INFLATION AND REHEATING

Juan Garcıa-BellidoDepartement de Physique Theorique, Universite de Geneve, 1211 Geneve 04, Switzerland

AbstractIn these lectures I first review the Standard Cosmological Model, based on thehot Big Bang Theory and the Inflationary Paradigm and then describe in detailthe event that marked the transition from one to the other, a process known as“reheating”, where a very rich phenomenology may allow us to open a newwindow into the early Universe. I will describe the recent developments inobservational cosmology, mainly the acceleration of the universe, the precisemeasurements of the microwave background anisotropies, and the formationof structure like galaxies and clusters of galaxies from tiny primodial densityfluctuations. Then I will address the main difficulties of the Big Bang Theorywith initial conditions and the surprisingly elegant resolution in terms of theInflationary Paradigm. I will go in detail over the ADM 3+1 splitting of cos-mological spacetimes and the Hamilton-Jacobi formalism behind the slow-rollapproximation of homogeneous scalar field cosmology during inflation. I willdescribe the process of particle production in quantum field theory in curvedS.T. and explain the origin of the primordial spectrum of density perturbationsand gravitational waves from scalar and tensor metric fluctuations. The re-heating of the Universe soon after the end of inflation is an extremely violentprocess where the dominant vacuum energy during inflation gets transferredto relativistic particles, which rescatter and backreact until they thermalize ata high temperature. The process is very rich - non-thermal phase transitions,production of topological defects, black holes and dark matter, baryogenesis,primordial magnetic fields and gravitational waves - and could have left theirsignature in the present Universe.

1 INTRODUCTION

The last ten years have seen the coming of age of Modern Cosmology, a mature branch of sciencebased on the hot Big Bang theory and the Inflationary Paradigm. In particular, we can now define ratherprecisely a Standard Model of Cosmology, where the basic parameters are determined within smalluncertainties, of just a few percent, thanks to a host of experiments and observations. This precisionera of cosmology has become possible thanks to important experimental developments in all fronts,from measurements of supernovae at high redshifts to the microwave background anisotropies, as wellas to the distribution of matter in galaxies and clusters of galaxies. In the near future we will havenew observational tools, like neutrino, high energy cosmic ray and gravitational wave astronomy to helpexplore the very Early Universe, where many new phenomena could have left their signature in thepresent Universe.

In these lecture notes I will first introduce the basic concepts and equations associated with the hotBig Bang cosmology, defining the main cosmological parameters and their corresponding relationships.Then I will address in detail the three fundamental observations that have shaped our present knowledge:the recent acceleration of the universe, the distribution of matter on large scales and the anisotropies inthe microwave background. Together these observations allow the precise determination of a handful ofcosmological parameters, in the context of the Λ cold dark matter paradigm.

I will then present the shortcomings of the hot Big Bang theory, mainly initial conditions problems,which will lead to the introduction of inflationary cosmology as an extremely elegant solution of those

problems. For a proper discussion of homogeneous scalar field dynamics in an expanding universe I willintroduce the Hamilton-Jaconi formalism in the context of the ADM 3+1 slicing, which will allow usto formulate the slow-roll approximation and introduce the gauge-invariant linear perturbation theory asthe origin of perturbations in the present Universe.

I will describe briefly the basic concepts of quantum field theory in curved space-times and recallthe “alarming phenomenon” of particle production in an expanding universe, envisioned in 1939 byErwin Shrodinger. I will describe in depth the transition from quantum fluctuations to classical metricperturbations and apply it to scalar and tensor fluctuations produced during inflation. I will deduce thepredicted primordial power spectra.

The next lecture will be devoted to the phenomenological signatures of inflation, with a descriptionof a generic inflationary model building based on high energy particle physics, and the predicted spectraltilts and tensor-to-scalar ratios, as well as the origin of non-gaussianities. Then I will concentrate on theobservational signatures of inflation in the CMB temperature and polarization anisotropies, specially theSachs-Wolfe plateau, the acoustic oscillations and the temperature-polarization cross-correlations. I willthen describe the transition from metric perturbations to large scale structures in our Universe, with em-phasis on the galaxy power spectrum, the concept of Jeans length and Baryon Acoustic Oscillation scale,to end with a discussion of the extraordinary agreement between inflationary predictions and presentcosmological observations.

Subsequent lectures will describe the violent process of reheating after inflation, with an explicitconnection to high energy particle physics, in the masses, coupling and decay rates of particles that arecreated directly or indirectly from the inflaton decays at the end of inflation. I will differentiate betweenthe perturbative decay of the inflaton, and a naive computation of the reheating temperature, and thenon-perturbative phenomena associated with the Bose condensate nature of the oscillating inflaton atthe end of inflation. I will present an account of QFT in background fields and in particular on theparametric amplification of the fields’ amplitudes, in the narrow and broad resonance regime. Then Iwill describe the violent phenomenon of particle production in hybrid or tachyonic preheating, associatedwith spinodal instabilities and symmetry breaking.

The final lecture will be devoted to the phenomenological signatures of reheating after inflation,with emphasis on the generation of entropy, the production of dark matter out of equilibrium, of non-thermal phase transitions and topological defects, of the possibility of electroweak baryogenesis, of thegeneration of primordial magnetic fields and a generic stochastic background of gravitational waves. Iwill make special emphasis on the specific signatures that this new and rich phenomenology might havein the present Universe which may allow us to open a new window into the physics of the early Universe.

2 FRIEDMANN-ROBERTSON-WALKER COSMOLOGY

Our present understanding of the universe is based upon the successful hot Big Bang theory, whichexplains its evolution from the first fraction of a second to our present age, around 13.6 billion yearslater. This theory rests upon four robust pillars, a theoretical framework based on general relativity,as put forward by Albert Einstein [1] and Alexander A. Friedmann [2] in the 1920s, and three basicobservational facts: First, the expansion of the universe, discovered by Edwin P. Hubble [3] in the 1930s,as a recession of galaxies at a speed proportional to their distance from us. Second, the relative abundanceof light elements, explained by George Gamow [4] in the 1940s, mainly that of helium, deuterium andlithium, which were cooked from the nuclear reactions that took place at around a second to a few minutesafter the Big Bang, when the universe was a few times hotter than the core of the sun. Third, the cosmicmicrowave background (CMB), the afterglow of the Big Bang, discovered in 1965 by Arno A. Penziasand Robert W. Wilson [5] as a very isotropic blackbody radiation at a temperature of about 3 degreesKelvin, emitted when the universe was cold enough to form neutral atoms, and photons decoupled frommatter, approximately 380,000 years after the Big Bang. Today, these observations are confirmed to

within a few percent accuracy, and have helped establish the hot Big Bang as the preferred model of theuniverse.

Modern Cosmology begun as a quantitative science with the advent of Einstein’s general rela-tivity and the realization that the geometry of space-time, and thus the general attraction of matter, isdetermined by the energy content of the universe [6]

Gµν ≡ Rµν −12gµνR+ Λ gµν = 8πGTµν . (1)

These non-linear equations are simply too difficult to solve without invoking some symmetries of theproblem at hand: the universe itself.

2.1 Homogeneity and isotropyOur models of the universe are based on the idea that the Universe is essentially identical at every space-time point; this is a generalization at the cosmological level of the Copernican Principle, otherwise knownas the Cosmological Principle (CP). It is obviously wrong: the center of the sun is very different fromintestellar space. We will however assume that the CP is applicable only on the largest scales, wherelocal variations in matter are averaged out. Its validity at large scales is confirmed by present observa-tions of diffuse gamma, X-rays, UV, visible, IR and radio backgrounds, but most importantly from theextraordinary isotropy of the cosmic microwave background (CMB). The irregularities (anisotropies) areonly of the order of one part in 105, as shown for the first time by the COsmic Background Explorer(COBE) of NASA en 1992.

From a mathematical point of view, the CP is related to two precise mathematical properties ofspace: isotropy and homogeneity. In differential geometry, isometries of the metric are related withconserved Killing vectors. A maximally symmetric space has the maximum allowed number of Killingvectors. For a space of n-dimensions the maximum number is n(n + 1)/2. In 4 dimensions this corre-sponds to the 10 generators of the Poincare algebra (4 translations, 3 rotations and 3 boosts). However,the Universe is not static and therefore we cannot impose time translation invariance. Therefore, weare left with the only requirement that spatial sections should be maximally symmetric, according to theCosmological Principle. This implies 6 Killing vectors, associated with 3 rotations and 3 translations(i.e. homogeneity and isotropy). In the context of GR, it is said that we can foliate the 4D manifold inlR×Σ, where lR represents the temporal direction and Σ is a maximally symmetric 3-space.

We can thus chose our metric to be of the form

ds2 = −dt2 + a2(t) γij(x) dxidxj , (2)

where t is a time-like coordinate which labels the cosmological events in the 3-surface Σ(x), and themetric γij is maximally symmetric in Σ. The function a(t) is known as the scale factor and gives usinformation about the relative size of spatial sections of Σ at time t. These coordinates, in which themetric has no cross-terms dt dxi, are known as comoving coordinates, and an observer which remainsat xi = constant, is known as a comoving observer. Only comoving observers see the Universe aroundthem as isotropic. In fact, no known astronomical object, like the Earth, the Sun or the Milky Waygalaxy, are exactly comoving; they move with peculiar velocities due to the gravitational attraction ofthe matter concentrations in their vicinity. In the case of the Earth, this peculiar velocity is believed to beresponsible for the observed dipolar anisotropy of the CMB, i.e. a conventional Doppler effect.

2.2 The Friedmann-Robertson-Walker metricLet us come back to the characterization of the Universe geometry. We are interested in maximallysymmetric 3-metrics γij . Such metrics satisfy

(3)Rijkl = k(γikγjl − γilγjk

), (3)

where we have introduced, for later convenience, a constant k = (3)R/6. The 3D Ricci tensor is

(3)Rij = 2k γij . (4)

If the space Σ is maximally symmetric it is necessarily spherically symmetric. For those spaces the mostgeneral metric can be written as

dσ2 = γij dxidxj = e2β(r)dr2 + r2

(dθ2 + sin2θ dφ2

). (5)

The components of the Ricci tensor associated with such a metric are

(3)R11 =2rβ′ , (6)

(3)R22 =(3)R33

sin2 θ= 1 + e−2β(rβ′ − 1) , (7)

(3)Rij = 0 , i 6= j . (8)

In order for them to be proportional to the metric, see (4), we must impose

β(r) = −12

ln(1− k r2) , (9)

with which the metric becomes

ds2 = −dt2 + a2(t)[

dr2

1− k r2+ r2

(dθ2 + sin2θ dφ2

)], (10)

which is the famous Friedmann-Robertson-Walker metric (FRW). Note that this metric is invariant underthe redefinition

k → k

|k|, r → r

√|k| , a→ a√

|k|, (11)

so that the only relevant parameter is K ≡ k/|k| = sign k. There are three possible cases. An openUniverse, with constant negative curvature, K = −1; a flat Universe of vanishing spatial curvature,K = 0; and a closed Universe, with constant positive curvature, K = +1, see Fig. 1.

Fig. 1: The 2D projections of the 3 possible curvatures in 3D space. From left to right: closed (K = +1), flat (K = 0) and

open (K = −1) space. Rather than “flat”, a spatial section with K = 0 should be called “Euclidean” since its distance element

is dσ2 = dx2 + dy2 + dz2.

These spaces Σ have a metric dσ2 = dχ2 + sinn2 χdΩ2, where

r = sinnχ ≡

sinhχ K = −1χ K = 0sinχ K = +1

(12)

With the FRW we can calculate the dynamics satisfied by the scale factor a(t), solving the Einsteinequations. For this we have to find first the non zero components of the affine connection,

Γ0ij =

a

agij , Γi0j =

a

aδij , Γ2

12 = Γ313 =

1r,

Γ111 =

K r

1−Kr2, Γ1

22 = −r(1−Kr2) =Γ1

33

sin2 θ, (13)

Γ233 = − sin θ cos θ , Γ3

23 = cot θ .

With which the non zero components of the Ricci tensor can be written as

R00 = −3a

a, Rij =

(aa+ 2a2 + 2K

)γij , (14)

and the 4D curvature scalarR =

6a2

(aa+ a2 +K

). (15)

2.2.1 The matter and energy content of the universe

The most general matter fluid consistent with the assumption of homogeneity and isotropy is a perfectfluid, one in which an observer comoving with the fluid would see the universe around it as isotropic. Theenergy momentum tensor associated with such a fluid can be written as [6]

Tµν = p gµν + (p+ ρ)uµuν , (16)

where p(t) and ρ(t) are the pressure and energy density of the fluid at a given time in the expansion, asmeasured by this comoving observer, and uµ is the comoving four-velocity, satisfying uµuµ = −1. Forsuch a comoving observer, the matter content looks isotropic (in its rest frame),

Tµν = diag(−ρ(t), p(t), p(t), p(t)) . (17)

Before substituting into the Einstein equations, let us consider the covariant conservation of the energy-momentum tensor in the presence of an expanding universe, Tµν;µ = 0. The ν = 0 component gives

0 = ∂µTµ0 + Γµµ0T

00 − Γλµ0T

µλ = −ρ− 3

a

a(ρ+ p) . (18)

In order to find explicit solutions, one has to supplement the conservation equation with an equa-tion of state relating the pressure and the density of the fluid, p = p(ρ). The most relevant fluids incosmology are barotropic, i.e. fluids whose pressure is linearly proportional to the density, p = w ρ, andtherefore the speed of sound is constant in those fluids.

We will restrict ourselves in these lectures to three main types of barotropic fluids:

• Radiation, with equation of state pR = ρR/3, associated with relativistic degrees of freedom (i.e.particles with temperatures much greater than their mass). In this case, the energy density ofradiation decays as ρR ∼ a−4 with the expansion of the universe.• Matter, with equation of state pM ' 0, associated with nonrelativistic degrees of freedom (i.e.

particles with temperatures much smaller than their mass). In this case, the energy density ofmatter decays as ρM ∼ a−3 with the expansion of the universe.• Vacuum energy, with equation of state pV = −ρV , associated with quantum vacuum fluctuations.

In this case, the vacuum energy density remains constant with the expansion of the universe.

This is all we need in order to solve the Einstein equations. Let us now write the equations ofmotion of observers comoving with such a fluid in an expanding universe. According to general relativity,these equations can be deduced from the Einstein equations (1), by substituting the FRW metric (10) andthe perfect fluid tensor (16). The µ = i, ν = j component of the Einstein equations, together with theµ = 0, ν = 0 component constitute the so-called Friedmann equations,(

a

a

)2

=8πG

3ρ+

Λ3− K

a2, (19)

a

a= − 4πG

3(ρ+ 3p) +

Λ3. (20)

These equations contain all the relevant dynamics, since the energy conservation equation (18) can beobtained from these.

2.3 The Cosmological ParametersI will now define the most important cosmological parameters. Perhaps the best known is the Hubbleparameter or rate of expansion today, H0 = a/a(t0). We can write the Hubble parameter in units of 100km s−1Mpc−1, which can be used to estimate the order of magnitude for the present size and age of theuniverse,

H0 ≡ 100h km s−1Mpc−1 , (21)

cH−10 = 3000h−1 Mpc , (22)

H−10 = 9.773h−1 Gyr . (23)

The parameter h was measured to be in the range 0.4 < h < 1 for decades, and only in the last few yearshas it been found to lie within 4% of h = 0.70. I will discuss those recent measurements in the nextSection.

Using the present rate of expansion, one can define a critical density ρc, that which correspondsto a flat universe,

ρc ≡3H2

0

8πG= 1.88h2 10−29 g/cm3 (24)

= 2.77h−1 1011 M/(h−1 Mpc)3 (25)

= 11.26h2 protons/m3 , (26)

where M = 1.989× 1033 g is a solar mass unit. The critical density ρc corresponds to approximately6 protons per cubic meter, certainly a very dilute fluid!

In terms of the critical density it is possible to define the density parameter

Ω0 ≡8πG3H2

0

ρ(t0) =ρ

ρc(t0) , (27)

whose sign can be used to determine the spatial (three-)curvature. The name of the critical density is dueto the fact that, using the above definition, the Friedmann equation (19) can be written as

Ω− 1 =K

a2H2, (28)

valid for all times. Thus, the sign of K determines whether Ω is greater, iqual or smaller 1,

ρ < ρc ⇔ Ω < 1 ⇔ K = −1 Σ open ,

ρ = ρc ⇔ Ω = 1 ⇔ K = 0 Σ flat ,

ρ > ρc ⇔ Ω > 1 ⇔ K = +1 Σ closed .

The Ω parameter, thus determines the geometry of our Universe. For example, from the observationsof the CMB anisotropies we can deduce that our Universe has spatial sections very close to Euclidean,K = 0, and therefore the total density of our Universe is very close to critical, Ω0 = 1.005± 0.005.

Moreover, we can define the individual ratios Ωi ≡ ρi/ρc, for matter, radiation, cosmologicalconstant and even curvature, as

ΩM =8πGρM

3H20

ΩR =8πGρR

3H20

(29)

ΩΛ =Λ

3H20

ΩK = − K

a20H

20

. (30)

For instance, we can evaluate today the radiation component ΩR, corresponding to relativistic parti-cles, from the density of microwave background photons, ρCMB = π2k4T 4

CMB/(15~3c3) = 4.5 ×10−34 g/cm3, which gives ΩCMB = 2.4 × 10−5 h−2. Three approximately massless neutrinos wouldcontribute a similar amount. Therefore, we can safely neglect the contribution of relativistic particles tothe total density of the universe today, which is dominated either by non-relativistic particles (baryons,dark matter or massive neutrinos) or by a cosmological constant, and write the rate of expansion in termsof its value today, as

H2(a) = H20

(ΩR

a40

a4+ ΩM

a30

a3+ ΩΛ + ΩK

a20

a2

). (31)

An interesting consequence of these definitions is that one can now write the Friedmann equation today,a = a0, as a cosmic sum rule,

1 = ΩM + ΩΛ + ΩK , (32)

where we have neglected ΩR today. That is, in the context of a FRW universe, the total fraction ofmatter density, cosmological constant and spatial curvature today must add up to one. For instance, if wemeasure one of the three components, say the spatial curvature, we can deduce the sum of the other two.

Looking now at the second Friedmann equation (20), we can define another basic parameter, thedeceleration parameter,

q0 = −a aa2

(t0) =4πG3H2

0

[ρ(t0) + 3p(t0)

], (33)

defined so that it is positive for ordinary matter and radiation, expressing the fact that the universe expan-sion should slow down due to the gravitational attraction of matter. We can write this parameter usingthe definitions of the density parameter for known and unknown fluids (with density Ωx and arbitraryequation of state wx) as

q0 = ΩR +12

ΩM − ΩΛ +12

∑x

(1 + 3wx) Ωx . (34)

Uniform expansion corresponds to q0 = 0 and requires a cancellation between the matter and vacuumenergies. For matter domination, q0 > 0, while for vacuum domination, q0 < 0. As we will see in amoment, we are at present probing the time dependence of the deceleration parameter and can determinewith some accuracy the moment at which the universe went from a decelerating phase, dominated bydark matter, into an acceleration phase at present, which seems to indicate the dominance of some kindof vacuum energy.

2.4 Cosmological solutions to the Einstein equationsGiven a certain matter and energy composition of the Universe, it is possible to solve the Einstein equa-tions for a(t). For example, for a flat Universe, K = 0, dominated by non-relativistic matter, ρ = ρM,(i.e. a matter dominated (MD) Universe) the Friedmann equation gives:(

a

a

)2

=8πG

3ρM =

8πG3

ρ(0)M

(a0

a

)3→ aM(t) ∝ t2/3 . (35)

While for a Universe dominated by relativistic matter, ρ = ρR, (i.e. a radiation dominated (RD) Universe)we have (

a

a

)2

=8πG

3ρR =

8πG3

ρ(0)R

(a0

a

)4→ aR(t) ∝ t1/2 . (36)

Note that, in both cases, ρ ∝ t−2, and thus H ∝ t−1, valid for any power law expansion, a(t) ∝ tp, forall p.

On the other hand, por a Universe dominated by vacuum energy, ρ = ρv, (i.e. a vacuum dominated(VD) Universe) (

a

a

)2

=8πG

3ρv =

Λ3

→ av(t) ∝ exp(t√

Λ/3) , (37)

and thus the scale factor grows exponentially while the rate of expansion is constant. Clearly in this case,the Universe accelerates: a cosmological constant is a gravitational repulsor. This may seem awkward,but in fact is a consequence of the second law of thermodynamics for adiabatic expansion, see Fig. 2.

Fig. 2: Ordinary matter dilutes as it expands. According to the second law of thermodynamics, its pressure on the piston walls

must be positive, since it exerts a force on them and makes work, thus loosing energy in the expansion. On the other hand,

the vacuum energy density is always the same, by definition, independently of the volume explored. Therefore, according

to the second law of thermodynamics, its pressure must be negative and equal in magnitude to the energy density. This

negative pressure makes the volume grow progressively more rapidly, which explains the exponential expansion of the Universe

dominated by a cosmological constant.

2.5 The (ΩM , ΩΛ) planeNow that we know that the universe is accelerating, one can parametrize the matter/energy content ofthe universe with just two components: the matter, characterized by ΩM , and the vacuum energy ΩΛ.Different values of these two parameters completely specify the universe evolution. It is thus natural toplot the results of observations in the plane (ΩM , ΩΛ), in order to check whether we arrive at a consistentpicture of the present universe from several different angles (different sets of cosmological observations).

Moreover, different regions of this plane specify different behaviors of the universe. The bound-aries between regions are well defined curves that can be computed for a given model. I will now describethe various regions and boundaries.

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3

!"

t 0H0 = 0.71.1 1.0 0.9 0.8

0.6

0.5

#

Bounce

Expansion

RecollapseClosedOpen

Accelerating

Decelerating

!M

Fig. 3: Parameter space (ΩM , ΩΛ). The green (dashed) line ΩΛ = 1−ΩM corresponds to a flat universe, ΩK = 0, separating

open from closed universes. The blue (dotted) line ΩΛ = ΩM/2 corresponds to uniform expansion, q0 = 0, separating

accelerating from decelerating universes. The violet (dot-dashed) line corresponds to critical universes, separating eternal

expansion from recollapse in the future. Finally, the red (continuous) lines correspond to t0H0 = 0.5, 0.6, . . . , ∞, beyond

which the universe has a bounce.

• Uniform expansion (q0 = 0). Corresponds to the line ΩΛ = ΩM/2. Points above this linecorrespond to universes that are accelerating today, while those below correspond to decelerat-ing universes, in particular the old cosmological model of Einstein-de Sitter (EdS), with ΩΛ =0, ΩM = 1. Since 1998, all the data from Supernovae of type Ia appear above this line, manystandard deviations away from EdS universes.• Flat universe (ΩK = 0). Corresponds to the line ΩΛ = 1 − ΩM . Points to the right of this line

correspond to closed universes, while those to the left correspond to open ones. In the last fewyears we have mounting evidence that the universe is spatially flat (in fact Euclidean).• Bounce (t0H0 = ∞). Corresponds to a complicated function of ΩΛ(ΩM ), normally expressed as

an integral equation, where

t0H0 =∫ 1

0da [1 + ΩM (1/a− 1) + ΩΛ(a2 − 1)]−1/2

is the product of the age of the universe and the present rate of expansion. Points above this linecorrespond to universes that have contracted in the past and have later rebounced. At present, theseuniverses are ruled out by observations of galaxies and quasars at high redshift (up to z = 10).

It is worth studying whether we can discard observationally the possibility that our Universesuffered in the past a contraction, before entering the present expansion phase. This case corre-sponds to a minimum of the scale factor (a = 0 and a > 0 in z = z∗),

x3ΩM + x2ΩK + ΩΛ = 0

2ΩΛ > x3ΩM

⇒ x3 − 3x+ 2 = z2

∗(z∗ + 3) ≤ 2ΩM

. (38)

In general, these ”bounce” cosmologies are ruled out by observations of galaxies and quasars athigh redshift. If we take the robust bound on the matter content, ΩM > 0.05, one finds z∗ < 3,while galaxies have been detected up to redshifts z ' 10. Moreover, the observation of themicrowave background, at zdec ' 1100, definitely rule out these bounce cosmologies.• Critical Universe (H = H = 0 and H > 0). Corresponds to the boundary between eternal

expansion in the future and recollapse. Since the scale factor is always positive, by definition,

if we take f(x) ≡ H2(a), with x = a0/a > 0, and using xd

dx=−1H

d

dt, we have that, for a

critical universe, xc, the function f(x) must have a minimum at cero, f(xc) = f ′(xc) = 0 andf ′′(xc) > 0, where

f(x) = x3ΩM + x2ΩK + ΩΛ = 0 , (39)

f ′(x) = 3x2ΩM + 2xΩK = 0xc = 0xc = −2ΩK/3ΩM > 0

, (40)

f ′′(x) = 6xΩM + 2ΩK =

+2ΩK > 0 xc = 0−2ΩK > 0 xc = 2|ΩK |/3ΩM

. (41)

Using now the cosmic sum rule, (ΩM − 1)/ΩM = 3(−ΩK/3ΩM) − 4(ΩΛ/4ΩM) and sometrigonometry, sin 3u = 3 sinu− 4 sin3 u with xc = 2 sinu, we find the solution

ΩΛ =

0 ΩM ≤ 1

4ΩM sin3[

13arcsin(1− Ω−1

M )]

ΩM ≥ 1. (42)

The first solution corresponds to the critical point xc = 0 (a =∞), and ΩK > 0, while the secondsolution corresponds to xc = 2|ΩK |/3ΩM and ΩK < 0. Expanding around ΩM = 1, we findΩΛ ' 4

27 (ΩM − 1)3/Ω2M, for ΩM ≥ 1. These critical solutions are asymptotic to the Einstein-de

Sitter model (ΩM = 1, ΩΛ = 0).

These boundaries, and the regions they delimit, can be seen in Fig. 3, together with the lines of equalt0H0 values.

In summary, the basic cosmological parameters that are now been hunted by a host of cosmologicalobservations are the following: the present rate of expansion H0; the age of the universe t0; the deceler-ation parameter q0; the spatial curvature ΩK ; the matter content ΩM ; the vacuum energy ΩΛ; the baryondensity ΩB; the neutrino density Ων , and many other that characterize the perturbations responsible forthe large scale structure (LSS) and the CMB anisotropies.

2.6 Geodesic movement in FRW metricIn order to determine in which epoch do we live, and learn about the evolution of the Universe, it isnecessary to measure distant processes via the properties of light and other particles (neutrinos, cosmicrays, gravitational waves) that reach us from far away, and thus deduce the different fractions of matterand energy that the Universe had in the past, at the time when the photon or the neutrino was emitted.That is, in order to interpret these measurements, we need to consider the geodesic movement of theseparticles in the (averaged) FRW metric.

Note that the FRW metric has 6 space-like Killing vectors (3 rotations y 3 translations, associatedwith the isometries of isotropy and homogeneity); However, there is no time-like Killing vector that maygive us a notion of conserved energy (in fact, energy is not conserved, it gets diluted in the expansion).

Let us consider a particle non-comoving with the fluid, with 4-velocity

uµ =dxµ

ds= (u0, ui) = (γ, γ vi) , (43)

and non vanishing peculiar velocity, vi = dxi

dt → |v|2 = gijv

ivj and γ = (1 − |v|2)−1/2. This particlewill follow a geodesic trajectory

duµ

ds+ Γµνλ u

νuλ = 0 . (44)

The µ = 0 component is du0

ds + Γ0νλ u

νuλ = 0. In the FRW metric, the only non-zero element of the affin

connection is Γ0ij =

a

agij , therefore

du0

ds+a

a|u|2 = 0 . (45)

Now, the normalizacion uµuµ = −1 leads us to u20 = 1 + |u|2, and thus u0du0 = |u| d|u|. Finally,

1u0

d|u|ds

+a

a|u| = 0 . (46)

Substituting u0 = dt/ds and integrating,

|u| ∝ 1a

⇒ |p| ∝ 1a, (47)

that is, the 3-moment magnitude of every particle evolves inversely proportional to the scale factor. For

a photon, ds2 = 0, we can use the Einstein-De Broglie relation, p =h

λ, and write

λ1

λ0=a(t1)a(t0)

⇒ z ≡ λ0 − λ1

λ1=a0

a1− 1 , (48)

which defines the cosmological redshift parameter, z, and its relation with the scale factor,

a(z) =a0

1 + z. (49)

An immediate question that arises is this: If photons loose energy in the expansion, where does it go?In the covariant conservation of energy, we saw that the total energy (matter-radiation plus geometry) isconserved, so what happens is that kinetic energy is converted into potential energy; i.e., the energy lostby photons and matter is used to expand the universe. OK, but how did the Universe start expanding inthe first place? That is a question we will have to leave for the next chapter of Inflation.

Fig. 4: Spectra of galaxies at different redshifts, from z = 0.107 to z = 2.401. The characteristic spectral lines correspond to

some elements like Hidrogen and Helium, which are more abundant, and some molecules, like CO, which allow us to determine

the cosmological redshift of each galaxy.

In Fig. 4 we show a series of galaxy spectra at different redshifts, together with the restframespectrum in the lab, with some characteristic lines that permit to compute the frequency shift due to theexpansion of the Universe. Note that, although the wavelengths get stretched like in the ordinary Dopplereffect, the cosmological redshift is not a Doppler shift: it is the expansion of space-time itself, not therelative velocity between observer and emitter, which is responsible for the redshift of wavelengths.Remember that in curved space-time we cannot perform a parallel transport of velocity vectors in orderto compare them, because this depends on the chosen path. However, the redshift is something we canmeasure. We know, from numerous observations in the laboratory, the wavelengths of a huge numberof spectral lines of atoms and molecules that generate the light we observe from distant galaxies, andtherefore we can compare and determine how much these wavelengths have changed in the course ofexpansion since they were emitted, λ1, until the moment we observe them here on Earth, λ2. Fromhere we deduce the ratio of the two scale factors, but we cannot determine the times at which they were

emitted: photon spectra do not contain enough information as to allow us to deduce how much coordinatetime has elapsed during its journey of hundreds of Mpc to Earth.

We are assuming, of course, that both the emitted and the restframe spectra are identical, so that wecan actually measure the effect of the intervening expansion, i.e. the growth of the scale factor from t1 tot0, when we compare the two spectra. Note however, that if the emitting galaxy and our own participatedin the expansion, i.e. if our measuring rods (our rulers) also expanded with the universe, we would seeno effect! The reason we can measure the redshift of light from a distant galaxy is because our galaxy isa gravitationally bounded object that has decoupled from the expansion of the universe. It is the distancebetween galaxies that changes with time, not the sizes of galaxies, nor the local measuring rods.

2.7 Kinematics in FRW: Hubble’s LawWe can now evaluate the relationship between physical distance and redshift as a function of the rate ofexpansion of the universe. In an approximate way, since a photon travels at the speed of light, its time offlight should be simply the distance covered, measured in lightseconds, but what is the “distance” of a faraway galaxy in an expanding Universe? Let us suppose that we are comoving observers with coordinates(r0, θ0, φ0). We can ask, From what coordinates (r, θ, φ) comes a light signal emitted at t = t1 anddetected at t = t0? The past light cone can be seen in Fig. 5.

Fig. 5: The past lightcone of an event at (t0, r0, θ0, φ0) goes through the point (t1, r1, θ0, φ0) where a source emits a photon,

which follows a null geodesic to us.

Because of homogeneity we can always choose our position to be at the origin r = 0 of ourspatial section. Imagine an object (a star) emitting light at time t1, at coordinate distance r1 from theorigin. Because of isotropy we can ignore the angular coordinates (θ, φ). Then the physical distance,to first order, will be d = a0 r1. Since light travels along null geodesics, ds2 = 0, we can write0 = −dt2 + a2(t) dr2/(1−Kr2), and therefore,∫ t0

t1

dt

a(t)=∫ r1

0

dr√1−Kr2

≡ f(r1) =

arcsin r1 K = 1r1 K = 0arcsinh r1 K = −1

(50)

If we now Taylor expand the scale factor to first order,

11 + z

=a(t)a0

= 1 +H0(t− t0) +O(t− t0)2 , (51)

we find, to first approximation,

r1 ≈ f(r1) =1a0

(t0 − t1) + · · · = z

a0H0+ . . .

Putting all together we find the famous Hubble law

H0 d = a0H0r1 = z ' vc , (52)

which is just a kinematical effect (we have not included yet any dynamics, i.e. the matter content ofthe universe). Note that at low redshift (z 1), one is tempted to associate the observed change inwavelength with a Doppler effect due to a hypothetical recession velocity of the distant galaxy. Thisis only an approximation. In fact, the redshift cannot be ascribed to the relative velocity of the distantgalaxy because in general relativity (i.e. in curved spacetimes) one cannot compare velocities throughparallel transport, since the value depends on the path! If the distance to the galaxy is small, i.e. z 1,the physical spacetime is not very different from Minkowsky and such a comparison is approximatelyvalid. As z becomes of order one, such a relation is manifestly false: galaxies cannot travel at speedsgreater than the speed of light; it is the stretching of spacetime which is responsible for the observedredshift.

Fig. 6: The Type Ia supernovae observed nearby show a relationship between their absolute luminosity and the timescale of

their light curve: the brighter supernovae are slower and the fainter ones are faster. A simple linear relation between the absolute

magnitude and a “stretch factor” multiplying the light curve timescale fits the data quite well. From Ref. [7].

Hubble’s law has been confirmed by observations ever since the 1920s, with increasing precision,which have allowed cosmologists to determine the Hubble parameter H0 with less and less systematicerrors. Nowadays, the best determination of the Hubble parameter was made by the Hubble SpaceTelescope Key Project [8], H0 = 72± 8 km/s/Mpc. This determination is based on objects at distancesup to 500 Mpc, corresponding to redshifts z ≤ 0.1.

Nowadays, we are beginning to probe much greater distances, corresponding to z ' 1, thanks totype Ia supernovae. These are white dwarf stars at the end of their life cycle that accrete matter froma companion until they become unstable and violently explode in a natural thermonuclear explosionthat out-shines their progenitor galaxy. The intensity of the distant flash varies in time, it takes aboutthree weeks to reach its maximum brightness and then it declines over a period of months. Althoughthe maximum luminosity varies from one supernova to another, depending on their original mass, theirenvironment, etc., there is a pattern: brighter explosions last longer than fainter ones. By studying thecharacteristic light curves, see Fig. 6, of a reasonably large statistical sample, cosmologists from theSupernova Cosmology Project [7] and the High-redshift Supernova Project [9], are now quite confidentthat they can use this type of supernova as a standard candle. Since the light coming from some of theserare explosions has travelled a large fraction of the size of the universe, one expects to be able to inferfrom their distribution the spatial curvature and the rate of expansion of the universe.

The connection between observations of high redshift supernovae and cosmological parametersis done via the luminosity distance, defined as the distance dL at which a source of absolute luminosity

Fig. 7: Upper panel: The Hubble diagram in linear redshift scale. Supernovae with ∆z < 0.01 of eachother have been

weighted-averaged binned. The solid curve represents the best-fit flat universe model, (ΩM = 0.25, ΩΛ = 0.75). Two other

cosmological models are shown for comparison, (ΩM = 0.25, ΩΛ = 0) and (ΩM = 1, ΩΛ = 0). Lower panel: Residuals of

the averaged data relative to an empty universe. From Ref. [7].

(energy emitted per unit time) L gives a flux (measured energy per unit time and unit area of the detector)F = L/4π d2

L. One can then evaluate, within a given cosmological model, the expression for dL as afunction of redshift [10],

H0 dL(z) =(1 + z)|ΩK |1/2

sinn

[∫ z

0

|ΩK |1/2 dz′√(1 + z′)2(1 + z′ΩM )− z′(2 + z′)ΩΛ

], (53)

where sinn(x) = x if K = 0; sin(x) if K = +1 and sinh(x) if K = −1, and we have used the cosmicsum rule (32).

Astronomers measure the relative luminosity of a distant object in terms of what they call theeffective magnitude, which has a peculiar relation with distance,

m(z) ≡M + 5 log10

[dL(z)Mpc

]+ 25 = M + 5 log10[H0 dL(z)] . (54)

Since 1998, several groups have obtained serious evidence that high redshift supernovae appear fainterthan expected for either an open (ΩM < 1) or a flat (ΩM = 1) universe, see Fig. 7. In fact, the universe

appears to be accelerating instead of decelerating, as was expected from the general attraction of matter,see Eq. (34); something seems to be acting as a repulsive force on very large scales. The most naturalexplanation for this is the presence of a cosmological constant, a diffuse vacuum energy that permeatesall space and, as explained above, gives the universe an acceleration that tends to separate gravitationallybound systems from each other. The best-fit results from the Supernova Cosmology Project [11] give alinear combination

0.8 ΩM − 0.6 ΩΛ = −0.16± 0.05 (1σ),

which is now many sigma away from an EdS model with Λ = 0. In particular, for a flat universe thisgives

ΩΛ = 0.71± 0.05 and ΩM = 0.29± 0.05 (1σ).

Surprising as it may seem, arguments for a significant dark energy component of the universe whereproposed long before these observations, in order to accommodate the ages of globular clusters, as wellas a flat universe with a matter content below critical, which was needed in order to explain the observeddistribution of galaxies, clusters and voids.

Taylor expanding the scale factor to third order,

a(t)a0

= 1 +H0(t− t0)− q0

2!H2

0 (t− t0)2 +j03!H3

0 (t− t0)3 +O(t− t0)4 , (55)

where

q0 = − a

aH2(t0) =

12

∑i

(1 + 3wi)Ωi =12

ΩM − ΩΛ , (56)

j0 = +...a

aH3(t0) =

12

∑i

(1 + 3wi)(2 + 3wi)Ωi = ΩM + ΩΛ , (57)

are the deceleration and “jerk” parameters. Substituting into Eq. (69) we find

H0 dL(z) = z +12

(1− q0) z2 − 16

(1− q0 − 3q20 + j0) z3 +O(z4) . (58)

This expression goes beyond the leading linear term, corresponding to the Hubble law, into the secondand third order terms, which are sensitive to the cosmological parameters ΩM and ΩΛ. It is only recentlythat cosmological observations have gone far enough back into the early universe that we can begin toprobe these terms, see Fig. 8.

This extra component of the critical density would have to resist gravitational collapse, otherwiseit would have been detected already as part of the energy in the halos of galaxies. However, if most of theenergy of the universe resists gravitational collapse, it is impossible for structure in the universe to grow.This dilemma can be resolved if the hypothetical dark energy was negligible in the past and only recentlybecame the dominant component. According to general relativity, this requires that the dark energy havenegative pressure, since the ratio of dark energy to matter density goes like a(t)−3p/ρ. This argumentwould rule out almost all of the usual suspects, such as cold dark matter, neutrinos, radiation, and kineticenergy, since they all have zero or positive pressure. Thus, we expect something like a cosmologicalconstant, with a negative pressure, p ≈ −ρ, to account for the missing energy.

However, if the universe was dominated by dark matter in the past, in order to form structure, andonly recently became dominated by dark energy, we must be able to see the effects of the transition fromthe deceleration into the acceleration phase in the luminosity of distant type Ia supernovae. This has beensearched for since 1998, when the first convincing results on the present acceleration appeared. However,only recently [12] do we have clear evidence of this transition point in the evolution of the universe. Thiscoasting point is defined as the time, or redshift, at which the deceleration parameter vanishes,

q(z) = −1 + (1 + z)d

dzlnH(z) = 0 , (59)

-1.0

-0.5

0.0

0.5

1.0

!(m

-M) (

mag

)

HST DiscoveredGround Discovered

0.0 0.5 1.0 1.5 2.0z

-1.0

-0.5

0.0

0.5

1.0

!(m

-M) (

mag

)

q(z)=q0+z(dq/dz)0

Coasting, q(z)=0

Constant Acceleration, q0=-, dq/dz=0 (j0=0)

Acceleration+Deceleration, q0=-, (dq/dz)0=++Acceleration+Jerk, q0=-, j0=++

Constant Deceleration, q0=+, dq/dz=0 (j0=0)

Fig. 8: The Supernovae Ia residual Hubble diagram. Upper panel: Ground-based discoveries are represented by diamonds,

HST-discovered SNe Ia are shown as filled circles. Lower panel: The same but with weighted averaged in fixed redshift bins.

Kinematic models of the expansion history are shown relative to an eternally coasting model q(z) = 0. From Ref. [12].

where

H(z) = H0

[ΩM (1 + z)3 + Ωx e

3R z0 (1+wx(z′)) dz′

1+z′ + ΩK(1 + z)2]1/2

, (60)

and we have assumed that the dark energy is parametrized by a density Ωx today, with a redshift-dependent equation of state, wx(z), not necessarily equal to −1. Of course, in the case of a true cosmo-logical constant, this reduces to the usual expression.

Let us suppose for a moment that the barotropic parameter w is constant, then the coasting redshiftcan be determined from

q(z) =12

[ ΩM + (1 + 3w) Ωx (1 + z)3w

ΩM + Ωx (1 + z)3w + ΩK(1 + z)−1

]= 0 , (61)

⇒ zc =(

(3|w| − 1)Ωx

ΩM

) 13|w|− 1 , (62)

which, in the case of a true cosmological constant, reduces to

zc =(2ΩΛ

ΩM

)1/3− 1 . (63)

When substituting ΩΛ ' 0.74 and ΩM ' 0.26, one obtains zc ' 0.8, in excellent agreement with recentobservations [12]. The plane (ΩM , ΩΛ) can be seen in Fig. 9, which shows a significant improvementwith respect to previous data.

Now, if we have to live with this vacuum energy, we might as well try to comprehend its origin.For the moment it is a complete mystery, perhaps the biggest mystery we have in physics today [13].We measure its value but we don’t understand why it has the value it has. In fact, if we naively predictit using the rules of quantum mechanics, we find a number that is many (many!) orders of magnitudeoff the mark. Let us describe this calculation in some detail. In non-gravitational physics, the zero-pointenergy of the system is irrelevant because forces arise from gradients of potential energies. However, we

Fig. 9: The recent supernovae data on the (ΩM , ΩΛ) plane. Shown are the 1-, 2- and 3-σ contours, as well as the data from

1998, for comparison. It is clear that the old EdS cosmological model at (ΩM = 1, ΩΛ = 0) is many standard deviations away

from the data. From Ref. [12].

know from general relativity that even a constant energy density gravitates. Let us write down the mostgeneral energy momentum tensor compatible with the symmetries of the metric and that is covariantlyconserved. This is precisely of the form T

(vac)µν = pV gµν = − ρV gµν , see Fig. 2. Substituting into

the Einstein equations (1), we see that the cosmological constant and the vacuum energy are completelyequivalent, Λ = 8πGρV , so we can measure the vacuum energy with the observations of the accelerationof the universe, which tells us that ΩΛ ' 0.7.

On the other hand, we can estimate the contribution to the vacuum energy coming from the quan-tum mechanical zero-point energy of the quantum oscillators associated with the fluctuations of all quan-tum fields,

ρthV =∑i

∫ ΛUV

0

d3k

(2π)3

12

~ωi(k) =~Λ4

UV

16π2

∑i

(−1)FNi +O(m2iΛ

2UV ) , (64)

where ΛUV is the ultraviolet cutoff signaling the scale of new physics. Taking the scale of quantum grav-ity, ΛUV = MPl, as the cutoff, and barring any fortuituous cancellations, then the theoretical expectation

(64) appears to be 120 orders of magnitude larger than the observed vacuum energy associated with theacceleration of the universe,

ρthV ' 1.4× 1074 GeV4 = 3.2× 1091 g/cm3 , (65)

ρobsV ' 0.7 ρc = 0.66× 10−29 g/cm3 = 2.9× 10−11 eV4 . (66)

Even if we assumed that the ultraviolet cutoff associated with quantum gravity was as low as the elec-troweak scale (and thus around the corner, liable to be explored in the LHC), the theoretical expectationwould still be 60 orders of magnitude too big. This is by far the worst mismatch between theory andobservations in all of science. There must be something seriously wrong in our present understanding ofgravity at the most fundamental level. Perhaps we don’t understand the vacuum and its energy does notgravitate after all, or perhaps we need to impose a new principle (or a symmetry) at the quantum gravitylevel to accommodate such a flagrant mismatch.

