632 20 The Very Early Universe

20.3 The Anthropic Cosmological Principle

There is certainly some truth in the fact that our ability to ask questionsabout the origin of the Universe says something about the sort of Universewe live in. The Cosmological Principle asserts that we do not live at anyspecial location in the Universe, and yet we are certainly privileged in beingable to make this statement at all. In this line of reasoning, there are onlycertain types of Universe in which life as we know it could have formed. Forexample, the stars must live long enough for there to be time for biologicallife to form and evolve into sentient beings. This line of reasoning is embodiedin the Anthropic Cosmological Principle, first expounded by Carter in 1974(Carter, 1974) and dealt with in extenso in the books by Barrow and Tiplerand by Gribbin and Rees (Barrow and Tipler, 1986; Gribben and Rees, 1989).Part of the problem stems from the fact that we have only one Universe tostudy – we cannot go out and investigate other Universes to see if they haveevolved in the same way as ours. There are a number of versions of thePrinciple, some of them stronger than others. In extreme interpretations, itleads to statements such as the strong form of the Principle enunciated byWheeler (Wheeler, 1977),

Observers are necessary to bring the Universe into being.

It is a matter of taste how seriously one wishes to take this line of reason-ing. To many cosmologists, it is not particularly appealing because it suggeststhat it will never be possible to find physical reasons for the initial condi-tions from which the Universe evolved, or for the values of the fundamentalconstants of nature. But some of these problems are really hard. Weinberg,for example, found it such a puzzle that the vacuum energy density ΩΛ isso very much smaller than the values expected according to current theoriesof elementary particles, that he invoked anthropic reasoning to account forits smallness (Weinberg, 1989, 1997). Another manifestation of this type ofreasoning is to invoke the range of possible initial conditions which mightcome out of the picture of chaotic inflation (Linde, 1983) and argue that, ifthere were at least 10120 of them, then we live in one of the few which hasthe right conditions for life to develop as we know it. Again, I leave it tothe reader how seriously these ideas should be taken. I worry about the issueof observational validation of these ideas. I prefer to regard the AnthropicCosmological Principle as the very last resort if all other physical approachesfail.

20.4 The Inflationary Universe – Historical Background

The most important conceptual development for studies of the very early Uni-verse can be dated to 1980 and the proposal by Guth of the inflationary modelfor the very early Universe (Guth, 1981). There had been earlier suggestions

20.4 The Inflationary Universe – Historical Background 633

foreshadowing his proposal. Zeldovich had noted in 1968 that there is a phys-ical interpretation of the cosmological constant Λ in terms of the zero-pointfluctuations in a vacuum (Zeldovich, 1968). Linde in 1974 and Bludman andRuderman in 1977 had shown that the scalar Higgs fields of particle physicshave similar properties to those which would result in a positive cosmologicalconstant (Linde, 1974; Bludman and Ruderman, 1977). A popular account ofthe history of the development of ideas about the inflation picture of the earlyUniverse is contained in Guth’s book The Inflationary Universe: The Questfor a New Theory of Cosmic Origins (Guth, 1997). The pedagogical reviewby Lineweaver can also be recommended. He adopts a somewhat scepticalattitude to the concept of inflation and our ability to test inflationary mod-els through confrontation with observations (Lineweaver, 2005). Nonetheless,for good reasons, this model dominates much of present-day cosmologicalthinking.

Guth realised that, if there were an early phase of exponential expansionof the Universe, this could solve the horizon problem and drive the Universetowards a flat spatial geometry, thus solving the flatness problem at thesame time. Suppose the scale factor, a, increased exponentially with timeas a ∝ et/T . Such exponentially expanding models were found in some ofthe earliest solutions of the Friedman equations, in the guise of empty deSitter models driven by what is now termed the vacuum energy density ΩΛ

(see Sect. 7.3.3) (Lanczos, 1922). Consider a tiny region of the early Universeexpanding under the influence of the exponential expansion. Particles withinthe region were initially very close together and in causal communication witheach other. Before the inflationary expansion began, the region had physicalscale less than the particle horizon, and so there was time for it to attaina uniform, homogeneous state. The region then expanded exponentially sothat neighbouring points were driven to such large distances that they couldno longer communicate by light signals – the causally-connected regions wereswept beyond their particle horizons by the inflationary expansion. At theend of the inflationary epoch, the Universe transformed into the standardradiation-dominated Universe and the inflated region continued to expand asa ∝ t1/2. More formal demonstrations of how the exponential expansion canresolve the problem of the particle horizon is given in Sects. 12.1 and 20.5.3.

In Guth’s original inflationary scenario, the exponential expansion wasassociated with the symmetry breaking of Grand Unified Theories of elemen-tary particles at very high energies through a first-order phase transition,only about 10−34 seconds after the Big Bang. Although this picture was soondemonstrated not to work, let us demonstrate to order of magnitude howthe argument runs. The time-scale 10−34 s is taken to be the characteristice-folding time for the exponential expansion. Over the interval from 10−34

seconds to 10−32 seconds, the radius of curvature of the Universe increasedexponentially by a factor of about e100 ≈ 1043. The horizon scale at the be-ginning of this period was only r ≈ ct ≈ 3× 10−26 m and this was inflated to

634 20 The Very Early Universe

a dimension of 3× 1017 m by the end of the inflationary era. This dimensionthen scaled as t1/2, as in the standard radiation-dominated Universe so thatthe region would have expanded to a size of 3 × 1042 m by the present day– this dimension far exceeds the present particle horizon r ≈ cT0 of the Uni-verse, which is about 1026 m. Thus, our present Universe would have arisenfrom a tiny region in the very early Universe which was much smaller than thehorizon scale at that time. This guaranteed that our present Universe wouldbe isotropic on the large scale, resolving the horizon problem. At the end ofthe inflationary era, there was an enormous release of energy associated withthe ‘latent heat’ of the phase transition and this reheated the Universe to avery high temperature indeed (Fig. 20.1).

Fig. 20.1. Comparison of the evolution of the scale factor and temperature in thestandard Big Bang and inflationary cosmologies.

The exponential expansion also had the effect of straightening out the ge-ometry of the early Universe, however complicated it may have been to begin

20.4 The Inflationary Universe – Historical Background 635

with. Suppose the tiny region of the early Universe had some complex ge-ometry. The radius of curvature the geometry Rc(t) scales as Rc(t) = < a(t),where < is the radius of curvature of the geometry at the present epoch t0,and so radius of curvature of the geometry is inflated to dimensions vastlygreater than the present size of the Universe, driving the geometry of theinflated region towards flat Euclidean geometry, Ωκ = 0, and consequentlythe Universe must have Ω0 + ΩΛ = 1. It is important that these two as-pects of the case for the inflationary picture can be made independently ofa detailed understanding of the physics of the inflation. There is also consid-erable freedom about the exact time when the inflationary expansion couldhave occurred, provided there are sufficient e-folding times to isotropise ourobservable Universe and flatten its geometry.

In Guth’s original proposal, the Universe was in a symmetric state, re-ferred to as a false vacuum state, at a very high temperature before theinflationary phase took place. As the temperature fell, spontaneous symmet-ric breaking took place through the process of barrier penetration from thefalse vacuum state and the Universe attained a lower energy state, the truevacuum. At the end of this period of exponential expansion, the phase tran-sition took place, releasing a huge amount of energy. The problem with thisrealisation was that it predicted ‘bubbles’ of true vacuum embedded in thefalse vacuum, with the result that huge inhomogeneities were predicted. An-other concern about the original proposal was that an excessive number ofmonopoles were created during the GUT phase transition. Kibble showedthat, when this phase transition took place, topological defects are expectedto be created, including point defects (or monopoles), line defects (or cosmicstrings) and sheet defects (or domain walls) (Kibble, 1976). Kibble showedthat one monopole is created for each correlation scale at that epoch. Sincethat scale cannot be greater than the particle horizon at the GUT phase tran-sition, it is expected that huge numbers of monopoles are created. Accordingto the simplest picture of the GUT phase transition, the mass density in thesemonopoles in the standard Big Bang picture would vastly exceed Ω0 = 1 atthe present epoch (Kolb and Turner, 1990).

The model was revised in 1982 by Linde and by Albrecht and Steinhardtwho proposed instead that, rather than through the process of barrier pen-etration, the transition took place through a second-order phase transitionwhich did not result in the formation of ‘bubbles’ and so excessive inhomo-geneities (Linde, 1982, 1983; Albrecht and Steinhardt, 1982). This picture,often referred to as new inflation, also eliminated the monopole problem sincethe likelihood of even one being present in the observable Universe was verysmall.

The original hope that a physical realisation for the inflationary expansioncould be found within the context of particle physics beyond the standardmodel has not been achieved, but the underlying concepts of the inflation-ary picture have been used to define the necessary properties of the inflaton

636 20 The Very Early Universe

potential needed to create the Universe as we know it. The successful realisa-tions are similar to those involved in the new inflationary picture. Once theinflationary expansion began at some stage in the early Universe, the changefrom the false to true vacuums states took place through a process of slow roll-over, meaning that the inflationary expansion took place over many e-foldingtimes before the huge energy release takes place. An excellent introductionto these concepts and the changing perspective on the inflationary picture ofthe early Universe is contained in the book Cosmological Inflation and Large-Scale Structure by Liddle and Lyth (Liddle and Lyth, 2000). Many differentversions of the inflationary picture of the early Universe have emerged, anamusing table of over 100 possibilities being presented by Shellard (Shellard,2003).

As a result, it cannot be claimed that there is a physical theory of the in-flationary Universe, but its basic concept resolves some of the basic problemslisted in Sect. 20.1. What is also does, and which gives it considerable appeal,is to suggest an origin for the spectrum of initial density perturbations asquantum perturbations on the scale of the particle horizon and that is thetopic we deal with next.

20.5 The Origin of the Spectrum of PrimordialPerturbations

In many ways, the story of inflation up to this point has been remarkablyphysics-free. All that has been stated is that an early period of rapid ex-ponential expansion can overcome a number of the fundamental problemsof cosmology. The next step involves real physics, but it is not the type ofphysics familiar to most astrophysical cosmologists. The key role is playedby scalar fields, which have quite different properties from the vector andtensor fields familiar in electrodynamics and general relativity. These fieldsare, however, common in theories of particle physics and so the particle the-orists are well prepared to take on the problem of putting real physics intothe inflationary paradigm.

The reason these ideas have to be taken seriously is that they suggest aremarkably natural origin for the spectrum of primordial perturbations withspectrum close to the Harrison-Zeldovich spectrum. The theory also makesquantitative predictions about the intensity and spectrum of primordial grav-itational waves which are accessible to experimental validation. According toLiddle and Lyth,

Although introduced to resolve problems associated with the ini-tial conditions needed for the Big Bang cosmology, inflation’s lastingprominence is owed to a property discovered soon after its introduc-tion: It provides a possible explanation for the initial inhomogeneitiesin the Universe that are believed to have led to all the structures we

20.5 The Origin of the Spectrum of Primordial Perturbations 637

see, from the earliest objects formed to the clustering of galaxies tothe observed irregularities in the microwave background.

There are now several recommendable books on this subject (Liddle andLyth, 2000; Dodelson, 2003; Mukhanov, 2005). For the standard ‘cosmologistin the street’, these do not make for particularly easy reading, largely becausethe reader must feel comfortable with many aspects of theoretical physicswhich lie outside the standard tools of the observational cosmologist. Theymay once have been understood as examination requirements in theoreticalphysics, but they rarely appear in the standard astrophysical literature –ladder operators, quantum field theory, zero point fluctuations in quantumfields, all of these developed within the framework of general relativity. Havingbattled with various degrees of success with the above books and many others,I found the essay by Baumann to be the most straightforward and accessibleaccount of the physical content of the theory (Baumann, 2007) – what followsis a ‘vulgarisation’ of his presentation. Developing the theory of the quantumorigin of density perturbations in detail cannot be carried out with modesteffort and is far beyond the ambitions of the present exposition.

Let us list some of the clues about the formulation of a successful theory.

