22. Big-Bang Cosmology

1

22. Big-Bang Cosmology
Revised August 2023 by K.A. Olive (Minnesota U.) and J.A. Peacock (Edinburgh U.).

22.1 Introduction to the standard big-bang model
The observed expansion of the Universe [1–3] is a natural (almost inevitable) result of any
homogeneous and isotropic cosmological model based on general relativity. However, by itself,
the Hubble expansion does not provide sufficient evidence for what we generally refer to as the
Big-Bang model of cosmology. While general relativity is in principle capable of describing thecosmology of any given distribution of matter, it is extremely fortunate that our Universe appears
to be homogeneous and isotropic on large scales. Together, homogeneity and isotropy allow us toextend the Copernican Principle to the Cosmological Principle, stating that all spatial positions in
the Universe are essentially equivalent.
The formulation of the Big-Bang model began in the 1940s with the work of George Gamow andhis collaborators, Ralph Alpher and Robert Herman. In order to account for the possibility thatthe abundances of the elements had a cosmological origin, they proposed that the early Universe
was once very hot and dense (enough so as to allow for the nucleosynthetic processing of hydrogen),and has subsequently expanded and cooled to its present state [4, 5]. In 1948, Alpher and Herman
predicted that a direct consequence of this model is the presence of a relic background radiation
with a temperature of order a few kelvin [6, 7]. Of course this radiation was observed 16 years lateras the cosmic microwave background (CMB) [8]. Indeed, it was the observation of this radiationthat singled out the Big-Bang model as the prime candidate to describe our Universe. Subsequent
work on Big-Bang nucleosynthesis further confirmed the necessity of our hot and dense past. See
Sec. 22.3.7 for a brief discussion of BBN, and Sec. 24 of this Review for a detailed discussion of
BBN. These relativistic cosmological models face severe problems with their initial conditions, towhich the best modern solution is inflationary cosmology, discussed in Sec. 22.3.5 and in – Sec. 23of this Review. If correct, these ideas would strictly render the term ‘Big Bang’ redundant, since
it was first coined by Hoyle to represent a criticism of the lack of understanding of the initialconditions.
22.1.1 The Robertson-Walker Universe
The observed homogeneity and isotropy enable us to describe the overall geometry and evolutionof the Universe in terms of two cosmological parameters accounting for the spatial curvature andthe overall expansion (or contraction) of the Universe. These two quantities appear in the mostgeneral expression for a space-time metric that has a (3D) maximally symmetric subspace of a 4D
space-time, known as the Robertson-Walker metric:
"

#

dr2
ds = dt − R (t)+ r2 (dθ2 + sin2 θ dφ2 ) .
1 − kr22

2

2(22.1)

Note that we adopt c = 1 throughout. By rescaling the radial coordinate, we can choose thecurvature constant k to take only the discrete values +1, −1, or 0 corresponding to closed, open,or spatially flat geometries. In this case, it is often more convenient to re-express the metric ashids2 = dt2 − R2 (t) dχ2 + Sk2 (χ) (dθ2 + sin2 θ dφ2 ) ,(22.2)

where the function Sk (χ) is (sin χ, χ, sinh χ) for k = (+1, 0, −1). The coordinate r [in Eq. (22.1)] andthe ‘angle’ χ [in Eq. (22.2)] are both dimensionless; the dimensions are carried by the cosmologicalscale factor, R(t), which determines proper distances in terms of the comoving coordinates. A
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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22. Big-Bang Cosmology

2common alternative is to define a dimensionless scale factor, a(t) = R(t)/R0 , where R0 ≡ R(t0 )is R at the present epoch. It is also sometimes convenient to define a dimensionless or conformal
time coordinate, η, by dη = dt/R(t). Along constant spatial sections, the proper time is defined by
the time coordinate, t. Similarly, for dt = dθ = dφ = 0, the proper distance is given by R(t)χ. For
standard texts on cosmological models see e.g., Refs. [9–16].
22.1.2 The redshift
The cosmological redshift is a direct consequence of the Hubble expansion, determined by R(t).
A local observer detecting light from a distant emitter sees a redshift in frequency. We can define
the redshift as
ν1 − ν2z≡' v12 ,
(22.3)
ν2where ν1 is the frequency of the emitted light, ν2 is the observed frequency, and v12 is the relative
velocity between the emitter and the observer. While the definition, z = (ν1 − ν2 )/ν2 is valid in
general, relating the redshift to a simple relative velocity is only correct on small scales (i.e., lessthan cosmological scales) such that the expansion velocity is non-relativistic. For light signals, we
can use the metric given by Eq. (22.1) and ds2 = 0 to write
v12 = Ṙ δr =ṘδR
R2 − R1
δt =
=
,
R
R
R1

(22.4)

where δr(δt) is the radial coordinate (temporal) separation between the emitter and observer.
Noting that physical distance, D, is Rδr or δt, Eq. (22.4) gives us Hubble’s law, v = HD. In
addition, we obtain a simple relation between the redshift and the scale factor:
1+z =

ν1
R2
=
.
ν2
R1

(22.5)

This result does not depend on the non-relativistic approximation.
22.1.3 The Friedmann equations of motion
The cosmological equations of motion are derived from Einstein’s equations:Rµν − 21 gµν R = 8πGN Tµν + Λgµν .

(22.6)

Gliner [17] and Zeldovich [18] pioneered the modern view, in which the Λ term is set on the rhs
and interpreted as an effective energy-momentum tensor Tµν for the vacuum of Λgµν /8πGN . It is
common to assume that the matter content of the Universe is a perfect fluid, for which
Tµν = −pgµν + (p + ρ) uµ uν ,(22.7)

where gµν is the space-time metric described by Eq. (22.1), p is the isotropic pressure, ρ is the energy
density and u = (1, 0, 0, 0) is the velocity vector for the isotropic fluid in co-moving coordinates.
With the perfect fluid source, Einstein’s equations lead to the Friedmann equations:
2

H ≡

Ṙ
R

!2

=

8π GN ρkΛ
− 2+ ,
3
R
3(22.8)andR̈
Λ 4πGN= −(ρ + 3p),R
3
3
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(22.9)

3

22. Big-Bang Cosmology

where H(t) is the Hubble parameter and Λ is the cosmological constant. The first of these issometimes called the Friedmann equation. Energy conservation via T µν;µ = 0, leads to a third usefulequation [which can also be derived from Eq. (22.8) and Eq. (22.9)]:ρ̇ = −3H (ρ + p) .

(22.10)Equation (22.10) can also be simply derived as a consequence of the first law of thermodynamics.
Equation (22.8) has a simple classical mechanical analog if we neglect (for the moment) thecosmological term Λ. By interpreting −k/R2 Newtonianly as a ‘total energy’, then we see that theevolution of the Universe is governed by a competition between the potential energy, 8πGN ρ/3, andthe kinetic term (Ṙ/R)2 . For Λ = 0, it is clear that the Universe must be expanding or contracting(except at the turning point prior to collapse in a closed Universe). The ultimate fate of the
Universe is determined by the curvature constant k. For k = +1, the Universe will recollapse in a
finite time, whereas for k = 0, −1, the Universe will expand indefinitely. These simple conclusions
can be altered when Λ 6= 0 or more generally with some component with (ρ + 3p) < 0.
22.1.4 Definition of cosmological parameters
In addition to the Hubble parameter, it is useful to define several other measurable cosmologicalparameters. The Friedmann equation can be used to define a critical density such that k = 0 when
Λ = 0:
3H 2
ρc ≡= 1.88 × 10−26 h2 kg m−3
8π GN(22.11)−5 2−3= 1.05 × 10 h GeV cm ,where the scaled Hubble parameter, h, is defined byH ≡ 100 h km s−1 Mpc−1⇒ H −1 = 9.778 h−1 Gyr−1= 2998 h(22.12)Mpc.

The cosmological density parameter Ωtot is defined as the energy density relative to the criticaldensity:Ωtot = ρ/ρc .
(22.13)Note that one can now rewrite the Friedmann equation as
k/R2 = H 2 (Ωtot − 1) .