In the meantime, one can at least parametrize our ignorance by making variations on the idea of aconstant vacuum energy. Let us assume that it actually evolves slowly with time. In that case, we do notexpect the equation of state p = −ρ to remain true, but instead we expect the barotropic parameter w(z)to depend on redshift. Such phenomenological models have been proposed, and until recently producedresults that were compatible withw = −1 today, but with enough uncertainty to speculate on alternativesto a truly constant vacuum energy. However, with the recent supernovae results [12], there seems to belittle space for variations, and models of a time-dependent vacuum energy are less and less favoured. Inthe near future, the SNAP satellite [14] will measure several thousand supernovae at high redshift andtherefore map the redshift dependence of both the dark energy density and its equation of state with greatprecision. This will allow a much better determination of the cosmological parameters ΩM and ΩΛ.

2.8 Cosmological distancesApart from the age of the Universe, it is possible to measure the main cosmological parameters bymeasuring various distances to far away objects and determining how the Universe has changed sincelight (or other messengers) was emitted by these objects. Perhaps the most used distance has been theluminosity distance, mentioned in the previous sections.

2.8.1 Luminosity distance

Suppose we have a source that emits light at a distance dL from a detector of area dA on Earth. Theluminosity distance L of the source is nothing more than the distance at which we would place a sourcethat emits that amount of energy per unit time. A standard candle is a luminous object which can becalibrated with some precision and whose absolute luminosity is known or reliably estimated withinsome systematic error. For example, the ”Cepheids” variable stars that Hubble used, or the type Iasupernovae used more recently, are considered by many astronomers as reasonably standard, i.e. whosecalibration errors are within acceptable limits. On the other hand, the energy flux F received in thephoton detector is a measure of the energy deposited in such a detector per unit time and unit area. Thenwe define the luminosity distance dL as the radius of a sphere centered on the source, for whom theabsolute luminosity of the source L would five the observed flux in the detector, F = L/(4π d2

L). In aUniverse described by a FRW metric, light travels along null geodesics, ds2 = 0, and thus

a0H0 dr√1 + a2

0H20 r

2 ΩK

=dz√

ΩM (1 + z)3 + ΩΛ + ΩK (1 + z)2, (67)

which determines the coordinate distance r = r(z,H0,ΩM,ΩΛ), as a function of redshift z and theother cosmological parameters. Let us consider now the effect of expansion on the flux coming fromthe distant source at a redshift z from us. First, the energy of the photon on its way to us has suffered aredshift and thus the observed energy should be E0 = E/(1 + z). Then, the rate at which photons arrive

at the detector has suffered time dilation with respect to that at the source, dt0 = (1 + z)dt. Finally, thesurface fraction of the 2-sphere centered at the source covered by the detector is dA/(4πa2

0 r2(z)). Thus

the total flux at the detector is

F =L

4πa20 (1 + z)2 r2(z)

≡ L4π d2

L

. (68)

The final expression of the luminosity distance dL as a function of redshift is given by

H0 dL(z) =(1 + z)|ΩK |1/2

sinn

[|ΩK |1/2

∫ z

0

dz′√ΩM (1 + z′)3 + ΩΛ + ΩK (1 + z′)2

]. (69)

Let us compute this distance in the two cases of previous sections. For an open universe without cosmo-logical constant, it is possible to compute the integral explicitly,

H0 dL(z) =1q2

0

[1− q0(1− z)− (1− q0)(1 + 2q0z)1/2

](70)

' z +12

(1− q0)z2 +12

(q0 − 1)q0 z3 +O(z4) , (71)

where the deceleration parameter is given by q0 = ΩM/2, and we have expanded to third order.

On the other hand, for a flat universe with cosmological constant, it is possible to write the integralin terms of hypergeometric functions,

H0 dL(z) =2(1 + z)√

ΩM

[F(1

6,12,76

;ΩM − 1

ΩM

)− 1√

1 + zF(1

6,12,76

;ΩM − 1

ΩM (1 + z)3

)](72)

' z +12

(1− q0)z2 +16

(3q20 + q0 − 2) z3 +O(z4) , (73)

where the deceleration parameter is in this case given by q0 = ΩM/2−ΩΛ. Both expressions go beyondfirst order, corresponding to Hubble law, see Eq. (52), and coincide at second order, being sensitive to thecosmological parameters ΩM y ΩΛ. Only recently has it been possible, with cosmological observationsof distant supernovae, to reach distances (and thus ages of the Universe) sufficiently far to be sensitive tohigher order terms in the expansion.

2.8.2 Angular diameter distance

Another distance often used in cosmology is the angular diameter distance, when the energy output ofa source is unknown (or too variable) like for distant galaxies and quasars, but for whom its physicaldimensions are known or estimated, like for example in the case of the baryon acoustic oscillation scale.Thus, suppose we know the proper diameter D of an extended object. The angle subtended by thatsource, seen from Earth, will be

θ =D

a(t1)r1(z), (74)

where t1 is the cosmological time at which the object emitted its light, and r1 is its coordinate distance(we occupy the position r = 0). Then, we define the angular diameter distance as

dA(z) ≡ D

θ= a(t1) r1(z) =

a0r1(z)1 + z

=dL(z)

(1 + z)2, (75)

which can be written in terms of the cosmological parameters ΩM and ΩΛ as

H0dA(z) =(1 + z)−1

|ΩK |1/2sinn

[|ΩK |1/2

∫ z

0

dz′√ΩM (1 + z′)3 + ΩΛ + ΩK (1 + z′)2

], (76)

and expanded to second order in redshift,

H0dA(z) = z − 12

(3 + q0)z2 +O(z3) . (77)

Note that this distance has a different redshift dependence than the luminosity distance. Its complemen-tarity can be used to determine the cosmological parameters ΩM and ΩΛ.

2.8.3 Proper motion distance

When neither the absolute flux nor the absolute size of an object is known or reliably estimated, one canstill use its proper motion to determine its distance from us. Suppose we know the absolute speed of anobject, u(t1), at the source’s position, and we measure locally the angular motion across the sky, θ. Thenthe proper motion distance is defined as

dM (z) ≡ u(t1)θ

= a(t0) r1(z) = (1 + z) dA(z) =dL(z)

(1 + z), (78)

which can be written in terms of the cosmological parameters ΩM and ΩΛ as

H0dM (z) =1

|ΩK |1/2sinn

[|ΩK |1/2

∫ z

0

dz′√ΩM (1 + z′)3 + ΩΛ + ΩK (1 + z′)2

], (79)

and expanded to second order in redshift,

H0dM (z) = z − 12

(1 + q0)z2 +O(z3) . (80)

Note that this distance has a different redshift dependence than the luminosity distance and the angulardiameter distance, and thus can be used to determine the cosmological parameters ΩM and ΩΛ.

2.8.4 Number counts of galaxies, quasars and clusters of galaxies

An extraordinarily simple (geometric) way of determining the cosmological parameters is counting ob-jects (galaxies or other luminous sources like quasars) at large distances. Suppose we have a comovingvolume element dVc(z), and let dNgal be the number of galaxies within that volume. Assuming we knowthe number density of galaxies per comoving volume element, as a function of time, nc(t), we have

dNgal = nc(t) dVc = nc(t)r2dr dΩ√1−Kr2

. (81)

Using (67), we can write the differential counting of galaxies as

dNgal

dz dΩ= (a0H0)−3 nc(z)

(a2

0H20r

2(z) a0H0dr

dz(1−Kr2)−1/2

)(82)

= (a0H0)−3 nc(z) z2(

1− 2(1 + q0)z + . . .). (83)

If we now suppose that nc(z) ' const. − i.e. that galaxies are neither created nor destroyed (which isa more or less reasonable approximation in the redshift range 0.2 < z < 6), we can use dNgal/dz dΩto determine q0. This way of measuring parameters can be used also with quasars, which allow one toreach even larger distances.

2.9 Cosmological Horizons and conformal timeSince the advent of special relativity we are aware of the fact that there can be events that are not acces-sible to us, simply because e.g. their distance to us is larger than what it takes light to travel in the timesince it was emitted. Since nothing travels faster than the speed of light, this determines the past and fu-ture light cones. The maximum distance accessible to us is called a horizon, in analogy with the physicalhorizon due to the curvature of the Earth, which prevented people on shore to see the approaching shipuntil it was above the horizon.

In the case of cosmological horizons, the situation is slightly more complicated because, contraryto the case of Minkowski, cosmological space-time (S.T.) is dynamical, and distances between objectsare not fixed but change over the course of cosmic time. For example, if the universe is decelerating,light can reach us from distances that are larger than in ordinary Minkowski S.T., see e.g. Fig. 10.

Fig. 10: The past light cone of an event at time t = t0 depends on the dynamics of space-time. In a FRW Universe dominated

by radiation, the particle horizon has twice the size of the Minkowski horizon.

2.9.1 Cosmological time as a function of redshift

Let us first compute the non-linear relation between redshift and time. While in Minkowski time andspace are univocally determined, this is not true in cosmology. For a known dynamics of the expansionof the Universe (i.e. once known the cosmological parameters and the matter-energy content of theUniverse), one can estimate the time at which a photon was emitted from a source at redshift z,

H0 t(z) =∫ (1+z)−1

0

H0 da

aH(a)=∫ (1+z)−1

0

da√ΩM/a+ ΩΛ a2 + ΩK

. (84)

For an open Universe without cosmological constant (ΩΛ = 0) we find

H0 t(z) =1

(1 + z)√

ΩK

√1 +

ΩM(1 + z)ΩK

− ΩM√Ω3K

sinh−1

(ΩK

ΩM(1 + z)

)1/2

, (85)

which in the limit ΩK → 0 becomes

H0 t(z) =2

3√

ΩM(1 + z)−3/2 . (86)

On the other hand, for a Universe with Euclidean spatial sections (ΩK = 0) we have

H0 t(z) =2

3√

ΩΛsinh−1

(ΩΛ

ΩM(1 + z)3

)1/2

, (87)

which has the same limit (86) as ΩΛ → 0.

For example, let us calculate the cosmic time in which light was emitted by a galaxy at redshiftz = 10, for a flat Universe with ΩM = 0.3. Using (87) we find H0 t = 0.03335, i.e. light was emittedwhen the Universe was 460 Myr old (after the Big Bang), approximately at 3% of the age of the Universe.

2.9.2 Particle Horizon

One of the most important distances in Cosmology is the particle horizon since it gives us the extent ofthe causal region we can access by the measurement of the most distant particles. The particle horizonis the proper distance travelled by a photon since the origin of the Universe until today,

dPH(z) = a(z)∫ rH

0

dr√1−Kr2

= a(t)∫ t

0

dt′

a(t′)=

a0

1 + z

∫ a

0

da′

a′2H(a′). (88)

For an open Universe without cosmological constant (ΩΛ = 0) we find

dPH(z) =2

H0(1 + z)|ΩK |−1/2sinn−1

(|ΩK |

ΩM(1 + z)

)1/2

. (89)

In the limit of flat EdS space, ΩK → 0, we have

dPH =2

H0

√ΩM

(1 + z)−3/2 = 3t , (90)

which is directly proportional to the distance travelled by light since the origin of the Universe at t = 0.The factor 3 (for the matter era) simply indicates that the expansion is decelerated and thus a photontravels more distance than it would in an otherwise flat Minkowski S.T., dH = ct, see Fig. 10.

On the other hand, for a Universe with Euclidean spatial sections (ΩK = 0) we have

dPH(z) =2

H0

√ΩM

(1 + z)−3/2F(1

6,12,76

;ΩM − 1

ΩM (1 + z)3

). (91)

In the limit of flat EdS space, ΩM → 1, we have

dPH =2

H0

√ΩM

(1 + z)−3/2 = 3t , (92)

which naturally coincides with the previous expression since in that limit the two models coincide withthe EdS model.

2.9.3 Visible Horizon

In universes like ours, in which the expansion is cooling the hot plasma and particles decouple one afteranother, as their interactions become slower than the rate of expansion (see Section 3), there is a horizonwhich is different from the particle horizon, although of similar origin. The visible horizon is the properdistance travelled by a particle, relativistic or not, since the moment it decoupled from the rest of theplasma,

dV H(t) = a(t)∫ t

tdec

dt′

a(t′), (93)

and depends on the particle, naturally. For example, photons decoupled when the Universe was approx-imately tdec,γ ∼ 380.000 yrs old, and since then have travelled unimpeded until they reached us. Dueto its interactions with the plasma, the dispersion of photons in the charged plasma before decouplingprevents us from reaching beyond this moment, in a similar way as their dispersion in the water vapor ofa cloud prevents us from seeing inside the cloud. The Visible Universe is that which comprises the whole

sphere of the visible horizon and its size differs from that of the particle horizon, since the latter takes intoaccount also the evolution of the Universe since its origin in the Big Bang until photon decoupling. Onthe other hand, neutrinos have a visible horizon much larger than that of photons since they decoupledwhen the Universe was approximately one second old; while gravitons (i.e. gravitational waves) havean even larger visible horizon since they decoupled from the plasma at Planck scale, tdec,gw ∼ 10−43 s.For the moment, the only visible horizon we detected was that of photons in the cosmic microwavebackground; perhaps in the future we may detect the cosmic neutrino background or even the primordialgravitational wave background.

2.9.4 Event Horizon

In universes with accelerated expansion, like today or during inflation, two particles that are separatedmore than a certain distance will never be in causal contact in the future: the expansion of space-timewill separate them irremissibly. That distance is known as event horizon and can be computed as

dEH(t) = a(t)∫ ∞t

dt′

a(t′). (94)

In the case of a Universe dominated by a cosmological constant, with constant rate of expansion, H =√Λ/3, the event horizon is also constant, dEH = H−1. Two particles in that Universe that are separated

more than a distance H−1 will never see each other again, unless such a period of exponential expansionends and a decelerating stage follows, which can allow events to enter the horizon again, as we believehappens after inflation, see Section 5.

2.9.5 Conformal time and Hubble radius

The horizon distance characterizes the size of a Universe at a given time. While in Minkowski thisdistance is given by the size of the past light cone, see Fig. 10, in a Friedmann-Robertson-Walker space-time, a(t) ∼ tp with p < 1, this distance, dH(t) = t/(1− p), is significantly larger than the Minkowskilight cone, dM = t. For example, for a flat RD universe, p = 1/2 and dH = 2t = H−1(t), whilefor a flat MD universe, p = 2/3 and dH = 3t = 2H−1(t). The peculiar time-dependence of thesedistances suggests we compare with the instantaneous rate of expansion, or what is usually called theHubble radius,

dH(t) ≡ H−1(t)

In the case of a RD universe, the horizon distance coincides with the Hubble radius, and this is the reasonone usually interchanges one with the other, but it is important to know the conceptual difference.

On the other hand, the horizon distance (88) is a physical distance. Its corresponding comovingdistance is called the conformal time

η =∫ t

0

dt′

a(t′), (95)

since the FRW metric, written in terms of this new time coordinate, is conformally equivalent (for flatuniverses) to the Minkowski metric,

ds2 = a2(η)[− dη2 + dx2

]. (96)

with the scale factor as conformal factor.

In summary, it is possible to define several cosmological distances: particle and event horizons,visible horizon and Hubble radius, etc. All these distances are of the order of the Hubble radius of theUniverse, dH ∼ H−1, and we will use them indistinguishably, unless stated otherwise.

3 HOT BIG BANG THEORY

The evolution of the Universe is nowadays understood in the context of general relativity, as we havedescribed in the previous chapter, together with a description of the matter and radiation content ofthe Universe since its origin in terms of perfect fluids with a given equation of state and in thermalequilibrium at a certain temperature. Most of the evolution of the Universe occurred at a rate sufficientlyslow that the main components of the fluid expanded adiabatically, except during those exceptionalperiods (probably the most interesting ones) in which the Universe went through phase transitions, orout of equilibrium processes, accompanied by entropy production. This ensures that the various particlespecies can be described to a very good approximation as corresponding to an equilbrium distribution, inwhich the characteristic momentum suffers a redshift displacement as the universe expands, see Eq. (47).

3.1 Thermodynamics of an expanding plasmaIn this section I will describe the main concepts associated with ensembles of particles in thermal equi-librium and the brief periods in which the universe fell out of equilibrium. To begin with, let me makecontact between the covariant energy conservation law (18) and the second law of thermodynamics,

T dS = dU + p dV , (97)

where U = ρ V is the total energy of the fluid, and p = w ρ is its barotropic pressure. Taking a comovingvolume for the universe, V = a3, we find

TdS

dt=

d

dt(ρ a3) + p

d

dt(a3) = 0 , (98)

where we have used (18). Therefore, entropy is conserved during the expansion of the universe, dS = 0;i.e., the expansion is adiabatic even in those epochs in which the equation of state changes, like in thematter-radiation transition (not a proper phase transition). Using (18), we can write

d

dtln(ρ a3) = −3H w . (99)

Thus, our universe expands like a gaseous fluid in thermal equilibrium at a temperature T . This tem-perature decreases like that of any expanding fluid, in a way that is inversely proportional to the cubicroot of the volume. This implies that in the past the universe was necessarily denser and hotter. As wego back in time we reach higher and higher temperatures, which implies that the mean energy of plasmaparticles is larger and thus certain fundamental reactions are now possible and even common, giving riseto processes that today we can only attain in particle physics accelerators. That is the reason why it isso important, for the study of early universe, to know the nature of the fundamental interactions at highenergies, and the basic connection between cosmology and high energy particle physics. However, Ishould clarify a misleading statement that is often used: “high energy particle physics colliders repro-duce the early universe” by inducing collisions among relativistic particles. Although the energies ofsome of the interactions at those collisions reach similar values as those attained in the early universe,the physical conditions are rather different. The interactions within the detectors of the great particlephysics accelerators occur typically in the perturbative regime, locally, and very far from thermal equi-librium, lasting a minute fraction of a second; on the other hand, the same interactions occurred withina hot plasma in equilibrium in the early universe while it was expanding adiabatically and its durationcould be significantly larger, with a distribution in energy that has nothing to do with those associatedwith particle accelerators. What is true, of course, is that the fundamental parameters corresponding tothose interactions−masses and couplings− are assumed to be the same, and therefore present terrestrialexperiments can help us imagine what it could have been like in the early universe, and make predictionsabout the evolution of the universe, in the context of an expanding plasma a high temperatures and highdensities, and in thermal equilibrium.

3.1.1 Fluids in thermal equilibrium

In order to understand the thermodynamical behaviour of a plasma of different species of particles at hightemperatures we will consider a gas of particles with g internal degrees of freedom weakly interacting.The degrees of freedom corresponding to the different particles can be seen in Table 1. For example,leptons and quarks have 4 degrees of freedom since they correspond to the two helicities for both particleand antiparticle. However, the nature of neutrinos is still unknown. If they happen to be Majoranafermions, then they would be their own antiparticle and the number of degrees of freedom would reduceto 2. For photons and gravitons (without mass) their 2 d.o.f. correspond to their states of polarization.The 8 gluons (also without mass) are the gauge bosons responsible for the strong interaction betwenquarks, and also have 2 d.o.f. each. The vector bosons W± and Z0 are massive and thus, apart from thetransverse components of the polarization, they also have longitudinal components.

Particle Spin Degrees of freedom (g) Nature

Higgs 0 1 Massive scalarphoton 1 2 Massless vectorgraviton 2 2 Massless tensorgluon 1 2 Massless vectorW y Z 1 3 Massive vectorleptons & quarks 1/2 4 Dirac Fermionneutrinos 1/2 4 (2) Dirac (Majorana) Fermion

Table 1: The internal degrees of freedom of various fundamental particles.

For each of these particles we can compute the number density n, the energy density ρ and thepressure p, in thermal equilibrium at a given temperature T ,

n = g

∫d3p

(2π)3f(p) , (100)

ρ = g

∫d3p

(2π)3E(p) f(p) , (101)

p = g

∫d3p

(2π)3

|p|2

3Ef(p) , (102)

where the energy is given by E2 = |p|2 + m2 and the momentum distribution in thermal (kinetic)equilibrium is

f(p) =1

e(E−µ)/T ± 1

−1 Bose− Einstein+1 Fermi−Dirac

(103)

The chemical potential µ is conserved in these reactions if they are in chemical equilibrium. For example,for reactions of the type i + j ←→ k + l , we have µi + µj = µk + µl. For example, the chemicalpotencial of the photon vanishes µγ = 0, and thus particles and antiparticles have opposite chemicalpotentials.

From the equilibrium distributions one can obtain the number density n, the energy ρ and the

pressure p, of a particle of mass m with chemical potential µ at the temperature T ,

n =g

2π2

∫ ∞m

dEE(E2 −m2)1/2

e(E−µ)/T ± 1, (104)

ρ =g

2π2

∫ ∞m

dEE2(E2 −m2)1/2

e(E−µ)/T ± 1, (105)

p =g

6π2

∫ ∞m

dE(E2 −m2)3/2

e(E−µ)/T ± 1. (106)

For a non-degenerate (µ T ) relativistic gas (m T ), we find

n =g

2π2

∫ ∞0

E2 dE

eE/T ± 1=

ζ(3)π2

g T 3 Bosons

34ζ(3)π2

g T 3 Fermions, (107)

ρ =g

2π2

∫ ∞0

E3 dE

eE/T ± 1=

π2

30g T 4 Bosons

78π2

30g T 4 Fermions

, (108)

p =13ρ , (109)

where ζ(3) = 1.20206 . . . is the Riemann Zeta function. For relativistic fluids, the energy density perparticle is

〈E〉 ≡ ρ

n=

π4

30ζ(3)T ' 2.701T Bosons

7π4

180ζ(3)T ' 3.151T Fermions

(110)

For relativistic bosons or fermions with µ < 0 and |µ| < T , we have

n =g

π2T 3 eµ/T , (111)

ρ =3gπ2

T 4 eµ/T , (112)

p =13ρ . (113)

For a bosonic particle, a positive chemical potential, µ > 0, indicates the presence of a Bose-Einsteincondensate, and should be treated separately from the rest of the modes.

On the other hand, for a non-relativistic gas (m T ), with arbitrary chemical potential µ, we find

n = g

(mT

2π

)3/2

e−(m−µ)/T , (114)

ρ = mn , (115)

p = nT ρ . (116)

The average energy density per particle is

〈E〉 ≡ ρ

n= m+

32T . (117)

Note that, at any given temperature T , the contribution to the energy density of the universe comingfrom non-relativistic particles in thermal equilibrium is exponentially supressed with respect to that ofrelativistic particles, therefore we can write

ρR =π2

30g∗ T

4 , pR =13ρR , (118)

g∗(T ) =∑

bosons

gi

(TiT

)4

+78

∑fermions

gi

(TiT

)4

, (119)

where the factor 7/8 takes into account the difference between the Fermi and Bose statistics; g∗ is thetotal number of light d.o.f. (m T ), and we have also considered the possibility that particle species i(bosons or fermions) have an equilibrium distribution at a temperature Ti different from that of photons,as happens for example when a given relativistic species decouples from the thermal bath, as we willdiscuss later. This number, g∗, strongly depends on the temperature of the universe, since as it expandsand cools, different particles go out of equilibrium or become non-relativistic (m T ) and thus becomeexponentially suppressed from that moment on. A plot of the time evolution of g∗(T ) can be seen inFig. 11.

Fig. 11: the light degrees of freedom g∗ and g∗S as a function of the temperature of the universe. From Ref. [15].

For example, for T 1 MeV, i.e. after the time of primordial Big Bang Nucleosynthesis (BBN)and neutrino decoupling, the only relativistic species are the 3 light neutrinos and the photons; since thetemperature of the neutrinos is Tν = (4/11)1/3Tγ = 1.90 K, see below, we have g∗ = 2 + 3 × 7

4 ×(411

)4/3= 3.36, while g∗S = 2 + 3× 7

4 ×(

411

)= 3.91.

For 1 MeV T 100 MeV, i.e. between BBN and the phase transition from a quark-gluonplasma to hadrons and mesons, we have, as relativistic species, apart from neutrinos and photons, alsothe electrons and positrons, so g∗ = 2 + 3× 7

4 + 2× 74 = 10.75.

For T 250 GeV, i.e. above the electroweak (EW) symmetry breaking scale, we have one photon(2 polarizations), 8 gluons (massless), the W± and Z0 (massive), 3 families of quarks and leptons, aHiggs (still undiscovered), with which one finds g∗ = 427

4 = 106.75.

At temperatures well above the electroweak transition we ignore the number of d.o.f. of particles,since we have never explored those energies in particle physics accelerators. Perhaps in the near future,with the results of the Large Hadron Collider (LHC) at CERN, we may may predict the behaviour of theuniverse at those energy scales. For the moment we even ignore whether the universe was in thermalequilibrium at those temperatures. The highest energy scale at which we can safely say the universewas in thermal equilibrium is that of BBN, i.e. 1 MeV, due to the fact that we observe the presentrelative abundances of the light element produced at that time. For instance, we can’t even claim that theuniverse went through the quark-gluon phase transtion, at ∼ 200 MeV, since we have not observed yetany signature of such an event, not to mention the electroweak phase transition, at ∼ 1 TeV.

Let us now use the relation between the rate of expansion and the temperature of relativistic parti-cles to obtain the time scale of the universe as a function of its temperature,

H = 1.66 g1/2∗

T 2

MP=

12t

=⇒ t = 0.301 g−1/2∗

MP

T 2∼(

T

MeV

)−2

s , (120)

thus, e.g. at the EW scale (100 GeV) the universe was just 10−10 s old, while during the primordial BBN(1− 0.1 MeV), it was 1 s to 3 min old.

Temperature New particle thershold 4g∗(T )

mν < T < me γ’s + ν’s 29me < T < mµ e± 43mµ < T < mπ µ± 57mπ < T < Tc π± 69Tc < T < ms u, u, d, d+ g’s 205ms < T < mc s, s 247mc < T < mτ c, c 289mτ < T < mb τ± 303mb < T < mW,Z b, b 345mW,Z < T < mH W±, Z0 381mH < T < mt H0 385mt < T t, t 427

Table 2: The effective degrees of freedom g∗(T ) of the cosmic fluid as a function of the temperature and depending on the new

thresholds for particle production. The temperature Tc ' 200 MeV corresponds to the QCD transition between the quark-gluon

plasma and the hadronic phase. The jump in effective degrees of freedom can be seen in Fig. 11.

3.1.2 The entropy of the universe

During most of the history of the universe, the rates of reaction, Γint, of particles in the thermal bathare much bigger than the rate of expansion of the universe, H , so that local thermal equilibrium wasmantained. In this case, the entropy per comoving volume remained constant. In an expanding universe,the second law of thermodynamics, applied to the element of comoving volume, of unit coordinatevolume and physical volume V = a3, can be written as, see (97),

T dS = d(ρ V ) + p dV = d[(ρ+ p)V ]− V dp = . (121)

Using the Maxwell condition of integrability,∂2S

∂T∂V=

∂2S

∂V ∂T, we find that dp = (ρ+p)dT/T , so that

dS = d

[(ρ+ p)

V

T+ const.

], (122)

i.e. the entropy in a comoving volume is S = (ρ+ p)VT , except for a constant. Using now the first law,the covariant conservation of energy, Tµν;ν = 0, we have

d[(ρ+ p)a3

]= a3 dp =⇒ d

[(ρ+ p)

a3

T

]= 0 , (123)

and thus, in thermal equilibrium, the total entropy in a comoving volume, S = a3(ρ + p)/T , is con-served. During most of the evolution of the universe, this entropy was dominated by the contributionfrom relativistic particles,

S =2π2

45g∗S (aT )3 = const. , (124)

g∗S(T ) =∑

bosons

gi

(TiT

)3

+78

∑fermions

gi

(TiT

)3

, (125)

where g∗S is the number of “entropic” degrees of freedom, as we can see in Fig. 11. Above the electron-positron annihilation, all relativistic particles had the same temperature and thus g∗S = g∗. It may be alsouseful to realize that the entropy density, s = S/a3, is propotional to the number density of relativisticparticles, and in particular to the number density of photons, s = 1.80g∗S nγ ; today, s = 7.04nγ .However, since g∗S in general is a function of temperature, we can’t always interchange s and nγ .

The conservation of S implies that the entropy density satisfies s ∝ a−3, and thus the physical sizeof the comoving volume is a3 ∝ s−1; therefore, the number of particles of a given species in a comovingvolume, N = a3n, is proportional to the number density of that species over the entropy density s,

N ∼ n

s=

45ζ(3) g2π4 g∗S

T m, µ

45 g4π5√

2 g∗S

(mT

)3/2e−

m−µT T m

(126)

If this number does not change, i.e. if those particles are neither created nor destroyed, then n/s remainsconstant. As a useful example, we will consider the barionic number in a comoving volume,

nBs≡ nb − nb

s. (127)

As long as the interactions that violate barion number occur sufficiently slowly, the barionic number percomoving volume, nB/s, will remain constant. Although

η ≡ nBnγ

= 1.80 g∗SnBs, (128)

the ratio between barion and photon numbers it does not remain constant during the whole evolutionof the universe since g∗S varies; e.g. during the annihilation of electrons and positrons, the number ofphotons per comoving volume, Nγ = a3 nγ , grows a factor 11/4, and η decreases by the same factor.After this epoch, however, g∗ is constant so that η ' 7nB/s and nB/s can be used indistinctly.

Another consequence of Eq. (124) is that S = const. implies that the temperature of the universeevolves as

T ∝ g−1/3∗S a−1 . (129)

As long as g∗S remains constant, we recover the well known result that the universe cools as it expandsaccording to T ∝ 1/a. The factor g−1/3

∗S appears because when a species becomes non-relativistic (whenT ≤ m), and effectively disappears from the energy density of the universe, its entropy is transferredto the rest of the relativistic particles in the plasma, making T decrease not as quickly as 1/a, until g∗Sagain becomes constant.

From the observational fact that the universe expands today one can deduce that in the past it musthave been hotter and denser, and that in the future it will be colder and more dilute. Since the ratio ofscale factors is determined by the redshift parameter z, we can obtain (to very good approximation) thetemperature of the universe in the past with

T = T0 (1 + z) . (130)

This expression has been spectacularly confirmed thanks to the absorption spectra of distant quasars [16].These spectra suggest that the radiation background was acting as a thermal bath for the molecules in theinterstellar medium with a temperature of 9 K at a redshift z ∼ 2, and thus that in the past the photonbackground was hotter than today. Furthermore, observations of the anisotropies in the microwave back-ground confirm that the universe at a redshift z = 1089 had a temperature of 0.3 eV, in agreement withEq. (130).

3.2 The thermal evolution of the universeIn a strict mathematical sense, it is impossible for the universe to have been always in thermal equilibriumsince the FRW model does not have a timelike Killing vector. In practice, however, we can say thatthe universe has been most of its history very close to thermal equilibrium. Of course, those periodsin which there were deviations from thermal equilibrium have been crucial for its evolution thereafter(e.g. baryogenesis, QCD transition, primordial nucleosynthesis, recombination, etc.); without these theuniverse today would be very different and probably we would not be here to tell the story.

The key to understand the thermal history of the universe is the comparison between the rates ofinteraction between particles (microphysics) and the rate of expansion of the universe (macrophysics).Ignoring for the moment the dependence of g∗ on temperature, the rate of change of T is given directlyby the rate of expansion, T /T = −H . As long as the local interactions − necessary in order that theparticle distribution function adjusts adiabatically to the change of temperature − are sufficiently fastcompared with the rate of expansion of the universe, the latter will evolve as a succession of states veryclose to thermal equilibrium, with a temperature proportional to a−1. If we evaluate the interaction ratesas

Γint ≡ 〈nσ |v|〉 , (131)

where n(t) is the number density of target particles, σ is the cross section on the interaction and v is therelative velocity of the reaction, all averaged on a thermal distribution; then a rule of thumb for ensuringthat thermal equilibrium is maintained is

Γint ∼> H . (132)

This criterium is understandable. Suppose, as often occurs, that the interaction rate in thermal equilib-rium is Γint ∝ Tn, with n > 2; then, the number of interactions of a particle after time t is

Nint =∫ ∞t

Γint(t′)dt′ =1

n− 2Γint

H(t) , (133)

therefore the particle interacts less than once from the moment in which Γint ≈ H . If Γint ∼> H , thespecies remains coupled to the thermal plasma. This doesn’t mean that, necessarily, the particle is outof local thermal equilibrium, since we have seen already that relativistic particles that have decoupledretain their equilibrium distribution, only at a different temperature from that of the rest of the plasma.

In order to obtain an approximate description of the decoupling of a particle species in an expand-ing universe, let us consider two types of interaction:

i) interactions mediated by massless gauge bosons, like for example the photon. In this case, the crosssection for particles with significant momentum transfer can be written as σ ∼ α2/T 2, with α = g2/4πthe coupling constant of the interaction. Assuming local thermal equilibrium, n(t) ∼ T 3 and thus theinteraction rate becomes Γ ∼ nσ |v| ∼ α2 T . Therefore,

ΓH∼ α2 MP

T, (134)

so that for temperatures of the universe T ∼< α2MP ∼ 1016 GeV, the reactions are fast enough and theplasma is in equilibrium, while for T ∼> 1016 GeV, reactions are too slow to maintain equilibrium andit is said that they are “frozen-out”. An important consequence of this result is that the universe couldnever have been in thermal equilibrium above the grand unification (GUT) scale.

ii) interactions mediated by massive gauge bosons, e.g. like the W± and Z0, or those responsible for theGUT interactions, X and Y . We will generically call them X bosons. The cross section depends ratherstrongly on the temperature of the plasma,

σ ∼

G2XT

2 T MX

α2

T 2T MX

(135)

where GX ∼ α/M2X is the effective coupling constant of the interaction at energies well below the mass

of the vector boson, analogous to the Fermi constant of the electroweak interaction,GF = g2/(4√

2M2W )

at tree level. Note that for T MX we recover the result for massless bosons, so we will concentratehere on the other case. For T ≤MX , the rate of thermal interactions is Γ ∼ nσ |v| ∼ G2

X T5. Therefore,

ΓH∼ G2

XMP T3 , (136)

such that at temperatures in the range

MX ∼> T ∼> G−2/3X M

−1/3P ∼

(MX

100 GeV

)4/3

MeV , (137)

reactions occur so fast that the plasma is in thermal equilibrium, while for T ∼< (MX/100 GeV)4/3 MeV,those reactions are too slow for maintaining equilibrium and they effective freeze-out, see Eq. (133).

3.2.1 The decoupling of relativistic particles

Those relativistic particles that have decoupled from the thermal bath do not participate in the transfer ofentropy when the temperature of the universe falls below the mass thershold of a given species T ' m;in fact, the temperature of the decoupled relativistic species falls as T ∝ 1/a, as we will now show.Suppose that a relativistic particle is initially in local thermal equilibrium, and that it decouples at atemperature TD and time tD. The phase space distribution at the time of decoupling is given by theequilibrium distribution,

f(p, tD) =1

eE/TD ± 1. (138)

After decoupling, the energy of each massless particle suffers redshift, E(t) = ED (aD/a(t)). Thenumber density of particles also decreases, n(t) = nD (aD/a(t))3. Thus, the phase space distribution ata time t > tD is

f(p, t) =dn

d3p= f(p

a

aD, tD) =

1eEa/aDTD ± 1

=1

eE/T ± 1, (139)

so that we conclude that the distribution function of a particle that has decoupled while being relativisticremains self-similar as the universe expands, with a temperature that decreases as

T = TDaDa∝ a−1 , (140)

and not as g−1/3∗S a−1, like the rest of the plasma in equilibrium (129).

3.2.2 The decoupling of non-relativistic particles

Those particles that decoupled from the thermal bath when they were non-relativistic (m T ) behavedifferently. Let us study the evolution of the distribution function of a non-relativistic particle that wasin local thermal equilibrium at a time tD, when the universe had a temperature TD. The moment ofeach particle suffers redshift as the universe expands, |p| = |pD| (aD/a), see Eq. (48). Therefore, theirkinetic energy satisfies E = ED (aD/a)2. On the other hand, the particle number density also varies,n(t) = nD (aD/a(t))3, so that a decoupled non-relativistic particle will have an equilibrium distributionfunction characterized by a temperature

T = TDa2D

a2∝ a−2 , (141)

and a chemical potential

µ(t) = m+ (µD −m)T

TD, (142)

whose variation is precisely that which is needed for the number density of particle to decrease as a−3.

In summary, a particle species that decouples from the thermal bath follows an equilibrium dis-tribution function with a temperature that decreases like TR ∝ a−1 for relativistic particles (TD m)or like TNR ∝ a−2 for non-relativistic particles (TD m). On the other hand, for semi-relativisticparticles (TD ∼ m), its phase space distribution does not maintain an equilibrium distribution function,and should be computed case by case.

3.2.3 Brief thermal history of the universe

I will briefly summarize here the thermal history of the Universe, from the Planck era to the present, seeFig. 12. As we go back in time, the universe becomes hotter and hotter and thus the amount of energyavailable for particle interactions increases. As a consequence, the nature of interactions goes from thosedescribed at low energy by long range gravitational and electromagnetic physics, to atomic physics,nuclear physics, all the way to high energy physics at the electroweak scale, gran unification (perhaps),and finally quantum gravity. The last two are still uncertain since we do not have any experimentalevidence for those ultra high energy phenomena, and perhaps Nature has followed a different path.

The way we know about the high energy interactions of matter is via particle accelerators, whichare unravelling the details of those fundamental interactions as we increase in energy. However, oneshould bear in mind that the physical conditions that take place in our high energy colliders are verydifferent from those that occurred in the early universe. These machines could never reproduce theconditions of density and pressure in the rapidly expanding thermal plasma of the early universe. Nev-ertheless, those experiments are crucial in understanding the nature and rate of the local fundamentalinteractions available at those energies. What interests cosmologists is the statistical and thermal proper-ties that such a plasma should have, and the role that causal horizons play in the final outcome of the earlyuniverse expansion. For instance, of crucial importance is the time at which certain particles decoupledfrom the plasma, i.e. when their interactions were not quick enough compared with the expansion of theuniverse, and they were left out of equilibrium with the plasma.

One can trace the evolution of the universe from its origin till today. There is still some speculationabout the physics that took place in the universe above the energy scales probed by present colliders.

Fig. 12: A sketch of the thermal history of the Universe.

Nevertheless, the overall layout presented here is a plausible and hopefully testable proposal. Accordingto the best accepted view, the universe must have originated at the Planck era (1019 GeV, 10−43 s)from a quantum gravity fluctuation. Needless to say, we don’t have any experimental evidence for sucha statement: Quantum gravity phenomena are still in the realm of physical speculation. However, itis plausible that a primordial era of cosmological inflation originated then. Its consequences will bediscussed below. Soon after, the universe may have reached the Grand Unified Theories (GUT) era (1016

GeV, 10−35 s). Quantum fluctuations of the inflaton field most probably left their imprint then as tinyperturbations in an otherwise very homogenous patch of the universe. At the end of inflation, the hugeenergy density of the inflaton field was converted into particles, which soon thermalized and became theorigin of the hot Big Bang as we know it. Such a process is called reheating of the universe. Sincethen, the universe became radiation dominated. It is probable (although by no means certain) that theasymmetry between matter and antimatter originated at the same time as the rest of the energy of theuniverse, from the decay of the inflaton. This process is known under the name of baryogenesis sincebaryons (mostly quarks at that time) must have originated then, from the leftovers of their annihilationwith antibaryons. It is a matter of speculation whether baryogenesis could have occurred at energiesas low as the electroweak scale (100 GeV, 10−10 s). Note that although particle physics experiments

have reached energies as high as 100 GeV, we still do not have observational evidence that the universeactually went through the EW phase transition. If confirmed, baryogenesis would constitute another“window” into the early universe. As the universe cooled down, it may have gone through the quark-gluon phase transition (102 MeV, 10−5 s), when baryons (mainly protons and neutrons) formed fromtheir constituent quarks.

The furthest window we have on the early universe at the moment is that of primordial nucleosyn-thesis (1 − 0.1 MeV, 1 s – 3 min), when protons and neutrons were cold enough that bound systemscould form, giving rise to the lightest elements, soon after neutrino decoupling: It is the realm of nuclearphysics. The observed relative abundances of light elements are in agreement with the predictions of thehot Big Bang theory. Immediately afterwards, electron-positron annihilation occurs (0.5 MeV, 1 min)and all their energy goes into photons. Much later, at about (1 eV, ∼ 105 yr), matter and radiation haveequal energy densities. Soon after, electrons become bound to nuclei to form atoms (0.3 eV, 3 × 105

yr), in a process known as recombination: It is the realm of atomic physics. Immediately after, photonsdecouple from the plasma, travelling freely since then. Those are the photons we observe as the cosmicmicrowave background. Much later (∼ 1−10 Gyr), the small inhomogeneities generated during inflationhave grown, via gravitational collapse, to become galaxies, clusters of galaxies, and superclusters, char-acterizing the epoch of structure formation. It is the realm of long range gravitational physics, perhapsdominated by a vacuum energy in the form of a cosmological constant. Finally (3K, 13 Gyr), the Sun,the Earth, and biological life originated from previous generations of stars, and from a primordial soupof organic compounds, respectively.

I will now review some of the more robust features of the Hot Big Bang theory of which we haveprecise observational evidence.

3.2.4 Primordial nucleosynthesis and light element abundance

In this subsection I will briefly review Big Bang nucleosynthesis and give the present observationalconstraints on the amount of baryons in the universe. In 1920 Eddington suggested that the sun mightderive its energy from the fusion of hydrogen into helium. The detailed reactions by which stars burnhydrogen were first laid out by Hans Bethe in 1939. Soon afterwards, in 1946, George Gamow realizedthat similar processes might have occurred also in the hot and dense early universe and gave rise to thefirst light elements [4]. These processes could take place when the universe had a temperature of aroundTNS ∼ 1 − 0.1 MeV, which is about 100 times the temperature in the core of the Sun, while the densityis ρNS = π2

30 g∗T4NS∼ 82 g cm−3, about the same density as the core of the Sun. Note, however, that

although both processes are driven by identical thermonuclear reactions, the physical conditions in starand Big Bang nucleosynthesis are very different. In the former, gravitational collapse heats up the core ofthe star and reactions last for billions of years (except in supernova explosions, which last a few minutesand creates all the heavier elements beyond iron), while in the latter the universe expansion cools the hotand dense plasma in just a few minutes. Nevertheless, Gamow reasoned that, although the early period ofcosmic expansion was much shorter than the lifetime of a star, there was a large number of free neutronsat that time, so that the lighter elements could be built up quickly by succesive neutron captures, startingwith the reaction n+ p→ D + γ. The abundances of the light elements would then be correlated withtheir neutron capture cross sections, in rough agreement with observations [6, 17].

Nowadays, Big Bang nucleosynthesis (BBN) codes compute a chain of around 30 coupled nuclearreactions [18], to produce all the light elements up to beryllium-7. 1 Only the first four or five elementscan be computed with accuracy better than 1% and compared with cosmological observations. Theselight elements are H, 4He,D, 3He, 7Li, and perhaps also 6Li. Their observed relative abundance tohydrogen is [1 : 0.25 : 3 · 10−5 : 2 · 10−5 : 2 · 10−10] with various errors, mainly systematic. The BBNcodes calculate these abundances using the laboratory measured nuclear reaction rates, the decay rate of

1The rest of nuclei, up to iron (Fe), are produced in heavy stars, and beyond Fe in novae and supernovae explosions.

Fig. 13: The relative abundance of light elements to Hidrogen. Note the large range of scales involved. From Ref. [17].

the neutron, the number of light neutrinos and the homogeneous FRW expansion of the universe, as afunction of only one variable, the number density fraction of baryons to photons, η ≡ nB/nγ . In fact,the present observations are only consistent, see Fig. 13 and Ref. [17, 18, 19], with a very narrow rangeof values of

η10 ≡ 1010 η = 6.15± 0.25 . (143)

Such a small value of η indicates that there is about one baryon per 109 photons in the universe today.Any acceptable theory of baryogenesis should account for such a small number. Furthermore, the presentbaryon fraction of the critical density can be calculated from η10 as

ΩBh2 = 3.6271× 10−3 η10 = 0.0224± 0.0018 (95% c.l.) (144)

Clearly, this number is well below closure density, so baryons cannot account for all the matter in theuniverse, as I shall discuss below.

