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## 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 the cosmology 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 to extend 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 and his collaborators, Ralph Alpher and Robert Herman. In order to account for the possibility that the 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 later as the cosmic microwave background (CMB) [8]. Indeed, it was the observation of this radiation that 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, to which the best modern solution is inflationary cosmology, discussed in Sec. 22.3.5 and in – Sec. 23 of 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 initial conditions.

#### 22.1.1 The Robertson-Walker Universe

The observed homogeneity and isotropy enable us to describe the overall geometry and evolution of the Universe in terms of two cosmological parameters accounting for the spatial curvature and the overall expansion (or contraction) of the Universe. These two quantities appear in the most general expression for a space-time metric that has a (3D) maximally symmetric subspace of a 4D space-time, known as the Robertson-Walker metric:

$$ds^2 = dt^2 - R^2(t) \left[ \frac{dr^2}{1 - kr^2} + r^2 (d\theta^2 + \sin^2\theta d\phi^2) \right]. \quad (22.1)$$

Note that we adopt $c = 1$ throughout. By rescaling the radial coordinate, we can choose the curvature 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 as

$$ds^2 = dt^2 - R^2(t) [d\chi^2 + S_k^2(\chi)(d\theta^2 + \sin^2\theta d\phi^2)], \quad (22.2)$$

where the function $S_k(\chi)$ is $(\sin\chi, \chi, \sinh\chi)$ for $k = (+1, 0, -1)$. The coordinate $r$ [in Eq. (22.1)] and the ‘angle’ $\chi$ [in Eq. (22.2)] are both dimensionless; the dimensions are carried by the cosmological scale factor, $R(t)$, which determines proper distances in terms of the comoving coordinates. A common alternative is to define a dimensionless scale factor, $a(t) = R(t)/R_0$, where $R_0 \equiv R(t_0)$ is $R$ at the present epoch. It is also sometimes convenient to define a dimensionless or conformal time coordinate, $\eta$, by $d\eta = dt/R(t)$. Along constant spatial sections, the proper time is defined by the time coordinate, $t$. Similarly, for $dt = d\theta = d\phi = 0$, the proper distance is given by $R(t)\chi$. 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

$$z \equiv \frac{\nu_1 - \nu_2}{\nu_2} \approx v_{12}, \quad (22.3)$$

where $\nu_1$ is the frequency of the emitted light, $\nu_2$ is the observed frequency, and $v_{12}$ is the relative velocity between the emitter and the observer. While the definition, $z = (\nu_1 - \nu_2)/\nu_2$ is valid in general, relating the redshift to a simple relative velocity is only correct on small scales (i.e., less than cosmological scales) such that the expansion velocity is non-relativistic. For light signals, we can use the metric given by Eq. (22.1) and $ds^2 = 0$ to write

$$v_{12} = \dot{R} \delta r = \frac{\dot{R}}{R} \delta t = \frac{\delta R}{R} = \frac{R_2 - R_1}{R_1}, \quad (22.4)$$

where $\delta r(\delta t)$ is the radial coordinate (temporal) separation between the emitter and observer. Noting that physical distance, $D$, is $R\delta r$ or $\delta t$, Eq. (22.4) gives us Hubble’s law, $v = HD$. In addition, we obtain a simple relation between the redshift and the scale factor:

$$1 + z = \frac{\nu_1}{\nu_2} = \frac{R_2}{R_1}. \quad (22.5)$$

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:

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = 8\pi G_N T_{\mu\nu} + \Lambda g_{\mu\nu}. \quad (22.6)$$

Gliner [17] and Zeldovich [18] pioneered the modern view, in which the $\Lambda$ term is set on the rhs and interpreted as an effective energy-momentum tensor $T_{\mu\nu}$ for the vacuum of $\Lambda g_{\mu\nu}/8\pi G_N$. It is common to assume that the matter content of the Universe is a perfect fluid, for which

$$T_{\mu\nu} = -pg_{\mu\nu} + (p + \rho) u_{\mu} u_{\nu}, \quad (22.7)$$

where $g_{\mu\nu}$ is the space-time metric described by Eq. (22.1), $p$ is the isotropic pressure, $\rho$ 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:

$$H^2 = \left(\frac{\dot{R}}{R}\right)^2 = \frac{8\pi G_N \rho}{3} - \frac{k}{R^2} + \frac{\Lambda}{3}, \quad (22.8)$$

and

$$\frac{\dot{R}}{R} = \frac{\Lambda}{3} - \frac{4\pi G_N}{3} (\rho + 3p), \quad (22.9)$$

where $H(t)$ is the Hubble parameter and $\Lambda$ is the cosmological constant. The first of these is sometimes called the Friedmann equation. Energy conservation via $T^{\mu\nu} = 0$, leads to a third useful equation [which can also be derived from Eq. (22.8) and Eq. (22.9)]:

$$\dot{\rho} = -3H(\rho + p). \quad (22.10)$$

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) the cosmological term $\Lambda$. By interpreting $-k/R^2$ Newtonianly as a ‘total energy’, then we see that the evolution of the Universe is governed by a competition between the potential energy, $8\pi G_N\rho/3$, and the kinetic term $(\dot{R}/R)^2$. For $\Lambda = 0$, 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 $\Lambda \neq 0$ or more generally with some component with $(\rho + 3p) < 0$.
### 22.1.4 Definition of cosmological parameters

In addition to the Hubble parameter, it is useful to define several other measurable cosmological parameters. The Friedmann equation can be used to define a critical density such that $k = 0$ when $\Lambda = 0$:

$$ \begin{aligned} \rho_c \equiv \frac{3H^2}{8\pi G_N} &= 1.88 \times 10^{-26} h^2 \text{ kg m}^{-3} \\ &= 1.05 \times 10^{-5} h^2 \text{ GeV cm}^{-3}, \end{aligned} \qquad (22.11) $$

where the scaled Hubble parameter, $h$, is defined by

$$ \begin{aligned} H &\equiv 100 h \text{ km s}^{-1} \text{ Mpc}^{-1} \\ \Rightarrow H^{-1} &= 9.778 h^{-1} \text{ Gyr} \\ &= 2998 h^{-1} \text{ Mpc.} \end{aligned} \qquad (22.12) $$

The cosmological density parameter $\Omega_{tot}$ is defined as the energy density relative to the critical density:

$$ \Omega_{tot} = \rho/\rho_c. \qquad (22.13) $$

Note that one can now rewrite the Friedmann equation as

$$ k/R^2 = H^2(\Omega_{tot} - 1). \qquad (22.14) $$

From Eq. (22.14), one can see that when $\Omega_{tot} > 1$, $k = +1$ and the Universe is closed, when $\Omega_{tot} < 1$, $k = -1$ and the Universe is open, and when $\Omega_{tot} = 1$, $k = 0$, and the Universe is spatially flat.

It is often necessary to distinguish different contributions to the density. It is therefore convenient to define present-day density parameters for pressureless matter ($\Omega_m$) and relativistic particles ($\Omega_r$), plus the quantity $\Omega_\Lambda = \Lambda/3H^2$. 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 $\Omega_v$. The Friedmann equation then becomes

$$ k/R_0^2 = H_0^2(\Omega_m + \Omega_r + \Omega_v - 1), \qquad (22.15) $$

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 the quantity $-k/R_0^2H_0^2$ is sometimes referred to as $\Omega_K$. 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.

### 22.1.5 Standard Model solutions

Much of the history of the Universe in the standard Big-Bang model can be easily described by assuming that either matter or radiation dominates the total energy density. During inflation and again 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 $w = p/\rho$. 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 is constant. In this case, Eq. (22.10) can be written as $\dot{\rho} = -3(1+w)\rho\dot{R}/R$ and is easily integrated to yield

$$ \rho \propto R^{-3(1+w)}. \qquad (22.16) $$

Note that at early times when $R$ is small, the less singular curvature term $k/R^2$ in the Friedmann equation 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 $w \neq -1$, one can insert this result into the Friedmann equation Eq. (22.8), and if one neglects the curvature and cosmological constant terms, it is easy to integrate the equation to obtain

$$ R(t) \propto t^{2/[3(1+w)]}. \qquad (22.17) $$

#### 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 $\rho \propto R^{-4}$. The ‘extra’ factor of $1/R$ is due to the cosmological redshift; not only is the number density of particles in the radiation background decreasing as $R^{-3}$ since volume scales as $R^3$, but in addition each particle’s energy is decreasing as $E \propto \nu \propto R^{-1}$. Similarly, one can substitute $w = 1/3$ into Eq. (22.17) to obtain

$$ R(t) \propto t^{1/2}; \qquad H = 1/2t. \qquad (22.18) $$

#### 22.1.5.3 A Matter-dominated Universe

At relatively late times, non-relativistic matter eventually dominates the energy density over radiation [see Eq. (22.3.8)]. A presureless gas ($w = 0$) leads to the expected dependence $\rho \propto R^{-3}$ from Eq. (22.16) and, if $k = 0$, we obtain

$$ R(t) \propto t^{2/3}; \qquad H = 2/3t. \qquad (22.19) $$

#### 22.1.5.4 A Universe dominated by vacuum energy

If there is a dominant source of vacuum energy, $V_0$, it would act as a cosmological constant with $\Lambda = 8\pi G_N V_0$ 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:

$$ R(t) \propto e^{\sqrt{\Lambda}/3t}. \qquad (22.20) $$

More generally we could write

$$ a(t) = \sinh^{2/3}(\sqrt{3\Lambda}t/2), \qquad (22.21) $$

which describes a flat Universe containing both matter and vacuum energy, with $a(t)$ being the scale factor normalized to unity when both components are equal.