(22.14)

From Eq. (22.14), one can see that when Ωtot > 1, k = +1 and the Universe is closed, when
Ωtot < 1, k = −1 and the Universe is open, and when Ωtot = 1, k = 0, and the Universe is spatiallyflat.
It is often necessary to distinguish different contributions to the density. It is therefore convenient to define present-day density parameters for pressureless matter (Ωm ) and relativistic particles(Ωr ), plus the quantity ΩΛ = Λ/3H 2 . In more general models, we may wish to drop the assumption that the vacuum energy density is constant, and we therefore denote the present-day density
parameter of the vacuum by Ωv . The Friedmann equation then becomes
k/R02 = H02 (Ωm + Ωr + Ωv − 1),(22.15)where the subscript 0 indicates present-day values. Thus, it is the sum of the densities in matter,
relativistic particles, and vacuum that determines the overall sign of the curvature. Note that thequantity −k/R02 H02 is sometimes referred to as ΩK . This usage is unfortunate: it encourages one
to think of curvature as a contribution to the energy density of the Universe, which is not correct.
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22. Big-Bang Cosmology

22.1.5 Standard Model solutions
Much of the history of the Universe in the standard Big-Bang model can be easily described byassuming that either matter or radiation dominates the total energy density. During inflation andagain today the expansion rate for the Universe is accelerating, and we should allow for domination
by a cosmological constant or some other form of dark energy. In the following, we shall delineate
the solutions to the Friedmann equation when a single component dominates the energy density.
Each component is distinguished by an equation of state parameter w = p/ρ. We concentrate on
solutions that expand at early times, although the Friedmann equation also permits a time-reversed
contracting solution.
22.1.5.1 Solutions for a general equation of state
Let us first assume a general equation of state parameter for a single component, w, which isconstant. In this case, Eq. (22.10) can be written as ρ̇ = −3(1 + w)ρṘ/R and is easily integrated
to yieldρ ∝ R−3(1+w) .
(22.16)Note that at early times when R is small, the less singular curvature term k/R2 in the Friedmannequation can be neglected so long as w > −1/3. Curvature domination occurs at rather late times(if a cosmological constant term does not dominate sooner). For w 6= −1, one can insert this result
into the Friedmann equation Eq. (22.8), and if one neglects the curvature and cosmological constantterms, it is easy to integrate the equation to obtain
R(t) ∝ t2/[3(1+w)] .

(22.17)22.1.5.2 A Radiation-dominated Universe
In the early hot and dense Universe, it is appropriate to assume an equation of state corresponding to a gas of radiation (or relativistic particles) for which w = 1/3. In this case, Eq. (22.16)
becomes ρ ∝ R−4 . The ‘extra’ factor of 1/R is due to the cosmological redshift; not only is thenumber density of particles in the radiation background decreasing as R−3 since volume scalesas R3 , but in addition each particle’s energy is decreasing as E ∝ ν ∝ R−1 . Similarly, one cansubstitute w = 1/3 into Eq. (22.17) to obtain
R(t) ∝ t1/2 ;H = 1/2t.

(22.18)22.1.5.3 A Matter-dominated Universe
At relatively late times, non-relativistic matter eventually dominates the energy density overradiation [see Eq. (22.3.8)]. A pressureless gas (w = 0) leads to the expected dependence ρ ∝ R−3from Eq. (22.16) and, if k = 0, we obtain
R(t) ∝ t2/3 ;H = 2/3t.

(22.19)22.1.5.4 A Universe dominated by vacuum energy
If there is a dominant source of vacuum energy, V0 , it would act as a cosmological constant with
Λ = 8πGN V0 and equation of state w = −1. In this case, the solution to the Friedmann equation
when curvature is neglected is particularly simple and leads to an exponential expansion of the
Universe:√
R(t) ∝ e Λ/3 t .
(22.20)More generally we could write√a(t) = sinh2/3 ( 3Λt/2),
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(22.21)

5

22. Big-Bang Cosmology

which describes a flat Universe containing both matter and vacuum energy, with a(t) being thescale factor normalized to unity when both components are equal.
A key parameter is the equation of state of the vacuum, w ≡ p/ρ; this need not be the w = −1of Λ, and may not even be constant [19–21]. There is much interest in the more general possibilityof a dynamically evolving vacuum energy, for which the name ‘dark energy’ has become commonly
used. A variety of techniques exist whereby the vacuum density as a function of time may bemeasured, usually expressed as the value of w as a function of epoch [22, 23]. The best current
measurement for the equation of state (assumed constant, but without assuming zero curvature)is w = −1.028 ± 0.031 [24]. Unless stated otherwise, we will assume that the vacuum energy is acosmological constant with w = −1 exactly.
The presence of vacuum energy can dramatically alter the fate of the Universe. For example, ifΛ < 0, the Universe will eventually recollapse independent of the sign of k. For large values of Λ > 0(larger than the Einstein static value needed to halt any cosmological expansion or contraction),even a closed Universe will expand forever. One way to quantify this is the deceleration parameter,
q0 , defined as
RR̈1(1 + 3w)
q0 = − 2 = Ω m + Ω r +Ωv .
(22.22)
2
Ṙ 0 2This equation shows us that w < −1/3 for the vacuum may lead to an accelerating expansion. To
the continuing astonishment of cosmologists, such an effect has been observed: one piece of direct
evidence is the supernova Hubble diagram [25–30] (see Fig. 22.1 below). Current data indicatethat vacuum energy is indeed the largest contributor to the cosmological density budget, with
Ωv = 0.685 ± 0.007 and Ωm = 0.315 ± 0.007 if k = 0 is assumed [24].
The existence of this constituent is without doubt the greatest puzzle raised by the current
cosmological model; the final section of this review discusses some of the ways in which the vacuumenergy problem is being addressed. For more details, see Dark Energy – Sec. 28 of this Review.

22.2 Introduction to observational cosmology
22.2.1 Fluxes, luminosities, and distances
The key quantities for observational cosmology can be deduced quite directly from the metric.
(1) The proper transverse size of an object seen by us to subtend an angle dψ is its comoving
size dψ Sk (χ) times the scale factor at the time of emission:d` = dψ R0 Sk (χ)/(1 + z).

(22.23)(2) The apparent flux density of an object is deduced by allowing its photons to flow througha sphere of current radius R0 Sk (χ); but photon energies and arrival rates are redshifted, and the
bandwidth dν is reduced. The observed photons at frequency ν0 were emitted at frequency ν0 (1+z),
so the flux density is the luminosity at this frequency, divided by the total area, divided by 1 + z:Sν (ν0 ) =Lν ([1 + z]ν0 ).
4πR02 Sk2 (χ)(1 + z)

(22.24)

These relations lead to the following common definitions:angular-diameter distance

DA = (1 + z)−1 R0 Sk (χ);

luminosity distance DL = (1 + z) R0 Sk (χ) .

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(22.25)

6

22. Big-Bang Cosmology

Figure 22.1: The type Ia supernova Hubble diagram, based on 1650 publicly available supernova
distance estimates [28–30]. The first panel shows that for z  1 the large-scale Hubble flow is
indeed linear and uniform; the second panel shows an expanded scale, with the linear trend divided
out, and with the redshift range extended to show how the Hubble law becomes nonlinear (Ωr = 0
is assumed). Larger points with errors show median values in redshift bins. Comparison with theprediction of Friedmann models favors a vacuum-dominated Universe.

These distance-redshift relations are expressed in terms of observables by using the equation of
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22. Big-Bang Cosmology

7

a null radial geodesic (R(t)dχ = dt) plus the Friedmann equation:
R0 dχ =

1
1
[ (1 − Ωm − Ωv − Ωr )(1 + z)2
dz =
H(z)
H0
+ Ωv (1 + z)3+3w + Ωm (1 + z)3+ Ωr (1 + z)4

i−1/2(22.26)dz.

The main scale for the distance here is the Hubble length, 1/H0 . The flux density is the product of
the specific intensity Iν and the solid angle dΩ subtended by the source: Sν = Iν dΩ. Combining
the angular size and flux-density relations thus gives the relativistic version of surface-brightness
conservation:Iν (ν0 ) =Bν ([1 + z]ν0 ),(1 + z)3(22.27)

where Bν is surface brightness (luminosity emitted into unit solid angle per unit area of source).
We can integrate over ν0 to obtain the corresponding total or bolometric formula:
Itot =

Btot.
(1 + z)4(22.28)This cosmology-independent form expresses Liouville’s Theorem: photon phase-space density isconserved along rays.
22.2.2 Distance data and geometrical tests of cosmology
In order to confront these theoretical predictions with data, we have to bridge the divide between
two extremes. Nearby objects may have their distances measured quite easily, but their radial
velocities are dominated by deviations from the ideal Hubble flow, which typically have a magnitudeof several hundred km s−1 . On the other hand, objects at redshifts z >∼ 0.01 will have observed<
recessional velocities that differ from their ideal values by ∼ 10%, but absolute distances are muchharder to supply in this case. The traditional solution to this problem is the construction of the
distance ladder: an interlocking set of methods for obtaining relative distances between various
classes of object, which begins with absolute distances at the 10 to 100 pc level, and terminateswith galaxies at significant redshifts. This is discussed in the article on Cosmological Parameters –Sec. 25 of this Review.
One of the key developments in this area has been the use of type Ia supernovae (SNe), which
now allow measurement of relative distances with 5% precision. In combination with improved
Cepheid data from the HST plus improved measurements of the distance to the LMC (or alternatively a direct geometrical distance to the maser galaxy NGC4258), SNe results extend the distanceladder to the point where deviations from uniform expansion are negligible, leading to the best existing Cepheid-based value for H0 : (73.0±1.0) km s−1 Mpc−1 [31]. Better still, the analysis of high-z
SNe has allowed a simple and direct test of cosmological geometry to be carried out: as shown in
Fig. 22.1 and Fig. 22.2, supernova data and measurements of CMB anisotropies strongly favor ak = 0 model dominated by vacuum energy. It is worth noting that there is some tension (4.2σ)
between the Cepheid and CMB determinations of H0 (the latter is (67.4 ± 0.5) km s−1 Mpc−1 [24]).
While it is remarkable that the two very different methods give such similar results, the formal
disagreement shows that either there are unidentified systematic errors or that some new post-CDM
physics is required; there is no current consensus in the community on these alternatives. We donote that a recent analysis of SNe Ia with a calibration of the tip of the red-giant branch gives a
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22. Big-Bang Cosmology

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result close to that of the CMB: (69.8 ± 0.6 (stat.) ± 1.6 (sys.)) km s−1 Mpc−1 [32, 33]. See Cosmological Parameters – Sec. 25 of this Review for a more comprehensive review of Hubble parameter
determinations.