3.2.5 Neutrino decoupling

Just before the nucleosynthesis of the lightest elements in the early universe, weak interactions were tooslow to keep neutrinos in thermal equilibrium with the plasma, so they decoupled. We can estimate thetemperature at which decoupling occurred from the weak interaction cross section, σw ' G2

FT2 at finite

temperature T , where GF = 1.2× 10−5 GeV−2 is the Fermi constant. The neutrino interaction rate, via

W boson exchange in n+ ν ↔ p+ e− and p+ ν ↔ n+ e+, can be written as [15]

Γν = nν〈σw|v|〉 ' 2.1G2FT

5 , (145)

while the rate of expansion of the universe at that time (g∗ = 10.75) was H ' 5.4 T 2/MP, whereMP = 1.22 × 1019 GeV is the Planck mass. Neutrinos decouple when their interaction rate is slowerthan the universe expansion, Γν ≤ H or, equivalently, at Tν−dec ' 0.8 MeV. Below this temperature,neutrinos are no longer in thermal equilibrium with the rest of the plasma, and their temperature continuesto decay inversely proportional to the scale factor of the universe. Since neutrinos decoupled beforee+e− annihilation, the cosmic background of neutrinos has a temperature today lower than that of themicrowave background of photons. Let us compute the difference. At temperatures above the mass of theelectron, T > me = 0.511 MeV, and below 0.8 MeV, the only particle species contributing to the entropyof the universe are the photons (g∗ = 2) and the electron-positron pairs (g∗ = 4 × 7

8 ); total number ofdegrees of freedom g∗ = 11

2 . At temperatures T ' me, electrons and positrons annihilate into photons,heating up the plasma (but not the neutrinos, which had decoupled already). At temperatures T < me,only photons contribute to the entropy of the universe, with g∗ = 2 degrees of freedom. Therefore, fromthe conservation of entropy, we find that the ratio of Tγ and Tν today must be

TγTν

=(11

4

)1/3= 1.401 ⇒ Tν = 1.945 K , (146)

where I have used TCMB = 2.725 ± 0.002 K. We still have not measured such a relic background ofneutrinos, and probably will remain undetected for a long time, since they have an average energy oforder 10−4 eV, much below that required for detection by present experiments (of order GeV), preciselybecause of the relative weakness of the weak interactions. Nevertheless, it would be fascinating if, in thefuture, ingenious experiments were devised to detect such a background, since it would confirm one ofthe most robust features of Big Bang cosmology.

3.2.6 Matter-radiation equality

Relativistic species have energy densities proportional to the quartic power of temperature and thereforescale as ρR ∝ a−4, while non-relativistic particles have essentially zero pressure and scale as ρM ∝ a−3.Therefore, there will be a time in the evolution of the universe in which both energy densities are equalρR(teq) = ρM(teq). Since then both decay differently, and thus

1 + zeq =a0

aeq=

ΩM

ΩR= 3.1× 104 ΩMh

2 , (147)

where I have used ΩRh2 = ΩCMBh

2 + Ωνh2 = 3.24× 10−5 for three massless neutrinos at T = Tν . As

I will show later, the matter content of the universe today is below critical, ΩM ' 0.3, while h ' 0.73,and therefore (1 + zeq) ' 3900, or about teq = 1300 (ΩMh

2)−2yr ' 70, 000 years after the origin ofthe universe. Around the time of matter-radiation equality, the rate of expansion (31) can be written as(a0 ≡ 1)

H(a) = H0

(ΩR a

−4 + ΩM a−3)1/2

= H0 Ω1/2M a−3/2

(1 +

aeq

a

)1/2. (148)

The horizon size is the coordinate distance travelled by a photon since the beginning of the universe,dH ∼ H−1, i.e. the size of causally connected regions in the universe. The comoving horizon size isthen given by

dH =∫

c da

a2H(a)= 2cH−1

0 Ω−1/2M a1/2

eq

(√1 +

aeq

a− 1). (149)

Thus the horizon size at matter-radiation equality (a = aeq) is

dH(aeq) = 2cH−10 Ω−1/2

M a1/2eq (√

2− 1) ' 18 (ΩMh)−1 h−1Mpc . (150)

This scale plays a very important role in theories of structure formation.

3.2.7 Recombination and photon decoupling

As the temperature of the universe decreased, electrons could eventually become bound to protons toform neutral hydrogen. Nevertheless, there is always a non-zero probability that a rare energetic photonionizes hydrogen and produces a free electron. The ionization fraction of electrons in equilibrium withthe plasma at a given temperature is given by the Saha equation [15]

1−Xeqe

(Xeqe )2

=4√

2ζ(3)√π

η

(T

me

)3/2

eEion/T , (151)

where Eion = 13.6 eV is the ionization energy of hydrogen, and η is the baryon-to-photon ratio (143). Ifwe now use Eq. (130), we can compute the ionization fraction Xeq

e as a function of redshift z. Note thatthe huge number of photons with respect to electrons (in the ratio 4He : H : γ ' 1 : 4 : 1010) impliesthat even at a very low temperature, the photon distribution will contain a sufficiently large number ofhigh-energy photons to ionize a significant fraction of hydrogen. In fact, defining recombination as thetime at which Xeq

e ≡ 0.1, one finds that the recombination temperature is Trec = 0.296 eV Eion,for η10 ' 6. Comparing with the present temperature of the microwave background, we deduce thecorresponding redshift at recombination, (1 + zrec) ' 1260.

Photons remain in thermal equilibrium with the plasma of baryons and electrons through elasticThomson scattering, with cross section

σT =8πα2

3m2e

= 6.65× 10−25 cm2 = 0.665 barn , (152)

where α = 1/137.036 is the dimensionless electromagnetic coupling constant. The mean free path ofphotons λγ in such a plasma can be estimated from the photon interaction rate, λ−1

γ ∼ Γγ = neσT c. Fortemperatures above a few eV, the mean free path is much smaller that the causal horizon at that time andphotons suffer multiple scattering: the plasma is like a dense fog. Photons will decouple from the plasmawhen their interaction rate cannot keep up with the expansion of the universe and the mean free pathbecomes larger than the horizon size: the universe becomes transparent. We can estimate this momentby evaluating Γγ = H at photon decoupling. Using ne = Xe η nγ , one can compute the decouplingtemperature as Tdec = 0.26 eV, and the corresponding redshift as 1 + zdec ' 1100. Recently, WMAPmeasured this redshift to be 1 + zdec ' 1089± 1 [20]. This redshift defines the so called last scatteringsurface, when photons last scattered off protons and electrons and travelled freely ever since. Thisdecoupling occurred when the universe was approximately tdec = 1.5 × 105 (ΩMh

2)−1/2 ' 380, 000years old.

3.2.8 The microwave background

One of the most remarkable observations ever made my mankind is the detection of the relic backgroundof photons from the Big Bang. This background was predicted by George Gamow and collaboratorsin the 1940s, based on the consistency of primordial nucleosynthesis with the observed helium abun-dance. They estimated a value of about 10 K, although a somewhat more detailed analysis by Alpher andHerman in 1949 predicted Tγ ≈ 5 K. Unfortunately, they had doubts whether the radiation would havesurvived until the present, and this remarkable prediction slipped into obscurity, until Dicke, Peebles,Roll and Wilkinson [22] studied the problem again in 1965. Before they could measure the photon back-ground, they learned that Penzias and Wilson had observed a weak isotropic background signal at a radiowavelength of 7.35 cm, corresponding to a blackbody temperature of Tγ = 3.5 ± 1 K. They publishedtheir two papers back to back, with that of Dicke et al. explaining the fundamental significance of theirmeasurement [6].

Since then many different experiments have confirmed the existence of the microwave background.The most outstanding one has been the Cosmic Background Explorer (COBE) satellite, whose FIRAS

Fig. 14: The Cosmic Microwave Background Spectrum seen by the FIRAS instrument on COBE. The left panel corresponds

to the monopole spectrum, T0 = 2.725± 0.002 K, where the error bars are smaller than the line width. The right panel shows

the dipole spectrum, δT1 = 3.372± 0.014 mK. From Ref. [21].

instrument measured the photon background with great accuracy over a wide range of frequencies (ν =1− 97 cm−1), see Ref. [21], with a spectral resolution ∆ν

ν = 0.0035. Nowadays, the photon spectrum isconfirmed to be a blackbody spectrum with a temperature given by [21]

TCMB = 2.725± 0.002 K (systematic, 95% c.l.) ± 7 µK (1σ statistical) (153)

In fact, this is the best blackbody spectrum ever measured, see Fig. 14, with spectral distortions belowthe level of 10 parts per million (ppm).

Moreover, the differential microwave radiometer (DMR) instrument on COBE, with a resolutionof about 7 in the sky, has also confirmed that it is an extraordinarily isotropic background. The devia-tions from isotropy, i.e. differences in the temperature of the blackbody spectrum measured in differentdirections in the sky, are of the order of 20µK on large scales, or one part in 105, see Ref. [23]. Thereis, in fact, a dipole anisotropy of one part in 103, δT1 = 3.372 ± 0.007 mK (95% c.l.), in the directionof the Virgo cluster, (l, b) = (264.14 ± 0.30, 48.26 ± 0.30) (95% c.l.). Under the assumption that aDoppler effect is responsible for the entire CMB dipole, the velocity of the Sun with respect to the CMBrest frame is v = 371± 0.5 km/s, see Ref. [21].2 When subtracted, we are left with a whole spectrumof anisotropies in the higher multipoles (quadrupole, octupole, etc.), δT2 = 18 ± 2 µK (95% c.l.), seeRef. [23] and Fig. 15.

Soon after COBE, other groups quickly confirmed the detection of temperature anisotropies ataround 30µK and above, at higher multipole numbers or smaller angular scales. As I shall discuss below,these anisotropies play a crucial role in the understanding of the origin of structure in the universe.

3.2.9 Large-scale structure formation

Although the isotropic microwave background indicates that the universe in the past was extraordinarilyhomogeneous, we know that the universe today is not exactly homogeneous: we observe galaxies, clus-ters and superclusters on large scales. These structures are expected to arise from very small primordialinhomogeneities that grow in time via gravitational instability, and that may have originated from tinyripples in the metric, as matter fell into their troughs. Those ripples must have left some trace as temper-ature anisotropies in the microwave background, and indeed such anisotropies were finally discovered

2COBE even determined the annual variation due to the Earth’s motion around the Sun – the ultimate proof of Copernicus’hypothesis.

Fig. 15: The Cosmic Microwave Background Spectrum seen by the DMR instrument on COBE. The top figure corresponds to

the monopole, T0 = 2.725 ± 0.002 K. The middle figure shows the dipole, δT1 = 3.372 ± 0.014 mK, and the lower figure

shows the quadrupole and higher multipoles, δT2 = 18 ± 2 µK. The central region corresponds to foreground by the galaxy.

From Ref. [23].

by the COBE satellite in 1992. The reason why they took so long to be discovered was that they appearas perturbations in temperature of only one part in 105.

3.2.10 Newtonian linear perturbation theory

Suppose we start with a perfect fluid with pressure p and energy density ρ, in the presence of a weakgravitational field described by a Newtonian potential φ. The metric in that case can be written, to firstorder, as

g00 = −1− 2φ, g0i = 0, gij = (1− 2φ)δij , (154)

and the energy-momentum tensor as

T 00 = ρ, T 0i = (ρ+ p) vi, T ij = (ρ+ p) vivj + p δij . (155)

In this case, the covariant conservation of the energy momentum tensor becomes

Tµν;ν =1√g∂ν(√gTµν) + ΓµνλT

νλ =∂p

∂xνgµν +

1√g∂ν

(√g uµuν(p+ ρ)

)+ Γµνλ(p+ ρ)uνuλ = 0 .

(156)Let us now assume that the dominant cosmic fluid during the period of structure formation (after photondecoupling) is a non-relativistic fluid with zero pressure p ρ, but whose pressure gradients can beimportant (e.g. to prevent gravitational collapse). Let us also assume that we write the comoving 4-velocity as uµ = (1,v), with peculiar velocity v. In this case, the energy conservation equation, togetherwith the Poisson equation determine the fundamental hydrodynamical equations of the fluid,

∂0ρ+∇(ρv) = 0 , (continuity eq.)

∂0v + (v · ∇)v +∇φ = −∇pρ, (Euler eq.) (157)

∇2φ = 4πGρ . (Poisson eq.)

As long as we only consider scales that are much smaller than the size of the horizon at the time ofstructure formation (z ∼ 20), these equations will give a reasonable description and we can ignore therelativistic effects of horizons, etc.

Since the universe is expanding, the physical coordinates can be written as r = a(t) x, where xare comoving coordinates. The proper fluid velocity can then be separated into the Hubble flow pluspeculiar velocities, u ≡ x,

v = a (H x + u) . (158)

Spatial gradients are now given by∇r = a−1∇x. On the other hand, energy density and pressure can bedecomposed into a background average plus a perturbation,

ρ(t,x) = ρ(t)(

1 + δ(t,x)), p(t,x) = 0 + δp(t,x) , (159)

where we have defined the density contrast as

δ(t,x) ≡ ρ(t,x)− ρ(t)ρ(t)

, (160)

with ρ(a) = ρ0 a−3 the average density of the Universe. If we make a change of coordinates from r to

x, and we linearize the equations, we find

∂0δ +∇xu = 0 , (continuity) (161)

∂0u + 2H u +1a2∇xφ+

∇xp

a2ρ= 0 , (Euler) (162)

1a2∇2

xφ− 4πG ρ δ = 0 . (Poisson) (163)

We can now combine the three equations to obtain a single second order PDE equation for the densitycontrast, which written in Fourier space

δ(t,x) ≡∫d3k δk(t) eik·x , (164)

becomesδk + 2H δk + (c2

s k2ph − 4πG ρ)δk = 0 , (165)

where c2s = dp/dρ is the speed of sound of the fluid (assuming adiabatic perturbations, δp/δρ = dp/dρ)

and kph = k/a is the wavenumber, the inverse of the physical wavelength of the perturbation. Thisequation is that of a damped harmonic oscillator. The null frequency oscillator determines the Jeanswavenumber, and its associated wavelength,

kJ =

√4πGc2s

, λJ =2πkJ

= cs

√π

Gρ, (166)

which separate stable perturbations from gravitationally unstable ones. For modes with wavenumberk kJ , the perturbation amplitude δk will grow exponentially on a dynamical timescale τgrav =Imω−1

k = (4πGρ)−1/2, which is the typical scale of gravitational collapse in this problem. If we definethe pressure response time of the fluid as the size of the perturbation over the sound speed, τpres ∼ λ/cs,then, when τpres > τgrav, gravitational collapse of the perturbation occurs before the pressure forces canrespond restoring hydrostatic equilibrium. On the other hand, if τpres < τgrav, thermal pressure preventsgravitational collapse and the fluid enters a series of damped acoustic oscillations.

3.2.11 The mater power spectrum and future galaxy surveys

While the predicted anisotropies have finally been seen in the CMB, not all kinds of matter and/or evo-lution of the universe can give rise to the structure we observe today. If we define the density contrastas [24]

δ(x, a) ≡ ρ(x, a)− ρ(a)ρ(a)

=∫d3k δk(a) eik·x , (167)

where ρ(a) = ρ0 a−3 is the average cosmic density, we need a theory that will grow a density contrast

with amplitude δ ∼ 10−5 at the last scattering surface (z = 1100) up to density contrasts of the order ofδ ∼ 102 for galaxies at redshifts z 1, i.e. today. This is a necessary requirement for any consistenttheory of structure formation [25].

Furthermore, the anisotropies observed by the COBE satellite correspond to a small-amplitudescale-invariant primordial power spectrum of inhomogeneities

P (k) = 〈|δk|2〉 ∝ kn , with n = 1 , (168)

where the brackets 〈·〉 represent integration over an ensemble of different universe realizations. Theseinhomogeneities are like waves in the space-time metric. When matter fell in the troughs of those waves,it created density perturbations that collapsed gravitationally to form galaxies and clusters of galaxies,with a spectrum that is also scale invariant. Such a type of spectrum was proposed in the early 1970s byEdward R. Harrison, and independently by the Russian cosmologist Yakov B. Zel’dovich, see Ref. [26],to explain the distribution of galaxies and clusters of galaxies on very large scales in our observableuniverse.

Today various telescopes – like the Hubble Space Telescope, the twin Keck telescopes in Hawaiiand the European Southern Observatory telescopes in Chile – are exploring the most distant regions ofthe universe and discovering the first galaxies at large distances. The furthest galaxies observed so far areat redshifts of z ' 10 (at a distance of 13.7 billion light years from Earth), whose light was emitted whenthe universe had only about 3% of its present age. Only a few galaxies are known at those redshifts, butthere are at present various catalogs like the CfA and APM galaxy catalogs, and more recently the IRASPoint Source redshift Catalog, see Fig. 16, and Las Campanas redshift surveys, that study the spatialdistribution of hundreds of thousands of galaxies up to distances of a billion light years, or z < 0.1,or the 2 degree Field Galaxy Redshift Survey (2dFGRS) and the Sloan Digital Sky Survey (SDSS),which reach z < 0.5 and study millions of galaxies. These catalogs are telling us about the evolutionof clusters and superclusters of galaxies in the universe, and already put constraints on the theory ofstructure formation. From these observations one can infer that most galaxies formed at redshifts of the

Fig. 16: The IRAS Point Source Catalog redshift survey contains some 15,000 galaxies, covering over 83% of the sky up to

redshifts of z ≤ 0.05. We show here the projection of the galaxy distribution in galactic coordinates. From Ref. [27].

order of 2 − 6; clusters of galaxies formed at redshifts of order 1, and superclusters are forming now.That is, cosmic structure formed from the bottom up: from galaxies to clusters to superclusters, and notthe other way around. This fundamental difference is an indication of the type of matter that gave rise tostructure.

We know from Big Bang nucleosynthesis that all the baryons in the universe cannot account forthe observed amount of matter, so there must be some extra matter (dark since we don’t see it) to accountfor its gravitational pull. Whether it is relativistic (hot) or non-relativistic (cold) could be inferred fromobservations: relativistic particles tend to diffuse from one concentration of matter to another, thus trans-ferring energy among them and preventing the growth of structure on small scales. This is excluded byobservations, so we conclude that most of the matter responsible for structure formation must be cold.How much there is is a matter of debate at the moment. Some recent analyses suggest that there is notenough cold dark matter to reach the critical density required to make the universe flat. If we want tomake sense of the present observations, we must conclude that some other form of energy permeates theuniverse. In order to resolve this issue, 2dFGRS and SDSS started taking data a few years ago. The firsthas already been completed, but the second one is still taking data up to redshifts z ' 5 for quasars, overa large region of the sky. These important observations will help astronomers determine the nature of thedark matter and test the validity of the models of structure formation.

Before COBE discovered the anisotropies of the microwave background there were serious doubtswhether gravity alone could be responsible for the formation of the structure we observe in the universetoday. It seemed that a new force was required to do the job. Fortunately, the anisotropies were foundwith the right amplitude for structure to be accounted for by gravitational collapse of primordial inho-mogeneities under the attraction of a large component of non-relativistic dark matter. Nowadays, thestandard theory of structure formation is a cold dark matter model with a non vanishing cosmologicalconstant in a spatially flat universe. Gravitational collapse amplifies the density contrast initially throughlinear growth and later on via non-linear collapse. In the process, overdense regions decouple fromthe Hubble expansion to become bound systems, which start attracting eachother to form larger boundstructures. In fact, the largest structures, superclusters, have not yet gone non-linear.

The primordial spectrum (168) is reprocessed by gravitational instability after the universe be-

comes matter dominated and inhomogeneities can grow. Linear perturbation theory shows that the grow-ing mode 3 of small density contrasts go like [24, 25]

δ(a) ∝ a1+3ω =a2 , a < aeq

a , a > aeq(169)

in the Einstein-de Sitter limit (ω = p/ρ = 1/3 and 0, for radiation and matter, respectively). There areslight deviations for a aeq, if ΩM 6= 1 or ΩΛ 6= 0, but we will not be concerned with them here.The important observation is that, since the density contrast at last scattering is of order δ ∼ 10−5, andthe scale factor has grown since then only a factor zdec ∼ 103, one would expect a density contrasttoday of order δ0 ∼ 10−2. Instead, we observe structures like galaxies, where δ ∼ 102. So how canthis be possible? The microwave background shows anisotropies due to fluctuations in the baryonicmatter component only (to which photons couple, electromagnetically). If there is an additional mattercomponent that only couples through very weak interactions, fluctuations in that component could growas soon as it decoupled from the plasma, well before photons decoupled from baryons. The reason whybaryonic inhomogeneities cannot grow is because of photon pressure: as baryons collapse towards denserregions, radiation pressure eventually halts the contraction and sets up acoustic oscillations in the plasmathat prevent the growth of perturbations, until photon decoupling. On the other hand, a weakly interactingcold dark matter component could start gravitational collapse much earlier, even before matter-radiationequality, and thus reach the density contrast amplitudes observed today. The resolution of this mismatchis one of the strongest arguments for the existence of a weakly interacting cold dark matter componentof the universe.

CDM n = 1

HDM n = 1 MDM

n = 1

0.001 0.01 0.1 1 10

k ( h Mpc )-1

5

P ( k

) (

h

Mpc

)-3

3

10

410

1000

100

10

1

0.1

TCDM n = .8

COBE

d ( h Mpc )-1

1000 100 10 1Microwave Background Superclusters Clusters Galaxies

Fig. 17: The power spectrum for cold dark matter (CDM), tilted cold dark matter (TCDM), hot dark matter (HDM), and mixed

hot plus cold dark matter (MDM), normalized to COBE, for large-scale structure formation. From Ref. [28].

How much dark matter there is in the universe can be deduced from the actual power spectrum (theFourier transform of the two-point correlation function of density perturbations) of the observed largescale structure. One can decompose the density contrast in Fourier components, see Eq. (160). This isvery convenient since in linear perturbation theory individual Fourier components evolve independently.A comoving wavenumber k is said to “enter the horizon” when k = d−1

H (a) = aH(a). If a certainperturbation, of wavelength λ = k−1 < dH(aeq), enters the horizon before matter-radiation equality, thefast radiation-driven expansion prevents dark-matter perturbations from collapsing. Since light can onlycross regions that are smaller than the horizon, the suppression of growth due to radiation is restrictedto scales smaller than the horizon, while large-scale perturbations remain unaffected. This is the reason

3The decaying modes go like δ(t) ∼ t−1, for all ω.

why the horizon size at equality, Eq. (150), sets an important scale for structure growth,

keq = d−1H (aeq) ' 0.083 (ΩMh)h Mpc−1 . (170)

The suppression factor can be easily computed from (169) as fsup = (aenter/aeq)2 = (keq/k)2. In otherwords, the processed power spectrum P (k) will have the form:

P (k) ∝

k , k keq

k−3 , k keq

(171)

This is precisely the shape that large-scale galaxy catalogs are bound to test in the near future, see Fig. 17.Furthermore, since relativistic Hot Dark Matter (HDM) transfer energy between clumps of matter, theywill wipe out small scale perturbations, and this should be seen as a distinctive signature in the matterpower spectra of future galaxy catalogs. On the other hand, non-relativistic Cold Dark Matter (CDM)allow structure to form on all scales via gravitational collapse. The dark matter will then pull in thebaryons, which will later shine and thus allow us to see the galaxies.

Naturally, when baryons start to collapse onto dark matter potential wells, they will convert a largefraction of their potential energy into kinetic energy of protons and electrons, ionizing the medium. As aconsequence, we expect to see a large fraction of those baryons constituting a hot ionized gas surroundinglarge clusters of galaxies. This is indeed what is observed, and confirms the general picture of structureformation.

4 DETERMINATION OF COSMOLOGICAL PARAMETERS

In this Section, I will restrict myself to those recent measurements of the cosmological parameters bymeans of standard cosmological techniques, together with a few instances of new results from recentlyapplied techniques. We will see that a large host of observations are determining the cosmologicalparameters with some reliability of the order of 10%. However, the majority of these measurements aredominated by large systematic errors. Most of the recent work in observational cosmology has beenthe search for virtually systematic-free observables, like those obtained from the microwave backgroundanisotropies, and discussed in Section 4.4. I will devote, however, this Section to the more ‘classical’measurements of the following cosmological parameters: The rate of expansion H0; the matter contentΩM; the cosmological constant ΩΛ; the spatial curvature ΩK , and the age of the universe t0.

4.1 The rate of expansion H0

Over most of last century the value of H0 has been a constant source of disagreement [29]. Around1929, Hubble measured the rate of expansion to be H0 = 500 km s−1Mpc−1, which implied an age ofthe universe of order t0 ∼ 2 Gyr, in clear conflict with geology. Hubble’s data was based on Cepheidstandard candles that were incorrectly calibrated with those in the Large Magellanic Cloud. Later on,in 1954 Baade recalibrated the Cepheid distance and obtained a lower value, H0 = 250 km s−1Mpc−1,still in conflict with ratios of certain unstable isotopes. Finally, in 1958 Sandage realized that the bright-est stars in galaxies were ionized HII regions, and the Hubble rate dropped down to H0 = 60 km s−1

Mpc−1, still with large (factor of two) systematic errors. Fortunately, in the past 15 years there hasbeen significant progress towards the determination of H0, with systematic errors approaching the 10%level. These improvements come from two directions. First, technological, through the replacement ofphotographic plates (almost exclusively the source of data from the 1920s to 1980s) with charged coupledevices (CCDs), i.e. solid state detectors with excellent flux sensitivity per pixel, which were previouslyused successfully in particle physics detectors. Second, by the refinement of existing methods for mea-suring extragalactic distances (e.g. parallax, Cepheids, supernovae, etc.). Finally, with the developmentof completely new methods to determine H0, which fall into totally independent and very broad cate-gories: a) Gravitational lensing; b) Sunyaev-Zel’dovich effect; c) Extragalactic distance scale, mainly

Cepheid variability and type Ia Supernovae; d) Microwave background anisotropies. I will review herethe first three, and leave the last method for Section 4.4, since it involves knowledge about the primordialspectrum of inhomogeneities.

4.1.1 Gravitational lensing

Imagine a quasi-stellar object (QSO) at large redshift (z 1) whose light is lensed by an interveninggalaxy at redshift z ∼ 1 and arrives to an observer at z = 0. There will be at least two differentimages of the same background variable point source. The arrival times of photons from two differentgravitationally lensed images of the quasar depend on the different path lengths and the gravitationalpotential traversed. Therefore, a measurement of the time delay and the angular separation of the differentimages of a variable quasar can be used to determine H0 with great accuracy. This method, proposed in1964 by Refsdael [30], offers tremendous potential because it can be applied at great distances and it isbased on very solid physical principles [31].

Unfortunately, there are very few systems with both a favourable geometry (i.e. a known massdistribution of the intervening galaxy) and a variable background source with a measurable time delay.That is the reason why it has taken so much time since the original proposal for the first results to comeout. Fortunately, there are now very powerful telescopes that can be used for these purposes. The bestcandidate to-date is the QSO 0957 + 561, observed with the 10m Keck telescope, for which there is amodel of the lensing mass distribution that is consistent with the measured velocity dispersion. Assuminga flat space with ΩM = 0.25, one can determine [32]

H0 = 72± 7 (1σ statistical) ± 15% (systematic) km s−1Mpc−1 . (172)

The main source of systematic error is the degeneracy between the mass distribution of the lens andthe value of H0. Knowledge of the velocity dispersion within the lens as a function of position helpsconstrain the mass distribution, but those measurements are very difficult and, in the case of lensing bya cluster of galaxies, the dark matter distribution in those systems is usually unknown, associated with acomplicated cluster potential. Nevertheless, the method is just starting to give promising results and, inthe near future, with the recent discovery of several systems with optimum properties, the prospects formeasuring H0 and lowering its uncertainty with this technique are excellent.

4.1.2 Sunyaev-Zel’dovich effect

As discussed in the previous Section, the gravitational collapse of baryons onto the potential wells gen-erated by dark matter gave rise to the reionization of the plasma, generating an X-ray halo around richclusters of galaxies, see Fig. 18. The inverse-Compton scattering of microwave background photonsoff the hot electrons in the X-ray gas results in a measurable distortion of the blackbody spectrumof the microwave background, known as the Sunyaev-Zel’dovich (SZ) effect. Since photons acquireextra energy from the X-ray electrons, we expect a shift towards higher frequencies of the spectrum,(∆ν/ν) ' (kBTgas/mec

2) ∼ 10−2. This corresponds to a decrement of the microwave background tem-perature at low frequencies (Rayleigh-Jeans region) and an increment at high frequencies, see Ref. [33].

Measuring the spatial distribution of the SZ effect (3 K spectrum), together with a high resolutionX-ray map (108 K spectrum) of the cluster, one can determine the density and temperature distributionof the hot gas. Since the X-ray flux is distance-dependent (F = L/4πd2

L), while the SZ decrement isnot (because the energy of the CMB photons increases as we go back in redshift, ν = ν0(1 + z), andexactly compensates the redshift in energy of the photons that reach us), one can determine from therethe distance to the cluster, and thus the Hubble rate H0.

The advantages of this method are that it can be applied to large distances and it is based onclear physical principles. The main systematics come from possible clumpiness of the gas (which would

Fig. 18: The Coma cluster of galaxies, seen here in an optical image (left) and an X-ray image (right), taken by the recently

launched Chandra X-ray Observatory. From Ref. [34].

reduce H0), projection effects (if the clusters are prolate, H0 could be larger), the assumption of hy-drostatic equilibrium of the X-ray gas, details of models for the gas and electron densities, and possiblecontaminations from point sources. Present measurements give the value [33]

H0 = 60± 10 (1σ statistical) ± 20% (systematic) km s−1Mpc−1 , (173)

compatible with other determinations. A great advantage of this completely new and independent methodis that nowadays more and more clusters are observed in the X-ray, and soon we will have high-resolution2D maps of the SZ decrement from several balloon flights, as well as from future microwave backgroundsatellites, together with precise X-ray maps and spectra from the Chandra X-ray observatory recentlylaunched by NASA, as well as from the European X-ray satellite XMM launched a few months ago byESA, which will deliver orders of magnitude better resolution than the existing Einstein X-ray satellite.

4.1.3 Cepheid variability

Cepheids are low-mass variable stars with a period-luminosity relation based on the helium ionizationcycles inside the star, as it contracts and expands. This time variability can be measured, and the star’sabsolute luminosity determined from the calibrated relationship. From the observed flux one can thendeduce the luminosity distance, see Eq. (69), and thus the Hubble rate H0. The Hubble Space Telescope(HST) was launched by NASA in 1990 (and repaired in 1993) with the specific project of calibrating theextragalactic distance scale and thus determining the Hubble rate with 10% accuracy. The most recentresults from HST are the following [35]

H0 = 71± 4 (random) ± 7 (systematic) km s−1Mpc−1 . (174)

The main source of systematic error is the distance to the Large Magellanic Cloud, which provides thefiducial comparison for Cepheids in more distant galaxies. Other systematic uncertainties that affect thevalue of H0 are the internal extinction correction method used, a possible metallicity dependence of theCepheid period-luminosity relation and cluster population incompleteness bias, for a set of 21 galaxieswithin 25 Mpc, and 23 clusters within z ∼< 0.03.

With better telescopes already taking data, like the Very Large Telescope (VLT) interferometerof the European Southern Observatory (ESO) in the Chilean Atacama desert, with 8 synchronized tele-scopes, and others coming up soon, like the Next Generation Space Telescope (NGST) proposed byNASA for 2008, and the Gran TeCan of the European Northern Observatory in the Canary Islands, for2010, it is expected that much better resolution and therefore accuracy can be obtained for the determi-nation of H0.

4.2 Dark MatterIn the 1920s Hubble realized that the so called nebulae were actually distant galaxies very similar to ourown. Soon afterwards, in 1933, Zwicky found dynamical evidence that there is possibly ten to a hundredtimes more mass in the Coma cluster than contributed by the luminous matter in galaxies [36]. However,it was not until the 1970s that the existence of dark matter began to be taken more seriously. At that timethere was evidence that rotation curves of galaxies did not fall off with radius and that the dynamicalmass was increasing with scale from that of individual galaxies up to clusters of galaxies. Since then,new possible extra sources to the matter content of the universe have been accumulating:

ΩM = ΩB, lum (stars in galaxies) (175)

+ ΩB, dark (MACHOs?) (176)

+ ΩCDM (weakly interacting : axion, neutralino?) (177)

+ ΩHDM (massive neutrinos?) (178)

The empirical route to the determination of ΩM is nowadays one of the most diversified of allcosmological parameters. The matter content of the universe can be deduced from the mass-to-light ratioof various objects in the universe; from the rotation curves of galaxies; from microlensing and the directsearch of Massive Compact Halo Objects (MACHOs); from the cluster velocity dispersion with the useof the Virial theorem; from the baryon fraction in the X-ray gas of clusters; from weak gravitationallensing; from the observed matter distribution of the universe via its power spectrum; from the clusterabundance and its evolution; from direct detection of massive neutrinos at SuperKamiokande; from directdetection of Weakly Interacting Massive Particles (WIMPs) at CDMS, DAMA or UKDMC, and finallyfrom microwave background anisotropies. I will review here just a few of them.

4.2.1 Rotation curves of spiral galaxies

The flat rotation curves of spiral galaxies provide the most direct evidence for the existence of largeamounts of dark matter. Spiral galaxies consist of a central bulge and a very thin disk, stabilized againstgravitational collapse by angular momentum conservation, and surrounded by an approximately spher-ical halo of dark matter. One can measure the orbital velocities of objects orbiting around the disk as afunction of radius from the Doppler shifts of their spectral lines.

The rotation curve of the Andromeda galaxy was first measured by Babcock in 1938, from thestars in the disk. Later it became possible to measure galactic rotation curves far out into the disk, anda trend was found [38, 39]. The orbital velocity rose linearly from the center outward until it reached atypical value of 200 km/s, and then remained flat out to the largest measured radii. This was completelyunexpected since the observed surface luminosity of the disk falls off exponentially with radius [38],I(r) = I0 exp(−r/rD). Therefore, one would expect that most of the galactic mass is concentratedwithin a few disk lengths rD, such that the rotation velocity is determined as in a Keplerian orbit, vrot =(GM/r)1/2 ∝ r−1/2. No such behaviour is observed. In fact, the most convincing observations comefrom radio emission (from the 21 cm line) of neutral hydrogen in the disk, which has been measuredto much larger galactic radii than optical tracers. A typical case is that of the spiral galaxy NGC 6503,where rD = 1.73 kpc, while the furthest measured hydrogen line is at r = 22.22 kpc, about 13 disklengths away. Nowadays, thousands of galactic rotation curves are known, see Fig. 19, and all suggestthe existence of about ten times more mass in the halos of spiral galaxies than in the stars of the disk. Theconnection with dark matter halos was emphasized in Ref. [40]. Recent numerical simulations of galaxyformation in a CDM cosmology [41] suggest that galaxies probably formed by the infall of material inan overdense region of the universe that had decoupled from the overall expansion.

The dark matter is supposed to undergo violent relaxation and create a virialized system, i.e.in hydrostatic equilibrium. This picture has led to a simple model of dark-matter halos as isothermalspheres, with density profile ρ(r) = ρc/(r2

c + r2), where rc is a core radius and ρc = v2∞/4πG, with

Fig. 19: The rotation curves of several hundred galaxies. Upper panel: As a function of their radii in kpc. Middle panel: The

central 5 kpc. Lower panel: As a function of scale radius.

v∞ equal to the plateau value of the flat rotation curve. This model is consistent with the universalrotation curves seen in Fig. 19. At large radii the dark matter distribution leads to a flat rotation curve.The question is for how long. In dense galaxy clusters one expects the galactic halos to overlap andform a continuum, and therefore the rotation curves should remain flat from one galaxy to another.However, in field galaxies, far from clusters, one can study the rotation velocities of substructures (likesatellite dwarf galaxies) around a given galaxy, and determine whether they fall off at sufficiently largedistances according to Kepler’s law, as one would expect, once the edges of the dark matter halo havebeen reached. These observations are rather difficult because of uncertainties in distinguishing betweentrue satellites and interlopers. Recently, a group from the Sloan Digital Sky Survey Collaboration claimthat they have seen the edges of the dark matter halos around field galaxies by confirming the fall-offat large distances of their rotation curves [42]. These results, if corroborated by further analysis, wouldconstitute a tremendous support to the idea of dark matter as a fluid surrounding galaxies and clusters,while at the same time eliminates the need for modifications of Newtonian of even Einstenian gravity atthe scales of galaxies, to account for the flat rotation curves.

That’s fine, but how much dark matter is there at the galactic scale? Adding up all the matter ingalactic halos up to a maximum radii, one finds

Ωhalo ' 10 Ωlum ≥ 0.03− 0.05 . (179)

Of course, it would be extraordinary if we could confirm, through direct detection, the existence of darkmatter in our own galaxy. For that purpose, one should measure its rotation curve, which is much more

difficult because of obscuration by dust in the disk, as well as problems with the determination of reliablegalactocentric distances for the tracers. Nevertheless, the rotation curve of the Milky Way has beenmeasured and conforms to the usual picture, with a plateau value of the rotation velocity of 220 km/s.For dark matter searches, the crucial quantity is the dark matter density in the solar neighbourhood, whichturns out to be (within a factor of two uncertainty depending on the halo model) ρDM = 0.3 GeV/cm3.We will come back to direct searched of dark matter in a later subsection.

4.2.2 Baryon fraction in clusters

Since large clusters of galaxies form through gravitational collapse, they scoop up mass over a largevolume of space, and therefore the ratio of baryons over the total matter in the cluster should be rep-resentative of the entire universe, at least within a 20% systematic error. Since the 1960s, when X-raytelescopes became available, it is known that galaxy clusters are the most powerful X-ray sources in thesky [43]. The emission extends over the whole cluster and reveals the existence of a hot plasma withtemperature T ∼ 107 − 108 K, where X-rays are produced by electron bremsstrahlung. Assuming thegas to be in hydrostatic equilibrium and applying the virial theorem one can estimate the total mass inthe cluster, giving general agreement (within a factor of 2) with the virial mass estimates. From theseestimates one can calculate the baryon fraction of clusters

fBh3/2 = 0.08 ⇒ ΩB

ΩM≈ 0.14 , for h = 0.70 . (180)

Since Ωlum ' 0.002 − 0.006, the previous expression suggests that clusters contain far more baryonicmatter in the form of hot gas than in the form of stars in galaxies. Assuming this fraction to be repre-sentative of the entire universe, and using the Big Bang nucleosynthesis value of ΩB = 0.04± 0.01, forh = 0.7, we find

ΩM = 0.3± 0.1 (statistical) ± 20% (systematic) . (181)

This value is consistent with previous determinations of ΩM . If some baryons are ejected from the clusterduring gravitational collapse, or some are actually bound in nonluminous objects like planets, then theactual value of ΩM is smaller than this estimate.

4.2.3 Weak gravitational lensing

Since the mid 1980s, deep surveys with powerful telescopes have observed huge arc-like features ingalaxy clusters. The spectroscopic analysis showed that the cluster and the giant arcs were at very differ-ent redshifts. The usual interpretation is that the arc is the image of a distant background galaxy whichis in the same line of sight as the cluster so that it appears distorted and magnified by the gravitationallens effect: the giant arcs are essentially partial Einstein rings. From a systematic study of the clus-ter mass distribution one can reconstruct the shear field responsible for the gravitational distortion [44].This analysis shows that there are large amounts of dark matter in the clusters, in rough agreement withthe virial mass estimates, although the lensing masses tend to be systematically larger. At present, theestimates indicate ΩM = 0.2− 0.3 on scales ∼< 6h−1 Mpc.

4.2.4 Large scale structure formation and the matter power spectrum

Although the isotropic microwave background indicates that the universe in the past was extraordinarilyhomogeneous, we know that the universe today is far from homogeneous: we observe galaxies, clustersand superclusters on large scales. These structures are expected to arise from very small primordial inho-mogeneities that grow in time via gravitational instability, and that may have originated from tiny ripplesin the metric, as matter fell into their troughs. Those ripples must have left some trace as temperatureanisotropies in the microwave background, and indeed such anisotropies were finally discovered by the

Fig. 20: The 2 degree Field Galaxy Redshift Survey contains some 250,000 galaxies, covering a large fraction of the sky up to

redshifts of z ≤ 0.25. From Ref. [45].

COBE satellite in 1992. However, not all kinds of matter and/or evolution of the universe can give riseto the structure we observe today. If we define the density contrast as

δ(~x, a) ≡ ρ(~x, a)− ρ(a)ρ(a)

=∫d3~k δk(a) ei~k·~x , (182)

where ρ(a) = ρ0 a−3 is the average cosmic density, we need a theory that will grow a density contrast

with amplitude δ ∼ 10−5 at the last scattering surface (z = 1100) up to density contrasts of the order ofδ ∼ 102 for galaxies at redshifts z 1, i.e. today. This is a necessary requirement for any consistenttheory of structure formation.

Furthermore, the anisotropies observed by the COBE satellite correspond to a small-amplitudescale-invariant primordial power spectrum of inhomogeneities

P (k) = 〈|δk|2〉 ∝ kn , with n = 1 , (183)

These inhomogeneities are like waves in the space-time metric. When matter fell in the troughs of thosewaves, it created density perturbations that collapsed gravitationally to form galaxies and clusters ofgalaxies, with a spectrum that is also scale invariant. Such a type of spectrum was proposed in the early1970s by Edward R. Harrison, and independently by the Russian cosmologist Yakov B. Zel’dovich [26],to explain the distribution of galaxies and clusters of galaxies on very large scales in our observableuniverse, see Fig. 20.

Since the primordial spectrum is very approximately represented by a scale-invariant Gaussianrandom field, the best way to present the results of structure formation is by working with the 2-pointcorrelation function in Fourier space, the so-called power spectrum. If the reprocessed spectrum of in-homogeneities remains Gaussian, the power spectrum is all we need to describe the galaxy distribution.Non-Gaussian effects are expected to arise from the non-linear gravitational collapse of structure, andmay be important at small scales. The power spectrum measures the degree of inhomogeneity in themass distribution on different scales, see Fig. 21. It depends upon a few basic ingredientes: a) the pri-mordial spectrum of inhomogeneities, whether they are Gaussian or non-Gaussian, whether adiabatic(perturbations in the energy density) or isocurvature (perturbations in the entropy density), whether theprimordial spectrum has tilt (deviations from scale-invariance), etc.; b) the recent creation of inhomo-geneities, whether cosmic strings or some other topological defect from an early phase transition areresponsible for the formation of structure today; and c) the cosmic evolution of the inhomogeneity,

Fig. 21: The measured power spectrum P (k) as a function of wavenumber k. From observations of the Sloan Digital Sky

Survey, CMB anisotropies, cluster abundance, gravitational lensing and Lyman-α forest. From Ref. [46].

whether the universe has been dominated by cold or hot dark matter or by a cosmological constant sincethe beginning of structure formation, and also depending on the rate of expansion of the universe.

The working tools used for the comparison between the observed power spectrum and the pre-dicted one are very precise N-body numerical simulations and theoretical models that predict the shapebut not the amplitude of the present power spectrum. Even though a large amount of work has goneinto those analyses, we still have large uncertainties about the nature and amount of matter necessary forstructure formation. A model that has become a working paradigm is a flat cold dark matter model witha cosmological constant and ΩM ∼ 0.3. This model is now been confronted with the recent very precisemeasurements from 2dFGRS [45] and SDSS [46].