20.5.1 The equation of state

We know from our analysis of the physical significance of the cosmologicalconstant Λ in Sect. 7.3 that exponential growth of the scale factor is foundif the dark energy has a negative pressure equation of state p = −%c2. Moregenerally, inspection of (7.1) shows that exponential growth of the scale factoris found provided the strong energy condition is violated, that is, if p <− 1

3%c2. To be effective in the very early Universe, the mass density of thescalar field has to be vastly greater than the value of ΩΛ we measure today.

20.5.2 The duration of the inflationary phase

In the example of the inflationary expansion given in Sect. 20.4, we arbitrarilyassumed that 100 e-folding times would take place during the inflationaryexpansion. A more careful calculation shows that there must have been atleast 60 e-folding times and these took place in the very early Universe,much earlier than those which have been explored experimentally by particlephysics experiments. It is customary to assume that inflation got seriouslyunderway not long after the Planck era, but there is quite a bit of room formanoevre.

20.5.3 The shrinking Hubble sphere

If these precepts are accepted, there is a natural way of understnading howfluctuations can be generated from processes in the very early Universe. It

638 20 The Very Early Universe

Fig. 20.2. (a) A repeat of conformal diagram Fig. 12.2(c) in which conformal timeis plotted against comoving radial distance coordinate. Now, the last scatteringsurface at the epoch of recombination is shown as well as the past light cone fromthe point at which our past light cone intersects the last scattering surface. (b) Anextended conformal diagram now showing the inflationary era. The time coordinateis set to zero at the end of the inflationary era and evolution of the Hubble sphereand the past light cone at recombination extrapolated back to the inflationary era.

20.5 The Origin of the Spectrum of Primordial Perturbations 639

is helpful to visit again the conformal diagrams for world models which werediscussed in Sect. 12.2, in particular, Fig. 12.2c. Recall that these diagrams areexact in the sense that the comoving radial distance coordinate and conformaltime are worked out for the reference model with Ω0 = 0.3 and ΩΛ = 0.7,the units in the abscissa being in c/H0 and the ordinate in H−1

0 . The effectof using conformal time is to stretch out time in the past and shrink it intothe future.

There are two additions to Fig. 12.2c in Fig. 20.2a. The redshift of 1000 isshown corresponding to the last scattering surface of the Cosmic MicrowaveBackground Radiation. The intersection with our past light cone is shownand then a past light cone from the last scattering surface to the singularityat t = 0 is shown as a shaded triangle. This is another way of demonstratingthe horizon problem – the region of causal contact is very small comparedwith moving an angle of 180 over the sky which would correspond to twicethe distance between the origin and the comoving radial distance coordinateat 3.09.

Let us now add the inflationary era to Fig. 20.2a. Baumann’s insights arehelpful in constructing Fig. 20.2b. He makes the important point that it isuseful to regard the end of the inflation era as the zero of time for the standardBig Bang and then to extend the diagram back to negative conformal times.In other words, we shift the zero of conformal time very slightly to, say, 10−32

s and then we can extend the light cones back through the entire inflationaryera.

This construction provides another way of understanding how the infla-tionary picture resolves the causality problem. The light cones have unit slopein the conformal diagram and so we draw light cones from the ends of theelement of comoving radial distance at t = 0 from the last scattering surface.These are shown in the diagram and it can be seen that projecting far enoughback in time, the light cones from opposite directions on the sky overlap, rep-resented by the dark grey shaded area in Fig. 20.2b. This is another way ofunderstanding how the inflationary picture results in causal contact in theearly Universe.

There is, however, an even better way of understanding what is going on.We were at pains to distinguish between the Hubble sphere and the particlehorizon in Sect. 12.2, but now this distinction becomes important. The parti-cle horizon is defines the maximum distance over which causal contact couldhave been made from the time of the singularity to a given epoch. In otherwords, it is not just what happened at a particular epoch which is important,but the history along the past light cone. In contrast, the Hubble radius is thedistance of causal contact at a particular epoch. It is the distance at whichthe velocity in the velocity-distance relation at that epoch is equal to thespeed of light. As shown in (12.29), the Hubble sphere has proper radius

rHS =ac

a. (20.3)

640 20 The Very Early Universe

Writing the exponential inflationary expansion of the scale factor as a =a0 exp[H(t − ti)], where a0 is the scale factor when the inflationary expan-sion began at ti, rHS = c/H and the comoving Hubble sphere has radiusrHS(com) = c/(Ha). Since H is a constant throughout most of the infla-tionary era, it follows that the comoving Hubble sphere decreases as theinflationary expansion proceeds.

We now need to join this evolution of the comoving Hubble sphere ontoits behaviour after the end of inflation, that is, join it onto Fig. 20.2a. Theexpression for conformal time during the inflationary era is

τ =∫

da

aa, (20.4)

and so, integrating and using the expression for rHS(com), we find

τ = constant− rHS(com)c

. (20.5)

This solution for rHS(com) is joined on to the standard result at the end ofthe inflationary epoch, as illustrated in Fig. 20.2b. The complete evolution ofthe Hubble sphere is indicated by the heavy line labelled ‘Hubble sphere’ inthat diagram.

Fig. 20.2b illustrates very beautifully how the inflationary pardigm solvesthe horizon problem. It will be noticed that the point at which the Hubblesphere crosses the comoving radial distance coordinate of the last scatteringsurface, exactly corresponds to the time when the past light cones from op-posite directions on the sky touch at conformal time −3. This is not a coinci-dence – they are different ways of stating that opposite regions of the CosmicMicrowave Background were in causal contact at conformal time t = −3.

But we learn a lot more. Because any object preserves its comoving ra-dial distance coordinate for all time, as represented by the vertical lines inFig. 20.2b, it can be seen that, in the early Universe, objects lie within theHubble sphere, but during the inflationary expansion, they pass through itand remain outside it for the rest of the inflationary expansion. Only whenthe Universe transforms back into the standard Friedman model does theHubble sphere begin to expand again and objects can then ‘re-enter the hori-zon’. Consider, for example, the region of the Universe out to redshift z = 0.5which corresponds to one of the comoving coordinate lines in Fig. 20.2b. Itremained within the Hubble sphere during the inflationary era until confor-mal time τ = −0.4 after which it was outside the horizon. It then re-enteredthe Hubble sphere at conformal time τ = 0.8. This behaviour occurs for allscales and masses of interest in understanding the origin of structure in thepresent Universe.

Since causal connection is no longer possible on scales greater than theHubble sphere, it follows that objects ‘freeze out’ when they pass throughthe Hubble sphere during the inflationary era, but they come back in again

20.5 The Origin of the Spectrum of Primordial Perturbations 641

and regain causal contact when they recross the Hubble sphere. This is oneof the key ideas behind the idea that the perturbations from which galaxiesformed were created in the early Universe, froze out on crossing the Hubblesphere and then grew again on re-entering it at conformal times τ > 0.

Notice that, at the present epoch, we are entering a phase of evolutionof the Universe when the comoving Hubble sphere about us has begun toshrink again. This can be seen in the upper part of Fig. 20.2b and is entirelydue to the fact that the dark energy is now dominating the expansion and itsdynamics are precisely another exponential expansion. In fact, the Hubblesphere tends asymptotically to the line labelled ‘event horizon’ in Fig. 20.2a.

20.5.4 Scalar Fields

As Baumann notes, there are three equivalent conditions necessary to producean inflationary expansion (Baumann, 2007):

• The decreasing of the Hubble sphere during the early expansion of theUniverse;

• An accelerated expansion;• Violation of the strong energy condition, specifically, p < −%c2/3.

How can this be achieved physically? It is simplest to quote Baumann’s words:

Answer: scalar field with special dynamics! Although no fundamentalscalar field has yet been detected in experiments, there are fortunatelyplenty of such fields in theories beyond the standard model of particlephysics. In fact, in string theory for example there are numerous scalarfields (moduli), but it proves very challenging to find just one withthe right characteristics to serve as an inflaton candidate.

At this point, I simply quote the results of calculations of the propertiesof the scalar field φ(t) which is assumed to be homogeneous at a given epoch.There are a kinetic energy φ2/2 and a potential energy, or self-interactionenergy, V (φ) associated with the field. Putting these through the machineryof field theory results in expressions for the density and pressure of the scalarfield:

%φ =12φ2 + V (φ) (20.6)

pφ =12φ2 − V (φ) (20.7)

Clearly the scalar field can result in a negative pressure equation of state,provided the potential energy of the field is very much greater than its kineticenergy. In the limit in which the kinetic energy is neglected, we obtain theequation of state p = −%c2, where I have restored the c2 which is set equalto one by professional field theorists.

642 20 The Very Early Universe

To find the time evolution of the scalar field, we need to combine theproperties (20.6) and (20.7) with the Einstein equations. The results are

H2 =13

(12φ2 + V (φ)

)(20.8)

φ + 3Hφ + V (φ),φ = 0 . (20.9)

Thus, to obtain the inflationary expansion over many e-folding times, thekinetic energy term must be very small compared with the potential energyand the potential energy term must be very slowly varying with time. This isformalised by requiring the two slow-roll parameters ε(φ) and η(φ) to be verysmall during the inflationary expansion. These parameters set constraintsupon the dependence of the potential energy function upon the field φ andare formally written:

ε(φ) ≡ 12

(V,φ

V

)2

; η(φ) ≡ V,φφ

Vwith ε(φ), |η(φ)| ¿ 1 . (20.10)

Under these conditions, we obtain what we need for inflation, namely,

H2 =13V (φ) = constant and a(t) ∝ eHt . (20.11)

At this stage, it may appear that we have not really made much progress sincewe have adjusted the theory of the scalar field to produce what we know weneed. The bonus comes when we consider fluctuations in the scalar field andtheir role in the formation of the spectrum of primordial perturbations.

20.5.5 The Quantised Harmonic Oscillator

The key result can be derived from the elementary quantum mechanics of aharmonic oscillator. The solutions of Schrodinger’s equation for a harmonicpotential have quantised energy levels

E =(n + 1

2

)hω (20.12)

and the wavefunctions of these stationary states are

ψn = Hn(ξ) exp(− 1

2ξ2)

, (20.13)

where Hn(ξ) is the Hermite polynomial of order n and ξ =√

βx. For thesimple harmonic oscillator, β2 = am/h2, where a is the constant in theexpression for the harmonic potential V = 1

2ax2 and m is the reduced massof the oscillator. Then, the angular frequency ω =

√a/m is exactly the same

as is found for the classical harmonic oscillator.We are interested in fluctuations about the zero-point energy, that is, the

stationary state with n = 0. The zero-point energy and Hermite polynomialof order n = 0 are

20.5 The Origin of the Spectrum of Primordial Perturbations 643

E = 12 hω and H0(ξ) = A = constant . (20.14)

The first expression is the well-known result that the oscillator has to havefinite kinetic energy in the ground state. The underlying cause of this is theneed to satisfy Heisenberg’s uncertainty principle.

Part of the package of quantum mechanics is that there must be quantumfluctuations in the stationaary states, again because of the need to satisfyHeisenberg’s uncertainty principle. It is straightforward to work out the vari-ance of the position coordinate x of the oscillator. First, we need to normalisethe wavefunction so that

∫ +∞

−∞ψψ∗ dx = 1 . (20.15)

Since (20.13) is real, it is straightforward to show that

ψ =(

am

h2π2

)1/8

exp(− 1

2ξ2)

. (20.16)

To find the variance of the position coordinate of the oscillator, we form thequantity

〈x2〉 =∫ +∞

−∞ψψ∗ x2 dx . (20.17)

Carrying out this integral, we find the important result

〈x2〉 =h

2√

am=

h

2ωm. (20.18)

This result is identical to that derived by Baumann who sets the particlemass m = 1 ‘for convenince’. The reason for this is that the analogy with thenext part of the calculation is clearest for the case of an oscillator with thismass. In this case, we find

〈x2〉 =h

2ω. (20.19)

These are the fluctuations which must necessarily accompany the zero-pointenergy of the vacuum fields.