Figure 22.2: Likelihood-based probability densities over the plane ΩΛ (i.e., Ωv assuming w = −1)
versus Ωm . The colored locus derives from Planck [34] and shows that the CMB alone requires aflat Universe Ωv +Ωm ' 1 if the Hubble constant is not too high. The SNe Ia results [35] very nearly
constrain the orthogonal combination Ωv − Ωm , and the intersection of these constraints directly
favors a flat model with Ωm ' 0.3, as does the measurement of the baryon acoustic oscillation
lengthscale (for which a joint constraint is shown on this plot). The CMB alone is capable ofbreaking the degeneracy with H0 by using the measurements of gravitational lensing that can bemade with modern high-resolution CMB data.
22.2.3 Age of the Universe
The most striking conclusion of relativistic cosmology is that the Universe has not existed
forever. The dynamical result for the age of the Universe may be written as
Z ∞H0 t0 =
0dz
(1 + z)H(z)

Z ∞

=
0

dz

(22.29),(1 + z) [(1 + z)2 (1 + Ωm z) − z(2 + z)Ωv ]1/2<
where we have neglected Ωr and chosen w = −1. Over the range of interest (0.1 <∼ Ωm ∼ 1,<|Ωv | ∼ 1), this exact answer may be approximated to a few per cent accuracy by
H0 t0 ' 23 (0.7Ωm + 0.3 − 0.3Ωv )−0.3 .
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(22.30)922. Big-Bang Cosmology

For the special case that Ωm + Ωv = 1, the integral in Eq. (22.29) can be expressed analytically as√
1 + Ωv
2√√
H0 t0 =ln(Ωm < 1).
(22.31)3 Ωv
1 − Ωv
The most accurate means of obtaining ages for astronomical objects is based on the naturalclocks provided by radioactive decay. The use of these clocks is complicated by a lack of knowledge ofthe initial conditions of the decay. In the Solar System, chemical fractionation of different elements
helps pin down a precise age for the pre-Solar nebula of 4.6 Gyr, but for stars it is necessary toattempt an a priori calculation of the relative abundances of nuclei that result from supernova
explosions. In this way, a lower limit for the age of stars in the local part of the Milky Way of
about 11 Gyr is obtained [36, 37].
The other major means of obtaining cosmological age estimates is based on the theory ofstellar evolution. In principle, the main-sequence turnoff point in the color-magnitude diagram of a
globular cluster should yield a reliable age. But these have been controversial, owing to theoreticaluncertainties in the evolution model – as well as observational uncertainties in the distance, dust
extinction, and metallicity of clusters. The present consensus favors ages for the oldest clusters of
about 13 Gyr [38].
These methods are all consistent with the age deduced from studies of structure formation,using the microwave background and large-scale structure: t0 = (13.80 ± 0.02) Gyr [24], where theextra accuracy comes at the price of assuming the simple 6-parameter ΛCDM model to be true.
22.2.4 Horizon, isotropy, flatness problems
For photons, the radial equation of motion is just c dt = R dχ. How far can a photon get in agiven time? The answer is clearly
Z t2

∆χ =
t1

dt
≡ ∆η,
R(t)

(22.32)

i.e., just the intervalof conformal time. We can replace dt by dR/Ṙ, which the Friedmann equationpsays is ∝ dR/ ρR2 at early times. Thus, this integral converges if ρR2 → ∞ as t1 → 0, otherwise
it diverges. Provided the equation of state is such that ρ changes faster than R−2 , light signals
can only propagate a finite distance between the Big Bang and the present; there is then said tobe a particle horizon. Such a horizon therefore exists in conventional Big-Bang models, which are
dominated by radiation (ρ ∝ R−4 ) at early times.
At late times, the integral for the horizon is largely determined by the matter-dominated phase,for whichZ t(z)
dt
6000 −1
DH = R0 χH ≡ R0'√
h Mpc (z  1).
(22.33)R(t)Ωm z0The horizon at the time of formation of the microwave background (‘last scattering’: z ' 1100)
was thus of order 100 M pc in size, subtending an angle of about 1◦ . Why then are the large numberof causally disconnected regions we see on the microwave sky all at the same temperature? The
Universe is very nearly isotropic and homogeneous, even though the initial conditions appear not
to permit such a state to be constructed.
A related problem is that the Ω = 1 Universe is unstable:Ω(a) − 1 =Ω−1
1 − Ω + Ωv a2 + Ωm a−1 + Ωr a−2,(22.34)

where Ω with no subscript is the total density parameter, and a(t) = R(t)/R0 . This requires Ω(t)to be unity to arbitrary precision as the initial time tends to zero; a Universe of non-zero curvature
today requires very finely tuned initial conditions.
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22.3 The Hot Thermal Universe
22.3.1 Thermodynamics of the early Universe
As alluded to above, we expect that much of the early Universe can be described by a radiationdominated equation of state. In addition, through much of the radiation-dominated period, thermal
equilibrium is established by the rapid rate of particle interactions relative to the expansion rateof the Universe (see Sec. 22.3.3 below). In equilibrium, it is straightforward to compute the thermodynamic quantities, ρ, p, and the entropy density, s. In general, the energy density for a given
particle type i can be written asZ
ρi =Ei dnqi ,(22.35)gi(exp[(Eqi − µi )/Ti ] ± 1)−1 qi2 dqi ,2π 2(22.36)

with the density of states given bydnqi =

where gi counts the number of degrees of freedom for particle type i, Eq2i = m2i + qi2 , µi is thechemical potential, and the ± corresponds to either Fermi or Bose statistics. Similarly, we candefine the pressure of a perfect gas asqi2
dnqi .
Ei(22.37)dnqi ,(22.38)ρi + pi − µi ni.
Ti(22.39)

pi =

1
3

Z

The number density of species i is simplyZni =and the entropy density issi =In the Standard Model, a chemical potential is often associated with baryon number, and since
the net baryon density relative to the photon density is known to be very small (of order 10−9 ), we
can neglect any such chemical potential when computing total thermodynamic quantities.
For photons, we can compute all of the thermodynamic quantities rather easily. Taking gi = 2
for the 2 photon polarization states, we have (in units where ~ = kB = 1)
ργ =

π2 4
T ,
15

1
pγ = ργ ,
3

sγ =

4ργ
,
3T

nγ =

2ζ(3) 3T ,π2(22.40)with 2ζ(3)/π 2 ' 0.2436. Note that Eq. (22.10) can be converted into an equation for entropy
conservation. Recognizing that ṗ = sṪ , Eq. (22.10) becomesd(sR3 )/dt = 0.

(22.41)

For radiation, this corresponds to the relationship between expansion and cooling, T ∝ R−1 in an
adiabatically expanding Universe. Note also that both s and nγ scale as T 3 .
22.3.2 Radiation content of the Early Universe
At the very high temperatures associated with the early Universe, massive particles are pair
produced, and are part of the thermal bath. If for a given particle species i we have T  mi ,then we can neglect the mass in Eq. (22.35) to Eq. (22.39), and the thermodynamic quantities are
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22. Big-Bang Cosmology

11

easily computed as in Eq. (22.40). In general, we can approximate the energy density (at hightemperatures) by including only those particles with mi  T . In this case, we have

ρ=

7X
gB +
gF
8 F
B

X

!

π2 4 π2T ≡
N (T ) T 4 ,3030(22.42)where gB(F) is the number of degrees of freedom of each boson (fermion) and the sum runs over allboson and fermion states with m  T . The factor of 7/8 is due to the difference between the Fermi
and Bose integrals. Eq. (22.42) defines the effective number of degrees of freedom, N (T ), by taking
into account new particle degrees of freedom as the temperature is raised. This quantity, calculated
from high temperature lattice QCD, is plotted in Fig. 22.3 [39]. Near the QCD transition, there
is a slight difference between the coefficient of T 4 for ρ and the coefficient of T 3 for the entropy
density s = (2π 2 /45)Ns (T )T 3 [40], as seen in the figure.

Figure 22.3: The effective numbers of relativistic degrees of freedom as a function of temperature.
The sharp drop corresponds to the quark-hadron transition. The bottom panel shows the relative
ratio between the number of degrees of freedom characterizing the energy density and the entropy.