4.2.5 The new redshift catalogs, 2dF and Sloan Digital Sky Survey

Our view of the large-scale distribution of luminous objects in the universe has changed dramaticallyduring the last 25 years: from the simple pre-1975 picture of a distribution of field and cluster galax-ies, to the discovery of the first single superstructures and voids, to the most recent results showing analmost regular web-like network of interconnected clusters, filaments and walls, separating huge nearlyempty volumes. The increased efficiency of redshift surveys, made possible by the development of spec-trographs and – specially in the last decade – by an enormous increase in multiplexing gain (i.e. theability to collect spectra of several galaxies at once, thanks to fibre-optic spectrographs), has allowedus not only to do cartography of the nearby universe, but also to statistically characterize some of itsproperties. At the same time, advances in theoretical modeling of the development of structure, withlarge high-resolution gravitational simulations coupled to a deeper yet limited understanding of how toform galaxies within the dark matter halos, have provided a more realistic connection of the models tothe observable quantities. Despite the large uncertainties that still exist, this has transformed the study of

Fig. 22: The observed cosmic matter components as functions of the Hubble expansion parameter. The luminous matter

component is given by 0.002 ≤ Ωlum ≤ 0.006; the galactic halo component is the horizontal band, 0.03 ≤ Ωhalo ≤ 0.05,

crossing the baryonic component from BBN, ΩB h2 = 0.0244± 0.0024; and the dynamical mass component from large scale

structure analysis is given by ΩM = 0.3 ± 0.1. Note that in the range H0 = 70 ± 7 km/s/Mpc, there are three dark matter

problems, see the text. From Ref. [47].

cosmology and large-scale structure into a truly quantitative science, where theory and observations canprogress together.

4.2.6 Summary of the matter content

We can summarize the present situation with Fig. 22, for ΩM as a function of H0. There are four bands,the luminous matter Ωlum; the baryon content ΩB , from BBN; the galactic halo component Ωhalo, andthe dynamical mass from clusters, ΩM . From this figure it is clear that there are in fact three dark matterproblems: The first one is where are 90% of the baryons? Between the fraction predicted by BBN andthat seen in stars and diffuse gas there is a huge fraction which is in the form of dark baryons. They couldbe in small clumps of hydrogen that have not started thermonuclear reactions and perhaps constitute thedark matter of spiral galaxies’ halos. Note that although ΩB and Ωhalo coincide at H0 ' 70 km/s/Mpc,this could be just a coincidence. The second problem is what constitutes 90% of matter, from BBNbaryons to the mass inferred from cluster dynamics? This is the standard dark matter problem and couldbe solved in the future by direct detection of a weakly interacting massive particle in the laboratory. Andfinally, since we know from observations of the CMB that the universe is flat, the rest, up to Ω0 = 1,must be a diffuse vacuum energy, which affects the very large scales and late times, and seems to beresponsible for the present acceleration of the universe, see Section 3. Nowadays, multiple observationsseem to converge towards a common determination of ΩM = 0.25± 0.08 (95% c.l.), see Fig. 23.

4.2.7 Massive neutrinos

One of the ‘usual suspects’ when addressing the problem of dark matter are neutrinos. They are the onlycandidates known to exist. If neutrinos have a mass, could they constitute the missing matter? We knowfrom the Big Bang theory, see Section 2.6.5, that there is a cosmic neutrino background at a temperatureof approximately 2K. This allows one to compute the present number density in the form of neutrinos,which turns out to be, for massless neutrinos, nν(Tν) = 3

11 nγ(Tγ) = 112 cm−3, per species of neutrino.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.1 1 10 100 1000 10000

!M

d (Mpc)

1 2

3

4

5

6

78

9

10

11 12

13

Fig. 23: Different determinations of ΩM as a function of distance, from various sources: 1. peculiar velocities; 2. weak

gravitacional lensing; 3. shear autocorrelation function; 4. local group of galaxies; 5. baryon mass fraction; 6. cluster mass

function; 7. virgocentric flow; 8. mean relative velocities; 9. redshift space distortions; 10. mass power spectrum; 11. integrated

Sachs-Wolfe effect; 12. angular diameter distance: SNe; 13. cluster baryon fraction. While a few years ago the dispersion

among observed values was huge and strongly dependent on scale, at present the observed value of the matter density parameter

falls well within a narrow range, ΩM = 0.25± 0.07 (95% c.l.) and is essentially independent on scale, from 100 kpc to 5000

Mpc. Adapted from Ref. [48].

If neutrinos have mass, as recent experiments seem to suggest, see Fig. 24, the cosmic energy density inmassive neutrinos would be ρν =

∑nνmν = 3

11 nγ∑mν , and therefore its contribution today,

Ωνh2 =

∑mν

93.2 eV. (184)

The discussion in the previous Sections suggest that ΩM ≤ 0.4, and thus, for any of the three familiesof neutrinos, mν ≤ 40 eV. Note that this limit improves by six orders of magnitude the present boundon the tau-neutrino mass [19]. Supposing that the missing mass in non-baryonic cold dark matter arisesfrom a single particle dark matter (PDM) component, its contribution to the critical density is boundedby 0.05 ≤ ΩPDMh

2 ≤ 0.4, see Fig. 23.

I will now go through the various logical arguments that exclude neutrinos as the dominant compo-nent of the missing dark matter in the universe. Is it possible that neutrinos with a mass 4 eV ≤ mν ≤ 40eV be the non-baryonic PDM component? For instance, could massive neutrinos constitute the darkmatter halos of galaxies? For neutrinos to be gravitationally bound to galaxies it is necessary that theirvelocity be less that the escape velocity vesc, and thus their maximum momentum is pmax = mν vesc.How many neutrinos can be packed in the halo of a galaxy? Due to the Pauli exclusion principle,the maximum number density is given by that of a completely degenerate Fermi gas with momen-tum pF = pmax, i.e. nmax = p3

max/3π2. Therefore, the maximum local density in dark matter

neutrinos is ρmax = nmaxmν = m4ν v

3esc/3π

2, which must be greater than the typical halo densityρhalo = 0.3 GeV cm−3. For a typical spiral galaxy, this constraint, known as the Tremaine-Gunn limit,gives mν ≥ 40 eV, see Ref. [50]. However, this mass, even for a single species, say the tau-neutrino,gives a value for Ωνh

2 = 0.5, which is far too high for structure formation. Neutrinos of such a lowmass would constitute a relativistic hot dark matter component, which would wash-out structure belowthe supercluster scale, against evidence from present observations, see Fig. 24. Furthermore, apply-ing the same phase-space argument to the neutrinos as dark matter in the halo of dwarf galaxies gives

Fig. 24: The neutrino parameter space, mixing angle against ∆m2, including the results from the different solar and atmospheric

neutrino oscillation experiments. Note the threshold of cosmologically important masses, cosmologically detectable neutrinos

(by CMB and LSS observations), and cosmologically excluded range of masses. Adapted from Refs. [49] and [94].

mν ≥ 100 eV, beyond closure density (184). We must conclude that the simple idea that light neutrinoscould constitute the particle dark matter on all scales is ruled out. They could, however, still play a roleas a sub-dominant hot dark matter component in a flat CDM model. In that case, a neutrino mass of order1 eV is not cosmological excluded, see Fig. 24.

Another possibility is that neutrinos have a large mass, of order a few GeV. In that case, their num-ber density at decoupling, see Section 2.5.1, is suppressed by a Boltzmann factor, ∼ exp(−mν/Tdec).For masses mν > Tdec ' 0.8 MeV, the present energy density has to be computed as a solutionof the corresponding Boltzmann equation. Apart from a logarithmic correction, one finds Ωνh

2 '0.1(10 GeV/mν)2 for Majorana neutrinos and slightly smaller for Dirac neutrinos. In either case, neu-trinos could be the dark matter only if their mass was a few GeV. Laboratory limits for ντ of around 18MeV [19], and much more stringent ones for νµ and νe, exclude the known light neutrinos. However,there is always the possibility of a fourth unknown heavy and stable (perhaps sterile) neutrino. If itcouples to the Z boson and has a mass below 45 GeV for Dirac neutrinos (39.5 GeV for Majorana neu-trinos), then it is ruled out by measurements at LEP of the invisible width of the Z. There are two logicalalternatives, either it is a sterile neutrino (it does not couple to the Z), or it does couple but has a largermass. In the case of a Majorana neutrino (its own antiparticle), their abundance, for this mass range,is too small for being cosmologically relevant, Ωνh

2 ≤ 0.005. If it were a Dirac neutrino there couldbe a lepton asymmetry, which may provide a higher abundance (similar to the case of baryogenesis).However, neutrinos scatter on nucleons via the weak axial-vector current (spin-dependent) interaction.For the small momentum transfers imparted by galactic WIMPs, such collisions are essentially coherentover an entire nucleus, leading to an enhancement of the effective cross section. The relatively largedetection rate in this case allowes one to exclude fourth-generation Dirac neutrinos for the galactic darkmatter [51]. Anyway, it would be very implausible to have such a massive neutrino today, since it wouldhave to be stable, with a life-time greater than the age of the universe, and there is no theoretical reason

to expect a massive sterile neutrino that does not oscillate into the other neutrinos.

Of course, the definitive test to the possible contribution of neutrinos to the overall density ofthe universe would be to measure directly their mass in laboratory experiments. There are at presenttwo types of experiments: neutrino oscillation experiments, which measure only differences in squaredmasses, and direct mass-searches experiments, like the tritium β-spectrum and the neutrinoless double-βdecay experiments, which measure directly the mass of the electron neutrino. The former experimentsgive a bound mνe ∼< 2.3 eV (95% c.l.) [52], while the latter claim [53] they have a positive evidencefor a Majorana neutrino of mass mν = 0.05 − 0.89 eV (95% c.l.), although this result still awaitsconfirmation by other experiments. Neutrinos with such a mass could very well constitute the HDMcomponent of the universe, ΩHDM ∼< 0.15. The oscillation experiments give a range of possibilitiesfor ∆m2

ν = 0.3 − 3 eV2 from LSND (not yet confirmed by Miniboone), to the atmospheric neutrinooscillations from SuperKamiokande (∆m2

ν ' 2.2 ± 0.5 × 10−3 eV2 , tan2 θ = 1.0 ± 0.3) and thesolar neutrino oscillations from KamLAND and the Sudbury Neutrino Observatory (∆m2

ν ' 8.2 ±0.3 × 10−5 eV2 , tan2 θ = 0.39 ± 0.05), see Ref. [49]. Only the first two possibilities would becosmologically relevant, see Fig. 24. Thanks to recent observations by WMAP, 2dFGRS and SDSS, wecan put stringent limits on the absolute scale of neutrino masses, see below.

-0.1

-0.05

0

0.05

0.1

500 1000 1500 2000 2500Time (day)

Res

idua

ls (c

pd/k

g/ke

V)

I II III IV V VI VII

2-4 keV

-0.1

-0.05

0

0.05

0.1

500 1000 1500 2000 2500Time (day)

Res

idua

ls (c

pd/k

g/ke

V)

I II III IV V VI VII

2-5 keV

-0.1

-0.05

0

0.05

0.1

500 1000 1500 2000 2500Time (day)

Res

idua

ls (c

pd/k

g/ke

V)

I II III IV V VI VII

2-6 keV

Fig. 25: The annual-modulation signal accumulated over 7 years is consistent with a neutralino of mass of mχ = 59 +17−14 GeV

and a proton cross section of ξσp = 7.0 +0.4−1.2 × 10−6 pb, according to DAMA. From Ref. [54].

4.2.8 Weakly Interacting Massive Particles

Unless we drastically change the theory of gravity on large scales, baryons cannot make up the bulkof the dark matter. Massive neutrinos are the only alternative among the known particles, but they areessentially ruled out as a universal dark matter candidate, even if they may play a subdominant role asa hot dark matter component. There remains the mystery of what is the physical nature of the dominantcold dark matter component. Something like a heavy stable neutrino, a generic Weakly InteractingMassive Particle (WIMP), could be a reasonable candidate because its present abundance could fallwithin the expected range,

ΩPDMh2 ∼ G3/2T 3

0 h2

H20 〈σannvrel〉

=3× 10−27 cm3s−1

〈σannvrel〉. (185)

Here vrel is the relative velocity of the two incoming dark matter particles and the brackets 〈·〉 denote athermal average at the freeze-out temperature, Tf ' mPDM/20, when the dark matter particles go outof equilibrium with radiation. The value of 〈σannvrel〉 needed for ΩPDM ≈ 1 is remarkably close towhat one would expect for a WIMP with a mass mPDM = 100 GeV, 〈σannvrel〉 ∼ α2/8πmPDM ∼3 × 10−27 cm3s−1. We still do not know whether this is just a coincidence or an important hint on thenature of dark matter.

There are a few theoretical candidates for WIMPs, like the neutralino, coming from supersymme-tric extensions of the standard model of particle physics,4 but at present there is no empirical evidence thatsuch extensions are indeed realized in nature. In fact, the non-observation of supersymmetric particlesat current accelerators places stringent limits on the neutralino mass and interaction cross section [55].If WIMPs constitute the dominant component of the halo of our galaxy, it is expected that some maycross the Earth at a reasonable rate to be detected. The direct experimental search for them rely onelastic WIMP collisions with the nuclei of a suitable target. Dark matter WIMPs move at a typicalgalactic “virial” velocity of around 200 − 300 km/s, depending on the model. If their mass is in therange 10 − 100 GeV, the recoil energy of the nuclei in the elastic collision would be of order 10 keV.Therefore, one should be able to identify such energy depositions in a macroscopic sample of the target.There are at present three different methods: First, one could search for scintillation light in NaI crystalsor in liquid xenon; second, search for an ionization signal in a semiconductor, typically a very puregermanium crystal; and third, use a cryogenic detector at 10 mK and search for a measurable temperatureincrease of the sample. The main problem with such a type of experiment is the low expected signal rate,with a typical number below 1 event/kg/day. To reduce natural radioactive contamination one mustuse extremely pure substances, and to reduce the background caused by cosmic rays requires that theseexperiments be located deeply underground.

The best limits on WIMP scattering cross sections come from some germanium experiments, likethe Criogenic Dark Matter Search (CDMS) collaboration at Stanford and the Soudan mine [56], as wellas from the NaI scintillation detectors of the UK dark matter collaboration (UKDMC) in the Boulby saltmine in England [57], and the DAMA experiment in the Gran Sasso laboratory in Italy [54]. Currentexperiments already touch the parameter space expected from supersymmetric particles, see Fig. 26,and therefore there is a chance that they actually discover the nature of the missing dark matter. Theproblem, of course, is to attribute a tentative signal unambiguously to galactic WIMPs rather than tosome unidentified radioactive background.

One specific signature is the annual modulation which arises as the Earth moves around the Sun.5

Therefore, the net speed of the Earth relative to the galactic dark matter halo varies, causing a modulationof the expected counting rate. The DAMA/NaI experiment has actually reported such a modulationsignal, from the combined analysis of their 7-year data, see Fig. 25 and Ref. [54], which provides aconfidence level of 99.6% for a neutralino mass of mχ = 52 +10

−8 GeV and a proton cross section of

4For a review of Supersymmetry (SUSY), see Kazakov’s contribution to these Proceedings.5The time scale of the Sun’s orbit around the center of the galaxy is too large to be relevant in the analysis.

101 10210!43

10!42

10!41

10!40

WIMP Mass [GeV]

Cros

s!se

ctio

n [c

m2 ] (

norm

alise

d to

nuc

leon

)

Fig. 26: Exclusion range for the spin-independent WIMP scattering cross section per nucleon from the NaI experiments and

the Ge detectors. The blue lines come from the CDMS experiment, which exclude the DAMA region at more than 3 sigma.

Also shown in yellow and red is the range of expected counting rates for neutralinos in the MSSM. From Ref. [56].

ξσp = 7.2 +0.4−0.9 × 10−6 pb, where ξ = ρχ/0.3 GeV cm−3 is the local neutralino energy density in

units of the galactic halo density. There has been no confirmation yet of this result from other darkmatter search groups. In fact, the CDMS collaboration claims an exclusion of the DAMA region atthe 3 sigma level, see Fig. 26. Hopefully in the near future we will have much better sensitivity atlow masses from the Cryogenic Rare Event Search with Superconducting Thermometers (CRESST)experiment at Gran Sasso. The CRESST experiment [58] uses sapphire crystals as targets and a newmethod to simultaneously measure the phonons and the scintillating light from particle interactions insidethe crystal, which allows excellent background discrimination. Very recently there has been also theproposal of a completely new method based on a Superheated Droplet Detector (SDD), which claims tohave already a similar sensitivity as the more standard methods described above, see Ref. [59].

There exist other indirect methods to search for galactic WIMPs [60]. Such particles could self-annihilate at a certain rate in the galactic halo, producing a potentially detectable background of highenergy photons or antiprotons. The absence of such a background in both gamma ray satellites andthe Alpha Matter Spectrometer [61] imposes bounds on their density in the halo. Alternatively, WIMPstraversing the solar system may interact with the matter that makes up the Earth or the Sun so that a smallfraction of them will lose energy and be trapped in their cores, building up over the age of the universe.Their annihilation in the core would thus produce high energy neutrinos from the center of the Earth orfrom the Sun which are detectable by neutrino telescopes. In fact, SuperKamiokande already covers alarge part of SUSY parameter space. In other words, neutrino telescopes are already competitive withdirect search experiments. In particular, the AMANDA experiment at the South Pole [62], which hasapproximately 103 Cherenkov detectors several km deep in very clear ice, over a volume ∼ 1 km3, iscompetitive with the best direct searches proposed. The advantages of AMANDA are also directional,since the arrays of Cherenkov detectors will allow one to reconstruct the neutrino trajectory and thus itssource, whether it comes from the Earth or the Sun. AMANDA recently reported the detection of TeVneutrinos [62].

4.3 The age of the universe t0The universe must be older than the oldest objects it contains. Those are believed to be the stars in theoldest clusters in the Milky Way, globular clusters. The most reliable ages come from the applicationof theoretical models of stellar evolution to observations of old stars in globular clusters. For about 30years, the ages of globular clusters have remained reasonable stable, at about 15 Gyr [63]. However,recently these ages have been revised downward [64].

During the 1980s and 1990s, the globular cluster age estimates have improved as both new obser-vations have been made with CCDs, and since refinements to stellar evolution models, including opaci-ties, consideration of mixing, and different chemical abundances have been incorporated [65]. From thetheory side, uncertainties in globular cluster ages come from uncertainties in convection models, opac-ities, and nuclear reaction rates. From the observational side, uncertainties arise due to corrections fordust and chemical composition. However, the dominant source of systematic errors in the globular clus-ter age is the uncertainty in the cluster distances. Fortunately, the Hipparcos satellite recently providedgeometric parallax measurements for many nearby old stars with low metallicity, typical of glubular clus-ters, thus allowing for a new calibration of the ages of stars in globular clusters, leading to a downwardrevision to 10 − 13 Gyr [65]. Moreover, there were very few stars in the Hipparcos catalog with bothsmall parallax erros and low metal abundance. Hence, an increase in the sample size could be critical inreducing the statatistical uncertaintites for the calibration of the globular cluster ages. There are alreadyproposed two new parallax satellites, NASA’s Space Interferometry Mission (SIM) and ESA’s mission,called GAIA, that will give 2 or 3 orders of magnitude more accurate parallaxes than Hipparcos, downto fainter magnitude limits, for several orders of magnitude more stars. Until larger samples are avail-able, however, distance errors are likely to be the largest source of systematic uncertainty to the globularcluster age [29].

The supernovae groups can also determine the age of the universe from their high redshift observa-tions. The high confidence regions in the (ΩM,ΩΛ) plane are almost parallel to the contours of constantage. For any value of the Hubble constant less than H0 = 70 km/s/Mpc, the implied age of the universeis greater than 13 Gyr, allowing enough time for the oldest stars in globular clusters to evolve [65]. In-tegrating over ΩM and ΩΛ, the best fit value of the age in Hubble-time units is H0t0 = 0.93 ± 0.06 orequivalently t0 = 14.1± 1.0 (0.65h−1) Gyr, see Ref. [7]. Furthermore, a combination of 8 independentrecent measurements: CMB anisotropies, type Ia SNe, cluster mass-to-light ratios, cluster abundanceevolution, cluster baryon fraction, deuterium-to-hidrogen ratios in quasar spectra, double-lobed radiosources and the Hubble constant, can be used to determine the present age of the universe [66]. Theresult is shown in Fig. 27, compared to other recent determinations. The best fit value for the age of theuniverse is, according to this analysis, t0 = 13.4 ± 1.6 Gyr, about a billion years younger than otherrecent estimates [66].

4.4 Cosmic Microwave Background AnisotropiesThe cosmic microwave background has become in the last five years the Holy Grail of Cosmology, sinceprecise observations of the temperature and polarization anisotropies allow in principle to determine theparameters of the Standard Model of Cosmology with very high accuracy. Recently, the WMAP satellitehas provided with a very detailed map of the microwave anisotropies in the sky, see Fig. 28, and indeedhas fulfilled our expectations, see Table 2.

The physics of the CMB anisotropies is relatively simple [67]. The universe just before re-combination is a very tightly coupled fluid, due to the large electromagnetic Thomson cross sectionσT = 8πα2/3m2

e ' 0.7 barn. Photons scatter off charged particles (protons and electrons), and carryenergy, so they feel the gravitational potential associated with the perturbations imprinted in the metricduring inflation. An overdensity of baryons (protons and neutrons) does not collapse under the effect ofgravity until it enters the causal Hubble radius. The perturbation continues to grow until radiation pres-sure opposes gravity and sets up acoustic oscillations in the plasma, very similar to sound waves. Since

Fig. 27: The recent estimates of the age of the universe and that of the oldest objects in our galaxy. The last three points

correspond to the combined analysis of 8 different measurements, for h = 0.64, 0.68 and 7.2, which indicates a relatively weak

dependence on h. The age of the Sun is accurately known and is included for reference. Error bars indicate 1σ limits. The

averages of the ages of the Galactic Halo and Disk are shaded in gray. Note that there isn’t a single age estimate more than

2σ away from the average. The result t0 > tgal is logically inevitable, but the standard EdS model does not satisfy this unless

h < 0.55. From Ref. [66].

overdensities of the same size will enter the Hubble radius at the same time, they will oscillate in phase.Moreover, since photons scatter off these baryons, the acoustic oscillations occur also in the photon fieldand induces a pattern of peaks in the temperature anisotropies in the sky, at different angular scales, seeFig. 29. There are three different effects that determine the temperature anisotropies we observe in theCMB. First, gravity: photons fall in and escape off gravitational potential wells, characterized by Φ in thecomoving gauge, and as a consequence their frequency is gravitationally blue- or red-shifted, δν/ν = Φ.If the gravitational potential is not constant, the photons will escape from a larger or smaller potentialwell than they fell in, so their frequency is also blue- or red-shifted, a phenomenon known as the Rees-Sciama effect. Second, pressure: photons scatter off baryons which fall into gravitational potential wellsand the two competing forces create acoustic waves of compression and rarefaction. Finally, velocity:baryons accelerate as they fall into potential wells. They have minimum velocity at maximum compres-sion and rarefaction. That is, their velocity wave is exactly 90 off-phase with the acoustic waves. Thesewaves induce a Doppler effect on the frequency of the photons. The temperature anisotropy induced bythese three effects is therefore given by [67]

δT

T(r) = Φ(r, tdec) + 2

∫ t0

tdec

Φ(r, t)dt +13δρ

ρ− r · v

c. (186)

Metric perturbations of different wavelengths enter the horizon at different times. The largest wave-lengths, of size comparable to our present horizon, are entering now. There are perturbations with wave-lengths comparable to the size of the horizon at the time of last scattering, of projected size about 1

Fig. 28: The anisotropies of the microwave background measured by the WMAP satellite with 10 arcminute resolution. It

shows the intrinsic CMB anisotropies at the level of a few parts in 105. The galactic foreground has been properly subtracted.

The amount of information contained in this map is enough to determine most of the cosmological parameters to few percent

accuracy. From Ref. [20].

in the sky today, which entered precisely at decoupling. And there are perturbations with wavelengthsmuch smaller than the size of the horizon at last scattering, that entered much earlier than decoupling, allthe way to the time of radiation-matter equality, which have gone through several acoustic oscillationsbefore last scattering. All these perturbations of different wavelengths leave their imprint in the CMBanisotropies.

The baryons at the time of decoupling do not feel the gravitational attraction of perturbations withwavelength greater than the size of the horizon at last scattering, because of causality. Perturbations withexactly that wavelength are undergoing their first contraction, or acoustic compression, at decoupling.Those perturbations induce a large peak in the temperature anisotropies power spectrum, see Fig. 29.Perturbations with wavelengths smaller than these will have gone, after they entered the Hubble scale,through a series of acoustic compressions and rarefactions, which can be seen as secondary peaks inthe power spectrum. Since the surface of last scattering is not a sharp discontinuity, but a region of∆z ∼ 100, there will be scales for which photons, travelling from one energy concentration to another,will erase the perturbation on that scale, similarly to what neutrinos or HDM do for structure on smallscales. That is the reason why we don’t see all the acoustic oscillations with the same amplitude, but infact they decay exponentialy towards smaller angular scales, an effect known as Silk damping, due tophoton diffusion [68, 67].

From the observations of the CMB anisotropies it is possible to determine most of the parametersof the Standard Cosmological Model with few percent accuracy, see Table 3. However, there are manydegeneracies between parameters and it is difficult to disentangle one from another. For instance, as men-tioned above, the first peak in the photon distribution corresponds to overdensities that have undergonehalf an oscillation, that is, a compression, and appear at a scale associated with the size of the horizon atlast scattering, about 1 projected in the sky today. Since photons scatter off baryons, they will also feelthe acoustic wave and create a peak in the correlation function. The height of the peak is proportional tothe amount of baryons: the larger the baryon content of the universe, the higher the peak. The position ofthe peak in the power spectrum depends on the geometrical size of the particle horizon at last scattering.Since photons travel along geodesics, the projected size of the causal horizon at decoupling depends onwhether the universe is flat, open or closed. In a flat universe the geodesics are straight lines and, bylooking at the angular scale of the first acoustic peak, we would be measuring the actual size of the hori-

Fig. 29: The Angular Power Spectrum of CMB temperature anisotropies, compared with the cross-correlation of temperature-

polarization anisotropies. From Ref. [20].

zon at last scattering. In an open universe, the geodesics are inward-curved trajectories, and therefore theprojected size on the sky appears smaller. In this case, the first acoustic peak should occur at higher mul-tipoles or smaller angular scales. On the other hand, for a closed universe, the first peak occurs at smallermultipoles or larger angular scales. The dependence of the position of the first acoustic peak on the spatialcurvature can be approximately given by lpeak ' 220 Ω−1/2

0 , where Ω0 = ΩM + ΩΛ = 1−ΩK . Presentobservations by WMAP and other experiments give Ω0 = 1.005± 0.006 at one standard deviation [20].

The other acoustic peaks occur at harmonics of this, corresponding to smaller angular scales. Sincethe amplitude and position of the primary and secondary peaks are directly determined by the soundspeed (and, hence, the equation of state) and by the geometry and expansion of the universe, they canbe used as a powerful test of the density of baryons and dark matter, and other cosmological parameters.With the joined data from WMAP, VSA, CBI and ACBAR, we have rather good evidence of the existenceof the second and third acoustic peaks, which confirms one of the most important predictions of inflation− the non-causal origin of the primordial spectrum of perturbations−, and rules out cosmological defectsas the dominant source of structure in the universe [69]. Moreover, since the observations of CMBanisotropies now cover almost three orders of magnitude in the size of perturbations, we can determinethe much better accuracy the value of the spectral tilt, n = 0.96 ± 0.03, which is compatible with theapproximate scale invariant spectrum needed for structure formation, and is a prediction of the simplestmodels of inflation. Soon after the release of data from WMAP, there was some expectation at the claimof a scale-dependent tilt. Nowadays, with better resolution in the linear matter power spectrum fromSDSS [70], we can not conclude that the spectral tilt has any observable dependence on scale.

Table 3: The parameters of the standard cosmological model. The standard model of cosmology has about 20 different

parameters, needed to describe the background space-time, the matter content and the spectrum of metric perturbations. We

include here the present range of the most relevant parameters (with 1σ errors), as recently determined by WMAP, and the error

with which the Planck satellite will be able to determine them in the near future. The rate of expansion is written in units of

H = 100h km/s/Mpc.

physical quantity symbol WMAP5+SDSS Plancktotal density Ω0 1.005± 0.006 0.7%baryonic matter ΩBh

2 0.0227± 0.0006 0.6%cosmological constant ΩΛ 0.723± 0.029 0.5%cold dark matter ΩCDM 0.231± 0.026 0.6%hot dark matter Ωνh

2 < 0.0065 (95% c.l.) 1.0%sum of neutrino masses

∑mν < 0.3 eV (95% c.l.) 1.0%

CMB temperature T0 (K) 2.725± 0.002 0.1%baryon to photon ratio η = nB/nγ (6.15± 0.25)× 10−10 0.5%baryon to matter ratio ΩB/ΩM 0.17± 0.01 1.0%spatial curvature |ΩK| < 0.005 (95% c.l.) 0.5%rate of expansion h 0.704± 0.024 0.8%age of the universe t0 (Gyr) 13.72± 0.14 0.1%age at decoupling tdec (kyr) 379± 8 0.5%age at reionization tr (Myr) 180± 100 5.0%spectral amplitude A 0.93± 0.06 0.1%spectral tilt ns 0.96± 0.03 0.2%spectral tilt variation dns/d ln k −0.034± 0.028 0.5%tensor-scalar ratio r < 0.36 (95% c.l.) 5.0%rms matter fluctuation σ8 0.811± 0.032 1.0%reionization optical depth τ 0.083± 0.016 5.0%redshift of equality zeq 3262± 137 5.0%redshift of decoupling zdec 1088± 1 0.1%width of decoupling ∆zdec 195± 2 1.0%redshift of reionization zr 10.8± 1.4 2.0%

The microwave background has become also a testing ground for theories of particle physics. Inparticular, it already gives stringent constraints on the mass of the neutrino, when analysed together withlarge scale structure observations. Assuming a flat ΛCDM model, the 2-sigma upper bounds on the sumof the masses of light neutrinos is

∑mν < 1.0 eV for degenerate neutrinos (i.e. without a large hierachy

between them) if we don’t impose any priors, and it comes down to∑mν < 0.34 eV if one imposes

the bounds coming from the HST measurements of the rate of expansion and the supernova data on thepresent acceleration of the universe [71]. The final bound on the neutrino density can be expressed asΩν h

2 =∑mν/93.2 eV ≤ 0.01. In the future, both with Planck and with the Atacama Cosmology

Telescope (ACT) we will be able to put constraints on the neutrino masses down to the 0.1 eV level.

Moreover, the present data is good enough that we can start to put constraints on the models of in-flation that give rise to structure. In particular, multifield models of inflation predict a mixture of adiabaticand isocurvature perturbations,6 and their signatures in the cosmic microwave background anisotropiesand the matter power spectrum of large scale structure are specific and perfectly distinguishable. Nowa-

6This mixture is generic, unless all the fields thermalize simultaneously at reheating, just after inflation, in which case theentropy perturbations that would give rise to the isocurvature modes disappear.

Fig. 30: The (ΩM , ΩΛ) plane with the present data set of cosmological observations − the acceleration of the universe, the

large scale structure and the CMB anisotropies − as well as the future determinations by SNAP and Planck of the fundamental

parameters which define our Standard Model of Cosmology.

days, thanks to precise CMB, LSS and SNIa data, one can put rather stringent limits on the relativefraction and correlation of the isocurvature modes to the dominant adiabatic perturbations [72].

We can summarize this Section by showing the region in parameter space where we stand nowa-days, thanks to the recent cosmological observations. We have plotted that region in Fig. 30. One couldalso superimpose the contour lines corresponding to equal t0H0 lines, as a cross check. It is extraordi-nary that only in the last few months we have been able to reduce the concordance region to where itstands today, where all the different observations seem to converge. There are still many uncertainties,mainly systematic; however, those are quickly decreasing and becoming predominantly statistical. In thenear future, with precise observations of the anisotropies in the microwave background temperature andpolarization anisotropies, thanks to Planck satellite, we will be able to reduce those uncertainties to thelevel of one percent. This is the reason why cosmologists are so excited and why it is claimed that welive in the Golden Age of Cosmology.

5 THE INFLATIONARY PARADIGM

The hot Big Bang theory is nowadays a very robust edifice, with many independent observational checks:the expansion of the universe; the abundance of light elements; the cosmic microwave background; apredicted age of the universe compatible with the age of the oldest objects in it, and the formation of

structure via gravitational collapse of initially small inhomogeneities. Today, these observations areconfirmed to within a few percent accuracy, and have helped establish the hot Big Bang as the preferredmodel of the universe. All the physics involved in the above observations is routinely tested in thelaboratory (atomic and nuclear physics experiments) or in the solar system (general relativity).

However, this theory leaves a range of crucial questions unanswered, most of which are initialconditions’ problems. There is the reasonable assumption that these cosmological problems will besolved or explained by new physical principles at high energies, in the early universe. This assumptionleads to the natural conclusion that accurate observations of the present state of the universe may shedlight onto processes and physical laws at energies above those reachable by particle accelerators, presentor future. We will see that this is a very optimistic approach indeed, and that there are many unresolvedissues related to those problems. However, there might be in the near future reasons to be optimistic.

5.1 Shortcomings of Big Bang CosmologyThe Big Bang theory could not explain the origin of matter and structure in the universe; that is, theorigin of the matter–antimatter asymmetry, without which the universe today would be filled by a uniformradiation continuosly expanding and cooling, with no traces of matter, and thus without the possibilityto form gravitationally bound systems like galaxies, stars and planets that could sustain life. Moreover,the standard Big Bang theory assumes, but cannot explain, the origin of the extraordinary smoothnessand flatness of the universe on the very large scales seen by the microwave background probes and thelargest galaxy catalogs. It cannot explain the origin of the primordial density perturbations that gave riseto cosmic structures like galaxies, clusters and superclusters, via gravitational collapse; the quantity andnature of the dark matter that we believe holds the universe together; nor the origin of the Big Bang itself.

A summary [10] of the problems that the Big Bang theory cannot explain is:

• The global structure of the universe.- Why is the universe so close to spatial flatness?- Why is matter so homogeneously distributed on large scales?• The origin of structure in the universe.

- How did the primordial spectrum of density perturbations originate?• The origin of matter and radiation.

- Where does all the energy in the universe come from?- What is the nature of the dark matter in the universe?- How did the matter-antimatter asymmetry arise?• The initial singularity.

- Did the universe have a beginning?- What is the global structure of the universe beyond our observable patch?

Let me discuss one by one the different issues:

5.1.1 The Flatness Problem

The Big Bang theory assumes but cannot explain the extraordinary spatial flatness of our local patch ofthe universe. In the general FRW metric, the parameter K that characterizes spatial curvature is a freeparameter. There is nothing in the theory that determines this parameter a priori. However, it is directlyrelated, via the Friedmann equation, to the dynamics, and thus the matter content, of the universe,

K =8πG

3ρa2 −H2a2 =

8πG3

ρa2(Ω− 1

Ω

). (187)

We can therefore define a new variable,

x ≡ Ω− 1Ω

=const.ρa2

, (188)

whose time evolution is given by

x′ =dx

dN= (1 + 3ω)x , (189)

where N = ln(a/ai) characterizes the number of e-folds of universe expansion (dN = Hdt) andwhere we have used Eq. (18) for the time evolution of the total energy, ρa3, which only depends on thebarotropic ratio ω. It is clear from Eq. (189) that the phase-space diagram (x, x′) presents an unstablecritical (saddle) point at x = 0 for ω > −1/3, i.e. for the radiation (ω = 1/3) and matter (ω = 0) eras.A small perturbation from x = 0 will drive the system towards x = ±∞. Since we know the universewent through both the radiation era (because of primordial nucleosynthesis) and the matter era (becauseof structure formation), tiny deviations from Ω = 1 would have grown since then, such that today

x0 =Ω0 − 1

Ω0= xin

(Tin

Teq

)2(1 + zeq) . (190)

In order that today’s value be in the range |ΩK | < 0.005, or x0 ≈ 10−3, it is required that at, say,primordial nucleosynthesis (TNS ' 106 Teq) its value be

Ω(tNS) = 1± 10−18 , (191)

which represents a tremendous finetuning. Perhaps the universe indeed started with such a peculiarinitial condition, but it is epistemologically more satisfying if we give a fundamental dynamical reasonfor the universe to have started so close to spatial flatness. These arguments were first used by RobertDicke in the 1960s, much before inflation. He argued that the most natural initial condition for thespatial curvature should have been the Planck scale curvature, (3)R = 6K/l2P, where the Planck lengthis lP = (~G/c3)1/2 = 1.62 × 10−33 cm, that is, 60 orders of magnitude smaller than the present sizeof the universe, a0 = 1.38 × 1028 cm. A universe with this immense curvature would have collapsedwithin a Planck time, tP = (~G/c5)1/2 = 5.39 × 10−44 s, again 60 orders of magnitude smaller thanthe present age of the universe, t0 = 4.1 × 1017 s. Therefore, the flatness problem is also related to theAge Problem, why is it that the universe is so old and flat when, under ordinary circumstances (based onthe fundamental scale of gravity) it should have lasted only a Planck time and reached a size of order thePlanck length? As we will see, inflation gives a dynamical reason to such a peculiar initial condition.

5.1.2 The Homogeneity Problem

An expanding universe has particle horizons, that is, spatial regions beyond which causal communica-tion cannot occur. The horizon distance can be defined as the maximum distance that light could havetravelled since the origin of the universe [15],

dH(t) ≡ a(t)∫ t

0

dt′

a(t′)∼ H−1(t) , (192)

which is proportional to the Hubble scale.7 For instance, at the beginning of nucleosynthesis the horizondistance is a few light-seconds, but grows linearly with time and by the end of nucleosynthesis it is afew light-minutes, i.e. a factor 100 larger, while the scale factor has increased only a factor of 10. Thefact that the causal horizon increases faster, dH ∼ t, than the scale factor, a ∼ t1/2, implies that at anygiven time the universe contains regions within itself that, according to the Big Bang theory, were neverin causal contact before. For instance, the number of causally disconnected regions at a given redshift zpresent in our causal volume today, dH(t0) ≡ a0, is

NCD(z) ∼(a(t)dH(t)

)3

' (1 + z)3/2 , (193)

which, for the time of decoupling, is of order NCD(zdec) ∼ 105 1.7For the radiation era, the horizon distance is equal to the Hubble scale. For the matter era it is twice the Hubble scale.

T1 T1 = T2

T2

Tdec = 0.3 eV

Our Hubble radius at

decoupling

T0 = 3 K

Universe expansion (z = 1100)

Our observable

universe today

Fig. 31: Perhaps the most acute problem of the Big Bang theory is explaining the extraordinary homogeneity and isotropy of the

microwave background, see Fig. 15. At the time of decoupling, the volume that gave rise to our present universe contained many

causally disconnected regions (top figure). Today we observe a blackbody spectrum of photons coming from those regions and

they appear to have the same temperature, T1 = T2, to one part in 105. Why is the universe so homogeneous? This constitutes

the so-called horizon problem, which is spectacularly solved by inflation. From Ref. [73].

This phenomenon is particularly acute in the case of the observed microwave background. Infor-mation cannot travel faster than the speed of light, so the causal region at the time of photon decouplingcould not be larger than dH(tdec) ∼ 3× 105 light years across, or about 1 projected in the sky today. Sowhy should regions that are separated by more than 1 in the sky today have exactly the same tempera-ture, to within 10 ppm, when the photons that come from those two distant regions could not have beenin causal contact when they were emitted? This constitutes the so-called horizon problem, see Fig. 31,and was first discussed by Robert Dicke in the 1970s as a profound inconsistency of the Big Bang theory.

5.2 Cosmological InflationIn the 1980s, a new paradigm, deeply rooted in fundamental physics, was put forward by Alan H.Guth [74], Andrei D. Linde [75] and others [76, 77, 78], to address these fundamental questions. Ac-cording to the inflationary paradigm, the early universe went through a period of exponential expansion,driven by the approximately constant energy density of a scalar field called the inflaton. In modernphysics, elementary particles are represented by quantum fields, which resemble the familiar electric,magnetic and gravitational fields. A field is simply a function of space and time whose quantum oscil-lations are interpreted as particles. In our case, the inflaton field has, associated with it, a large potentialenergy density, which drives the exponential expansion during inflation, see Fig. 32. We know from gen-eral relativity that the density of matter determines the expansion of the universe, but a constant energydensity acts in a very peculiar way: as a repulsive force that makes any two points in space separate atexponentially large speeds. (This does not violate the laws of causality because there is no informationcarried along in the expansion, it is simply the stretching of space-time.)

This superluminal expansion is capable of explaining the large scale homogeneity of our observ-able universe and, in particular, why the microwave background looks so isotropic: regions separatedtoday by more than 1 in the sky were, in fact, in causal contact before inflation, but were stretched to

Fig. 32: The inflaton field can be represented as a ball rolling down a hill. During inflation, the energy density is approximately

constant, driving the tremendous expansion of the universe. When the ball starts to oscillate around the bottom of the hill,

inflation ends and the inflaton energy decays into particles. In certain cases, the coherent oscillations of the inflaton could

generate a resonant production of particles which soon thermalize, reheating the universe. From Ref. [73].

cosmological distances by the expansion. Any inhomogeneities present before the tremendous expansionwould be washed out. This explains why photons from supposedly causally disconneted regions haveactually the same spectral distribution with the same temperature, see Fig. 31.

Moreover, in the usual Big Bang scenario a flat universe, one in which the gravitational attractionof matter is exactly balanced by the cosmic expansion, is unstable under perturbations: a small deviationfrom flatness is amplified and soon produces either an empty universe or a collapsed one. As we dis-cussed above, for the universe to be nearly flat today, it must have been extremely flat at nucleosynthesis,deviations not exceeding more than one part in 1015. This extreme fine tuning of initial conditions wasalso solved by the inflationary paradigm, see Fig. 33. Thus inflation is an extremely elegant hypothesisthat explains how a region much, much greater that our own observable universe could have becomesmooth and flat without recourse to ad hoc initial conditions. Furthermore, inflation dilutes away any“unwanted” relic species that could have remained from early universe phase transitions, like monopoles,cosmic strings, etc., which are predicted in grand unified theories and whose energy density could be solarge that the universe would have become unstable, and collapsed, long ago. These relics are diluted bythe superluminal expansion, which leaves at most one of these particles per causal horizon, making themharmless to the subsequent evolution of the universe.

The only thing we know about this peculiar scalar field, the inflaton, is that it has a mass anda self-interaction potential V (φ) but we ignore everything else, even the scale at which its dynamicsdetermines the superluminal expansion. In particular, we still do not know the nature of the inflaton fielditself, is it some new fundamental scalar field in the electroweak symmetry breaking sector, or is it justsome effective description of a more fundamental high energy interaction? Hopefully, in the near future,experiments in particle physics might give us a clue to its nature. Inflation had its original inspiration inthe Higgs field, the scalar field supposed to be responsible for the masses of elementary particles (quarksand leptons) and the breaking of the electroweak symmetry. Such a field has not been found yet, and itsdiscovery at the future particle colliders would help understand one of the truly fundamental problems inphysics, the origin of masses. If the experiments discover something completely new and unexpected, itwould automatically affect the idea of inflation at a fundamental level.

Fig. 33: The exponential expansion during inflation made the radius of curvature of the universe so large that our observable

patch of the universe today appears essentialy flat, analogous (in three dimensions) to how the surface of a balloon appears

flatter and flatter as we inflate it to enormous sizes. This is a crucial prediction of cosmological inflation that will be tested to

extraordinary accuracy in the next few years. From Ref. [77, 73].

6 The Arnowitt-Deser-Misner formalism

The Arnowitt–Deser–Misner formalism gives a (3+1)-splitting of space-time, a foliation in which thefour dimensional metric gµν is parametrized by the three-metric hij and the lapse and shift functions,N and N i, which describe the evolution of time-like hypersurfaces, with proper interval ds, betweenxα = (t, xi) and xα + dxα = (t+ dt, xi + dxi), given by

ds2 = −(Ndt)2 + hij(dxi +N idt)(dxj +N jdt) .