A key parameter is the equation of state of the vacuum, $w \equiv p/\rho$; this need not be the $w = -1$ of $\Lambda$, and may not even be constant [19–21]. There is much interest in the more general possibility of 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 be measured, 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 $w = -1.028 \pm 0.031$ [24]. Unless stated otherwise, we will assume that the vacuum energy is a cosmological constant with $w = -1$ exactly.

The presence of vacuum energy can dramatically alter the fate of the Universe. For example, if $\Lambda < 0$, the Universe will eventually recollapse independent of the sign of $k$. For large values of $\Lambda > 0$ (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, $q_0$, defined as

$$ q_0 = - \left. \frac{R \dot{R}}{\dot{R}^2} \right|_0 = \frac{1}{2} \Omega_m + \Omega_r + \frac{(1+3w)}{2} \Omega_v. \qquad (22.22) $$

This 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 indicate that vacuum energy is indeed the largest contributor to the cosmological density budget, with $\Omega_v = 0.685 \pm 0.007$ and $\Omega_m = 0.315 \pm 0.007$ 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 vacuum-energy problem is being addressed. For more details, see Dark Energy – Sec. 28 of this Review.
<figure><img src="image_128.png" /><figcaption>**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 $z \ll 1$ 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 ($\Omega_r = 0$ is assumed). Larger points with errors show median values in redshift bins. Comparison with the prediction of Friedmann models favors a vacuum-dominated Universe.</figcaption></figure>

## 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 $d\psi$ is its comoving size $d\psi S_k(\chi)$ times the scale factor at the time of emission:

$$d\ell = d\psi R_0 S_k(\chi)/(1+z). \qquad (22.23)$$

(2) The apparent flux density of an object is deduced by allowing its photons to flow through a sphere of current radius $R_0 S_k(\chi)$; but photon energies and arrival rates are redshifted, and the bandwidth $d\nu$ is reduced. The observed photons at frequency $\nu_0$ were emitted at frequency $\nu_0(1+z)$, so the flux density is the luminosity at this frequency, divided by the total area, divided by $1+z$:

$$S_{\nu}(\nu_0) = \frac{L_{\nu}([1+z]\nu_0)}{4\pi R_0^2 S_k^2(\chi)(1+z)}. \qquad (22.24)$$

These relations lead to the following common definitions:

$$\text{angular-diameter distance} \quad D_A = (1+z)^{-1} R_0 S_k(\chi); \qquad (22.25)$$

$$\text{luminosity distance} \quad D_L = (1+z) R_0 S_k(\chi).$$

These distance-redshift relations are expressed in terms of observables by using the equation of a null radial geodesic ($R(t)d\chi = dt$) plus the Friedmann equation:

$$R_0 d\chi = \frac{1}{H(z)} dz = \frac{1}{H_0} \left[ (1 - \Omega_m - \Omega_v - \Omega_r)(1+z)^2 + \Omega_v(1+z)^{3+3w} + \Omega_m(1+z)^3 + \Omega_r(1+z)^4 \right]^{-1/2} dz. \qquad (22.26)$$

The main scale for the distance here is the Hubble length, $1/H_0$. The flux density is the product of the specific intensity $I_\nu$ and the solid angle $d\Omega$ subtended by the source: $S_\nu = I_\nu d\Omega$. Combining the angular size and flux-density relations thus gives the relativistic version of surface-brightness conservation:

$$I_{\nu}(\nu_0) = \frac{B_{\nu}([1+z]\nu_0)}{(1+z)^3}, \qquad (22.27)$$

where $B_\nu$ is surface brightness (luminosity emitted into unit solid angle per unit area of source). We can integrate over $\nu_0$ to obtain the corresponding total or bolometric formula:

$$I_{\text{tot}} = \frac{B_{\text{tot}}}{(1+z)^4}. \qquad (22.28)$$

This cosmology-independent form expresses Liouville’s Theorem: photon phase-space density is conserved 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 magnitude of several hundred km s⁻¹. On the other hand, objects at redshifts $z \gtrsim 0.01$ will have observed recessional velocities that differ from their ideal values by $\lesssim 10\%$, but absolute distances are much harder 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 terminates with 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 distance ladder to the point where deviations from uniform expansion are negligible, leading to the best existing Cepheid-based value for $H_0$: (73.0 ± 1.0) km s⁻¹Mpc⁻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 a $k=0$ model dominated by vacuum energy. It is worth noting that there is some tension (4.2σ) between the Cepheid and CMB determinations of $H_0$ (the latter is (67.4±0.5) km s⁻¹Mpc⁻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 do note that a recent analysis of SNe Ia with a calibration of the tip of the red-giant branch gives a result close to that of the CMB: (69.8 ± 0.6 (stat.) ± 1.6 (sys.)) km s⁻¹Mpc⁻¹ [32, 33]. See Cosmological Parameters – Sec. 25 of this Review for a more comprehensive review of Hubble parameter determinations.

### 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
<figure><img src="image_129.png" /><figcaption>**Figure 22.2:** Likelihood-based probability densities over the plane $\Omega_{\Lambda}$ (i.e., $\Omega_{v}$ assuming $w = -1$) versus $\Omega_{m}$. The colored locus derives from Planck [34] and shows that the CMB alone requires a flat Universe $\Omega_{v} + \Omega_{m} \simeq 1$ if the Hubble constant is not too high. The SNe Ia results [35] very nearly constrain the orthogonal combination $\Omega_{v} - \Omega_{m}$, and the intersection of these constraints directly favors a flat model with $\Omega_{m} \simeq 0.3$, 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 of breaking the degeneracy with $H_{0}$ by using the measurements of gravitational lensing that can be made with modern high-resolution CMB data.</figcaption></figure>

the age of the Universe may be written as

$$ H_0 t_0 = \int_0^\infty \frac{dz}{(1+z)H(z)} \\ = \int_0^\infty \frac{dz}{(1+z) [(1+z)^2(1+\Omega_m z) - z(2+z)\Omega_v]^{1/2}}, \quad (22.29) $$

where we have neglected $\Omega_{r}$ and chosen $w = -1$. Over the range of interest ($0.1 \lesssim \Omega_{m} \lesssim 1$, $|\Omega_{v}| \lesssim 1$), this exact answer may be approximated to a few per cent accuracy by

$$ H_0 t_0 \approx \frac{2}{3} (0.7\Omega_m + 0.3 - 0.3\Omega_v)^{-0.3}. \quad (22.30) $$

For the special case that $\Omega_{m} + \Omega_{v} = 1$, the integral in Eq. (22.29) can be expressed analytically as

$$ H_0 t_0 = \frac{2}{3\sqrt{\Omega_v}} \ln \frac{1 + \sqrt{\Omega_v}}{\sqrt{1 - \Omega_v}} \quad (\Omega_m < 1). \quad (22.31) $$

The most accurate means of obtaining ages for astronomical objects is based on the natural clocks provided by radioactive decay. The use of these clocks is complicated by a lack of knowledge of the 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 to attempt 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 of stellar 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 theoretical uncertainties 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: $t_0 = (13.80 \pm 0.02)$ Gyr [24], where

the extra accuracy comes at the price of assuming the simple 6-parameter $\Lambda$CDM model to be true.

### 22.2.4 Horizon, isotropy, flatness problems

For photons, the radial equation of motion is just $cdt = R d\chi$. How far can a photon get in a given time? The answer is clearly

$$ \Delta\chi = \int_{t_1}^{t_2} \frac{dt}{R(t)} \equiv \Delta\eta, \quad (22.32) $$

i.e., just the interval of conformal time. We can replace $dt$ by $dR/\dot{R}$, which the Friedmann equation says is $\propto dR/\sqrt{\rho R^2}$ at early times. Thus, this integral converges if $\rho R^2 \to \infty$ as $t_1 \to 0$, otherwise it diverges. Provided the equation of state is such that $\rho$ changes faster than $R^{-2}$, light signals can only propagate a finite distance between the Big Bang and the present; there is then said to be a particle horizon. Such a horizon therefore exists in conventional Big-Bang models, which are dominated by radiation ($\rho \propto R^{-4}$) at early times.

At late times, the integral for the horizon is largely determined by the matter-dominated phase, for which

$$ D_H = R_0 \chi_H \equiv R_0 \int_0^{t(z)} \frac{dt}{R(t)} \approx \frac{6000}{\sqrt{\Omega_m z}} h^{-1} \text{Mpc} \quad (z \gg 1). \quad (22.33) $$

The horizon at the time of formation of the microwave background ('last scattering': $z \simeq 1100$) was thus of order 100 Mpc in size, subtending an angle of about $1^{\circ}$. Why then are the large number of 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 $\Omega = 1$ Universe is unstable:

$$ \Omega(a) - 1 = \frac{\Omega - 1}{1 - \Omega + \Omega_v a^2 + \Omega_m a^{-1} + \Omega_r a^{-2}}, \quad (22.34) $$

where $\Omega$ with no subscript is the total density parameter, and $a(t) = R(t)/R_0$. This requires $\Omega(t)$ 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.