The value of N (T ) at any given temperature depends on the particle physics model. In thestandard SU(3) × SU(2) × U(1) model, we can specify N (T ) up to temperatures of O(100) GeV.
The change in N (ignoring mass effects) can be seen in the table below.
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22. Big-Bang Cosmology

12

Temperature
T < meme < T < m µmµ < T < m π
m π < T < Tc †Tc < T < mstrange
ms < T < mcharmmc < T < m τmτ < T < mbottom
mb < T < mW,Z
mW,Z < T < mHiggs
mH < T < mtop
mt < T

New Particles
γ’s + ν’se±µ±
π’s
π’s + u, ū, d, d¯ + gluons
s, s̄
c, c̄
τ±
b, b̄
W ±, Z
H0t, t̄

4N (T )
29
43
57
69
205
247
289
303
345
381
385
427

† T corresponds to the confinement-deconfinement transition between quarks and hadrons.
cAt higher temperatures, N (T ) will be model-dependent. For example, in the minimal SU(5)model, one needs to add 24 states to N (T ) for the charged and colored X and Y gauge bosons,another 24 from the adjoint Higgs, and another six scalar degrees of freedom (in addition to thefour associated with the complex Higgs doublet already counted in the longitudinal components of
W ± and Z, and in H) from the 5 of Higgs. Hence for T > mX in minimal SU(5), N (T ) = 160.75.
In a supersymmetric model this would at least double.
In the radiation-dominated epoch, Eq. (22.10) can be integrated (neglecting the T -dependenceof N ), giving us a relationship between the age of the Universe and its temperature:


t=

90
3
32π GN N (T )

1/2

T −2 .

(22.43)This can be expressed more conveniently in the form2t TMeV= 2.4[N (T )]−1/2 ,(22.44)

where t is measured in seconds and TMeV in units of MeV.
22.3.3 Neutrinos and equilibrium
Due to the expansion of the Universe, certain rates may be too slow to either establish or
maintain equilibrium. Quantitatively, for each particle i, a minimal condition for equilibrium
requires that the interaction rate Γ i involving that type of particle should be larger than the
expansion rate of the Universe:
Γ i > H.
(22.45)
Recalling that the age of the Universe is determined by H −1 , this condition is equivalent to requiringthat on average, at least one interaction has occurred over the lifetime of the Universe.
A good example for a process that goes in and out of equilibrium is the weak interaction ofneutrinos. On dimensional grounds, one can estimate the thermally averaged scattering crosssection:
hσvi ∼ O(10−2 )T 2 /m4W
(22.46)
3<
for T ∼ mW . Recalling that the number density of leptons is n ∝ T , we can compare the weakinteraction rate, Γwk ∼ nhσvi, with the expansion rate and so obtainH=

8πGN ρ
3

1/2

=

8π 3
N (T )
90' 1.66N (T )31st May, 2024

!1/2

1/2

T 2 /MP2

T /MP ,

(22.47)

13

22. Big-Bang Cosmology−1/2

where the Planck mass MP = GN
= 1.22 × 1019 GeV.
Neutrinos will be in equilibrium when Γwk > H or
T > (500 m4W /MP )1/3 ∼ 1 MeV.

(22.48)

However, this condition assumes T  mW ; for higher temperatures, we should write hσvi ∼−2O(10−2 )/T 2 , so
√ that Γ ∼ 10 T . Thus, in the very early stages of expansion, at temperatures−2>
T ∼ 10 MP / N , equilibrium will not have been established.
Having attained a quasi-equilibrium stage, the Universe then cools further to the point where
the interaction and expansion timescales match once again. The temperature at which these rates
are equal is commonly referred to as the neutrino decoupling or freeze-out temperature and isdefined by Γwk (Td ) = H(Td ). For T < Td , neutrinos drop out of equilibrium. The Universe
becomes transparent to neutrinos and their momenta simply redshift with the cosmic expansion.
The effective neutrino temperature will simply fall with T ∝ 1/R.
Soon after decoupling, e± pairs in the thermal background begin to annihilate (when T <∼ me ).
Because the neutrinos are decoupled, the energy released due to annihilation heats up the photon
background relative to the neutrinos. The change in the photon temperature can be easily computedfrom entropy conservation. The neutrino entropy must be conserved separately from the entropy
of interacting particles. A straightforward computation yields
Tν = (4/11)1/3 Tγ ' 1.9 K.

(22.49)The total entropy density is therefore given by the contribution from photons and three flavors ofneutrinos:




4 π2
21
4 π2
21
(22.50)s=
2 + (Tν /Tγ )3 Tγ3 =2+
Tγ3 = 7.04 nγ .
3 30
4
3 30
11
Similarly, the total relativistic energy density is given by
π2
21
ρr =
2 + (Tν /Tγ )4 Tγ4 ' 1.68ργ .
30
4




(22.51)

In practice, a small correction is needed to this, since neutrinos are not totally decoupled at e±annihilation: the effective number of massless neutrino species is 3.044, rather than 3 [41].
This expression ignores neutrino rest masses, but current oscillation data require at least one
neutrino eigenstate to have a mass exceeding 0.06 eV. In this minimal case, Ων h2 = 6 × 10−4 ,so the neutrino contribution to the matter budget would be negligibly small (which is our normal
assumption). However, a nearly degenerate pattern of mass eigenstates could allow larger densities,
since oscillation experiments only measure differences in m2 values. Note that a 0.06-eV neutrino
has Tν = mν at z ' 357, so the above expression for the total present relativistic density is really
only an extrapolation. However, neutrinos are almost certainly relativistic at all epochs where theradiation content of the Universe is dynamically significant. Combining Planck data with datafrom large-scale structure surveys such as DES [42] provides an upper limit to the sum of neutrino
masses, Σmν < 0.13 eV, and may provide the first measurement of a neutrino mass (as opposed to adifference of neutrino mass2 from oscillation data) from future surveys. See Neutrinos in Cosmology– Sec. 26 of this Review for a detailed discussion of neutrinos in cosmology.
22.3.4 Field theory and phase transitions
It is very likely that the Universe has undergone one or more phase transitions during thecourse of its evolution [43–46]. Our current vacuum state is described by SU(3)c × U(1)em , whichin the Standard Model is a remnant of an unbroken SU(3)c × SU(2)L × U(1)Y gauge symmetry.
31st May, 2024

14

22. Big-Bang Cosmology

Symmetry breaking occurs when a non-singlet gauge field (the Higgs field in the Standard Model)
picks up a non-vanishing vacuum expectation value, determined by a scalar potential. For example,
a simple (non-gauged) potential describing symmetry breaking is V (φ) = 14 λφ4 − 12 µ2 φ2 + V (0).
√The resulting expectation value is simply hφi = µ/ λ.
In the early Universe, finite temperature radiative corrections typically add terms to the potential of the form φ2 T 2 . Thus, at very high temperatures, the symmetry is restored and hφi = 0.
As the Universe cools, depending on the details of the potential, symmetry breaking will occur
via a first-order phase transition in which the field tunnels through a potential barrier, or via asecond-order transition in which the field evolves smoothly from one state to another (as would be
the case for the above example potential).
The evolution of scalar fields can have a profound impact on the early Universe. The equationof motion for a scalar field φ can be derived from the energy-momentum tensor:
1Tµν = ∂µ φ∂ν φ − gµν ∂ρ φ∂ ρ φ − gµν V (φ).
2(22.52)

By associating ρ = T00 and p = R−2 (t)Tii we have11
ρ = φ̇2 + R−2 (t)(∇φ)2 + V (φ) ;22
1 2 1 −2p = φ̇ − R (t)(∇φ)2 − V (φ) ,26(22.53)and from Eq. (22.10) we can write the equation of motion as follows (by considering a homogeneous
region, we can ignore the gradient terms):φ̈ + 3H φ̇ = −∂V /∂φ.

