0 by,

δ0Φ(x) := Φ′(x)− Φ(x) = Φ′(x′)− Φ(x′). (3.8)

Expanding the function to first order in δxj as

Φ(x′) = Φ(x) + δxj ∂jΦ(x), (3.9)

we obtain

δΦ(x) = Φ′(x′)− Φ(x) = Φ′(x′)− Φ(x′) + Φ(x′)− Φ(x) = δ0Φ(x) + δxj ∂jΦ(x), (3.10)

orδ0Φ(x) = δΦ(x) − δxj∂jΦ(x). (3.11)

The advantage to have the fixed point variation is that δ0 commutes with ∂j :

δ0∂jΦ(x) = ∂jδ0Φ(x). (3.12)

For Φ(x) = χ(x), we haveδχ = δ0χ+ δxi∂iχ, (3.13)

andδ∂iχ = ∂i(δ0χ)− ∂(δxj)∂iχ. (3.14)


Using the fixed point variation in the integrand of (3.6) gives

δI =

∫

Ω

[δ0L(x) + ∂j(δx

j L(x))]d4x. (3.15)

If we require the action integral defined over any arbitrary region Ω be invariant, thatis, δI = 0, then we must have

δL + L∂j(δxj) = δ0L+ ∂j(Lδxj) = 0. (3.16)

If ∂j(δxj) = 0, then δL = 0, that is, the Lagrangian density L is invariant. In general,

however, ∂j(δxj) 6= 0, and L transforms like a scalar density. In other words, L is a

Lagrangian density unless ∂j(δxj) = 0.

For convenience, let us introduce a function h(x) that behaves like a scalar density,namely

δh + h∂j(δxj) = 0. (3.17)

We further assume L(χ, ∂jχ;x) = h(x)L(χ, ∂jχ;x). Then we see that

δL + L∂j(δxj) = hδL. (3.18)

Hence the action integral remains invariant if

δL = 0. (3.19)

The newly introduced function L(χ, ∂jχ;x) is the scalar Lagrangian of the system.Let us calculate the integrand of (3.15) explicitly. The fixed point variation of L(x)

is a consequence of a fixed point variation of the field χ(x),

δ0L =∂L∂χ

δ0χ+∂L

∂(∂jχ)δ0(∂jχ) (3.20)

which can be cast into the form,

δ0L = [L]χδ0χ+ ∂j

(∂L

∂(∂jχ)δ0χ

)(3.21)

where

[L]χ ≡∂L∂χ− ∂j

(∂L

∂(∂jχ)

). (3.22)

Consequently, we have the action integral in the form

δI =

∫

Ω

[L]χδ0χ+ ∂j

(∂L

∂(∂jχ)δχ− T j

k δxk

)d4x, (3.23)

where

T jk :=

∂L∂(∂jχ)

∂kχ− δjk L (3.24)

3.3. GLOBAL POINCAR E INVARIANCE 79

is the canonical energy-momentum tensor density. If the variations are chosen in sucha way that δxj = 0 over Ω and δ0χ vanishes on the boundary of Ω, then δI = 0 gives usthe Euler-Lagrange equation,

[L]χ =∂L∂χ− ∂j

(∂L

∂(∂jχ)

)= 0. (3.25)

On the other hand, if the field variables obey the Euler-Lagrange equation, [L]χ = 0,then we have

∂j

(∂L

∂(∂jχ)δχ− T j

k δxk

)= 0, (3.26)

which gives rise, considering also the Noether theorem, to conservation laws. These verystraightforward considerations are at the basis of our following discussion.

3.3 Global Poincare Invariance

As standard, we assert that our spacetime in the absence of gravitation is a Minkowskispace M4. The isometry group of M4 is the group of Poincare transformations (PT)which consists of the Lorentz group SO(3, 1) and the translation group T (3, 1). ThePoincare transformations of coordinates are

xi PT→ x′i = aijx

j + bi, (3.27)

where aij and bi are real constants, and ai

j satisfy the orthogonality conditions aika

kj = δi

j .For infinitesimal variations,

δx′i = χ′(x′)− χ(x) = εi jxj + εi (3.28)

where εij + εji = 0. While the Lorentz transformation forms a six parameter group,the Poincare group has ten parameters. The Lie algebra for the ten generators of thePoincare group is

[Ξij, Ξkl] = ηik Ξjl + ηjl Ξik − ηjk Ξil − ηil Ξjk,

(3.29)

[Ξij , Tk] = ηjkTi − ηikTj , [Ti, Tj ] = 0,

where Ξij are the generators of Lorentz transformations, and Ti are the generators offour-dimensional translations. Obviously, ∂i(δx

i) = 0 for the Poincare transformations(3.27). Therefore, our Lagrangian density L, which is the same as L with h(x) = 1 inthis case, is invariant; namely, δL = δL = 0 for δI = 0.

Suppose that the field χ(x) transforms under the infinitesimal Poincare transforma-tions as

δχ =1

2εijSijχ, (3.30)


where the tensors Sij are the generators of the Lorentz group, satisfying

Sij = −Sji, [Sij , Skl] = ηik Sjl + ηjl Sik − ηjk Sil − ηil Sjk. (3.31)

Correspondingly, the derivative of χ transforms as

δ(∂kχ) =1

2εijSij∂kχ− εi k∂iχ. (3.32)

Since the choice of infinitesimal parameters εi and εij is arbitrary, the vanishing variationof the Lagrangian density δL = 0 leads to the identities,

∂L∂χ

Sijχ+∂L

∂(∂kχ)(Sij∂kχ+ ηki∂jχ− ηkj∂iχ) = 0. (3.33)

We also obtain the following conservation laws

∂j Tjk = 0, ∂k

(Sk

ij − xiTk

j + xjTk

i

)= 0, (3.34)

where

Skij := − ∂L

∂(∂kχ)Sijχ. (3.35)

These conservation laws imply that the energy-momentum and angular momentum,respectively

Pl =

∫T 0

l d3x, Jij =

∫ [S0

ij −(xiT

0j − xjT

0i

)]d3x, (3.36)

are conserved. This means that the system invariant under the ten parameter symmetrygroup has ten conserved quantities. This is an example of Noether symmetry. The firstterm of the angular momentum integral corresponds to the spin angular momentumwhile the second term gives the orbital angular momentum. The global Poincare invari-ance of a system means that, for the system, the spacetime is homogeneous (all spacetimepoints are equivalent) as dictated by the translational invariance and is isotropic (all di-rections about a spacetime point are equivalent) as indicated by the Lorentz invariance.It is interesting to observe that the fixed point variation of the field variables takes theform

δ0χ =1

2εj kΞj

k χ+ εj Tj χ, (3.37)

where

Ξjk = Sj

k +(xj∂k − xk∂j

), Tj = −∂j. (3.38)

We remark that Ξjk are the generators of the Lorentz transformation and Tj are those

of the translations.

3.4. LOCAL POINCAR E INVARIANCE 81

3.4 Local Poincare Invariance

As next step, let us consider a modification of the infinitesimal Poincare transformations(3.28) by assuming that the parameters εjk and εj are functions of the coordinates andby writing them altogether as

δxµ = εµ ν(x)xν + εµ(x) = ξµ, (3.39)

which we call the local Poincare transformations (or the general coordinate transforma-tions). In order to make a distinction between the global transformation and the localtransformation, we use the Latin indices (j, k = 0, 1, 2, 3) for the former and the Greekindices (µ, ν = 0, 1, 2, 3) for the latter. The variation of the field variables χ(x) definedat a point x is still the same as that of the global Poincare transformations,

δχ =1

2εijS

ij χ. (3.40)

The corresponding fixed point variation of χ takes the form,

δ0χ =1

2εijS

ijχ− ξν∂νχ. (3.41)

Differentiating both sides of (3.41) with respect to xµ, we have

δ0∂µχ =1

2εijSij∂µχ+

1

2(∂µε

ij)Sijχ− ∂µ(ξν∂νχ). (3.42)

By using these variations, we obtain the variation of the Lagrangian L,

δL + ∂µ(δxµ)L = hδL = δ0L+ ∂ν(Lδxν) = −1

2(∂µε

ij)Sµij − ∂µξ

ν T µν , (3.43)

which is no longer zero unless the parameters εij and ξν become constants. Accordingly,the action integral for the given Lagrangian density L is not invariant under the localPoincare transformations. We notice that while ∂j(δx

j) = 0 for the local Poincaretransformations, ∂µξ

µ does not vanish under local Poincare transformations. Hence, asexpected L is not a Lagrangian scalar but a Lagrangian density. As mentioned earlier,in order to define the Lagrangian L, we have to select an appropriate non-trivial scalarfunction h(x) satisfying

δh+ h∂µξµ = 0. (3.44)

Now we consider a minimal modification of the Lagrangian so as to make the actionintegral invariant under the local Poincare transformations. It is rather obvious that ifthere is a covariant derivative ∇kχ which transforms as

δ(∇kχ) =1

2εijSij∇kχ− εi k∇iχ, (3.45)


then a modified Lagrangian L′(χ, ∂kχ, x) = L(χ, ∇kχ, x), obtained by replacing ∂kχ ofL(χ, ∂kχ, x) by ∇kχ, remains invariant under the local Poincare transformations, thatis

δL′ =∂L′

∂χδχ+

∂L′

∂(∇kχ)δ(∇kχ) = 0. (3.46)

To find such a k-covariant derivative, we introduce the gauge fields Aijµ = −Aji

µ anddefine the µ-covariant derivative

∇µχ := ∂µχ+1

2Aij

µSijχ, (3.47)

in such a way that the covariant derivative transforms as

δ0∇µχ =1

2Sij∇µχ− ∂µ(ξν∇νχ). (3.48)

The transformation properties of Aabµ are determined by ∇µχ and δ∇µχ. Making use

of

δ∇µχ =1

2εij ,µ Sijχ+

1

2εijSij∂µχ− (∂µξ

ν) ∂νψ+1

2δAij

µSijχ+1

4Aij

µSijεklSklχ (3.49)

and comparing with (3.47) we obtain,

δAijµSijχ+ εij ,µ Sijχ+

1

2

(Aij

µεkl − εijAkl

µ

)SijSklχ+ (∂µξ

ν)AijνSijχ = 0. (3.50)

Using the antisymmetry in ij and kl to rewrite the term in parentheses on the RHS of(3.50) as [Sij , Skl]A

ijµεklχ, we see the explicit appearance of the commutator [Sij , Skl].

Using the expression for the commutator of Lie algebra generators

[Sij, Skl] =1

2c[ef ]

[ij][kl]Sef , (3.51)

where c[ef ]

[ij][kl] (the square brackets denote anti-symmetrization) is the structure con-

stants of the Lorentz group (deduced below), we have

[Sij, Skl]Aij

µεkl =

1

2

(Aic

µ εjc −Acj

µ εic

)Sij . (3.52)

The substitution of this equation and the consideration of the antisymmetry of ε bc = −εbc

enables us to write

δAijµ = εi kA

kjµ + εj kA

ikµ − (∂µξ

ν)Aijν − ∂µε

ij . (3.53)

We require the k-derivative and µ-derivative of χ to be linearly related as

∇kχ = Vkµ(x)∇µχ, (3.54)

3.5. SPINORS, VECTORS AND TETRADS 83

where the coefficients Vkµ(x) are position-dependent and behave like a new set of field

variables. From (3.54) it is evident that ∇kχ varies as

δ∇kχ = δV µk ∇µχ+ V µ

k δ∇µχ. (3.55)

Comparing with δ∇kχ = 12ε

abSab∇kχ− εjk∇jχ we obtain,

V kα δV

µk ∇µχ− ξν ,α∇νχ+ V k

α εjk∇jχ = 0. (3.56)

Exploiting δ(V k

α Vµk

)= 0 we find the quantity Vk

µ transforms according to

δVkµ = Vk

ν∂νξµ − Vi

µεi k. (3.57)

It is also important to recognize that the inverse of det(Vkµ) transforms like a scalar

density as h(x) does. For our minimal modification of the Lagrangian density, we utilizethis available quantity for the scalar density h; namely, we let

h(x) = [det(Vkµ)]−1. (3.58)

In the limiting case, when we consider Poincare transformations, that are not spacetimedependent, Vk

µ → δµk so that h(x) → 1. This is a desirable property. Then we replace

the Lagrangian density L(χ, ∂kχ, x), invariant under the global Poincare transforma-tions, by a Lagrangian density

L(χ, ∂µχ; x)→ h(x)L(χ, ∇kχ). (3.59)

The action integral with this modified Lagrangian density remains invariant under thelocal Poincare transformations. Since the local Poincare transformations δxµ = ξµ(x)are nothing else but generalized coordinate transformations, the newly introduced gaugefields V λ

i and Aijµ can be interpreted, respectively, as the tetrad (vierbein) fields which

set the local coordinate frame and as a local affine connection with respect to the tetradframe (see also [245]).

3.5 Spinors, Vectors and Tetrads

Let us consider first the case where the multiplet field χ is the Dirac field ψ(x) whichbehaves like a four-component spinor under the Lorentz transformations and transformsas

ψ(x)→ ψ′(x′) = S(Λ)ψ(x), (3.60)

where S(Λ) is an irreducible unitary representation of the Lorentz group. Since thebilinear form vk = iψγkψ is a vector, it transforms according to

vj = Λjkv

k, (3.61)


where Λji is a Lorentz transformation matrix satisfying

Λij + Λji = 0. (3.62)

The invariance of vi (or the covariance of the Dirac equation) under the transformationψ(x)→ ψ′(x′) leads to

S−1(Λ)γµS(Λ) = Λµνγ

ν , (3.63)

where the γ′s are the Dirac γ-matrices satisfying the anticommutator,

γiγj + γjγi = ηij1. (3.64)

Furthermore, we notice that the γ-matrices have the following properties:

(γ0)† = −γ0,

(γ0)2

= (γ0)2 = −1, γ0 = −γ0 and γ0γ

0 = 1

(γk)† = γk ,

(γk)2

= (γk)2 = 1; (k = 1, 2, 3) and γk = γk

(γ5)† = −γ5, (γ5)

2 = −1 and γ5 = γ5.

(3.65)

We assume the transformation S(Λ) can be put into the form S(Λ) = eΛµνγµν. Ex-

panding S(Λ) about the identity and only retaining terms up to the first order in theinfinitesimals and expanding Λµν to the first order in εµν

Λµν = δµν + εµν , εij + εji = 0, (3.66)

we get

S(Λ) = 1 +1

2εijγij . (3.67)

In order to determine the form of γij , we substitute (3.66) and (3.67) into (3.63) toobtain

1

2εij

[γij , γk

]= ηkiεjiγ

j . (3.68)

Rewriting the RHS of (3.68) using the antisymmetry of εij as

ηkiεjiγj =

1

2εij

(ηkiγj − ηkjγi

), (3.69)

yields [γk, γij

]= ηkiγj − ηkjγi. (3.70)

Assuming the solution to have the form of an antisymmetric product of two matrices,we obtain the solution

γij :=1

2

[γi, γj

]. (3.71)


If χ = ψ, the group generator Sij appearing in (3.31) is identified with

Sij ≡ γij =1

2(γiγj − γjγi). (3.72)

To be explicit, the Dirac field transforms under Lorentz transformations (LT) as

δψ(x) =1

2εijγijψ(x). (3.73)

The Pauli conjugate of the Dirac field is denoted ψ and defined by

ψ(x) := iψ†(x) γ0, i C. (3.74)

The conjugate field ψ transforms under LTs as,

δψ = −ψ1

2εijψγij. (3.75)

Under local LTs, εab(x) becomes a function of spacetime. Now, unlike ∂µψ(x),the derivative of ψ′(x′) is no longer homogenous due to the occurrence of the termγab [∂µεab(x)]ψ(x) in ∂µψ

′(x′), which is non-vanishing unless εab is constant. Whengoing from locally flat to curved spacetime, we must generalize ∂µ to the covariantderivative ∇µ to compensate for this extra term, allowing to gauge the group of LTs.Thus, by using ∇µ, we can preserve the invariance of the Lagrangian for arbitrary localLTs at each spacetime point

∇µψ′(x′) = S(Λ(x))∇µψ(x). (3.76)

To determine the explicit form of the connection belonging to∇µ, we study the derivativeof S(Λ(x)). The transformation S(Λ(x)) is given by

S(Λ(x)) = 1 +1

2εab(x)γ

ab. (3.77)

Since εab(x) is only a function of spacetime for local Lorentz coordinates, we express thisinfinitesimal LT in terms of general coordinates only by shifting all spacetime dependenceof the local coordinates into tetrad fields as

εab(x) = V λa (x)V ν

b(x)ελν . (3.78)

Substituting this expression for εab(x), we obtain

∂µεab(x) = ∂µ

[V λ

a (x)V νb(x)ελν

]. (3.79)

However, since ελν has no spacetime dependence, this reduces to

∂µεab(x) = V λa (x)∂µVbλ(x)− V ν

b (x)∂µVaν(x). (3.80)


Lettingωµba := V ν

b (x)∂µVaν(x), (3.81)

the first and second terms in Eq.(3.80) become V λa (x)∂µVbλ(x) = 1

2ωµab and V νb (x)∂µVaν(x) =

12ωµba respectively. Using the identification

∂µεab(x) = ωµab, (3.82)

we write

∂µS(Λ(x)) = −1

2γabωµab. (3.83)

According to (3.47), the covariant derivative of the Dirac spinor is

∇µψ = ∂µψ +1

2Aij

µγijψ. (3.84)

Correspondingly, the covariant derivative of ψ is given by

∇µψ = ∂µψ −1

2Aij

µψγij . (3.85)

Using the covariant derivatives of ψ and ψ, we can show that

∇µvj = ∂µvj −Aijµvi. (3.86)

The same covariant derivative should be used for any covariant vector vk under theLorentz transformation. Since ∇µ(viv

i) = ∂µ(vivi), the covariant derivative for a con-

travariant vector vi must be

∇µvi = ∂µv

i +Aijµv

j . (3.87)

Since the tetrad Viµ is a covariant vector under Lorentz transformations, its covari-

ant derivative must transform according to the same rule. Using ∇a = V µa (x)∇µ, the

covariant derivatives of a tetrad in local Lorentz coordinates read

∇νViµ = ∂νVi

µ −AkiνVk

µ, ∇νViµ = ∂νV

iµ +Ai

kνVk

µ. (3.88)

The inverse of Viµ is denoted by V i

µ and satisfies

V iµVi

ν = δµν , V i

µVjµ = δi

j . (3.89)

To allow the transition to curved spacetime, we take account of the general coordinatesof objects that are covariant under local Poincare transformations. Here we define thecovariant derivative of a quantity vλ which behaves like a contravariant vector underthe local Poincare transformation. Namely

Dνvλ ≡ Vi

λ∇νvi = ∂νv

λ + Γλµνv

µ, Dνvµ ≡ V iµ∇νvi = ∂νvµ − Γλ

µνvλ, (3.90)


whereΓλ

µν := Viλ∇νV

iµ ≡ −V i

µ∇νViλ. (3.91)

The definition of Γλµν implies

DνViλ = ∇νVi

λ + ΓλµνVi

µ = ∂νViλ −Ak

iνVkλ + Γλ

µνViµ = 0, (3.92)

DνViµ = ∇νV

iµ − Γλ

µνViλ = ∂νV

iµ +Ai

kνVk

µ − ΓλµνV

iλ = 0.