## 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 radiation-dominated 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 rate of the Universe (see Sec. 22.3.3 below). In equilibrium, it is straightforward to compute the thermodynamic quantities, $\rho$, $p$, and the entropy density, $s$. In general, the energy density for a given particle type $i$ can be written as

$$ \rho_i = \int E_i \, dn_{q_i}, \quad (22.35) $$

with the density of states given by

$$ dn_{q_i} = \frac{g_i}{2\pi^2} \left( \exp\left[ \frac{(E_{q_i} - \mu_i)}{T_i} \right] \pm 1 \right)^{-1} q_i^2 dq_i, \quad (22.36) $$

where $g_i$ counts the number of degrees of freedom for particle type $i$, $E_{q_i}^2 = m_i^2 + q_i^2$, $\mu_i$ is the chemical potential, and the $\pm$ corresponds to either Fermi or Bose statistics. Similarly, we can define the pressure of a perfect gas as

$$ p_i = \frac{1}{3} \int \frac{q_i^2}{E_i} dn_{q_i}. \quad (22.37) $$

The number density of species $i$ is simply

$$ n_i = \int dn_{q_i}, \quad (22.38) $$
and the entropy density is

$$s_i = \frac{\rho_i + p_i - \mu_i n_i}{T_i}. \qquad (22.39)$$

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 $g_i = 2$ for the 2 photon polarization states, we have (in units where $\hbar = k_B = 1$)

$$\rho_{\gamma} = \frac{\pi^2}{15} T^4, \quad p_{\gamma} = \frac{1}{3} \rho_{\gamma}, \quad s_{\gamma} = \frac{4\rho_{\gamma}}{3T}, \quad n_{\gamma} = \frac{2\zeta(3)}{\pi^2} T^3, \qquad (22.40)$$

with $2\zeta(3)/\pi^2 \simeq 0.2436$. Note that Eq. (22.10) can be converted into an equation for entropy conservation. Recognizing that $\dot{p} = s\dot{T}$, Eq. (22.10) becomes

$$d(sR^3)/dt = 0. \qquad (22.41)$$

For radiation, this corresponds to the relationship between expansion and cooling, $T \propto R^{-1}$ in an adiabatically expanding Universe. Note also that both $s$ and $n_\gamma$ 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 $T \gg m_i$, then we can neglect the mass in Eq. (22.35) to Eq. (22.39), and the thermodynamic quantities are easily computed as in Eq. (22.40). In general, we can approximate the energy density (at high temperatures) by including only those particles with $m_i \ll T$. In this case, we have

$$\rho = \left( \sum_B g_B + \frac{7}{8} \sum_F g_F \right) \frac{\pi^2}{30} T^4 = \frac{\pi^2}{30} N(T) T^4, \qquad (22.42)$$

where $g_{B(F)}$ is the number of degrees of freedom of each boson (fermion) and the sum runs over all boson and fermion states with $m \ll T$. 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 $\rho$ and the coefficient of $T^3$ for the entropy density $s = (2\pi^2/45)N_s(T)T^3$ [40], as seen in the figure.

The value of $N(T)$ at any given temperature depends on the particle physics model. In the standard 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.

<table><thead><tr><th>Temperature</th><th>New Particles</th><th>4N(T)</th></tr></thead><tbody><tr><td>T &lt; m<sub>e</sub></td><td>γ's + ν's</td><td>29</td></tr><tr><td>m<sub>e</sub> &lt; T &lt; m<sub>μ</sub></td><td>e<sup>±</sup></td><td>43</td></tr><tr><td>m<sub>μ</sub> &lt; T &lt; m<sub>π</sub></td><td>μ<sup>±</sup></td><td>57</td></tr><tr><td>m<sub>π</sub> &lt; T &lt; T<sub>c</sub><sup>†</sup></td><td>π's</td><td>69</td></tr><tr><td>T<sub>c</sub> &lt; T &lt; m<sub>strange</sub></td><td>π's + u, ū, d, ¯d + gluons</td><td>205</td></tr><tr><td>m<sub>s</sub> &lt; T &lt; m<sub>charm</sub></td><td>s, ¯s</td><td>247</td></tr><tr><td>m<sub>c</sub> &lt; T &lt; m<sub>τ</sub></td><td>c, ¯c</td><td>289</td></tr><tr><td>m<sub>τ</sub> &lt; T &lt; m<sub>bottom</sub></td><td>τ<sup>±</sup></td><td>303</td></tr><tr><td>m<sub>b</sub> &lt; T &lt; m<sub>W,Z</sub></td><td>b, ¯b</td><td>345</td></tr><tr><td>m<sub>W,Z</sub> &lt; T &lt; m<sub>Higgs</sub></td><td>W<sup>±</sup>, Z</td><td>381</td></tr><tr><td>m<sub>H</sub> &lt; T &lt; m<sub>top</sub></td><td>H<sup>0</sup></td><td>385</td></tr><tr><td>m<sub>t</sub> &lt; T</td><td>t, ¯t</td><td>427</td></tr></tbody></table>

†$T_c$ corresponds to the confinement-deconfinement transition between quarks and hadrons.

At 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,

<figure><img src="image_130.png" /><figcaption>**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.</figcaption></figure>

another 24 from the adjoint Higgs, and another six scalar degrees of freedom (in addition to the four associated with the complex Higgs doublet already counted in the longitudinal components of $W^\pm$ and $Z$, and in $H$) from the $\bar{5}$ of Higgs. Hence for $T > m_X$ 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 $T$-dependence of $N$), giving us a relationship between the age of the Universe and its temperature:

$$t = \left( \frac{90}{32\pi^3 G_N N(T)} \right)^{1/2} T^{-2}. \qquad (22.43)$$

This can be expressed more conveniently in the form

$$t T_{\text{MeV}}^2 = 2.4[N(T)]^{-1/2}, \qquad (22.44)$$

where $t$ is measured in seconds and $T_{\text{MeV}}$ 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 $\Gamma_i$ involving that type of particle should be larger than the expansion rate of the Universe:

$$\Gamma_i > H. \qquad (22.45)$$

Recalling that the age of the Universe is determined by $H^{-1}$, this condition is equivalent to requiring that 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 of neutrinos. On dimensional grounds, one can estimate the thermally averaged scattering cross-section:

$$\langle \sigma v \rangle \sim O(10^{-2}) T^2 / m_W^4 \qquad (22.46)$$

for $T \lesssim m_W$. Recalling that the number density of leptons is $n \propto T^3$, we can compare the weak interaction rate, $\Gamma_{\text{wk}} \sim n\langle \sigma v \rangle$, with the expansion rate and so obtain

$$H = \left( \frac{8\pi G_N \rho}{3} \right)^{1/2} = \left( \frac{8\pi^3 N(T)}{90} \right)^{1/2} T^2 / M_P \qquad (22.47) \\ \approx 1.66 N(T)^{1/2} T^2 / M_P,$$

where the Planck mass $M_P = G_N^{-1/2} = 1.22 \times 10^{19}$ GeV.

Neutrinos will be in equilibrium when $\Gamma_{\text{wk}} > H$ or

$$T > (500 m_W^4 / M_P)^{1/3} \sim 1 \text{ MeV}. \qquad (22.48)$$
However, this condition assumes $T \ll m_W$; for higher temperatures, we should write $\langle \sigma v \rangle \sim O(10^{-2})/T^2$, so that $\Gamma \sim 10^{-2}T$. Thus, in the very early stages of expansion, at temperatures $T \gtrsim 10^{-2}M_P/\sqrt{N}$, 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 is defined by $\Gamma_{\text{wk}}(T_d) = H(T_d)$. For $T < T_d$, 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 $T \propto 1/R$.

Soon after decoupling, $e^{\pm}$ pairs in the thermal background begin to annihilate (when $T \lesssim m_e$). 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 computed from entropy conservation. The neutrino entropy must be conserved separately from the entropy of interacting particles. A straightforward computation yields

$$T_{\nu} = (4/11)^{1/3} T_{\gamma} \approx 1.9 \text{ K.} \qquad (22.49)$$

The total entropy density is therefore given by the contribution from photons and three flavors of neutrinos:

$$s = \frac{4}{3} \frac{\pi^2}{30} \left( 2 + \frac{21}{4} \left( \frac{T_{\nu}}{T_{\gamma}} \right)^3 \right) T_{\gamma}^3 = \frac{4}{3} \frac{\pi^2}{30} \left( 2 + \frac{21}{11} \right) T_{\gamma}^3 = 7.04 n_{\gamma}. \qquad (22.50)$$

Similarly, the total relativistic energy density is given by

$$\rho_r = \frac{\pi^2}{30} \left[ 2 + \frac{21}{4} \left( \frac{T_{\nu}}{T_{\gamma}} \right)^4 \right] T_{\gamma}^4 \approx 1.68 \rho_{\gamma}. \qquad (22.51)$$

In practice, a small correction is needed to this, since neutrinos are not totally decoupled at $e^{\pm}$ 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, $\Omega_{\nu}h^2 = 6 \times 10^{-4}$, 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 $m^2$ values. Note that a 0.06-eV neutrino has $T_{\nu} = m_{\nu}$ at $z \simeq 357$, 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 the radiation content of the Universe is dynamically significant. Combining Planck data with data from large-scale structure surveys such as DES [42] provides an upper limit to the sum of neutrino masses, $\Sigma m_{\nu} < 0.13$ eV, and may provide the first measurement of a neutrino mass (as opposed to a difference 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 the course of its evolution [43–46]. Our current vacuum state is described by $SU(3)_c \times U(1)_em$, which in the Standard Model is a remnant of an unbroken $SU(3)_c \times SU(2)_L \times U(1)_Y$ gauge symmetry. 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 $V(\phi) = \frac{1}{4}\lambda\phi^4 - \frac{1}{2}\mu^2\phi^2 + V(0)$. The resulting expectation value is simply $\langle \phi \rangle = \mu/\sqrt{\lambda}$.