(22.54)22.3.5 Inflation
In Sec. 22.2.4, we discussed some of the problems associated with the standard Big-Bang model.
However, during a phase transition, our assumptions of an adiabatically expanding Universe are
generally not valid. If, for example, a phase transition occurred in the early Universe such thatthe field evolved slowly from the symmetric state to the global minimum, the Universe may havebeen dominated by the vacuum energy density associated with the potential near φ ' 0. During
this period of slow evolution, the energy density due to radiation will fall below the vacuum energy
density, ρ  V (0). When this happens, the expansion rate will be dominated by the constantV(0), and we obtain the exponentially expanding solution given in Eq. (22.20). When the fieldevolves towards the global minimum it will begin to oscillate about that minimum, energy will be
released during its decay, and a hot thermal Universe will be restored. If released fast enough,it will produce radiation at a temperature N TR 4 <∼ V (0). In this reheating process, entropy has
been created and the final value of R T is greater than the initial value of R T . Thus, we see that,during a phase transition, the relation R T ∼ constant need not hold true. This is the basis of theinflationary Universe scenario [47–49].
If, during the phase transition, the value of R T changed by a factor of O(1029 ), the cosmologicalproblems discussed above would be solved. The observed isotropy would be generated by theimmense expansion; one small causal region could get blown up, and thus our entire visible Universe
would have been in thermal contact some time in the past. In addition, the density parameter Ω
would have been driven to 1 (with exponential precision). Density perturbations will be stretchedby the expansion, λ ∝ R(t). Thus it will appear that λ  H −1 or that the perturbations have left
the horizon, where in fact the size of the causally connected region is now no longer simply H −1 .
31st May, 2024

15

22. Big-Bang Cosmology

However, not only does inflation offer an explanation for large scale perturbations, it also offers asource for the perturbations themselves through quantum fluctuations.
Problems with early models of inflation based on either a first-order [50] or second-order [51,52]
phase transition of a Grand Unified Theory led to models invoking a completely new scalar field:
the inflaton, φ. The potential of this field, V (φ), needs to have a very low gradient and curvaturein order to match observed metric fluctuations. For a more thorough discussion of the problems ofearly models and a host of current models being studied see the review on inflation – Sec. 23 of thisReview. In most current inflation models, reheated bubbles typically do not percolate, so inflation is‘eternal’ and continues with exponential expansion in the region outside the bubbles. These causally
disconnected bubble Universes constitute a ‘multiverse’, where low-energy physics can vary between
different bubbles. This has led to a controversial ‘anthropic’ approach to cosmology [53–55], whereobserver selection within the multiverse can be introduced as a means of understanding e.g. why
the observed level of vacuum energy is so low (because larger values suppress growth of structure).
22.3.6 Baryogenesis
The Universe appears to be populated exclusively with matter rather than antimatter [56, 57].
Indeed antimatter is only detected in accelerators or in cosmic rays. However, the presence ofantimatter in the latter is understood to be the result of collisions of primary particles in theinterstellar medium. There is in fact strong evidence against primary forms of antimatter in the
Universe. Furthermore, the density of baryons compared to the density of photons is extremely
small, η ∼ 10−9 .
The production of a net baryon asymmetry requires baryon number violating interactions, C
and CP violation, and a departure from thermal equilibrium [58]. The first two of these ingredientsare expected to be contained in Grand Unified Theories (GUTs) as well as in the non-perturbative
sector of the Standard Model; the third can be realized in an expanding Universe where, as we haveseen, interactions come in and out of equilibrium.
There are several interesting and viable mechanisms for the production of the baryon asymmetry.
While we cannot review any of them here in any detail, we mention some of the important scenarios.
In all cases, all three ingredients listed above are incorporated. One of the first mechanisms was
based on the out of equilibrium decay of a massive particle such as a superheavy GUT gauge or Higgs
boson [59, 60]. A novel mechanism involving the decay of flat directions in supersymmetric models
is known as the Affleck-Dine scenario [61]. There is also the possibility of generating the baryon
asymmetry at the electroweak scale using the non-perturbative interactions of sphalerons [62].
Because these interactions conserve the sum of baryon and lepton number, B + L, it is possible
to first generate a lepton asymmetry (e.g., by the out-of-equilibrium decay of a superheavy righthanded neutrino), which is converted to a baryon asymmetry at the electroweak scale [63]. Thismechanism is known as lepto-baryogenesis, or simply leptogenesis.
22.3.7 Nucleosynthesis
An essential element of the standard cosmological model is Big-Bang nucleosynthesis (BBN),
the theory that predicts the abundances of the light element isotopes D, 3 He, 4 He, and 7 Li. Nucleosynthesis takes place at a temperature scale of order 1 MeV. The nuclear processes lead primarily
to 4 He, with a primordial mass fraction of about 25%. Lesser amounts of the other light elements
are produced: about 10−5 of D and 3 He and about 10−10 of 7 Li by number relative to H. The
abundances of the light elements depend almost solely on one key parameter, the baryon-to-photon
ratio, η. The nucleosynthesis predictions can be compared with observational determinations of theabundances of the light elements. Consistency between theory and observations, driven primarily
31st May, 2024

22. Big-Bang Cosmology

16

by recent D/H measurements [64, 65], leads to a range of [66]5.8 × 10−10 < η < 6.3 × 10−10 .

(22.55)The value of η is related to the baryon density as follows:Ωb = 3.66 × 107 η h−2 ,(22.56)or 1010 η = 274Ωb h2 . The Planck result [24] for Ωb h2 of 0.0224 ± 0.0002 translates into a value ofη = 6.12±0.04. This result can be used to ‘predict’ the light element abundances, which can in turn
be compared with observation [67]. The resulting D/H abundance is in excellent agreement withthat found in quasar absorption systems. It is in reasonable agreement with the helium abundance
observed in extragalactic HII regions (once systematic uncertainties are accounted for). The BBN
predicted Li abundance is systematically higher than the Li abundance observed in the atmospheresof halo dwarf stars [68], but there is mounting evidence that effects of stellar depletion cannot beignored so that the stellar abundances of Li may not correspond to the primordial abundance [69].
See BBN – Sec. 24 of this Review for a detailed discussion of BBN, in addition to references [66,70–74].
22.3.8 The transition to a matter-dominated Universe
In the Standard Model, the temperature (or redshift) at which the Universe undergoes a transition from a radiation-dominated to a matter-dominated Universe is determined by the amountof dark matter. Assuming three nearly massless neutrinos, the energy density in radiation attemperatures T  1 MeV, is given by
"

π2
21
ρr =
2+
30
4



4
11

4/3 #

T 4.

(22.57)In the absence of non-baryonic dark matter, the matter density can be written as
ρm = mN η nγ ,(22.58)where mN is the nucleon mass. Recalling that nγ ∝ T 3 [cf. Eq. (22.40)], we can solve for thetemperature or redshift at the matter-radiation equality when ρr = ρm :Teq = 0.22 mN η

(1 + zeq ) = 0.22 η

or

mN
,
T0

(22.59)where T0 is the present temperature of the microwave background. For η = 6.1 × 10−10 , thiscorresponds to a temperature Teq ' 0.13 eV or (1 + zeq ) ' 550. A transition this late would be
problematic for structure formation (see Sec. 22.4.5).
The redshift of matter domination can be pushed back significantly if non-baryonic dark matteris present. If instead of Eq. (22.58), we writeρm = Ωm ρc

T
T0

3

,

(22.60)

we find that
Teq = 0.9

Ωm ρc
T03

or

(1 + zeq ) = 2.4 × 104 Ωm h2 .

31st May, 2024

(22.61)

17

22. Big-Bang Cosmology

22.4 The Universe at late times
22.4.1 The CMB
One form of the infamous Olbers’ paradox says that, in Euclidean space, surface brightness is
independent of distance. Every line of sight will terminate on matter that is hot enough to be
3
ionized and so scatter photons: T >∼ 10 K, and the sky should therefore shine as brightly as thesurface of the Sun. The reason the night sky is dark is entirely due to the expansion, which cools theradiation temperature to 2.73 K. This gives a Planck function peaking at around 1 mm to produce
the CMB.
The CMB spectrum is a very accurate match to a Planck function [75]; see also CMB – Sec. 29of this Review. The COBE estimate of the temperature is [76]T = (2.7255 ± 0.0006) K.

(22.62)The lack of any distortion of the Planck spectrum is a strong physical constraint. It is very
difficult to account for in any expanding Universe other than one that passes through a hot stage.
Alternative schemes for generating the radiation, such as thermalization of starlight by dust grains,inevitably generate a superposition of temperatures. What is required in addition to thermal
equilibrium is that T ∝ 1/R, so that radiation from different parts of space arrive at an observerwith the same apparent temperature.
Although it is common to speak of the CMB as originating at ‘recombination’, a more accurate terminology is the era of ‘last scattering’. In practice, this takes place at z ' 1100, almost
independently of the main cosmological parameters, at which time the fractional ionization is verysmall. This occurred when the age of the Universe was about 370,000 years. But the CMB photons
themselves were not generated at this point, and were the result of thermalization at z ∼ 107 . See
CMB – Sec. 29 of this Review for a full discussion of the CMB.
22.4.2 Matter in the Universe
One of the main tasks of cosmology is to measure the density of the Universe, and how this is<divided between dark matter and baryons. The baryons consist partly of stars, with 0.002 <∼ Ω∗ ∼
0.003 [77] but mainly inhabit the intergalactic medium (IGM). One powerful way in which this canbe studied is via the absorption of light from distant luminous objects such as quasars. Even verysmall amounts of neutral hydrogen can absorb rest-frame UV photons (the Gunn-Peterson effect),and should suppress the continuum by a factor exp(−τ ), where"
4.62 −1