From (3.92) we find,

Aikν = V i

λ∂νVkλ + Γλ

µνViλVk

µ = −Vkλ∂νV

iλ + Γλ

µνViλVk

µ. (3.93)

or, equivalently, in terms of ω defined in (3.81),

Aikν = ωi

νk + ΓλµνV

iλVk

µ = −ω ikν + Γλ

µνViλVk

µ. (3.94)

Using this in (3.84), we may write

∇µψ = (∂µ − Γµ)ψ, (3.95)

where

Γµ =1

4

(ωi

jµ − ΓλµνV

iλVj

ν)γi

j , (3.96)

which is known as the Fock-Ivanenko connection.We now study the transformation properties of Aµab. Recall ωµab = V λ

a (x)∂µVβλ(x)and since ∂µηab = 0, we write

Λaaηab∂µΛ b

b= Λa

a∂µΛab. (3.97)

Note that barred indices are equivalent to the primed indices used above. Hence, thespin connection transforms as

Aabc = Λ aa Λ b

bΛ c

c Aabc + Λ aa Λ c

c V µa(x)∂µΛbc. (3.98)

To determine the transformation properties of

Γabc = Aabc − [V µa(x)∂µV

νb(x)]Vνc(x), (3.99)

we consider the local LT of [V µa (x)∂µV

νb(x)]Vνc(x) which is,

[V µ

a(x)∂µVνb

]Vνc(x) = Λ a

a Λ bb

Λ cc [Aν

abVνc(x)] + Λ aa Λ c

c V µa(x)∂µΛcb. (3.100)

From this result, we obtain the following transformation law,

Γabc = Λ aa Λ b

bΛ c

c Γabc. (3.101)


We now explore the consequence of the antisymmetry of ωabc in bc. Recalling theequation for Γabc, exchanging b and c and adding the two equations, we obtain

Γabc + Γacb = −V µa(x) [(∂µV

νb(x)) Vνc(x) + (∂µV

νc(x))Vνb(x)] . (3.102)

We know however, that

∂µ [V νb(x)Vνc(x)] = Vνc(x)∂µV

νb(x) + Vλb(x)∂µV

λc(x) + V ν

b (x)V λc(x)∂µgλν . (3.103)

Letting λ→ ν and exchanging b and c, we obtain

∂µ [V νb(x)Vνc(x)] = −V λ

b (x)V νc(x)∂µgνλ, (3.104)

so that, finally,

Γabc + Γacb = V µa (x)V λ

b (x)V νc (x)∂µgνλ. (3.105)

This, however, is equivalent to

Γabc + Γa cb = V µa(x)V

λb(x)V ν

c(x)∂µgνλ, (3.106)

and then

Γµλν + Γµνλ = ∂µgνλ, (3.107)

which we recognize as the general coordinate connection. It is known that the covariantderivative for general coordinates is

∇µAλ

ν = ∂µAλ

ν + ΓλµσA

σν − Γσ

µνAλ

σ . (3.108)

In a Riemannian manifold, the connection is symmetric under the exchange of µν, thatis, Γλ

µν = Γλνµ. Using the fact that the metric is a symmetric tensor we can now

determine the form of the Christoffel connection by cyclically permuting the indices ofthe general coordinate connection equation (3.107) yielding

Γµνλ =1

2(∂µgνλ + ∂νgλµ − ∂λgµν) . (3.109)

Since Γµνλ = Γνµλ is valid for general coordinate systems, it follows that a similarconstraint must hold for local Lorentz transforming coordinates as well, so we expectΓabc = Γbac. Recalling the equation for Γabc and exchanging a and b, we obtain

ωabc − ωbac = Vνc(x)[V µ

a(x)∂µVνb(x)− V µ

b(x)∂µVνa(x)

]. (3.110)

We now define the objects of anholonomicity as

Ωcab := Vνc(x)[V µ

a(x)∂µVνb(x)− V µ

b(x)∂µVνa(x)

]. (3.111)

3.6. CURVATURE, TORSION AND METRIC 89

Using Ωcab = −Ωcba, we permute indices in a similar manner as was done for the deriva-tion of the Christoffel connection above yielding,

ωabµ =1

2[Ωcab + Ωbca − Ωabc]V

cµ ≡ ∆abµ. (3.112)

For completeness, we determine the transformation law of the Christoffel connection.Making use of Γλ

µνeλ = ∂µeν where

∂µeν = XµµX

νν∂µeν +Xµ

µ (∂µXνν) eν , (3.113)

we can showΓλ

µ ν = XµµX

ννX

λλ Γλ

µν +XµµX

λν Xν

µν , (3.114)

whereXν

µν ≡ ∂µ∂νxν . (3.115)

In the light of the above considerations, we may regard infinitesimal local gaugetransformations as local rotations of basis vectors belonging to the tangent space [243,248] of the manifold. For this reason, given a local frame on a tangent plane to thepoint x on the base manifold, we can obtain all other frames on the same tangent planeby means of local rotations of the original basis vectors. Reversing this argument, weobserve that by knowing all frames residing in the horizontal tangent space to a point xon the base manifold enables us to deduce the corresponding gauge group of symmetrytransformations.

3.6 Curvature, Torsion and Metric

From the definition of the Fock-Ivanenko covariant derivative, we can find the secondorder covariant derivative

DνDµψ = ∂ν∂µψ +1

2Scd

(ψ∂νA

cdµ +A cd

µ ∂νψ)

+ ΓρµνDρψ +

1

2SefA

efν ∂µψ

+1

4SefScdA

efν A cd

µ ψ. (3.116)

Recalling DνVcµ = 0, we can solve for the spin connection in terms of the Christoffel

connectionA cd

µ = −V dλ∂µV

cλ − Γ cdµ . (3.117)

The derivative of the spin connection is then

∂µAcd

ν = −V dλ∂µ∂νV

cλ −(∂νV

cλ)∂µV

dλ − ∂µΓcd

ν . (3.118)

Noting that the Christoffel connection is symmetric and partial derivatives commute,we find

[Dµ, Dν ]ψ =1

2Scd

[(∂νA

cdµ − ∂µA

cdν

)ψ]

+1

4SefScd

[(Aef

νAcd

µ −AefµA

cdν

)ψ],

(3.119)


where∂νA

cdµ − ∂µA

cdν = ∂µΓcd

ν − ∂νΓcdµ. (3.120)

Relabeling running indices, we can write

1

4SefScd

(Aef

νAcd

µ −AefµA

cdν

)ψ =

1

4[Scd, Sef ]Aef

µAcd

νψ. (3.121)

Using γa, γb = 2ηab to deduce

γa, γb γcγd = 2ηabγcγd, (3.122)

we find that the commutator of bi-spinors is given by

[Scd, Sef ] =1

2

[ηceδ

adδ

bf − ηdeδ

ac δ

bf + ηcfδ

ae δ

bd − ηdfδ

ae δ

bc

]Sab. (3.123)

Clearly the terms in brackets on the RHS of (3.123) are antisymmetric in cd and efand also antisymmetric under the exchange of pairs of indices cd and ef . Since thealternating spinor is antisymmetric in ab, so it must be the terms in brackets: thismeans that the commutator does not vanish. Hence, the term in brackets is totallyantisymmetric under interchange of indices ab, cd and ef and exchange of these pairs ofindices. We identify this as the structure constant of the Lorentz group [249]

[ηceδ

adδ

bf − ηdeδ

ac δ

bf + ηcfδ

ae δ

bd − ηdf δ

ae δ

bc

]= c[cd][ef ]

[ab] = c[ab][cd][ef ], (3.124)

with the aid of which we can write

1

4[Scd, Sef ]Aef

µAcd

νψ =1

2Sab

[Aa

eνAeb

µ −AbeνA

aeµ

]ψ, (3.125)

whereAa

eνAeb

µ −AbeνA

aeµ = Γa

νeΓeb

µ − ΓbνeΓ

eaµ. (3.126)

Combining these results, the commutator of two µ-covariant differentiations gives

[∇µ,∇ν ]χ = −1

2Rij

µνSijχ, (3.127)

whereRi

jµν = ∂νAijµ − ∂µA

ijν +Ai

kνAk

jµ −AikµA

kjν. (3.128)

Using the Jacobi identities for the commutator of covariant derivatives, it follows thatthe field strength Ri

jµν satisfies the Bianchi identity

∇λRijµν +∇µR

ijνλ +∇νR

ijλµ = 0. (3.129)

Permuting indices, this can be put into the cyclic form

εαβρσ∇βRij

ρσ = 0, (3.130)

3.6. CURVATURE, TORSION AND METRIC 91

where εαβρσ is the Levi-Civita alternating symbol. Furthermore, Rijµν = ηjkRi

kµν isantisymmetric with respect to both pairs of indices,

Rijµν = −Rji

µν = Rjiνµ = −Rij

νµ. (3.131)

This condition is known as the first curvature tensor identity.

To determine the analogue of [∇µ,∇ν ]χ in local coordinates, we start from ∇kψ =V µ

k∇µψ. From ∇kψ we obtain,

∇l∇kψ = V νl

(∇νV

µk

)∇µψ + V ν

lVµk∇ν∇µψ. (3.132)

Permuting indices and recognizing

V aµ ∇νV

µk = −V µ

k ∇νVaµ, (3.133)

(which follows from ∇ν

(V a

µ Vµk

)= 0), we arrive at

V νl

(∇νV

µk

)∇µψ − V µ

k (∇µVνl)∇νψ =

(V µ

lVνk − V µ

kVνl

) (∇νV

aµ

)∇aψ. (3.134)

Defining

Cakl :=

(V µ

kVνl − V µ

lVνk

)∇νV

aµ , (3.135)

the commutator of the k-covariant differentiations takes the final form [233]

[∇k,∇l]χ = −1

2Rij

klSijχ+ Cikl∇iχ, (3.136)

where

Rijkl = Vk

µVlνRij

µν . (3.137)

As done for Rijµν using the Jacobi identities for the commutator of covariant derivatives,

we find the Bianchi identity in Einstein-Cartan spacetime [255, 247]

εαβρσ∇βRijρσ = εαβρσC λ

βρ Rijσλ. (3.138)

The second curvature identity

Rk[ρσλ] = 2∇[ρC

kσλ] − 4C b

[ρσ Ck

λ]b (3.139)

leads to,

εαβρσ∇βCk

ρσ = εαβρσRkjρσV

jβ. (3.140)

Notice that if

Γλµν = V λ

i ∇νViµ = −V i

µ ∇νVλi, (3.141)

then

Γλµν − Γλ

νµ = V λi

(∇νV

iµ −∇µV

iν

). (3.142)


Contracting by V µk V

νl , we obtain [233],

Cakl = V µ

k V νl V a

λ

(Γλ

µν − Γλνµ

). (3.143)

We therefore conclude that Cakl is related to the antisymmetric part of the affine con-

nectionΓλ

[µν] = V kµ V l

ν V λa Ca

kl ≡ T λµν , (3.144)

which is usually interpreted as spacetime torsion T λµν . Considering ∆abµ defined in

(3.112), we see that the most general connection in the Poincare gauge approach togravitation is

Aabµ = ∆abµ −Kabµ + ΓλνµVaλVb

ν , (3.145)

whereKabc = −

(T λ

νµ − T λνµ + T λ

µ ν

)VaλVb

νV µc , (3.146)

is the contorsion tensor. Now, the quantity Rρσµν = Vi

ρRiσµν may be expressed as

Rρσµν = ∂νΓρ

σµ − ∂µΓρσν + Γρ

λνΓλσµ − Γρ

λµΓλσν . (3.147)

Therefore, we can regard Rρσµν as the curvature tensor with respect the affine connec-

tion Γλµν . By using the inverse of the tetrad, we define the metric of the spacetime

manifold bygµν = V i

µVj

νηij . (3.148)

From (3.92) and the fact that the Minkowski metric is constant, it is obvious that themetric so defined is covariantly constant, that is,

Dλgµν = 0. (3.149)

The spacetime thus specified by the local Poincare transformation is said to be metric.It is not difficult to show that

√−g = [detV iµ] = [detVi

µ]−1, (3.150)

where g = det gµν . Hence we may take√−g for the density function h(x).

3.7 Field Equations for Gravity

Finally, we are able to deduce the field equations for the gravitational field. From thecurvature tensor Rρ

σµν , given in (3.147), the Ricci tensor follows

Rσν = Rµσµν . (3.151)

and the scalar curvature

R = Rνν =

LR+ ∂iK

iaa − T bc

a K abc (3.152)

3.7. FIELD EQUATIONS FOR GRAVITY 93

whereLR denotes the usual Ricci scalar of General Relativity. Using this scalar curvature

R, we choose the Lagrangian density for free Einstein-Cartan gravity

LG =1

2κ

√−g(

LR+ ∂iK

iaa − T bc

a K abc − 2Λ

), (3.153)

where κ is a gravitational coupling constant, and Λ is the cosmological constant. These

considerations can be easily extended to any function ofLR as in [246]. Observe that the

second term is a divergence and may be ignored. The field equation can be obtainedfrom the total action,

S =

∫ Lfield(χ, ∂µχ, Vi

µ, Aijµ) + LG

d4x, (3.154)

where the matter Lagrangian density is taken to be

Lfield =1

2

[ψγaDaψ −

(Daψ

)γaψ

]. (3.155)

Modifying the connection to include Christoffel, spin connection and contorsion contri-butions so as to operate on general, spinoral arguments, we have

Γµ =1

4gλσ

(∆σ

µρ −LΓ σ

ρµ −Kσρµ

)γλρ. (3.156)

It is important to keep in mind that ∆σµρ act only on multi-component spinor fields,

whileLΓ σ

ρµ act on vectors and arbitrary tensors. The gauge covariant derivative for aspinor and adjoint spinor is then given by

Dµψ = (∂µ − Γµ)ψ, Dµψ = ∂µψ − ψΓµ. (3.157)

The variation of the field Lagrangian is

δLfield = ψ (δγµDµ + γµδΓµ)ψ. (3.158)

We know that the Dirac gamma matrices are covariantly vanishing, so

Dκγι = ∂κγι − Γµικγµ +

[γι, Γκ

]= 0. (3.159)

The 4 × 4 matrices Γκ are real matrices used to induce similarity transformations onquantities with spinor transformation [254] properties according to

γ′i = Γ−1γiΓ. (3.160)

Solving for Γκ leads to,

Γκ =1

8[(∂κγι) γ

ι − Γµικγµγ

ι] . (3.161)


Taking the variation of Γκ,

δΓκ =1

8

[(∂κδγι) γ

ι + (∂κγι) δγι − (δΓµ

ικ) γµγι

−Γµικ ((δγµ) γι + γµδγ

ι)

](3.162)

=1

8[(∂κδγι) γ

ι − (δΓµικ) γµγ

ι] .

Since we require the anticommutator condition on the gamma matrices γµ, γν = 2gµν

to hold, the variation of the metric gives

2δgµν = δγµ, γν+ γµδγν. (3.163)

One solution to this equation is,

δγν =1

2γσδγ

σν . (3.164)

With the aid of this result, we can write

(∂κδγι) γι =

1

2∂κ (γνδgνι) γ

ι. (3.165)

Finally, exploiting the anti-symmetry in γµν we obtain

δΓκ =1

8

[gνσδΓ

σµκ − gµσδΓ

σνκ

]γµν . (3.166)

The field Lagrangian defined in the Einstein-Cartan spacetime can be written [250, 251,255, 388, 247] explicitly in terms of its Lorentzian and contorsion components as

Lfield =1

2

[(LDµψ

)γµψ − ψγµ

LDµψ

]− ~c

8Kµαβψ

γµ, γαβ

ψ. (3.167)

Using the following relations

−14Kµαβψ

γµ, γαβ

ψ = 1

4Kµαβψγβαγµψ − 1

4Kµαβψγµγαβψ,

γµγνγλεµνλσ =γµ, γνλ

εµνλσ = 3!γσγ5,

γµ, γνλ

= γ[µγνγλ],

(3.168)

we obtain

Kµαβψγµ, γαβ

ψ =

1

2iKµαβε

αβµν(ψγ5γνψ

). (3.169)

Here we define the contorsion axial vector

Kν :=1

3!εαβµνKαβµ. (3.170)

Multiplying through by the axial current j5ν = ψγ5γνψ, we obtain

(ψγ5γνψ

)εαβµνKµαβ = −6ij5νK

ν . (3.171)


Thus, the field Lagrangian density becomes

Lfield =1

2

[(LDµψ

)γµψ − ψγµ

LDµψ

]+

3i~c

8Kµj

µ5 . (3.172)

The total action reads

δI = δ

∫LG√−gd4x+ δ

∫Lfield

√−gd4x (3.173)

=

∫(δLG + δLfield)

√−gd4x.

Writing the metric in terms of the tetrads gµν = V µi V

νi, we observe

δ√−g = −1

2

√−g(δV µ

i Vi

µ + VνiδVνi). (3.174)

By usingδV νi = δ

(ηijV ν

j

)= ηijδV ν

j , (3.175)

we are able to deduceδ√−g = −√−gV i

µ δVµ

i . (3.176)

For the variation of the Ricci tensor Riν = V µi Rµν we have

δLRiν = δV µ

i

LRµν + V µ

i δLRµν . (3.177)

In an inertial frame, the Ricci tensor reduces to

LRµν = ∂ν

LΓ β

βµ − ∂β

LΓ β

νµ, (3.178)

so that

δLRiν = δV µ

i

LRµν + V µ

i

(∂νδ

LΓ β

βµ − ∂βδLΓ β

νµ

). (3.179)

The second term can be converted into a surface term, so it may be ignored. Collectingour results, we have

δgµν = −gµρgνσδgρσ ,δ√−g = −1

2

√−ggµνδgµν = −√−gV i

µ δVµ

i ,

δRµν = gρµ

(∇λδΓ

λρν −∇νδΓ

λρλ

)+ T ρ

λµ δΓλρν , δ

LRiν = δV µ

i

LRµν

δR =LR µνδgµν + gµν

(∇λδ

LΓ λ

µν −∇νδLΓ λ

µλ

)− T bc

a δK abc .

(3.180)

From the above results, we obtain

δIG =1

16π

∫ (R µ

i − 12V

µi R− V µ

i Λ)δV i

µ + 2gρλT σµλ δΓµ

ρσ

+gµν

(∇λδ

LΓ λ

µν −∇νδLΓ λ

µλ

)√−gd4x. (3.181)


The last term in the action can be converted into a surface term, so it may be ignored.Using the four-current vµ introduced earlier, the action for the matter fields read [254]

δIfield =

∫ [ψδγµ∇µψ + ψγµδΓµψ

]√−gd4x (3.182)

=

∫

[12g

µνψγi (∇νψ) + T µρσT

ρσi − δµ

i TλρσTλρσ]δV i

µ

+18 (gρνvµ − gρµvν)

(gµσδ

LΓ σ

νρ − gνσδLΓ σ

µρ

)√−gd4x.

Removing the derivatives of variations of the metric appearing in δΓσνρ via partial

integration, and equating to zero the coefficients of δgµν and δT σνρ in the variation of

the action integral, we obtain

0 =1

16π

(Rµν −

1

2gµνR− gµνΛ

)+

(1

2ψγν∇µψ −

1

4∇µvν

)(3.183)

+∇σTσ

µν + TµρσTρσ

ν − gµνTλρσTλρσ

and

Tρσλ = 8πτρσλ. (3.184)

Eqs.(3.183) have the form of Einstein equations

Gµν − gµνΛ = 8πΣµν , (3.185)

where the Einstein tensor and non-symmetric energy-momentum tensors are

Gµν = Rµν −1

2gµνR, (3.186)

Σµν = Θµν + Tµν , (3.187)

respectively. Here we identify Θµν as the canonical energy-momentum

Θµν =

∂Lfield

∂(∇µχ)∇νχ− δµ

νLfield, (3.188)

while Tµν is the stress-tensor form of the non-Riemannian manifold. For the case ofspinor fields being considered here the explicit form of the energy-momentum compo-nents [253] are (after symmetrization of corresponding canonical source terms in theEinstein equation),

Θµν = −[ψγµ∇νψ −

(∇νψ

)γµψ + ψγν∇µψ −

(∇µψ

)γνψ

](3.189)

and by using the second field equation (3.184), we determine

Tµν = ∇σTσ

µν + Tµρστρσ

ν − gµνTλρστλρσ, (3.190)


where τ σµν is the so-called spin - energy potential [251, 255]

τ σµν :=

∂Lfield

∂(∇σχ)γµνχ. (3.191)

Explicitly, the spin energy potential reads τµνσ = ψγ[µγνγσ]ψ. The equation of motionobtained from the variation of the action with respect to ψ reads [251, 255]

γµ∇µψ +3

8Tµνσγ

[µγνγσ]ψ = 0. (3.192)

It is interesting to observe that this generalized curved spacetime Dirac equation can berecast into the nonlinear equation of the Heisenberg-Pauli type

γµ∇µψ +3

8ε(ψγµγ5ψ

)γµγ5ψ = 0. (3.193)

Although the gravitational field equation is similar in form to the Einstein fieldequation, it differs from the original Einstein equations because the curvature tensor,containing spacetime torsion, is non-Riemannian. Assuming that the Euler-Lagrangeequations for the matter fields are satisfied, we obtain the following conservation lawsfor the angular - momentum and energy - momentum

V µiV

νjΣ[µν] = ∇ντ

νij ,

V kµ ∇νΣ

νκ = Σν

κTkµν + τν

ijRij

µν .