This elementary calculation sweeps an enormous number of technical is-sues under the carpet. Baumann’s clear presentation of the proper calcula-tion, which involves the definition of the action, the introduction of canonicalquantisation and of creation and annihilation operators and so on, as well asworrying about the issues of applying the formalism in curved space-time,can be warmly recommended. It is reassuring that his final answer agrees ex-actly with the results (20.12) and (20.19) for the one-dimensional harmonicoscillator.

644 20 The Very Early Universe

20.5.6 The Spectrum of Fluctuations in the Scalar Field

We have almost gone as far as is reasonable without becoming involved inseriously heavy calculation. We need only one more equation – the expressionfor the evolution of the vacuum fluctuations in the inflationary expansion.The inflaton field is decomposed into a uniform homogeneous backgroundand a perturbed component δφ which is the analogue of the deviation x ofthe zero point oscillations of the harmonic oscillator. We need to work outthe spectrum of these fluctuations and so we consider the amplitude of theperturbation associated with a particular wavenumber k, δφk. If k is takento be the comoving wavenumber and λ0 the wavelength at the present epoch,the proper wavelength of the perturbation is λ = aλ0 ∼ a/k and the properwavenumber at scale factor a is kprop = k/a. Then, the evolution of δφk isgiven by the differential equation

¨δφk + 3H ˙δφk +k2

a2δφk = 0 , (20.20)

where H = a/a. The derivation of this equation is outlined by Baumann,who also warns of the many technical complexities which need to be dealtwith in a rigorous treatment.

The equation (20.20) bears a strong resemblance to (11.24) and (11.75),which we derived for the case of the density contrast ∆ of non-relativisticand relativistic material respectively in the standard world models. We recallthat (11.24) reads

d2∆

dt2+ 2

(a

a

)d∆

dt= ∆(4πG%0 − k2c2

s ) ,

where k is the proper wavenumber and cs is the speed of sound. The analogyis very close when we realise that the dynamics are dominated by the vacuumfields and so we can neglect the term in G in the right-hand side. The speedof sound cs is the speed of light, which according to Baumann’s conventions,is set equal to unity in (20.20). One might worry that ∆ refers to the densitycontrast rather than to δ%, but, because of the assumption that the equationof state is very close to p = −%c2, the density % is a constant and so thesimilarity between the equations is really remarkably close. In fact, the onlyreal difference is the factor 3, rather than 2, in the damping term in H. Thus,the evolution equation for δφk has a really familiar look. The big advantageof (20.20) is that it can also be applied on superhorizon scales as well as forthose within the horizon.

We recognise that (20.20) is the equation of motion for a damped har-monic oscillator. If the ‘damping term’ 3H ˙δφk is set equal to zero, we findharmonic oscillations, just as in the case of the Jeans’ analysis of Sect. 11.3.On the other hand, for scales much greater than the radius of the Hubblesphere, λ À c/H, an order of magnitude calculation shows that the damping

20.5 The Origin of the Spectrum of Primordial Perturbations 645

term dominates and the velocity ˙δφk tends exponentially to zero, correspond-ing to the ‘freezing’ of the fluctuations on superhorizon scales.

We now use the results of Sect. 20.5.5. Both x and δφk have zero pointfluctuations in the ground state. In the case of the harmonic oscillator, wefound 〈x2〉 ∝ ω−1. In exactly the same way, we expect the fluctuations in δφk

to be inversely proportional to the ‘angular frequency’ in (20.20), that is,

〈(δφk)2〉 ∝ 1k/a

∝ λ , (20.21)

recalling that λ is the proper wavelength. Since λ ∝ a, the ‘noise-power’〈(δφk)2〉 increases linearly proportional to the scale factor until the wave-length is equal to the dimensions of the Hubble sphere when the noise-powerstops growing. Therefore, the power spectrum is given by the power withinthe horizon when λ = c/H, that is, when k = a∗H∗ where a∗ and H∗ arethe values of the scale factor and Hubble’s constant when the wavelength isequal to the radius of the Hubble sphere. Therefore, per unit volume, theprimordial power spectrum on superhorizon scales is expected to have theform

〈(δφk)2〉 ∝ 1a3∗(k/a∗)

∝ H2∗

k3. (20.22)

In the simplest approximation, H∗ = H = constant throughout the inflation-ary era. Now, (20.22) is the power-spectrum in Fourier space and to convertit into a real space power spectrum we need to integrate over wavenumber k.

〈(δφ)2〉 ∝∫ kf

ki

k2〈(δφk)2〉 dk ∝ H2 ln(

kf

ki

). (20.23)

Thus, we obtain the important result

〈(δφ)2〉 ∝ H2 . (20.24)

At the end of the inflationary expansion, the scalar field is assumed todecay into the types of particles which dominate our Universe at the presentepoch, releasing a vast amount of energy which reheats the contents of theUniverse to a very high temperature as illustrated schematically in Fig. 20.1.The final step in the calculation is to relate the fluctuations in φ to thosein the highly relativistic plasma in the post-inflation era. In the simplestpicture, we can think of this transition as occurring abruptly between the erawhen p = −%c2 and the scale factor increases exponentially with time, as inthe de Sitter metric, to that in which the standard relativistic equation ofstate p = 1

3%c2 applies with associated variation of the inertial mass densitywith cosmic time % ∝ H2 ∝ t−2 (see (9.7)). Guth and Pi introduced what itknown as the time-delay formalism which enables the density perturbationto be related to the inflation parameters (Guth and Pi, 1982).

The idea is that the presence of the perturbation in the scalar field δφresults in a time delay

646 20 The Very Early Universe

δt =δφ

φ. (20.25)

This should be evaluated at the time the fluctuation in φ is frozen in athorizon crossing. At the end of the inflationary era, this time delay translatesinto a perturbation in the density in the radiation-dominated era. Since % ∝t−2 and H ∝ t−1,

δ%

%∝ H δt . (20.26)

Since Hubble’s constant must be continuous across the discontinuity at theend of the inflationary era, and must have roughly the same value at horizoncrossing, it follows that

δ%

%∝ H2

∗φ∗

. (20.27)

This order of magnitude calculation illustrates how quantum fluctuations inthe scalar field φ can result in density fluctuations in the matter which allhave the same amplitude when they passed through the horizon in the veryearly Universe. They then remained frozen in until they re-entered the horizonvery much later in the radiation-dominated era, as illustrated in Fig. 20.2b.

This schematic calculation is only intended to illustrate why the inflationparadigm is taken so seriously by theorists. It results remarkably naturallyin the Harrison-Zeldovich spectrum for the spectrum of primordial perturba-tions. The above calculation is a gross simplification of the many complexitiesinvovled in the full calculation and these are nicely presented by Baumann(Baumann, 2007).

In the full theory of the origin of the perturbations, the values of thesmall parameters ε and η defined by (20.10) cannot be neglected and theyhave important consequences for the spectrum of the perturbations and theexistence of primordial gravitational waves. Specifically, the spectral index ofthe perturbations on entering the horizon is predicted to be

nS − 1 = 2η − 6ε . (20.28)

Furthermore, tensor perturbations, corresponding to gravitational waves, arealso expected to be excited during the inflationary era and their spectralindex is predicted to be

nT − 1 = −2ε , (20.29)

where scale-invariance corresponds to nT = 0. The tensor-to-scalar ratio isdefined as

r =∆2

T

∆2S

= 16ε , (20.30)

where ∆2T and ∆2

S are the power spectra of gravitational and matter pertur-bations respectively.

20.6 Baryogenesis 647

These results illustrate why the deviations of the spectral index of theobserved perturbations from the value nS = 1 are so important. They sug-gest that there may well be a background of primordial gravitational waves,as was discussed in Sects. 15.4.3 and 15.8.4. These are really very great ob-servational challenges, but they provide a remarkably direct link to processeswhich may have occurred during the inflationary epoch. To many cosmolo-gists, this would be the ‘smoking gun’ which sets the seal on the inflationarymodel of the early Universe.

Whilst the above calculation is a considerable triumph for the inflation-ary scenario, we should remember that there is as yet no physical realisationof the scalar field. Although the scale-invariant spectrum is a remarkableprediction, the amplitude of the perturbation psectrum is model dependent.There are literally hundreds of possible inflationary models depending uponthe particular choice of the inflationary potential. We should also not neglectthe possibility that there are other sources of perturbations which could haveresulted from various types of topological defect, such as cosmic strings, do-main walls, textures and so on (Shellard, 2003). Granted all these caveats,startling success of inflationary ideas in accounting for the observed spectrumof fluctuations in the Cosmic Microwave Background Radiation has made itthe model of choice for studies of the early Universe.

20.6 Baryogenesis

A key contribution of particle physics to studies of the early Universe concernsthe baryon-asymmetry problem, a subject referred to as baryogenesis. In aprescient paper of 1967, Sakharov enunciated the three conditions necessaryto account for the baryon-antibaryon asymmetry of the Universe (Sakharov,1967). Sakharov’s rules for the creation of non-zero baryon number from aninitially baryon symmetric state are:

• Baryon number must be violated;• C (charge conjugation) and CP (charge conjugation combined with parity)

must be violated;• The asymmetry must be created under non-equilibrium conditions.

The reasons for these rules can be readily appreciated from simple argu-ments (Kolb and Turner, 1990). Concerning the first rule, it is evident that,if the baryon asymmetry developed from a symmetric high temperature state,baryon number must have been violated at some stage – otherwise, the baryonasymmetry would have to be built into the model from the very beginning.The second rule is necessary in order to ensure that a net baryon number iscreated, even in the presence of interactions which violate baryon conserva-tion. The third rule is necessary because baryons and antibaryons have thesame mass and so, thermodynamically, they would have the same abundances

648 20 The Very Early Universe

in thermodynamic equilibrium, despite the violation of baryon number andC and CP invariance.

There is evidence that all three rules can be satisfied in the early Universefrom a combination of theoretical ideas and experimental evidence from par-ticle physics. Thus, baryon number violation is a generic feature of GrandUnified Theories which unify the strong and electroweak interactions – thesame process is responsible for the predicted instability of the proton. C andCP violation have been observed in the decay of the neutral K0 and K0

mesons. The K0 meson should decay symmetrically into equal numbers ofparticles and antiparticles but, in fact, there is a slight preference for matterover antimatter, at the level of 10−3, very much greater than the degree ofasymmetry necessary for baryogenesis, ∼ 10−8. The need for departure fromthermal equilibrium follows from the same type of reasoning which led tothe primordial synthesis of the light elements (Sects. 10.3 and 10.6). As inthat case, so long as the time-scales of the interactions which maintainedthe various constituents in thermal equilibrium were less than the expan-sion time-scale, the number densities of particles and antiparticles of thesame mass would be the same. In thermodynamic equilibrium, the numberdensities of different species did not depend upon the cross-sections for theinteractions which maintain the equilibrium. It is only after decoupling, whennon-equilibrium abundances were established, that the number densities de-pended upon the specific values of the cross-sections for the production ofdifferent species.

In a typical baryogenesis scenario, the asymmetry is associated with somevery massive boson and its antiparticle, X, X, which are involved in theunification of the strong and electroweak forces and which can decay into finalstates which have different baryon numbers. Kolb and Turner provided a cleardescription of the principles by which the observed baryon asymmetry can begenerated at about the epoch of grand unification or soon afterwards, whenthe very massive bosons can no longer be maintained in equilibrium(Kolb andTurner, 1990). Although the principles of the calculations are well defined,the details are not understood, partly because the energies at which theyare likely to be important are not attainable in laboratory experiments, andpartly because predicted effects, such as the decay of the proton, have notbeen observed. Thus, although there is no definitive evidence that this line ofreasoning is secure, well-understood physical processes of the type necessaryfor the creation of the baryon-antibaryon asymmetry exist. The importance ofthese studies goes well beyond their immediate significance for astrophysicalcosmology. As Kolb and Turner remark,

. . . in the absence of direct evidence for proton decay, baryogenesismay provide the strongest, albeit indirect, evidence for some kind ofunification of the quarks and the leptons.

338 ll Inflcrtiottar.l'cosrrro/ogl'

ln this scnse. especialll'rvitl.r 1l.re appearance ot'the Planck scale as the miuitnum requrreficld yalr-rc. it is not clcar tl'rat thc aim of r-ealizins inflation in a classical wav distin.fiom quantum gravity'h:rs bcen 1ìlfillcd.