In the early Universe, finite temperature radiative corrections typically add terms to the potential of the form $\phi^2 T^2$. Thus, at very high temperatures, the symmetry is restored and $\langle \phi \rangle = 0$. 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 a second-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 equation of motion for a scalar field $\phi$ can be derived from the energy-momentum tensor:

$$T_{\mu\nu} = \partial_{\mu}\phi\partial_{\nu}\phi - \frac{1}{2}g_{\mu\nu}\partial_{\rho}\phi\partial^{\rho}\phi - g_{\mu\nu}V(\phi). \qquad (22.52)$$

By associating $\rho = T_{00}$ and $p = R^{-2}(t)T_{ii}$ we have

$$\begin{aligned} \rho &= \frac{1}{2}\dot{\phi}^2 + \frac{1}{2}R^{-2}(t)(\nabla\phi)^2 + V(\phi); \\ p &= \frac{1}{2}\dot{\phi}^2 - \frac{1}{6}R^{-2}(t)(\nabla\phi)^2 - V(\phi), \end{aligned} \qquad (22.53)$$

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):

$$\ddot{\phi} + 3H\dot{\phi} = -\partial V / \partial \phi. \qquad (22.54)$$

### 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 that the field evolved slowly from the symmetric state to the global minimum, the Universe may have been dominated by the vacuum energy density associated with the potential near $\phi \simeq 0$. During this period of slow evolution, the energy density due to radiation will fall below the vacuum energy density, $\rho \ll V(0)$. When this happens, the expansion rate will be dominated by the constant $V(0)$, and we obtain the exponentially expanding solution given in Eq. (22.20). When the field evolves 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 $N T_R^4 \lesssim V(0)$. In this reheating process, entropy has been created and the final value of $RT$ is greater than the initial value of $RT$. Thus, we see that, during a phase transition, the relation $RT \sim$ constant need not hold true. This is the basis of the inflationary Universe scenario [47–49].

If, during the phase transition, the value of $RT$ changed by a factor of $O(10^{29})$, the cosmological problems discussed above would be solved. The observed isotropy would be generated by the immense 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 $\Omega$ would have been driven to 1 (with exponential precision). Density perturbations will be stretched by the expansion, $\lambda \propto R(t)$. Thus it will appear that $\lambda \gg H^{-1}$ or that the perturbations have left the horizon, where in fact the size of the causally connected region is now no longer simply $H^{-1}$. However, not only does inflation offer an explanation for large scale perturbations, it also offers a source 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, $\phi$. The potential of this field, $V(\phi)$, needs to have a very low gradient and curvature in order to match observed metric fluctuations. For a more thorough discussion of the problems of early models and a host of current models being studied see the review on inflation – Sec. 23 of this Review. 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], where observer 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 of antimatter in the latter is understood to be the result of collisions of primary particles in the interstellar 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, $\eta \sim 10^{-9}$.

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 ingredients are 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 have seen, 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 right-handed neutrino), which is converted to a baryon asymmetry at the electroweak scale [63]. This mechanism 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, $\eta$. The nucleosynthesis predictions can be compared with observational determinations of the abundances of the light elements. Consistency between theory and observations, driven primarily by recent D/H measurements [64, 65], leads to a range of [66]

$$5.8 \times 10^{-10} < \eta < 6.3 \times 10^{-10}. \qquad (22.55)$$

The value of $\eta$ is related to the baryon density as follows:

$$\Omega_b = 3.66 \times 10^7 h^{-2}, \qquad (22.56)$$

or $10^{10}\eta = 274\Omega_b h^2$. The Planck result [24] for $\Omega_b h^2$ of 0.0224 ± 0.0002 translates into a value of $\eta = 6.12 \pm 0.04$. 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 with that 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 atmospheres of halo dwarf stars [68], but there is mounting evidence that effects of stellar depletion cannot be ignored 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 amount of dark matter. Assuming three nearly massless neutrinos, the energy

density in radiation at temperatures $T \ll 1$ MeV, is given by

$$\rho_r = \frac{\pi^2}{30} \left[ 2 + \frac{21}{4} \left( \frac{4}{11} \right)^{4/3} \right] T^4. \qquad (22.57)$$

In the absence of non-baryonic dark matter, the matter density can be written as

$$\rho_m = m_N \eta n_\gamma, \qquad (22.58)$$

where $m_N$ is the nucleon mass. Recalling that $n_\gamma \propto T^3$ [cf. Eq. (22.40)], we can solve for the temperature or redshift at the matter-radiation equality when $\rho_r = \rho_m$:

$$T_{\text{eq}} = 0.22 m_N \eta \quad \text{or} \quad (1+z_{\text{eq}}) = 0.22 \eta \frac{m_N}{T_0}, \qquad (22.59)$$

where $T_0$ is the present temperature of the microwave background. For $\eta = 6.1 \times 10^{-10}$, this corresponds to a temperature $T_{\text{eq}} \simeq 0.13$ eV or $(1+z_{\text{eq}}) \simeq 550$. 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 matter is present. If instead of Eq. (22.58), we write

$$\rho_m = \Omega_m \rho_c \left(\frac{T}{T_0}\right)^3, \qquad (22.60)$$

we find that

$$T_{\text{eq}} = 0.9 \frac{\Omega_m \rho_c}{T_0^3} \quad \text{or} \quad (1+z_{\text{eq}}) = 2.4 \times 10^4 \Omega_m h^2. \qquad (22.61)$$

## 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 ionized and so scatter photons: $T \gtrsim 10^3$ K, and the sky should therefore shine as brightly as the surface of the Sun. The reason the night sky is dark is entirely due to the expansion, which cools the radiation 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. 29 of this Review. The COBE estimate of the temperature is [76]

$$T = (2.7255 \pm 0.0006) \text{ K}. \qquad (22.62)$$

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 $T \propto 1/R$, so that radiation from different parts of space arrive at an observer with 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 $z \simeq 1100$, almost independently of the main cosmological parameters, at which time the fractional ionization is very small. 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 $z \sim 10^7$. 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 $\lesssim \Omega_*$ $\lesssim 0.003$ [77] but mainly inhabit the intergalactic medium (IGM). One powerful way in which this can be studied is via the absorption of light from distant luminous objects such as quasars. Even very small amounts of neutral hydrogen can
absorb rest-frame UV photons (the Gunn-Peterson effect), and should suppress the continuum by a factor $\exp(-\tau)$, where

$$ \tau \approx 10^{4.62} h^{-1} \left[ \frac{n_{\text{HI}}(z)/\text{m}^{-3}}{(1+z)\sqrt{1+\Omega_{\text{m}}z}} \right], \quad (22.63) $$

and this expression applies while the Universe is matter dominated ($z \gtrsim 1$ in the $\Omega_{\text{m}} = 0.3$ $\Omega_{\text{v}} = 0.7$ model). At $z < 6$, the dominant effect on quasar spectra is a 'forest' of narrow absorption lines, which produce a mean $\tau = 1$ in the Ly$\alpha$ forest at about $z = 3$, and so we have $\Omega_{\text{HI}} \simeq 10^{-6.7} h^{-1}$. This is such a small number that the IGM must be very highly ionized at these redshifts, apart from a few high-density clumps. But at $z > 6$ there is good evidence for a 'reionization' era at which 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 Ly$\alpha$ 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, each absorption system in the Ly$\alpha$ forest is accompanied by an offset deuterium line. By careful selection of 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 determinations with the theory of primordial nucleosynthesis yields a baryon density of $\Omega_{\text{b}}h^2 = 0.021-0.023$ (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. Studies of this phenomenon allow us to achieve an accounting of the total baryonic material in clusters. Within the central $\simeq 1$ Mpc, the masses in stars, X-ray emitting gas, and total dark matter can be determined with reasonable accuracy (perhaps 20% rms), and this allows a minimum baryon fraction to be determined [79, 80]:

$$ \frac{M_{\text{baryons}}}{M_{\text{total}}} \gtrsim 0.009 + (0.066 \pm 0.003) h^{-3/2}. \quad (22.64) $$

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% (for reasonable $h$), which is too high for $\Omega_{\text{m}} = 1$ if we take $\Omega_{\text{b}}h^2 \simeq 0.02$ from nucleosynthesis. This is therefore one of the more robust arguments in favor of $\Omega_{\text{m}} \simeq 0.3$; for more details, see Cosmological Parameters – Sec. 25 of this Review. This argument is also consistent with the inference on $\Omega_{\text{m}}$ that can be made from Fig. 22.2.