(3.194)

Chapter 4

Space-time deformations and

conformal transformations

towards extended theories of

gravity

4.1 Deformation and conformal transformations, how?

The issue to consider a general way to deform the space-time metrics is not new. Ithas been posed in different ways and is related to several physical problems rangingfrom the spontaneous symmetry breaking of unification theories up to gravitationalwaves, considered as space-time perturbations. In cosmology, for example, one faces theproblem to describe an observationally lumpy universe at small scales which becomesisotropic and homogeneous at very large scales according to the Cosmological Principle.In this context, it is crucial to find a way to connect background and locally perturbedmetrics [256]. For example, McVittie [257] considered a metric which behaves as aSchwarzschild one at short ranges and as a Friedman-Lemaitre-Robertson-Walker metricat very large scales. Gautreau [258] calculated the metric generated by a Schwarzschildmass embedded in a Friedman cosmological fluid trying to address the same problem. Onthe other hand, the post-newtonian parameterization, as a standard, can be consideredas a deformation of a background, asymptotically flat Minkowski metric.

In general, the deformation problem has been explicitly posed by Coll and collab-orators [259, 260, 261] who conjectured the possibility to obtain any metric from thedeformation of a space-time with constant curvature. The problem was solved onlyfor 3-dimensional spaces but a straightforward extension should be to achieve the sameresult for space-times of any dimension.

In principle, new exact solutions of the Einstein field equations can be obtained bystudying perturbations. In particular, dealing with perturbations as Lorentz matrices

99

1004. SPACE-TIME DEFORMATIONS AND CONFORMAL TRANSFORMATIONS TOWARDS EXTENDED

THEORIES OF GRAVITY

of scalar fields ΦAC reveals particularly useful. Firstly they transform as scalars with

respect the coordinate transformations. Secondly, they are dimensionless and, in eachpoint, the matrix ΦA

C behaves as the element of a group. As we shall see below, such anapproach can be related to the conformal transformations giving an ”extended” inter-pretation and a straightforward physical meaning of them (see [347, 210] and referencestherein for a comprehensive review). Furthermore scalar fields related to space-timedeformations have a straightforward physical interpretation which could contribute toexplain several fundamental issues as the Higgs mechanism in unification theories, theinflation in cosmology and other pictures where scalar fields play a fundamental role indynamics. In this chapter, we are going to discuss the properties of the deforming ma-trices ΦA

C and we will derive, from the Einstein equations, the field equations for them,showing how them can parameterize the deformed metrics, according to the boundaryand initial conditions and to the energy-momentum tensor. The layout is the following,we define the space-time perturbations in the framework of the metric formalism givingthe notion of first and second deformation matrices. When are devoted to the mainproperties of deformations. particular, we discuss how deformation matrices can besplit in their trace, traceless and skew parts. We derive the contributions of deformationto the geodesic equation and, starting from the curvature Riemann tensor, the generalequation of deformations. We discuss the notion of linear perturbations under the stan-dard of deformations. In particular, we recast the equation of gravitational waves andthe transverse traceless gauge under the standard of deformations. After we discuss theaction of deformations on the Killing vectors. The result consists in achieving a notionof approximate symmetry.

4.2 Generalities on space-time deformations

In order to start our considerations, let us take into account a metric g on a space-timemanifold M. Such a metric is assumed to be an exact solution of the Einstein fieldequations. We can decompose it by a co-tetrad field ωA(x)

g = ηABωAωB. (4.1)

Let us define now a new tetrad field ω = ΦAC(x)ωC , with ΦA

C(x) a matrix of scalar

fields. Finally we introduce a space-time M with the metric g defined in the followingway

g = ηABΦACΦB

D ωCωD = γCD(x)ωCωD, (4.2)

where also γCD(x) is a matrix of fields which are scalars with respect to the coordinatetransformations.

If ΦAC(x) is a Lorentz matrix in any point ofM, then

g ≡ g (4.3)

4.3. PROPERTIES OF DEFORMING MATRICES 101

otherwise we say that g is a deformation of g and M is a deformedM. If all the functionsof ΦA

C(x) are continuous, then there is a one - to - one correspondence between the

points of M and the points of M.In particular, if ξ is a Killing vector for g on M, the corresponding vector ξ on M

could not necessarily be a Killing vector.A particular subset of these deformation matrices is given by

ΦAC(x) = Ω(x) δA

C . (4.4)

which define conformal transformations of the metric,

g = Ω2(x)g . (4.5)

In this sense, the deformations defined by Eq. (4.2) can be regarded as a generaliza-tion of the conformal transformations.

We call the matrices ΦAC(x) first deformation matrices, while we can refer to

γCD(x) = ηABΦAC(x)ΦB

D(x). (4.6)

as the second deformation matrices, which, as seen above, are also matrices of scalarfields. They generalize the Minkowski matrix ηAB with constant elements in the defini-tion of the metric. A further restriction on the matrices ΦA

C comes from the theoremproved by Riemann by which an n-dimensional metric has n(n−1)/2 degrees of freedom(see [260] for details). With this definitions in mind, let us consider the main propertiesof deforming matrices.

4.3 Properties of deforming matrices

Let us take into account a four dimensional space-time with Lorentzian signature. Afamily of matrices ΦA

C(x) such that

ΦAC(x) GL(4)∀x, (4.7)

are defined on such a space-time.These functions are not necessarily continuous and can connect space-times with

different topologies. A singular scalar field introduces a deformed manifold M with aspace-time singularity.

As it is well known, the Lorentz matrices ΛAC leave the Minkowski metric invariant

and then

g = ηEF ΛEAΛF

BΦACΦB

D ωCωD = ηABΦA

CΦBD ω

CωD. (4.8)

It follows that ΦAC give rise to right cosets of the Lorentz group, i.e. they are the ele-

ments of the quotient groupGL(4,R)/SO(3, 1). On the other hand, a right-multiplicationof ΦA

C by a Lorentz matrix induces a different deformation matrix.


THEORIES OF GRAVITY

The inverse deformed metric is

gab = ηCDΦ−1ACΦ−1B

DeaAe

bB (4.9)

where Φ−1ACΦC

B = ΦACΦ−1C

B = δAB .

Let us decompose now the matrix ΦAB = ηAC ΦCB in its symmetric and antisym-

metric parts

ΦAB = Φ(AB) + Φ[AB] = Ω ηAB + ΘAB + ϕAB (4.10)

where Ω = ΦAA, ΘAB is the traceless symmetric part and ϕAB is the skew symmetric

part of the first deformation matrix respectively. Then standard conformal transforma-tions are nothing else but deformations with ΘAB = ϕAB = 0 [263].

Finding the inverse matrix Φ−1AC in terms of Ω, ΘAB and ϕAB is not immediate,

but as above, it can be split in the three terms

Φ−1AC = αδA

C + ΨAC + ΣA

C (4.11)

where α, ΨAC and ΣA

C are respectively the trace, the traceless symmetric part and theantisymmetric part of the inverse deformation matrix. The second deformation matrix,from the above decomposition, takes the form

γAB = ηCD(Ω δCA + ΘC

A + ϕCA)(Ω δD

B + ΘDB + ϕD

B) (4.12)

and then

γAB = Ω2 ηAB + 2Ω ΘAB + ηCD ΘCA ΘD

B + ηCD (ΘCA ϕ

DB

+ϕCA ΘD

B) + ηCD ϕC

A ϕD

B . (4.13)

In general, the deformed metric can be split as

gab = Ω2gab + γab (4.14)

whereγab =

(2Ω ΘAB + ηCD ΘC

A ΘDB + ηCD (ΘC

A ϕD

B + ϕCA ΘD

B)

+ηCD ϕC

A ϕD

B

)ωA

a ωBb (4.15)

In particular, if ΘAB = 0, the deformed metric simplifies to

gab = Ω2gab + ηCD ϕCA ϕ

DBω

Aaω

Bb (4.16)

and, if Ω = 1, the deformation of a metric consists in adding to the background metrica tensor γab. We have to remember that all these quantities are not independent as, bythe theorem mentioned in [260], they have to form at most six independent functions ina four dimensional space-time.

4.3. PROPERTIES OF DEFORMING MATRICES 103

Similarly the controvariant deformed metric can be always decomposed in the fol-lowing way

gab = α2gab + λab (4.17)

Let us find the relation between γab and λab. By using gabgbc = δca, we obtain

α2Ω2δca + α2γc

a + Ω2λca + γabλ

bc = δca (4.18)

if the deformations are conformal transformations, we have α = Ω−1, so assuming sucha condition, one obtain the following matrix equation

α2γca + Ω2λc

a + γabλbc = 0 , (4.19)

and(δb

a + Ω−2γba)λ

cb = −Ω−4γc

a (4.20)

and finallyλc

b = −Ω−4(δ + Ω−2γ)−1abγ

ca (4.21)

where (δ + Ω−2γ)−1 is the inverse tensor of (δba + Ω−2γb

a).To each matrix ΦA

B, we can associate a (1,1)-tensor φab defined by

φab = ΦA

BωBb e

aA (4.22)

such thatgab = gcdφ

caφ

db (4.23)

which can be decomposed as in Eq.(4.16). Vice-versa from a (1,1)-tensor φab, we can

define a matrix of scalar fields as

φAB = φa

bωAa e

bB . (4.24)

The Levi Civita connection corresponding to the metric (4.14) is related to theoriginal connection by the relation (see the NOTE for details)

Γcab = Γc

ab + Ccab (4.25)

(see [263]), where

Ccab = 2gcdgd(a∇b)Ω− gabg

cd∇dΩ +1

2gcd (∇aγdb +∇bγad −∇dγab) . (4.26)

Therefore, in a deformed space-time, the connection deformation acts like a force thatdeviates the test particles from the geodesic motion in the unperturbed space-time. Asa matter of fact the geodesic equation for the deformed space-time

d 2xc

dλ2+ Γc

ab

dxa

dλ

dxb

dλ= 0 (4.27)


THEORIES OF GRAVITY

becomes

d 2xc

dλ2+ Γc

ab

dxa

dλ

dxb

dλ= −Cc

ab

dxa

dλ

dxb

dλ. (4.28)

The deformed Riemann curvature tensor is then

R dabc = R d

abc +∇bCdac −∇aC

dbc + Ce

acCdbe −Ce

bcCdae, (4.29)

while the deformed Ricci tensor obtained by contraction is

Rab = Rab +∇dCdab −∇aC

ddb + Ce

abCdde − Ce

dbCdae (4.30)

and the curvature scalar

R = gabRab = gabRab + gab[∇dC

dab −∇aC

ddb + Ce

abCdde − Ce

dbCdae

](4.31)

From the above curvature quantities, we obtain finally the equations for the defor-mations. In the vacuum case, we simply have

Rab = Rab +∇dCdab −∇aC

ddb + Ce

abCdde − Ce

dbCdae = 0 (4.32)

where Rab must be regarded as a known function. In presence of matter, we considerthe equation

Rab +∇dCdab −∇aC

ddb + Ce

abCdde − Ce

dbCdae = Tab −

1

2gabT (4.33)

we are assuming, for the sake of simplicity 8πG = c = 1. This last equation can beimproved by considering the Einstein field equations

Rab = Tab −1

2gabT (4.34)

and then

∇dCdab −∇aC

ddb +Ce

abCdde − Ce

dbCdae = Tab −

1

2gabT −

(Tab −

1

2gabT

)(4.35)

is the most general equation for deformations.

4.4 Metric deformations as perturbations and gravitational

waves

Metric deformations can be used to describe perturbations. To this aim we can simplyconsider the deformations

4.4. METRIC DEFORMATIONS AS PERTURBATIONS AND GRAVITATION AL WAVES 105

ΦAB = δA

B + ϕAB (4.36)

with|ϕA

B | ≪ 1, (4.37)

together with their derivatives| ∂ϕA

B | ≪ 1 . (4.38)

With this approximation, immediately we find the inverse relation

(Φ−1)AB ≃ δAB − ϕA

B . (4.39)

As a remarkable example, we have that gravitational waves are generally described, inlinear approximation, as perturbations of the Minkowski metric

gab = ηab + γab. (4.40)

In our case, we can extend in a covariant way such an approximation. If ϕAB is anantisymmetric matrix, we have

gab = gab + γab (4.41)

where the first order terms in ϕAB vanish and γab is of second order

γab = ηABϕA

CϕB

DωC

aωD

b. (4.42)

Consequentlygab = gab + γab (4.43)

whereγab = ηAB(ϕ−1)CA(ϕ−1)DBe

aC e b

D . (4.44)

Let us consider the background metric gab, solution of the Einstein equations in thevacuum

Rab = 0. (4.45)

We obtain the equation of perturbations considering only the linear terms in Eq.(4.32)and neglecting the contributions of quadratic terms. We find

Rab = ∇dCdab −∇aC

ddb = 0 , (4.46)

and, by the explicit form of Cdab, this equation becomes

(∇d∇aγ

db +∇d∇bγ

da −∇d∇dγab

)−(∇a∇dγ

db +∇a∇bγ

dd −∇a∇dγd

b

)= 0 . (4.47)

Imposing the transverse traceless gauge on γab , i.e. the standard gauge conditions

∇aγab = 0 (4.48)


THEORIES OF GRAVITY

and

γ = γaa = 0 (4.49)

Eq.(4.47) reduces to

∇b∇bγac − 2R bac

dγbd = 0 , (4.50)

see also [263]. In our context, this equation is a linearized equation for deformations andit is straightforward to consider perturbations and, in particular, gravitational waves,as small deformations of the metric. This result can be immediately translated into theabove scalar field matrix equations. Note that such an equation can be applied to theconformal part of the deformation, when the general decomposition is considered.

As an example, let us take into account the deformation matrix equations appliedto the Minkowski metric, when the deformation matrix assumes the form (4.36). In thiscase, the equations (4.47), become ordinary wave equations for γab. Considering thedeformation matrices, these equations become, for a tetrad field of constant vectors,

∂d∂dϕC

AϕCB + 2 ∂dϕC

A∂dϕCB + ϕC

A∂d∂dϕCB = 0 . (4.51)

The above gauge conditions are now

ϕABϕBA = 0 (4.52)

and

edD[∂dϕCAϕ

CB + ϕCA∂dϕ

CB

]= 0 . (4.53)

This result shows that the gravitational waves can be fully recovered starting from thescalar fields which describe the deformations of the metric. In other words, such scalarfields can assume the meaning of gravitational wave modes.

4.5 Approximate Killing vectors

Another important issue which can be addressed starting from space-time deformationsis related to the symmetries. In particular, they assume a fundamental role in describingwhen a symmetry is preserved or broken under the action of a given field. In GeneralRelativity, the Killing vectors are always related to the presence of given space-timesymmetries [263].

Let us take an exact solution of the Einstein equations, which satisfies the Killingequation

(Lξg)ab = 0 (4.54)

where ξ, being the generator of an infinitesimal coordinate transformation, is a Killingvector. If we take a deformation of the metric with the scalar matrix

ΦAB = δA

B + ϕAB (4.55)

4.5. APPROXIMATE KILLING VECTORS 107

with|ϕA

B | ≪ 1 , (4.56)

and(Lξ g)ab 6= 0 , (4.57)

being

(LξeA)a = 0 , (4.58)

we have(Lξϕ)AB = ξa∂aϕ

AB 6= 0 . (4.59)

If there is some region D of the deformed space-time Mdeformed where

| (Lξϕ)AB| ≪ 1 (4.60)

we say that ξ is an approximate Killing vector on D. In other words, these approximateKilling vectors allow to ”control” the space-time symmetries under the action of a givendeformation. 1

1We can calculate the modified connection Γcab in many alternative ways. Let us introduce the tetrad

eA and cotetrad ωB satisfying the orthogonality relation

ieAω

B = δBA (4.61)

and the non-integrability condition (anholonomy)

dωA =

1

2ΩA

BCωB∧ ω

C. (4.62)

The corresponding connection is

ΓABC =

1

2

“

ΩABC − η

AA′

ηBB′ΩB′

A′C − ηAA′

ηCC′ΩC′

A′B

”

(4.63)

If we deform the metric as in (4.2), we have two alternative ways to write this expression: either writingthe “deformation” of the metric in the space of tetrads or “deforming” the tetrad field as in the followingexpression

g = ηABΦACΦB

D ωC

ωD = γAB ω

Aω

B = ηAB ωA

ωB

. (4.64)

In the first case, the contribution of the Christoffel symbols, constructed by the metric γAB, appears

ΓABC =

1

2

“

ΩABC − γ

AA′

γBB′ΩB′

A′C − γAA′

γCC′ΩC′

A′B

”

+1

2γ

AA′ `

ieCdγBA′ − ieB

dγCA′ − ieA′

dγBC

´

(4.65)

In the second case, using (4.62), we can define the new anholonomy objects CABC .

dωA =

1

2ΩA

BC ωB∧ ω

C. (4.66)

After some calculations, we have

ΩABC = ΦA

EΦ−1D

BΦ−1F

C ΩEDF + 2ΦA

F eaG

“

Φ−1G

[B∂aΦ−1F

C]

”

(4.67)


THEORIES OF GRAVITY

4.6 Deformations in f(R) -Theories

Resuming the equation 4.71, i.e

g = Ω2(x)g . (4.71)

we can recast the conformal factor with f ′(R)

As we are assuming a constant metric in tetradic space, the deformed connection is

ΓABC =

1

2

“

ΩABC − η

AA′

ηBB′ ΩB′

A′C − ηAA′

ηCC′ ΩC′

A′B

”

. (4.68)

Substituting (4.67) in (4.68), the final expression of ΓABC , as a function of ΩA

BC , ΦAB , Φ−1D

C and eaG

is

ΓABC = ∆DEFABC

»

1

2ηDG ΦG

G′Φ−1E′

EΦ−1F ′

F ΩG′

E′F ′ + ηDKΦKHe

aGΦ−1G

[E∂|a|Φ−1H

F ]

–

(4.69)

where∆DEF

ABC = δDA δ

EC δ

FB − δ

DB δ

ECδ

FA + δ

DC δ

EAδ

FB . (4.70)

Chapter 5

Probing the Minkowskian limit:

Gravitational waves in

f (R)-Theories

5.1 Why the gravitational waves in f(R)-Theories?

Recently, the data analysis of interferometric GWs detectors has been started (for thecurrent status of GWs interferometers see [312, 313, 314, 315, 316, 317, 318, 319]) and thescientific community aims at a first direct detection of GWs in next years. The designand the construction of a number of sensitive detectors for gravitational waves (GWs)is underway today. There are some laser interferometers like the VIRGO detector,built in Cascina, near Pisa, Italy, by a joint Italian-French collaboration, the GEO600 detector built in Hanover, Germany, by a joint Anglo-German collaboration, thetwo LIGO detectors built in the United States (one in Hanford, Washington and theother in Livingston, Louisiana) by a joint Caltech-Mit collaboration, and the TAMA300 detector, in Tokyo, Japan. Many bar detectors are currently in operation too, andseveral interferometers and bars are in a phase of planning and proposal stages (for thecurrent status of gravitational waves experiments see [402, 403]). The results of thesedetectors will have a fundamental impact on astrophysics and gravitation physics andwill be important for a better knowledge of the Universe and either to confirm or rulingout the physical consistency of General Relativity or any other theory of gravitation[320, 321, 322, 323, 324, 325].