11.5 Relic fluctuations from inflation

N,IOTIVATIO\ We harte seen that de Sitter slace crìutirirls l tnle evcnt horizon.propcr size t'f II . This suggcsts tl.rat there u'ill be thclmal fluctuations plesent. irs uiLblack hole. tbr rihich the Halvking temperature is /iT,, - Ì1t'l(4nr,\. This ar.ralo-ey is clL',

but imperlèct. ar.rd tl.re characterisLic temperature o1'dc Sittcl space is a tìLctor'2 hi,rrh;

/,I,1..i"., : 'lf tl ì :

_ft

(scc chzrptcr 8). This cristcnce of thcrmal fluctuations is one piecc olintuitire tnoli\rrlbl erpecting fluctuations in the cluantLrn.r fields th:rt ale pl'esent in c1e Sittel sp,,.

but is not so usefìrl in detail. Tn practice" rve need a nìorc basic calculatior.r. ol L

the zero-point fluctnations in small-scale quantum moclcs lieeze out as classic:rÌ clel,flLrctuations once tire modes have been inflated to super-horizcl.r scllcs.

The details ol this calculation are gi"'cn belorv. Horvever. \\rc can ir.nmedr., .

nole that a natulal predicliou ri'ill be iì spectnnx ol perturbations that are nearlr ..irtrtriuut. This means that tl-re metric fluctuations ol spacetirnc receive eclual lerc.'distortion fi'om each decade ol pertr,rrbation 'nvavelen-eth. rLnd r.nar, be quantificd in r:of'the rr.r.rs lluctuations. o. in Nen'tonian gravitational potcntial. tD (i' : 1):

. ì r/ol(tD),)- - A.

,1 lnA - uorì\l:rlll.

Thc notatior.r ò,, ariscs bccause tÌrc potential peltr-rlbation is ol the same oLder- -,

dcnsitl,'fluctuatior.r on thc scalc of thc horizon rt lnv siler timc (see chiìpter 15).

It is cornmonlv alguccl that thc prcdiction o['scale invariance ar'ìses bectL.,Sittcl spacc is inr,ariant r:nder time translatior-r: therc is no natural origin ol tinre ..

erponenlial expansion. At a given tin,e. the only length scale in thc moclel is the hi'size L'lH. so it is inel'itable that thc fluctuations that erisl on this scitle are the srr:

all tin-res. Aficr inflation ccascs. thc rcsulting flì.rctLruti'rrìs (i-rt cLrll\1rìrì1 amplitucle.scalc olthc holizon) gir,c us thc Zeldovich or scale-invariant spcctlum. The plobler:-this ar,qur.nent is tl.rat it ignores the issLre ol- hou' the perturbations cr,olve ri'l.rilc th:outside the holizon:',ve havc onl1'r'eally'calculatecl the amplitude fbr the last gen.' .

olflr-rctr-rations i.e. those that ale on the scale of the horizor.r i,Lt the time inflatir.r: .

Fhrctuutior.rs gcneratcd at eallier times nill be inflatcd outsidc tl.re cle Sitter horiz..lvill re-entel the FRW holizon at solne time a1ìel inllation has ceased.

The evolution durine this period is a topic where some care is nccdccl. sjr,descliption ofthese large-scale pertnrbiìtions is sensitive to the gauge h'cedom in ..relatir,itl'. A technìcal discussi,'rn i: giren ìn e.g. Nfukhanov. Feldman & Brander:.(1991). but thcre is uo space to do thisjustice l.rere. Ciiaptcr' 15 plesents a f iÌthilintuitive discussior-r olthe gaLrge issue:lcrr the plesent. u'e shall r'el1'on simplv mor.the inflationarv rcsult. rvhich is that potcntiai pcrtrLlbations re-entcr the holrz,'thc sar.nc amplitude thcl'had on lcaving. This rnal'be macle leasonable in trt.

,rLrbatictr-rs outsic, any lar-ue-scalc

-iro\\r or declinc.:urbations preser.be ne-ulected an

-.."- that the inflar, :calc-invariant,og length-scale.

.,rrbations that a

.zon evolution isiÌ'thc expansion

-: torvards thc enr-:;tcd.

To anticipate- rrntplitude

.r can be unclers. .lilTcrent parts r

l otds. tve are c

-.Ì at ciillereut tir

Lrrtrverses rvill tl,ts (fìgr-re 11.,11

It-. 11131 the unir

.hc last step Llse

_:rren by É112n.

.:;nsional gronnr

LL'C]TUATION SI

-:;tail (sce Lidcll...rtiou of motior

Lrn iS

,. seek the colre:: rrith slig1irl1, c

.:.- tbnn ol- a cor..Ìc !: dd : ,l g

may be treatec

repro-tsrg oqt sfoqo .lg ptrU peqrnlred oql ueqt '1uu1suoc e s€ peteorl eq ,eur ,/ leqlos 'perrnsse osle ere suorlrprioc llor-,^ó,ols eql y'(nllry - x. rp)dxe y: Q?:y epnlrlduepuu ) roqrunue,re,tt Sur,r.oruoJ Jo uorl€qjn1:ed e,te,u-eue1d Sur,rouoc e Jo tuto-J eql soìetuorleqrnlred srql esoddnS 'seceld ]uo.r3Jtp ul Ó Jo sanle^ luoreJtp .11q3r1s qli.\ uoreuur3ur1;e1s,q peurelgo @g uorleg.rnl;ed eql rog uorlunbe Surpuodserroc oql ìeos e,\ pur

(l7r r)'o : kil,A -t Q en - Q at +,1,-

sr Lrorloru -Iouorlenbe crs?q aqJ- 'plou uolr?gur eLJl ur suorleqJnlred ,,(q pofeqo uorlo(u go uorlenbe aqrreprsuoc'1srrg (tueurl€art leelc,(1-reincrl.red e ro.l 866 I r1il.1 ry elppr1 ees) Irutop eroulreqler uI llnsor [u]r^ srrll ro^o o3 o1 peeu,\ou eIA W1àIJEdS No[vnIJnId EI,t

'(e;nleredurel Jo suorsuolurp oql seq plog eql 's1run 1e;n1eu ur) spunor8 leuorsueurp uoelqeuoseor ,(lelurpourutr sr lr ]nq ',\olèq pa^rrop eq lll.^A llnser srqf .xZ/

H [q ue,rr8 sr Er,srur orll ]eql s,es qcrq,tr 'froeql pleg unluenb 3o lndut Ietcntc eql sesn <ie1s 1se1 eq1 orsq \

's\È,r\ o,\l ur eJqEr

lttr\ uozrroq eql Iiurle,trlour fldrurs:loru JèqleJ u sluorai-requepuerg ry

r

'l:aue8 ur ruopoor.J

:i1 Ocurs 'pepeou s

'truP ruozrJoq Jellrsspua uorleuur eujrlrotterouo8 lsEI eql::e ,eql olrq,r o^[o

1rr.r urelqord eqg .

:rt uo èpnlrldure 1r

:E O[res 0q1 oJB eleIrlzIJot Oql Sl Iepot:rpun olut] 3o ur8rr,:p Osneceg sosrJ€ o

'(91 reldeqc I

:ql ss ieplo eruEs

-r I r)

.( I :t:u.loi ur pegrluenb r

,r'r slèAOI lenbe e,tre:

:-rus f1;eau ele leqrleterparurur uec e,

.SOI

'.rrsuep iecrsslllc se l'^ori Jo 'uor]€lncluc:ceds .re11tg ep urjùrl?^rloru 0ArlInluI

9t ì I)

.roqSrqZrolcuJ€sr':solJ sr ,(6o1eue srq.l: qlr,r\ sc 'lusserd su

-Lr 'uozlJotl luo^e èt

:Jurlsrp,u,u lecrssel:Do:rnbor runullurru oL

(orr r)

:uorl€gur Jo pue eql 3ur.uo1103 pepuedxe o^eq sosJe^run erll 1€g] slunourlluareJrp oq1 .(g ue,tr8 sr epnlldrue,(lrsuep a1eJs-uozrroq aqJ_.(t.II ern8g) serlrsu:pf3reue ur peerds E 01 Surpeal 'seurr] JuoreJ]p le uorleuul qslug ueqt IIr,\ sosre^run osoqf

(oE rr)aEA

seurl luero.lJrp le po,\at 1

lnq '(l)0 -rnoi,teqeq-Surllor orues eql go serdoc snorr€^ qtr.u Buqeep ore e,\A, ,spro.u ;eqltu1 @q lunorue ue.rq peqrnged olt leql spleg e^eq esro^run eql;o sged uereJrp oìeuroJ sr suor]€nlcnu Jo ]coJe ureu aql €qt eur8eul 's,\olloJ su poolsropun oq uec qcrq\\

(sE rr)

epnlrldure elers-uozrJorl e Jo st uorlcrperd freuorlegur oql 'luourluell polrulep eql eledrcilue o1

'pelcedr-'0q ol lou sr Ocu€rr€^ur Oleos lc€xo leq] r€alc sr Jr os .uorleuur

Jo puo eql spJe,rol lsl\:op flgrsneyd suorlerlep qcns ]eql uoes o^eq e,u ilerlueuodxo-uou si uorsuedxe eqj3r,1u.-ocu€IJe^ul olecs Jo uotlctpe.rd sltl] Jell€ o1 elqrssod sr lI 'onssr ue lou sr rrorlnlo^o uozrJou-:edns leq] os 'uorlegur Jo pue oq] l€ uozrroq eql Sur,reel lsnl er€ leql suorteqrnl:adaql o'r slecs euo uo opnlrldue l€q] olelncl€c ol soculns uèql 1I 'e1ecs-q13ue1 3o1 rad,ss0urllurJ,r\, otues oql s€q teql crJJOru B o1 puodser:oc suorlenlcnu luerJe^uf-olecs leqlesuos eqì ur eìll-lelce4 sr leql osre^run e secnpord ssecord,.reuorlegur eql leq] en8:rololeJorll ol& 'uozrroq sq1 ol spuodserJoc elecs IpcrlrlJ srql l€ql pue pelce18eu eg u3--slJoJo eJnsse.rd ereq,u sèl€îs uo e,re ,eql 1r:q1 pepr,tord 'enl€^ lreq e,tteserd suorleqrnl:edierluelod Ilelus lEql 91 relduqc ur ,\oqs IIErs e,\ ',(11eruro3 ero] I .eutlcep ro .,\\or8 orpesoddns se,\\ lr Jeqloq,\ ,,\\ouì, plnoc eurloceds ur sseuluH-uou ol€os-o8re1 ,(ue ,uoq00s oi pJeLI sr Jr os 'slceJo l€sn€c ol ounlutur oJB uozrroq eql eprslno suorl€qJnuèd

6î.tuotlog[ut uLo.tf suoqnnlnl/' c4ay g'y 1

óuz

i:rgll="9

340 1 1 Infationurl, cosmologl,

Figure 11.4. This plot shows how fluctuations in the scalar field translormthemselr,es into density fluctuations at the end ol inflation. Differentpoints ol the universe inflate from points on the potential perturbedby a fluctuation òry', like two ba1ls rolling from different starting points.Inflation finishes at times separated by ór in time fbr these two points,inducing a density fluctuation ò : Hòt.

perturbation of the equation of motion for the main fie1d:

taól + 3Hlò4)l + (kla)21òó):0, (tl.ctwhich is a standard wave equation for a massless fìe1d evolving in an expanding universe.

Having seen that the inflaton perturbation behaves in this way, it is not muchwork to obtain the quantum fluctuations that result in the field at late times (i.e. on scalesmuch larger than the de Sitter horizon). First consider the flrictuations in flat space: thefield would be expanded as

tt'1 : o1ra1. -l af a!r,

and the field variance would be

(ol dr12 lo) : ,r,i2

To solve the general problem, we only need to find how the amplitude (rÀ changes as

the universe expands. The idea is to start from the situation where we are well inside thehorizon (kla>> Il), in which case flat-space quantum theory will apply, and end ar thepoint of interest outside the horizon (where kla << H).