This method is much more robust than the older classical technique for weighing the Universe: '$L \times M/L$. 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 $\Omega_{\text{m}}$ – 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):

$$ \alpha(D_{\text{L}}\theta_{\text{I}}) = \frac{D_{\text{S}}}{D_{\text{LS}}} (\theta_{\text{I}} - \theta_{\text{S}}), \quad (22.65) $$

where the angles $\theta_{\text{I}}$, $\theta_{\text{S}}$, and $\alpha$ are in general two-dimensional vectors on the sky. The distances $D_{\text{LS}}$ etc. are given by an extension of the usual distance-redshift formula:

$$ D_{\text{LS}} = \frac{R_0 S_k (\chi_{\text{S}} - \chi_{\text{L}})}{1 + z_{\text{S}}}. \quad (22.66) $$

This is the angular-diameter distance for objects on the source plane as perceived by an observer on 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 radius is the Einstein radius:

$$ \begin{aligned} \theta_{\text{E}} &= \left(4GM \frac{D_{\text{LS}}}{D_{\text{L}}D_{\text{S}}}\right)^{1/2} \\ &= \left(\frac{M}{10^{11.09} \text{M}_{\odot}}\right)^{1/2} \left(\frac{D_{\text{L}}D_{\text{S}}/D_{\text{LS}}}{\text{Gpc}}\right)^{-1/2} \text{arcsec}. \end{aligned} \quad (22.67) $$

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 with other 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 cosmological structure 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 in measuring cosmological parameters, based on survey data from only $\sim 10^3$ deg$^2$ [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 the poorly-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 this gives the CMB greater internal power [85]. The main difficulty of principle with lensing is that part of 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

$$ \delta(\mathbf{x}) \equiv \frac{\rho(\mathbf{x}) - \langle\rho\rangle}{\langle\rho\rangle}. \quad (22.68) $$

A critical feature of the $\delta$ field is that it inhabits a Universe that is isotropic and homogeneous in its large-scale properties. This suggests that the statistical properties of $\delta$ should also be statistically homogeneous – i.e., it is a stationary random process.

It is often convenient to describe $\delta$ as a Fourier superposition:

$$ \delta(\mathbf{x}) = \sum_{\mathbf{k}} \delta_{\mathbf{k}} e^{-i\mathbf{k} \cdot \mathbf{x}}. \quad (22.69) $$

We avoid difficulties with an infinite Universe by applying periodic boundary conditions in a cube of 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:

$$ \langle\delta^2\rangle = \sum_{\mathbf{k}} |\delta_{\mathbf{k}}|^2 \equiv \sum_{\mathbf{k}} P(k). \quad (22.70) $$

Note that the statistical nature of the fluctuations must be isotropic, so we write $P(k)$ rather than $P(\mathbf{k})$. The $\langle\dots\rangle$ average here is a volume average. Cosmological density fields are an example of an ergodic process, in which the average over a large
volume tends to the same answer as the average over a statistical ensemble.

The statistical properties of discrete objects sampled from the density field are often described in 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, the correlation function is readily shown to be identical to the autocorrelation function of the $\delta$ field: $\xi(r) = \langle\delta(x)\delta(x+r)\rangle$.

The power spectrum and correlation function are Fourier conjugates, and thus are equivalent descriptions of the density field (similarly, k-space equivalents exist for the higher-order correlations). It is convenient to take the limit $V \to \infty$ 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 $\Delta^2(k) = d\langle\delta^2\rangle/d\ln k =Vk^3P(k)/2\pi^2$:

$$ \xi(r) = \int \Delta^2(k) \frac{\sin kr}{kr} d\ln k; \quad \Delta^2(k) = \frac{2}{\pi} k^3 \int_0^\infty \xi(r) \frac{\sin kr}{kr} r^2 dr. \qquad (22.71) $$

For many years, an adequate approximation to observational data on galaxies was $\xi = (r/r_0)^{-\gamma}$, with $\gamma \simeq 1.8$ and $r_0 \simeq 5h^{-1}$ 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 is required). 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:

$$ \delta \propto \begin{cases} a^2(t) & (\text{radiation domination}; \Omega_r = 1); \\ a(t) & (\text{matter domination}; \Omega_m = 1). \end{cases} \qquad (22.72) $$

For low-density Universes, the growth is slower:

$$ d \ln \delta / d \ln a \simeq \Omega_m^\gamma (a), \qquad (22.73) $$

where the parameter $\gamma$ 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 the total 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 in the 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 that decays 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 $z_{eq}$ is therefore an important scale:

$$ D_H(z_{eq}) = \frac{2(\sqrt{2} - 1)}{(\Omega_m z_{eq})^{1/2} H_0} = \frac{16.0}{\Omega_m h^2} \text{ Mpc.} \qquad (22.74) $$

2. At early times, dark matter particles will undergo free streaming at the speed of light, and so erase all scales up to the horizon – a process that only ceases when the particles go nonrelativistic. For light massive neutrinos, this happens at $z_{eq}$; all structure up to the horizon-scale power-spectrum

break is in fact erased. Hot(cold) dark matter models are thus sometimes dubbed large(small)-scale damping models.

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

<figure><img src="image_131.png" /><figcaption>**Figure 22.4:** A plot of transfer functions for various models. For adiabatic models, $T_k \to 1$ at small $k$, whereas the opposite is true for isocurvature models. For dark-matter models, the characteristic wavenumber scales proportional to $\Omega_m h^2$. The scaling for baryonic models does not obey this exactly; the plotted cases correspond to $\Omega_m = 1, h = 0.5$.</figcaption></figure>

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 the primordial scalar-mode power law, times the square of the transfer function:

$$ P(k) \propto k^{n_s} T_k^2. \qquad (22.75) $$

The most generic power-law index is $n_s = 1$: the ‘Zeldovich’ or ‘scale-invariant’ spectrum. Inflationary models tend to predict a small ‘tilt’: $|n_s - 1| \lesssim 0.05$ [12, 13]. On the assumption that the dark matter is cold, the power spectrum then depends on five parameters: $n_s, h, \Omega_b, \Omega_c$ ($\equiv \Omega_m - \Omega_b$), and an overall amplitude. The latter is often specified as $\sigma_8$, the linear-theory fractional rms in density when a spherical filter of radius $8h^{-1}$ Mpc is applied in linear theory. This scale can be probed directly via weak gravitational lensing, and also via its effect on the abundance of rich galaxy clusters. The normalization derived from samples of X-ray clusters has been given variously as [93, 94]

$$ \sigma_8 = [0.746 \pm 0.012(\text{stat.}) \pm 0.022(\text{sys.})] (\Omega_m/0.3)^{-0.47}; \qquad (22.76) $$

$$ \sigma_8 = [0.81 \pm 0.03] (\Omega_m/0.3)^{-0.17}. \qquad (22.77) $$

The higher figure is well consistent with Planck, whereas the lower is rather similar to the normalization inferred from weak lensing [95]:

$$ \sigma_8 \simeq [0.790^{+0.018}_{-0.014}] (\Omega_m/0.3)^{-0.5}. \qquad (22.78) $$

This figure is in 1.9$\sigma$ tension with the Planck values of $(\sigma_8, \Omega_m) = (0.811 \pm 0.006, 0.315 \pm 0.007)$. If real, such a discrepancy could indicate interesting new physics; but the current evidence is not strong enough to make such a claim. However, it is worth noting that the CMB requires an almost perfect degeneracy, $\Omega_m h^3 = \text{constant}$ 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, $\delta n/n$,
may differ from the mass fluctuations, $\delta\rho/\rho$. It is commonly assumed that the two fields obey some proportionality on large scales where the fluctuations are small, $\delta n/n = b\delta\rho/\rho$, but even this is not guaranteed [96].

The main shape of the transfer function is a break around the horizon scale at $z_{eq}$, which depends just on $\Omega_m h$ 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 can be 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 $\Omega_m h \simeq 0.20$ and a baryon fraction of about 0.15 for $n_s \simeq 1$ (see Fig. 22.5).

In principle, accurate data over a wide range of $k$ could determine both $\Omega_m h$ and $n_s$, but in practice there is a strong degeneracy between these. In order to constrain $n_s$ itself, it is necessary to examine data on anisotropies in the CMB.

<figure><img src="image_132.png" /><figcaption>**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 $\Lambda$CDM model with $\Omega_b h^2 \simeq 0.02$ and $\Omega_m h \simeq 0.2$. 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 the matter content of the Universe is dominated by collisionless dark matter.</figcaption></figure>

### 22.4.6 CMB anisotropies

The CMB has a clear dipole anisotropy, of magnitude $1.23 \times 10^{-3}$. This is interpreted as being due to the Earth's motion, which is equivalent to a peculiar velocity for the Milky Way of

$$v_{\text{MW}} \approx 600 \text{ km s}^{-1} \text{ towards } (\ell, b) \approx (270^{\circ}, 30^{\circ}). \quad (22.79)$$

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 (of order $10^{-5}$), and interpreted as signatures of density fluctuations at last scattering ($\simeq 1100$).

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 defined as

$$\mathcal{T}^2(\ell) = \frac{\ell(\ell + 1)}{2\pi} C_{\ell}. \quad (22.80)$$

This function encodes information from the three distinct mechanisms that cause CMB anisotropies:

* (1) Gravitational (Sachs-Wolfe) perturbations. Photons from high-density regions at last scattering have to climb out of potential wells, and are thus redshifted.