Moreover as we write in the above Chapters the emergence of issues as dark matterand dark energy cloud be related to the need of revising the theory of gravitation atastrophysical and cosmological scales and/or at strong field regimes. However, also con-sidering the recent flurry of papers on the argument, a comprehensive effective theory ofgravity, acting consistently at any scale, is far, up to now, to be found out, and demandsan improvement of observational datasets and the search for experimentally testable

109

110 5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES I N F (R)-THEORIES

theories. A more pragmatic point of view could be to “reconstruct” the suitable theoryof gravity starting from data. The main issues of this “inverse scattering” approachis matching consistently observations at different scales and taking into account wideclasses of gravitational theories where “ad hoc” hypotheses are avoided. In principle, themost popular dark energy models can be achieved considering f(R) theories of gravity[320] and the same track can be followed, at completely different scales, to match galacticdynamics [316]. Here, f(R) is a generic analytic function of the Ricci curvature scalarR. As we show in the above Chapter 4 the deformations can be described as extendedconformal transformations and this fact gives a straightforward physical interpretationof conformal transformations because conformally related metrics can be seen as the”background” and the ”perturbed” metrics. Then space-time metric deformations canbe immediately recast in terms of perturbation theory allowing a completely covariantapproach to the problem of gravitational waves (GW). In this Chapter, we want to facethe problem of how the GW stochastic background and f(R) gravity can be relatedshowing, vice-versa, that a revealed stochastic GW signal could be a powerful probe fora given effective theory of gravity. Our goal is to show that the conformal treatment ofGWs can be used to parameterize in a natural way f(R) theories.

5.2 Stochastic background of gravitational waves ”tuned”

by f(R) gravity

GWs are the perturbations hµν of the metric gµν which transform as 3-tensors. Following[332], the GW-equations in the transverse-traceless gauge are

hji = 0 (5.1)

where ≡ (−g)−1/2∂µ(−g)1/2gµν∂ν is the usual d’Alembert operator and these equa-tions are derived from the Einstein field equations deduced from the Hilbert-Lagrangiandensity L = R. Clearly matter perturbations do not appear in (5.1) since scalar andvector perturbations do not couple with tensor perturbations in Einstein equations. TheLatin indexes run from 1 to 3 the Greek ones from 0 to 3. Our task is now to derive theanalog of Eqs. (5.1) assuming a generic theory of gravity given by the action

A =1

2k

∫d4x√−gf(R) (5.2)

where, for the sake of simplicity, we have discarded matter contributions. A conformalanalysis will help to this goal. In fact, assuming the conformal transformation

gµν = e2Φgµν with e2Φ = f ′(R) (5.3)

where the prime indicates the derivative with respect to the Ricci scalar R and Φ is the“conformal scalar field”, we obtain the conformally equivalent Hilbert-Einstein action

A =1

2k

∫ √−gd4x

[R+ L

(Φ,Φ;µ

)](5.4)

5.2. STOCHASTIC BACKGROUND OF GRAVITATIONAL WAVES ”TUNED” BY F (R) GRAVITY 111

where L(Φ,Φ;µ

)is the conformal scalar field contribution derived from

Rµν = Rµν + 2

(Φ;µΦ;ν − gµνΦ;δΦ

;δ −Φ;µν −1

2gµνΦ;δ

;δ

)(5.5)

andR = e−2Φ

(R− 6Φ− 6Φ;δΦ

;δ)

(5.6)

In any case, as we will see, the L(Φ,Φ;µ

)-term does not affect the GW-tensor equations

so it will not be considered any longer1.Starting from the action (5.4) and deriving the Einstein-like conformal equations,

the GW-equations arehj

i = 0 (5.7)

expressed in the conformal metric gµν . Since no scalar perturbation couples to the tensorpart of gravitational waves, we have

hji = gljδgil = e−2Φglje2Φδgil = hj

i (5.8)

which means that hji is a conformal invariant.

As a consequence, the plane-wave amplitude hji = h(t)eji exp(ikix

i), where eji is thepolarization tensor, are the same in both metrics. In any case, the d’Alembert operatortransforms as

= e−2Φ( + 2Φ;λ∂;λ

)(5.9)

and this means that the background is changing while the tensor wave amplitude not.In order to study the cosmological stochastic background, the operator (5.9) can be

specified for a Friedmann-Robertson-Walker metric and then Eq. (5.7) becomes

h+(3H + 2Φ

)h+ k2a−2h = 0 (5.10)

being =∂

∂t2+ 3H

∂

∂t, a(t) the scale factor and k the wave number.

It is worth stressing that Eq. (5.10) applies to any f(R) theory whose conformaltransformation can be defined as e2Φ = f ′(R). The solution, i.e. the GW amplitude,depends on the specific cosmological background (i.e. a(t)) and the specific theoryof gravity (i.e. Φ(t)). For example, if we assume power law behaviors for a(t) andΦ(t) = 1

2 ln f ′(R(t)), that is

Φ(t) = f ′(R) = f ′0 (t/t0)m , a(t) = a0 (t/t0)

n (5.11)

it is easy show that general relativity is recovered for m = 0 while

n =m2 +m− 2

m+ 2(5.12)

1Actually a scalar component in gravitational radiation is often considered [322, 324] but here we aretaking into account only the genuine tensor part of stochastic background.


is the relation between the parameters for a generic f(R) = f0Rs where s = 1− m

2 withs 6= 1 [325]. Eq. (5.10) can be recast in the form

h+ (3n+m) t−1h+ k2a0 (t0/t))2n h = 0 (5.13)

whose general solution is

h(t) =

(t

t0

)β

[C1Jα(x) + C2J−α(x)] (5.14)

Jα’s are Bessel functions and

α =1− 3n−m2(n − 1)

, β =1− 3n−m

2, x =

kt1−n

1− n (5.15)

while t0, C1, C2 are constants related to the specific values of n and m. In Fig. (5.1),some examples are given. The plots are labelled by the set of parameters m, n, swhich assign the time evolution of Φ(t) and a(t) with respect to a given power-lawtheory f(R) = f0R

s.The time units are in terms of the Hubble radius H−1; n = 1/2 is a radiation-like

evolution; n = 2/3 is a dust-like evolution, n = 2 labels power-law inflationary phasesand n = −5 is a pole-like inflation. From Eq. (5.12), a singular case is for m = −2 ands = 2. It is clear that the conformally invariant plane-wave amplitude evolution of thetensor GW strictly depends on the background.

Let us now take into account the issue of the production of GWs contributing tothe stochastic background. Several mechanism can be considered as cosmological pop-ulations of astrophysical sources [319], vacuum fluctuations, phase transitions [322] andso on. In principle, we could seek for contributions due to every high-energy physicalprocess in the early phases of the Universe evolution.

It is important to distinguish processes coming from transitions like inflation, wherethe Hubble flow emerges in the radiation dominated phase and process, like the earlystar formation rates, where the production takes place during the dust dominated era.In the first case, stochastic GW background is strictly related to the cosmological model.This is the case we are considering here which is, furthermore, also connected to thespecific theory of gravity. In particular, one can assume that the main contribution tothe stochastic background comes from the amplification of vacuum fluctuations at thetransition between an inflationary phase and the radiation dominated era. However, inany inflationary model, we can assume that the GWs generated as zero-point fluctuationduring the inflation undergo adiabatically damped oscillations (∼ 1/a) until they reachthe Hubble radius H−1. This is the particle horizon for the growth of perturbations.On the other hand, any other previous fluctuation is smoothed away by the inflationaryexpansion. The GWs freeze aut for a/k ≫ H−1 and reenter the H−1 radius afterthe reheating in the Friedmann era (see also [285, 286]). The reenter in the radiation-dominated or in the dust-dominated era depends on the scale of the GW. After the


100 200 300 400 500t

-0.3

-0.2

-0.1

0.1

0.2

0.3

hHtL n=12 , m=0 , s=1

100 200 300 400 500t

-0.1

-0.05

0.05

0.1

0.15

hHtL n=23 , m=0 , s=1

1000 2000 3000 4000 5000t

-2·1013

2·1013

4·1013

hHtL n=23 , m=-6 , s=4

0.05 0.1 0.15 0.2 0.25t

-10000

-5000

5000

10000

hHtL n=2 , m=2 , s=-12

1 2 3 4 5t

-20

-10

10

20

hHtL n=-2 , m=-1 , s=32

1.2 1.4 1.6 1.8 2 2.2 2.4t

-40

-20

20

40

hHtL n=-5 , m=-4 , s=3

Figure 5.1: Evolution of the GW amplitude for some power-law behaviors of a(t), Φ(t)and f(R). The scales of time and amplitude strictly depend on the cosmological back-ground giving a ”signature” for the model.


reenter, Gws can be detected by their Sachs-Wolfe effect on the temperature anisotropyT/T at the decoupling [328]. When Φ acts as the inflaton [329] we have Φ≪ H duringthe inflation. Considering also the conformal time dη = dt/a, Eq. (5.10) reads

h′′ + 2χ′

χh′ + k2h = 0 (5.16)

where χ = aeΦ and derivation is with respect to η. Inflation means that a(t) =a0 exp(Ht) and then η =

∫dt/a = (aH)−1 and χ′/χ = −η−1 . The exact solution

of (5.16) is

h(η) = k−3/2√

2/k [C1 (sin kη − cos kη) + C2 (sin kη + cos kη)] (5.17)

Inside the H−1 radius we have kη ≫ 1. Furthermore considering the absence of gravi-tons in the initial vacuum state, we have only negative-frequency modes and then theadiabatic behavior is

h = k1/2√

2/π1

aHC exp(−ikη) . (5.18)

At the first horizon crossing (aH = k), the averaged amplitude Ah = (k/2π)3/2 |h|of the perturbation is

Ah =1

2π2C (5.19)

when the scale a/k grows larger than the Hubble radius H−1, the growing mode of evo-lution is constant, that is it is frozen. This situation corresponds to the limit −kη ≪ 1in Eq. (5.17). Since Φ acts as the inflaton field, it is Φ ∼ 0 at reenter (after the endof inflation). Then the amplitude Ah of the wave is preserved until the second horizoncrossing after which it can be observed, in principle, as an anisotropy perturbation onthe CMBR. It can be shown that T/T . Ah as an upper limit to Ah since other effectscan contribute to the background anisotropy [330]. From this consideration, it is clearthat the only relevant quantity is the initial amplitude C in Eq. (5.18) which is conserveduntil the reenter. Such an amplitude directly depends on the fundamental mechanismgenerating perturbations. Inflation gives rise to processes capable of producing pertur-bations as zero-point energy fluctuations. Such a mechanism depends on the adoptedtheory of gravitation and then (T/T ) could constitute a further constraint to selecta suitable f(R)-theory. Considering a single graviton in the form of a monochromaticwave, its zero-point amplitude is derived through the commutation relations:

[h(t, x), πh(t, y)] = iδ3(x− y) (5.20)

calculated at a fixed time t, where the amplitude h is the field and πh is the conjugatemomentum operator. Writing the Lagrangian for h

L =1

2

√−ggµνh;µh;ν (5.21)


in the conformal FRW metric gµν (h is conformally invariant), we obtain

πh =∂L∂h

= e2Φa3h (5.22)

The Eq. (5.20) becomes

[h(t, x), h(y, y)

]= i

δ3(x− y)a3e2Φ

(5.23)

and the fields h and h can be expanded in terms of creation and annihilation operators

h(t, x) =1

(2π)3/2

∫d3k

[h(t)e−ikx + h∗(t)e+ikx

], (5.24)

h(t, x) =1

(2π)3/2

∫d3k

[h(t)e−ikx + h∗(t)e+ikx

]. (5.25)

The commutation relations is conformal time are then

[hh′∗ − h∗h′

]=i(2π)3

a3e2Φ(5.26)

Inserting (5.18) and (5.19), we obtain C =√

2π2He−Φ where H and Φ are calculatedat the first horizon-crossing and then

Ah =

√2

2He−Φ (5.27)

which means that the amplitude of GWs produced during inflation directly depends onthe given f(R) theory being Φ = 1

2 ln f ′(R). Explicitly, it is

Ah =H√

2f ′(R). (5.28)

This result deserves some discussion and can be read in two ways. From one side theamplitude of GWs produced during inflation depends on the given theory of gravity that,if different from general relativity, gives extra degrees of freedom which assume the roleof inflaton field in the cosmological dynamics [329]. On the other hand, the Sachs-Wolfeeffect related to the CMBR temperature anisotropy could constitute a powerful tool totest the true theory of gravity at early epochs, i.e. at very high redshift. This probe,related with data at medium [320] and low redshift [331], could strongly contribute i) toreconstruct cosmological dynamics at every scale; ii) to further test general relativity orto rule out it against alternative theories, iii) to give constraints on the GW-stochasticbackground, if f(R) theories are independently probed at other scales.

In summary, we have shown that amplitudes of tensor GWs are conformally invariantand their evolution depends on the cosmological background. Such a background is


tuned by conformal scalar field which is not present in the standard general relativity.Assuming that primordial vacuum fluctuations produce stochastic GWS, beside scalarperturbations, kinematical distortions and so on, the initial amplitude of these ones is afunction of the f(R)-theory of gravity and then the stochastic background can be, in acertain sense “tuned” by the theory. Vice versa, data coming from the Sachs-Wolfe effectcould contribute to select a suitable f(R) theory which can be consistently matched withother observations. However, further and accurate studies are needed in order to testthe relation between Sachs-Wolfe effect and f(R) gravity. This goal could be achievedvery soon through the forthcoming space (LISA) and ground-based (VIRGO, LIGO)interferometers.

5.3 Massive gravitational waves from f(R) theories of grav-

ity: Potential detection with LISA

Now, we will analyse the general case, i.e.

S =

∫d4x√−gf(R) + Lm, (5.29)

where f(R) is a generic high order theory of gravity.Of course, the cases which have been analysed in [327] and in [330] are particular

cases of the more general case that we are going to analyse now.As we will interact with gravitational waves, i.e. the linearized theory in vacuum,

Lm = 0 will be put and the pure curvature action

S =

∫d4x√−gf(R) (5.30)

will be considered.By varying the action (5.30) in respect to gµν (see refs. [327, 328, 330] for a parallel

computation) the field equations are obtained (note that in this thesis we work withG = 1, c = 1 and ~ = 1):

f ′(R)Rµν −1

2f(R)gµν − f ′(R);µ;ν + gµνf ′(R) = 0 (5.31)

which are the modified Einstein field equations. f ′(R) is the derivative of f in respectto the Ricci scalar. Writing down, exlplicitly, the Einstein tensor, eqs. (5.31) become

Gµν =1

f ′(R)12gµν [f(R)− f ′(R)R] + f ′(R);µ;ν − gµνf ′(R). (5.32)

Taking the trace of the field equations (5.32) one gets

3f ′(R) +Rf ′(R)− 2f(R) = 0, (5.33)

5.3. MASSIVE GRAVITATIONAL WAVES FROM F (R) THEORIES OF GRAVITY: POTENTIALDETECTION WITH LISA 117

and, with the identifications [334]

Φ→ f ′(R) and dVdΦ →

2f(R)−Rf ′(R)3

(5.34)

a Klein - Gordon equation for the effective Φ scalar field is obtained:

Φ =dV

dΦ. (5.35)

To study gravitational waves, the linearized theory has to be analyzed, with a littleperturbation of the background, which is assumed given by a a Minkowskian backgroundplus Φ = Φ0,i.e. we are linearizing into a background with constant curvature [330, 335].We also assume Φ0 to be a minimum for V :

V ≃ 1

2αδΦ2 ⇒ dV

dΦ≃ m2δΦ, (5.36)

and the constant m has mass dimension.Putting

gµν = ηµν + hµν

Φ = Φ0 + δΦ.(5.37)

to first order in hµν and δΦ, calling Rµνρσ , Rµν and R the linearized quantitywhich correspond to Rµνρσ , Rµν and R, the linearized field equations are obtained[327, 328, 330]:

Rµν − eR2 ηµν = (∂µ∂νhf − ηµνhf )

hf = m2hf ,

(5.38)

where

hf ≡δΦ

Φ0. (5.39)

Rµνρσ and eqs. (5.38) are invariants for gauge transformations [327, 328, 330]

hµν → h′µν = hµν − ∂(µεν)

δΦ→ δΦ′ = δΦ;

(5.40)

then

hµν ≡ hµν −h

2ηµν + ηµνhf (5.41)

can be defined, and, considering the transformation for the parameter εµ


εν = ∂µhµν , (5.42)

a gauge parallel to the Lorenz one of electromagnetic waves can be choosen:

∂µhµν = 0. (5.43)

In this way, field equations read like

hµν = 0 (5.44)

hf = m2hf (5.45)

Solutions of eqs. (5.44) and (5.45) are plan waves [327, 328, 330]:

hµν = Aµν(−→p ) exp(ipαxα) + c.c. (5.46)

hf = a(−→p ) exp(iqαxα) + c.c. (5.47)

where

kα ≡ (ω,−→p ) ω = p ≡ |−→p |

qα ≡ (ωm,−→p ) ωm =

√m2 + p2.

(5.48)

In eqs. (5.44) and (5.46) the equation and the solution for the standard wavesof General Relativity [332, 333] have been obtained, while eqs. (5.45) and (5.47) arerespectively the equation and the solution for the massive mode (see also [327, 328, 330]).

The fact that the dispersion law for the modes of the massive field hf is not linearhas to be emphatized. The velocity of every “ordinary” (i.e. which arises from GeneralRelativity) mode hµν is the light speed c, but the dispersion law (the second of eq.(5.48)) for the modes of hf is that of a massive field which can be discussed like a wave-packet [327, 328, 330]. Also, the group-velocity of a wave-packet of hf centered in −→pis

−→vG =−→pω, (5.49)

which is exactly the velocity of a massive particle with mass m and momentum −→p .

From the second of eqs. (??) and eq. (5.49) it is simple to obtain:

vG =

√ω2 −m2

ω. (5.50)

Then, wanting a constant speed of the wave-packet, it has to be [327, 328, 330]


m =√

(1− v2G)ω. (5.51)

Now, the analysis can remain in the Lorenz gauge with trasformations of the typeεν = 0; this gauge gives a condition of transversality for the ordinary part of the field:kµAµν = 0, but does not give the transversality for the total field hµν . From eq. (5.41)it is

hµν = hµν −h

2ηµν + ηµνhf . (5.52)

At this point, if being in the massless case [327, 328, 330], it could been put

εµ = 0

∂µεµ = − h

2 + hf ,

(5.53)

which gives the total transversality of the field. But in the massive case this isimpossible. In fact, applying the Dalembertian operator to the second of eqs. (5.53)and using the field equations (5.44) and (5.45) it results

εµ = m2hf , (5.54)

which is in contrast with the first of eqs. (5.53). In the same way, it is possible toshow that it does not exist any linear relation between the tensorial field hµν and themassive field hf . Thus a gauge in wich hµν is purely spatial cannot be chosen (i.e. itcannot be put hµ0 = 0, see eq. (5.52)) . But the traceless condition to the field hµν canbe put :

εµ = 0

∂µεµ = − h

2 .

(5.55)

These equations imply

∂µhµν = 0. (5.56)

To save the conditions ∂µhµν and h = 0 transformations like

εµ = 0

∂µεµ = 0

(5.57)

can be used and, taking −→p in the z direction, a gauge in which only A11, A22, andA12 = A21 are different to zero can be chosen. The condition h = 0 gives A11 = −A22.Now, putting these equations in eq. (5.52), it results


hµν(t, z) = A+(t− z)e(+)µν +A×(t− z)e(×)

µν + hf (t− vGz)ηµν . (5.58)

The term A+(t− z)e(+)µν +A×(t− z)e(×)

µν describes the two standard polarizations ofgravitational waves which arise from General Relativity, while the term hf (t− vGz)ηµν

is the massive field arising from the generic high order f(R) theory. In other words, thefunction f ′(R) of the Ricci scalar generates a third massive polarization for gravitationalwaves which is not present in standard General Relativity. Note that the line element(5.58) has been obtained in both of references [327] and [330] . Here we have shown thatsuch a line element is characteristic of every f(R) theory of gravity.