Before finishing the calculation, note the critical assumption that the initial staieis the vacuum:in the modes that will eventually be relevant for observational cosmolo,s\,-we start with not even one quantùm of the inflaton field present. ls this smuggling finetuning of the initial conditions in again through the back door? Given our ignoranceol the exact conditions in the primordial chaos from which the inflationary phase is

supposed to emerge, it is something ol a matter ol taste wl.rether this is seen as beinga problem. Certainly, if the initial state is close to equilibrium at temperature I, this i..

easily understood, since all initial scales are given in terms of 7. In natural units. theenergy density tn V($) and radiation will be - Ia, and the proper size of the horizl.nwill be - T-2.In the initierl state, the inflaton occupation number will be 1/2 for veqr

long wavelengths, and will lall for proper wavelengths : T-t .Now, remember that fis in units ol the Planck temperature, so that T - 10 a for GUT-scale inflation. Tharmeans that perturbations of scale smaller than T times the horizon would start n'irl

(11.4-1i

( 1 1.4+,

l-0forathermaoccupation numbertravelength. Thus, a.remains inside the hrmotivation lbr a theof energy density. Ifdo this il the inflatoradiation density woihat inflates as being

It is also worin principle affect thrandom-walk elemenby V(rhl In cases whtlong, it is possible forinflation in a self'-sus1986, 1989) [probtem

Returning nov.-hanges as the wavelrr;sult from chapter 7,

fhe powers ol the sc

;avenumbers /<. TheI pfoper volume; hen.roking at this is thar I

:rpands. With this bor:he lollowing expressio

::member that È1 is a c-re horizon is much lar-:strll. ercept that the r

i:.]ce / lt |k uH 1 : _,: r ol the desired /<r/a-

.t'l -'. so .r is effectir e,i -rctuations dominate.

At the opposite" lr at the value

- ìe initial quantum ze:_nstant classical fluctu:-ie fluctuations in ry' de: : nstant :

d(òd1

( 8t'il )

ere :pEcsp .roci suorlunlcllu eqÌ lcql iE,r 1l _itl.ts ur )'3.r l1l Jr1.r])' clll Js -03.r'u J sll -+losì r lssj [rc ut,(1ien lue,ro neru ol poqt,nsrrujÌ rreoq 0^uLI plag orll ur suorlllllltllu

- o - rt1, o J1') -- Y LII /'

,\xl -tdrt'l/'f "c \

.\ ru i:JUlllSUrra

uo puecl:p rfi ur suorlunlJtip erilÌEq1 uorlr:rrucnu Iustssllls lLmlsrio.lurod-o,rez runluunb [Btlrur orl]

,Jll\\.l-rEls pltlo^\.l]rll LrollEllur 0i

/, ltrll roqluoltle.ra,r .ro1 Z/l "q I

,,trzrJoLÌ 0ql Jo 3t

rtlÌ'slru1't Iu.rltlPL-l slql 'J erutclo:iureq su uoJs st

:r è:^trld ruuorr,:liu.rotr8r ino r-re

:uq ;-u1133nrus .^ir

.ioIitmsoc 1utro11:l1rÌs Iur]tril èrlj lr

.:L1t 1u pur puu " lLIl sprslrr [Jer\\ 0-n

.r 53f9p113 '/6.1 3p

rr' l t.)

jir)

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..ì[1]JS uo 'o'l) sourr

. lrìr_u lou sr 1r 'itr!.ia,\rurl 3ur;luedr

"r'rr)

'slrrrcld r.t.n.y

'slurod Surlpcc1,rn1:ed

Irrè,reJrc 'r

LLrJOlSLrU,r'l p

(tv'tt)

01tlt?r Jr[l 11? t11irrczo-rJ soruorecl cpnlrldr-uÙ uorlllrllJliu eLIÌ'l << 1/IJreurejlre olrsoclclo oql .IV

'O.lllLrrlrrop sLrorlù nlan -[.

unluunb e,rcq,'n qrode cqÌ-lo,\\or^.10 lurocl sLJl ruor.+ lrmrsrros u,[1o,tr1legr sI, os.r l1 >>sI uorll?[r]so.;o por,rccl eql 'serilJ r(i.rue ,,rer.o lllxil èql Lrl .LU,ro.

/,,/ty pejrsrp orl.+o j lr.eouepuedcp Éurpuel u slrr[ ttr.ràl i.lirIrlllrsrr .1rll tl]rll ers:,\\'rl_y - (pD1l\tp/p) cJtrr.'ie.\c.\\oH '(11,/t[-)clxa iUmq'pp. el]]rl B sìool:orrcpueclcp orurl Jill iuql 1clcrxc,11ns:..eruds-]r:g oql sr )úr os purr >> .liftfi'qr;uelc,\u.\\ rq] uurl.r reir.,1 rJc1Ll sr rrozr.Tolf orl-ucq'rr'seturl '1'reo Ìy'.,!e -HD - t2H : (HnlQplp) l1lrrl .,s'ru'rsLroc':^r H l'rr.rscrLlrorLr:.1

(9t t r):(Ii Ht)tI Il 11,,,,11 a, , (nlI-) _ , D:1i)

:uctrlunbc uorlt1lolc orll sJtIst]cs uctrsse-rcjrc iur_,no1111 eq:llrlll Lrorlnlrrsqns iq ìc3rlc ol p,rr:,tro3iq. rr:;1s 1r'uorlrpuor,(.rupuuoq srql qllÀ\.:^pLllcl\:ès'I3r\tull etll sll . ir se seoÉ suollluul.lo .lrsrrep -teqlLillu -redord eql leLll sr s1l1 lg Fur16o-3o .(e'u rcrllouv 'l: ,1 Sur,ror-r-roo esn èr\ -Jr '.rotru.I -.- u oqr oJLroLT :JrunJ.,r -reclo.id l.

!u1eq I ; r ,l .lo.roÌJrr3 iurzrluru.lou 1l srrrllluor eptì1tl,lunr plcu eL1 .l s-roqurnuo,\11 \,,8ur'rouroc ul plo!I èr[] F'rprrilclxJ,ro-i,\\oJI1l tsnl'(i)r,.,o1.rr1 .1,'r... aql -Jo s.,eoo,l eq1

#-(0,)d,o)

(st'r I)

ecr:ds-1ug

epnlrldr-uu

,,,y1 d, r QllZ)-, t) : \1i1

sll srrLrll Srr,rouroc ur Lrèlrr-r,\\e, cq uBJ r-lJrL[.o'l .re1c.J.:qc u]o-r[ ]lnsi.rJql e,rl?q e'''

"(1|:r1rU1 'uozr,roq eqr qSno-rql sessucl qr.3r-rs1o,\l].\\ cqì sr: so8uui1..

Jpou eql ,r\oLJ .roLrì ol lull,\\ e,rr .uorltlrlJluo orJl o.l ,uou iurn-ru1eg'[g 11 Luelqo.rd] (CSOt '986Ì

opull,) uollslJul JllsuqJols.lo ldecuoc crìt sr srrJl .ssaoo,rd iunrrulsls_11as u ur uotlpurl'l3Lil'In.Ì 'iul1t:a'tr lno -I3LIlln.+ rf qsncl ol sLrorlulrlJlll; urnluunb clll -roJ elqr>^socl sr 1r ,iu.iilluangns,roj tsrs-red ol uortutlut,rog ur8r-ro arll ol JsoJJ ool si rf.i èjarl,r soseJ Lrl .(rl)_,r ,rctpoLluop t13no-r1 3tl.l tl\\op plorl ,I1?luts eql .1o 3ur11o-r TuJrssr?Tc crll ol tuortrelo )llr\\-LLroplrll,ru Srrrppc su .1o lqinor-11 oq rruc fer-11 11cs1r Lrorttgrrr 1o sse.rfio.rcl oql lrr.Uu elclpur,rd urrmJ plaq JpTllJS srll ur suorlunJJnLI assLll ier11 Snrssr:d ur 3ur10u ql-ro,\\ oslu sr ll 'cJllls runlìJlll il s,reluo,,(Jpidr:l lllLIl eLro Fureq su so.lllgrrr ]t:qlsttlls Illlllul lttu luelÌ ol olquuosuo,l )^lllOcs SllLIl .lJ ri.re,rrp rrcqì pll1o,\\ 1rsuo1-r uor1,r,p,,.,o^r1co1lo oL[] osnBcocl ql8ueJc,re.,r oJOZ o.l u.,\\op lslxo suorllìr.ìlcnU .rolpgur oLfl lr srqt oplottLIuJ ]I put 'oJl?Lrrruop lsrlLrr (,1)l ,ttc ir.l ricdcluq ot sr uortllllur 31 r(lrsuep .,i-reu: -1pstulel ut Opultr èq ucc lutocl lllllLu olll tnq'ìco.'\\ sf olul:^ lellrul lllnr-rcrll u.ro.+ Lrorlu,\rloltr3ql'o:^lnoJ-JO ieqLunu uotludullo o-lcz pe.ttuber Jtll o,\Bq JIl,\\ Llozr.loq aLllJprsLtr sllpruc.r .llrLJl

opotu,t.re,re'(.ue1 u flrrr'r'r'r) u()rllrUrrr.:lrì siurplo.l-, (1111n1.rèuu,snqf .q1iuc1e,rr.1Sttl,touttll ue,rtF u.loJ luulstroJ sr lllql lrrur,rll,rrrr Jrlrqprpu Lrp èq llr,\\.rcq111nu uortcdnrocrIEL[] loèdxo è,\\ puu 'olEls lr,rlrrrr oql ,luo sr snll 'rJ^O,\\oH .eluls

lpur.rf,qÌ E ,roJ 0 _ ll

uoty:tLfiu trro.rf sttorlnnTtrtlf t4ay 1. 1 1

-\

Itr

342 11 Inftationctryt,osntologv

(the lactor (2n) 3 comes from the Fourier translorm; 4nk2 rtk:4nk3 ri lnk comes fromthe k-space volume element). This con-rpletes the argument. The rms value of fluctuationsin / can be used as above to deduce the power spectrum of mass fluctuations well afterinflation is over. In terms olthe variance per lnk in potential perturbations, the answer is

Inflation thusthe initial <iensity pespectrum. With sufficperturbations can becarefully adjusting ttGaussian character c& Vishniac 1993), esa fluctuation in field-senerated by cosmic r

is an inevitable result

flve anr predìoive prnflatlon. As we shall r

state.

I\FLATON COUPLINGlmit on the inflation pr-rl e-foldings ol inflatio

N:Suppose Z(r/) takes the--:n then be erpressed a

:-nce N ] 60, the obser.

ternattvely, in the case- mrt'2lml - t0-5. Si

' -gives

These constraintsr-.3 to rise the theory to

.-rrnstrain the theory. T" .::rolo-ey, and it is vital trì.":r an expianation does;r : '.', ould have

-' :rlt'r l0t. ln lact, the si:r : -tLrrizon-scale amplitr_rd,

.H1)' -A,p(k):^.-t.t1T (p r

H, _Y V

-l rif3Hó -- -V"

( r 1.49)

where we have also written once again the exact relation betrveen H and y' and theslow-ro1l condition, since manipulation of thesc three equations is often required inderivations.

This result calls for a number of comments. First, if H and ,/ are bothconstant then the predicted spectrum is exactly scale invariant, with some characteristicinhomogeneity on the scale ol the horizon. As we have seen, exact de Sitter space withconstant É1 will not be strictly correct fbr most inflationary potentials; nevertheless, inmost cases the main points ol the analysis still go through. The fluctuations in @ startas normal flat-space fluctuations (and so not specific to de Sitter space), which changetheir character as they are advected beyond the horizon and become frozen-out classicalfluctuations. All that matters is that the Hubble parameter is roughly constant for thelerv e-foldings that are required f'or this transition to happen. If 11 does change with time.the number to use is the value at the time that a mode of given /< crosses thc horizon.Even if 11 were to be precisely constant, there remains the dependence on y'r. uhich agairwill change as different scales cross the horizon. This means that different inflationar'models display different characteristic deviations from a nearly scale-inr atiant spectruLand this is discussed in more detail below.