τ ' 10hnHI (z)/m−3√(1 + z) 1 + Ωm z#,

(22.63)and this expression applies while the Universe is matter dominated (z >∼ 1 in the Ωm = 0.3 Ωv = 0.7
model). At z < 6, the dominant effect on quasar spectra is a ‘forest’ of narrow absorption lines,
which produce a mean τ = 1 in the Lyα forest at about z = 3, and so we have ΩHI ' 10−6.7 h−1 .
This is such a small number that the IGM must be very highly ionized at these redshifts, apartfrom a few high-density clumps. But at z > 6 there is good evidence for a ‘reionization’ era atwhich the general IGM is not so strongly ionized [78]. As discussed below, this ionized IGM at low
z is also detectable via the secondary Compton scattering of CMB photons.
The Lyα forest is of great importance in pinning down the abundance of deuterium. Because
electrons in deuterium differ in reduced mass by about 1 part in 4000 compared to hydrogen, eachabsorption system in the Lyα forest is accompanied by an offset deuterium line. By careful selectionof systems with an optimal HI column density, a measurement of the D/H ratio can be made. This
has now been done with high accuracy in 10 quasars, with consistent results [64]. Combining these
31st May, 2024

22. Big-Bang Cosmology

18

determinations with the theory of primordial nucleosynthesis yields a baryon density of Ωb h2 =0.021–0.023 (95% confidence) in excellent agreement with the Planck result. For more details, see
BBN – Sec. 24 of this Review.
Ionized IGM can also be detected in emission when it is densely clumped, via bremsstrahlung
radiation. This generates the spectacular X-ray emission from rich clusters of galaxies. Studiesof this phenomenon allow us to achieve an accounting of the total baryonic material in clusters.
Within the central ' 1 Mpc, the masses in stars, X-ray emitting gas, and total dark matter canbe determined with reasonable accuracy (perhaps 20% rms), and this allows a minimum baryon
fraction to be determined [79, 80]:
Mbaryons >0.009 + (0.066 ± 0.003) h−3/2 .
Mtotal ∼(22.64)

Because clusters are the largest collapsed structures, it is reasonable to take this as applying to
the Universe as a whole. This equation implies a minimum baryon fraction of perhaps 12% (forreasonable h), which is too high for Ωm = 1 if we take Ωb h2 ' 0.02 from nucleosynthesis. This istherefore one of the more robust arguments in favor of Ωm ' 0.3; for more details, see CosmologicalParameters – Sec. 25 of this Review. This argument is also consistent with the inference on Ωmthat can be made from Fig. 22.2.
This method is much more robust than the older classical technique for weighing the Universe:‘L × M/L’. The overall light density of the Universe is reasonably well determined from redshift
surveys of galaxies, so that a good determination of mass M and luminosity L for a single object
suffices to determine Ωm – but only if the mass-to-light ratio were universal.
22.4.3 Gravitational lensing
A robust method for determining masses in cosmology is to use gravitational light deflection.
Most systems can be treated as a geometrically thin gravitational lens, where the light bending
is assumed to take place only at a single distance. Simple geometry then determines a mapping
between the coordinates in the intrinsic source plane (S) and the observed image plane (I):α(DL θI ) =DS(θI − θS ),DLS(22.65)where the angles θI , θS , and α are in general two-dimensional vectors on the sky. The distances DLS
etc. are given by an extension of the usual distance-redshift formula:DLS =R0 Sk (χS − χL ).
1 + zS(22.66)This is the angular-diameter distance for objects on the source plane as perceived by an observeron the lens.
Solutions of this equation divide into weak lensing, where the mapping between source plane
and image plane is one-to-one, and strong lensing, in which multiple imaging is possible. For
circularly-symmetric lenses, an on-axis source is multiply imaged into a ‘caustic’ ring, whose radiusis the Einstein radius:DLSθE = 4GM
DL DS




=

M
1011.09 M

1/2

1/2 DL DS /DLS
Gpc

31st May, 2024

(22.67)−1/2arcsec .

19

22. Big-Bang Cosmology

The observation of ‘arcs’ (segments of near-perfect Einstein rings) in rich clusters of galaxies has
thus given very accurate masses for the central parts of clusters – generally in good agreement withother indicators, such as analysis of X-ray emission from the cluster IGM [81, 82].
Gravitational lensing has also developed into a particularly promising probe of cosmologicalstructure on 10-Mpc to 100-Mpc scales. Weak image distortions manifest themselves as an additional ellipticity of galaxy images (‘shear’), which can be observed by averaging many images
together (the corresponding flux amplification is less readily detected). The result is a ‘cosmic
shear’ field of order 1% ellipticity, coherent over scales of around 30 arcmin, which is directly related to the cosmic mass field. For this reason, weak lensing is seen as potentially the cleanest
probe of matter fluctuations, next to the CMB. Already, impressive results have been obtained inmeasuring cosmological parameters, based on survey data from only ∼ 103 deg2 [83, 84]. A particular strength of lensing is its ability to measure the amplitude of mass fluctuations; this can be
deduced from the amplitude of CMB fluctuations, but only with low precision on account of thepoorly-known optical depth due to Compton scattering after reionization. However, the effect of
weak lensing on the CMB map itself can be detected via the induced non-Gaussian signal, and thisgives the CMB greater internal power [85]. The main difficulty of principle with lensing is that partof the signal is generated by small-scale density fluctuations; thus a model is required for nonlinear
evolution, including astrophysical effects that separate baryons and dark matter. In this respect,
the CMB is a cleaner probe of the primordial fluctuations.
22.4.4 Density fluctuations
The overall properties of the Universe are very close to being homogeneous; and yet telescopes
reveal a wealth of detail on scales varying from single galaxies to large-scale structures of size
exceeding 100 Mpc. The existence of these structures must be telling us something important
about the initial conditions of the Big Bang, and about the physical processes that have operated
subsequently. This motivates the study of the density perturbation field, defined as
δ(x) ≡

ρ(x) − hρi.
hρi(22.68)A critical feature of the δ field is that it inhabits a Universe that is isotropic and homogeneous in its
large-scale properties. This suggests that the statistical properties of δ should also be statisticallyhomogeneous – i.e., it is a stationary random process.
It is often convenient to describe δ as a Fourier superposition:
δ(x) =

Xδk e−ik·x .

(22.69)We avoid difficulties with an infinite Universe by applying periodic boundary conditions in a cubeof some large volume V . The cross-terms vanish when we compute the variance in the field, which
is just a sum over modes of the power spectrum:hδ 2 i =X|δk |2 ≡X

P (k).

(22.70)Note that the statistical nature of the fluctuations must be isotropic, so we write P (k) rather than
P (k). The h. . . i average here is a volume average. Cosmological density fields are an example of anergodic process, in which the average over a large volume tends to the same answer as the averageover a statistical ensemble.
The statistical properties of discrete objects sampled from the density field are often describedin terms of N -point correlation functions, which represent the excess probability over random
for finding one particle in each of N boxes in a given configuration. For the 2-point case, the31st May, 2024

22. Big-Bang Cosmology

20

correlation function is readily shown to be identical to the autocorrelation function of the δ field:
ξ(r) = hδ(x)δ(x + r)i.
The power spectrum and correlation function are Fourier conjugates, and thus are equivalentdescriptions of the density field (similarly, k-space equivalents exist for the higher-order correlations). It is convenient to take the limit V → ∞ and use k-space integrals, defining a dimensionless
power spectrum, which measures the contribution to the fractional variance in density per unit
logarithmic range of scale, as ∆2 (k) = dhδ 2 i/d ln k = V k 3 P (k)/2π 2 :Zξ(r) =

sin kr∆ (k)d ln k;kr
2

2∆ (k) = k 3
π
2

Z ∞

ξ(r)
0

sin kr 2
r dr.
kr(22.71)

For many years, an adequate approximation to observational data on galaxies was ξ = (r/r0 )−γ ,with γ ' 1.8 and r0 ' 5 h−1 Mpc. Modern surveys are now able to probe into the large-scale linear
regime where unaltered traces of the curved post-recombination spectrum can be detected [86–88].
22.4.5 Formation of cosmological structure
The simplest model for the generation of cosmological structure is gravitational instability acting
on some low-amplitude initial fluctuations (for the origin of which a theory such as inflation isrequired). If the perturbations are adiabatic (i.e., fractionally perturb number densities of photons
and matter equally), the linear growth law for matter fluctuations is simple:(δ∝a2 (t)a(t)(radiation domination; Ωr = 1);(matter domination; Ωm = 1) .