The analysis of the two standard polarization is well known in the literature [313,314, 332, 333]. For a the pure polarization arising from the f(R) theory eq. (5.101) canbe rewritten as

hµν(t− vGz) = hf (t− vGz)ηµν (5.59)

and the corrispondent line element is the conformally flat one

ds2 = [1 + hf (t− vGz)](−dt2 + dz2 + dx2 + dy2). (5.60)

In [330] it has been shown that in this kind of line element the effect of the massis the generation of a longitudinal force (in addition to the transverse one) while in thelimit m→ 0 the longitudinal force vanishes.

Now, before starting the analysis, it has to be discussed if there are fenomenogicallimitations to the mass of the GW [330, 335]. A strong limitation arises from the fact thatthe GW needs a frequency which falls in the frequency-range for both of earth basedand space based gravitational antennas, that is the interval 10−4Hz ≤ f ≤ 10KHz[312, 315, 316, 317, 318, 319, 337, 338]. For a massive GW, from [323, 325, 327, 330] itis:

2πf = ω =√m2 + p2, (5.61)

were p is the momentum. Thus, it needs

0eV ≤ m ≤ 10−11eV. (5.62)

A stronger limitation is given by requirements of cosmology and Solar System testson extended theories of gravity. In this case it is [335]

0eV ≤ m ≤ 10−33eV. (5.63)

For these light scalars, their effect can be still discussed as a coherent GW.

The frequency-dependent response function, for a massive mode of gravitationalradiation, has been obtained in [330] for the particular case f(R) = R+R−1. Here the


computation will be performed with another treatment and the results will be applliedto LISA, following the advice in [335].

Eq. (5.60) can be rewritten as

(dt

dτ)2 − (

dx

dτ)2 − (

dy

dτ)2 − (

dz

dτ)2 =

1

(1 + hf ), (5.64)

where τ is the proper time of the test masses.

From eqs. (5.60) and (5.64) the geodesic equations of motion for test masses (i.e.the beam-splitter and the mirrors of the interferometer), can be obtained

d2xdτ2 = 0

d2ydτ2 = 0

d2tdτ2 = 1

2∂t(1+hf )(1+hf )2

d2zdτ2 = −1

2∂z(1+hf )

(1+hf )2.

(5.65)

The first and the second of eqs. (5.65) can be immediately integrated obtaining

dx

dτ= C1 = const. (5.66)

dy

dτ= C2 = const. (5.67)

In this way eq. (5.64) becomes

(dt

dτ)2 − (

dz

dτ)2 =

1

(1 + hf ). (5.68)

If we assume that test masses are at rest initially we get C1 = C2 = 0. Thus we seethat, even if the GW arrives at test masses, we do not have motion of test masses withinthe x−y plane in this gauge. We could understand this directly from eq. (5.60) becausethe absence of the x and of the y dependences in the metric implies that test massesmomentum in these directions (i.e. C1 and C2 respectively) is conserved. This results,for example, from the fact that in this case the x and y coordinates do not esplicitlyenter in the Hamilton-Jacobi equation for a test mass in a gravitational field [313].

Now we will see that, in presence of the GW, we have motion of test masses in thez direction which is the direction of the propagating wave. An analysis of eqs. (5.65)shows that, to simplify equations, we can introduce the retarded and advanced timecoordinates (u, v):


u = t− vGz

v = t+ vGz.(5.69)

From the third and the fourth of eqs. (5.65) we have

d

dτ

du

dτ=∂v[1 + hf (u)]

(1 + hf (u))2= 0. (5.70)

This equation can be integrated obtaining

du

dτ= α, (5.71)

where α is an integration constant. From eqs. (5.68) and (5.71), we also get

dv

dτ=

β

1 + hf(5.72)

where β ≡ 1α , and

τ = βu+ γ, (5.73)

where the integration constant γ correspondes simply to the retarded time coordinatetranslation u. Thus, without loss of generality, we can put it equal to zero. Now let ussee what is the meaning of the other integration constant β. We can write the equationfor z from eqs. (5.71) and (5.72):

dz

dτ=

1

2β(

β2

1 + hf− 1). (5.74)

When it is hf = 0 (i.e. before the GW arrives at the test masses) eq. (5.74) becomes

dz

dτ=

1

2β(β2 − 1). (5.75)

But this is exactly the initial velocity of the test mass, thus we have to choose β = 1because we suppose that test masses are at rest initially. This also imply α = 1.

To find the motion of a test mass in the z direction we see that from eq. (5.73) wehave dτ = du, while from eq. (5.72) we have dv = dτ

1+hf. Because it is vGz = v−u

2 we

obtain

dz =1

2vG(

dτ

1 + hf− du), (5.76)

which can be integrated as


z = z0 + 12vG

∫( du1+hf

− du) =

= z0 − 12vG

∫ t−vGz−∞

hf (u)1+hf (u)du,

(5.77)

where z0 is the initial position of the test mass. Now the displacement of the testmass in the z direction can be written as

∆z = z − z0 = − 12vG

∫ t−vGz0−vG∆z−∞

hf (u)1+hf (u)du

≃ − 12vG

∫ t−vGz0

−∞hf (u)

1+hf (u)du.

(5.78)

We can also rewrite the results in function of the time coordinate t:

x(t) = x0

y(t) = y0

z(t) = z0 − 12vG

∫ t−vGz0

−∞hf (u)

1+hf (u)d(u)

τ(t) = t− vGz(t),

(5.79)

Calling l and L+ l the unperturbed positions of the beam-splitter and of the mirrorand using the third of eqs. (5.79) the varying position of the beam-splitter and of themirror are given by

zBS(t) = l − 12vG

∫ t−vGl−∞

hf (u)1+hf (u)d(u)

zM (t) = L+ l − 12vG

∫ t−vG(L+l)−∞

hf (u)1+hf (u)d(u)

(5.80)

But we are interested in variations in the proper distance (time) of test masses, thus,in correspondence of eqs. (5.80), using the fourth of eqs. (5.79) we get

τBS(t) = t− vGl − 12

∫ t−vGl−∞

hf (u)1+hf (u)d(u)

τM (t) = t− vGL− vGl − 12

∫ t−vG(L+l)−∞

hf (u)1+hf (u)d(u).

(5.81)

Then the total variation of the proper time is given by

τ(t) = τM (t)− τBS(t) = vGL−1

2

∫ t−vG(L+l)

t−vGl

hf (u)

1 + hf (u)d(u). (5.82)


In this way, recalling that in the used units the unperturbed proper distance (time)is T = L, the difference between the total variation of the proper time in presence andthe total variation of the proper time in absence of the GW is

δτ(t) ≡ τ(t)− L = −L(vG + 1)− 1

2

∫ t−vG(L+l)

t−vGl

hf (u)

1 + hf (u)d(u). (5.83)

This quantity can be computed in the frequency domain, defining the Fourier trans-form of hf as

hf (ω) =

∫ ∞

−∞dt hf (t) exp(iωt). (5.84)

and using the translation and derivation Fourier theorems, obtaining

δτ (ω) = L(1− v2G) exp[iωL(1 + vG)] + L

2ωL(v2G−1)2

[exp[2iωL](vG + 1)3(−2i+ ωL(vG − 1) + 2L exp[iωL(1 + vG)]

(6ivG + 2iv3G − ωL+ ωLv4

G) + L(vG + 1)3(−2i+ ωL(vG + 1))]hR.

(5.85)

A “signal” can be also defined:

S(ω) ≡ δeτ(ω)L = (1− v2

G) exp[iωL(1 + vG)] + 12ωL(v2

G−1)2

[exp[2iωL](vG + 1)3(−2i+ ωL(vG − 1) + 2 exp[iωL(1 + vG)]


G) + (vG + 1)3(−2i+ ωL(vG + 1))]hR.

(5.86)

Then the function

Υl(ω) ≡ (1− v2G) exp[iωL(1 + vG)] + 1

2ωL(v2G−1)2

[exp[2iωL](vG + 1)3(−2i+ ωL(vG − 1) + 2 exp[iωL(1 + vG)]


G) + (vG + 1)3(−2i+ ωL(vG + 1))],

(5.87)

is the response function of an arm of the interferometer located in the z-axis, dueto the longitudinal component of the massive gravitational wave arising from the highorder gravity theory and propagating in the same direction of the axis.

For vG → 1 it is Υl(ω) → 0. Such a response function has been obtained in [330]too, but with a different kind of analysis.

In figures 1 and 2 are shown the response functions (5.87) for an arm of LISA(L = 5∗106Km) [337, 338] for vG = 0.1 (non-relativistic case) and vG = 0.9 (relativistic


0.002 0.004 0.006 0.008 0.01f

1.025

1.05

1.075

1.1

1.125

Hz

ÈHHfLÈ

Figure 5.2: the longitudinal response function (5.87) of an arm of LISA for vG = 0.1(non-relativistic case)

0.002 0.004 0.006 0.008 0.01f

0.19

0.195

0.205

0.21

0.215

Hz

ÈHHfLÈ

Figure 5.3: the longitudinal response function (5.87) of an arm of LISA for vG = 0.9(relativistic case)


case). We see that in the non-relativistic case the signal is stronger as it could beexpected (for m→ 0 we expectΥl(ω)→ 0).

It is very important to emphasize that, differently from the response functions ofmassless gravitational waves, this longitudinal response function increases with fre-quency, .i.e , the presence of the mass prevents signal to drop off the regime in thehigh-frequency portion of the sensitivity band. Thus, considering such a high-frequencyportion of the sensitivity band becomes fundamental if LISA would detect massive GWsarising from f(R) theories of gravity which are not banned by requirements of Cosmologyand Solar System tests [335, 336].

5.4 Stochastic background of relic scalar gravitational waves

from scalar-tensor gravity

In this section consider the framework of scalar-tensor gravity and we discuss the scalarmodes of gravitational waves and the primordial production of this scalar componentwhich is generated beside tensorial one. Then the Scalar-tensor gravity theories are aparticular case of Extended Theories of Gravity which are revealing a useful paradigmto deal with several problems in cosmology, astrophysics and fundamental physics (fora comprehensive discussion see, for example [304, 305]). In the most general case,considering only the Ricci scalar among the curvature invariants, they arises from theaction

S =

∫d4x√−g

[F (R,R,2R,kR,φ)− ε

2gµνφ;µφ;ν + Lm

], (5.88)

where F is an unspecified function of curvature invariants and of a scalar field φ and is the D’Alembert operator. The term Lm is the minimally coupled ordinary mattercontribution. Scalar-tensor gravity, is recovered from (5.88) through the choice

F (R,φ) = f(φ)R− V (φ), ε = −1 . (5.89)

Considering (5.89), a general action for scalar-tensor gravity in four dimensions is

S =

∫d4x√−g

[f(φ)R+

1

2gµνφ;µφ;ν − V (φ) + Lm

], (5.90)

which can be recast in a Brans-Dicke-like form [306] by

ϕ = f(φ) , ω(ϕ) = f(φ)2′f(φ) , W (ϕ) = V (φ(ϕ))

, ,(5.91)

and then

S =

∫d4x√−g

[ϕR− ω(ϕ)

ϕgµνϕ;µϕ;ν −W (ϕ) + Lm

]. (5.92)

5.4. STOCHASTIC BACKGROUND OF RELIC SCALAR GRAVITATIONAL W AVES FROMSCALAR-TENSOR GRAVITY 127

By varying the action (5.92) with respect to gµν , we obtain the field equations

Gµν = −4πGϕ T

(m)µν + ω(ϕ)

ϕ2 (ϕ;µϕ;ν − 12gµνg

αβϕ;αϕ;β)+

+ 1ϕ(ϕ;µν − gµνϕ) + 1

2ϕgµνW (ϕ)

(5.93)

while the variation with respect to ϕ gives the Klein - Gordon equation

ϕ =1

2ω(ϕ) + 3

[−4πGT (m) + 2W (ϕ) + ϕW ′(ϕ) +

dω(ϕ)

dϕgµνϕ;µϕ;ν

]. (5.94)

We are assuming physical units G = 1, c = 1 and ~ = 1. T(m)µν is the matter stress-

energy tensor and G is a dimensional, strictly positive, gravitational coupling constant[294, 295]. The Newton constant is replaced by the effective coupling

Geff = − 1

2ϕ, (5.95)

which is, in general, different from G. General Relativity is recovered for

ϕ = ϕ0 = −1

2. (5.96)

In order to study gravitational waves, we assume first-order, small perturbations in

vacuum (T(m)µν = 0). This means

gµν = ηµν + hµν , ϕ = ϕ0 + δϕ (5.97)

and

W ≃ 1

2αδϕ2 ⇒W ′ ≃ αδϕ (5.98)

for the self-interacting, scalar-field potential. These assumptions allow to derive the”linearized” curvature invariants Rµνρσ , Rµν and R which correspond to Rµνρσ , Rµν

and R, and then the linearized field equations [295, 392]

Rµν − eR2 ηµν = −∂µ∂νΦ + ηµνΦ

Φ = m2Φ,

(5.99)

where

Φ ≡ −δϕϕ0

, m2 ≡ αϕ0

2ω + 3. (5.100)

The case ω = const and W = 0 has been analyzed in [295] considering the so-called“canonical” linearization [392]. In particular, the transverse-traceless (TT) gauge (see[392]) can be generalized to scalar-tensor gravity obtaining the total perturbation of a


GW incoming in the z+ direction in this gauge as we have see in the above section 5.3as

hµν(t− z) = A+(t− z)e(+)µν +A×(t− z)e(×)

µν + Φ(t− z)e(s)µν . (5.101)

The term A+(t − z)e(+)µν + A×(t − z)e

(×)µν describes the two standard (i.e. tensorial)

polarizations of a gravitational wave arising from General Relativity in the TT gauge

[392], while the term Φ(t − z)e(s)µν is the extension of the TT gauge to the scalar case.This means that, in scalar-tensor gravity, the scalar field generates a third componentfor the tensor polarization of GWs. This is because three different degrees of freedom arepresent (see Eq.(32) of [295]), while only two are present in standard General Relativity.

Let us now take into account the primordial physical process which gave rise to acharacteristic spectrum Ωsgw for the early stochastic background of relic scalar GWs.The production physical process has been analyzed, for example, in [393, 394, 395]but only for the first two tensorial components of eq. (5.101) due to standard GeneralRelativity. Actually the process can be improved considering also the third scalar-tensorcomponent.

Before starting with the analysis, it has to be emphasized that, considering a stochas-tic background of scalar GWs, it can be described in terms of the scalar field Φ andcharacterized by a dimensionless spectrum (see the analogous definition for tensorialwaves in [393, 396, 384, 394])

Ωsgw(f) =1

ρc

dρsgw

d ln f, (5.102)

where

ρc ≡3H2

0

8πG(5.103)

is the (actual) critical energy density of the Universe, H0 the today observed Hubbleexpansion rate, and dρsgw is the energy density of the scalar part of the gravitationalradiation contained in the frequency range f to f+df . We are considering now standardunits.

The existence of a relic stochastic background of scalar GWs is a consequence ofgeneral assumptions. Essentially it derives from basic principles of Quantum Field The-ory and General Relativity. The strong variations of gravitational field in the earlyUniverse amplifies the zero-point quantum fluctuations and produces relic GWs. It iswell known that the detection of relic GWs is the only way to learn about the evolu-tion of the very early universe, up to the bounds of the Planck epoch and the initialsingularity [393, 396, 384, 394, 301]. It is very important to stress the unavoidable andfundamental character of such a mechanism. It directly derives from the inflationaryscenario [397, 398], which well fit the WMAP data in particular good agreement withalmost exponential inflation and spectral index ≈ 1, [399, 400].

A remarkable fact about the inflationary scenario is that it contains a natural mecha-nism which gives rise to perturbations for any field. It is important for our aims that such


a mechanism provides also a distinctive spectrum for relic scalar GWs. These pertur-bations in inflationary cosmology arise from the most basic quantum mechanical effect:the uncertainty principle. In this way, the spectrum of relic GWs that we could detecttoday is nothing else but the adiabatically-amplified zero-point fluctuations [393, 394].The calculation for a simple inflationary model can be performed for the scalar fieldcomponent of eq. (5.101). Let us assume that the early Universe is described an infla-tionary de Sitter phase emerging in a radiation dominated phase [393, 396, 394]. Theconformal metric element is

ds2 = a2(η)[−dη2 + d−→x 2 + hµν(η,−→x )dxµdxν ], (5.104)

where, for a purely scalar GW the metric perturbation (5.101) reduces to

hµν = Φe(s)µν , (5.105)

Following [393, 394], in the de Sitter phase, we have:

η < η1 conformal time

P = −ρ equation of state

η21η

−10 (2η − η)−1 scale factor

Hds = cη0/η21 Hubble constant

while, in the radiation dominated phase we have, respectively,

η > η1 conformal time

P = ρ/3 equation of state

η/η0 scale factor

H = cη0/η2 Hubble constant

η1 is the inflation-radiation transition conformal time and η0 is the value of conformaltime today. If we express the scale factor in terms of comoving time cdt = a(t)dη, wehave

a(t) ∝ exp(Hdst), a(t) ∝√t (5.106)

for the de Sitter and radiation phases respectively. In order to solve the horizon and

flatness problems, the conditiona(η0)

a(η1)> 1027 has to be satisfied. The relic scalar-tensor

GWs are the weak perturbations hµν(η,−→x ) of the metric (5.105) which can be writtenin the form

hµν = e(s)µν (k)X(η) exp(−→k · −→x ), (5.107)


in terms of the conformal time η where−→k is a constant wavevector. From Eq.(5.107),

the scalar component is

Φ(η,−→k ,−→x ) = X(η) exp(

−→k · −→x ). (5.108)

Assuming Y (η) = a(η)X(η), from the Klein-Gordon equation in the FRW metric, onegets

Y ′′ + (|−→k |2 − a′′

a)Y = 0 (5.109)

where the prime ′ denotes derivative with respect to the conformal time. The solutionsof Eq. (5.109) can be expressed in terms of Hankel functions in both the inflationaryand radiation dominated eras, that is:For η < η1

X(η) =a(η1)

a(η)[1 +Hdsω

−1] exp−ik(η − η1), (5.110)

for η > η1

X(η) =a(η1)

a(η)[α exp−ik(η − η1) + β exp ik(η − η1), (5.111)

where ω = ck/a is the angular frequency of the wave (which is function of the time

being k = |−→k | constant), α and β are time-independent constants which we can obtaindemanding that both X and dX/dη are continuous at the boundary η = η1 between theinflationary and the radiation dominated eras. By this constraint, we obtain

α = 1 + i

√HdsH0

ω− HdsH0

2ω2, β =

HdsH0

2ω2(5.112)

In Eqs. (5.112), ω = ck/a(η0) is the angular frequency as observed today, H0 = c/η0 isthe Hubble expansion rate as observed today. Such calculations are referred in literatureas the Bogoliubov coefficient methods [393, 394].