Two other characteristics of the perturbations are more general: they will brGaussian and adiabatic in nature. A Gaussian density field is one for which the joir:probability distribution ol the density at any given number ol poirrts is a multivar.iar;Gaussian. The easiest way lor this to arise in practice is for the density fie1d to b-constructed as a superposition ol Fourier modes with independent random phases: thGaussian property then follows lrom the central limit theorem (see chapter 16). lt :;easy to see in the case of inflation that this requirement will be satisfied: the quantu:commutation relations only apply to modes ol the same /<, so that modes of differe::wavelength behave independently and have independent zero-point fluctuations. Final-rthe principal resr-rlt of the inflationary fluctuations in their late-time classical guise is ',:a perturbation to curvature, and it is not easy to see how to produce the separati;,rin behaviour between photon and matter pertulbations that is needed for isocurvatu:*modes. Towards the end of inflation. the universe contains nothing but scalar fi:r,iand whatever mechanisms that generate the matter antimatter asymmetry have 1er::,operate. When they do, the result will be a universal ratio ol photor.rs to baryons. b-nwith a total density modulated by the residual inflationary fluctuations adiabatic inir"dconditions. in short.

- -)!ru .. , . _ "Qz ltA

sr epnlqdute olecs-uozrror oqìSursserdxeJo,e,n Joqoue ocurs'relrrurs sr uorlll!:ur ur uorJ€nlrs eq 'itc€J u1 '(91 :eldeqrl

(rE rr),(ol/tn't?) - "g

0^€g plno,\ 0_\\

ereq,u 's1co3op 1ecr3o1odo1 uo posr?q sorJoeql JoJ lsrxe ol rueos soop uotleut?1dxe ue qcng'epnlru8eru slr roJ uorl€ueldxe oldurrs e sr eJerll Jeqlorl,^A ,\\ouì ol lelr^ sr 1r pue ',Soloursocur sroqrunu 1ue1;odurr isoru eql Jo ouo sl "g Jo epnlldure eqa ,,rooqt oql urerlsuoc ot

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(ss'rr)

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,AtQp cHE | :+,rr, [ :

344 11 Inflationnry cosn'tology

We have argued that inflation will end with e of order unity; if the potential were tohave the characteristic value Z - EÍr, then this rvould give the same prediction lorò,r as in delect theories. The appearance of a tunable 'knob' in inflation theories reallyarises because we need to satisly ó - m, (fbr sufficient inflation), while dealing withthe characteristic value v - E!,u, (to be fair, this is likely to apply only at the startof inflation, but the potential at 60 e-fo1ds from the end ol inflation will not be verydifferent lrom its starting value unless the total number of e-folds is >> 60). It is therelorereasonable to say that a much smaller horizon-scale amplitude would need v 11 Er!,r,r,

i.e. a smaller Eo' than the conventional value.This section has demonstrated the cul-de-sac in which inflationary models now

find themselves: the field that drives inflation must be very weakly couplecl ancleffectively undetectable in the laboratory. Instead ol Guth's original heroic vision ola theory motivated by particle physics, we have had to introduce a new entity intoparticle physics that exists only for cosmological purposes. In a sense, then, inflation isa failure. However, the hope of a consistent scheme cventr-rally emerging (plus the lackol any alternative), means that inflationary models continue to be explored with greatvigour.

GRA.VITY WAVES AND TILT The density perturbations lelt behind as a residue of thequantum fluctuations in the inflaton field during inflation are an important relic ol thatepoch, but are not the only one. In principle, a lurther important test ol the inflationarymodel is that it also prcdicts a background of gravitational waves, whose propertiescoupie with those of the density fluctuations.

It is easy to see in principle how such waves arise. In linear theory, any quantumfield is expanded in a similar way into a sum ol oscillators with the usual creationand annihilation operators; the above analysis o1- quantr-rm fluctuations in a scalar fieldis thus readily adapted to show that analogous fluctuations will be generated in otherfìelds during inflation. In fact, the linearized contribution of a gravity wave, ftr,, to theLagrangian looks like a scalar field @ : (nt,l4rli) /ru" fproblem 11.5], so the expectedrms gravity-wave amplitude is

hrr.rr-Hfmr. (11.56t

The flr-rctuations in / are transmuted into density fluctuations, but gravity waves willsurvive to the present day, albeit redshifted.

This redshifting produces a break in the spectrum of waves. Prior to horizon entrr'.the gravity w:rves produce a scale-invariant spectrum of metric distortions, with amplitude/r'-, per ln/'. These distortions are observable via the large-scale CMB anisotropies.where the tensor modes produce a spectrum with the same scale dependence as theSachs Wol1è gravitational redshift from scalar metric perturbations. In the scalar case(discussed in chapter 18), we have òrlr - $13c2, i.e. of the ordcr o1'the Newtonianmetric perturbation; similarly, the tensor effect is

(T).- - /"-' s ò* - 1o 5 (11.5-

where the second step follows because the tensor modes car.r constitute no more thac100% of the observed CMB anisotropy. The energy density olthe waves i. p,,," - ntiÉk:"

where ft - H(arnt,therefore expect

After horizon entrpresent-day energywhile the universeWhat is the densitlQ,,*' - ft1,r. at thedominated, with e,maintains a constanpresent-day density

The gravity-wave r

fluctuations on supe.\s discussed in chalperturbations in darianisotropies, we mus

A gravity_warcosmic strings (see se

ri here p is the mass 1

so f),;r, - 10-7 is exprThe part ol

r'mission front pulsarthis modulation sets ,i ery long way fi:om thol this level ol relic gr:nergy density ol grav

I

:rr that the typical st,ia\res, as opposed to i-:rconceivable that spac.-r search for kHz_peri.:nsitivity. A direct detr:.r much the same for:i Penzias and Wilson

An alternative w. r ia the ratio between-,i'olfe

effect, as first ar. :itctional temperature

20.7 The Planck Era 649

20.7 The Planck Era

Enormous progress has been made in understanding the types of physicalprocess necessary to resolve the basic problems of cosmology but it is notclear how independent evidence for them can be found. The methodologicalproblem with these ideas is that they are based upon extrapolations to en-ergies vastly exceeding those which can be tested in terrestrial laboratories.Cosmology and particle physics come together in the early Universe and theyboot-strap their way to a self-consistent solution. This may be the best thatwe can hope for but it would be preferable to have independent constraintsupon the theories.

Fig. 20.3. A schematic diagram illustrating the evolution of the Universe fromthe Planck era to the present time. The shaded area to the right of the diagramindicates the regions of known physics.

A representation of the evolution of the Universe from the Planck era tothe present day is shown in Fig. 20.3. The Planck era is that time in the

650 20 The Very Early Universe

very remote past when the energy densities were so great that a quantumtheory of gravity is needed. On dimensional grounds, this era must haveoccurred when the Universe was only about tPl ∼ (hG/c5)1/2 ∼ 10−43 sold. Despite enormous efforts on the part of theorists, there is no quantumtheory of gravity and so we can only speculate about the physics of theseextraordinary eras.

Being drawn on a logarithmic scale, Fig. 20.2 encompasses the evolutionof the whole of the Universe, from the Planck area at t ∼ 10−43 s to thepresent age of the Universe which is about 4× 1017 s or 13.5× 109 years old.Halfway up the diagram, from the time when the Universe was only abouta millisecond old, to the present epoch, we can be reasonably confident thatthe Big Bang scenario is the most convincing framework for astrophysicalcosmology.

At times earlier than about 1 millisecond, we quickly run out of knownphysics. This has not discouraged theorists from making bold extrapolationsacross the huge gap from 10−3 s to 10−43 s using current understanding of par-ticle physics and concepts from string theories. Some impression of the typesof thinking involved in these studies can be found in the ideas expoundedin the excellent volume The Future of Theoretical Physics, celebrating the60th birthday of Stephen Hawking (Gibbons et al., 2003). Maybe many ofthese these ideas will turn out to be correct, but there must be some con-cern that some fundamentally new physics will emerge at higher and higherenergies before we reach the GUT era at t ∼ 10−36 s and the Planck era att ∼ 10−43 s. This is why the particle physics experiments to be carried withthe Large Hadron Collider at CERN are of such importance for astrophysicsand cosmology, as well as for particle physics. It is fully expected that definiteevidence will be found for the Higgs’ boson. In addition, there is the possibil-ity of discovering new types of particles, such as the lightest supersymmetricparticle or new massive ultra-weakly interacting particles, as the accessiblerange of particle energies increases from about 100 GeV to 1 TeV. These ex-periments should provide clues to the nature of physics beyond the standardmodel of particle physics and will undoubtedly feed back into understandingof the physics of the early Universe.

It is certain is that at some stage a quantum theory of gravity is neededwhich may help resolve the problems of singularities in the early Universe.The singularity theorems of Penrose and Hawking show that, according toclassical theories of gravity under very general conditions, there is inevitablya physical singularity at the origin of the Big Bang, that is, as t → 0, theenergy density of the Universe tends to infinity. However, it is not clear thatthe actual Universe satisfies the various energy conditions required by thesingularity theorems, particularly if the negative pressure equation of statep = −%c2 holds true in the very early Universe. All these considerations showthat new physics is needed if we are to develop a convincing physical pictureof the very early Universe.

206 C o s mol o g ic al Pe rturb ation s

4.4.4 Linear Perturbation Sp e ctrum

As discussed above, the power spectrum P(k) is an important quantity characterizing a randomfìeld. In fact it is the only quantity required to specify a homogeneous and isotropic Gaussianrandom field. As we have seen in $4.3, because different Fourier modes evolve independently ofeach other in the linear regime, the linear power spectrum at any given time can be simply relatedto the initial power spectrum via the linear transfer function. We now take a more detailed lookat the initial power spectrum.

(a) The Initial Power Spectrum In the absence of a complete theory for the origin of thedensity perturbations, the initial (untransferred) perturbation spectrum is commonly assumed tobe a power law,

Pi(k) * k'' , (4.267)

where n is usually called the spectral index. As we will show in 94.5, the power spectra predictedby inflation models generally have this form.

It is often useful to define the dimensionless quantity,

which expresses the contribution to the variance by the power in a unit logarithmic interval of kln terms of A2(/.) we have that

L2(kt - l-t'pt,:'.2n2

L21k1 * 1,t+n '

The corresponding quantity for the gravitational potential is

(4.268)

(4.269)

(4.270)

:o determine theiasks of observat

One historical.rl the galaxy dis,. ariance of the d

',,. here

:s the Fourier trat

The value of o(..2.7.1). Thus, o

6R):latR:.inceo'(R)-11rormalization is-n the mass distr.nigh density regithe galaxy distrildìstribution, ther

where b : constmass density fie

Since an accurastill uncertain. Iproperties, such

To accuratellby nonlinear evdistribution. Inthe power spect

power spectrunreasons, the am

It is importantpresent time acr

have gone nonl:actual, present-(

Table 4.1 lisobtained fiom tfj6.7 for details)

^?*(k) : )rt e*p,; * k 4

^2 &) n k -r,

which is independent of ft for n : L Thus, for the special case of n : 1, which is called theHarrison-Zel'dovich spectrum or scale-invariant spectrum, the gravitational potential is fìniteon both large and small scales. This is clearly desirable, because divergence of the gravita-tional potential on small or large scales would lead to perturbations on these scales that are toolarge.

As we will see in $4.5, in inflation models the metric (potential) perturbations are generatedby quantum fluctuations during inflation. At the end of inflation, all perturbations become super-horizon because of the huge amount of expansion caused by inflation. Since metric perturbationsremain roughly constant during super-horizon evolution, the amplitude of a metric perturbation atthe time when it re-enters the horizon should be approximately the same as the initial amplitude.Thus, the amplitude of Aa(/r) evaluated at the time of horizon re-entry is proportional to k"-1 ,

which is independent of ft for a scale-invariant spectrum.