The components of the metric thus become

g00 = −N2 + hijNiNj , g0i = gi0 = Ni, gij = hij , (194)

and inverse metric

g00 = −N−2, g0i = gi0 = N−2N i, gij = hij −N−2N iN j , (195)

where the 3-metric is used to raise and lower spatial indices, N i = hijNj , with hikhkj = δij . Thissplitting corresponds to a 3-hypersurface Σ and a timelike unit vector normal to it, with components

nα = (−N, 0) , nα = (N−1,−N−1N i) ,

satisfying nαnα = −1. We can then define an intrinsic curvature to the 3-surface, (3)Rij , written in termsof the 3-metric hij , as well as an extrinsic curvature, related to the normal vector,

Kij = −ni|j = −NΓ0ij =

12N

(2N(i|j) − ∂0hij

), (196)

where bars denote 3-space covariant derivatives with connections derived from hij , and subindices inparenthesis denote symmetrization, 2A(ij) = Aij + Aji, while brakets denote antisymmetrization,2A[ij] = Aij − Aji. The traceless part of a tensor is denoted by an overbar. In particular, the traceand traceless parts of the extrinsic curvature are

Kij = Kij −13Khij , K = K i

i =1N

[Ni|i − ∂0 ln

√h]. (197)

The trace K is a generalization of the Hubble parameter, as will be shown below.

Instead of the coordinate basis (e0 = ∂0 , ei = ∂i), with 1-forms (dt ,dxi), we will use a basiswith the normal 3-vector n instead of the time vector,

vectors 1− forms

en =1N

(∂0 −N i∂i) wn = (n · n)n = Ndt (198)

ei = ∂i wi = dxi +N idt . (199)

In this case, for instance, the kinetic term of a scalar field is written as

− ∂µφ∂µφ = (Πφ)2 − (∂iφ)2 , (200)

where Πφ is the scalar-field’s conjugate momentum

Πφ =1N

(φ−N iφ|i) . (201)

The gravitational Lagrangian can be written as

LG =√−gR = N

√h(

(3)R+KijKij −K2

)(202)

from which we can obtain the conjugate momentum of the metric

Πij =∂LG∂hij

= −√h(Kij −Khij), (203)

with trace and traceless parts given by

Π = 2√hK , Πij = −

√h Kij . (204)

After some algebra it can be shown that the gravitational Lagrangian (202) can be written as

LG = N√h(3)R+

N√h

(ΠijΠij − 1

2Π2)

= Πij hij −NH−NiHi − 2∇i(ΠijNj) ,

where the lapse and shift functions appear as Lagrange multipliers, and

H(hij , Πij) = −√h(3)R+

1√h

(ΠijΠij − 1

2Π2), (205)

Hi(hij , Πij) = −2Πij|j , (206)

The gravitational Hamiltonian then becomes

HG = Πij hij − LG = −N√h(3)R+

N√h

(ΠijΠij − 1

2Π2)

+ 2ΠijN(i|j) . (207)

and the Hamiltonian and momentum constraints,

δHGδN

= H = 0 , (208)

δHGδNi

= Hi = 0 . (209)

While the Hamiltonian evolution equations for the independent variables hij and Πij can be written as

hij =δHGδΠij

= −2N Kij + 2N(i|j) , (210)

Πij = − δHGδhij

= −N√h(

(3)Rij − 12

(3)Rhij)

+N

2√hhij(

ΠklΠkl − 12

Π2)

(211)

−2N√h

(ΠikΠ j

k −12

Π Πij)

+√h(N |ij − hij N |k|k

)(212)

+(NkΠij

)|k− 2Πk(iN

j)|k . (213)

With these equations we can evaluate the derivative of the trace Π,

Π = Πij hij + Πij hij =12N√h (3)R+

32N√h(KijK

ij −K2)− 2√hN|i|i + 2

√h(KN i)|i (214)

while from (204) we have

Π = 2√hK +

√hK hij hij = 2

√hK − 2N

√hK2 + 2

√hK N i

|i ,

and therefore the derivate of the trace of the extrinsic curvature is

K −N iK|i = −N |i|i +N(1

4(3)R+

34KijK

ij +12K2). (215)

Now we can evaluate the derivative of the traceless part Πij . Using the identity

2N√h

(ΠikΠ j

k −12

Π Πij)

= 2N√h(KikK j

k −13K Kij − 2

9K2hij

),

and 2NKikK jk = Kik

(N j|k +N

|jk

)−(

23NK Kij + Kikhklh

lj)

, we have, after some algebra,

˙Πij = Πij − 13

Πhij − 13

Π hij = −N√h(3)Rij − 2N

√hKikK j

k −23NK Kij

+√h(N |ij − 1

3N|k|k h

ij)−√hNk Kij

|k −√hKijNk

|k + 2√hKk(iN

j)|k . (216)

Also, from Πij = −√h Kij , we obtain

˙Πij = −√h ˙Kij +N

√hK Kij −

√hKij Nk

|k (217)

and therefore, comparing the two expressions, we deduce

˙Kij −NkKi

j|k +N i|kK

kj −Nk

|jKik = −N |i|j +

13N|k|k δ

ij +N

((3)Rij +K Ki

j

). (218)

Let us consider now the matter content and write the gravitational action for a scalar field withpotential V (φ) in the ADM formalism as

S =∫d4x√−g[ 1

2κ2R− 1

2(∂φ)2 − V (φ)

]=∫d4xN

√h[ 1

2κ2

((3)R+ KijK

ij − 23K2)

+12

[(Πφ)2 − φ|iφ|i

]− V (φ)

], (219)

Variation of the action with respect to N and N i yields the energy and momentum constraint equationsrespectively

−(3)R+ KijKij − 2

3K2 + 2κ2 T00 = 0 , (220)

Kji|j −

23K|i + κ2 T 0

i = 0 . (221)

Variation with respect to hij gives the dynamical gravitational field equations, which can be separatedinto the trace and traceless parts

K −N iK|i = − N |i|i +N

(14

(3)R+34KijK

ij +12K2 +

κ2

2T

), (222)

˙Kij −NkKi

j|k +N i|kK

kj −Nk

|jKik = −N |i|j +

13N|k|k δ

ij +N

((3)Rij +K Ki

j − κ2T ij

). (223)

The matter energy-momentum tensor is

T00 =12

(Πφ)2 +12φ|kφ

|k + V (φ) , (224)

T 0i = Πφφ|i , (225)

T ij = φ|iφ|j + δij

[12

(Πφ)2 − 12φ|kφ

|k − V (φ)], (226)

T ij = φ|iφ|j −13φ|kφ

|kδij , (227)

T =32

(Πφ)2 − 12φ|kφ

|k − 3V (φ) (228)

Variation with respect to φ gives the scalar-field’s equation of motion

1N

(Πφ −N iΠφ|i)− KΠφ − 1

NN|iφ

|i − φ |i|i +∂V

∂φ= 0 . (229)

It is extremely difficult to solve these highly nonlinear coupled equations in a cosmological sce-nario without making some approximations. The usual approach is to assume homogeneity of the fieldsto give a background solution and then linearize the equations to study deviations from spatial uniformity.The smallness of cosmic microwave background anisotropies gives some justification for this perturba-tive approach at least in our local part of the Universe. However, there is no reason to believe it willbe valid on much larger scales. In fact, the stochastic approach to inflation suggests that the Universeis extremely inhomogeneous on very large scales. Fortunately, in this framework one can coarse-grainover a horizon distance and separate the short- from the long-distance behavior of the fields, where theformer communicates with the latter through stochastic forces. The equations for the long-wavelengthbackground fields are obtained by neglecting large-scale gradients, leading to a self-consistent set ofequations, as we will discuss in the next section.

6.1 Spatial gradient expansionIt is reasonable to expand in spatial gradients whenever the forces arising from time variations of thefields are much larger than forces from spatial gradients. In linear perturbation theory one solves theperturbation equations for evolution outside of the horizon: a typical time scale is the Hubble timeH−1, which is assumed to exceed the gradient scale a/k, where k is the comoving wave number of theperturbation. Since we are interested in structures on scales larger than the horizon, it is reasonable toexpand in k/(aH). In particular, for inflation this is an appropriate parameter of expansion since spatialgradients become exponentially negligible after a few e-folds of expansion beyond horizon crossing,k = aH .

It is therefore useful to split the field φ into coarse-grained long-wavelength background fieldsφ(t, xj) and residual short-wavelength fluctuating fields δφ(t, xj). There is a preferred timelike hyper-surface within the stochastic inflation approach in which the splitting can be made consistently, but thedefinition of the background field will depend on the choice of hypersurface, i.e. the smoothing is notgauge invariant. For stochastic inflation the natural smoothing scale is the comoving Hubble length(aH)−1 and the natural hypersurfaces are those on which aH is constant. In that case a fundamentaldifference between φ and δφ is that the short-wavelength components are essentially uncorrelated at dif-ferent times, while long-wavelength components are deterministically correlated through the equationsof motion.

In order to solve the equations for the background fields, we will have to make suitable approx-imations. The idea is to expand in the spatial gradients of φ and to treat the terms that depend onthe fluctuating fields as stochastic forces describing the connection between short- and long-wavelengthcomponents. In this Section we will neglect the stochastic forces due to quantum fluctuations of thescalar fields and will derive the approximate equation of motion for the background fields. We retainonly those terms that are at most first order in spatial gradients, neglecting such terms as φ |i|i , φ|iφ|i, (3)R,(3)Rij , and T ij .

We will also choose the simplifying gaugeN i = 0 [Note that for the evolution during inflation thisis a consequence of the rapid expansion, more than a gauge choice]. The evolution equation (223) forthe traceless part of the extrinsic curvature is then ˙Ki

j = NKKij . Using NK = −∂t ln

√h from (197),

we find the solution Kij ∝ h−1/2, where h is the determinant of hij . During inflation h−1/2 ≡ a−3,

with a the overall expansion factor, therefore Kij decays extremely rapidly and can be set to zero in the

approximate equations. The most general form of the three-metric with vanishing Kij is

hij = a2(t, xk) γij(xk), a(t, xk) ≡ eα(t,xk), (230)

where the time-dependent conformal factor is interpreted as a space-dependent expansion factor. Thetime-independent three-metric γij , of unit determinant, describes the three-geometry of the conformallytransformed space. Since a(t, xk) is interpreted as a scale factor, we can substitute the trace K of theextrinsic curvature for the local space-dependent Hubble parameter

H(t, xi) ≡ 1N(t, xi)

α(t, xi) = −13K(t, xi). (231)

The energy and momentum constraint equations, (220) and (221), can now be written as

H2 =κ2

3

[12

(Πφ)2 + V (φ)], (232)

H|i = −κ2

2Πφφ|i , (233)

together with the evolution equation (222)

− 1NH =

32H2 +

κ2

6T =

κ2

2(Πφ)2 , (234)

where T = 32(Πφ)2 − 3V (φ).

In general, H is a function of the scalar field and time, H(t, xi) ≡ H(φ(t, xi), t). From themomentum constraint (233) we find that the scalar-field’s momentum must obey

Πφ = − 2κ2

(∂H

∂φ

)t

. (235)

Comparing Eq. (234) with the time derivative of H(φ, t),

1N

(∂H

∂t

)x

= Πφ

(∂H

∂φ

)t

+1N

(∂H

∂t

)φ

= −κ2

2(Πφ)2 +

1N

(∂H

∂t

)φ

, (236)

we find(∂H

∂t

)φ

= 0. In fact, we should not be surprised since this is actually a consequence of the

general covariance of the theory.

On the other hand, the scalar field’s equation (229) can be written to first order in spatial gradientsas

1N

Πφ + 3HΠφ +∂V

∂φ= 0 . (237)

We can also show that the conjugate momentum Πφ does not depend explicitly on time, its onlydependence comes through φ. For this, differentiate Eq. (232) w.r.t. φ to obtain

Πφ

(∂Πφ

∂φ

)t

+ 3H Πφ +∂V

∂φ= 0

and compare with (237), where

1N

Πφ = Πφ

(∂Πφ

∂φ

)t

+(∂Πφ

∂t

)φ

, (238)

which implies(∂Πφ

∂t

)φ

= 0.

6.2 Hamilton-Jacobi formalismWe can now summarise what we have learned. The evolution of a general foliation of space-time inthe presence of a scalar field fluid can be described solely in terms of the rate of expansion, which is afunction of the scalar field only, H ≡ H(φ(t, xi)), satisfying the Hamiltonian constraint equation:

3H2(φ) =2κ2

(∂H

∂φ

)2

+ κ2 V (φ) , (239)

together with the momentum constraint and the evolution of the scale factor,

1Nφ = − 2

κ2

(∂H

∂φ

)= Πφ (240)

1Nα = H(φ) , (241)

as well as the dynamical gravitational and scalar field evolution equations

1NH = − 2

κ2

(∂H

∂φ

)2

= − κ2

2(Πφ)2 , (242)

1N

Πφ = − 3H Πφ − V ′(φ) . (243)

Therefore, H(φ) is all you need to specify (to second order in field gradients) the evolution of thescale factor and the scalar field during inflation.

These equations are still too complicated to solve for arbitrary potentials V (φ). In the next sectionwe will find solutions to them in the slow-roll approximation.

6.3 Slow-roll approximation and attractorGiven the complete set of constraints (232)-(233) and evolution equations (234)-(237), we can constructthe following parameters,

ε ≡ − H

H2=

2κ2

(H ′(φ)H(φ)

)2

= − ∂ lnH∂ ln a

, (244)

δ ≡ − φ

Hφ=

2κ2

(H ′′(φ)H(φ)

)= − ∂ lnH ′

∂ ln a, (245)

in terms of which we can define the number of e-folds Ne as

Ne ≡ lnaend

a(t)=∫ tend

tHdt = −κ

2

2

∫ φend

φ

H(φ)dφH ′(φ)

. (246)

In order for inflation to be predictive, you need to ensure that inflation is independent of initialconditions. That is, one should ensure that there is an attractor solution to the dynamics, such thatdifferences between solutions corresponding to different initial conditions rapidly vanish.

Let H0(φ) be an exact, particular, solution of the constraint equation (239), either inflationary ornot. Add to it a homogeneous linear perturbation δH(φ), and substitute into (239). The linear perturba-tion equation reads H ′0(φ) δH ′(φ) = (3κ2/2)H0δH , whose general solution is

δH(φ) = δH(φi) exp(

3κ2

2

∫ φ

φi

H0(φ)dφH ′0(φ)

)= δH(φi) exp(−3∆N) , (247)

where ∆N = Ni −N > 0, and we have used (246) with the particular solution H0(φ). This means thatany deviation from the attractor dies away exponentially fast. This ensures that we can effectively reduceour two-dimensional space (φ,Πφ) to just a single trajectory in phase space.

As a consequence, regardless of the initial condition, the attractor behaviour implies that late-timesolutions are the same up to a constant time shift, which cannot be measured.

6.3.1 An example: Power-law inflation

An exponential potential is a particular case where the attractor can be found explicitly and one can studythe approach to it, for an arbitrary initial condition.

Consider the inflationary potential

V (φ) = V0 e−βκφ , (248)

with β 1 for inflation to proceed. A particular solution to the Hamiltonian constraint equation (239)is

Hatt(φ) = H0 e− 1

2βκφ , (249)

H20 =

κ2

3V0

(1− β2

6

)−1

. (250)

This model corresponds to an inflationary universe with a scale factor that grows like

a(t) ∼ tp , p =2β2 1 . (251)

The slow-roll parameters are both constant,

ε =2κ2

(H ′(φ)H(φ)

)2

=β2

2=

1p 1 , (252)

δ =2κ2

(H ′′(φ)H(φ)

)=β2

2=

1p 1 , (253)

ξ =4κ4

(H ′H ′′′(φ)H2(φ)

)=β4

4=

1p2 1 . (254)

All trajectories tend to the attractor (249), while we can also write down the solution correspondingto the slow-roll approximation, ε = δ = 0,

H2SR(φ) =

κ2

3V0 e

−βκφ , (255)

which differs from the actual attractor by a tiny constant factor, 3p/(3p − 1) ' 1, responsible for aconstant time-shift which cannot be measured.

7 Homogeneous scalar field dynamics

In this subsection I will describe the theoretical basis for the phenomenon of inflation. Consider a scalarfield φ, a singlet under any given interaction, with an effective potential V (φ). The Lagrangian for sucha field in a curved background is

Sinf =∫

d4x√−gLinf , Linf = −1

2gµν∂µφ∂νφ− V (φ) , (256)

whose evolution equation in a Friedmann-Robertson-Walker metric and for a homogeneous field φ(t) isgiven by

φ+ 3Hφ+ V ′(φ) = 0 , (257)

where H is the rate of expansion, together with the Einstein equations,

H2 =κ2

3

(12φ2 + V (φ)

), (258)

H = −κ2

2φ2 , (259)

where κ2 ≡ 8πG. The dynamics of inflation can be described as a perfect fluid with a time dependentpressure and energy density given by

ρ =12φ2 + V (φ) , (260)

p =12φ2 − V (φ) . (261)

The field evolution equation (257) can then be written as the energy conservation equation,

ρ+ 3H(ρ+ p) = 0 . (262)

If the potential energy density of the scalar field dominates the kinetic energy, V (φ) φ2, then we seethat

p ' −ρ ⇒ ρ ' const. ⇒ H(φ) ' const. , (263)

which leads to the solution

a(t) ∼ exp(Ht) ⇒ a

a> 0 accelerated expansion . (264)

Using the definition of the number of e-folds, N = ln(a/ai), we see that the scale factor grows expo-nentially, a(N) = ai exp(N). This solution of the Einstein equations solves immediately the flatnessproblem. Recall that the problem with the radiation and matter eras is that Ω = 1 (x = 0) is an un-stable critical point in phase-space. However, during inflation, with p ' −ρ ⇒ ω ' −1, we havethat 1 + 3ω ≥ 0 and therefore x = 0 is a stable attractor of the equations of motion, see Eq. (189).As a consequence, what seemed an ad hoc initial condition, becomes a natural prediction of inflation.Suppose that during inflation the scale factor increased N e-folds, then

x0 = xin e−2N

(Trh

Teq

)2(1 + zeq) ' e−2N 1056 ≤ 1 ⇒ N ≥ 65 , (265)

where we have assumed that inflation ended at the scale Vend, and the transfer of the inflaton energydensity to thermal radiation at reheating occurred almost instantaneously8 at the temperature Trh ∼V

1/4end ∼ 1015 GeV. Note that we can now have initial conditions with a large uncertainty, xin ' 1, and

still have today x0 ' 1, thanks to the inflationary attractor towards Ω = 1. This can be understood veryeasily by realizing that the three curvature evolves during inflation as

(3)R =6Ka2

= (3)Rin e−2N −→ 0 , for N 1 . (266)

Therefore, if cosmological inflation lasted over 65 e-folds, as most models predict, then today the uni-verse (or at least our local patch) should be exactly flat, a prediction that has been tested with great (2%)accuracy by WMAP from observations of temperature anisotropies in the microwave background.

Furthermore, inflation also solves the homogeneity problem in a spectacular way. First of all, dueto the superluminal expansion, any inhomogeneity existing prior to inflation will be washed out,

δk ∼(k

aH

)2

Φk ∝ e−2N −→ 0 , for N 1 . (267)

Moreover, since the scale factor grows exponentially, while the horizon distance remains essentiallyconstant, dH(t) ' H−1 = const., any scale within the horizon during inflation will be stretched by thesuperluminal expansion to enormous distances, in such a way that at photon decoupling all the causallydisconnected regions that encompass our present horizon actually come from a single region duringinflation, about 65 e-folds before the end. This is the reason why two points separated more than 1

in the sky have the same backbody temperature, as observed by the COBE satellite: they were actuallyin causal contact during inflation. There is at present no other proposal known that could solve thehomogeneity problem without invoquing an acausal mechanism like inflation.

8There could be a small delay in thermalization, due to the intrinsic inefficiency of reheating, but this does not changesignificantly the required number of e-folds.

Finally, any relic particle species (relativistic or not) existing prior to inflation will be diluted bythe expansion,

ρM ∝ a−3 ∼ e−3N −→ 0 , for N 1 , (268)

ρR ∝ a−4 ∼ e−4N −→ 0 , for N 1 . (269)

Note that the vacuum energy density ρv remains constant under the expansion, and therefore, very soonit is the only energy density remaining to drive the expansion of the universe.

7.1 The slow-roll approximationIn order to simplify the evolution equations during inflation, we will consider the slow-roll approximation(SRA). Suppose that, during inflation, the scalar field evolves very slowly down its effective potential,then we can define the slow-roll parameters,

ε ≡ − H

H2=

κ2

2φ2

H2 1 , (270)

δ ≡ − φ

Hφ 1 , (271)

ξ ≡...φ

H2φ− δ2 1 . (272)

It is easy to see that the condition

ε < 1 ⇐⇒ a

a> 0 (273)

characterizes inflation: it is all you need for superluminal expansion, i.e. for the horizon distance to growmore slowly than the scale factor, in order to solve the homogeneity problem, as well as for the spatialcurvature to decay faster than usual, in order to solve the flatness problem.

The number of e-folds during inflation can be written with the help of Eq. (270) as

N = lnaend

ai=∫ te

ti

Hdt =∫ φe

φi

κdφ√2ε(φ)

, (274)

which is an exact expression in terms of ε(φ).

In the limit given by Eqs. (270), the evolution equations (257) and (258) become

H2(

1− ε

3

)' H2 =

κ2

3V (φ) , (275)

3Hφ(

1− δ

3

)' 3Hφ = −V ′(φ) . (276)

Note that this corresponds to a reduction of the dimensionality of phase-space from two to one dimen-sions, H(φ, φ) → H(φ). In fact, it is possible to prove a theorem, for single-field inflation, whichstates that the slow-roll approximation is an attractor of the equations of motion, and thus we can al-ways evaluate the inflationary trajectory in phase-space within the SRA, therefore reducing the numberof initial conditions to just one, the initial value of the scalar field. If H(φ) only depends on φ, then

H ′(φ) = −κ2φ/2 and we can rewrite the slow-roll parameters (270) as

ε =2κ2

(H ′(φ)H(φ)

)2

' 12κ2

(V ′(φ)V (φ)

)2

≡ εV 1 , (277)

δ =2κ2

H ′′(φ)H(φ)

' 1κ2

V ′′(φ)V (φ)

− 12κ2

(V ′(φ)V (φ)

)2

≡ ηV − εV 1 , (278)

ξ =4κ4

H ′(φ)H ′′′(φ)H2(φ)

' 1κ4

V ′(φ)V ′′′(φ)V 2(φ)

− 32κ4

V ′′(φ)V (φ)

(V ′(φ)V (φ)

)2

+3

4κ4

(V ′(φ)V (φ)

)4

≡ ξV − 3ηV εV + 3ε2V 1 . (279)

These expressions define the new slow-roll parameters εV , ηV and ξV . The number of e-folds can alsobe rewritten in this approximation as

N '∫ φe

φi

κdφ√2εV (φ)

= κ2

∫ φe

φi

V (φ) dφV ′(φ)

, (280)

a very useful expression for evaluating N for a given effective scalar potential V (φ).

8 The origin of density perturbations

If cosmological inflation made the universe so extremely flat and homogeneous, where did the galaxiesand clusters of galaxies come from? One of the most astonishing predictions of inflation, one that was noteven expected, is that quantum fluctuations of the inflaton field are stretched by the exponential expansionand generate large-scale perturbations in the metric. Inflaton fluctuations are small wave packets ofenergy that, according to general relativity, modify the space-time fabric, creating a whole spectrum ofcurvature perturbations. The use of the word spectrum here is closely related to the case of light wavespropagating in a medium: a spectrum characterizes the amplitude of each given wavelength. In thecase of inflation, the inflaton fluctuations induce waves in the space-time metric that can be decomposedinto different wavelengths, all with approximately the same amplitude, that is, corresponding to a scale-invariant spectrum. These patterns of perturbations in the metric are like fingerprints that unequivocallycharacterize a period of inflation. When matter fell in the troughs of these waves, it created densityperturbations that collapsed gravitationally to form galaxies, clusters and superclusters of galaxies, witha spectrum that is also scale invariant. Such a type of spectrum was proposed in the early 1970s (beforeinflation) by Harrison and Zel’dovich [26], to explain the distribution of galaxies and clusters of galaxieson very large scales in our observable universe. Perhaps the most interesting aspect of structure formationis the possibility that the detailed knowledge of what seeded galaxies and clusters of galaxies will allowus to test the idea of inflation.

8.1 Linear perturbations on a homogeneous backgroundWe will now choose a different kind of expansion. Instead of assuming that gradients are small on largescales, we will expand in the amplitude of the metric and scalar field perturbations. Then the linearizedequations will be valid both on scales larger and smaller than the horizon, as long as the amplitude ofthe perturbations is not too large. This approximation will fail both in the limit of very large scales,where the stochastic approach to inflation suggest that curvature perturbations become of order unity oreven larger, and in the limit of very small scales, well inside the horizon, where non-linear gravitationalcollapse makes the density contrast of matter increase beyond perturbation theory to form collapsedtructures like stars and galaxies.

The rapid expansion during inflation quickly makes the comoving trajectories orthogonal to thespatial hypersurfaces Σ. This means that the shift function quickly becomes negligible, and we can

choose g0i = 0. In that case, we can write the other metric components to first order in perturbations as

g00 = −(

1 + 2Φ(t,x)), (281)

gij =(

1− 2Ψ(t,x))a2(t)δij , (282)

φ = φ0 + δφ(t,x) , (283)

where we are assuming a vanishing homogeneus 3-curvature, K = 0; i.e. the universe is spatially flat onlarge scales due to the tremendous expansion of the scale factor. These metric components correspond toa lapse function and a metric with intrinsic and extrinsic curvatures given to first order in perturbationsby

N = 1 + Φ ,√h = (1− 3Ψ) a3 , (284)

K = − 1N∂t ln

√h = −3H(1− Φ) + 3Ψ , (285)

Kij = −Φ Ψa2 δij = 0 , (286)

(3)R =4a2

Ψ|i|i ,(3)Rij =

1a2

(Ψ|i|j −

13

Ψ|k|kδij

), (287)

Πφ = (1− Φ)(φ0 + ˙δφ) = φ0 + ˙δφ− Φφ0 . (288)

The energy momentum tensor to first order, and taking into account all gradient terms is

T00 =12φ2

0 + φ0˙δφ− φ2

0 Φ + V (φ0) + V ′(φ) δφ , (289)

T 0i = φ0 δφ|i , (290)

T ij = δφ|iδφ|j −13δφ|kδφ

|kδij = 0 , (291)

T =32φ2

0 + 3φ0˙δφ− 3φ2

0 Φ− 3V (φ0)− 3V ′(φ) δφ (292)

With these ingredients, we can compute the constraint and evolution equations. Let us start withthe energy constraint equation (220), to first order in perturbation theory, K2/3 + (3)R/2 = κ2T00, with(3)R = (4/a2)Ψ|i|i,

1a2

Ψ|i|i − 3HΨ− HΦ− 3H2Φ =κ2

2

(φ0

˙δφ+ V ′(φ)δφ)

(293)

where we have used the zero-th order (homogeneous) equations

H2 =κ2

3

(12φ2

0 + V (φ0)), (294)

H = −κ2

2φ2

0 . (295)

In a similar way, the evolution equation for K, (222), gives

Ψ + 3HΨ +HΦ + HΦ + 3H2Φ =1

3a2(Ψ− Φ)|i|i +

κ2

2

(φ0

˙δφ− V ′(φ)δφ). (296)

The momentum constrain equation (221) reads, to first order,

(Ψ +HΦ)|i =κ2

2φ0δφ|i . (297)

While the evolution equation for the traceless part Kij , (223), is

(Ψ− Φ)|i|j −13

(Ψ− Φ)|k|kδij = κ2a2T ij = 0 ⇒ Φ = Ψ , (298)

which constitutes an important constraint, implying that the two Newtonian potentials do not differ,to first order in perturbation theory, a result that agrees with the PPN formalism of general relativity.Substituting into (293) and (296), we end up with the metric perturbations’ equations

Φ + 4HΦ + HΦ + 3H2Φ =κ2

2

(φ0

˙δφ− V ′(φ)δφ),

− 1a2

Φ|i|i + 3HΦ + HΦ + 3H2Φ = −κ2

2

(φ0

˙δφ+ V ′(φ)δφ), (299)

Φ +HΦ =κ2

2φ0δφ ,

together with the evolution equation for the scalar field perturbation,

δφ+ 3H ˙δφ− 1a2δφ|i|i + V ′′(φ0)δφ = 4φ0 Φ− 2V ′(φ0) Φ , (300)

where we have used the homogeneous field equation

φ0 + 3Hφ0 + V ′(φ0) = 0 . (301)

8.2 Gauge invariant linear perturbation theoryThe unperturbed (background) FRW metric can be described by a scale factor a(t) and a homogeneousdensity field ρ(t),

ds2 = a2(η)[−dη2 + γij dxidxj ] , (302)

where η is the conformal time η =∫

dt

a(t)and the background equations of motion can be written as

H2 = a2H2 =κ2

3a2 ρ−K , (303)

H′ −H2 = a2H = K − κ2

2a2(ρ+ p) , (304)

whereH = aH .

The most general line element, in linear perturbation theory, with both scalar, vector and tensormetric perturbations, is given by

ds2 = a2(η)−(1+2φ)dη2+2(B|i−Si)dxidη+

[(1−2ψ)γij+2E|ij+2F(i|j)+hij

]dxidxj

. (305)

The indices i, j label the three-dimensional spatial coordinates with metric γij , and the |i denotescovariant derivative with respect to that metric. The vector perturbation is transverse γijSi|j = γijFi|j =0, and the tensor perturbation hij corresponds to a symmetric transverse traceless gravitational wave,∇ihij = γijhij = 0. In total, these correspond to n(n + 1)/2 = 10 independent degrees of freedom,4 scalars, 2 vectors (3 components − 1 transverse condition each) and 2 tensor (6 components − 3transverse conditions − 1 trace condition).

8.3 General coordinate transformations and perturbation theoryNot all 10 degrees of freedom are physical. As we know, there is a gauge invariance of the theory undergeneral coordinate transformations. The homogeneity of the FRW space-time gives a natural choice ofcoordinates in the absence of perturbations; however, the presence of first-order perturbations allow ageneral coordinate (gauge) transformation,

xµ → xµ = xµ + ξµ

η = η + ξ0(η, xk) ,

xi = xi + γij ξ|j(η, xk) + ξi(η, xk) ,(306)

with arbitrary scalar functions ξ0 and ξ. The vector function ξi is a transverse field, ξi|i = 0. After

this gauge transformation, ξ0 determines the choice of constant−η hypersurfaces, while ξ|i and ξi selectthe spatial coordinates within these hypersurfaces. The choice of coordinates is arbitrary to first orderand definitions of first-order metric and matter perturbations are thus coordinate (gauge)-dependent.The result of the gauge transformation (306) acting on any tensor Q is that of the Lie derivative of thebackground value Q0 of that physical quantity,

δQ = δQ−£ξQ0 . (307)

Alternatively, we can obtain the transformed metric components by perturbing the line element,

dη = dη − ξ0′dη − ξ0|i dx

i , (308)

dxi = dxi − γij(ξ′|j + ξ′j) dη − (ξ|i|j + ξi|j) dξj . (309)

Substituting them into the line element and using a(η) = a(η)− a′(η)ξ0, we get the line element in thenew coordinate system, to first order in metric and coordinate transformations,

ds2 = a2(η)−(

1 + 2(φ−Hξ0 − ξ0′))dη2 + 2

((B − ξ0 − ξ′)|i − (Si + ξ′i)

)dηdxi

+[(

1− 2(ψ +Hξ0))γij + 2(E − ξ)|ij + 2

(F(i|j) − ξ(i|j)

)+ hij

]dxidxj

.

Since ds2 = ds2 is invariant under general coordinate transformations, we can read off the transforma-tion equations for the metric perturbations by writing down the new line element with the new metricperturbations as

ds2 = a2(η)−(1+2φ)dη2+2(B|i−Si)dηdxi+

[(1−2ψ) γij+2E|ij+2F(i|j)+hij

]dxidxj

. (310)

Thus, the gauge transformation of scalar perturbations becomes

φ = φ−Hξ0 − ξ0′ , B = B + ξ0 − ξ′ , (311)

ψ = ψ +Hξ0 , E = E − ξ , (312)

that of vector perturbations is

Si = Si + ξ′i , (313)

Fi = Fi − ξi , (314)

and finally, tensor perturbations remain invariant

hij = hij . (315)

Alternatively, these metric transformations could have been obtained from the general expression hµν =hµν − ∂µξν − ∂νξµ for a coordinate change (306).

8.4 Time slicing and spatial hypersurfacesEventually we will have to relate metric perturbations to physical observable quantities. To linear pertur-bation theory, the unit time-like vector field orthogonal to constant−η hypersurfaces nµ is

nµ = (N−1,−N−1N i) =1a

(1− φ,−B|i + Si) , (316)

nµ = (−N,0) = −a(1 + φ,0) , (317)

and therefore, the lapse and shift functions, to linear order, are given by

N = a(1 + φ) , N i = B|i − Si . (318)

With these, we can decompose the congruence of worldlines into

Θµν = nµ;ν =13

ΘPµν + σµν + ωµν − aµnν , (319)

where the projection operator, Pµν = δµν + nµnν , projects any tensor into a plane orthogonal to nµ.Note that in general nµ is not necessarily identical to uµ = dxµ/dτ , the tangent vector to the worldline,and thus nµ does not follow a geodesic. The tensor Θµν measures the extent to which neighbouringworldlines deviate from remaining parallel.

Imagine a sphere of test particles centered at some point, and describe the evolution of the particleswith respect to the central worldline. The 4 components of the congruence have physical meaning. Thetrace gives the expansion rate of the congruence,

Θ = nµ;µ , (320)

and characterizes the growth in the volume of the sphere; it’s a scalar. The traceless symmetric part,

σµν =12Pαµ P

βν (nα;β + nβ;α)− 1

3ΘPµν , (321)

is the shear tensor of the congruence, and represents the distortion in the shape of the set of test particles;it describes how the sphere is deformed into an ellipsoid. The antisymmetric part,

ωµν =12Pαµ P

βν (nα;β − nβ;α) , (322)

is the vorticity tensor, and describes the rotation of the sphere of test particles around a center axis. Inour case, nµ = (−N, 0), the congruence is always orthogonal by definition to a family of hypersurfacesand therefore it cannot sustain vorticity.

Finally, if the normal vectors do not follow geodesics of the metric, then there is also an accelera-tion,

aµ = nνnµ;ν . (323)

These geometric quantities allow one to obtain the evolution of the congruence. By computing thecovariant derivative of Θµν along the worldlines, and taking the trace, ones finds the famous Raychaud-hury equation,

dΘdτ

= −13

Θ2 − σµνσµν + ωµνωµν −Rµνnµnν + aµ;µ . (324)

Note that on spatial hypersurfaces, the shear, vorticity, expansion and acceleration coincide withtheir Newtonian counterparts. Moreover, with these definitions one can obtain the extrinsic curvature tothe spatial hypersurface, Kµν = −Pαµ P

βν nα;β .

Let us come back to linear perturbation theory over a FRW space-time. The scalar perturbationsgive

Θ =1a

[3H(1− φ)− 3ψ′ −∇2(B − E′)

], (325)

ai = φ|i , a0 = 0 , (326)

σij = a(σ|ij −

13γij∇2σ

), σ = E′ −B , (327)

where σ is the scalar shear. The intrinsic spatial curvature on hypersurfaces of constant conformal timeη is

(3)R =6Ka2

+12Ka2

ψ +4a2∇2ψ . (328)

For a perturbation with wavenumber k,∇2ψ = −k2ψ, we have

δ(3)R =4a2

(3K − k2)ψ . (329)

The vector perturbations have a(v)µ = 0, σ(v)

00 = σ(v)0i = 0, Θ(v) = 0. The only nonzero first order

quantity is the shear,τij ≡ σ(v)

ij = a(S(i|j) + F ′(i|j)

). (330)

There is no tensor contribution to the expansion, acceleration and vorticity. There is only a nonzerocontribution to shear,

σ(t)ij =

a

2h′ij (331)

As an aside, already at this purely geometrical level, we can obtain an evolution equation for thecurvature perturbation ψ. Multiplying by (1 +φ) the expression of the scalar expansion Θ and writing itin terms of coordinate time t, we find

Θ ≡ (1 + φ)θ = 3H − 3ψ +∇2σ , (332)

where σ = E −B/a. Writing the perturbation in the expansion as δΘ ≡ Θ− 3H , we find

ψ = −13δΘ +

13∇2σ , (333)

which is independent of the field equations (follows simply from geometry). However, on large scales,∇2σ → 0, the curvature perturbation satisfies ψ = −δΘ/3, a very interesting relation, which will allowus later to relate the curvature perturbation on a comoving hypersurface to the change in e-folds betweendifferent slicings.

8.5 Perturbations in MatterWe will consider here the stress-energy tensor for a matter fluid including anisotropic stresses. Like themetric, the coordinate representation of matter fields is gauge-dependent. The energy-momentum tensorof a fluid with density, isotropic pressure and 4-velocity uµ is

Tµν = (ρ+ p)uµuν + p δµν + Πµν . (334)

The anisotropic stress-tensor Πµν decomposes into a trace-free part, Π, a vector part, Πi, and a tensorpart, Πij ,

Πij = Π|i|j −

13∇2Π δij +

12

(Πi|j + Π|ij

)+ Πi

j . (335)

The anisotropic stress-tensor has only spatial coordinates and is therefore gauge-invariant, since thebackground value is zero, Πij = Πij .

The linearly perturbed velocity can be written as

uµ =1a

((1− φ), v|i + vi

),

uµ = a(− (1 + φ), (v +B)|i + vi − Si

),

(336)

satisfying uµuµ = −1. We have introduced a scalar velocity potential, v, for irrotational flows, as wellas a transverse vector field, vi|i = 0. We can then write the components of the energy-momentum tensoras

T 00 = −(ρ0 + δρ) ,

T 0i = (ρ0 + p0)(B|i + v|i + vi − Si) ,

T i0 = −(ρ0 + p0)(v|i + vi) ,

T ij = (p0 + δp) δij + Πij .

(337)

Coordinate transformations affect the splitting between spatial and temporal components of matter fieldsand thus quantities like density, pressure and 3-velocity are gauge dependent. We have seen that the ten-sor perturbations are gauge independent, but let us see how scalar and vector matter fields change with acoordinate transformation xµ = xµ+ξµ. Any scalar function q which is homogeneous in the backgroundFRW space-time can be written as q(η,x) = q0(η) + δq(η,x). Thus the perturbed quantity transformsas δq = δq− ξ0q′0. Physical scalars measured on spatial hypersurfaces like curvature, acceleration, shearor density perturbations, only depend on the choice of temporal gauge, ξ0, but are independent of thecoordinates within 3-dimensional hypersurfaces determined by ξ. That is, the spatial gauge ξ can onlyaffect the components of 3-vectors or 3-tensors on spatial hypersurfaces, but not 3-scalars. For example,the scalar shear transforms as

σ = σ − ξ0 . (338)

Vector quantities that are derived from a potential, such as the velocity potential v, only dependon the shift ξ within a hypersurface, and are independent of ξ0. We thus find

v = v + ξ′ . (339)

The vector function ξi only affects components of divergence-free (transverse) 3-vectors and 3-tensorsin spatial hypersurfaces such as the velocity perturbation,

vi = vi + ξi′. (340)

8.6 Gauge invariant gravitational potentials and matter variablesThe gauge dependence of metric perturbations had bothered people for a long time until Bardeen sug-gested the construction of gauge invariant combinations of perturbations. Out of 10 independent com-ponents of the metric perturbations we should be able to reduce them to 6 gauge independent degrees offreedom: 2 scalars, 2 (transverse) vectors and 2 (transverse traceless) tensors.

Let us construct first the scalars. The two scalar gauge functions, ξ0 and ξ, allow two of the fourscalars to be eliminated, resulting 2 possible combinations

Φ ≡ φ+H(B − E′) + (B − E′)′ ,Ψ ≡ ψ −H(B − E′) ,

(341)

which are related to those proposed by Bardeen (80), Φ = ΦAQ(0) and Ψ = −ΦHQ

(0).

These functions coincide with the metric perturbations in particular gauges, like the orthogonalzero-shear gauge, the conformal Newtonian gauge and the longitudinal gauge, but these are by no meanspreferred gauge choices, as we will see. In fact, any unambiguous choice of time slicing can be used todefine gauge-invariant perturbations.

In general, for a gauge-dependent scalar quantity q, whose perturbation transforms like δq =δq − q′0 ξ0, we can find a gauge-invariant combination as δq(g.i.) = δq + q′0 (B − E′). For example, thescalar field perturbation with background value ϕ0(η) has a gauge-independent expression

δϕ(g.i.) = δϕ+ ϕ′0(η)(B − E′) . (342)

Let us now construct the gauge invariant vector combinations. There is one transverse vectorgauge function ξi, and there are two transverse vector metric perturvations, so we can construct a singlegauge invariant transverse perturbation,

Σi ≡ Si + F ′i , (343)

which is directly related to the gauge invariant vector shear,

τij = aΣ(i|j) . (344)

Finally the tensors. For all tensor fields with vanishing or constant background contributions, suchthat £ξQ0 = 0, we can construct gauge invariant perturbation equations in terms of gauge-independentquantities. Since we can always write general relativity equations as Q = 0, it is always possible toconstruct these equations, which are then free from spurious gauge modes.

Before describing specific time-slicings, let us emphasize that the choice of Eq. (341) is not unique.Any combination of gauge invariant quantities will also be gauge invariant. For example, the followingquantity will play a crucial role for testing the predictions of inflation,

ζ ≡ ψ −H(v +B) , (345)

which coincides with the scalar curvature perturbation on comoving hypersurfaces, and can be writtenas a function of Φ and Ψ as

ζ = Ψ +H

H2 −H′(Ψ′ +HΦ) . (346)

8.6.1 Gauge invariant matter variables

Let us now construct the gauge invariant matter variables. Again, the tensor matter fields are automat-ically gauge invariant, but the scalar and vector fields are not. If we write the density and pressureperturbations as

ρ = ρ0(1 + δ) , (347)

p = p0(1 + πL) , (348)

where δ and πL are the density and pressure contrasts, then we can readily construct a gauge-invariantquantity,

Γ ≡ πL −c2s

wδ , (349)

which measures the intrinsic non-adiabaticity of the matter content, where

w =p

ρ, c2

s =p′

ρ′, (350)

are the barotropic index and the sound speed of the fluid. Actually, Γ is related to the entropy productionrate. If the pressure of the fluid is a function of local energy density only, p = p(ρ), then

δp

δρ=p′

ρ′= c2

s ⇒ Γ = 0 .

For instance, in the case of a single barotropic fluid, with p = wρ, we have Γ = 0. Values different fromzero may arise from the relative entropy (particle number) of a mixture of several fluid components.

Another scalar quantity, the density contrast, does not have a unique gauge-invariant expressionbecause it requires the use of the metric variables, and that depends on the specific choice of slicing. Forexample, three different gauge invariant density contrasts, related to three difference gauge choices, are

Dl ≡ δ − 3(1 + w)H(B − E′) longitudinal gauge (351)

Df ≡ δ − 3(1 + w)ψ flat slicing gauge (352)

Dc ≡ δ − 3(1 + w)H(v +B) comoving gauge (353)

The distinction between them is only important on superhorizon scales, since on small (subhorizon)scales they reduce to the same variable, the Newtonian density contrast.