(22.72)For low-density Universes, the growth is slower:d ln δ/d ln a ' Ωγm (a),(22.73)where the parameter γ is close to 0.55 independent of the vacuum density [89, 90].
The alternative perturbation mode is isocurvature: only the equation of state changes, and thetotal density is initially unperturbed. These modes perturb the total entropy density, and thus
induce additional large-scale CMB anisotropies [91]. Although the character of perturbations inthe simplest inflationary theories are purely adiabatic, correlated adiabatic and isocurvature modes
are predicted in many models; the simplest example is the curvaton, which is a scalar field thatdecays to yield a perturbed radiation density. If the matter content already exists at this time,
the overall perturbation field will have a significant isocurvature component. Such a prediction
is inconsistent with current CMB data [92], and most analyses of CMB and large-scale structure(LSS) data assume the adiabatic case to hold exactly.
Linear evolution preserves the shape of the power spectrum. However, a variety of processes
mean that growth actually depends on the matter content.
1. Pressure opposes gravity effectively for wavelengths below the horizon length while the Universe is radiation dominated. The comoving horizon size at zeq is therefore an importantscale:√2( 2 − 1)
16.0
DH (zeq ) ==Mpc .
(22.74)1/2Ωm h2(Ωm zeq ) H0
2. At early times, dark matter particles will undergo free streaming at the speed of light, andso erase all scales up to the horizon – a process that only ceases when the particles go
nonrelativistic. For light massive neutrinos, this happens at zeq ; all structure up to thehorizon-scale power-spectrum break is in fact erased. Hot(cold) dark matter models are thus
sometimes dubbed large(small)-scale damping models.
31st May, 2024

21

22. Big-Bang Cosmology
3. A further important scale arises where photon diffusion can erase perturbations in the matter– radiation fluid; this process is named Silk damping.

Figure 22.4: A plot of transfer functions for various models. For adiabatic models, Tk → 1
at small k, whereas the opposite is true for isocurvature models. For dark-matter models, thecharacteristic wavenumber scales proportional to Ωm h2 . The scaling for baryonic models does notobey this exactly; the plotted cases correspond to Ωm = 1, h = 0.5.
The overall effect is encapsulated in the transfer function, which gives the ratio of the late-time
amplitude of a mode to its initial value (see Fig. 22.4). The overall power spectrum is thus theprimordial scalar-mode power law, times the square of the transfer function:
P (k) ∝ k ns Tk2 .

(22.75)The most generic power-law index is ns = 1: the ‘Zeldovich’ or ‘scale-invariant’ spectrum. Inflationary models tend to predict a small ‘tilt:’ |ns − 1| <∼ 0.05 [12, 13]. On the assumption that the dark
matter is cold, the power spectrum then depends on five parameters: ns , h, Ωb , Ωc (≡ Ωm − Ωb ),and an overall amplitude. The latter is often specified as σ8 , the linear-theory fractional rms indensity when a spherical filter of radius 8 h−1 Mpc is applied in linear theory. This scale can beprobed directly via weak gravitational lensing, and also via its effect on the abundance of richgalaxy clusters. The normalization derived from samples of X-ray clusters has been given variously
as [93, 94]
σ8 = [0.746 ± 0.012(stat.) ± 0.022(sys.)](Ωm /0.3)−0.47 ;(22.76)
31st May, 2024

2222. Big-Bang Cosmology
σ8 = [0.81 ± 0.03](Ωm /0.3)−0.17 .

(22.77)

The higher figure is well consistent with Planck, whereas the lower is rather similar to the normalization inferred from weak lensing [95]:−0.5
σ8 ' [0.790+0.018.
−0.014 ](Ωm /0.3)(22.78)This figure is in 1.9σ tension with the Planck values of (σ8 , Ωm ) = (0.811 ± 0.006, 0.315 ± 0.007).
If real, such a discrepancy could indicate interesting new physics; but the current evidence is notstrong enough to make such a claim. However, it is worth noting that the CMB requires an almost
perfect degeneracy, Ωm h3 = constant for flat models, so that raising h as far as allowed by the CMB
would require a lower density, which would reduce the tension with the amplitude of the lensing
measurements.
A direct measure of mass inhomogeneity is valuable, since the galaxies inevitably are biased
with respect to the mass. This means that the fractional fluctuations in galaxy number, δn/n, maydiffer from the mass fluctuations, δρ/ρ. It is commonly assumed that the two fields obey some
proportionality on large scales where the fluctuations are small, δn/n = bδρ/ρ, but even this is notguaranteed [96].
The main shape of the transfer function is a break around the horizon scale at zeq , which depends
just on Ωm h when wavenumbers are measured in observable units (h Mpc−1 ). For reasonable baryon
content, weak oscillations in the transfer function are also expected, and these BAOs (baryon
acoustic oscillations) have been clearly detected [97, 98]. As well as directly measuring the baryon
fraction, the scale of the oscillations directly measures the acoustic horizon at decoupling; this canbe used as an additional standard ruler for cosmological tests, and the BAO signature has become
one of the most important applications of large galaxy surveys. Overall, current power-spectrum
data [86–88] favor Ωm h ' 0.20 and a baryon fraction of about 0.15 for ns ' 1 (see Fig. 22.5).
In principle, accurate data over a wide range of k could determine both Ωm h and n, but inpractice there is a strong degeneracy between these. In order to constrain ns itself, it is necessary
to examine data on anisotropies in the CMB.
22.4.6 CMB anisotropies
The CMB has a clear dipole anisotropy, of magnitude 1.23 × 10−3 . This is interpreted as being
due to the Earth’s motion, which is equivalent to a peculiar velocity for the Milky Way ofvMW ' 600 km s−1

towards(`, b) ' (270◦ , 30◦ ).

(22.79)A continuing challenge in cosmology is to demonstrate that this dipole is indeed kinematic, as
opposed to representing a violation of large-scale isotropy. Galaxy surveys have attempted to
identify the accelerating superclusters responsible for the motion [99], but there are also claims
that the observed matter distribution contains an intrinsic dipole, which would be incompatible
with the standard interpretation [100], but see also Ref. [101].
All higher-order multipole moments of the CMB are however much smaller than the dipole (oforder 10−5 ), and interpreted as signatures of density fluctuations at last scattering (' 1100). To
analyze these, the sky is expanded in spherical harmonics as explained in the review on the CMB– Sec. 29 of this Review. The dimensionless power per ln k or ‘bandpower’ for the CMB is definedas`(` + 1)T 2 (`) =C` .
(22.80)2πThis function encodes information from the three distinct mechanisms that cause CMB anisotropies:
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22. Big-Bang Cosmology

Figure 22.5: The galaxy power spectrum from the SDSS BOSS survey [88]. The solid points
with error bars show the power estimate. The solid line shows a standard ΛCDM model withΩb h2 ' 0.02 and Ωm h ' 0.2. The inset amplifies the region where BAO features are visible. The
fact that these perturb the power by ∼ 20% rather than order unity is direct evidence that thematter content of the Universe is dominated by collisionless dark matter.
• (1) Gravitational (Sachs – Wolfe) perturbations. Photons from high-density regions at lastscattering have to climb out of potential wells, and are thus redshifted.
• (2) Intrinsic (adiabatic) perturbations. In high-density regions, the coupling of matter andradiation can compress the radiation also, giving a higher temperature.
• (3) Velocity (Doppler) perturbations. The plasma has a non-zero velocity at recombination,
which leads to Doppler shifts in frequency and hence shifts in brightness temperature.
Because the potential fluctuations obey Poisson’s equation, ∇2 Φ/a2 = 4πGρ̄δ, and the comoving
velocity field satisfies the continuity equation ∇ · u = −δ̇, the resulting different powers of k ensurethat the Sachs-Wolfe effect dominates on large scales and adiabatic effects on small scales.
The relation between angle and comoving distance on the last-scattering sphere requires the
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22. Big-Bang Cosmology

comoving angular-diameter distance to the last-scattering sphere; because of its high redshift, thisis effectively identical to the horizon size at the present epoch, DH :
2
Ωm H0
2
DH ' 0.4
Ωm H0

DH =

(Ωv = 0);(22.81)(flat : Ωm + Ωv = 1) .

These relations show
√ how the CMB is strongly sensitive to curvature: the horizon length at lastscattering is ∝ 1/ Ωm , so that this subtends an angle that is virtually independent of Ωm for a flatmodel. Observations of a peak in the CMB power spectrum at relatively large scales (` ' 221) arethus strongly inconsistent with zero-Λ models with low density: current CMB + BAO + lensing
data require Ωm + Ωv = 0.999 ± 0.004 (95%) [24]. (See e.g., Fig. 22.2).
In addition to curvature, the CMB encodes information about several other key cosmologicalparameters. Within the compass of simple adiabatic CDM models, there are nine of these:ωc , ωb , Ωtot , h, τ, ns , nt , r, Q.