(b) The Amplitude of the Linear Power Spectrum So far we have only discussed the shape ofthe linear power spectrum. To completely specify P(k), we also need to fix its overall amplitude.Because we do not yet have a refined theory for the origin of the cosmological perturbations,the amplitude of P(ic) is not predicted a priori but rather has to be fixed by observations. Evenfor inflation models, where we can make detailed predictions for the shape of the initial powerspectrum, the current theory has virtually no predictive power regarding the amplitude (see nextsection).

For a power spectrum with a given shape, the amplitude is fixed if we know the value ofP(k) at any ft, or the value of any statistic that depends only on P(k). Not surprisingly, manyobservational results can be used to normalize P(ft). DifTerent observations may probe the powerspectrum at different scales, providing additional constraints on the shape of P(k). In fact, trying

1.1 Sttrti,stit'nl Propertie s

to deterntine the shape and aniplitude of the linea| porrcr spectrrÌm is one of the most importanttasks of observational cosmology (see Chapter 6).

One hisktrical prescription fbr normalizing a theoretical power spectrum involves the varianceof the -lalaxy dish'ibution when samplecl with randomly placed spheres of raclii R. The preclicteclvariance of the density fìeld is t'elated to the power spectrunt by

201

where

is the For-rrier transfbrm of the spherical top-hat window finction

.1torrRr .'_, lp\k 1,1,,/i ,A Ar.lA.

Watkt .,,'-. *in,fR, /<Reo.rARrl^^ l-

1 .t 1+z.Rr 1 it'r RYYA(/') - < .." t 0 olhe rri isc

r. - o..,1r8/r lMpe.l I

o,,, lÍì/r 'Mpc\ ,

. ,

(1.211)

(4.272)

aJ )75 r

(1,213)

The value ol o(R) clerived fiom the distr-ibution of galaxies is abour uniry fbrR - Sft rMpc(\2.7.1). Thus. one ccluld in principle not'malize the theoretical power spectrum by requirilgo(À) - I at R : 8 /z lMpc. However. there are several problerns with this apploach. First of alisince o(R) - I we are not accurately probing the linearrcgime fbr which ò << l. Seconclly. thisnormalization is based on the assumption that galaxies iìre accurale tracers of the fluctuationsin the mass distribution. This nray not be true if, for example, galaxies fbrmed preferentiaÌly inhigh dcnsity regions. Indeed. if we adopt the less restrictive assurnption thar the fluctuations inthe -ualaxy distribution are proportional (but not necessarily eqr.ral) to the fluctuations in the massdistribution. then

òo"t - b òr. (4.211)

where ó : corlstant is a bias paronteter whose value depencls on how galaxies have fbrrned in themass clensity field. In this case

Since an accurate theory firr galaxy forrnatiolr is still lackirrg at the present time, the value of b isstill uncertain. ln fact, as we will see in Chapter l-5, b is fbund to be a function of various galaxyproperties. such as luminosity and color.

To accurately non.nalize the linear power spectrum thus recluires a mcthod that is not aflèctedby nonlinear evolution and that does not depend on the assumption of galaxies tracing the massdistribution. In Chapter 6 we will describe various statistical measlrres that can be usecl to probethe power spectrum. Some of these methocls are much better suitecl fot' normalizing the linearpower spech'unt than ogrr (8li lMpc) is. Howc\er. as a eonvcntion. ancl largely fbr historicall'easons, the arnplitude of a power spectrum is usually represented by the valr-re

os = err(S/i Itf,lp.) (1.216)

It is important to t'ealize thirt o3 is evalr-rated from the initial power spectrum evolved to thepresent tit.ne accordittg to linear theory. Since pertr-rrbations or.r scales of - 8/i lMpc may wellhave gone nonlinear by the present tinre. this is not necessarily the same as the variance of theactual. present-day mass distribution.

Table '1. I lists a number of adiabatic models and their power spectrun-i normalization asobtainecl fiorn the terìlperature fluctr-rations in the CMB obtained by COBE on lalge scales (see

!6.7 lbr details) and fiorn the abundance of rich clusters of galaxies (sec [ì7.2.5 fbl cletails). The

208 C o smolo gíc al Pe rturb ation s

Table 4.1. The values of os in different cosmogonies.

Q-,0 Qu.(J C),t,0 os(COBE) os(cluster)

SCDMHDMMDMOCDMACDM

0.6

0.50.91.0

0010

0.3 0000 0.7

0.30.3

0.5 n:0.5 n:0.5 n:0.'/ n:0.7 n:

1.31.30.50.51.0

Having discusBroadly speal

transitions. Inscalar field. A

with a close t<

are certainly 1

that originatemodels typica

As discussed irelated to theexperiences arfield. called ttexpected to br

scales by the r

structure form

(a) Heuristicunderstand so.

ing inflation. '

the Universe istructures thatthe horizon si;

ating the initiinegligible selffluctuations ofbations are exlinflation phase

reheating con\,

any segregatioisentropic. FinSitter space), t

This fìnal pin $3.6, in orc

long enough. '

inflaton must I

value (E), anc

Therefore. if s,

these perturbatsome physicalscale ,2";. TheseBecause ofthetimes are inflaland t2, the ratir

-4 -3 -2 -1 0

log k lMpc-11

Fig. 4.4. The power spectra, P(ft), as a function of k for various cosmogonic models. The initial powerspectrxm is assumed to be scale invariant (.t.e. n - 1), and the spectra are normalized to reproduce theCOBE observations of CMB anisotropies.

COBE-normalized, linear, dark matter power spectra are shown in Fig.4.4. Not all models matchat small ft, despite the fact that they are all normalized at these scales. This arises because the

observed temperature fluctuations are in angular scales and the conversion from angles to dis-tances is cosmology-dependent. Note that the COBE and cluster-abundance normalizations are

only in agreement with each other for the ACDM model and the MDM model. Since they have

been obtained from measurements of the power spectrum amplitude at vastly different scales, a

discrepancy in the infèrred values of o3 signals that the shape of the model power spectrum isinconsistent with the data. Indeed, as we shall see in Chapter 6, stringent constraints on the shape

and amplitude of the linear power spectrum can now be obtained fiom a variety of observations,and the model that is currently favored is the ACDM model with parameters similar, but notidentical, to those listed in Table 4. 1.

COBE normalization

z/o(9

4.5 The Origùt oJ Cosnutlogical Perturbtttittns

4.5 The Origin of Cosmological Perturbations

Having discussed the (linear) evolution of cosmologicaÌ perturbations we now turn to their origin.

Broadly speaking, two ditfèrent mechanisms have been proposed. narnely inflation and phase

transitions. ln inflation models the perturbations arise from quanturn fluctuations of the inflaton

scalar fìeld. As we will see, these models typically predict isentropic. Gar-rssian perturbations

with a close to scale-invariant power-spectrum (although deviations fiom this typical prediction

are certainly possible). In the alternative model, perturbations arise tiom cosmological defects

that originate fiom phase transitions in the early Universe. Contrary to inflation models. these

models typically predict non-Gaussian perturbations.

1.5.I Perturbations from Inflation

As discussed in [3.6, the concept of inflation is introduced to solve a number of nagging problems

related to the initial conditions of the standtrrd cosmology. In inflation models. the Universe

experiences an early period of exponential expansion, driven by the false vacuum state of a scalar

fìeld, called the inflaton. Because of quantum fluctuations the energy density of the inflaton isexpected to be jnhon.rogeneous. These inhomogeneities are initially inflated to super-horizon

scales by the exponential expansion, but re-enter the horizon after the inflation is over to seed

strucîure fìrrmation.

(a) Heuristic Arguments Without going into details, we may use some simple arguments to

understancl some of the most important properties of the density perturbations generated dur-

ing inflation. The arguments consist of the fbllowin-e fbur important points. First of all. since

the Universe is assumed to have gonc through a phase of very fast expansion (inflation). allstructures that we observe in the present-day Universe had sizes that u''ere much smaller than

the horizon size befbre inflation started. Therefore, inflation provides a mechanisnt for gener-

ating the initial perturbations in a causal way. Second, if the scalar fìeld driving inflation has

negiigible self-coLrpling (as is assumed in most models), then difl'erent modes in the quantum

fluctuations of the fìeld should be independent ol each other. Consequently, the density pertur-

bations are expected to tbllow Gaussian statistics. Thircl, the perturbations produced during the

inflation phase are perturbations in the energy density ofthe scalar field. At the end of inflation,

reheating converts this energy density into photons and other particles. Since we clo not expect

any segregation between difTerent particle species, the resulting perturbations are expected to be

isentropic. Finally, since during inflation space is invaliant under time translation (i.e. it is a de

Sitter space), the perturbations generated by inflation are expectecl to be sc:ale-invarianî.

This final point is not very straightfìrrward, ar.rcl requires some mol'e discussion. As showrt

in 13.6. in orcler tor inflation to solve the horizon problcm, the perìod ol inflation tnust last

long enough. This translates into a slow-roll condition, which states that the potential of the

inflaton must be sufîciently flat. As a result. the Hubble constant 11, the inflaton expectation

value (E), and the potential ener-qy V(E), are al1 rou-ehly tirne-independent dutìng inflation.

Therefore, il'some physical process can generate perturbations in the inflaton, the properties ofthe se pertulbations should be independent of their tirne of generation. Asstrme, fbr example. that

some physical process generates perturbations at all times (during inflation) on a fìxed physical

scale .1.;. These pertr-rrbations will then all be generated with approxirnately the same arnplitude.

Because ofthe (exponential) increase ofthe scale factor. rz(r), pcrturbations gcnerated at di11èrent

times are inflated into diflèrent scales. For cxample, fbr two perturbations generated at times 1r

ancl /2, the ratio between their scales at any later timc during the exponential expansion is

Htr, r' t (1.211)

209

)"1

),.

2r0 C o smo I o g ic al Pe rt urb ations

The time /H when a perturbation of comoving scale 2" reaches the horizon is given by )" xc lalal(L)lUlts(L)l Since É1 is approximately consranr, we can wrire

,l ' =

o k" (1, )] _ ,H .rsr ).2 , ra, ).1 i).2 oltsQq)l

Comparing Eqs. (4.277) and (4.218), we see that

,, zuttli +

-rlri + k'yrr :9.'a

ts(h) tz : tuQq) - tt. (4.27e)

Thus, the time between generation and horizon-exit is the same for all perturbations. Since theproperties of space change little during inflation, the amplitudes of all perturbations should beapproximately the same at horizon-exit, without depending on their scale at that time. Note thatthe above discussion makes no reference to the value of ,1,;, and so the conclusion is true even ifperturbations with a range of physical scales are generated at each time. Thus, the expectationvalues of the perturbation amplitudes are scale-independent at horizon-exit. When such a pertur-bation re-enters the horizon some time well after inflation has ended, its amplitude will be aboutthe same as it was at horizon-exit, since causal physics cannot act on super-horizon perturba-tions to produce observable consequences.l The perturbations are therefore scale-independent athorizon-reentry. Thus, inflation generically predicts scale-invariant perturbations.

(b) Some Detailed Considerations To make detailed predictions for the density perturbationsin an inflation model, one needs to consider a specific model for the scalar field, which we denoteby E(x,l). We write

E$,t): Eo(r)+ V(x,r), (4.280)

where rpe(r) is the background field and r4(x,r) (with ly] << E0l) is rhe perturbation. The dynam-ical properties of the scalar field are described in $3.6.3. Under the slow-roll condition, which isrequired for inflation to occur, the evolution of E is given by

titlluE-a )V2E-0. (4.28t)

Thus the equation of motion for the Fourier modes of the perturbation field, ì/k, can bewritten as

(4.218)

(4.284)

(4.28s)

where (yp : 6

[f"dal(ua2) r

have set c: C

2H2az ;:2f rzQ: e ik'(r -

After inflation.

The typical amThe question istructure forma

A convenienis to consider tl

where the secorlows from Eq. (

tensor definedthe metric pertis that it is corthe deflnition t

kr << l. Since iation dominaterBecause the ar

horizon evolutiin the post-inflrthe horizon aftrtions are still nperturbations ar

where the subs<

mode exits the I

where the secoEq. (4.287). Thscale-invari ant.