The velocity perturbations can be written also in gauge-invariant form

V ≡ v + E′ , (354)

Vi ≡ vi + F ′i . (355)

Some useful relations between gauge invariant quantities are

Dl = Df + 3(1 + w)Ψ , (356)

Dc = Dl − 3(1 + w)HV , (357)

Dc = Df + 3(1 + w)ζ , (358)

which are further related byζ = Ψ−HV (359)

8.7 Different time slices and their gauge choices8.7.1 The longitudinal gauge

It is also known as the conformal Newtonian gauge or the zero-shear gauge. It is defined by the conditionσ = 0 in Eq. (338). If we start from arbitrary coordinates and perform a gauge transformation with

ξ0l = E′ −B , (360)

then we indeed find σl = 0. This is sufficient to find the two scalar metric perturbations, or any otherscalar quantity on these hypersurfaces. However, in addition, we must supply the vector gauge transfor-mation. The longitudinal gauge is completely determined by the spatial gauge choice

ξl = E , ξil′ = −Si , (361)

and thus El = Bl = 0, as well as Sil = 0. The remaining scalar metric perturbations and densityperturbations become

φl = φ+H(B − E′) + (B − E′)′ , (362)

ψl = ψ −H(B − E′) , (363)

δρl = δρ+ ρ′0(B − E′) , (364)

where the two scalar metric perturbations coincide with the Bardeen potentials (341). In this gauge theline element becomes

ds2 = a(η)2[−(1 + 2Φ)dη2 + (1− 2Ψ)γijdxidxj

](365)

which is a cosmological (conformal) generalization of the Schwarzschild metric, where Φ plays the roleof the first-order Newtonian potential φ, and Ψ gives the Newtonian curvature perturbation. This is thereason why it is sometimes called the conformal Newtonian gauge.

8.7.2 The flat-slicing gauge

Also known as the uniform curvature gauge or the off-diagonal gauge. It is defined purely by localmetric quantities. In this gauge one selects spatial hypersurfaces on which the induced 3-metric is leftunperturbed: ψ = E = 0. This corresponds to a gauge transformation

ξ0f = −ψ

H, ξf = E . (366)

The definitions of the remaining gauge invariant metric perturbations are

φf = φ+ ψ +(ψH

)′, (367)

Bf = B − E′ − ψ

H, (368)

δρf = δρ+ ρ′0ψ

H. (369)

The first two quantities coincide with the two potentials, A and B, defined by Kodama and Sasaki (84).Note also that in this gauge, σf = −Bf , while the difference between different choices of slicings givesgauge invariant metric perturbations, ξ0

f − ξ0l = Bf = −ψl/H, and thus

σf =ψlH,

i.e. the shear perturbation in the uniform-curvature gauge is equal to the curvature perturbation in thezero-shear (longitudinal) gauge over the rate of expansion.

In this gauge, the line element can be written as

ds2 = a(η)2[−(1 + 2φf )dη2 + 2Bf

|idxidη + γijdx

idxj]

(370)

In some cases, it is more convenient to use gauge invariant quantities like φf and Bf , rather than Φ and Ψwhen the background space-time behaves singularly (e.g. in a collapsing universe like in pre-Big-Bangcosmology), since they remain small.

Notice also that the scalar field perturbations in uniform-curvature (flat-slicing) hypersurfaces isthe gauge invariant scalar field perturbation defined by Mukhanov (85),

δϕf = δϕ+ ϕ′0ψ

H=u

a. (371)

8.7.3 The synchronous gauge

It is defined by the condition that the time vector is orthogonal to constant-time 3-space hypersurfaces:φ = B = 0 and Bi = 0. The problem is that this gauge choice does not determine the time-slicing

unambiguously: there is a residual gauge freedom and it is not possible to define gauge-invariant quan-tities in general using this gauge condition. One could always, as Lifshitz did in 1963, eliminate theunphysical degrees of freedom by symmetry arguments.

The metric in this gauge becomes

ds2 = a(η)2[−dη2 + (γij + hij)dxidxj

], (372)

with the perturbation

hij(η,x) ≡ h(η,x)|ij +(∇i∇j −

13γij∇2

)6τ(η,x) , (373)

written in terms of two arbitrary functions h(η,x) and τ(η,x). The gauge transformations that take anarbitrary set of coordinates to the synchronous gauge are

ξ0 = −1a

∫dη aφ+

α(x)a

, (374)

ξ =∫dη (ξ0 −B) + β(x) , (375)

ξi = −∫dη Bi + βi(x) , (376)

with βi|i = 0. It has a residual gauge freedom: 4 integration constants (α, β, βi) that correspond todifferent choices of spatial hypersurfaces, that give rise to fictitious “gauge modes” in the perturbationequations, and must be removed by physical considerations.

8.7.4 The comoving orthogonal gauge

This gauge is defined by choosing spatial coordinates such that the 3-velocity of the cosmic fluid van-ishes, v = 0. Orthogonality of the constant−η hypersurfaces to the 4-velocity uµ then requires v+B = 0,which implies that these hypersurfaces have zero vorticity.

The gauge transformations needed to go to this gauge are

ξ0c = −(v +B) , (377)

ξc = −∫dη v + ξ(x) , (378)

where ξ is a residual gauge freedom corresponding to a constant shift of spatial coordinates. All 3-scalarslike curvature, expansion, acceleration and shear, are independent of ξ. The scalar perturbations in thecomoving orthogonal gauge are then

φc = φ+H(v +B) + (v +B)′ , (379)

ψc = ψ −H(v +B) , (380)

Ec = E +∫dη v − ξ . (381)

Apart from the residual gauge dependence of Ec, these variables are gauge invariant under transforma-tions of their component parts.

Note that the curvature perturbation in the comoving gauge, ψc has the same expression as thevariable ζ (345), used for the first time by Lukash (80) and later called R by Lyth (85). The densityperturbation on comoving orthogonal hypersurfaces is given in gauge invariant form by

δρc = δρ+ ρ′0 (v +B) , (382)

and is equivalent to the expression εmρ0Q(0) of Bardeen (80). The gauge-invariant scalar density pertur-

bation of Ellis and Bruni (89) is ∆ = 1ρ0δρ|ic |i. The velocity perturvation in this gauge is vc is equivalent

to the gauge-invariant velocity perturvation V .

One can always write these quantities in terms of metric perturbations, rather than the velocitypotential, using the Einstein equations, to give

v +B =Ψ′ +HΦ−K(B − E′)

H′ −H2 −K. (383)

In particular, one can write the comoving curvature perturbation ψc in terms of longitudinal gauge in-variant quantities,

ψc = Ψ− H(Ψ′ +HΦ)H′ −H2 −K

, (384)

which coincides (for K = 0) with the variable ζ in Eq. (346), defined for the first time by Bardeen,Steinhardt and Turner (83), and later by Mukhanov, Feldman and Brandenberger (92).

8.7.5 The comoving total matter gauge

This gauge extends the previous one to the case of a multi-component fluid. Here the total momentumvanishes,

(ρ+ p)(v + B) ≡∑α

(ρα + pα)(vα + Bα) = 0 . (385)

Orthogonality of constant−η hypersurfaces to the 4-velocity again requires B = 0. Note that the gauge-invariant scalar velocity perturbation, V (the velocity in the longitudinal gauge), coincides in the totalmatter gauge with the shear σc of constant−η hypersurfaces.

8.7.6 The uniform density gauge

This gauge is defined by imposing that the matter density remains unperturbed, δρ = 0, in these hyper-surfaces. The coordinate transformation that takes us there is

ξ0d =

δρ

ρ′0. (386)

On these uniform-density hypersurfaces, the gauge-invariant curvature perturbation is

ζ ≡ ψd = ψ +Hδρρ′0. (387)

There is another degree of freedom left inside spatial hypersurfaces and we can choose either B, E or vto vanish. If we choose B = v = 0, then the gauge function is

ξd = −∫dη v . (388)

Note that the gauge-invariant curvature perturbation in the uniform density gauge is equal to the gauge-invariant curvature perturbation in the comoving gauge.

8.8 Perturbed Einstein field equationsWe will now write down the Einstein field equations, Gµν = κ2 Tµν and separate between the back-ground Einstein equations, G(0)

µν = κ2 T(0)µν , and the first order perturbed equations, δGµν = κ2 δTµν .

These equations can be written in specific gauges as well as in a gauge invariant form.

Einstein equations can be separated into the Hamiltonian and momentum contraint equations,(220) and (221), together with the evolution equation, which itself can be separated into the trace (222)and traceless (223) parts. The background equations give the Friedmann equations,

H2 +K =κ2

3a2 ρ , Hamiltonian constraint (389)

0 = 0 , Momentum constraint (390)

H′ = −κ2

6a2 (ρ+ 3p) , Evolution equation (trace) (391)

0 = 0 , Evolution equation (traceless) (392)

All the background equations are scalar equations since the background fields are also scalars. However,the perturbed equations can be separated into scalar, vector and tensor perturbation equations. We willwrite first the gauge-dependent equations, and then their gauge-invariant forms, in terms of the gauge-invariant potentials and matter fields.

The scalar perturbation equations can be separated into the energy and momentum constraintequations, plus the trace and traceless evolution equations,

3H(ψ′ +Hφ)− (∇2 + 3K)ψ −H∇2σ = −κ2

2a2 δρ , Hamiltonian constraint (393)

ψ′ +Hφ+Kσ = −κ2

2a2 (ρ+ p)(v +B) , Momentum constraint (394)

ψ′′ + 2Hψ′ −Kψ +Hφ′ + (2H′ +H2)φ =κ2

2a2(δp+

23∇2Π

), Trace equation (395)

σ′ + 2Hσ + ψ − φ = κ2 a2 Π , Traceless equation (396)

where σ = E′ − B is the scalar shear. The gauge-invariant scalar equations for the 4 gauge invariantscalar quantities (Φ,Ψ, V,D ≡ Dc) is

(∇2 + 3K)Ψ =κ2

2a2 ρ0D , Poisson equation (397)

Ψ′ +HΦ = −κ2

2a2 (ρ0 + p0)V , Constraint equation (398)

HU ′ + (2H′ +H2)U =κ2

2a2(ρ0(c2

sDf + wΓ) +23∇2Π

), Evolution equation (399)

Ψ− Φ = κ2 a2 Π , Anisotropic stress (400)

where δp = ρ0(c2sδ + wΓ), and we have defined a new variable U ≡ εζ, with ε = (1 − H′/H2), and

Df = D−3(1+w)ζ. In order to solve these coupled equations we need to specify the matter content, i.e.w, c2

s, Γ and Π. For instance, for a perfect fluid without anisotropic stresses, Γ = Π = 0 automaticallyimplies that the two Newtonian potentials are identical, Ψ = Φ.

The vector perturbation equations can be separated into a momentum constraint equation and anevolution equation

(∇2 + 2K)(Si + F ′i ) = 2κ2 a2 (ρ0 + p0)(Si − vi) , Constraint equation (401)

τ ′ij +Hτij =κ2

2a3 Π(i|j) , Evolution equation (402)

written in terms of the vector shear (330).

The gauge-invariant vector perturbation equations can be written in terms of the gauge-invariantvector fields, Σi and Vi, (343) and (354),(

∇2 + 2K − 4εH2)

Σi = −2κ2 a2 (ρ0 + p0)Vi , Constraint equation (403)

Σ′i + 2HΣi = κ2 a2 Πi , Evolution equation (404)

For a perfect fluid with Πi = 0, these equations, on large scales and for a flat universe, give

Σi = Vi ∝1a2→ 0 , (405)

which is equivalent to δu(g.i.)i ∝ 1/a3, and therefore, in the case of a matter fluid dominated e.g. by a

scalar field, the vector perturbations are quickly driven to zero, even if initially there were anisotropicsources, as long as in the late time evolution there are no sources of vorticity (like due to magnetic fieldsor gravitational waves).

The tensor perturbation equations are already written in gauge-invariant form and arise form the(ij) components of the Einstein equations,

h′′ij + 2Hhij + (2K −∇2)hij = 2κ2 a2 Πij . (406)

Since there is no tensor conservation equation for the matter variables, the tensor metric perturbationis solely determined by the Hubble parameter and the scale factor, and is only sourced by the matteranisotropic tensor perturbation. For scalar and vector matter fields, Πij = 0 for linear perturbations, andthus the evolution equation is source-free and homogeneous.

On superhorizon scales and zero curvature, the homogeneous equation in the radiation era (H =1/η) has a decaying (hij ∝ 1/η and a growing (hij ∝ const.) solution. As the mode enters the horizon,the oscillatory behaviour takes over, and the gravitational wave propagates with frequency k2 + 2K, andamplitude damped as h ∝ 1/a.

In the absence of anisotropic stresses (Πij = 0) in a flat universe (K = 0), the general solutionfor hij = h(η,x) εij , with εij the polarization tensor associated with the wave, in Fourier space is

h(η, k) =1

(kη)p−1

[Ajp−1(kη) +B np−1(kη)

], (407)

with jν and nν the spherical Bessel and Von Neumann functions of order ν. The evolution of the scalefactor is given by a ∝ ηp.

8.9 Conservation of energy-momentum tensorThe covariant conservation of the energy-momentum tensor, Tµν;ν = 0, gives the homogeneous conser-vation of energy in the isoentropic expansion, T 0ν

;ν = 0,

ρ′0 = −3H(ρ0 + p0) , or d(ρ0a3) + p0 d(a3) = 0 . (408)

There is no background momentum conservation equation, T iν;ν = 0, since the momentum is zero byassumption of homogeneity.

The perturbed conservation equations follow from the Bianchi identity, δTµν;ν = 0, and providewith evolution equations for the physical degrees of freedom that are often simpler and easier to dealwith than the perturbed Einstein equations, since they are first order, rather than second order, as thelatter usually are.

The first order scalar perturbation equation for the energy density perturbation is

δρ′ + 3H(δρ+ δp) = (ρ0 + p0)[3ψ′ −∇2(v + E′)] , (409)

[(ρ0 + p0)(v +B)]′ + δp+23

(∇2 + 3K)Π = −(ρ0 + p0)[φ+ 4H(v +B)] , (410)

together with the evolution equation for the scalar momentum perturbation. These equations can bewritten in gauge-invariant form as two equations, one for the density contrast Df (in the flat-slicinggauge) and the other for the velocity perturbation V , using δp = ρ0(wΓ + c2

sδ) and w′ = −3H(1 +w)(c2

s − w),

D′f + 3H(c2s − w)Df = −3HwΓ− (1 + w)∇2V , (411)

V ′ +H(1− 3c2s)V = −Φ− 3c2

sΨ−1

1 + w

[wΓ + c2

sDf +2w3

(∇2 + 3K)Π], (412)

where we have redefined Π = p0Π = wρ0Π. Sometimes it is more convenient to write them in terms ofthe density contrast in the comoving gauge, D ≡ Dc,

D′ − 3wHD = (∇2 + 3K)[2HΠ− (1 + w)V

], (413)

V ′ +HV = −Φ− 11 + w

[wΓ + c2

sD +2w3

(∇2 + 3K)Π]. (414)

There is no energy conservation equation in the case of vector perturbations since energy is a scalarin the spatial hypersurfaces, but we do have a momentum conservation equation for vector perturbationssince momentum is a 3-vector,

[(ρ0 + p0)(vi − Si)]′ + 4H(ρ0 + p0)(vi − Si) = −(∇2 + 2K)Πi . (415)

This equation can be put in gauge-invariant form by defining a new variable, the vorticity vector Ωi ≡Σi − Vi,

Ω′i +H(1− 3c2s)Ωi =

w

2(1 + w)∇2Πi . (416)

If the anisotropic stress vanishes, Πi = 0, then there is a solution

Ωi ∝ a3c2s−1

Ωi ∝ const. for radiation eraΩi ∝ a−1 for matter era

(417)

There is no energy nor momentum conservation equation for tensor matter variables and thus theyare decoupled from the other tensor quantities.

8.9.1 Three different gauge choices

To illustrate the gauge invariant approach, we will take the gauge-dependent equations and write themin terms of gauge-invariant quantities, which coincide with physical degrees of freedom in particulartime-slicings.

The evolution equations in the longitudinal gauge arise when imposing B = E = 0, and thusσ = 0, as well as Si = 0. The gauge-invariant equations of motion in this gauge are the following.

The conservation of the energy-momentum tensor gives the continuity equation and the constraintequation,

δρ′l + 3H(δρl + δpl) = (ρ0 + p0)[3ψ′l −∇2v′l] , (418)

[(ρ0 + p0)vl]′ + δpl +23

(∇2 + 3K)Πl = −(ρ0 + p0)[φ′l + 4Hvl] . (419)

The scalar metric perturbation evolution equations (trace and traceless parts) are

ψ′′l + 2Hψ′l −Kψl +Hφ′l + (2H′ +H2)φl =κ2

2a2(δpl +

23∇2Πl

), (420)

ψl − φl = κ2a2 Πl . (421)

The energy and momentum constraint equations are

3H(ψ′l +Hφl)− (∇2 + 3K)φl = −κ2

2a2 δρl , (422)

ψ′l +Hφl = −κ2

2a2 (ρ0 + p0)vl . (423)

The case Πl = 0 for a perfect fluid, or a scalar field, gives ψl = φl, which simplifies the equations.

The evolution equations in the uniform density gauge arise when imposing δρ = 0, and also fixingspatial hypersurfaces with B = 0, so that σd = Ed. The gauge-invariant equations of motion in thisgauge are the following.

The conservation of the energy-momentum tensor gives the continuity equation and the constraintequation,

3H(ρ0 + p0)

δpd = 3ψ′d −∇2(vd + σd) , (424)

[(ρ0 + p0)vd]′ + δpd +23

(∇2 + 3K)Πd = −(ρ0 + p0)[φ′d + 4Hvd] . (425)

The first equation is crucial for understanding the behaviour of perturbations on large scales, when thesecond term in the R.H.S. vanishes, and thus ψd = constant on large scales for vanishing non-adiabaticpressure perturbations.

The Einstein evolution equations (trace and traceless parts) give

ψ′′d + 2Hψ′d −Kψd +Hφ′d + (2H′ +H2)φd =κ2

2a2(δpd +

23∇2Πd

), (426)

σ′d + 2Hσd + ψd − φd = κ2a2 Πd . (427)

The energy and momentum constraint equations are

3H(ψ′d +Hφd)− (∇2 + 3K)φd −H∇2σd = 0 , (428)

ψ′d +Hφd +Kσd = −κ2

2a2 (ρ0 + p0)vd . (429)

The case Πd = 0 for a perfect fluid or a scalar field does not imply in this case ψd = φd.

The evolution equations in the comoving gauge are defined by a vanishing 3-velocity v = 0, andalso fixing spatial hypersurfaces with B = 0. The gauge-invariant equations of motion in this gaugebecome as follows.

The conservation of the energy-momentum tensor gives the continuity and the constraint equation,

δρ′c + 3H(δρc + δpc) = (ρ0 + p0)[3ψ′c −∇2E′c] , (430)

δpc +23

(∇2 + 3K)Πc = −(ρ0 + p0) φ′c . (431)

The evolution equations (trace and traceless parts) give

ψ′′c + 2Hψ′c −Kψc +Hφ′c + (2H′ +H2)φc =κ2

2a2(δpc +

23∇2Πc

), (432)

σ′c + 2Hσc + ψc − φc = κ2a2 Πc . (433)

The energy and momentum constraint equations are

3H(ψ′c +Hφc)− (∇2 + 3K)φc −H∇2σc = −κ2

2a2 δρc , (434)

ψ′d +Hφd +Kσd = 0 . (435)

8.9.2 The Bardeen equation

It is often convenient to have an evolution equation for the Bardeen potential written in terms of thetotal matter content. By combining the matter conservation equation (413) with the Poisson (397) andconstraint (398) equations we obtain a second order equation for Ψ,

Ψ′′ + 3H(1 + c2s)Ψ

′ + [3(c2s − w)H2 − (1 + 3c2

s)K − c2s∇2]Ψ = (436)

= κ2a2[HΠ′ +

(2H′ + 3H2

w(w − c2

s))

Π +12∇2Π +

12p0Γ]. (437)

This equation can be recast (using the variable U = εζ) into a first order evolution equation for the gaugeinvariant curvature perturbation ζ = Ψ −HV . For hydrodynamical matter with Π = 0, and flat spatialsections (K = 0), we find

ζ ′ =1εH

[c2s∇2Ψ +

32H2wΓ] , (438)

which is extremely useful when discussing the evolution of curvature and entropy perturbations in thecase of multiple fluids.

8.10 Multiple fluidsPrevious expressions assume the universe is filled with, or dominated by, a single fluid with energy den-sity ρ and pressure p. We know the universe is filled during most of its evolution by more than one fluid.In particular, during the epoch of photon decoupling, there are four different components, two relativistic(photons and neutrinos) and two non-relativistic (baryons and cold dark matter). Therefore, we shouldtake into account these different components with different equations of state, and in some cases withexchange of energy between them, as occurs between baryons and photons in the tight-coupling regimebefore recombination.

If there is more than one matter component, then the total perturbation variables are actuallyweighted averages of the individual perturbation variables,

δ ≡∑α

ραρδα , vi ≡

∑α

(ρα + pαρ+ p

)viα , Πij ≡

∑α

Πijα . (439)

The equation of state and adiabatic sound speed are defined for each component,

wα ≡pαρα

, c2α ≡

p′αρ′α

, ρ′α + 3Hρα(1 + wα) = 0 , (440)

where we have assumed for the moment that there is no exchange of energy between different compo-nents and thus the total energy on each component is conserved in the expansion. For the overall mixture,we have

w ≡ p

ρ, c2

s ≡p′

ρ′, ρ′ + 3Hρ(1 + w) = 0 . (441)

The transformation properties under general coordinate reparametrizations are the same as for total mat-ter variables, e.g. δα = δα − 3H(1 + wα)ξ0. For each matter component we can define gauge invariantquantities,

Γα ≡ πL,α −c2α

wαδα , (442)

Vα ≡ vα + E′ , (443)

Dl,α ≡ δα − 3(1 + wα)H(B − E′) , longitudinal gauge (444)

Df,α ≡ δα − 3(1 + wα)ψ , flat slicing gauge (445)

Dc,α ≡ δα − 3(1 + wα)H(vα +B) . comoving gauge (446)

For multiple matter components, it is often useful to work with gauge-invariant density contrasts,

∆α ≡ δα − 3(1 + wα)H(B + v) , (447)

which corresponds to the density contrast in the gauge where the total matter is at rest (the total comovinggauge). It is also related to the density contrast in the flat slicing gauge,

∆α = Df,α + 3(1 + wα)ζ , (448)

a very useful expression since it includes the gauge-invariant curvature perturbation ζ, which remainsconstant on superhorizon scales, for adiabatic perturbations.

8.10.1 Entropy perturbations

When more than one component is present, entropy perturbations can arise even for amixture of perfectuncoupled fluids. Clearly a multicomponent fluid can exchange not only energy, but also particle number(entropy). The non-adiabaticity of the mixture is given by Γ, see (349). Using definitions of mattervariables, this can be expressed as

pΓ = pΓint +∑α

δαρα(c2α − c2

s) = p(Γint + Γrel) , (449)

where the total intrinsic entropy isΓint =

∑α

pαp

Γα , (450)

while the relative entropy is given by

pΓrel =12

∑α,β

(1 + wα)(1 + wβ)ραρβ(1 + w)ρ

(c2α − c2

β)(

δα1 + wα

−δβ

1 + wβ

). (451)

Note that we have assumed that the components have decoupled from each other, i.e. Qνα = 0. We willdescribe later the case for interacting fluids.

The entropy perturbation betwen α and β components is defined as

Sαβ ≡δα

1 + wα−

δβ1 + wβ

= δ lnnαnβ

, (452)

That is, the fluctuation in the local number density of different components. In fact, the entropy perturba-tion is itself a gauge-invariant quantity that can be expressed in terms of gauge invariant density contrastsas

Sαβ =∆α

1 + wα−

∆β

1 + wβ. (453)

Since both the number density and the entropy density evolve with temperature as T 3, we have δnr/nr =(3/4)δρr/ρr, while for matter δnm/nm = δρm/ρm. Therefore,

Srm =34δρrρr− δρm

ρm. (454)

A non zero entropy density thus corresponds to a spatial inhomogeneity in the relative number density,or equivalently, to spatial variations of the equation of state.

9 Quantum field theory in curved space-time

We will describe in this section the formalism used to describe the origin of metric fluctuations duringinflation.

9.1 Scalar field perturbed equationsConsider the action (256) with line element

ds2 = a(η)2[−(1 + 2Φ)dη2 + (1− 2Φ)dx2

]in the Longitudinal gauge, where Φ is the gauge-invariant gravitational potential (341). Then the gauge-invariant equations for the perturbations on comoving hypersurfaces (constant energy density hypersur-faces) are

Φ′′ + 3HΦ′ + (H′ + 2H2)Φ =κ2

2[φ′δφ′ − a2V ′(φ)δφ] , (455)

−∇2Φ + 3HΦ′ + (H′ + 2H2)Φ = −κ2

2[φ′δφ′ + a2V ′(φ)δφ] , (456)

Φ′ +HΦ =κ2

2φ′δφ , (457)

δφ′′ + 2Hδφ′ −∇2δφ+ a2V ′′(φ)δφ = 4φ′Φ′ − 2a2V ′(φ)Φ .

This system of equations seem too difficult to solve at first sight.However, there is a gauge invariant combination of Mukhanov variables

u ≡ aδφ+ zΦ ,

z ≡ aφ′

H.

for which the above equations simplify enormously,

u′′ −∇2u− z′′

zu = 0 ,

∇2Φ =κ2

2Ha2

(zu′ − z′u) , (458)(a2ΦH

)′=κ2

2zu . (459)

From these, we can find a solution u(z), which can be integrated to give Φ(z), and together allow us toobtain δφ(z).

9.2 Canonical quantization in perturbation theoryUntil now we have treated the perturbations as classical, but we should in fact consider the perturbationsΦ and δφ as quantum fields. Note that the perturbed action for the scalar mode u can be written as

δS =12

∫d3x dη

[(u′)2 − (∇u)2 +

z′′

zu2]. (460)

In order to quantize the field u in the curved background defined by the metric (302), we can write theoperator

u(η,x) =∫

d3k(2π)3/2

[uk(η) ak eik·x + u∗k(η) a†k e

−ik·x], (461)

where the creation and annihilation operators satisfy the commutation relation of bosonic fields, and thescalar field’s Fock space is defined through the vacuum condition,

[ak, a†k′ ] = δ3(k− k′) , (462)

ak|0〉 = 0 . (463)

Note that we are not assuming that the inflaton is a fundamental scalar field, but that is can be written asa quantum field with its commutation relations (as much as a pion can be described as a quantum field atlow energies).

If we impose the equal-time commutation relations on the fields themselves,

[u(η,x), Π†u(η,x′)] = i~δ3(x− x′) ,

we find a normalization condition on the modes uk

uk u∗k′ − u′k u∗k = i , (464)

that coincides with the Wronskian of the mode equation,

u′′k +(k2 − z′′

z

)uk = 0 . (465)

Note that the modes decouple in linear perturbation theory. The ratio U(η) = z′′/z acts like a time-dependent potential for this 1D Schrodinger like equation (with time↔ space),

−u′′k + U(η)uk = k2 uk .

In order to find exact solutions to the mode equation, we will use the slow-roll parameters (270),

ε = 1− H′

H2=κ2

2z2

a2, (466)

δ = 1− φ′′

Hφ′= 1 + ε− z′

Hz, (467)

ξ = −(

2− ε− 3δ + δ2 − φ′′′

H2φ′

). (468)

In terms of these parameters, the conformal time and the effective potential for the uk mode can bewritten as

η =−1H

+∫εda

aH, (469)

z′′

z= H2

[(1 + ε− δ)(2− δ) +H−1(ε′ − δ′)

]. (470)

Note that the slow-roll parameters, (466) and (467), can be taken as constant,9 to order O(ε2),

ε′ = 2H(ε2 − εδ

)= O(ε2) ,

δ′ = H(εδ − ξ

)= O(ε2) .

(471)

In that case, for constant slow-roll parameters, we can write

η =−1H

11− ε

, (472)

z′′

z=

1η2

(ν2 − 1

4

), where ν =

1 + ε− δ1− ε

+12. (473)

9.3 Exact solutionsWe are now going to search for approximate solutions of the mode equation (465), where the effectivepotential (470) is of order z′′/z ' 2H2 in the slow-roll approximation. In quasi-de Sitter there is acharacteristic scale given by the (event) horizon size or Hubble scale during inflation, H−1. There willbe modes uk with physical wavelengths much smaller than this scale, k/a H , that are well withinthe de Sitter horizon and therefore do not feel the curvature of space-time. On the other hand, therewill be modes with physical wavelengths much greater than the Hubble scale, k/a H . In these twoasymptotic regimes, the solutions can be written as

uk =1√2k

e−ikη k aH , (474)

uk = C1(k) z k aH . (475)

In the limit k aH the modes behave like ordinary quantum modes in Minkowsky space-time, ap-propriately normalized, while in the opposite limit, u/z becomes constant on superhorizon scales. Forapproximately constant slow-roll parameters one can find exact solutions to (465), with the effectivepotential given by (473), that interpolate between the two asymptotic solutions,

uk(η) =√π

2ei(ν+ 1

2)π

2 (−η)1/2H (1)ν (−kη) , (476)

where H (1)ν (z) is the Hankel function of the first kind (e.g. H (1)

3/2(x) = −eix√

2/πx(1 + i/x)), and ν isgiven by (473) in terms of the slow-roll parameters. In the limit kη → 0, the solution becomes

|uk| =2ν−

32

√2k

Γ(ν)Γ(3

2)(−kη)

12−ν ≡ C(ν)√

2k

( k

aH

) 12−ν, (477)

C(ν) = 2ν−32

Γ(ν)Γ(3

2)(1− ε)ν−

12 ' 1 for ε, δ 1 . (478)

We can now compute Φ and δφ from the super-Hubble-scale mode solution (475), for k aH .Substituting into Eq. (459), we find

Φ = C1

(1− H

a2

∫a2dη

)+ C2

Ha2, (479)

δφ

φ′=C1

a2

∫a2dη − C2

a2. (480)

9For instance, there are models of inflation, like power-law inflation, a(t) ∼ tp, where ε = δ = 1/p < 1, that give constantslow-roll parameters.

The term proportional to C1 corresponds to the growing solution, while that proportional to C2 corre-sponds to the decaying solution, which can soon be ignored. These quantities are gauge invariant butevolve with time outside the horizon, during inflation, and before entering again the horizon during theradiation or matter eras. We would like to write an expression for a gauge invariant quantity that isalso constant for superhorizon modes. Fortunately, in the case of adiabatic perturbations, there is such aquantity:

ζ ≡ Φ +1εH

(Φ′ +HΦ) =u

z, (481)

which is constant, see Eq. (475), for k aH . In fact, this quantity ζ is identical, for superhorizonmodes, to the gauge invariant curvature metric perturbation Rc on comoving (constant energy density)hypersurfaces,

ζ = Rc +1εH2∇2Φ . (482)

Using Eq. (458) we can write the evolution equation for ζ = uz as ζ ′ = 1

εH ∇2Φ, which confirms

that ζ is constant for (adiabatic) superhorizon modes, k aH , but fails for entropy or isocurvatureperturbations. Therefore, we can evaluate the Newtonian potential Φk when the perturbation reenters thehorizon during radiation/matter eras in terms of the curvature perturbation Rk when it left the Hubblescale during inflation,

Φk =(

1− Ha2

∫a2dη

)Rk =

3 + 3ω5 + 3ω

Rk =

23 Rk radiation era ,

35 Rk matter era .

(483)

These expressions will be of special importance later.

9.4 Gravitational wave perturbationsLet us now compute the tensor or gravitational wave metric perturbations generated during inflation. Theperturbed action for the tensor mode can be written as

δS =12

∫d3x dη

a2

2κ2

[(h′ij)

2 − (∇hij)2], (484)

with the tensor field hij considered as a quantum field,

hij(η,x) =∫

d3k(2π)3/2

∑λ=1,2

[hk(η) eij(k, λ) ak,λ eik·x + h.c.

], (485)

where eij(k, λ) are the two polarization tensors, satisfying symmetric, transverse and traceless conditions

eij = eji , kieij = 0 , eii = 0 , (486)

eij(−k, λ) = e∗ij(k, λ) ,∑λ

e∗ij(k, λ)eij(k, λ) = 4 , (487)

while the creation and annihilation operators satisfy the usual commutation relation of bosonic fields,Eq. (462). We can now redefine our gauge invariant tensor amplitude as

vk(η) =a√2κ

hk(η) , (488)

which satisfies the following evolution equation, decoupled for each mode vk(η) in linear perturbationtheory,

v′′k +(k2 − a′′

a

)vk = 0 . (489)

The ratio a′′/a acts like a time-dependent potential for this Schrodinger like equation, analogous to theterm z′′/z for the scalar metric perturbation. For constant slow-roll parameters, the potential becomes

a′′

a= 2H2

(1− ε

2

)=

1η2

(µ2 − 1

4

), (490)

µ =1

1− ε+

12. (491)

We can solve equation (489) in the two asymptotic regimes,

vk =1√2k

e−ikη k aH , (492)

vk = C3(k) a k aH . (493)

In the limit k aH the modes behave like ordinary quantum modes in Minkowsky space-time, ap-propriately normalized, while in the opposite limit, the metric perturbation hk becomes constant onsuperhorizon scales. For constant slow-roll parameters one can find exact solutions to (489), with effec-tive potential given by (490), that interpolate between the two asymptotic solutions. These are identicalto Eq. (476) except for the substitution ν → µ. In the limit kη → 0, the solution becomes

|vk| =C(µ)√

2k

( k

aH

) 12−µ. (494)

Since the mode hk becomes constant on superhorizon scales, we can evaluate the tensor metric pertur-bation when it reentered during the radiation or matter era directly in terms of its value during inflation.

9.5 Power spectra of scalar and tensor metric perturbationsNot only do we expect to measure the amplitude of the metric perturbations generated during inflationand responsible for the anisotropies in the CMB and density fluctuations in LSS, but we should also beable to measure its power spectrum, or two-point correlation function in Fourier space. Let us considerfirst the scalar metric perturbationsRk, which enter the horizon at a = k/H . Its correlator is given by

〈0|R∗kRk′ |0〉 =|uk|2

z2δ3(k− k′) ≡ PR(k)

4πk3(2π)3 δ3(k− k′) , (495)

PR(k) =k3

2π2

|uk|2

z2=κ2

2ε

(H2π

)2 ( k

aH

)3−2ν≡ A2

S

( k

aH

)ns−1, (496)

where we have used Rk = ζk = ukz and Eq. (477). This last equation determines the power spectrum in

terms of its amplitude at horizon-crossing, AS , and a tilt,

ns − 1 ≡ d lnPR(k)d ln k

= 3− 2ν = 2(δ − 2ε

1− ε

)' 2ηV − 6εV , (497)

see Eqs. (277), (278). Note from this equation that it is possible, in principle, to obtain from inflation ascalar tilt which is either positive (n > 1) or negative (n < 1). Furthermore, depending on the particularinflationary model, we can have significant departures from scale invariance.

Note that at horizon entry kη = −1, and thus we can alternatively evaluate the tilt as

ns − 1 ≡ − d lnPRd ln η

= −2ηH[(1− ε)− (ε− δ)− 1

]= 2(δ − 2ε

1− ε

)' 2ηV − 6εV , (498)

and the running of the tilt

dnsd ln k

= − dnsd ln η

= −ηH(

2ξ + 8ε2 − 10εδ)' 2ξV + 24ε2V − 16ηV εV , (499)

where we have used Eqs. (471).

Let us consider now the tensor (gravitational wave) metric perturbation, which enter the horizonat a = k/H ,

∑λ

〈0|h∗k,λhk′,λ|0〉 = 42κ2

a2|vk|2δ3(k− k′) ≡ Pg(k)

4πk3(2π)3 δ3(k− k′) , (500)

Pg(k) = 8κ2(H

2π

)2 ( k

aH

)3−2µ≡ A2

T

( k

aH

)nT, (501)

where we have used Eqs. (488) and (494). Therefore, the power spectrum can be approximated by apower-law expression, with amplitude AT and tilt

nT ≡d lnPg(k)d ln k

= 3− 2µ =−2ε1− ε

' −2εV < 0 , (502)

which is always negative. In the slow-roll approximation, ε 1, the tensor power spectrum is scaleinvariant.

Alternatively, we can evaluate the tensor tilt by

nT ≡ −d lnPgd ln η

= −2ηH[(1− ε)− 1

]=−2ε1− ε

' −2εV , (503)

and its running by

dnTd ln k

= − dnTd ln η

= −ηH(

4ε2 − 4εδ)' 8ε2V − 4ηV εV , (504)

where we have used Eqs. (471).

9.6 Massless minimally coupled scalar field fluctuationsThe fluctuations of a massless minimally-coupled scalar field φ during inflation (quasi de Sitter) arequantum fields in a curved background. We will redefine y(x, t) = a(t) δφ(x, t), whose action is

S =12

∫d3x dη

[(y′)2 − (∇y)2 +

a′′

ay2

], (505)

where primes denote derivatives w.r.t. conformal time η =∫dt/a(t) = −1/(aH), withH the constant

rate of expansion during inflation. Using the identity (y′)2 + a′′

a y = (y′− a′

a y)2 +(a′

a y2)′, we can define

the conjugate momentum as p = ∂L∂y′ = y′ − a′

a y, and write the corresponding Hamiltonian as

H =12

∫d3x

[p2 + (∇y)2 + 2

a′

ap y

]. (506)

We can now Fourier transform: Φ(k, η) =∫

d3x(2π)3/2

Φ(x, η) e−ix·k

all the fields and momenta. Since the scalar field is assumed real, we have: y(k, η) = y†(−k, η) andp(k, η) = p†(−k, η), and the Hamiltonian becomes

H =12

∫d3k

[p(k, η) p†(k, η) + k2 y(k, η) y†(k, η)

+a′

a

(y(k, η) p†(k, η) + p(k, η) y†(k, η)

)].

(507)

As we will see later, it is the last term which is responsible for squeezing.

The Euler-Lagrange equations for this field can be written in terms of the field eigenmodes as aseries of uncoupled oscillator equations:

p′ = −i [p,H] = −k2 y − a′

ap

y′ = −i [y,H] = p+a′

ay .

y′′(k, η) +(k2 − a′′

a

)y(k, η) = 0 . (508)

9.7 Heisenberg picture: The field operatorsWe can now treat each mode as a quantum oscillator, and introduce the corresponding creation andannihilation operators:

a(k, η) =

√k

2y(k, η) + i

1√2k

p(k, η) , (509)

a†(−k, η) =

√k

2y(k, η)− i 1√

2kp(k, η) , (510)

which can be inverted to give

y(k, η) =1√2k

[a(k, η) + a†(−k, η)

], (511)

p(k, η) = −i√k

2

[a(k, η)− a†(−k, η)

]. (512)

The usual equal-time commutation relations for fields (~ = 1 here and throughout),[y(x, η), p(x′, η)

]= i δ3(x− x′) , (513)

becomes a commutation relation for the creation and annihilation operators,[y(k, η), p†(k′, η)

]= i δ3(k− k′)⇒

[a(k, η), a†(k′, η)

]= δ3(k− k′) . (514)

In terms of these operators, the Hamiltonian becomes:

H =12

∫d3k

[k[a(k, η) a†(k, η) + a†(k, η) a(k, η)

]+ i

a′

a

[a†(k, η) a†(k, η) + a(k, η) a(k, η)

]].

(515)

It is the last (non-diagonal) term which is responsible for squeezing.

The evolution equation can be written as(a′(k)

a†′(−k)

)=

(−ik a′

a

a′

a ik

)(a(k)

a†(−k)

), (516)

whose general solution is, in terms of the initial conditions a(k, η0),

a(k, η) = uk(η) a(k, η0) + vk(η) a†(−k, η0) , (517)

a†(−k, η) = u∗k(η) a†(−k, η0) + v∗k(η) a(k, η0) , (518)

which correspond to a Bogoliubov transformation of the creation and annihilation operators, and charac-terizes the time evolution of the system of harmonic oscillators in the Heisenberg representation.

The commutation relation (514) is preserved under the unitary evolution if

|uk(η)|2 − |vk(η)|2 = 1 , (519)

which gives a normalization condition for these functions.

We can write the quantum fields y and p in terms of these as,

y(k, η) = fk(η) a(k, η0) + f∗k (η) a†(−k, η0) , (520)

p(k, η) = −i[gk(η) a(k, η0)− g∗k(η) a†(−k, η0)

], (521)

where the functions

fk(η) =1√2k

[uk(η) + v∗k(η)] , (522)

gk(η) =

√k

2[uk(η)− v∗k(η)] , (523)

are the field and momentum modes, respectively, satisfying the following equations and initial conditions,

f ′′k +(k2 − a′′

a

)fk = 0 , fk(η0) = 1√

2k, (524)

gk = i

(f ′k −

a′

afk

), gk(η0) =

√k2 , (525)

as well as the Wronskian condition,

i (f ′k f∗k − f ′k

∗fk) = gk f

∗k + g∗k fk = 1 . (526)

9.8 Squeezing parametersSince we have two complex functions, fk and gk, plus a constraint (526), we can write these in terms ofthree real functions in the standard parametrization for squeezed states,

uk(η) = e−i θk(η) cosh rk(η) , (527)

vk(η) = ei θk(η)+2i φk(η) sinh rk(η) , (528)

where rk is the squeezing parameter, φk the squeezing angle, and θk the phase.

We can also write its relation to the usual Bogoliubov formalism in terms of the functions αk, βk,

uk = αk e−ikη , v∗k = βk e

ikη , (529)

which is useful for the adiabatic expansion, and allows one to write the average number of particles andother quantities,

nk = |βk|2 = |vk|2 =12k

∣∣∣gk − k fk∣∣∣2 = sinh2 rk , (530)

σk = 2Re(α∗kβk e

2ikη)

= 2Re (u∗k v∗k) = cos 2φk sinh 2rk , (531)

τk = 2Im(α∗kβk e

2ikη)

= 2Im (u∗k v∗k) = − sin 2φk sinh 2rk . (532)

We can invert these expressions to give (rk, θk, φk) as a function of uk and vk,

sinh rk =√

Rev2k + Imv2

k , cosh rk =√

Reu2k + Imu2

k , (533)

tan θk = − ImukReuk

, tan 2φk =ImvkReuk + ImukRevkRevkReuk − ImukImvk

. (534)

We can now write Eqs. (520) and (521) in terms of the initial values,

y(k, η) =√

2k fk1(η) y(k, η0)−√

2kfk2(η) p(k, η0) , (535)

p(k, η) =

√2kgk1(η) p(k, η0) +

√2k gk2(η) y(k, η0) , (536)

where subindices 1 and 2 correspond to real and imaginary parts, fk1 ≡ Re fk and fk2 ≡ Im fk, andsimilarly for the momentum mode gk.

9.9 The Schrodinger picture: The vacuum wave functionLet us go now from the Heisenger to the Schrodinger picture, and compute the initial state vacuumeigenfunction Ψ0(η = η0). The initial vacuum state |0, η0〉 is defined through the condition

∀k , a(k, η0)|0, η0〉 =

[√k

2yk(η0) + i

1√2k

pk(η0)

]|0, η0〉 = 0 ,[

y0k +

1k

∂

∂y0k∗

]Ψ0

(y0k, y

0k∗, η0

)= 0 ⇒ Ψ0

(y0k, y

0k∗, η0

)= N0 e

−k |y0k|

2

where we have used the position representation, yk(η0) = y0k , pk(η0) = −i ∂

∂y0k∗ , and N0 gives the

corresponding normalization.

We will now study the time evolution of this initial wave function using the unitary evolutionoperator S = S(η, η0), i.e. the state evolves in the Schrodinger picture as |0, η〉 = S|0, η0〉. Now,inverting (520) and (521)

a(k, η0) = g∗k(η) y(k, η) + i f∗k (η) p(k, η) , (537)

which, acting on the initial state becomes, ∀k ,∀η,

S

[y(k, η) + i

f∗k (η)g∗k(η)

p(k, η)]S−1S|0, η0〉 = 0

⇒[yk(η0) + i

f∗k (η)g∗k(η)

pk(η0)]|0, η〉 = 0 ,

⇒ Ψ0

(y0k, y

0k∗, η)

=1√

π |fk(η)|e−Ωk(η) |y0

k|2, (538)

where

Ωk(η) =g∗k(η)f∗k (η)

= ku∗k − vku∗k + vk

=1− 2i Fk(η)

2|fk(η)|2, (539)

Fk(η) = Im(f∗k gk) = Im(uk vk) =12

sin 2φk sinh 2rk . (540)

We see that the unitary evolution preserves the Gaussian form of the wave functional. The wave function(538) is called a 2-mode squeezed state.