In an inflationary scenario, every classical or macroscopic perturbation is dampedout by the inflation, i.e. the minimum allowed level of fluctuations is that requiredby the uncertainty principle. The solution (5.110) corresponds to a de Sitter vacuumstate. If the period of inflation is long enough, the today observable properties ofthe Universe should be indistinguishable from the properties of a Universe started inthe de Sitter vacuum state. In the radiation dominated phase, the eigenmodes whichdescribe particles are the coefficients of α, and these which describe antiparticles arethe coefficients of β (see also [401]). Thus, the number of particles created at angularfrequency ω in the radiation dominated phase is given by

Nω = |βω |2 =

(HdsH0

2ω2

)2

. (5.113)


Now it is possible to write an expression for the energy density of the stochastic scalarrelic gravitons background in the frequency interval (ω, ω + dω) as

dρsgw = 2~ω

(ω2dω

2π2c3

)Nω =

~H2dsH

20

4π2c3dω

ω=

~H2dsH

20

4π2c3df

f, (5.114)

where f , as above, is the frequency in standard comoving time. Eq. (5.114) can berewritten in terms of the today and de Sitter value of energy density being

H0 =8πGρc

3c2, Hds =

8πGρds

3c2. (5.115)

Introducing the Planck density ρP lanck =c7

~G2the spectrum is given by

Ωsgw(f) =1

ρc

dρsgw

d ln f=

f

ρc

dρsgw

df=

16

9

ρds

ρP lanck. (5.116)

At this point, some comments are in order. First of all, such a calculation works for asimplified model that does not include the matter dominated era. If also such an erais also included, the redshift at equivalence epoch has to be considered. Taking intoaccount also results in [395], we get

Ωsgw(f) =16

9

ρds

ρP lanck(1 + zeq)

−1, (5.117)

for the waves which, at the epoch in which the Universe becomes matter dominated,have a frequency higher than Heq, the Hubble parameter at equivalence. This situationcorresponds to frequencies f > (1 + zeq)

1/2H0. The redshift correction in Eq.(5.117) isneeded since the today observed Hubble parameter H0 would result different without amatter dominated contribution. At lower frequencies, the spectrum is given by [393, 394]

Ωsgw(f) ∝ f−2. (5.118)

As a further consideration, let us note that the results (5.116) and (5.117), which are notfrequency dependent, does not work correctly in all the range of physical frequencies.For waves with frequencies less than today observed H0, the notion of energy density hasno sense, since the wavelength becomes longer than the Hubble scale of the Universe.In analogous way, at high frequencies, there is a maximal frequency above which thespectrum rapidly drops to zero. In the above calculation, the simple assumption that thephase transition from the inflationary to the radiation dominated epoch is instantaneoushas been made. In the physical Universe, this process occurs over some time scale ∆τ ,being

fmax =a(t1)

a(t0)

1

∆τ, (5.119)


which is the redshifted rate of the transition. In any case, Ωsgw drops rapidly. The twocutoffs at low and high frequencies for the spectrum guarantee that the total energydensity of the relic scalar gravitons is finite. For GUT energy-scale inflation it is of theorder [393]

ρds

ρP lanck≈ 10−12. (5.120)

These results can be quantitatively constrained considering the recent WMAP release.In fact, it is well known that WMAP observations put strongly severe restrictions on thespectrum. In Fig. 5.4 the spectrum Ωsgw is mapped : considering the ratio ρds/ρP lanck,the relic scalar GW spectrum seems consistent with the WMAP constraints on scalarperturbations. Nevertheless, since the spectrum falls off ∝ f−2 at low frequencies, thismeans that today, at LIGO-VIRGO and LISA frequencies (indicated in fig. 5.4), onegets

Ωsgw(f)h2100 < 2.3 × 10−12. (5.121)

It is interesting to calculate the corresponding strain at ≈ 100Hz, where interferometerslike VIRGO and LIGO reach a maximum in sensitivity. The well known equation forthe characteristic amplitude [393, 394, 301] adapted to the scalar component of GWscan be used:

Φc(f) ≃ 1.26 × 10−18(1Hz

f)√h2

100Ωsgw(f), (5.122)

and then we obtainΦc(100Hz) < 2× 10−26. (5.123)

Then, since we expect a sensitivity of the order of 10−22 for the above interferometersat ≈ 100Hz, we need to gain four order of magnitude. Let us analyze the situation alsoat smaller frequencies. The sensitivity of the VIRGO interferometer is of the order of10−21 at ≈ 10Hz and in that case it is

Φc(100Hz) < 2× 10−25. (5.124)

The sensitivity of the LISA interferometer will be of the order of 10−22 at ≈ 10−3Hzand in that case it is

Φc(100Hz) < 2× 10−21. (5.125)

This means that a stochastic background of relic scalar GWs could be, in principle,detected by the LISA interferometer.


-15 -10 -5 5 10Hz H Graphics L

-18

-16

-14

-12

-10

-6

Energy

LISA LIGO

Figure 5.4: The spectrum of relic scalar GWs in inflationary models is flat over a widerange of frequencies. The horizontal axis is log10 of frequency, in Hz. The verticalaxis is log10 Ωgsw. The inflationary spectrum rises quickly at low frequencies (wavewhich re-entered in the Hubble sphere after the Universe became matter dominated) andfalls off above the (appropriately redshifted) frequency scale fmax associated with thefastest characteristic time of the phase transition at the end of inflation. The amplitudeof the flat region depends only on the energy density during the inflationary stage;we have chosen the largest amplitude consistent with the WMAP constrains on scalarperturbations. This means that at LIGO and LISA frequencies, Ωsgw < 2.3 ∗ 10−12

Chapter 6

Further probe: Parametrized

Post Newtonian limit

6.1 f(R) gravity constrained by PPN parameters and stochas-

tic background of gravitational waves

The idea that Einstein gravity should be extended or corrected at large scales (infraredlimit) or at high energies (ultraviolet limit) is suggested by several theoretical and ob-servational issues. Quantum field theory in curved spacetimes, as well as the low-energylimit of String/M theory, both imply semi-classical effective actions containing higher-order curvature invariants or scalar-tensor terms. In addition, GR has been definitelytested only at Solar System scales while it may show several shortcomings if checked athigher energies or larger scales. Besides, the Solar System experiments are, up to now,not so conclusive to state that the only viable theory of gravity is GR: for example,the limits on PPN parameters should be greatly improved to fully remove degeneracies[342].

Of course, modifying the gravitational action asks for several fundamental chal-lenges. These models can exhibit instabilities [343] or ghost - like behavior [344], while,on the other hand, they have to be matched with observations and experiments in theappropriate low energy limit.

Despite of all these issues, in the last years, some interesting results have beenachieved in the framework of the so called f(R)-gravity at cosmological, Galactic andSolar System scales. Here f(R) is a general (analytic) function of the Ricci scalar R (seeRefs. [345, 346, 347] for review).

For example, there exist cosmological solutions that give the accelerated expansionof the universe at late times [348, 349, 350, 351]. In addition, it has been discoveredthat some stability conditions can lead to avoid ghost and tachyon solutions. Further-more there exist viable f(R) models which satisfy both background cosmological con-straints and stability conditions [353, 355, 352, 356, 357, 358, 359, 360] and results have

135

136 6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT

been achieved in order to place constraints on f(R) cosmological models by CMBRanisotropies and galaxy power spectrum [361, 362, 363]. Moreover, some of such vi-able models lead to the unification of early-time inflation with late-time acceleration[358, 359, 360].

On the other hand, by considering f(R)-gravity in the low energy limit, it is possibleto obtain corrected gravitational potentials capable of explaining the flat rotation curvesof spiral galaxies or the dynamics of galaxy clusters without considering huge amountsof dark matter [364, 365, 366, 367, 368, 369].

Furthermore, several authors have dealt with the weak field limit of fourth ordergravity, in particular considering the PPN limit [371, 372, 373, 374, 375, 376, 377, 383]and the spherically symmetric solutions [378, 379, 380, 382].

This great deal of work needs an essential issue to be pursued: we need to compareexperiments and probes at local scales (e.g. Solar System) with experiments and probesat large scales (Galaxy, extragalactic scales, cosmology) in order to achieve self-consistentf(R) models. Some work has been done in this direction (see e.g. [355]) but thelarge part of efforts has been devoted to address single data sets (observations at agiven redshift) by a single model which, several time, is not working at other scalesthan the one considered. In particular, a given f(R) model, evading Solar Systemtests, should be not simply extrapolated at extragalactic and cosmological scales onlyrequiring accelerated cosmological solutions but it should be confronted with data andprobes coming from cosmological observations. Reliable models are then those matchingdata at very different scales (and redshifts).

In order to constrain further viable f(R)-models, one could take into account alsothe stochastic background of gravitational waves (GW) which, together with cosmicmicrowave background radiation (CMBR), would carry a huge amount of information onthe early stages of the Universe evolution. In fact, if detected, such a background couldconstitute a further probe for these theories at very high red-shift [401]. On the otherhand, a key role for the production and the detection of the relic gravitational radiationbackground is played by the adopted theory of gravity [384, 385]. This means that theeffective theory of gravity should be probed at zero, intermediate and high redshifts tobe consistent at all scales and not simply extrapolated up to the last scattering surface,as in the case of GR.

The aim of this chapter is to discuss the PPN Solar-System constraints and theGW stochastic background considering some recently proposed f(R) gravity models[352, 353, 355, 358, 359, 360] which satisfy both cosmological and stability conditionsmentioned above. Using the definition of PPN-parameters γ and β in terms of f(R)-models [377] and the definition of scalar GWs [386], we compare and discuss if it ispossible to search for parameter ranges of f(R)-models working at Solar System andGW stochastic background scale. This phenomenological approach is complementaryto the one proposed, e.g. in [355, 363] where also galactic and cosmological scales havebeen considered to constraint the models.

6.2. F (R) GRAVITY 137

6.2 f(R) gravity

Let us start from the following action

S = Sg + Sm =1

k2

∫d4x√−g [R+ f(R) + Lm] , (6.1)

where we have considered the gravitational and matter contributions and k2 ≡ 16πG.The non-linear f(R) term has been put in evidence with respect to the standard Hilbert-Einstein term R and Lm is the perfect-fluid matter Lagrangian. The field equations are

1

2gµνF (R)−RµνF

′(R)− gµνF ′(R) +∇µ∇νF′(R) = −k

2

2T (m)

µν . (6.2)

Here F (R) = R+f(R) and T(m)µν is the matter energy - momentum tensor. By introducing

the auxiliary field A, one can rewrite the gravitational part in the Action (6.1) as

Sg =1

k2

∫d4x√−g

(1 + f ′(A)

)(R−A) +A+ f(A)

. (6.3)

As it is clear from Eq.(6.3), if F ′(R) = 1 + f ′(R) < 0, the coupling k2eff = k2/F ′(A)

becomes negative and the theory enters the anti-gravity regime. Note that it is not thecase for the standard GR.

Action (6.3) can be recast in a scalar-tensor form. By using the conformal scaletransformation gµν → eσgµν with σ = − ln (1 + f ′(A)), the action can be written in theEinstein frame as follows [345]:

SE =1

k2

∫d4x√−g

(R− 3

2gρσ∂ρσ∂σσ − V (σ)

), (6.4)

where

V (σ) = eσg(e−σ

)− e2σf

(g(e−σ

))=

A

F ′(A)− F (A)

F ′(A)2. (6.5)

The form of g (e−σ) is given by solving σ = − ln (1 + f ′(A)) = lnF ′(A) as A =g (e−σ). The transformation gµν → eσgµν induces a coupling of the scalar field σ withmatter.

In general, an effective mass for σ is defined as [360]

m2σ ≡

1

2

d2V (σ)

dσ2=

1

2

[A

F ′(A)− 4F (A)

(F ′(A))2+

1

F ′′(A)

], (6.6)

which, in the weak field limit, could induce corrections to the Newton law. This allows, asit is well known, to deal with the extra degrees of freedom of f(R)-gravity as an effectivescalar field which reveals particularly useful in considering ”chameleon” models [354].This ”parameterization” will be particularly useful to deal with the scalar componentof GWs.


6.3 f(R) viable models

Let us consider now a class of f(R) models which do not contain cosmological constantand are explicitly designed to satisfy cosmological and Solar-System constraints in givenlimits of the parameter space. In practice, we choose a class of functional forms off(R) capable of matching, in principle, observational data (see [350] for the generalapproach). Firstly, the cosmological model should reproduce the CMBR constraintsin the high-redshift regime (which agree with the presence of an effective cosmologicalconstant). Secondly, it should give rise to an accelerated expansion, at low redshift,according to the ΛCDM model. Thirdly, there should be sufficient degrees of freedomin the parameterization to encompass low redshift phenomena (e.g. the large scalestructure) according to the observations [363]. Finally, small deviations from GR shouldbe consistent with Solar System tests. All these requirements suggest that we can assumethe limits

limR→∞

f(R) = constant, (6.7)

limR→0

f(R) = 0, (6.8)

which are satisfied by a general class of broken power law models, proposed in [355],which are

fI(R) = −m2 c1(

Rm2

)n

c2(

Rm2

)n+ 1

(6.9)

or otherwise written as

FI(R) = R− λRc

(RRc

)2n

(RRc

)2n+ 1

(6.10)

where m is a mass scale and c1,2 are dimensionless parameters.Besides, another viable class of models was proposed in [352]

FII(R) = R+ λRc

[(1 +

R2

R2c

)−p

− 1

]. (6.11)

Since F (R = 0) = 0, the cosmological constant has to disappear in a flat spacetime.The parameters n, p, λ, Rc are constants which should be determined by experimentalbounds.

Other interesting models with similar features have been studied in [360, 358, 356,357, 359]. In all these models, a de-Sitter stability point, responsible for the late-timeacceleration, exists for R = R1 (> 0), where R1 is derived by solving the equationR1f,R(R1) = 2f(R1) [381]. For example, in the model (6.11), we have R1/Rc = 3.38for λ = 2 and p = 1. If λ is of the unit order, R1 is of the same order of Rc. The

6.3. F (R) VIABLE MODELS 139

stability conditions, f,R > 0 and f,RR > 0, are fulfilled for R > R1 [352, 357]. Moreoverthe models satisfy the conditions for the cosmological viability that gives rise to thesequence of radiation, matter and accelerated epochs [357].

In the region R≫ Rc both classes of models (6.9) and (6.11) behave as

FIII(R) ≃ R− λRc

[1− (Rc/R)2s

], (6.12)

where s is a positive constant. The model approaches ΛCDM in the limit R/Rc →∞.

Finally, let also consider the class of models [353, 370, 362]

FIV (R) = R− λRc

(R

Rc

)q

. (6.13)

Also in this case λ, q and Rc are positive constants (note that n, p, s and q have toconverge toward the same values to match the observations). We do not consider themodels whit negative q, because they suffer for instability problems associated withnegative F,RR [387, 361]. In Fig.(6.1), we have plotted some of the selected models as

function ofR

Rcfor suitable values of p, n, q, s, λ .

Let us now estimate mσ for the models discussed above. For Model I [355], whenthe curvature is large, we find

fI(R) ∼ −m2c1c2

+m2+2nc1c22R

n+ · · · , (6.14)

and obtain the following expression:

m2σ ∼

m2c222n(n+ 1)c1

(R

m2

)n+2

. (6.15)

Here the order of the mass-dimensional parameter m2 should be m2 ∼ 10−64 eV2. Thenin Solar System, where R ∼ 10−61 eV2, the mass is given by m2

σ ∼ 10−58+3n eV2 whileon the Earth atmosphere, where R ∼ 10−50 eV2, it has to be m2

σ ∼ 10−36+14n eV2. The

order of the radius of the Earth is 107 m ∼(10−14 eV

)−1. Therefore the scalar field σ

is enough heavy if n ≫ 1 and the correction to the Newton law is not observed, beingextremely small. In fact, if we choose n = 10, the order of the Compton length ofthe scalar field σ becomes that of the Earth radius. On the other hand, in the Earthatmosphere, if we choose n = 10, for example, we find that the mass is extremely large:

mσ ∼ 1043 GeV ∼ 1029 ×MPlanck . (6.16)

Here MPlanck is the Planck mass. Hence, the Newton law correction should be extremelysmall.


In Model II

fII(R) = −λR0

[1−

(1 +

R2

R20

)−p], (6.17)

if R is large compared with R0, whose order of magnitude is that of the curvature inthe present universe, we find

fII(R) = −λR0 + λR2p+1

0

R2p+ · · · . (6.18)

By comparing Eq.(6.18) with Eq. (6.14), if the curvature is large enough when com-pared with R0 or m2, as in the Solar System or on the Earth, we can set the followingidentifications:

λR0 ↔m2c1c2

, λR2p+10 ↔ m2+2nc1

c22, 2p↔ n . (6.19)

We have 41m2 ∼ R0. Then, if p is large enough, there is no correction to the Newtonlaw as in Model I given by Eq.(6.10).

Let us now discuss the instability of fluid matter proposed in [387], which may appearif the matter-energy density (or the scalar curvature) is large enough when comparedwith the average density the Universe, as it is inside the Earth. Considering the traceof the above field equations and with a little algebra, one obtains

R+F (3)(R)

F (2)(R)∇ρR∇ρR+

F ′(R)R

3F (2)(R)− 2F (R)

3F (2)(R)=

κ2

6F (2)(R)T . (6.20)

Here T is the trace of the matter energy-momentum tensor: T ≡ T (m)ρρ . We also denote

the derivative dnF (R)/dRn by F (n)(R). Let us now consider the perturbation of theEinstein gravity solutions. We denote the scalar curvature, given by the matter densityin the Einstein gravity, by Rb ∼ (κ2/2)ρ > 0 and separate the scalar curvature R intothe sum of Rb (background) and the perturbed part Rp as R = Rb + Rp (|Rp| ≪ |Rb|).Then Eq.(6.20) leads to the perturbed equation:

0 = Rb +F (3)(Rb)

F (2)(Rb)∇ρRb∇ρRb +

F ′(Rb)Rb

3F (2)(Rb)

− 2F (Rb)

3F (2)(Rb)− Rb

3F (2)(Rb)+ Rp + 2

F (3)(Rb)

F (2)(Rb)∇ρRb∇ρRp + U(Rb)Rp .(6.21)

Here the potential U(Rb) is given by

U(Rb) ≡(F (4)(Rb)

F (2)(Rb)− F (3)(Rb)

2

F (2)(Rb)2

)∇ρRb∇ρRb +

Rb

3

−F(1)(Rb)F

(3)(Rb)Rb

3F (2)(Rb)2− F (1)(Rb)

3F (2)(Rb)+

2F (Rb)F(3)(Rb)

3F (2)(Rb)2− F (3)(Rb)Rb

3F (2)(Rb)2(6.22)


It is convenient to consider the case where Rb and Rp are uniform and do not depend onthe spatial coordinates. Hence, the d’Alembert operator can be replaced by the secondderivative with respect to the time, that is: Rp → −∂2

tRp. Eq.(6.22) assumes thefollowing structure:

0 = −∂2tRp + U(Rb)Rp + const . (6.23)

If U(Rb) > 0, Rp becomes exponentially large with time, i.e. Rp ∼ e√

U(Rb)t, and thesystem becomes unstable.

In the 1/R-model, considering the background values, we find

U(Rb) = −Rb +R3

b

6µ4∼ R3

0

µ4∼(10−26sec

)−2(

ρm

g cm−3

)3

,

Rb ∼(103sec

)−2(

ρm

g cm−3

). (6.24)

Here the mass parameter µ is of the order

µ−1 ∼ 1018sec ∼(10−33eV

)−1. (6.25)

Eq.(6.24) tells us that the model is unstable and it would decay in 10−26 sec (consideringthe Earth size). In Model I, however, U(Rb) is negative:

U(R0) ∼ −(n+ 2)m2c22c1n(n+ 1)

< 0 . (6.26)

Therefore, there is no matter instability.For Model (6.17), as it is clear from the identifications (6.19), there is no matter

instability too.In order to study the stability of the de Sitter solution, let us proceed as follows.

From the field equations (6.2), we obtain the trace

f ′(R) =1

3

[R− f ′(R)R+ 2f(R) + κ2T

]. (6.27)

Here, as above, F (R) is F (R) = R+ f(R) and T ≡ gµνT(m)µν .