* (2) Intrinsic (adiabatic) perturbations. In high-density regions, the coupling of matter and radiation 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, $\nabla^2\Phi/a^2 = 4\pi G\bar{\rho}\delta$, and the comoving velocity field satisfies the continuity equation $\nabla \cdot \mathbf{u} = -\dot{\delta}$, the resulting different powers of $k$ ensure that 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 comoving angular-diameter distance to the last-scattering sphere; because of its high redshift, this is effectively identical to the horizon size at the present epoch, $D_H$:

$$ \begin{aligned} D_H &= \frac{2}{\Omega_m H_0} && (\Omega_v = 0); \\ D_H &\approx \frac{2}{\Omega_m^{0.4} H_0} && (\text{flat} : \Omega_m + \Omega_v = 1). \end{aligned} \quad (22.81) $$

These relations show how the CMB is strongly sensitive to curvature: the horizon length at last scattering is $\propto 1/\sqrt{\Omega_m}$, so that this subtends an angle that is virtually independent of $\Omega_m$ for a flat model. Observations of a peak in the CMB power spectrum at relatively large scales ($\ell \simeq 221$) are thus strongly inconsistent with zero-$\Lambda$ models with low density: current CMB + BAO + lensing data require $\Omega_m + \Omega_v = 0.999 \pm 0.004$ (95%) [24]. (See e.g., Fig. 22.2).

In addition to curvature, the CMB encodes information about several other key cosmological parameters. Within the compass of simple adiabatic CDM models, there are nine of these:

$$\omega_c, \omega_b, \Omega_{\text{tot}}, h, \tau, n_s, n_t, r, Q. \quad (22.82)$$

The symbol $\omega$ denotes the physical density, $\Omega h^2$: the transfer function depends only on the densities of CDM ($\omega_c$) and baryons ($\omega_b$). Transcribing the power spectrum at last scattering into an angular power spectrum brings in the total density parameter ($\Omega_{\text{tot}} \equiv \Omega_m + \Omega_v = \Omega_c + \Omega_b + \Omega_v$) and $h$: there is a near-exact geometrical degeneracy [102] between these that keeps the angular-diameter distance to 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-sight information 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 Cosmological Parameters – 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 ($\tau$). Adding a large tensor contribution reduces the contrast between low $\ell$ and the peak at $\ell \simeq 221$ (because the tensor spectrum has no acoustic component). The previous relative height of the peak can be recovered by increasing $n_s$ to increase the small-scale power in the scalar
component; this in turn over-predicts the power at $\ell \sim 1000$, but this effect can be 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 dimensionless parameters:

$$ \epsilon \equiv \frac{M_{\mathrm{P}}^2}{16\pi} \left(\frac{V'}{V}\right)^2; \quad \eta \equiv \frac{M_{\mathrm{P}}^2}{8\pi} \left(\frac{V''}{V}\right), \qquad (22.83) $$

where 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 $r \simeq 16\epsilon$, and the spectral indices are $n_s = 1-6\epsilon+2\eta$ and $n_t = -2\epsilon$. The natural expectation of inflation is that the quasi-exponential phase ends once the magnitudes of the slow-roll parameters become of order unity, so that both $n_s \neq 1$ and a significant 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 therefore stands at a fascinating point, given that the most recent CMB data reject the zero-tensor $n_s = 1$ model at more than $8\sigma$: $n_s = 0.965 \pm 0.004$ [24]. This rejection is strong enough that it is also able to break the tensor degeneracy, so that no model with $n_s = 1$ is acceptable, whatever the value of $r$.

The current limit on $r$ is < 0.036 at 95% confidence [104, 105]. In conjunction with the measured value of $n_s$, this upper limit sits close to the prediction of a linear potential (i.e. $|\eta| \ll |\epsilon|$). Any further reduction in the limit on $r$ will force $\eta$ 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 $V(\phi) = m^2\phi^2$ or $\lambda\phi^4$, which are now excluded. Examples of models that are currently in excellent agreement with the Planck results are the Starobinsky model of $R+R^2$ gravity [106], or the Higgs-inflation model where the Higgs field is non-minimally coupled [107]. Assuming 55 e-foldings of inflation, these models predict $n_s = 0.965$ 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 is damped when the Universe becomes $\Lambda$-dominated, and this is an independent way of detecting $\Lambda$ [108]. The potential also causes gravitational lensing of the CMB: structures at $z \simeq 1-2$ displace features on the CMB sky by about 2 arcmin over coherent degree-scale patches. Detection of these distortions allows a map to be made of overdensity projected from $z=0$ to 1100 [85]. This is 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 the blurring of photon energies by hot gas; and the kinetic Sunyaev-Zeldovich effect is sensitive to the bulk velocity of the gas. Both these effects start to dominate over the intrinsic CMB fluctuations at multipoles $\ell \gtrsim 2000$ [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 $(2.2 \text{ meV})^4$ (for $w = -1$, $\Omega_v = 0.68$, $h = 0.67$), or roughly $10^{-123} M_P^4$, and such an unnaturally small number is hard to understand. 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 this presumably occurs at $> 1$ TeV scales, not meV; thus the apparent energy scale of the vacuum is at least

$10^{15}$ 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 $\Lambda$ 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 $\Lambda$ before any data strongly indicated a non-zero value.

But it may be that the phenomenon of dark energy is entirely illusory. The necessity for this 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 to distinguish 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 an inhomogeneous Universe [112, 113]. Most would argue that a standard Newtonian treatment of such 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 means that it is endowed with two thermodynamic properties that astronomers can try to measure: the bulk equation of state, and the sound speed. If the sound speed is close to the speed of light, the effect 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 of change of the vacuum density: $d\ln\rho_v/d\ln a = -3(1+w)$, so it can be accessed via the evolving expansion rate, $H(a)$. This can be measured most cleanly by using the inbuilt natural ruler of large-scale structure: the BAO horizon scale [117]:

$$ D_{\mathrm{BAO}} \simeq 147.0 (\Omega_{\mathrm{m}}h^2/0.1424)^{-0.253} (\Omega_{\mathrm{b}}h^2/0.0224)^{-0.128} \mathrm{Mpc}. \qquad (22.84) $$

Observing the equivalent acoustic scale at last scattering in the CMB yields the precise geometrical degeneracy $\Omega_m h^3 = 0.09633 \pm 0.00029$ [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 $\ddot{\delta} + 2H(a)\dot{\delta} = 4\pi G\rho_0 \delta$. Thus, both the scale and amplitude of density fluctuations are sensitive to $w(a)$ – but only weakly. These observables typically change by only $\sim 0.2\%$ 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 to a few parts in 1000.

Testing whether theories of gravity require revision can also be done using data on cosmological inhomogeneities. Two separate issues arise, concerning the metric perturbation potentials $\Psi$ and $\Phi$, 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: $\nabla^2\Phi/a^2 = 4\pi G\bar{\rho}\delta$. Empirically, modifications of gravity require us to explore a change with scale and with time of the ‘slip’ ($\Psi/\Phi$) 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 addressed on 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: $d\ln\delta/d\ln a = \Omega_m(a)$ [89, 90]. The parameter $\gamma$ 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 ($\Omega_m(a) \to 1$ as $a \to 0$), but it is a convenient way of capturing the power of various experiments. Current data are consistent with standard $\Lambda$CDM [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 Telescope in 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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## 23. Inflation

Revised August 2023 by J. Ellis (King's Coll. London; CERN) and D. Wands (Portsmouth U.).

### 23.1 Motivation and Introduction

The standard Big-Bang model of cosmology provides a successful framework in which to understand the thermal history of our Universe and the growth of cosmic structure, but it is essentially incomplete. As described in Sec. 22.2.4, Big-Bang cosmology requires very specific initial conditions. It postulates a uniform cosmological background, described by a spatially-flat, homogeneous and isotropic Robertson-Walker (RW) metric (Eq. (22.1) in “Big Bang Cosmology” review), with scale factor $R(t)$. Within this setting, it also requires an initial almost scale-invariant distribution of primordial density perturbations as seen, for example, in the cosmic microwave background (CMB) radiation (described in Chap. 29, “Cosmic Microwave Background” review), on scales far larger than the causal horizon at the time the CMB photons last scattered.

The Hubble expansion rate, $H \equiv \dot{R}/R$, in a RW cosmology is given by the Friedmann constraint equation (Eq. (22.8) in “Big Bang Cosmology” review)

$$H^2 = \frac{8\pi\rho}{3M_P^2} + \frac{\Lambda}{3} - \frac{k}{R^2}, \quad (23.1)$$

where $k/R^2$ is the intrinsic spatial curvature. We use natural units such that the speed of light $c = 1$ and hence we have the Planck mass $M_P = G_N^{-1/2} \simeq 10^{19}$ GeV (see “Astrophysical Constants and Parameters”). A cosmological constant, $\Lambda$, of the magnitude required to accelerate the Universe today (see Chap. 28, “Dark Energy” review) would have been completely negligible in the early Universe where the energy density $\rho \gg M_P^2 \Lambda \sim 10^{-12}$ (eV)$^4$. The standard early Universe cosmology, described in Sec. 22.1.5 in “Big Bang Cosmology” review, is thus dominated by non-relativistic matter ($p_m \ll \rho_m$) or radiation ($p_r = \rho_r/3$ for an isotropic distribution). This leads to a decelerating expansion with $\dot{R} < 0$.

The hypothesis of inflation [1,2] postulates a period of accelerated expansion, $\dot{R} > 0$, in the very early Universe, preceding the standard radiation-dominated era, which offers a physical model for the origin of these initial conditions, as reviewed in [3-7]. Such a period of accelerated expansion (i) drives a curved Robertson-Walker spacetime (with spherical or hyperbolic spatial geometry) towards spatial flatness, and (ii) it also expands the causal horizon beyond the present Hubble length, so as to encompass all the scales relevant to describe the large-scale structure observed in our Universe today, via the following two mechanisms.