The normalized probability distribution,

P0 (y(k, η0), y(−k, η0), η) =1

π |fk(η)|2exp

(−|y(k, η0)|2

|fk(η)|2

), (541)

is a Gaussian distribution, with dispersion given by |fk|2.

In fact, we can compute the vacum expectation values,

〈∆y(k, η) ∆y†(k′, η)〉 ≡ ∆y2(k) δ3(k− k′) = |fk|2 δ3(k− k′) , (542)

〈∆p(k, η) ∆p†(k′, η)〉 ≡ ∆p2(k) δ3(k− k′) = |gk|2 δ3(k− k′) , (543)

and therefore the Heisenberg uncertainty principle reads

∆y2(k) ∆p2(k) = |fk|2 |gk|2 = F 2k (η) +

14≥ 1

4. (544)

It is clear that for η = η0, Ωk(η0) = k and Fk(η0) = 0, and thus we have initially a minimum wavepacket, ∆y∆p = 1

2 . However, through its unitary evolution, the function Fk grows exponentially, see(540), and we quickly find ∆y∆p 1

2 , corresponding to the semiclassical regime, as we will soondemonstrate rigorously.

9.10 The squeezing formalismLet us now use the squeezing formalism to describe the evolution of the wave function. The equationsof motion for the squeezing parameters follow from those of the field and momentum modes,

r′k =a′

acos 2φk , (545)

φ′k = −k − a′

acoth 2rk sin 2φk , (546)

θ′k = k +a′

atanh 2rk sin 2φk . (547)

As we will see, the evolution is driven towards large rk ∝ N 1, the number of e-folds during inflation.Thus, in that limit,

(θk + φk)′ = −a′

a

sin 2φksinh 2rk

→ 0 ,

and therefore θk + φk → const. We can always choose this constant to be zero, so that the real andimaginary components of the field and momentum modes become

fk1 =1√2k

erk cosφk , fk2 =1√2k

e−rk sinφk , (548)

gk1 =

√k

2e−rk cosφk , gk2 =

√k

2erk sinφk . (549)

It is clear that, in the limit of large squeezing (rk → ∞), the field mode fk becomes purely real, whilethe momentum mode gk becomes pure imaginary.

This means that the field (535) and momentum (536) operators become, in that limit,

y(k, η) →√

2k fk1(η) y(k, η0)

p(k, η) →√

2k gk2(η) y(k, η0)

⇒ p(k, η) ' gk2(η)

fk1(η)y(k, η) . (550)

As a consequence of this squeezing, information about the initial momentum p0 distribution islost, and the positions (or field amplitudes) at different times commute,[

y(k, η1), y(k, η2)]' 1

2e−2rk cos2 φk ≈ 0 . (551)

This result defines what is known as a quantum non-demolition (QND) variable, which means that onecan perform succesive measurements of this variable with arbitrary precision without modifying the wavefunction. Note that y = aδφ is the amplitude of fluctuations produced during inflation, so what we havefound is: first, that the amplitude is distributed as a classical Gaussian random field with probability(541); and second that we can measure its amplitude at any time, and as much as we like, withoutmodifying the distribution function.

In a sense, this problem is similar to that of a free non-relativistic quantum particle, describedinitially by a minimum wave packet, with initial expectation values 〈x〉0 = x0 and 〈p〉0 = p0, whichbecomes broader by its unitary evolution, and at late times (t mx0/p0) this Gaussian state becomesan exact WKB state,

Ψ(x) = Ω−1/2R exp(−Ωx2/2),

with ImΩ ReΩ (i.e. high squeezing limit). In that limit, [x, p] ≈ 0, and we have lost informationabout the initial position x0 (instead of the initial momentum like in the inflationary case), x(t) 'p(t) t/m =p0 t/m and p(t) = p0. Therefore, not only [p(t1), p(t2)] = 0, but also, at late times, [x(t1), x(t2)] ≈ 0.

9.11 The Wigner functionThe Wigner function is the best candidate for a probability density of a quantum mechanical system inphase-space. Of course, we know from QM that such a probability distribution function cannot exist, butthe Wigner function is just a good approximation to that distribution. Furthermore, for a Gaussian state,this function is in fact positive definite.

Consider a quantum state described by a density matrix ρ. Then the Wigner function can be writtenas

W (y0k, y

0k∗, p0k, p

0k∗) =

∫ ∫dx1 dx2

(2π)2e−i(p1 x1+p2 x2)

⟨y − x

2, η∣∣∣ ρ ∣∣∣y +

x

2, η⟩.

If we substitute for the state our vacuum initial condition ρ = |Ψ0〉〈Ψ0|, with Ψ0 given by (538), we canperform the integration explicitly to obtain

W0(y0k, y

0k∗, p0k, p

0k∗) =

1π2

exp

(− |y|

2

|fk|2− 4|fk|2

∣∣∣∣p− Fk|fk|2

y

∣∣∣∣2)

≡ Φ(y1, p1) Φ(y2, p2) (552)

Φ(y1, p1) =1π

exp−(

y21

|fk|2+ 4|fk|2 p2

1

), (553)

p1 ≡ p1 −Fk|fk|2

y1 .

In general, W0 describes an asymmetric Gaussian in phase space, whose 2σ contours satisfy

y21

|fk|2+ 4|fk|2 p2

1 ≤ 1 . (554)

For instance, at time η = η0, we have y01 = 1√

2k= |fk(η0)| , p0

1 =√

k2 = 1/2|fk(η0)|, and Fk(η0) = 0,

so that p01 = p0

1, and the 2σ contours become

y21

y01

2 +p2

1

p01

2 ≤ 1 ,

which is a circle in phase space.

On the other hand, for time η η0, we have

|fk| →1√2k

erk ∼ y0k e

N , growing mode , (555)

12|fk|

→√k

2e−rk ∼ p0

k e−N , decaying mode , (556)

so that the ellipse (554) becomes highly “squeezed”.

Note that Liouville’s theorem implies that the volume of phase space is conserved under Hamil-tonian (unitary) evolution, so that the area within the ellipse should be conserved. As the probabilitydistribution compresses (squeezes) along the p-direction, it expands along the y-direction. At late times,the Wigner function is highly concentrated around the region

p2 =(p− Fk|fk|2

y

)2

<1

4|fk|2∼ e−2N 1 . (557)

We can thus take the above squeezing limit in the Wigner function (552) and write the exponential termas a Dirac delta function,

W0(y, p) rk→∞−→ 1π2

exp− |y|

2

|fk|2

δ

(p− Fk|fk|2

y

). (558)

In this limit we have

pk(η) =Fk|fk|2

yk(η) ' gk2(η)fk1(η)

yk(η) , (559)

so we recover the previous result (550). This explains why we can treat the system as a classical Gaussianrandom field: the amplitude of the field y is uncertain with probability distribution (541), but once ameasurement of y is performed, we can automatically asign to it a definite value of the momentum,according to (550).

Note that the condition F 2k 1 is actually a condition between operators and their commuta-

tors/anticommutators. The Heisenberg uncertainty principle states that

∆ΨA∆ΨB ≥12

∣∣∣〈Ψ|[A, B]|Ψ〉∣∣∣,

for any two hermitian operators (observables) in the Hilbert space of the wave function Ψ. In our case,and in Fourier space, this corresponds to (544)

∆y2(k) ∆p2(k) = F 2k (η) +

14≥ 1

4

∣∣∣〈Ψ|[yk(η), p†k(η)]|Ψ〉∣∣∣2 , (560)

with |Ψ〉 = |0, η〉 the evolved wave function.

On the other hand, the phase Fk can be written as

Fk = − i

2(gk f∗k − fk g∗k) = − i

2

(gkfk|fk|2 − |fk|2

g∗kf∗k

)=

=12〈Ψ|p(k, η) y†(k, η) + y(k, η) p†(k, η)|Ψ〉 , (561)

and we have used that, in the semiclassical limit, we can write〈Ψ||yk(η)|2|Ψ〉 = |fk|2, as well as p(k, η) = − i gkfk y(k, η), see (550).

The above relation just indicates that, for any state Ψ, the condition of classicality (Fk 1) issatisfied whenever, for that state,

yk(η), p†k(η) |[yk(η), p†k(η)]| = ~ ,

which is an interesting condition.

9.12 Massless scalar field fluctuations on superhorizon scalesThe gauge invariant tensor fluctuations (gravitational waves) act as a minimally-coupled massless scalarfield during inflation, so we will study here the generation of its fluctuations during quasi de Sitter.

Let us consider here the exact solutions to the equation of motion of a minimally-coupled masslessscalar field during inflation or quasi de Sitter, with scale factor a = −1/Hη,

fk =1√2k

e−ikη(

1− i

kη

), (562)

gk = i

(f ′k −

a′

afk

)=

√k

2e−ikη , (563)

which satisfy the Wronskian condition, gk f∗k + g∗k fk = 1. The eigenmodes become

uk = e−ikη(

1− i

2kη

)= e−ikη−iδk cosh rk , (564)

vk = eikηi

2kη= eikη+iπ

2 sinh rk , (565)

which comparing with (527) and (528) provides the squeezing parameter, the angle and the phase, asinflation proceeds towards kη → 0−,

sinh rk = tan δk =1

2kη→ −∞ , (566)

θk = kη + arctan1

2kη→ −π

2, φk =

π

4− 1

2arctan

12kη→ π

2, (567)

while the imaginary part of the phase of the wave function becomes

Fk(η) =12

sin 2φk sinh 2rk =1

2kη→ −∞ . (568)

The number of scalar field particles produced during inflation grow exponentially, nk = |βk|2 =sinh2 rk = (2kη)−2 →∞.

Thus, through unitary evolution, the fluctuations will very soon enter the semiclasical regimedue to a highly squeezed wave function. The question which remains is when do fluctuations becomeclassical?

9.13 Hubble crossingAs we will see, the field fluctuation modes will become semiclassical as their wavelength becomes largerthan the only physical scale in the problem, the de Sitter horizon scale, λphys = 2πa/k H−1.

Therefore, let us consider the general solution to Eq. (524) for the superhorizon modes (k aH),

fk(η) = C1(k) a+ C2(k) a∫ η dη′

a2(η′)= C1(k) a− C2(k)

1a2H

. (569)

We can always choose C1(k) to be real, while C2(k) will be complex in general. The first term corre-sponds to the growing mode, while the second term is the decaying mode.

Integrating out gk from (525), one finds

gk(η) = i C2(k)1a− i C1(k) k2 1

a

∫a2 dη = i C2(k)

1a− i C1(k)

k2

H, (570)

where we have added a k2 term for completeness. To second order in k2, the Wronskian becomes

C1(k) ImC2(k)(

1 +k2

a2H2

)' C1(k) ImC2(k) = −1

2. (571)

Comparing with the exact solutions (562), we find, to first order,

C1(k) =Hk√2k3

, C2(k) = − i k3/2

√2Hk

, (572)

where Hk is the Hubble rate at horizon crossing, kη = −1, i.e. when the perturbation’s physical wave-length becomes of the same order as the de Sitter horizon size, k = aH = H.

We are now prepared to answer the question of classicality of the modes. Let us compute the wavefunction phase shift

|Fk| = |Im(f∗k gk)| =∣∣∣∣C2

1 (k)k2a

H+ |C2(k)|2 1

a3H(573)

− C1(k) ReC2(k)(

1 +k2

a2H2

)∣∣∣∣ . (574)

Since only the first term remains after kη → 0, we see that |Fk| 1 whenever

C21 (k) =

H2k

2k3 H

k2a⇒ λphys =

2πak λHC =

2πHk

. (575)

Therefore, we confirm that modes that start as Minkowski vacuum well inside the de Sitter horizon arestretched by the expansion and become semiclassical soon after horizon crossing, and their amplitudecan be described as a classical Gaussian random variable.

Furthermore, the fact that the momentum is immediately defined once the amplitude for a givenwavelength is known, implies that there is a fixed temporal phase coherence for all perturbations with thesame wavelength. As we know, this implies that inflationary perturbations will induce coherent acousticoscillations in the plasma just before decoupling, which should be seen in the microwave backgroundanisotropies as acoustic peaks in the angular power spectrum.

10 Anisotropies of the microwave background

The Universe just before recombination is a very tightly coupled fluid, due to the large electromagneticThomson cross section. Photons scatter off charged particles (protons and electrons), and carry energy,so they feel the gravitational potential associated with the perturbations imprinted in the metric duringinflation. An overdensity of baryons (protons and neutrons) does not collapse under the effect of gravityuntil it enters the causal Hubble radius. The perturbation continues to grow until radiation pressureopposes gravity and sets up acoustic oscillations in the plasma. Since overdensities of the same size willenter the Hubble radius at the same time, they will oscillate in phase. Moreover, since photons scatteroff these baryons, the acoustic oscillations occur also in the photon field and induces a pattern of peaksin the temperature anisotropies in the sky, at different angular scales.

Three different effects determine the temperature anisotropies we observe in the microwave back-ground:

Gravity: photons fall in and escape off gravitational potential wells, characterized by Φ in thecomoving gauge, and as a consequence their frequency is gravitationally blue- or red-shifted, δν/ν = Φ.If the gravitational potential is not constant, the photons will escape from a larger or smaller potentialwell than they fell in, so their frequency is also blue- or red-shifted, a phenomenon known as the Rees-Sciama effect.

Pressure: photons scatter off baryons which fall into gravitational potential wells, and radiationpressure creates a restoring force inducing acoustic waves of compression and rarefaction.

Velocity: baryons accelerate as they fall into potential wells. They have minimum velocity atmaximum compression and rarefaction. That is, their velocity wave is exactly 90 off-phase with theacoustic compression waves. These waves induce a Doppler effect on the frequency of the photons.

The temperature anisotropy induced by these three effects is therefore given by

δT

T(r) = Φ(r, tdec) + 2

∫ t0

tdec

Φ(r, t)dt +13δρ

ρ(r, tdec) −

r · vc

. (576)

Metric perturbations of different wavelengths enter the horizon at different times. The largest wave-lengths, of size comparable to our present horizon, are entering now. There are perturbations with wave-lengths comparable to the size of the horizon at the time of last scattering, of projected size about 1

in the sky today, which entered precisely at decoupling. And there are perturbations with wavelengthsmuch smaller than the size of the horizon at last scattering, that entered much earlier than decoupling,during the radiation era, which have gone through several acoustic oscillations before last scattering. Allthese perturbations of different wavelengths leave their imprint in the CMB anisotropies.

The baryons at the time of decoupling do not feel the gravitational attraction of perturbations withwavelength greater than the size of the horizon at last scattering, because of causality. Perturbations withexactly that wavelength are undergoing their first contraction, or acoustic compression, at decoupling.Those perturbations induce a large peak in the temperature anisotropies power spectrum. Perturbationswith wavelengths smaller than these will have gone, after they entered the Hubble scale, through a seriesof acoustic compressions and rarefactions, which can be seen as secondary peaks in the power spectrum.Since the surface of last scattering is not a sharp discontinuity, but a region of ∆z ∼ 100, there will bescales for which photons, travelling from one energy concentration to another, will erase the perturbationon that scale, similarly to what neutrinos or HDM do for structure on small scales. That is the reason whywe don’t see all the acoustic oscillations with the same amplitude, but in fact they decay exponentialytowards smaller angular scales, an effect known as Silk damping, due to photon diffusion.

10.1 The Sachs-Wolfe effectThe anisotropies corresponding to large angular scales are only generated via gravitational red-shift anddensity perturbations through the Einstein equations, δρ/ρ = −2Φ (for adiabatic perturbations); wecan ignore the Doppler contribution, since the perturbation is non-causal. In that case, the temperatureanisotropy in the sky today is given by

δT

T(θ, φ) =

13

Φ(ηLS)Q(η0, θ, φ) + 2∫ η0

ηLS

drΦ′(η0 − r)Q(r, θ, φ) , (577)

where η0 is the coordinate distance to the surface of last scattering, i.e. the present conformal time, whileηLS ' 0 determines its comoving hypersurface. The Sachs-Wolfe effect (577) contains two parts, theintrinsic and the Integrated Sachs-Wolfe (ISW) effect, due to the integration along the line of sight oftime variations in the gravitational potential.

In linear perturbation theory, the scalar metric perturbations can be separated into Φ(η,x) ≡Φ(η)Q(x), where Q(x) are the scalar harmonics, eigenfunctions of the Laplacian in three dimensions,

∇2Qklm(r, θ, φ) = −k2Qklm(r, θ, φ).

These functions have the general form

Qklm(r, θ, φ) = Πkl(r)Ylm(θ, φ) , (578)

where Ylm(θ, φ) are the usual spherical harmonics, and the radial parts can be written (in a flat Universe)

in terms of spherical Bessel functions, Πkl(r) =√

2π k jl(kr). On the other hand, the time evolution of

the metric perturbation during the matter era is given by

Φ′′ + 3HΦ′ + a2Λ Φ− 2K Φ = 0 . (579)

In the case of a flat universe (K = 0) without cosmological constant, the Newtonian potential Φ remainsconstant during the matter era and only the intrinsic SW effect contributes to δT/T . In case of a non-vanishing Λ, since its contribution is negligible in the past, most of the photon’s trajectory towards us isunperturbed. We will consider here the approximation Φ ' constant during the matter era.

The growing mode solution of the metric perturbation that left the Hubble scale during inflationcontributes to the temperature anisotropies on large scales as

δT

T(θ, φ) =

13

Φ(ηLS)Q =15RQ(η0, θ, φ) ≡

∞∑l=2

l∑m=−l

alm Ylm(θ, φ) , (580)

where we have used the fact that, at horizon reentry during the matter era, the gauge-invariant Newtonianpotential Φ = 3

5R is related to the curvature perturbationR at Hubble-crossing during inflation.

We can now compute the two-point correlation function or angular power spectrum, C(θ), of theCMB anisotropies on large scales, defined as an expansion in multipole number,

C(θ) =⟨δT

T

∗(n)

δT

T(n′)

⟩n·n′=cos θ

=1

4π

∞∑l=2

(2l + 1)Cl Pl(cos θ) , (581)

where Pl(z) are the Legendre polynomials, and we have averaged over different universe realizations.Since the coefficients alm are isotropic (to first order), we can compute the Cl = 〈|alm|2〉 as

C(S)l =

4π25

∫ ∞0

dk

kPR(k) j2

l (kη0) . (582)

In the case of scalar metric perturbation produced during inflation, the scalar power spectrum at reentryis given by PR(k) = A2

S(kη0)n−1, in the power-law approximation. In that case, one can integrate (582)to give

C(S)l =

2π25

A2S

Γ[32 ] Γ[1− n−1

2 ] Γ[l + n−12 ]

Γ[32 −

n−12 ] Γ[l + 2− n−1

2 ], (583)

l(l + 1)C(S)l

2π=A2S

25= constant , for n = 1 . (584)

This last expression corresponds to what is known as the Sachs-Wolfe plateau, and is the reason why thecoefficients Cl are always plotted multiplied by l(l + 1).

10.2 The tensor perturbations Sachs-Wolfe effectTensor metric perturbations also contribute with an approximately constant angular power spectrum,l(l + 1)Cl. The Sachs-Wolfe effect for a gauge-invariant tensor perturbation is given by

δT

T(θ, φ) =

∫ η0

ηLS

dr h′(η0 − r)Qrr(r, θ, φ) , (585)

where Qrr is the rr-component of the tensor harmonic along the line of sight. The tensor perturbationhk(η) during the matter era satisfies

h′′k + 2H h′k + (k2 + 2K)hk = 0 , (586)

which depends on the wavenumber k, contrary to what happens with the scalar modes, see Eq. (579). Fora flat (K = 0) universe, the solution to this equation is hk(η) = hGk(η), where h is the constant tensormetric perturbation at horizon crossing and Gk(η) = 3 j1(kη)/kη, normalized so that Gk(0) = 1 at thesurface of last scattering. The radial part of the tensor harmonic Qrr in a flat universe can be written as

Qrrkl (r) =[

(l − 1)l(l + 1)(l + 2)πk2

]1/2 jl(kr)r2

. (587)

The tensor angular power spectrum can finally be expressed as

C(T )l =

9π4

(l − 1)l(l + 1)(l + 2)∫ ∞

0

dk

kPg(k) I2

kl , (588)

Ikl =∫ x0

0dx

j2(x0 − x)jl(x)(x0 − x)x2

, (589)

where x ≡ kη, and Pg(k) is the primordial tensor spectrum. For a scale invariant spectrum, nT = 0, wecan integrate (588) to give

l(l + 1)C(T )l =

π

36

(1 +

48π2

385

)A2T Bl , (590)

with Bl = (1.1184, 0.8789, . . . , 1.00) for l = 2, 3, . . . , 30. Therefore, l(l + 1)C(T )l also becomes

constant for large l. Beyond l ∼ 30, the Sachs-Wolfe expression is not a good approximation and thetensor angular power spectrum decays very quickly at large l.

10.3 The consistency conditionIn spite of the success of inflation in predicting a homogeneous and isotropic background on which toimprint a scale-invariant spectrum of inhomogeneities, it is difficult to test the idea of inflation. Beforethe 1980s anyone would have argued that ad hoc initial conditions could have been at the origin of thehomogeneity and flatness of the universe on large scales, while most cosmologists would have agreedwith Harrison and Zel’dovich that the most natural spectrum needed to explain the formation of structurewas a scale-invariant spectrum. The surprise was that inflation incorporated an understanding of both theglobally homogeneous and spatially flat background, and the approximately scale-invariant spectrum ofperturbations in the same formalism. But that could have been a coincidence.

What is unique to inflation is the fact that inflation determines not just one but two primordialspectra, corresponding to the scalar (density) and tensor (gravitational waves) metric perturbations, froma single continuous function, the inflaton potential V (φ). In the slow-roll approximation, one determines,from V (φ), two continuous functions, PR(k) and Pg(k), that in the power-law approximation reducesto two amplitudes, AS and AT , and two tilts, n and nT . It is clear that there must be a relation between

the four parameters. Indeed, one can see from Eqs. (590) and (584) that the ratio of the tensor to scalarcontribution to the angular power spectrum is proportional to the tensor tilt,

R ≡C

(T )l

C(S)l

=259

(1 +

48π2

385

)2ε ' −2π nT . (591)

This is a unique prediction of inflation, which could not have been postulated a priori. If we finallyobserve a tensor spectrum of anisotropies in the CMB, or a stochastic gravitational wave background inlaser interferometers like LIGO or VIRGO, with sufficient accuracy to determine their spectral tilt, onemight have some chance to test the idea of inflation, via the consistency relation (591).

For the moment, observations of the microwave background anisotropies suggest that the Sachs-Wolfe plateau exists, but it is still premature to determine the tensor contribution. Perhaps in the nearfuture, from the analysis of polarization as well as temperature anisotropies, with the CMB satellitesMAP and Planck, we might have a chance of determining the validity of the consistency relation.

Assuming that the scalar contribution dominates over the tensor on large scales, i.e. R 1, onecan actually give a measure of the amplitude of the scalar metric perturbation from the observations ofthe Sachs-Wolfe plateau in the angular power spectrum,[

l(l + 1)C(S)l

2π

]1/2

=AS5

= (1.03± 0.07)× 10−5 , (592)

n = 0.96± 0.03 . (593)

These measurements can be used to normalize the primordial spectrum and determine the parameters ofthe model of inflation. In the near future these parameters will be determined with much better accuracy.

10.4 The acoustic peaksBefore decoupling, the photons and the baryons are tightly coupled via Thomson scattering. The dynamisof the photon-baryon fluid is described by a forced and damped harmonic oscillator equation for thebaryon density contrast,

δ′′k +H R

1 +Rδ′k + k2 c2

s δk = F (Φk) , (594)

where R = 3ρB/4ργ is the baryon-to-photon ratio, c2s = c2/3(1 +R) is the sound speed of the plasma,

and F (Φk) is the external force due to the gravitational effect of dark matter and neutrinos. Baryonstend to collapse due to self-gravitation, while radiation pressure provides the restoring force, settingup acoustic oscillations in the plasma. Because of tight coupling, δk = 3Θ0(k, η), and the baryonoscillations give rise to oscillations in the temperature fluctuations Θ0. The higher the baryon fractionR, the higher the amplitude of the oscillations. The external gravitational force displaces the zero-pointof oscillations, which makes higher the amplitude of compressions versus rarefactions.

At decoupling there is a freeze out of the oscillations. The microwave background is like a snap-shot of the instant of last scattering, where each mode k is at a different stage of oscillation,

Θ0(k, η) ∝ (1 +R)−1/4

ζk(0) cos krs adiabatic ,Sk(0) k sin krs isocurvature ,

(595)

where rs =∫ ηdec

0 csdη ' cs ηdec is the sound horizon at decoupling. These fluctuations induce acous-tic peaks in the Angular Power Spectrum that correspond to maxima and minima of oscillations. Foradiabatic and isocurvature perturbations, the harmonic peaks appear at wavenumber k(A)

n = nπ/csηdec

and k(I)n = (n + 1/2)π/csηdec, respectively. In particular, the angle subtended by the sound horizon at

decoupling, θs = rs/dA, corresponds to a multipole number (e.g. for adiabatic perturbations)

ln ≈nπ

θs=nπ

2cs

((1 + zdec)ΩM

|ΩK|

)1/2

sinn∫ zdec

0

|ΩK|1/2dz[ΩΛ + ΩM(1 + z)3 + ΩK(1 + z)2

]1/2

10.4.1 The new microwave anisotropy satellites, WMAP and Planck

The large amount of information encoded in the anisotropies of the microwave background is the reasonwhy both NASA and the European Space Agency have decided to launch two independent satellites tomeasure the CMB temperature and polarization anisotropies to unprecendented accuracy. The WilkinsonMicrowave Anisotropy Probe [91] was launched by NASA at the end of 2000, and has fulfilled most ofour expectation, while Planck [92] is expected to be lanched by ESA in 2007. There are at the momentother large proposals like CMB Pol [98], ACT [99], etc. which will see the light in the next few years,see Ref. [93].

As we have emphasized before, the fact that these anisotropies have such a small amplitude allowfor an accurate calculation of the predicted anisotropies in linear perturbation theory. A particular cosmo-logical model is characterized by a dozen or so parameters: the rate of expansion, the spatial curvature,the baryon content, the cold dark matter and neutrino contribution, the cosmological constant (vacuumenergy), the reionization parameter (optical depth to the last scattering surface), and various primordialspectrum parameters like the amplitude and tilt of the adiabatic and isocurvature spectra, the amount ofgravitational waves, non-Gaussian effects, etc. All these parameters can now be fed into very fast CMBcodes called CMBFAST [96] and CAMB [97], that compute the predicted temperature and polarizationanisotropies to better than 1% accuracy, and thus can be used to compare with observations.

These two satellites will improve both the sensitivity, down to µK, and the resolution, down to arcminutes, with respect to the previous COBE satellite, thanks to large numbers of microwave horns of var-ious sizes, positioned at specific angles, and also thanks to recent advances in detector technology, withhigh electron mobility transistor amplifiers (HEMTs) for frequencies below 100 GHz and bolometersfor higher frequencies. The primary advantage of HEMTs is their ease of use and speed, with a typicalsensitivity of 0.5 mKs1/2, while the advantage of bolometers is their tremendous sensitivity, better than0.1 mKs1/2, see Ref. [100]. This will allow cosmologists to extract information from around 3000 mul-tipoles! Since most of the cosmological parameters have specific signatures in the height and position ofthe first few acoustic peaks, the higher the resolution, the more peaks one is expected to see, and thus thebetter the accuracy with which one will be able to measure those parameters, see Table 2.

Although the satellite probes were designed for the accurate measurement of the CMB temperatureanisotropies, there are other experiments, like balloon-borne and ground interferometers [93]. Probablythe most important objective of the future satellites (beyond WMAP) will be the measurement of theCMB polarization anisotropies, discovered by DASI in November 2002 [101], and confirmed a fewmonths later by WMAP with greater accuracy [20], see Fig. 29. These anisotropies were predicted bymodels of structure formation and indeed found at the level of microKelvin sensitivities, where the newsatellites were aiming at. The complementary information contained in the polarization anisotropiesalready provides much more stringent constraints on the cosmological parameters than from the temper-ature anisotropies alone. However, in the future, Planck and CMB pol will have much better sensitivities.In particular, the curl-curl component of the polarization power spectra is nowadays the only means wehave to determine the tensor (gravitational wave) contribution to the metric perturbations responsible fortemperature anisotropies, see Fig. 34. If such a component is found, one could constraint very preciselythe model of inflation from its spectral properties, specially the tilt [94].

Fig. 34: Theoretical predictions for the four non-zero CMB temperature-polarization spectra as a function of multipole moment,

together with the expectations from Planck. From Ref. [95].

10.5 From metric perturbations to large scale structureIf inflation is responsible for the metric perturbations that gave rise to the temperature anisotropies ob-served in the microwave background, then the primordial spectrum of density inhomogeneities inducedby the same metric perturbations should also be responsible for the present large scale structure [102].This simple connection allows for more stringent tests on the inflationary paradigm for the generationof metric perturbations, since it relates the large scales (of order the present horizon) with the smallestscales (on galaxy scales). This provides a very large lever arm for the determination of primordial spectraparameters like the tilt, the nature of the perturbations, whether adiabatic or isocurvature, the geometryof the universe, as well as its matter and energy content, whether CDM, HDM or mixed CHDM.

10.5.1 The galaxy power spectrum

As metric perturbations enter the causal horizon during the radiation or matter era, they create densityfluctuations via gravitational attraction of the potential wells. The density contrast δ can be deduced fromthe Einstein equations in linear perturbation theory, see Eq. (397),

δk ≡δρkρ

=(k

aH

)2 23

Φk =(k

aH

)2 2 + 2ω5 + 3ω

Rk , (596)

where we have assumed K = 0, and used Eq. (483). From this expression one can compute the powerspectrum, at horizon crossing, of matter density perturbations induced by inflation, see Eq. (495),

P (k) = 〈|δk|2〉 = A

(k

aH

)n, (597)

with n given by the scalar tilt (497), n = 1 + 2η − 6ε. This spectrum reduces to a Harrison-Zel’dovichspectrum (168) in the slow-roll approximation: η, ε 1.

Since perturbations evolve after entering the horizon, the power spectrum will not remain con-stant. For scales entering the horizon well after matter domination (k−1 k−1

eq ' 81 Mpc), the metricperturbation has not changed significantly, so that Rk(final) = Rk(initial). Then Eq. (596) determinesthe final density contrast in terms of the initial one. On smaller scales, there is a linear transfer functionT (k), which may be defined as [79]

Rk(final) = T (k)Rk(initial) . (598)

To calculate the transfer function one has to specify the initial condition with the relative abundanceof photons, neutrinos, baryons and cold dark matter long before horizon crossing. The most naturalcondition is that the abundances of all particle species are uniform on comoving hypersurfaces (withconstant total energy density). This is called the adiabatic condition, because entropy is conserved inde-pendently for each particle species X , i.e. δρX = ρXδt, given a perturbation in time from a comovinghypersurface, so

δρXρX + pX

=δρY

ρY + pY, (599)

where we have used the energy conservation equation for each species, ρX = −3H(ρX + pX), valid tofirst order in perturbations. It follows that each species of radiation has a common density contrast δr,and each species of matter has also a common density contrast δm, with the relation δm = 3

4δr.

Given the adiabatic condition, the transfer function is determined by the physical processes oc-curing between horizon entry and matter domination. If the radiation behaves like a perfect fluid, itsdensity perturbation oscillates during this era, with decreasing amplitude. The matter density contrastliving in this background does not grow appreciably before matter domination because it has negligibleself-gravity. The transfer function is therefore given roughly by, see Eq. (171),

T (k) =

1 , k keq

(k/keq)2 , k keq

(600)

The perfect fluid description of the radiation is far from being correct after horizon entry, becauseroughly half of the radiation consists of neutrinos whose perturbation rapidly disappears through freestreeming. The photons are also not a perfect fluid because they diffuse significantly, for scales belowthe Silk scale, k−1

S ∼ 1 Mpc. One might then consider the opposite assumption, that the radiation haszero perturbation after horizon entry. Then the matter density perturbation evolves according to

δk + 2Hδk + (c2s k

2ph − 4πGρ) δk = 0 , (601)

which corresponds to the equation of a damped harmonic oscillator. The zero-frequency oscillator definesthe Jeans wavenumber, kJ =

√4πGρ/c2

s. For k kJ , δk grows exponentially on the dynamicaltimescale, τdyn = Imω−1 = (4πGρ)−1/2 = τgrav, which is the time scale for gravitational collapse.One can also define the Jeans length,

λJ =2πkJ

= cs

√π

Gρ, (602)

which separates gravitationally stable from unstable modes. If we define the pressure response timescaleas the size of the perturbation over the sound speed, τpres ∼ λ/cs, then, if τpres > τgrav, gravitationalcollapse of a perturbation can occur before pressure forces can response to restore hydrostatic equilibrium(this occurs for λ > λJ ). On the other hand, if τpres < τgrav, radiation pressure prevents gravitationalcollapse and there are damped acoustic oscillations (for λ < λJ ).

We will consider now the behaviour of modes within the horizon during the transition from theradiation (c2

s = 1/3) to the matter era (c2s = 0). The growing mode solution increases only by a factor of

2 between horizon entry and the epoch when matter starts to dominate, i.e. y = 1. The transfer function istherefore again roughly given by Eq. (600). Since the radiation consists roughly half of neutrinos, whichfree streem, and half of photons, which either form a perfect fluid or just diffuse, neither the perfectfluid nor the free-streeming approximation looks very sensible. A more precise calculation is needed,including: neutrino free streeming around the epoch of horizon entry; the diffusion of photons aroundthe same time, for scales below Silk scale; the diffusion of baryons along with the photons, and theestablishment after matter domination of a common matter density contrast, as the baryons fall into thepotential wells of cold dark matter. All these effects apply separately, to first order in the perturbations, to

each Fourier component, so that a linear transfer function is produced. There are several parametrizationsin the literature, but the one which is more widely used is that of Ref. [103],

T (k) =[1 +

(ak + (bk)3/2 + (ck)2

)ν]−1/ν, ν = 1.13 , (603)

a = 6.4 (ΩMh)−1 h−1 Mpc , (604)

b = 3.0 (ΩMh)−1 h−1 Mpc , (605)

c = 1.7 (ΩMh)−1 h−1 Mpc . (606)

We see that the behaviour estimated in Eq. (600) is roughly correct, although the break at k = keq is notat all sharp, see Fig. 35. The transfer function, which encodes the soltion to linear equations, ceases tobe valid when the density contrast becomes of order 1. After that, the highly nonlinear phenomenon ofgravitational collapse takes place, see Fig. 35.

Fig. 35: The CDM power spectrum P (k) as a function of wavenumber k, in logarithmic scale, normalized to the local abun-

dance of galaxy clusters, for an Einstein-de Sitter universe with h = 0.5. The solid (dashed) curve shows the linear (non-linear)

power spectrum. While the linear power spectrum falls off like k−3, the non-linear power-spectrum illustrates the increased

power on small scales due to non-linear effects, at the expense of the large-scale structures. From Ref. [44].

10.5.2 The new redshift catalogs, 2dF and Sloan Digital Sky Survey

Our view of the large-scale distribution of luminous objects in the universe has changed dramaticallyduring the last 25 years: from the simple pre-1975 picture of a distribution of field and cluster galaxies,to the discovery of the first single superstructures and voids, to the most recent results showing an almostregular web-like network of interconnected clusters, filaments and walls, separating huge nearly emptyvolumes. The increased efficiency of redshift surveys, made possible by the development of spectro-graphs and – specially in the last decade – by an enormous increase in multiplexing gain (i.e. the abilityto collect spectra of several galaxies at once, thanks to fibre-optic spectrographs), has allowed us notonly to do cartography of the nearby universe, but also to statistically characterize some of its properties,see Ref. [104]. At the same time, advances in theoretical modeling of the development of structure, withlarge high-resolution gravitational simulations coupled to a deeper yet limited understanding of how toform galaxies within the dark matter halos, have provided a more realistic connection of the models to theobservable quantities [105]. Despite the large uncertainties that still exist, this has transformed the studyof cosmology and large-scale structure into a truly quantitative science, where theory and observationscan progress side by side.

I will concentrate on two of the new catalogs, which are taking data at the moment and whichhave changed the field, the 2-degree-Field (2dF) Catalog and the Sloan Digital Sky Survey (SDSS). Theadvantages of multi-object fibre spectroscopy have been pushed to the extreme with the construction ofthe 2dF spectrograph for the prime focus of the Anglo-Australian Telescope [45]. This instrument isable to accommodate 400 automatically positioned fibres over a 2 degree in diameter field. This impliesa density of fibres on the sky of approximately 130 deg−2, and an optimal match to the galaxy countsfor a magnitude bJ ' 19.5, similar to that of previous surveys like the ESP, with the difference thatwith such an area yield, the same number of redshifts as in the ESP survey can be collected in about 10exposures, or slightly more than one night of telescope time with typical 1 hour exposures. This is thebasis of the 2dF galaxy redshift survey. Its goal is to measure redshifts for more than 250,000 galaxieswith bJ < 19.5. In addition, a faint redshift survey of 10,000 galaxies brighter than R = 21 will be doneover selected fields within the two main strips of the South and North Galactic Caps. The survey hasnow finished, with a quarter of a million redshifts. The final result can be seen in Ref. [45].

The most ambitious and comprehensive galaxy survey currently in progress is without any doubtthe Sloan Digital Sky Survey [46]. The aim of the project is, first of all, to observe photometrically thewhole Northern Galactic Cap, 30 away from the galactic plane (about 104 deg2) in five bands, at limitingmagnitudes from 20.8 to 23.3. The expectation is to detect around 50 million galaxies and around 108

star-like sources. This has already led to the discovery of several high-redshift (z > 4) quasars, includingthe highest-redshift quasar known, at z = 5.0, see Ref. [46]. Using two fibre spectrographs carrying 320fibres each, the spectroscopic part of the survey will then collect spectra from about 106 galaxies withr′ < 18 and 105 AGNs with r′ < 19. It will also select a sample of about 105 red luminous galaxies withr′ < 19.5, which will be observed spectroscopically, providing a nearly volume-limited sample of early-type galaxies with a median redshift of z ' 0.5, that will be extremely valuable to study the evolution ofclustering. The data that is coming from these catalogs is so outstanding that already cosmologists areusing them for the determination of the cosmological parameters of the standard model of cosmology.The main outcome of these catalogs is the linear power spectrum of matter fluctuations that give rise togalaxies, and clusters of galaxies. It covers from the large scales of order Gigaparsecs, the realm of theunvirialised superclusters, to the small scales of hundreds of kiloparsecs, where the Lyman-α systemscan help reconstruct the linear power spectrum, since they are less sensitive to the nonlinear growth ofperturbations.

As often happens in particle physics, not always are observations from a single experiment suffi-cient to isolate and determine the precise value of the parameters of the standard model. We mentionedin the previous Section that some of the cosmological parameters created similar effects in the tem-perature anisotropies of the microwave background. We say that these parameters are degenerate withrespect to the observations. However, often one finds combinations of various experiments/observationswhich break the degeneracy, for example by depending on a different combination of parameters. This isprecisely the case with the cosmological parameters, as measured by a combination of large-scale struc-ture observations, microwave background anisotropies, Supernovae Ia observations and Hubble SpaceTelescope measurements. It is expected that in the near future we will be able to determine the param-eters of the standard cosmological model with great precision from a combination of several differentexperiments.

OBSERVATIONAL SIGNATURES OF INFLATION

11 Inflationary model building. The particle physics connection

For the moment, observations of the microwave background anisotropies suggest that the Sachs-Wolfeplateau exists, but it is still premature to determine the tensor contribution. Perhaps in the near future,from the analysis of polarization as well as temperature anisotropies, with the CMB satellites MAP andPlanck, we might have a chance of determining the validity of the consistency relation. Assuming thatthe scalar contribution dominates over the tensor on large scales, i.e. r 1, one can actually give a

measure of the amplitude of the scalar metric perturbation from the observations of the Sachs-Wolfeplateau in the angular power spectrum,[

l(l + 1)C(S)l

2π

]1/2

=AS5

= (1.03± 0.07)× 10−5 , (607)

n = 0.98± 0.03 . (608)

These measurements can be used to normalize the primordial spectrum and determine the parameters ofa particular model of inflation. In the near future these parameters will be determined with much betteraccuracy, to less than a percent.

In the next sections we will consider specific models of inflation. The formulae we will be usingare

ε =1

2κ2

(V ′

V

)2

η =1κ2

(V ′′

V

)ξ =

1κ4

(V ′V ′′′

V 2

)(609)

N =∫ φ

φend

κdφ√2ε

(610)

together with the formula for the amplitude and tilt of scalar and tensor anisotropies

AS =κ√2ε

H

2π, n = 1 + 2η − 6ε (611)

AT =√

2πκH , nT = −2ε , r = −2π nT (612)

11.1 Power-law inflationV (φ) = V0e

−βκφ β 1 for inflation

3H2(φ) =2κ2

(∂H

∂φ

)2

+ κ2V (φ)

H(φ) = H0e− 1

2βκφ ⇒ 1

H

∂H

∂φ= − 1

2βκ = const

H20 =

κ2

3V0

(1− β2

6

)−1

where V0 ≡M4 (613)

ε =2κ2

(H ′

H

)2

=12β2 < 1 (614)

δ =2κ2

(H ′′

H

)2

=12β2 < 1 (615)

ε = − H

H2=

12β2 ⇒ a ∝ tp ε=1/p−→ p =

2β2

(616)

ε = δ =1p

= const (617)

N =∫ φend

φ

κdφ√2ε

=κ

β(φend − φ) = 65 (618)

AS =κ√2ε

H

2π= 5× 10−5 ⇒ M ' 10−3MP (619)

n− 1 = 2(δ − 2ε1− ε

)= − 2

p− 1(620)

|n− 1| < 0.05 ⇒ p > 41 (621)

nT = − 2ε1− ε

= − 2p− 1

, r = −2π nT <π

10= 0.314 (622)

11.2 Chaotic inflation (m2φ2)

V (φ) =12m2φ2 ⇒ H2 ' κ2

6m2φ2 (623)

ε =1

2κ2

(V ′

V

)2

=2

κ2φ2= 1 ⇒ φend =

MP

2√π' MP

3.5(624)

η =1κ2

(V ′′

V

)=

2κ2φ2

= ε =1

2N(625)

N =∫ φ

φend

κdφ√2ε

=(κφ

2

)2∣∣∣∣∣φ

φend

≈ κ2φ2

4⇒ φ65 = 6.4MP (626)

AS =κm√

6κ2φ2

4π= N

√4

3πm

MP= 5× 10−5 ⇒ (627)

m = 1.2× 10−6MP = 1.4× 1013 GeV (628)

n = 1 + 2η − 6ε = 1− 2N≈ 0.97 (629)

AT =4√π

H

MP< 10−5 (630)

nT = −2ε = − 1N' −0.016 (631)

r =CTlCSl

=259

(1 +

48π2

385

)2ε ' −2π nT =

2πN' 0.1 (632)

11.3 Chaotic inflation (λφ4)

V (φ) =14λφ4 ⇒ H2 ' κ2

12λφ4 (633)

ε =1

2κ2

(V ′

V

)2

=8

κ2φ2= 1 ⇒ φend =

MP√π' MP

1.8(634)

η =1κ2

(V ′′

V

)=

12κ2φ2

=3ε2

=3

2N(635)

N =∫ φ

φend

κdφ√2ε

=(κφ

8

)2∣∣∣∣∣φ

φend

≈ κ2φ2

8⇒ φ65 = 4.5MP (636)

AS =

√λ

3κ3φ3

16π=

√λ

3(8N)3/2

16π= 5× 10−5 ⇒ (637)

λ = 1.3× 10−13 (638)

n = 1 + 2η − 6ε = 1− 3N≈ 0.95 (639)

AT =4√π

H

MP< 10−5 (640)

nT = −2ε = − 2N' −0.03 (641)

r =CTlCSl

=259

(1 +

48π2

385

)2ε ' −2π nT =

4πN' 0.2 (642)

11.4 New inflation

V (φ) =λ

4(φ2 − v2

)2 ⇒ H2 ' κ2

12λ(φ2 − v2

)2 (643)

complete here!