Now we consider the (in)stability around the de Sitter solution, where R = R0, andtherefore f(R0) and f ′(R0), are constants. Then since the l.h.s. in Eq.(6.27) vanishesfor R = R0, we find

R0 − f ′(R0)R0 + 2f(R0) + κ2T0 = 0 . (6.28)

Let us expand both sides of (6.28) around R = R0 as

R = R0 + δR . (6.29)

One obtains

f ′′(R0)δR =1

3

(1− f ′′(R0)R0 + f ′(R0)

)δR . (6.30)


Since

δR = −d2δR

dt2− 3H0

dδR

dt, (6.31)

in the de Sitter background, if

C(R0) ≡ limR→R0

1− f ′′(R)R+ f ′(R)

f ′′(R)> 0 , (6.32)

the de Sitter background is stable but, if C(R0) < 0, the de Sitter background is unstable.The expression for C(R0) could be valid even if f ′′(R0) = 0. More precisely, the solutionof (6.30) is given by

δR = A+eλ+t +A−eλ−t . (6.33)

Here A± are constants and

λ± =−3H0 ±

√9H2

0 − C(R0)

2. (6.34)

Then, if C(R0) < 0, λ+ is always positive and the perturbation grows up. This leads tothe instability. We have also to note that, when C(R0) is positive, if C(R0) > 9H2

0 , δRoscillates and the amplitude becomes exponentially small being:

δR = (A cos ω0t+B sinω0t) e−3H0t/2 , ω ≡√C(R0)− 9H2

0

2. (6.35)

Here A and B are constant. On the other hand, if C(R0) < 9H20 , there is no oscillation

in δR.Let us now consider the case where the matter contribution T can be neglected in the

de Sitter background and assume f ′(R) = 0 in the same background. We can assumethat there are two de Sitter background solutions satisfying f ′(R) = 0, for R = R1

and R = R2 as it could be the physical case if one asks for an inflationary and a darkenergy epoch. We also assume f ′(R) 6= 0 if R1 < R < R2 or R2 < R < R1. In thecase C(R1) < 0 and C(R2) > 0, the de Sitter solution, corresponding to R = R1, isunstable but the solution corresponding to R = R2 is stable. Then there should bea solution where the (nearly) de Sitter solution corresponding to R1 transits to the(nearly) de Sitter solution R2. Since the solution corresponding to R2 is stable, theuniverse remains in the de Sitter solution corresponding to R2 and there is no moretransition to any other de Sitter solution.

As an example, we consider Model I. For large curvature values, we find

fI(R) = −Λ +α

R2n+1. (6.36)

Here Λ and α are positive constants and n is a positive integer. Then we find

C(R) ∼ 1

f ′′(R)∼ R2n+2

2n(2n + 1)α> 0 . (6.37)


This means that the de Sitter solution in Model I can be stable. We have also tonote that C(R0) ∼ H4n+4

0 /m4n+2. Here m2 is the mass scale introduced in [355] andm2 ≪ H2

0 : this means that C(R0)≫ 9H20 and therefore there could be no oscillation.

We may also consider the model proposed in [358](here Model V):

fV (R) =αR2n − βRn

1 + γRn. (6.38)

Here α, β, and γ are positive constants and n is a positive integer. In Fig.6.2, weshow the behavior of Model V and of its first derivative. When the curvature is large(R→∞), f(R) behaves as a power law. Since the derivative of f(R) is given by

f ′V (R) =nRn−1

(αγR2n − 2αRn − β

)

(1 + γRn)2, (6.39)

we find that the curvature R0 in the present universe, which satisfies the conditionf ′(R0) = 0, is given by

R0 =

[1

γ

(1 +

√1 +

βγ

α

)]1/n

, (6.40)

and

f(R0) ∼ −2R0 =α

γ2

(1 +

(1− βγ/α)√

1 + βγ/α

2 +√

1 + βγ/α

). (6.41)

As shown in [358], the magnitudes of the parameters is given by

α ∼ 2R0R−2n0 , β ∼ 4R2

0R−2n0 Rn−1

I , γ ∼ 2R0R−2n0 Rn−1

I . (6.42)

Here RI is the curvature in the inflationary epoch and we have assumed f(RI) ∼(α/γ)Rn

I ∼ RI .C(R0) in (6.32) is given by

C(R0) ∼1

f ′′(R0)=

1 + γRn0

2n2αR2n−20 (γRn

0 − 1). (6.43)

By using the relations (6.42), we find

C(R0) ∼R2

0

4n2R0

, (6.44)

which is positive and therefore the de Sitter solution is stable. We notice that C(R0) <9H2

0 and therefore, there could occur oscillations as in (6.35).Furthermore, we can take into account the following model [359] (Model VI):

fV I(R) = −α[tanh

(b (R−R0)

2

)+ tanh

(bR0

2

)]= −α

[eb(R−R0) − 1

eb(R−R0) + 1+

ebR0 − 1

ebR0 + 1

]

(6.45)


where α and b are positive constants. When R→ 0, we find that

fV I(R)→ − αbR

2 cosh2(

bR02

) , (6.46)

and thus f(0) = 0. On the other hand, when R→ +∞,

fV I(R)→ −2Λeff ≡ −α[1 + tanh

(bR0

2

)]. (6.47)

If R ≫ R0, in the present universe, Λeff plays the role of the effective cosmologicalconstant. We also obtain

f ′V I(R) = − αb

2 cosh2(

b(R−R0)2

) , (6.48)

which has a minimum when R = R0, that is:

f ′V I(R0) = −αb2. (6.49)

Then in order to avoid anti-gravity, we find

0 < 1 + f ′V I(R0) = 1− αb

2. (6.50)

Beside the above model, we can consider a model which is able to describe, inprinciple, both the early inflation and the late acceleration epochs. The following two-step model [359] (Model VII):

fV II(R) = −α0

[tanh

(b0 (R−R0)

2

)+ tanh

(b0R0

2

)]−αI

[tanh

(bI (R−RI)

2

)+ tanh

(bIRI

2

)],

(6.51)could be useful to this goal. Let us assume

RI ≫ R0 , αI ≫ α0 , bI ≪ b0 , (6.52)

and

bIRI ≫ 1 . (6.53)

When R→ 0 or R≪ R0 ≪ RI , fV II(R) behaves as

fV II(R)→ −

α0b0

2 cosh2(

b0R02

) +αIbI

2 cosh2(

bIRI

2

)

R , (6.54)


and we find again fV II(0) = 0. When R≫ RI , we find

f(R)V II → −2ΛI ≡ −α0

[1 + tanh

(b0R0

2

)]−αI

[1 + tanh

(bIRI

2

)]∼ −αI

[1 + tanh

(bIRI

2

)].

(6.55)On the other hand, when R0 ≪ R≪ RI , we find

fV II(R)→ −α0

[1 + tanh

(b0R0

2

)]− αIbIR

2 cosh2(

bIRI

2

) ∼ −2Λ0 ≡ −α0

[1 + tanh

(b0R0

2

)].

(6.56)Here, we have assumed the condition (6.53). We also find

f ′V II(R) = − α0b0

2 cosh2(

b0(R−R0)2

) − αIbI

2 cosh2(

bI (R−RI )2

) , (6.57)

which has two minima for R ∼ R0 and R ∼ RI . When R = R0, we obtain

f ′V II(R0) = −α0b0 −αIbI

2 cosh2(

bI (R0−RI)2

) > −αIbI − α0b0 . (6.58)

On the other hand, when R = RI , we get

f ′V II(RI) = −αIbI −α0b0

2 cosh2(

b0(R0−RI )2

) > −αIbI − α0b0 . (6.59)

Then, in order to avoid the anti-gravity behavior, we find

αIbI + α0b0 < 1 . (6.60)

Let us now investigate the correction to the Newton potential and the matter insta-bility issue related to Models VI and VII. In the Solar System domain, on or inside theEarth, where R≫ R0, f(R) in Eq.(6.45) can be approximated by

fV I(R) ∼ −2Λeff + 2αe−b(R−R0) . (6.61)

On the other hand, since R0 ≪ R≪ RI , by assuming Eq. (6.53), f(R) in (6.51) can bealso approximated by

fV II(R) ∼ −2Λ0 + 2αe−b0(R−R0) , (6.62)

which has the same expression, after having identified Λ0 = Λeff and b0 = b. Then, wemay check the case of (6.61) only. In this case, the effective mass has the following form

m2σ ∼

eb(R−R0)

4αb2, (6.63)


which could be again very large. In fact, in the Solar System, we find R ∼ 10−61 eV2.Even if we choose α ∼ 1/b ∼ R0 ∼

(10−33 eV

)2, we find that m2

σ ∼ 101,000 eV2, whichis, ultimately, extremely heavy. Then, there will be no appreciable correction to theNewton law. In the Earth atmosphere, R ∼ 10−50 eV2, and even if we choose α ∼ 1/b ∼R0 ∼

(10−33 eV

)2again, we find that m2

σ ∼ 1010,000,000,000 eV2. Then, a correction tothe Newton law is never observed in such models. In this case, we find that the effectivepotential U(Rb) has the form

U(Re) = − 1

2αb

(2Λ +

1

b

)e−b(Re−R0) , (6.64)

which could be negative, what would suppress any instability.In order that a de Sitter solution exists in f(R)-gravity, the following condition has

to be satisfied:R = Rf ′(R)− 2f(R) . (6.65)

For the model (6.45), the r.h.s of (6.65) has the following form:

R = − bαR

2 cosh2(

b(R−R0)2

) + 2α

[tanh

(b (R−R0)

2

)+ tanh

(bR0

2

)]. (6.66)

For large R, the r.h.s. behaves as

− bαR

2 cosh2(

b(R−R0)2

) + 2α

[tanh

(b (R−R0)

2

)+ tanh

(bR0

2

)]→ 2α , (6.67)

although the l.h.s. goes to infinity. On the other hand, when R is small, the r.h.s.behaves as

− bαR

2 cosh2(

b(R−R0)2

) + 2α

[tanh

(b (R−R0)

2

)+ tanh

(bR0

2

)]→ bαR

2 cosh2(

bR02

) .

(6.68)Then if

bα

2 cosh2(

bR02

) > 1 , (6.69)

there is a de Sitter solution. Combining Eq.(6.69) with Eq.(6.50), we find

2 > αb >1

2 cosh2(

bR02

) . (6.70)

The stability, as above, is given by C(RdS), where RdS is the solution of (6.66). Theexpression is given by

C(RdS) = −RdS +2cosh3

(b(RdS−R0)

2

)

αb2 sinh(

b(RdS−R0)2

) − 1

b tanh(

b(RdS−R0)2

) . (6.71)


Let us now rewrite Eq.(6.66) as follows,

RdS = 2α

[tanh

(b (RdS −R0)

2

)+ tanh

(bR0

2

)]1 +

αb

2 cosh2(

b(RdS−R0)2

)

−1

.

(6.72)Then by using (6.72), we may rewrite (6.71) in the following form:

C(RdS) =−α2b2

(1− x2

) [(x− x0)

2 + 1− x20

]+ 4

αb2x (1− x2) [2 + αb (1− x2)], (6.73)

where

x = tanh

(b (RdS −R0)

2

), x0 = − tanh

(bR0

2

), (6.74)

and therefore we have−1 < x0 ≤ x < 1 , x0 < 0 . (6.75)

Let us now consider (6.66) in order to find a de Sitter solution. Since Eq.(6.66) isdifficult to solve in general, we assume 0 < RdS ≪ R0. Then we find

RdS =ε

bx0, ε ≡ 1−

2 cosh2(

bR02

)

αb= 1− 2

αb(1− x2

0

) . (6.76)

Eq.(6.69) tells that the parameter ε is positive and, by assumption, very small: 0 < ε≪1. Since ε is small, by using Eqs.(6.74), we find

x = x0 +

(1− x2

0

)

2x0ε+O

(ε2). (6.77)

Then by using the expression (6.73) for C(RdS), we find

C(RdS) ∼−α2b2

(1− x2

0

)2+ 4

αb2x0

(1− x2

0

) [2 + αb

(1− x2

0

)] . (6.78)

From the definition of ε in (6.76), we find

αb(1− x2

0

)= 2 + 2ε+O

(ε2), (6.79)

and then, from Eq.(6.79), Eq.(6.78) can be written as follows;

C(RdS) ∼ − ε

bx0. (6.80)

Since x0 < 0 in the condition (6.75), we find C(RdS) > 0 and therefore the de Sittersolution is stable.


Mercury perihelion Shift |2γ − β − 1| < 3× 10−3

Lunar Laser Ranging 4β − γ − 3 = (0.7± 1)× 10−3

Very Long Baseline Interferometer |γ − 1| < 4× 10−4

Cassini Spacecraft γ − 1 = (2.1 ± 2.3) × 10−5

Table 6.1: Solar System experimental constraints on the PPN parameters.

In Fig. 6.3, we have plotted the two models (6.45) and (6.51) written in theform F (R) = R + f(R). We have used the inequalities (6.52) assuming, RI ∼ ρg ∼10−24 g/cm3 for the Galactic density in the Solar vicinity and R0 ∼ ρg ∼ 10−29 g/cm3

for the present cosmological density. .

Our task is now to find reliable experimental bounds for such models working atsmall and large scales. To this goal, we shall take into account constraints comingfrom Solar System experiments (which, at present, are capable of giving upper limitson the PPN parameters) and constraints coming from interferometers, in particularthose giving limits on the (eventual) scalar components of GWs. If constraints (and inparticular the ranges of model parameters given by them) are comparable, this couldconstitute, besides other experimental and observational probes, a good hint to achievea self-consistent f(R) theory at very different scales.

6.4 Constraining f(R)-models by PPN parameters

The above models can be constrained at Solar System level by considering the PPNformalism. This approach is extremely important in order to test gravitational theoriesand to compare them with GR. As it is shown in [372, 377], one can derive the PPN-parameters γ and β in terms of a generic analytic function F (R) and its derivative

γ − 1 = − F ′′(R)2

F ′(R) + 2F ′′(R)2, (6.81)

β − 1 =1

4

[F ′(R) · F ′′(R)

2F ′(R) + 3F ′′(R)2

]dγ

dR. (6.82)

These quantities have to fulfill the constraints coming from the Solar System experi-mental tests summarized in Table I. They are the perihelion shift of Mercury [388], theLunar Laser Ranging [389], the upper limits coming from the Very Long Baseline Inter-ferometry (VLBI) [390] and the results obtained from the Cassini spacecraft mission inthe delay of the radio waves transmission near the Solar conjunction [391].

6.4. CONSTRAINING F (R)-MODELS BY PPN PARAMETERS 149

Let us take into account before the f(R)-models (6.10)-(6.13). Specifically, we wantto investigate the values or the ranges of parameters in which they match the Solar-System experimental constraints in Table 6.1. In other words, we use these modelsto search under what circumstances it is possible to significantly address cosmologicalobservations by f(R)-gravity and, simultaneously, evade the local tests of gravity.

By integrating Eqs. (6.81)-(6.82), one obtains f(R) solutions depending on β andγ which has to be confronted with βexp and γexp [377]. If we plug into such equationsthe models (6.10)-(6.13) and the experimental values of PPN parameters, we will obtainalgebraic constraints for the phenomenological parameters n, p, q, λ, s. This is theissue which we want to take into account in this section.

From Eq.(6.81), assuming F ′(R)+2F ′′(R)2 6= 0 and definingA =

∣∣∣∣1− γ2γ − 1

∣∣∣∣, we obtain

[F ′′(R)

]2 −AF ′(R) = 0 . (6.83)

The general solution of such an equation is a polynomial function [377].

Considering Model II given by (6.11), we obtain

1−

2pR(

R2

R2c

+ 1)−p−1

λ

Rc

∣∣∣∣γ − 1

2γ − 1

∣∣∣∣−4p2

(R2

R2c

+ 1)−2p

R2c

(R2

c − (2p + 1)R2)2λ2

(R2 +R2c)

4 = 0 .

(6.84)Our issue is now to find the values of λ, p, and R/Rc for which the Solar Systemexperimental constraints are satisfied. Some preliminary considerations are in order atthis point. Considering the de Sitter solution achieved from (6.11), we have R = const =R1 = x1Rc, and x1 > 0. It is straightforward to obtain

λ =x1

(1 + x2

1

)p+1

2[(

1 + x21

)p+1 − 1− (p+ 1) x21

] . (6.85)

On the other hand, the stability conditions F,R > 0 and F,RR > 0 give the inequality

(1 + x2

1

)p+2> 1 + (p+ 2) x2

1 + (p+ 1) (2p+ 1) x41 , (6.86)

which has to be satisfied. In particular, for p = 1, it is x1 >√

3 and then λ >8

3√

3= 1.5396.

In addition, the value of x1 satisfying the relation (6.86) is also the point where λ(x1),in Eq.(6.85), reaches its minimum.

To determine values of R compatible with PPN constraints, let us consider the traceof the field equations (6.2) and explicit solutions, given the density profile ρ(r), in theSolar vicinity. One can set the boundary condition considering F,R∞ = FRg


F,Rg = F,R(R = k2ρg) (6.87)

where ρg ∼ 10−24 g/cm3 is the observed Galactic density in the Solar neighborhoods.At this point, we can see when the relation (6.84) satisfies the constraints for very LongBaseline Interferometer (γ− 1 = 4× 10−4) and Cassini Spacecraft (γ− 1 = 2.1× 10−5).This allows to find out suitable values for p.

An important remark is in order at this point. These constraint equations work ifstability conditions hold. In the range

0 <R

Rc<

1√2p+ 1

(6.88)

F,RR is negative for the model (6.11) and then stability conditions are violated. Toavoid this range, we need, at least, R

Rc> 1. For example, we can choose R

Rc= 3.38,

corresponding to de Sitter behavior. Then we have p = 1 and λ = 2. On the otherhand, for 0.944 < λ < 0.966, we have p = 2 and R

Rc=√

3; finally, for R >> Rc, we haveλ = 2 and p = 1.5. For these values of parameters, the Solar System tests are evaded.

Let us consider now Model I, given by (6.9). Inserting it into the relation (6.83), weget

R3

[(RRc

)2n+ 1

]4[R

((RRc

)2n+ 1

)2

− 2n(

RRc

)2nRcλ

] ∣∣∣ γ−12γ−1

∣∣∣− 4n2

[(2n+ 1)

(RRc

)2n− 2n + 1

]2 (RRc

)4n

R4

[(RRc

)2n+ 1

]6

(6.89)Using the same procedure as above, λ is related to the de Sitter behavior. This means

λ =

(1 + x2n

1

)2

x2n−11

(2 + 2x2n

1 − 2n) , (6.90)

while, from the stability conditions, we get

2x41 − (2n− 1) (2n + 4) x2n

1 + (2n− 1) (2n− 2) ≥ 0 . (6.91)

For n = 1, one obtains x1 >√

3 , λ > 83√

3. In this model, F,RR is negative for

0 <R

Rc<

(2n − 1

2n + 1

) 12n

. (6.92)

The VLBI constraint is satisfied for n = 1 and λ = 2, while, for n = 1 and λ = 1.5,Cassini constraint holds.

6.4. CONSTRAINING F (R)-MODELS BY PPN PARAMETERS 151

By inserting Model III, given by Eq.(6.12), into the relation (6.83), we obtain

R3[R− 2sRc

(Rc

R

)2sλ] ∣∣∣ γ−1

2γ−1

∣∣∣− 4(2s2 + s

)2R2

c

(Rc

r

)4sλ2

R4= 0 . (6.93)

The de-Sitter point corresponds to

λ =x2s+1

1

2(x2s1 − s− 1)

. (6.94)

while the stability condition is x2s1 > 2s2 + 3s + 1. VLBI and Cassini constraints are

satisfied by the sets of values: s = 1, λ = 1.53, for RRc∼ 1; s = 2, λ = 0.95, for R

Rc=√

3,

; s = 1, λ = 2, for RRc

= 3.38.Finally let us consider Model VI, given by Eq.(6.45), and Model VII, given by

Eq.(6.51). Using Eq.(6.83) for (6.45), we get

−1

4bαsech2

(1

2b(R−R0)

)[b3αsech2

(1

2b(R−R0)

)tanh2

(1

2b(R −R0)

)− 2

∣∣∣∣γ − 1

2γ − 1

∣∣∣∣]

= 0 .