1. A spatially-flat Universe with vanishing spatial curvature, $k = 0$, has the dimensionless density parameter $\Omega_{tot} = 1$, where we define (Eq. (22.13) in “Big Bang Cosmology” review; see Chap. 25, “Cosmological Parameters” review for more complete definitions)

$$\Omega_{tot} \equiv \frac{8\pi\rho_{tot}}{3M_P^2 H^2}, \quad (23.2)$$

with $\rho_{tot} \equiv \rho + \Lambda M_P^2 / 8\pi$. If we re-write the Friedmann constraint (Eq. (23.1)) in terms of $\Omega_{tot}$ we have

$$1 - \Omega_{tot} = -\frac{k}{\dot{R}^2}. \quad (23.3)$$

Observations require $|1 - \Omega_{tot,0}| < 0.005$ today [8], where the subscript 0 denotes the present-day value. Taking the time derivative of Eq. (23.3) we obtain

$$\frac{d}{dt}(1 - \Omega_{tot}) = -2\frac{\ddot{R}}{R}(1 - \Omega_{tot}). \quad (23.4)$$

Thus in a decelerating expansion, $\dot{R} > 0$ and $\ddot{R} < 0$, any small initial deviation from spatial flatness grows, $(d/dt)|1 - \Omega_{tot}| > 0$. A small value such as $|1 - \Omega_{tot,0}| < 0.005$ today requires an even smaller value at earlier times, e.g., $|1 - \Omega_{tot}| < 10^{-5}$ at

the last scattering of the CMB, which appears unlikely, unless for some reason space is exactly flat. However, an extended period of accelerated expansion in the very early Universe, with $\dot{R} > 0$ and $\ddot{R} > 0$ and hence $(d/dt)|1 - \Omega_{tot}| < 0$, can drive $\Omega_{tot}$ sufficiently close to unity, so that $|1 - \Omega_{tot,0}|$ remains unobservable small today, even after the radiation- and matter-dominated eras, for a wide range of initial values of $\Omega_{tot}$.

2. The comoving distance (the present-day proper distance) traversed by light between cosmic time $t_1$ and $t_2$ in an expanding Universe can be written, (see Eq. (22.32) in “Big Bang Cosmology” review), as

$$D_0(t_1, t_2) = R_0 \int_{t_1}^{t_2} \frac{dt}{R(t)} = R_0 \int_{\ln R_1}^{\ln R_2} \frac{d(\ln R)}{\dot{R}}. \quad (23.5)$$

In standard decelerated (radiation- or matter-dominated) cosmology the integrand, $1/\dot{R}$, decreases towards the past, and there is a finite comoving distance traversed by light (a particle horizon) since the Big Bang ($R_1 \to 0$). For example, the comoving size of the particle horizon at the CMB last-scattering surface ($R_2 = R_{lss}$) corresponds to $D_0 \sim 100$ Mpc, or approximately 1° on the CMB sky today (see Sec. 22.2.4 in “Big Bang Cosmology” review). However, during a period of inflation, $1/\dot{R}$ increases towards the past, and hence the integral (Eq. (23.5)) diverges as $R_1 \to 0$, allowing an arbitrarily large causal horizon, dependent only upon the duration of the accelerated expansion. Assuming that the Universe inflates with a finite Hubble rate $H_*$ at $t_1 = t_*$, ending with $H_{end} < H_*$ at $t_2 = t_{end}$, we have

$$D_0(t_*, t_{end}) > \left( \frac{R_0}{R_{end}} \right) H_*^{-1} (e^{N_*} - 1), \quad (23.6)$$

where $N_* \equiv \ln(R_{end}/R_*)$ describes the duration of inflation, measured in terms of the logarithmic expansion (or “e-folds”) from $t_1 = t_*$ up to the end of inflation at $t_2 = t_{end}$, and $R_0/R_{end}$ is the subsequent expansion from the end of inflation to the present day. If inflation occurs above the TeV scale, the comoving Hubble scale at the end of inflation, $(R_0/R_{end})H_{end}^{-1}$, is less than one astronomical unit ($\sim 10^{11}$ m), and a causally-connected patch can encompass our entire observable Universe today, which has a size $D_0 > 30$ Gpc, if there were more than 40 e-folds of inflation ($N_* > 40$). If inflation occurs at the GUT scale ($10^{15}$ GeV) then we require more than 60 e-folds.

Producing an accelerated expansion in general relativity requires an energy-momentum tensor with negative pressure, $p < -\rho/3$ (see Eq. (22.9) in “Big Bang Cosmology” review and Chap. 28, “Dark Energy” review), quite different from the hot dense plasma of relativistic particles in the hot Big Bang. However a positive vacuum energy $V > 0$ does exert a negative pressure, $p_V = -\rho_V$. The work done by the cosmological expansion must be negative in this case so that the local vacuum energy density remains constant in an expanding Universe, $\dot{p}_V = -3H(\rho_V + p_V) = 0$. Therefore, a false vacuum state can drive an exponential expansion, corresponding to a de Sitter spacetime with a constant Hubble rate $H^2 = 8\pi\rho_V / 3M_P^2$ on spatially-flat hypersurfaces.

A constant vacuum energy $V$, equivalent to a cosmological constant $\Lambda$ in the Friedmann equation Eq. (23.1), cannot provide a complete description of inflation in the early Universe, since inflation must necessarily have come to an end in order for the standard Big-Bang cosmology to follow. A phase transition to the present true vacuum is required to release the false vacuum energy into the energetic plasma of the hot Big Bang and produce the large total entropy of our observed Universe today. Thus, we must necessarily study dynamical models of inflation, where the time-invariance of the false vacuum state is broken by a time-dependent field. A first-order phase transition would produce a very inhomogeneous Universe [9] unless a time-dependent scalar field leads to a rapidly changing percolation rate [10-12]. However,
a second-order phase transition [13, 14], controlled by a slowly-
rolling scalar field, can lead to a smooth classical exit from the
vacuum-dominated phase.

As a spectacular bonus, quantum fluctuations in that scalar
field could provide a source of almost scale-invariant density fluc-
tuations [15, 16], as detected in the CMB (see Chap. 29), which
are thought to be the origin of the structures seen in the Universe
today.

Accelerated expansion and primordial perturbations can also be
produced in some modified gravity theories (e.g., [1, 17]), which
introduce additional non-minimally coupled degrees of freedom.
Such inflation models can often be conveniently studied by trans-
forming variables to an ‘Einstein frame’ in which Einstein’s equa-
tions apply with minimally coupled scalar fields [18–20].

In the following we will review scalar field cosmology in general
relativity and the spectra of primordial fluctuations produced dur-
ing inflation, before studying selected inflation models.

## 23.2 Scalar Field Cosmology

The energy-momentum tensor for a canonical scalar field $\phi$ with self-interaction potential $V(\phi)$ is given in Eq. (22.52) in “Big Bang Cosmology” review. In a homogeneous background this corresponds to a perfect fluid with density

$$ \rho = \frac{1}{2} \dot{\phi}^2 + V(\phi), \qquad (23.7) $$

and isotropic pressure

$$ p = \frac{1}{2} \dot{\phi}^2 - V(\phi), \qquad (23.8) $$

while the 4-velocity is proportional to the gradient of the field,
$u^\mu \propto \nabla^\mu \phi$.

A field with vanishing potential energy acts like a stiff fluid with $p = \rho = \dot{\phi}^2/2$, whereas if the time-dependence vanishes we have $p = -\rho = -V$ and the scalar field is uniform in time and space. Thus a classical, potential-dominated scalar-field cosmology, with $p \simeq -\rho$, can naturally drive a quasi-de Sitter expansion; the slow time-evolution of the energy density weakly breaks the exact $O(1,3)$ symmetry of four-dimensional de Sitter spacetime down to a Robertson-Walker (RW) spacetime, where the scalar field plays the role of the cosmic time coordinate.

In a scalar-field RW cosmology the Friedmann constraint equa-
tion (Eq. (23.1)) reduces to

$$ H^2 = \frac{8\pi}{3M_P^2} \left( \frac{1}{2}\dot{\phi}^2 + V \right) - \frac{k}{R^2}, \qquad (23.9) $$

while energy conservation (Eq. (22.10) in “Big Bang Cosmol-
ogy” review) for a homogeneous scalar field reduces to the Klein-
Gordon equation of motion (Eq. (22.54) in “Big Bang Cosmology”
review)

$$ \ddot{\phi} = -3H\dot{\phi} - V'(\phi). \qquad (23.10) $$

The evolution of the scalar field is thus driven by the potential
gradient $V' = dV/d\phi$, subject to damping by the Hubble expan-
sion $3H\dot{\phi}$.

If we define the Hubble slow-roll parameter

$$ \epsilon_H \equiv -\frac{\dot{H}}{H^2}, \qquad (23.11) $$

then we see that inflation ($\ddot{R} > 0$ and hence $\dot{H} > -H^2$) requires $\epsilon_H < 1$. In this case the spatial curvature decreases relative to the scalar field energy density as the Universe expands. Hence, in the following we drop the spatial curvature and consider a spatially-flat RW cosmology, assuming that inflation has lasted sufficiently long that our observable Universe is very close to spatially flatness. However, we note that bubble nucleation, leading to a first-order phase transition during inflation, can lead to homogeneous hyper-surfaces with a hyperbolic (‘open’) geometry, effectively resetting the spatial curvature inside the bubble [21]. This is the basis of so-called open inflation models [22–24], where inflation inside the bubble has a finite duration, leaving a finite negative spatial curvature.