(22.82)The symbol ω denotes the physical density, Ωh2 : the transfer function depends only on the densitiesof CDM (ωc ) and baryons (ωb ). Transcribing the power spectrum at last scattering into an angularpower spectrum brings in the total density parameter (Ωtot ≡ Ωm +Ωv = Ωc +Ωb +Ωv ) and h: there
is a near-exact geometrical degeneracy [102] between these that keeps the angular-diameter distanceto last scattering invariant, so that models with substantial spatial curvature and large vacuum
energy cannot be ruled out without prior knowledge of the Hubble parameter. Alternatively, the
CMB alone cannot measure the Hubble parameter without taking into account the line-of-sightinformation from CMB lensing.
A further possible degeneracy involves the tensor contribution to the CMB anisotropies. These
are important at large scales (up to the horizon scales); for smaller scales, only scalar fluctuations(density perturbations) are important. Each of these components is characterized by a spectral
index, n, and a ratio between the power spectra of tensors and scalars (r). See CosmologicalParameters – Sec. 25 of this Review for a technical definition of the r parameter. Finally, the overall
amplitude of the spectrum must be specified (Q), together with the optical depth to Compton
scattering owing to recent reionization (τ ). Adding a large tensor contribution reduces the contrast
between low ` and the peak at ` ' 221 (because the tensor spectrum has no acoustic component).
The previous relative height of the peak can be recovered by increasing ns to increase the small-scale
power in the scalar component; this in turn over-predicts the power at ` ∼ 1000, but this effect canbe counteracted by raising the baryon density [103]. This approximate 3-way degeneracy is broken
as we increase the range of multipoles sampled.
The reason the tensor component is introduced, and why it is so important, is that it is the
only non-generic prediction of inflation. Slow-roll models of inflation involve two dimensionlessparameters: M2 V 0 2
M 2 V 00
≡ P
;
η≡ P
,
(22.83)16π V
8πVwhere V is the inflaton potential, and dashes denote derivatives with respect to the inflation field.
In terms of these, the tensor-to-scalar ratio is r ' 16, and the spectral indices are ns = 1 − 6 + 2ηand nt = −2. The natural expectation of inflation is that the quasi-exponential phase ends oncethe magnitudes of the slow-roll parameters become of order unity, so that both ns 6= 1 and asignificant tensor component are expected. These predictions can be avoided in some models, but
it is undeniable that observation of such features would be a great triumph for inflation. Cosmology
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22. Big-Bang Cosmology

therefore stands at a fascinating point, given that the most recent CMB data reject the zero-tensor
ns = 1 model at more than 8σ: ns = 0.965 ± 0.004 [24]. This rejection is strong enough that it isalso able to break the tensor degeneracy, so that no model with ns = 1 is acceptable, whatever thevalue of r.
The current limit on r is < 0.036 at 95% confidence [104,105]. In conjunction with the measured
value of ns , this upper limit sits close to the prediction of a linear potential (i.e. |η|  ||). Any
further reduction in the limit on r will force η to be negative – i.e. a convex potential at the point
where LSS scales were generated (sometimes called a ‘hilltop’), in contrast to simple early models
such as V (φ) = m2 φ2 or λφ4 , which are now excluded. Examples of models that are currently inexcellent agreement with the Planck results are the Starobinsky model of R + R2 gravity [106],or the Higgs-inflation model where the Higgs field is non-minimally coupled [107]. Assuming 55 efoldings of inflation, these models predict ns = 0.965 and r = 0.0035. Assuming that no systematic
error in the CMB data can be identified, cosmology has thus passed a critical hurdle in rejecting
scale-invariant fluctuations. The years ahead will be devoted to the task of searching for the tensor
fluctuations – for which the main tool will be the polarization of the CMB [14].
22.4.6.1 CMB foregrounds
As the quality of CMB data improves, there is a growing interest in effects that arise along
the line of sight. The CMB temperature is perturbed by dark-matter structures and by Compton
scattering from ionized gas. In the former case, we have the integrated Sachs-Wolfe effect, which
is sensitive to the time derivative of the gravitational potential. In the linear regime, this isdamped when the Universe becomes Λ-dominated, and this is an independent way of detecting
Λ [108]. The potential also causes gravitational lensing of the CMB: structures at z ' 1–2 displace
features on the CMB sky by about 2 arcmin over coherent degree-scale patches. Detection ofthese distortions allows a map to be made of overdensity projected from z = 0 to 1100 [85]. Thisis a very powerful calibration for direct studies of gravitational lensing using galaxies. Finally,
Comptonization affects the CMB in two ways: the thermal Sunyaev-Zeldovich effect measures theblurring of photon energies by hot gas; and the kinetic Sunyaev-Zeldovich effect is sensitive to thebulk velocity of the gas. Both these effects start to dominate over the intrinsic CMB fluctuations
at multipoles ` & 2000 [109].
22.4.7 Probing dark energy and the nature of gravity
The most radical element of our current cosmological model is the dark energy that accelerates
the expansion. The energy density of this component is approximately (2.2 meV)4 (for w = −1,Ωv = 0.68, h = 0.67), or roughly 10−123 MP4 , and such an unnaturally small number is hard tounderstand. Various quantum effects (most simply, zero-point energy) should make contributions
to the vacuum energy density. These may be truncated by new physics at high energy, but thispresumably occurs at > 1 TeV scales, not meV; thus the apparent energy scale of the vacuum is
at least 1015 times smaller than its natural value. A classic review of this situation is given by
Weinberg [53], which lists extreme escape routes – especially the multiverse viewpoint, according
to which low values of Λ are rare, but high values suppress the formation of structure and observers.
It is certainly impressive that Weinberg used such reasoning to predict the value of Λ before any
data strongly indicated a non-zero value.
But it may be that the phenomenon of dark energy is entirely illusory. The necessity forthis constituent arises from using the Friedmann equation to describe the evolution of the cosmic
expansion; if this equation is incorrect, it would require the replacement of Einstein’s relativistic
theory of gravity with some new alternative. A frontier of current cosmological research is todistinguish these possibilities [110, 111]. We also note that it has been suggested that dark energy
might be an illusion even within general relativity, owing to an incorrect treatment of averaging in
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an inhomogeneous Universe [112, 113]. Most would argue that a standard Newtonian treatment ofsuch issues should be adequate inside the cosmological horizon, but debate on this issue continues.
Dark energy can differ from a classical cosmological constant in being a dynamical phenomenon[114, 115], e.g., a rolling scalar field (sometimes dubbed ‘quintessence’). Empirically, this meansthat it is endowed with two thermodynamic properties that astronomers can try to measure: thebulk equation of state, and the sound speed. If the sound speed is close to the speed of light, theeffect of this property is confined to very large scales, and mainly manifests itself in the large-angle
multipoles of the CMB anisotropies [116]. The equation of state parameter governs the rate ofchange of the vacuum density: d ln ρv /d ln a = −3(1 + w), so it can be accessed via the evolving
expansion rate, H(a). This can be measured most cleanly by using the inbuilt natural ruler oflarge-scale structure: the BAO horizon scale [117]:
DBAO ' 147.0 (Ωm h2 /0.1424)−0.253 (Ωb h2 /0.0224)−0.128 Mpc.

(22.84)

Observing the equivalent acoustic scale at last scattering in the CMB yields the precise geometricaldegeneracy Ωm h3 = 0.09633 ± 0.00029 [24].
H(a) is measured by radial clustering, since dr/dz = c/H; clustering in the plane of the sky
measures the integral of this. The expansion rate is also measured by the growth of density
fluctuations, where the pressure-free growth equation for the density perturbation is δ̈ + 2H(a)δ̇ =
4πGρ0 δ. Thus, both the scale and amplitude of density fluctuations are sensitive to w(a) – but
only weakly. These observables typically change by only ∼ 0.2% for a 1% change in w. Current
constraints [24] place a constant w to within 5–10% of −1, depending on the data combination
chosen. A substantial improvement in this precision will require us to limit systematics in data toa few parts in 1000.
Testing whether theories of gravity require revision can also be done using data on cosmologicalinhomogeneities. Two separate issues arise, concerning the metric perturbation potentials Ψ andΦ, which affect respectively the time and space parts of the metric. In Einstein gravity, these potentials are both equal to the Newtonian gravitational potential, which satisfies Poisson’s equation:∇2 Φ/a2 = 4πGρ̄δ. Empirically, modifications of gravity require us to explore a change with scale
and with time of the ‘slip’ (Ψ/Φ) and the effective G on the rhs of the Poisson equation. The
former aspect can only be probed via gravitational lensing, whereas the latter can be addressedon 10–100 Mpc scales via the growth of clustering. Various schemes for parameterizing modified
gravity exist, but a practical approach is to assume that the growth rate can be tied to the density
parameter: d ln δ/d ln a = Ωγm (a) [89, 90]. The parameter γ is close to 0.55 for standard relativistic gravity, but can differ by around 0.1 from this value in many non-standard models. Clearly
this parameterization is incomplete, since it explicitly rejects the possibility of early dark energy(Ωm (a) → 1 as a → 0), but it is a convenient way of capturing the power of various experiments.
Current data are consistent with standard ΛCDM [118], and exclude variations in slip or effective
G of larger than a few times 10%.
Current planning envisages a set of satellite probes that, a decade hence, will have pursued
these fundamental tests via gravitational lensing measurements over thousands of square degrees,> 108 redshifts, and photometry of > 1000 supernovae (Euclid in Europe, Roman Space Telescopein the USA) [22, 23]. These experiments will measure both w and the perturbation growth rate
to an accuracy of around 1%. The outcome will be either a validation of the standard relativistic
vacuum-dominated Big Bang cosmology at a level of precision far beyond anything attempted
to date, or the opening of entirely new directions in cosmological models. For a more complete
discussion of dark energy and future probes see Dark Energy – Sec. 28 of this Review.
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31st May, 2024