A crucial re<

expands expon

(1.282)

where again a prime denotes derivative with respect to the conformal time r. If the expansion ofthe Universe is neglected (.i.e. at f a: 0), the above equation is that for a harmonic oscillator, andthe solutions are e- ikt I-. Upon quantization. ì/1 becomes an operaror:

Vt - Qtk.T)à -Q \k.T)ù+ . (4.283)

where d and ài are the operators annihilating and creating a particle, respectively, and Q:, k"

1vEE. The expected quantum fluctuation of rgp in the ground state, l0), ii then .ha.a.t.riredby the dispersion (0 EIVrtOl Using the properties of ó and rî'i', ir can be shown rhat

(l,lrult) = (0 ttti,Vt 0) : e(k,r\2 : t11ztc1.

When the expansion term is included, Eq.(4.282) can be converted into the form

/r' + (t' a" f al,pr : g,

I Note, however, that the amplituiles of super-horizon perturbations may appear to be evolving with time in some gauges(see $4.2.1). Such evolution, however, is completely geometrical and should not have any observable.onseqieni".such as changing the amplitude of a physicaÌ perturbation trt horizon entry.

1.5 Tlrc Origin of CosmtLogical pertttrbcrtiorts 211

where r1 - avr. Since during inflation H: a'ltf is roughly constant, we rrave that T:l,i"dal(Ha2) = -llHo. where a,, is thc expansion f'actor at tlte encl of the inflation ancl wehave set r - 0 at this time. Using the fact that cr aa during inflation, we have that cr,,f ct=2H2tt2 x 2/r2. lnserting this into Eq. (u1.285) one fìnds that its rwo solutions ar-e e ancl e* withQ: e it'' 0 i/ftr). Thus,

ù*r- Qrk.r, '--l it -l l21. L' tkr12]

After inflation, when kr << l. this sives

' I P2'Vt - = rr..r-.,: = 1,.,._^ \(/ r i- _^

, Ha7 ikiHul6T.),'i - 2tte, lcl +@( - cD - -- - :-' O -'''__ _: O.' A' k:\p P1 - .ì I rr.

(4.287)

Tlre typical arrplitude of y4 due to quantum fluctuarions in the ground stare is then a l(.t[2f3 ).The cluestion is then how sr-rch fluctuations generate the metric perturbations responsible forstructure fbnnation in the Universe.

A convenient way to tnake connections between (lVu]t) and the metric perturbations O(fr.r)is to consicler the fbllowing quantity:

lzt.l8ó.r

(4.289)

(.4.290)

(4.288)

where the second relation tbilows fronr the clefinition of g in Eq. (:1. l,l9) ancl the thircl relation fbl_lows lì'tlrl Eq. (4. lt16) with Y - (D. For the scalar fielcl consiclered here. the encrsv-monlentunrtensor defìned in ti3.6.3 leads to 0 - -kl\rklEI and so defìnecl above is a Jombinati.' ofthe nietr-ic pelturbation <D ancl the perturbation in the inflaton. One important plrperty of is that it is consern'ed, i.e. i':0. cluring super-hot.izon evoh-rtion. This can fr"'pr,ru.n ur;ngthe definition .f ( together with Eqs.(4.146) and (4.156) (again wirh y: @) in the timitkt << l. Since afier inflation all perturbations are super-horizon and the Universe becomes radi-ation dominated (so rhat x,- l/3), *. obtain ( -3A12 from rhe lasr relarion in Eq.(zl.2gg).Because the amplitudes of the metric perturbations renrain roughly constant durin-e super-horizon evolution. the amplitLrcle of ( at the time when the p".tu.boiiun re-enters the horizonin the post-inflation era shoulcl be cqual to its value at the time when the pertr-u-biition exitsthe horizon afier it is generated during inflation. At the time of horizon exit, rnetric perturba-titrt.ts are still negligible' ancl so - uHVlElt. It then fbllows that the post-inflation merricperturbations are

)2 ur,J@: I HLIL-l g).lh,,riz,,n-.rir

where the subsct'ipt 'horizon-exit' indicates that the quantity is evaluated at the tine when thenrocle exits the horizon. i.e. at 1 - olli. The post-inflation power spectrulì is therefbre

Por() l'n,*+l ,l [r]l[9 g,l J,,,,,;r,,,,-..,, 9f ' Lg,l J,,,,,,r,,,,_.r,,'

where the second equation fbllows fiom replacing V7.2 with its quantum expectation yalue inEq. (4.287). Thus, if Ht lrú is time-inclepcnclent, the ròsulting poit-inflarion pu*,.1. ,p..u-u- i.scale-invariant. wirl.r n - I lsee F,q. (.210)1.

A clucial requirernent 1'or obtaining the scale-invariant power spectmm is that the Uni'erseexpands exponentially during the periocl of inflation. Deviations fr:om a purely exponential

212 C o smolo g ic al Pe rturb at ions

expansion will cause deviations from a pure scale-invariant power spectrum. To see this, considerthe quantity

where we have used Eqs. (3.253) and (3.254). We define the tilt of the power spectrum as

, |., I H1 I /8ft \r v\-\r , 'k'Paikt - eF ,pl, = ,rnt (;f ,) w rap

1 _n-_q!4dlnk'

dv ldE d

SnVdE

(4.29t)

(4.292')

(4.2e3)

(4.294)

(4.29s)

(4.296)

The tensorbackground (t

sor modes pr<

importance of

where the nunthe contributi<inflation modtalso causes acalculations sl

So t-ar weturbations foramplitude of tvarying inflatrthe flatness o1

inflation is

where /s and /e

tact that lEo(r,model can be,

This model tht

that depends r

the observed r

small couplintrinflation provitruly viable mr

An alternativevided by topolUniverse. Acc,described in te

under certain tbreaking of th,

a phase transitIn fact, as discsyrnmetry brerconfiguration,phase transitiofield configura

Depending ,

ing fbrms: mot

so that it is zero for a scale-invariant spectrum. At the time when a perturbation exits the horizon,its wavelength is equal to the Hubble radius, and so af k - H 1 . Since 11 is nearly constant duringinflation. we can write

d d odalr* - Olt-- U arp

The tilt can then be written as

ú

where e and 4 are defined in Eqs. (3.258) and (3.259). Unless the potential V is perfectly flat, thepower spectrum is expected to be tilted slightly with respect to the scale-invariant f'orm. However,since the slow-roll condition demands that both e and 4 are small, the tilt is expected to be smallas well.

The above analysis ofquantum fluctuations in a scalar field is valid for all free quantum fieldswith dynamics similar to that of a free oscillator. The expected amplitude of the quantum fluctu-ations given by Eq.(4.287) is independent of the details of the fìeld in consideration and so allsuch fìelds are expected to have similar quantum fluctuations during inflation. In particular, underthe weak-field assumption, the gravitational Lagrangianis I : Rll6nG = Aahpu Aahp, f 32nG,where R is the perturbation in the curvature scalar and hu, is the perturbation in the metric tensor.

This suggests that hp' lyÍ6ftG can be effectively treated as a scalar field and so the amplitudeof the fluctuation in hpu athortzon crossing is given by

l-n:6t-2t1"

. k3 . 4HZA; --

( hl-\u d-) \ t t _ )LIL- tL mll

;,)fluu ).)-Pcu - ,:;,h-ilr,tl,u)

The propagation of the tensor mode of the metric perturbations produces gravitational waves,and so inflation models generically predict the existence of a background of gravitational waves.Note that this mode of perturbations does not generate density perturbations (which corespondto scalar mode), because these two modes evolve independently (see \1.2.2). At the time ofhorizon crossing, the energy density ofthe gravitational waves is

Once the mode re-enters the horizon, pcw evolves as a 4,like radiation. For modes that re-enterthe horizon while the Universe is dominated by radiation, pcw f p, is a constant, and so

Ocw - Q,(H f mp1)z - 10-1 (v l*fr) (4.2e1)

This relation follows from the fact that A, : Snprl(Hmpt)2 - 1 at t t"q, which gives p. -(H*pl)2 and (p6y7/p,) - (H l*rr)t.

4.5 The Origin ofCosmological Perturbations 213

The tensor metric perturbations generated during inflation can aff-ect the cosmic microwavebackground (CMB) anisotropies. For modes that re-enter the horizon after decoupling, the ten-

sor modes produce a CMB spectrum with the same scaie dependence as the scalar modes. Theimportance of the tensor (T) modes relative to the scalar (S) modes is represented by the ratio

where the numerical coeff,cient is based on the detailed calculations of Starobinsky (1985). Thus,

the contribution of the gravitational waves to the CMB anisotropy can become significant in an

inflation model that has a relatively large gradient in the inflaton potential. Since such a gradient

also causes a tilt in the power spectrum, the ratio A+lLl is related to the tilt I n. Detaileclcalculations show that L+ lA! = 7 (I - z) (e.8. Davis et al., 1992).

So far we have seen thart inflation is quite successful in predicting the initial density per-

turbations fbr structure formation. There is, however, a severe problem concerning the overallarnplitude of the perturbations predicted by such a scheme. As an illustration, consider a slowlyvarying inflaton potential, V(rp):Vo (.rcla)Ea, where r is a coupJing constant describingthe flatness of the potential. Using the slow-roll equation (3.253) the number of e-fbldings ofinflation is

4-4-10s,A. Aó

x: .l'"

ua, - 1,,'" #o^': J ,'" #kaE, - #t,),

.H6Afr--______-161/'- (dY lctr\o)'

(1.299)

where /, and /" are the times at which inflation starts and ends, respectively, and we have used the

f'act that ] qo (1, ) << | Eo (r. ) . Thus, the amplitudes of the metric perturbations predicted by this

model can be expressed as

(4.298)

(4.300)

This model therefore predicts an initial power spectrllm that is scale invariant, with an amplitudethat depends on the coupling constant rc. Since N 2 50 for a successful inflation (see !3.6.2),the observed amplitucle Ao - l0-5 requires that K S l0 l'5 according to Eq.(4.300). Such asmall coupling constant does not come naturally from current particle physics. Thus, althoughinflation provides an attractive scheme to explain the origin of the cosrnological density field, atruly viable model has yet to be found.

1.5.2 Perturbations from Topological Defects

An alternative class of mechanisms for generating cosrnological density perturbations is pro-vided by topological defects which can be produced during some phase transitions in the earlyUniverse. According to the current view of particle physics, matter in the very early Universe isdescribed in terms of f,elds, and the theory governing the motions of these fields is symmetricunder certain transformations. As the Universe expands and the temperature drops, spontaneous

breaking of these internal symmetries can occur. During a symmetry breaking, the field makes

a phase transition from its original configuration to some final configuration with lower energy.

In fact, as discussed in $3.6, the inflaton is an example of such a field undergoing spontaneous

symmetry breaking. If there are more than one topologically distinct vacuum states fbr the fìnalconfìguration, difTerent regions in the Universe can end up in different vaculrm states after the

phase transition. These regions are separated by topological defècts, which are still on the originalfìeld configuration. The energy trapped in such defècts can then produce density perturbations.

Depending on the symmetry of the field, the vacuum manifold can be in one of the follow-ing fbrms: monopoles, domain walls, cosmic strings, and textures. Monopoles and domain walls

214 C o s molo gic ctl Pe rturb at i o n s

can be ruled out immediately as the seeds for cosmic structure, since both would be produced

with an energy density that implies Qo )) 1. Studies of topological defects as the origin ofcosmological density perturbations have therefore focused on cosmic strings and textures. Incontrast to inflation, cosmic strings and textures in general generate non-Gaussian and isocurva-ture perturbations. However, the angular power spectrum of the temperature fluctuations in the

cosmic microwave background predicted by these models does not agree with current observa-tions (e.g. Pen et a1., 1997), so that virtually all present-day models focus on inflation as the mainmechanism to generate the primordial perturbations.

Grar

Many objectsdensities orde

are thus in the

formation in tquasi-linear re

overdensities r

In this chalsystems in wldynamics is dtions have tomade about tl5.3). Althougtproblem of gl

involved. In $1

ical models dsystem, and athese modelsconstrain theitphysical relaxrium configurtproperties ofreflection of tlunderstandingdescribed in thfield.

In the absencedensity perturb