11.5 Natural inflation

V (φ) = M2 f2

(1− cos

φ

f

)= 2M2 f2 sin2 φ

2f(644)

=12M2φ2 − 1

4!M2

f2φ4 +O(φ6) (645)

ε =1

2κ2f2

(sin φ

f

1− cos φf

)2

=cot2 φ

2f

2κ2f2 1 (646)

η =1

κ2f2

cos φf1− cos φf

= ε− 12κ2f2

1 (647)

N = 2κ2f2

∫ x

xend

dx tanx = −2κ2f2 ln cosφ

2f

∣∣∣∣φφend

(648)

ε = 1 ⇒ cosφend

2f=(

2κ2f2

1 + 2κ2f2

)1/2

< 1 (649)

cosφ

2f=

(2κ2f2

1 + 2κ2f2

)1/2

e− N

2κ2f2 (650)

ε65 =1

2κ2f2

(e

Nκ2f2 − 1

)−1

12κ2f2

(651)

η65 = ε65 −1

2κ2f2⇒ n ' 1− 1

κ2f2(652)

AS =

√23κM

2π(2κ2f2) sinh

N

2κ2f2(653)

If f = MP ⇒ M = 9× 10−7MP = 1013 GeV (654)

n = 1− 18π

= 0.96 (655)

r = −2π nT =2πκ2f2

(e

Nκ2f2 − 1

)−1

' 0.02 (656)

11.6 Starobinski inflation

Rµν −12gµν R = κ2 〈Tµν〉ren =

16M2

(1)Hµν +1H2

0

(3)Hµν , (657)

(1)Hµν = 2(∇µ∇ν − gµν∇2)R+ 2RRµν −12gµνR

2 (658)

(3)Hµν = R λµ Rλν −

23RRµν −

12gµνR

ρσRρσ +14gµνR

2 (659)

Substituting FRW metric and using the Slow Roll Approximation,

H = −M2

6

(1− H2

H20

). (660)

At first stage: H20 M2 ⇒ −H < M2/6 H2

0 ⇒ H ≈ H0 = const. However, H growsand becomes unstable. When H ∼ M inflation ends. Alternatively, one can study the evolution in theeffective action formalism, including higher derivatives,

Sg =∫d4x√−g 1

2κ2

(R− R2

6M2

)≡∫d4x√−g 1

2κ2f(R) (661)

which gives rise to Eq. (658). One can then write this action as the usual Einstein-Hilbert action plus ascalar field, making use of the conformal transformation

gµν = F (R) gµν ≡ eακφgµν ⇒√−g = e2ακφ√−g (662)

Rµν = Rµν −ακ

2(gµν∇2φ+ 2∇µ∇νφ

)(663)

R = e−ακφ[R− 3ακ∇2φ− 3

2α2κ2(∂φ)2

](664)

The scalar field φ will have canonical kinetic term for α2 = 2/3. From the equations of motion one findsthe relationship F (R) = f ′(R), and therefore the effective scalar potential becomes

V (φ) =1

2κ2

f(R)−Rf ′(R)(f ′(R))2

=R2

12κ2M2

(1− R

3M2

)−2

(665)

V (φ) =3M2

4κ2

(1− e−ακφ

)2=

12M2φ2 (1 + ακφ+ . . . ) (666)

ε =2α2

(eακφ − 1)2= 1 ⇒ φend =

√3MP

4√π

ln(

1 +2√3

)' MP

5.33⇒ Hend =

√3M

2√

2 +√

3

η =2α2(2− eακφ)

(eακφ − 1)2= 0 ⇒ φ∗ =

√3MP

4√π

ln 2 ' MP

5.90< φend

N =eακφ − ακφ

2α2

∣∣∣∣φφend

' 34eακφ ⇒ φ65 = 1.09MP (667)

ε65 '1

2α2N2η65 ' −

1N

(668)

ακφ65 = 4.46 1 ⇒ V (φ65) ' M2

2α2κ2⇒ H65 '

M

2(669)

AS =αN

2πκH = 5× 10−5 ⇒ M ' 2.4× 10−6MP (670)

n = 1− 2N' 0.97 (671)

AT =√

2π

H

MP=

2√π

M

MP= 2.7× 10−6 (672)

nT = −2ε ' − 32N2

= −1.6× 10−4 (673)

r = −2π nT ' 10−3 (674)

11.7 Hybrid inflation

V (φ, χ) =λ

4(χ2 − v2)2 +

12g2φ2χ2 +

12m2φ2 (675)

The effective Higgs mass in the false vacuum (χ = 0):

m2χ ≡

∂2V

∂χ2= g2φ2 − λv2 = 0 ⇒ φc ≡

M

g=

√λv

g(676)

For large values of the inflaton, the Higgs has a large mass and sits at its minimum, and therefore theffective potential during inflation is

V (φ) = V0 +12m2φ2 ≡ V0(1 + ξ) ' V0 = const. (677)

H0 '√

2π3Mv

MP(678)

ε =m2

κ2V0ξ η =

m2

κ2V0⇒ n = 1 +

2m2

κ2V0> 1 (679)

N =κ2V0

m2lnφ

φc⇒ φ = φc e

ηN (680)

Inflation ends not because of the end of slow-roll (ε = 1) but because of symmetry breaking bythe Higgs

AS =H2

2πφ=

gH

2πηMe−ηN = 5× 10−5 ⇒ (681)

g =

√3π8

(n− 1) 10−4 MP

ve(n−1)N

2 (682)

Negligible gravitational waves:

r = −2π nT = 4πε 2π (n− 1) (683)

Many possibilities of scales of inflation: e.g. GUT scale,

v = 10−3MP , λ = 0.1 , g = 0.01 , ⇒ n− 1 = 0.035 , (684)

M = 4× 1015 GeV , m = 1.3× 1012 GeV , r = 5× 10−4 (685)

11.8 Radiative corrections on SUSY hybrid inflationColeman-Weinberg potential

V1−loop =1

64π2

∑i

(−1)Fim4i ln

m2i

Λ2(686)

Supergravity hybrid model (units κ = 1)

W =√λΦ(ΣΣ− v2) (687)

V = λ∣∣σσ − v2

∣∣2 +λ

2φ2(|σ|2 + |σ|2) + D− term (688)

where φ =√

2 Φ is the canonically normalized field. The absolute minimum appears at φ = 0, σ =σ = v. For φ > φc =

√2 v, the fields σ, σ have a positive mass squared and stay at the origin. Inflation

takes place along that “flat” direction, which is lifted by radiative corrections. The masses of bosons arem2B = 1

2λ(φ2 ± 2v2), while that of the fermion is m2F = 1

2λφ2. The loop corrected potential along the

flat direction is

V1−loop(φ) =λ2

128π2

[(φ2 − 2v2)2 ln

(φ2 − 2v2

Λ2

)+ (φ2 + 2v2)2 ln

(φ2 + 2v2

Λ2

)− 2φ4 ln

(φ2

Λ2

)](689)

⇒ V (φ) ' λ v4

(1 +

λ

8π2lnφ

φc

), φ φc (690)

ε =λ2

128π4φ2=

λ

32π2N, η = − λ

8π2φ2= − 1

2N(691)

N =∫

dφ√2ε

=4π2φ2

λ(692)

AS =

√N

316π

v2

M2P

= 5× 10−5 ⇒ v = 5.6× 1015 GeV (693)

n = 1− 1N

= 0.98 , r = −2π nT = 4πε =λ

8πN 1 (694)

12 REHEATING AFTER INFLATION

One of the fundamental quests of cosmology is to understand the origin of all the matter and radiationpresent in the universe today. We have seen how inflation produces a homogeneous and flat backgroundspace-time, and imprints on top of it a set of scalar and tensor quantum fluctuations that become classicalGaussian random fields outside the horizon, with an approximately scale invariant spectrum.

Inflation also dilutes any relic species left from a hypothetical earlier period of the universe, suchthat at the end of inflation there remains only a homogeneous zero mode of the inflaton field with tinyfluctuations on the homogeneous metric. That is, the universe is empty and very cold: the entropy of theuniverse is exponentially small and the temperature can be taken to be zero, S = T = 0.

Therefore we are left with the puzzle: How does the large entropy and energy of our presenthorizon, S ∼ 1089 and M ∼ 1023M, arise from such a cold and empty universe? The answer seems tolie in the process by which the large potential energy density present during inflation gets converted intoradiation at the end of inflation, a process known as reheating of the universe.

This process was studied soon after the first models of inflation were proposed and consideredthe perturbative decay of the inflaton field into quanta of other fields to which it coupled, e.g. fermions,gauge fields, and other scalars. Such couplings exist during inflation but play no role (except for inducingradiative corrections, as we will discuss later), because even if those particles were produced duringinflation the exponential expansion would dilute them almost instantaneously, and nothing would be leftat the end of inflation.

Let us write down the most general Lagrangian with couplings of the inflaton to other fields andamong themselves,

L = − 12

(∂µφ)2 − V (φ)− 12

(∂µχ)2 − 12m2χχ

2 − 12ξχ2R

− ψ(iγµ∂µ −mψ)ψ − hφψψ − 12g2φ2χ2 − g2σφχ2 , (695)

where g, h, ξ, etc. are small couplings (to avoid large radiative corrections during inflation); σ is thepossibly finite vev of the inflaton, and we have shifted the inflaton potential by φ− σ → φ, such that theminimum is at φ = 0 and the potential can be expanded around the minimum as

V (φ) =12m2φ2 +O(φ4) , (696)

where m is the mass of the inflaton at the minimum. In chaotic inflation of the type m2φ2 or λφ4, thismass and self-coupling are bounded by observations of the CMB to be

m ∼ 1013 GeV , λ ∼< 10−13 . (697)

We will consider this mass to be much larger than that of the other fields to which it couples: m2 m2χ,m

2ψ g2σφ, hφ. Also, the end of inflation occurs in these models whenH ∼ m, and subsequently,

the rate of expansion decays as H ∼ 1/t < m.

Let us compute the evolution of the inflaton after inflation, whose amplitude satisfies the equation(we are neglecting here the couplings to other fields, but we will consider them later)

φ+ 3H(t)φ+m2φ = 0 , (698)

whose solution is oscillatory,φ(t) = Φ(t) sin mt , (699)

with the amplitude of oscillations decaying like Φ ∼ a−3/2, as we will prove now. Consider the averageenergy density and pressure of the homogeneous inflaton field over one period of oscillations,

〈ρ〉 =12〈φ2〉+

12m2〈φ2〉 =

12m2Φ2(t)

(〈cos2mt〉+ 〈sin2mt〉

), (700)

〈p〉 =12〈φ2〉 − 1

2m2〈φ2〉 =

12m2Φ2(t)

(〈cos2mt〉 − 〈sin2mt〉

)' 0 (701)

where we have neglected the change in Φ(t) due to the condition m H during reheating. The fact thatan oscillating homogeneous scalar field behaves like a pressureless fluid means that the universe duringthat period expands like a matter dominated universe,

ρ+ 3H(ρ+ p) = 0 ⇒ ρ =12m2Φ2(t) ∼ a−3 , (702)

and therefore Φ ∼ a−3/2 ∼ t−1. That is, a homogeneous scalar field oscillating with frequency equal toits mass can be considered as a coherent wave of φ particles with zero momenta and particle density

nφ = ρφ/m =12mΦ2 ∼ a−3 , (703)

oscillating coherently with the same phase.

Until now we have considered only the effects of expansion, and ignored the effects due to theproduction of particles from the inflaton. This can be accounted for by including, in the equation ofmotion, the denominator of the QFT propagator,

φ+ 3H(t)φ+(m2 + Π(ω)

)φ = 0 , (704)

where Π(ω) is the Minkowski space polarization operator for φ with four-momentum kµ = (ω,0),with ω = m. The real part of the polarization operator can be neglected (due to the small cou-plings), Re Π(ω) m2. However, due to phase space, if the frequency of oscillations satisfies ω min(2mχ, 2mψ), then the polarization operator acquires an imaginary part,

Im Π(m) = mΓφ , (705)

where Γφ is the total decay rate of the inflaton, and we have used the optical theorem (i.e. unitarity) torelate both quantities at the physical pole, ω = m.

The total decay rate can be written as a sum over partial decays,

Γφ =∑i

Γ(φ→ χiχi) +∑i

Γ(φ→ ψiψi) , (706)

Γ(φ→ χiχi) =g4i v

2

8πm, Γ(φ→ ψiψi) =

h2im

8π, (707)

Γφ ≡h2

effm

8π m, h2

eff =∑i

(h2i +

g4i v

2

m2

)(708)

The evolution of the inflaton during the period of oscillations after inflation can be describedthrough the phenomenological equation

φ+ 3H(t)φ+ Γφφ+m2φ = 0 , (709)

which includes the decay rate Γφ as a friction term giving rise to the damping of the oscillations due toinflaton particle decay. It assumes the inflaton condensate (the homogeneous zero mode) is composed ofvery many inflaton particles, each of these decaying into other particles to which it couples. The solutionto this equation is given by (699) with

Φ(t) = Φ0 e− 1

2

∫3Hdt

e−12

Γφt =Φ0

te−

12

Γφt , (710)

where we have used H = 2/3t.We can now compute the evolution of the energy and number density of the inflaton field under

the effect of particle production,

d

dt

(ρφa

3)

= −Γφ ρφa3 , (711)

d

dt

(nφa

3)

= −Γφ nφa3 , (712)

which simply states the usual exponential decay law for particles with decay rate Γ. Initially, the totaldecay rate is much smaller than the rate of expansion, Γφ 3H = 2/t m, and the total comovingenergy and total number of inflaton particles is conserved, their energy and number densities decayinglike a matter fluid, ρφ ' mnφ ∼ a−3.

Eventually, the universe expands sufficiently (this may take many many inflaton oscillations) thatthe decay rate becomes larger than the rate of expansion, or alternatively, the inflaton life-time, ηφ = Γ−1

φ ,becomes smaller than the age of the universe, ηφ < tU = H−1, and the inflaton decays suddenly (inless than one Hubble time), releasing all its energy density ρφ into relativistic particles χ and ψ, in anexponential burst of energy. Subsequently, the produced particles interact among themselves and soonthermalize to a common temperature. This process is responsible for the present abundance of matterand radiation energy, and could be associated with the Big Bang of the “old” cosmology.

At first sight it may seem paradoxical that the universe may have to “wait” until it is old enough forthe inflaton to decay, because we are accustomed to very rapid decays in our particle physics detectors,where life-times of order 10−17s are possible, while our universe is 10+17s old! However, if inflationtook place at energy densities of order the GUT scale, the Hubble time of a causal domain at the endof inflation would be of order 10−35s, which is many orders of magnitude smaller than even the fastestdecays of the inflaton, ∼ 10−25s. So the probability that the inflaton decays in such a short Hubble timeis negligible, and the universe has to wait until it is old enough that there is any probability of decay ofa single inflaton particle. Eventually, of course, once the universe is older than the inflaton life-time, it(the inflaton) will decay exponentially fast due to its constant decay rate Γφ.

Let us now compute the reheating temperature of the universe that arises from the thermalizationof the products of decay of the inflaton. Note that the process of reheating, once possible, is essentiallyinstantaneous and therefore the energy density at reheating can be estimated as that corresponding toa rate of expansion H = Γφ. Since all that energy density will be quickly converted into a plasma ofrelativistic particles, we can estimate

ρ(trh) =3Γ2

φM2P

8π=π2

30g(Trh)T 4

rh , (713)

⇒ Trh ' 0.1√

ΓφMP , (714)

where we have assumed g(Trh) ∼ 102 − 103. Let us estimate this temperature. If we substitute Γφ =h2

effm/8π with m ∼ 1013 GeV, we find

Trh ' 2× 1014 heff GeV ∼< 1011 GeV , (715)

where we have imposed the constraint heff ∼< 10−3 from radiative corrections in chaotic type models. Letus estimate it: if we consider the quantum loop corrections to the inflaton potential due to its coupling toother fields like in the Lagrangian described above, we find

V (φ) =12m2φ2

(1 +

3g4

16π4λ

)+λ

4φ4(

1 +3g4

16π4λ− h4

16π4λ

)+ . . . (716)

Therefore, the couplings of the inflaton to other fields cannot be very large otherwise they wouldmodify the amplitude of CMB anisotropies. If we impose that the mass and self-coupling of the inflatonsatisfy (697), then the other couplings are bound to

3g4 , h4 < 16π2λ ⇒ g , h ∼< 10−3 . (717)

For completeness, let us mention that in theories with only gravitational interactions, like e.g. in Starobin-sky model, the decay of the inflaton is induced via gravity only and

Γgrav ∼m3

M2P

∼ 10−18MP ⇒ Trh ∼ 109 GeV . (718)

All this indicates that, although the energy density at the end of inflation may be large, the finalreheating temperature Trh may not be higher than 1012 GeV, and thus the usually assumed thermal phasetransition at Grand Unification, which was the basis for most of the early universe phenomenology, likeproduction of topological defects, GUT baryogenesis, etc., could not have taken place.

We will see shortly that such phenomenology may be resucitated in the context of preheating andnon-thermal phase transitions, but for the moment let us focus our attention onto two well differentiatedand concrete cases of ordinary reheating:

12.1 Reheating in chaotic inflation modelsConsider a m2φ2 model of inflation, for which the value of the inflaton at the end of inflation is φend =MP/2

√π, and the corresponding energy density

ρend =32V (φend) =

3m2M2P

16π=(

6.5× 1015 GeV)4. (719)

On the other hand, CMB anisotropies require

AS = N

√4

3πm

MP= 5× 10−5 ⇒ m ' 1.4× 1013 GeV , (720)

while radiative corrections impose the constraint heff ∼< 10−3.

We are thus left with three time scales:

tosc ∼ m−1 ∼ 10−36 stexp ∼> H−1

end ∼ 10−35 stdec ∼ Γ−1

φ ∼ 10−25 s

⇒ tosc texp tdec , (721)

so there are several oscillations per Hubble time, and we also expect many oscillations of the inflatonfield before it decays. This result is typical of most high-scale models of inflation.

12.2 Reheating in low-scale hybrid inflation modelsIn this case, reheating occurs in very different circumstances. Most models of inflation occur at scales oforder the GUT scale, because their parameters are fixed by the amplitude of CMB anisotropies, δT/T ∼m/MP ∼ 10−5. However, in models of hybrid inflation, which end due to the symmetry breaking ofa field coupled to the inflaton, and not because of the end of slow-roll, it is possible to decouple theamplitude of CMB fluctuations from the scale of inflation. For instance, consider a hybrid model at theelectroweak scale, where the symmetry breaking field is the SM Higgs field, with a vev v = 246 GeV, arelatively strong coupling to the inflaton, g = 0.4, and a Higgs self-coupling λ = 0.12, giving rise to thefollowing masses in the true vacuum

minf = g v ∼ 100 GeV , mH =√

2λ v ∼ 120 GeV , (722)

which are much larger than the rate of expansion at the end of inflation

Hend =√π

3mHv

MP∼ 2× 10−5 eV mH , (723)

and therefore we can neglect it during the oscillations of the inflaton and Higgs fields around the mini-mum of their potential.

The energy density at the end of inflation is

ρend =18m2

Hv2 ∼ (102 GeV)4 , (724)

which is very low indeed.

The couplings of the Higgs to matter could be large, e.g. the top quark Yukawa ht ∼ 1, althoughfor such a low mass Higgs there is no phase space for top perturbative production. On the other hand,the inflaton may couple to other particles, so it is expected that their decay widths be similar and bothof order Γ ∼ 1 GeV. Naively, using (714), one would thus expect that the reheating temperature beTrh ∼ 109 GeV, but that is impossible because it would correspond to an energy density during inflationmuch above the actual false vacuum energy, ρend ∼ (102 GeV)4.

Actually, since the rate of expansion is so low compared with the other scales, we can ignore thedecay in energy due to the expansion of the universe, which was so important during chaotic inflation,and use energy conservation to estimate

ρend =λv4

4=π2

30g∗ T

4rh ⇒ Trh '

(15λ

2π2g∗

)1/4

v ∼ 42 GeV , (725)

where we have used g∗ = 106.75 as the effective number of degrees of freedom of the SM particles.Note that this temperature is rather low, but in fact we have no observational evidence that the universehas actually gone through a thermal period with a temperature above this.

We are thus left with three time scales:

tosc ∼ m−1H∼ 10−27 s

texp ∼ H−1 ∼ 10−10 stdec ∼ Γ−1 ∼ 10−23 s

⇒ tosc tdec texp , (726)

so there are many oscillations per Hubble time, but contrary to the case of chaotic inflation models, herethe decay time is much smaller than the expansion time, because the universe is already quite old, soonce the inflaton and Higgs start oscillating they decay very soon via their usual perturbative decay.

13 Preheating

The previous discussion falls under the name of perturbative reheating, because it assumes that thecoherently oscillating inflaton will decay as if it were composed on individual inflaton quanta, each onedecaying as described by ordinary QFT, with the perturbative decay rate computed above. This was thestandard lore during at least a decade since it was first proposed in 1982. However, it was soon realizedthat the inflaton at the end of inflation is actually a coherent wave, a zero mode, a condensate made outof many inflaton quanta, all oscillating with the same phase, and non-perturbative effects associated withthis condensate were bound to be important for the problem of reheating. In fact, a few years ago, in aseminal paper, Linde, Kofman and Starobinsky proposed a new picture of reheating, which has becomeknown as preheating. I will describe these new developments in the following sections. They make useof the well studied problem of particle production in the presence of strong background fields, whoseformalism we have already encountered for the analysis of the generation of metric fluctuations duringinflation. In this case, instead of a quantum field evolving in a rapidly changing gravitational field (likeduring inflation), we have a field coupled to the inflaton, which has a rapidly changing frequency or massdue to the inflaton oscillations.

We will first describe the Bogolyubov formalism for a single scalar field with a time-dependentmass and then particularize to the case of the inflaton oscillations after inflation. Later on, we will alsoextended the formalism to fermions, which can also be produced at preheating.

Consider a massive scalar field φ with Lagrangian density

L =12φ2 − 1

2(∇φ)2 − 1

2m2φ2 , (727)

which gives a canonically conjugate momentum π =δLδφ

= φ, and the Hamiltonian

H = πφ− L =12φ2 +

12

(∇φ)2 +12m2φ2 . (728)

We can treat the fields as quantum fields and define the usual equal time commutation relation

[φ(x, t), π(x′, t)] = i δ(3)(x− x′) , (729)

as well as expand in Fourier components,

φ(x, t) =∫

d3k

(2π)3/2φk(t) eik·x . (730)

The field mode φk(t) satisfies the harmonic oscillator equation

φk + ω2k φk = 0 , (731)

ω2k(t) = k2 +m2(t) , (732)

where the time dependence of the oscillation frequency comes through that of the mass. We will assumethat the field is real, so we should impose the constraint φk(t) = φ∗−k(t). Following the quantizationcondition (729), we can write the field and momentum operators in terms of time-dependent creation andannihilation operators,

φk(t) =1√2ωk

(ak(t) + a†−k(t)

),

πk(t) = −i√ωk2

(ak(t) + a†−k(t)

),

(733)

satisfying the usual commutation relation, ∀t,

[ak(t), a†k′(t)] = δ(3)(k− k′) ,

and in terms of which the Hamiltonian becomes

H =12

∫d3k

[πkπ

†k + ω2

k φkφ†k

]=

12

∫d3k ωk

(a†kak + aka

†k

)≡ Hpart +Hvac(t) , (734)

where

Hpart =∫d3k ωk a

†kak , (735)

Hvac(t) =V

(2π)3

∫d3k

ωk2. (736)

We can then define a number operator for these fields

N =∫d3k a†kak , (737)

and a Fock space with vacuum state defined as

ak(t)|0t〉 = 0 , 〈0t|0t〉 = 1 , (738)

and particle states |nk〉 ∝ (a†k)n|0t〉 satisfying

Hpart|nk〉 = nkωk|nk〉 ≡ Ek|nk〉 , (739)

N |nk〉 = nk|nk〉 . (740)

In the vacuum state |0t〉, the energy takes its lowest possible value, Hvac(t) = 〈0t|H|0t〉.We can compute the equations of motion as usual with

d

dtak =

∂ak∂t

+ i [H, ak]

where we can invert the relations (733)

ak(t) =√ωk2φk(t) +

i√2ωk

πk(t) ,

a†−k(t) =√ωk2φk(t)− i√

2ωkπk(t) .

(741)

In the Heisenberg picture, the original canonical operators φk, πkmay have no explicit time-dependence,but ωk is indeed time-dependent, so

d

dtak = −iωkak +

ωk2ωk

a†−k . (742)

The solution to the equations of motion is ak(t)

a†−k(t)

=

(uk(t) vk(t)

v∗k(t) u∗k(t)

) ak(0)

a†−k(0)

(743)

The unitary evolution preserves the commutation relation (729) iff

|uk|2 − |vk|2 = 1 , (744)

with initial condition : |uk|2 = 1 , |vk|2 = 0 . (745)

If the initial state is the vacuum, |0〉 ≡ |0t=0〉, then

ak(0)|0〉 = 0 ⇒ ak(t)|0〉 = vk(t) a†−k(0)|0〉 6= 0 (746)

In particular, the number density of particles created from the vacuum is

n(t) =1V〈0|N |0〉 =

∫d3k

(2π)3|vk|2(t) . (747)

In order to find the function n(t) explictly, we have to solve for uk and vk as a solution of

(uk(t)

v∗k(t)

)=

−iωkωk2ωk

ωk2ωk

iωk

(uk(t)

v∗k(t)

)(748)

It is customary to write the mode functions uk and vk is terms of the usual Bogolyubov coefficients,αk, βk,

uk = αk e−i

∫ t

ωkdt

, v∗k = βk e+i

∫ t

ωkdt

, (749)

then the evolution equations (748) become

αk =ωk2ωk

βk e+2i

∫ t

ωkdt

,

βk =ωk2ωk

αk e−2i

∫ t

ωkdt

,

(750)

which can be integrated in the adiabatic approximation, to give

n(t) =∫

d3k

(2π)3nk(t) =

∫d3k

(2π)3|βk|2(t) , (751)

the number density of particles produced due to the time-dependent background field.

Alternatively, one can introduce the |in〉 and |out〉 states, and make the field decomposition overtime-independent creation and annihilation operators ak, a†−k,

φ(x, t) =∫

d3k

(2π)3/2

[fk(t) ak eik·x + h.c.

]=

∫d3k

(2π)3/2

(fk(t) ak + f∗k (t) a†−k

)eik·x , (752)

π(x, t) =∫

d3k

(2π)3/2

(gk(t) ak + g∗k(t) a

†−k

)eik·x , (753)

where the mode functions fk(t) and gk(t) depend only on the modulus k = |k|, thanks to the homogene-ity and isotropy of the background fields. These functions satisfy the equations of motion

fk + ω2k fk = 0 , gk = ifk . (754)

Comparing with the former decomposition (733), we find the relation

uk =1√2ωk

(ωkfk + gk) ,

vk =1√2ωk

(ωkfk − gk) ,(755)

and viceversafk =

1√2ωk

(uk + v∗k) ,

gk =√ωk2

(uk − v∗k) ,(756)

which gives for the occupation number

nk(t) = |βk|2 =1

2ωk|fk|2 +

ωk2|fk|2 −

12, (757)

where we have used the Wronskian

i(fkf∗k − f∗kfk) = 2 Re (f∗kgk) = 1 ⇔ |uk|2 − |vk|2 = 1 . (758)

13.1 Diagonalization of the HamiltonianWith the above decomposition, we can write the Hamiltonian as

H =∫d3k

[Ek(t)

(a†k ak + ak a

†k

)+Fk(t) ak a−k + F ∗k (t) a†k a

†−k

], (759)

where

Ek(t) =12

(|fk|2 + ω2k|fk|2) = ωk

(nk +

12

), (760)

Fk(t) =12

(f2k + ω2

kf2k ) , (761)

E2k(t) − |Fk(t)|2 =

ω2k

4. (762)

Let us now introduce a canonical Bogolyubov transformation ak

a†−k

=

(uk(t) vk(t)

v∗k(t) u∗k(t)

) bk

b†−k

(763)

Then

nk = |βk|2 =2Ek − ωk

2ωk, (764)

ukvk

=2Ek + ωk

2F ∗k, (765)

and the Hamiltonian becomes diagonal

H =∫d3k

ωk2

(b†k bk + bk b

†k

), (766)

which can be decomposed into Hpart and Hvac, as before, see (735).

13.2 The Schrodinger pictureWe can define the unitary evolution operator U †(t) = U−1(t), where i~ ∂tU(t) = H U(t), such thattime evolution determines

ak(t) = U †(t)ak(0)U(t) . (767)

The solution of the Schrodinger equation is the squeezed state

|ψ(t)〉 = U(t)|ψ(0)〉 . (768)

The vacuum at time t is given by |0t〉 = U †(t)|0〉. Let |ψ(0)〉 be the initial vacuum state |0〉. Then theoperator

bk(t) = U(t)ak(0)U †(t) = u∗k ak(0)− vk a†−k(0) (769)

annihilates the state |ψ(t)〉. Now let us use (741) to evaluate

ak(0) =√ωk2φk(0) +

i√2ωk

πk(0) , (770)

and substitute into bk(t)|ψ(t)〉 = 0,

1√2ωk

[(u∗k − vk)ωk φk(0) + i(u∗k + vk)πk(0)] |ψ(t)〉 = 0.

Therefore, the evolved state satisfies the Schrodinger equation

[Ωk(t)φk(0) + iπk(0)]|ψ(t)〉 = 0 , (771)

Ωk(t) ≡ ωku∗k − vku∗k + vk

=g∗kf∗k

=1− 2iFk(t)

2|fk(t)|2, (772)

where we have used g∗kfk = Re (g∗kfk)− i Im (f∗kgk) = 12 − iRe (f∗k fk). Using the operator definition

πk = −i ∂∂φ−k

= −i ∂∂φ∗k

, we find the solution

ψ(φk, φ∗k, t) ∼ e−Ωk(t)|φk|2 , (773)

P (φk, φ∗k, t) = |ψ(φk, φ∗k, t)|2 ∼ e− 1|fk(t)|2

|φk|2. (774)

The phase Fk(t) = Re (f∗k fk) 1 quickly becomes very large during preheating, which ensures that thestate becomes a squeezed state, with large occupation numbers, described by the Gaussian distribution(774).

13.3 Parametric resonanceWe will consider here the case of a scalar field χ coupled to the inflaton φ with coupling 1

2g2φ2χ2, which

induces an oscillating mass termm2χ(t) = m2

χ + g2φ2(t) . (775)

The inflaton is assumed to oscillate like (699) with a frequency given by its mass m, not necessarilymuch larger than the “bare” mass of the field χ. In that case, the frequency can be written as

ω2k(t) = k2 +m2

χ + g2Φ2(t) sin2mt , (776)

and the mode equation (754) can be written as a Mathieu equation, where z = mt, and primes denotedifferentiation w.r.t. z,

f ′′k + (Ak − 2q cos 2z) fk = 0 , (777)

Ak =k2 +m2

χ

m2+ 2q , q =

g2Φ2(t)4m2

. (778)

The Mathieu equation is part of a large class of Hill equations (which includes also the Lame equationand many others) that present unstable solutions for certain values of the momenta for a given set ofparameters Ak, q, with A ≥ 2q. These solutions fall into bands of instability that are narrow for smallvalues of the resonant parameter q ≤ 1, but can be very broad for larger values of q.

The solutions to the Mathieu eq. have the form fk(z) = eµk z p(z), with µk the Floquet index,characterizing the exponential instability, and typically much smaller than one, although it could be aslarge as µmax = 0.28055; and where p(z) is a periodic function of z. The occupation number can thenbe computed to be

nk(t) ∼ e2µkmt , (779)

which can grow significantly in a few oscillations, if the growth index µk is not totally negligible.

The effect of parametric resonance is similar to the lasing effect (or light amplification by stim-ulated emission of radiation), where a large population of particles is produced from oscillations of acoherent source.

13.4 Narrow resonanceLet us consider first the case where mχ, gΦ m, or q 1. Then the Mathieu equation instabilitychart shows that the resonance occurs only in some narrow bands around Ak ' l2 , l = 1, 2, . . . , withwidths in momentum space of order ∆k ∼ mql; so, for q < 1, the most important band is the first one,Ak ∼ 1± q, centered around k = m/2.

The growth factor µk for the first instability band is given by

µk =

√(q2

)2−(

2km− 1)2

. (780)

The resonance occurs for k within the range m2 (1 ± q

2). The index µk vanishes at the edgesof the resonance band and takes its maximum value µk = q

2 at k = m2 . The corresponding modes

grow at a maximal rate χk ∼ exp(qz/2). This leads to a growth of the occupation numbers (779) asnk ∼ exp(qmt).

We can interpret this as follows. In the limit q 1, the effective mass of the χ particles ismuch smaller than m, and each decaying φ particle creates two χ particles with momentum k ∼ m/2.The difference with respect to the perturbative decay Γ(φ → χχ) is that, in the regime of parametricresonance, the rate of production of χ particles is proportional to the number of particles produced earlier(which gives rise to an exponential growth in time). This is a non-perturbative effect, as we will discusslater, and we could not have obtained it by using the methods described in the previous section, at anyfinite order of perturbation theory with respect to the interaction term g2Φ2 sin2mt. It is by solving themode equation (777) exactly that we have found this result.

Note that only a very narrow range of modes grow exponentially with time, so the spectrum ofparticles is dominated by these modes, while the rest are still in the vacuum, produced only through theordinary perturbative decay process. Of course, the exponential production does not last for ever: theuniverse expansion is going to affect the resonant production of particles in two ways, leading to the endof the narrow resonance regime.

First, the time-dependent amplitude of oscillations Φ(t), which determines q, see (778), not onlydecays (∼ t−1) due to the expansion of the universe, but also due to the perturbative decay of the inflatonfield, Φ(t) ∼ exp(Γφt/2). Therefore, the narrow resonance will end when the usual perturbative decaybecomes important, i.e. when q m < Γφ.

Second, in the evolution equation (777), the momenta k are actually physical momenta, whichredshift with the scale factor as kphys = k/a, and therefore, even if a given mode is initially within thenarrow band, ∆k ∼ q m, it will very quickly redshift away from it, within the time scale ∆t ∼ q H−1,

preventing its occupation numbers (779) from growing exponentially. Thus, the narrow resonance willend when q2m < H .

Therefore, if the amplitude of inflaton oscillations decays like Φ ∼ 1/t, there will always be atime (typically a dozen oscillations) for which one of the two conditions above will hold and the narrowresonance will end.

13.5 Broad resonanceIf the initial amplitude of oscillations is very large, like in models of chaotic inflation, in which Φ0 ∼MP/10 and m ∼ 10−6MP, then the initial q-parameter could be very large,

q0 =g2Φ2

0

4m2∼ g2 1010 ∼< 104 , (781)

where we have used the constraint due to radiative corrections (717). In this case, the χ particle produc-tion due to stimulated emission by the oscillating inflaton field can be very efficient as it enters into thebroad resonance regime.

Particles are produced only at the instances of maximum acceleration of the inflaton field, whenφ(t) ∼ 0, and ∣∣∣∣ ωkω2

k

∣∣∣∣ 1 , (782)

a relation known as the non-adiabaticity condition. When it holds, we cannot define a proper Fock spacefor the χ particles, and the occupation numbers of those particles grow very quickly. We thus associate(782) with particle production.

We will now describe how to compute the growth of modes and the Floquet index in this regime,using the formalism developed above. We can expand the quantum field χ in Fourier components fksatisfying the mode equation (754) with time-dependent frequency (776) and initial conditions

fk(0) =1√2ωk

e−iωkt , gk(0) = ifk(0) = ωk fk(0) , (783)

whose evolution in terms of the Bogolyubov coefficients is

fk(t) =αk(t)√

2ωke−i

∫ωkdt

+βk(t)√

2ωke

+i

∫ωkdt

, (784)

αk(0) = 1 , βk(0) = 0 . (785)

And the occupation numbers are

nk(t) = |βk(t)|2 =1

2ωk|fk|2 +

ωk2|fk|2 −

12. (786)

The inflaton field has maximum acceleration at t = tj = jπ/m, such that sinmtj = 0. Betweentj and tj+1, the amplitude φ(t) ≈ φ0 = const, so that the frequency ωk(t) is approximately constantbetween succesive zeros of the inflaton, and we can properly define a Fock space for χ. At tj , theamplitude changes rapidly, such that (782) is satisfied and we cannot define an adiabatic invariant likethe occupation number (786). Therefore, let us study the behaviour of the modes χk precisely at thoseinstances t = tj . We can expand the time-dependent frequency (776) around those points (where thefrequency has a minimum) as

ω2k(t) = ω2

k(tj) +12ω2k′′(tj)(t− tj)2 + · · · (787)

and make the change of variables

η ≡ [2ω2k′′(tj)]1/4(t− tj) , (788)

κ2 ≡ω2k(tj)√

2ω2k′′(tj)

=k2 +m2

χ

2gmΦ=Ak − 2q

4√q

. (789)

The mode equation (754) around t = tj then becomes

d2fkdη2

+(κ2 +

η2

4

)fk = 0 , (790)

which can be interpreted as a Schrodinger equation for a wave function scattering in an inverted parabolicpotential. The exact solutions are parabolic cylinder functions, W (−κ2,±η), whose asymptotic expres-sions are well known. Thus we have substituted the problem of parametric resonance after chaoticinflation with that of partial waves scattering off successive inverted parabolic potentials.

Let the wave fk(t) have the form of the adiabatic solution (784) before scattering at tj ,

f jk(t) =αjk√2ωk

e−i

∫ωkdt

+βjk√2ωk

e+i

∫ωkdt

, (791)

where the coefficients αjk, βjk are constant, for tj−1 < t < tj .

After scattering off the potential at tj , the wave fk(t) takes the form

f j+1k (t) =

αj+1k√2ωk

e−i

∫ωkdt

+βj+1k√2ωk

e+i

∫ωkdt

, (792)

where the coefficients αj+1k , βj+1

k are again constant, for tj < t < tj+1. These are essentially theasymptotic expressions for the incoming and the outgoing waves, scattered at tj . Therefore, the outgoingamplitudes αj+1

k , βj+1k can be expressed in terms of the incoming amplitudes αjk, β

jk with the help

of the reflection Rk and transmission Dk coefficients of scattering at tj , αj+1k e−iθ

jk

βj+1k e+iθjk

=

1Dk

R∗kD∗k

RkDk

1D∗k

αjke

−iθjk

βjke+iθjk

(793)

where θjk =∫ tj

0ωk(t)dt, and

Rk = −i e−iφk [1 + e2πκ2]−1/2 ,

Dk = e−iφk [1 + e−2πκ2]−1/2 ,

|Rk|2 + |Dk|2 = 1 . (794)

The k-dependent angle of scattering is

φk = Arg Γ[1

2+ iκ2

]+ κ2(1− lnκ2) . (795)

Simplifying (793), we find αj+1k

βj+1k

=

[1 + e−2πκ2]1/2eiφk i e−πκ

2+2iθk

−i e−πκ2−2iθk [1 + e−2πκ2]1/2e−iφk

αjk

βjk

(796)

and therefore, using njk = |βjk|2 and |αjk|

2|βjk|2 = njk(n

jk + 1), we have

nj+1k = e−2πκ2

+ (1 + 2e−2πκ2)njk

− 2e−πκ2[1 + e−2πκ2

]1/2[njk(njk + 1)]1/2 sin θjtot , (797)

where θjtot = 2θjk − φk + Arg βjk −Argαjk .

This expression is very enlightening. Let us describe its properties:

• Step-like. The number of created particles is a step-like function of time. The occupation numberbetween successive scatterings is constant. In the first scattering (when n0

k = 0), we have

nk = e−2πκ2= e− πk2

gmΦ < 1 . (798)

• Non-perturbative. The occupation number (798) cannot be expanded perturbatively, for smallcoupling, because the function e−1/g is non-analytical at g = 0. This is the form that most non-perturbative effects take in quantum field theory.• Infrared effect. For large momenta, the occupation number decays exponentially, so even if there

are bands at low momenta, i.e. in the IR region, the high momentum modes will not be populated,

κ2 π−1 ⇒ nj+1k ' njk ' 0 . (799)

• Non-linear. For small momenta one may have production of particles with mass greater than thatof the inflaton:

κ2 =k2 +m2

χ

2gmΦ0∼< π−1 ⇒ nk large if m2 < m2

χ gmΦ0 (800)

• Exponential boson production. In the case of bosons (we will discuss the fermionic case later),the occupation number can grow exponentially due to Bose-Einstein statistics, nk ∼ exp(2µkz)1,

nj+1k ' [(1 + 2e−2πκ2

)− 2e−πκ2[1 + e−2πκ2

]1/2 sin θjtot]njk ≡ e

2πµjk njk

which allows one to estimate the Floquet index µk.• Resonant production. Valid only for periodic sources. If scattering occurs in phase, the incoming

and outgoing waves add up constructively, and we can have resonant effects. This occurs whenθjtot is a semi-integer multiple of π. In that case, it is possible that, for some modes, nj+1

k > njk.This gives rise to a particular band structure.• Stochastic preheating. It may happen that the phase a mode has acquired in a given scattering ex-

actly compensates for the universe expansion in that interval and the phases destructively interfere,decreasing the number of particles in that mode. This gives rise to a stochastic growth of particles,where approximately 3/4 of the time the particle number increases.• Band structure. Different models of inflation give rise to different evolution laws for the ampli-

tude of inflaton oscillations, and therefore to different mode equations (754). The correspondingHill equations (linear second order differential equations with periodic coefficients) can have quitedifferent band structures, e.g. those of Mathieu or Lame equations.

Even if we compute the complete band structure of the Mathieu or Lame equation and we determine thegrowth factors µk with great accuracy, the universe expansion will shift any given mode from one bandto the next, as the mode redshifts and the amplitude of inflaton oscillations decreases: A mode starts in agiven band, its occupation numbers increase exponentially through several oscillations, and suddenly itfalls out of the band, until the expansion makes it fall into the next band, and so on until it reaches thenarrow resonance regime described above.

14 CONCLUSION

In the last ten years we have seen a true revolution in the quality and quantity of cosmological datathat has allowed cosmologists to determine most of the cosmological parameters with a few percentaccuracy and thus fix a Standard Model of Cosmology. The art of measuring the cosmos has developedso rapidly and efficiently that one may be temped of renaming this science as Cosmonomy, leaving theword Cosmology for the theories of the Early Universe. In summary, we now know that the stuff we aremade of − baryons − constitutes just about 4% of all the matter/energy in the Universe, while 25% isdark matter − perhaps a new particle species related to theories beyond the Standard Model of ParticlePhysics −, and the largest fraction, 70%, some form of diffuse tension also known as dark energy −perhaps a cosmological constant. The rest, about 1%, could be in the form of massive neutrinos.

Nowadays, a host of observations − from CMB anisotropies and large scale structure to the ageand the acceleration of the universe − all converge towards these values, see Fig. 30. Fortunately, wewill have, within this decade, new satellite experiments like Planck, CMBpol, SNAP as well as deepgalaxy catalogs from Earth, to complement and precisely pin down the values of the Standard Modelcosmological parameters below the percent level, see Table 1.

All these observations would not make much sense without the encompassing picture of the infla-tionary paradigm that determines the homogeneous and isotropic background on top of which it imprintsan approximately scale invariant gaussian spectrum of adiabatic fluctuations. At present all observationsare consistent with the predictions of inflation and hopefully in the near future we may have information,from the polarization anisotropies of the microwave background, about the scale of inflation, and thusabout the physics responsible for the early universe dynamics.

ACKNOWLEDGEMENTS

I would like to thank Misha Shaposhnikov and Ruth Durrer for inviting me to give this course at theTroisieme Cycle de Physique de la Suisse Romand. This work was supported in part by a CICYT projectAYA2009-13936-C06-06.

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