(6.95)As above, considering the stability conditions and the de Sitter behavior, we get theparameter ranges 0 < b < 2 and 0 < α ≤ 2 which satisfy both VLBI and Cassiniconstraints. Inserting now Model VII in (6.83), we have

1

2

∣∣∣∣γ − 1

2γ − 1

∣∣∣∣[bαsech2

(1

2b(R −R0)

)− bIαIsech

2

(1

2bI(R−RI)

)+ 2

]

− 1

4

[b2αsech2

(1

2b(R−R0)

)tanh

(1

2b(R −R0)

)− b2IαIsech

2

(1

2bI(R−RI)

)tanh

(1

2bI(R−RI)

)]2

= 0 .

(6.96)

From the stability condition, we have that F,R > 0 for R > 0, (see Fig.6.6) andF,RR < 0 for 0 < R < 2.35 in suitable units (see Fig.6.7). Observational constraintsfrom VLBI and Cassini experiments are fulfilled for

RI ≫ R0 , αI ≫ α , bI ≪ b . (6.97)

Plots for b = 2, bI = 0.5, α = 1.5 and αI = 2, verifying the constraints, are reported inFigs. 6.6 and 6.7.

Considering now the relation for β given by Eq. (6.82), one can easily verify that itis

dγ

dR= − d

dR

[F ′′(R)2

F ′(R) + 2F ′′(R)2

]= 0 , (6.98)


and this result implies

4(β − 1) = 0 . (6.99)

This means the complete compatibility of the f(R) solutions between the PPN-parametersβ and γ.

Now we want to see if the parameter values, obtained for these models, are compatiblewith bounds coming from the stochastic background of GWs achieved by interferometricexperiments.

6.5 Stochastic backgrounds of gravitational waves to con-

strain f(R)-gravity

As we said before, also the stochastic background of GWs can be taken into account in or-der to constrain models. This approach could reveal very interesting because productionof primordial GWs could be a robust prediction for any model attempting to describethe cosmological evolution at primordial epochs. However, bursts of gravitational radi-ation emitted from a large number of unresolved and uncorrelated astrophysical sourcesgenerate a stochastic background at more recent epochs, immediately following the onsetof galaxy formation. Thus, astrophysical backgrounds might overwhelm the primordialone and their investigation provides important constraints on the signal detectabilitycoming from the very early Universe, up to the bounds of the Planck epoch and theinitial singularity [384, 393, 394, 396].

It is worth stressing the unavoidable and fundamental character of such a mechanism.It directly derives from the inflationary scenario [397, 398], which well fits the WMAPdata with particular good agreement with almost exponential inflation and spectralindex ≈ 1, [399, 400].

The main characteristics of the gravitational backgrounds produced by cosmologicalsources depend both on the emission properties of each single source and on the sourcerate evolution with redshift. It is therefore interesting to compare and contrast theprobing power of these classes of f(R)-models at hight, intermediate and zero redshift[401].

To this purpose, let us take into account the primordial physical process whichgave rise to a characteristic spectrum Ωsgw for the early stochastic background of relicscalar GWs by which we can recast the further degrees of freedom coming from fourth-order gravity. This approach can greatly contribute to constrain viable cosmologicalmodels. The physical process related to the production has been analyzed, for example,in [393, 394, 395] but only for the first two tensorial components due to standard GeneralRelativity. Actually the process can be improved considering also the third scalar-tensorcomponent strictly related to the further f(R) degrees of freedom [386].

At this point, using the above LIGO, VIRGO and LISA upper bounds, calculatedfor the characteristic amplitude of GW scalar component, let us test the f(R)-gravity

6.5. STOCHASTIC BACKGROUNDS OF GRAVITATIONAL WAVES TO CONS TRAIN F (R)-GRAVITY153

models, considered in the previous sections, to see whether they are compatible bothwith the Solar System and GW stochastic background.

Before starting with the analysis, taking into account the discussion in the Chapter5, section 5.4. As above, for the considered models, we have to determine the valuesof the characteristic parameters which are compatible with both Solar System and GWstochastic background.

Let us start, for example, with the model (6.12). Starting from the definitions (??),it is straightforward to derive the scalar component amplitude

ΦIII =s(2s+ 1)

(Rc

R

)2s+1λ[

sRc

(Rc

R

)2sλ−R

]log[2− 2s

(Rc

R

)2s+1λ] . (6.100)

Such an equation satisfies the constraints in Table.4.26 for the values s = 0.5, RRc∼ 1,

λ = 1.53 and s = 1, RRc∼ 1, λ = 0.95 (LIGO); s = 2, R

Rc=√

3, λ = 2 (VIRGO); s = 1,

λ = 2 and RRc

= 3.38 (LISA).It is important to stress the nice agreement with the figures achieved from the PPN

constraints. In this case, we have assumed Rc ∼ ρc ∼ 10−29 g/cm3, where ρc is thepresent day cosmological density.

Considering the model (6.9), we obtain

ΦI = −n

[(2n + 1)

(RRc

)2n− 2n+ 1

](RRc

)2n−1λ

[(RRc

)2n+ 1

]R

[(RRc

)2n+ 1

]2

− n(

RRc

)2nRcλ

log

1−

2n“

RRc

”2n−1λ

„“RRc

”2n+1

«2

.

(6.101)The expected constraints for GW scalar amplitude are fulfilled for n = 1 and λ = 2 andfor n = 1 and λ = 1.5 when 0.3 < R

Rc< 1.

Furthermore, considering the model (6.11), one gets

ΦI = −2p(1 + R2

R2c

)−pRc

((1 + 2p)R2 −R2

c

)λ

(R2 −R2c)

2

2−

2p

„1+ R2

R2c

«−1−p

λ

Rc

ln

2−

2pR

„1+ R2

R2c

«−1−p

λ

Rc

. (6.102)

The LIGO upper bound is fulfilled for p = 1, RRc

>√

3, λ > 83√

3; the VIRGO one for

p = 1, RRc

= 3.38, λ = 2; finally, for LISA, we have p = 2, RRc

=√

3 and 0.944 < λ <0.966. Besides, considering LISA in the regime R >> Rc, we have λ = 2 and p = 1.5.

Finally, let us consider Models VI and VII. We have

ΦV I =b2α tanh

[12b(R−R0)

]

[bα+ cosh(b(R −R0)) + 1] ln[

bαcosh(b(R−R0))+1

] , (6.103)


and

ΦV II = log[0.5(bαsech2(0.5b(R −R0))− bIαIsech

2(0.5bI(R −RI)) + 2)]

×[bαsech2(0.5b(R −R0))− bIαIsech

2(0.5bI (R−RI)) + 4]

×[b2αsech2(0.5b(R −R0)) tanh(0.5b(R −R0))− b2IαIsech

2(0.5bI(R−RI)) tanh(0.5bI(R −RI))].

(6.104)

These equations satisfy the constraints for VIRGO, LIGO and LISA for b = 2, bI =0.5, α = 1.5 and αI = 2 with RI valued at Solar System scale and R0 at cosmologicalscale.


0 0.5 1 1.5−0.2

0

0.2

0.4

0.6

0.8

1

R/Rc

f(R

)/R

c

I modelII modelIII modelIV modelf(R)=R

Figure 6.1: Plots of four different F (R) models as function of RRc

. Model I in Eq. (6.9)with n = 1 and λ = 2 (dashed line). Model II in Eq.(6.11) with p = 2, λ = 0.95 (dashdotline). Model III in Eq.(6.12) with s = 0.5 and λ = 1.5 (dotted). Model IV in Eq.(6.13)with q = 0.5 and λ = 0.5 (solid line). We also plot F (R) = R (solid thick line) to seewhether or not the stability condition F,R > 0 is violated.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.5

0

0.5

1

1.5

R

f(R

)

V modelf’(R) of V model

R=0.64

R=1

Figure 6.2: Plots of Model V (6.38) (solid line) and its first derivative (dashed line).Here n = 2 and α, β, γ are assumed as in (6.42) with the value of R0 taken in the SolarSystem. f ′(R) is negative for 0 < R < 0.64. f(R) is given in the range 0 < R < 1 wherewe have adopted suitable units.


0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

R

f(R

)

VI modelVII model

Figure 6.3: Plots of Model VI (6.45) (solid line) and Model VII (6.51) (dashed line).Here b = 2 and bI = 0.5 with α = 1.5 and αI = 2. The value of RI is taken in the SolarSystem while R0 corresponds to the present cosmological value.


0 1 2 3 4 5 60

0.5

1

1.5

2

R

f’(R

f’(R) of VI modelf’(R) of VII model

Figure 6.6: Plots represent the first derivatives of functions (6.50) (solid line) and (6.51)(dashed line). Here, b = 2, bI = 0.5, α = 1.5 and αI = 2 with RI with the Solar Systemvalue and R0 the today cosmological value. It is F,R > 0 for R > 0.


0 1 2 3 4 5 6

−1

−0.5

0

0.5

1

1.5

R

f’’(R

)

f’’(R) of VI model f’’(R) of VII model

R=2.35

R=4

Figure 6.7: Second derivatives of Model VI (solid line) and VII (dashed line). Here F,RR

is negative in the range 0 < R < 4 for Model VI and in the range 0 < R < 2.35 forModel VII. As above, we have used b = 2, bI = 0.5, α = 1.5 and αI = 2 with the valueof RI taken in the Solar System and R0 for the today cosmological value.

Chapter 7

Future perspectives and

conclusions

7.1 Brief summary

Before concluding this thesis or discussing future perspectives of the work presentedhere, let us attempt to summarize in this section some of the results presented so far.The motivation of this thesis has been thoroughly discussed in Chapter 1 and a generaldiscussion about modifications of gravity was laid out in Chapter 2. In Chapter ??,a compact, self-contained approach to gravitation, based on the local Poincare gaugeinvariance, is proposed and we have shown as generate the gravitation by a simplesimmetry break. We discussed the Invariance Principle as our starting point in orderto derive gravity as a local Poincare gauge theory. Global Poincare invariance and thelocal one are discussed, showing how a local transformation is related to the gaugefields. Spinors, vectors and tetrads which transform under Lorentz transformations arediscussed. In particular we discuss the Fock - Ivanenko connection in the frameworkof the local Poincare transformations. Starting from the Fock - Ivanenko covariantderivative, curvature, torsion and metric tensors are derived after field equations forgravity are discussed in the framework of the present approach. In Chapter 4 in wedefined the space-time perturbations in the framework of the metric formalism givingthe notion of first and second deformation matrices. we do the main properties ofdeformations. In particular, we discuss how deformation matrices can be split in theirtrace, traceless and skew parts. We derive the contributions of deformation to thegeodesic equation and, starting from the curvature Riemann tensor, the general equationof deformations. In We discuss the notion of linear perturbations under the standardof deformations. In particular, we recast the equation of gravitational waves and thetransverse traceless gauge under the standard of deformations and discussed the actionof deformations on the Killing vectors. The result consists in achieving a notion ofapproximate symmetry. Finally in the Chapters 5 and 6 focused on the cosmological

163

164 7. FUTURE PERSPECTIVES AND CONCLUSIONS

and astrophysical aspects of these theories and on their viability. Infact we wanted toface the problem of how the GW stochastic background and f(R) gravity can be relatedshowing, vice-versa, that a revealed stochastic GW signal could be a powerful probe fora given effective theory of gravity. Our goal was to show that the conformal treatmentof GWs can be used to parameterize in a natural way f(R) theories.we review the fieldequations of f(R) gravity in the metric approach and their scalar-tensor representation,useful to compare the theory with observations. We review and discuss some viablef(R) models capable of satisfying both local gravity prescriptions as well as the observedcosmological behavior. In particular, we discuss their stability conditions and the fieldvalues which have to achieved to fulfill physical bounds. We derived the values of modelparameters in agreement with the PPN experimental constraints while, we deal with theconstraints coming from the stochastic background of GWs. These latter ones have tobe confronted with those coming from PPN parameterization. As a general remark, wefind out that bounds coming from the interferometric ground-based (VIRGO, LIGO) andspace (LISA) experiments could constitute a further probe for f(R) gravity if matchedwith bounds at other scales.

As mentioned in the Introduction, these theories were introduced as tools that couldhelp us to examine how much and in which ways one can deviate from General Relativity.Our intention was neither to tailor a model within the framework of extended theoriesthat would fit the data adequately nor to pick out a specific well-motivated low-energyeffective action from some fundamental theory and to confront it with observations.The task which we undertook was to consider theories that were easy to handle, each ofthem deviating from the framework of General Relativity , and to exploit them in orderto get a deeper understanding of the difficulties and limitations of modified gravity. Inthe light of this, it is probably preferable to provide here a qualitative summary of ourresults which summarizes the lessons learned from this procedure, instead of repeatingin detail the results already presented in the previous chapters.

Starting from the theoretical side we have shown that all the necessary ingredients fora theory of gravitation can be obtained from a gauge theory of local Poincare symmetry.Gauge fields have been obtained by requiring the invariance of the Lagrangian densityunder local Poincare transformations.

The resulting Einstein-Cartan theory describes a space endowed with non-vanishingcurvature and torsion. The lowest order gravitational action is one that is linear inthe curvature scalar while being quadratic in torsion. However, the scheme can beimmediately extended to more general gravitational theories as in [246].

The Dirac spinors can be introduced as matter sources and it has been found thatthey couple to gravity via the torsion stress form Tµν component of the total energy-momentum Σµν . The field equations obtained from the action by means of a stan-dard variational principle describe a nonlinear equation of the Heisenberg-Pauli typein the matter sector, gravitational field equations similar to the Einstein equationsand a constraint equation relating torsion to spin energy potential. The generalizedenergy-momentum tensor is comprised of the usual canonical energy-momentum tensor

7.1. BRIEF SUMMARY 165

of matter in addition to a torsion stress form. The stress form contains a torsion di-vergence term as well as a term similar to an external non-spinor source to gravity. Inview of the structure of the generalized energy-momentum tensor, we remark that thegravitational field equations here obtained are similar to the equations of motion foundin Einstein-Yang-Mills theory, the torsion tensor playing the role of the Yang-Mills fieldstrength.

The Bianchi identities of Einstein-Cartan gravity differ from those of General Rela-tivity since the Riemann curvature tensor characterizing the non-Riemannian geometrydoes not exhibit the usual symmetry properties. In the limit of vanishing torsion, theBianchi identities reduce to their usual form. The conservation laws for the angu-lar momentum and the energy-momentum has been obtained. From the former, it hasbeen found that the generalized energy-momentum tensor contains a non-vanishing anti-symmetric component proportional to the divergence of the spin-energy potential. Forthe latter, we found that the generalized energy-momentum tensor is only divergencelessin the limit of vanishing torsion.

We have proposed a novel definition of space-time metric deformations parameter-izing them in terms of scalar field matrices. The main result is that deformations canbe described as extended conformal transformations. This fact gives a straightforwardphysical interpretation of conformal transformations: conformally related metrics canbe seen as the ”background” and the ”perturbed” metrics. In other words, the relationsbetween the Jordan frame and the Einstein frame can be directly interpreted throughthe action of the deformation matrices contributing to solve the issue of what the truephysical frame is [347, ?].

Besides, space-time metric deformations can be immediately recast in terms of per-turbation theory allowing a completely covariant approach to the problem of gravita-tional waves.

The discussion about the cosmological and astrophysical aspects of the theories ex-amined here and the confrontation of the theories with cosmological, astrophysical andSolar System observations hopefully clarified that it is very difficult to construct a simpleviable model in an alternative theory of gravity.

In summary, we have shown that amplitudes of tensor GWs are conformally invariantand their evolution depends on the cosmological background. Such a background istuned by conformal scalar field which is not present in the standard general relativity.Assuming that primordial vacuum fluctuations produce stochastic GWS, beside scalarperturbations, kinematical distortions and so on, the initial amplitude of these ones is afunction of the f(R)-theory of gravity and then the stochastic background can be, in acertain sense “tuned” by the theory. Vice versa, data coming from the Sachs-Wolfe effectcould contribute to select a suitable f(R) theory which can be consistently matched withother observations. However, further and accurate studies are needed in order to testthe relation between Sachs-Wolfe effect and f(R) gravity. This goal could be achievedvery soon through the forthcoming space (LISA) and ground-based (VIRGO, LIGO)interferometers.


We have investigated the possibility that some viable f(R) models could be con-strained considering both Solar System experiments and upper bounds on the stochasticbackground of gravitational radiation. Such bounds come from interferometric ground-based (VIRGO and LIGO) and space (LISA) experiments. The underlying philosophy isto show that the f(R) approach, in order to describe consistently the observed universe,should be tested at very different scales, that is at very different redshifts. In otherwords, such a proposal could partially contribute to remove the unpleasant degeneracyaffecting the wide class of dark energy models, today on the ground.

Beside the request to evade the Solar System tests, new methods have been recentlyproposed to investigate the evolution and the power spectrum of cosmological pertur-bations in f(R) models [363]. The investigation of stochastic background, in particularof the scalar component of GWs coming from the f(R) additional degrees of freedom,could acquire, if revealed by the running and forthcoming experiments, a fundamentalimportance to discriminate among the various gravity theories [401]. These data (to-day only upper bounds coming from simulations) if combined with Solar System tests,CMBR anisotropies, LSS, etc. could greatly help to achieve a self-consistent cosmologybypassing the shortcomings of ΛCDM model.

Specifically, we have taken into account some broken power law f(R) models fulfillingthe main cosmological requirements which are to match the today observed acceleratedexpansion and the correct behavior in early epochs. In principle, the adopted parame-terization allows to fit data at extragalactic and cosmological scales [355]. Furthermore,such models are constructed to evade the Solar System experimental tests. Beside thesebroken power laws, we have considered also two models capable of reproducing the ef-fective cosmological constant, the early inflation and the late acceleration epochs [359].These f(R)-functions are combinations of hyperbolic tangents.

We have discussed the behavior of all the considered models. In particular, theproblem of stability has been addressed determining suitable and physically consistentranges of parameters. Then we have taken into account the results of the main SolarSystem current experiments. Such results give upper limits on the PPN parameterswhich any self-consistent theory of gravity should satisfy at local scales. Starting fromthese, we have selected the f(R) parameters fulfilling the tests. As a general remark,all the functional forms chosen for f(R) present sets of parameters capable of matchingthe two main PPN quantities, that is γexp and βexp. This means that, in principle,extensions of GR are not a priori excluded as reasonable candidates for gravity theories.

The interesting feature, and the main result of this thesis, is that such sets of parame-ters are not in conflict with bounds coming from the cosmological stochastic backgroundof GWs. In particular, some sets of parameters reproduce quite well both the PPN upperlimits and the constraints on the scalar component amplitude of GWs.

Far to be definitive, these preliminary results indicate that self-consistent modelscould be achieved comparing experimental data at very different scales without extrap-olating results obtained only at a given scale.

7.2. CONCLUDING REMARKS 167

7.2 Concluding remarks

To conclude, even though some significant progress has been made with developingalternative gravitation theories, one cannot help but notice that it is still unclear how torelate principles and experiments in practice, in order to form simple theoretical viabilitycriteria which are expressed mathematically. Our inability to express these criteria andalso several of our very basic definitions in a representation-invariant way seems to haveplayed a crucial role in the lack of development of a theory of gravitation theories. Thisis a critical obstacle to overcome if we want to go beyond a trial-and-error approach indeveloping alternative gravitation theories.

It is the author’s opinion that such an approach should be one of the main futuregoals in the field of modified gravity. This is not to say, of course, that efforts to proposeor use individual theories, such as f(R) gravity, in order to deepen our understandingabout the gravitational interaction should be abandoned or have less attention paid tothem. Such theories have proved to be excellent tools for this cause so far, and thereare still a lot of unexplored corners of the theories mentioned in this thesis, as well asin other alternative theories of gravity.

The motivation for modified gravity coming from High Energy Physics, Cosmologyand Astrophysics is definitely strong. Even though modifying gravity might not bethe only way to address the problems mentioned in Chapter 1, it is our hope that thereader is by now convinced that it should at least be considered very seriously as oneof the possible solutions and, therefore, given appropriate attention. The path to thefinal answer is probably long. However, this has never been a good enough reason forscientists to be discouraged.

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