In a scalar field-dominated cosmology (Eq. (23.11)) gives

$$ \epsilon_H = \frac{3\dot{\phi}^2}{2V + \ddot{\phi}^2}, \qquad (23.12) $$

in which case we see that inflation requires a potential-dominated
expansion, $\dot{\phi}^2 < V$.

### 23.2.1 Slow-Roll Inflation

It is commonly assumed that the field acceleration term, $\ddot{\phi}$, in (Eq. (23.10)) can be neglected, in which case one can give an approximate solution for the inflationary attractor [25]. This slow-roll approximation reduces the second-order Klein-Gordon equation (Eq. (23.10)) to a first-order system, which is over-damped, with the potential gradient being approximately balanced against to the Hubble damping:

$$ 3H\dot{\phi} \approx -V', \qquad (23.13) $$

and at the same time that the Hubble expansion (Eq. (23.9)) is
dominated by the potential energy

$$ H^2 \approx \frac{8\pi}{3M_P^2} V(\phi), \qquad (23.14) $$

corresponding to $\epsilon_H \ll 1$.

A necessary condition for the validity of the slow-roll approxi-
mation is that the potential slow-roll parameters

$$ \epsilon \equiv \frac{M_P^2}{16\pi} \left(\frac{V'}{V}\right)^2, \quad \eta \equiv \frac{M_P^2}{8\pi} \left(\frac{V''}{V}\right), \qquad (23.15) $$

are small, i.e., $\epsilon \ll 1$ and $|\eta| \ll 1$, requiring the potential to be correspondingly flat. If we identify $V''$ with the effective mass of the field, we see that the slow-roll approximation requires that the mass of the scalar field must be small compared with the Hubble scale. We note that the Hubble slow-roll parameter (Eq. (23.11)) coincides with the potential slow-roll parameter, $\epsilon_H \simeq \epsilon$, to leading order in the slow-roll approximation.

The slow-roll approximation allows one to determine the Hubble expansion rate as a function of the scalar field value, and vice versa. In particular, we can express, in terms of the scalar field value during inflation, the total logarithmic expansion, or number of “e-folds”:

$$ \begin{aligned} N_* &\equiv \ln \left( \frac{R_{end}}{R_*} \right) \\ &= \int_{t_*}^{t_{end}} H dt \approx - \int_{\phi_*}^{\phi_{end}} \sqrt{\frac{4\pi}{\epsilon}} \frac{d\phi}{M_P} \quad \text{for } V' > 0. \end{aligned} \qquad (23.16) $$

Given that the slow-roll parameters are approximately constant
during slow-roll inflation, $d\epsilon/dN \simeq 2\epsilon(\eta - 2\epsilon) = \mathcal{O}(\epsilon^2)$, we have

$$ N_* \approx \frac{4 \Delta \phi}{\sqrt{\epsilon} M_P}. \qquad (23.17) $$

Since we require $N > 40$ to solve the flatness, horizon and entropy problems of the standard Big Bang cosmology, we require either very slow roll, $\epsilon < 0.01$, or a large change in the value of the scalar field relative to the Planck scale, $\Delta\phi > M_P$.

### 23.2.2 Reheating

Slow-roll inflation can lead to an exponentially large Universe, close to spatial flatness and homogeneity, but the energy density is locked in the potential energy of the scalar field, and needs to be converted to particles and thermalised to recover a hot Big Bang cosmology at the end of inflation [26, 27]. This process is usually referred to as reheating, although there was not necessarily any preceding thermal era. Reheating can occur when the scalar field evolves towards the minimum of its potential, converting the potential energy first to kinetic energy. This can occur either through the breakdown of the slow-roll condition in single-field models, or due to an instability triggered by the inflaton reaching
a critical value, in multi-field models known as hybrid inflation models [28].

Close to a simple minimum, the scalar field potential can be described by a quadratic function, $V = m^2\phi^2/2$, where $m$ is the mass of the field. We can obtain slow-roll inflation in such a potential at large field values, $\phi \gg M_P$. However, for $\phi \ll M_P$ the field approaches an oscillatory solution:

$$ \phi(t) \approx \frac{M_P}{\sqrt{3\pi}} \frac{\sin(mt)}{mt}. \qquad (23.18) $$

For $|\phi| < M_P$ the Hubble rate drops below the inflaton mass, $H < m$, and the field oscillates many times over a Hubble time. Averaging over several oscillations, $\Delta t \gg m^{-1}$, we find $\langle\phi^2/2\rangle_{\Delta t} \simeq \langle m^2\phi^2/2\rangle_{\Delta t}$ and hence

$$ \langle\rho\rangle_{\Delta t} \approx \frac{M_P^2}{6\pi t^2}, \quad \langle p\rangle_{\Delta t} \approx 0. \qquad (23.19) $$

This coherent oscillating field corresponds to a condensate of non-relativistic massive inflaton particles, driving a matter-dominated era at the end of inflation, with scale factor $R \propto t^{2/3}$.

The inflaton condensate can lose energy through perturbative decays due to terms in the interaction Lagrangian, such as

$$ \mathcal{L}_{int} \subset -\lambda_i \sigma \phi \chi_i^2 - \lambda_j \phi \bar{\psi}_j \psi_j \qquad (23.20) $$

that couple the inflation to scalar fields $\chi_i$ or fermions $\psi_j$, where $\sigma$ has dimensions of mass and the $\lambda_i$ are dimensionless couplings. When the mass of the inflaton is much larger than the decay products, the decay rate is given by [29]

$$ \Gamma_i = \frac{\lambda_i^2 \sigma^2}{8\pi m}, \quad \Gamma_j = \frac{\lambda_j^2 m}{8\pi}. \qquad (23.21) $$

These decay products must in turn thermalise with Standard Model particles before we recover conventional hot Big Bang cosmology. An upper limit on the reheating temperature after inflation is given by [27]

$$ T_{rh} = 0.2 \left( \frac{100}{g_*} \right)^{1/4} \sqrt{M_P \Gamma_{tot}}, \qquad (23.22) $$

where $g_*$ is the effective number of degrees of freedom and $I_{tot}$ is the total decay rate for the inflaton, which is required to be less than $m$ for perturbative decay.

The baryon asymmetry of the Universe must be generated after the main release of entropy during inflation, which is an important constraint on possible models. Also, the fact that the inflaton mass is much larger than the mass scale of the Standard Model opens up the possibility that it may decay into massive stable or metastable particles that could be connected with dark matter, constraining possible models. For example, in the context of supergravity models the reheat temperature is constrained by the requirement that gravitinos are not overproduced, potentially destroying the successes of Big Bang nucleosynthesis. For a range of gravitino masses one must require $T_{rh} < 10^9$ GeV [30, 31].

The process of inflaton decay and reheating can be significantly altered by interactions leading to space-time dependences in the effective masses of the fields. In particular, parametric resonance can lead to explosive, non-perturbative decay of the inflaton in some cases, a process often referred to as preheating [26, 32]. For example, an interaction term of the form

$$ \mathcal{L}_{int} \subset -\lambda^2 \phi^2 \chi^2, \qquad (23.23) $$

leads to a time-dependent effective mass for the $\chi$ field as the inflaton $\phi$ oscillates. This can lead to non-adiabatic particle production if the bare mass of the $\chi$ field is small for large couplings or for rapid changes of the inflaton field. The process of preheating is highly model-dependent, but it highlights the possible role of non-thermal particle production after and even during inflation.

## 23.3 Primordial Perturbations from Inflation

Although inflation was originally discussed as a solution to the problem of initial conditions required for homogeneous and isotropic hot Big Bang cosmology, it was soon realised that inflation also offered a mechanism to generate the inhomogeneous initial conditions required for the formation of large-scale structure [15–17, 33].

### 23.3.1 Metric Perturbations

In a homogeneous classical inflationary cosmology driven by a scalar field, the inflaton field is uniform on constant-time hypersurfaces, $\phi = \phi_0(t)$. However, quantum fluctuations inevitably break the spatial symmetry leading to an inhomogeneous field:

$$ \phi(t, x^i) = \phi_0(t) + \delta\phi(t, x^i). \qquad (23.24) $$

At the same time, one should consider inhomogeneous perturbations of the RW spacetime metric (see, e.g., [34–36]):

$$ ds^2 = (1+2A)dt^2 - 2RB_i dtdx^i - R^2 [(1+2C)\delta_{ij} + \partial_i \partial_j E + h_{ij}] dx^i dx^j, \qquad (23.25) $$

where A, B, E and C are scalar perturbations while $h_{ij}$ represents transverse and tracefree, tensor metric perturbations. Vector metric perturbations can be eliminated using Einstein constraint equations in a scalar field cosmology.

The tensor perturbations remain invariant under a temporal gauge transformation $t \to t + \delta t(t, x^i)$, but both the scalar field and the scalar metric perturbations transform. For example, we have

$$ \delta\phi \to \delta\phi - \dot{\phi}_0\delta t, \quad C \to C - H\delta t. \qquad (23.26) $$

However, there are gauge invariant combinations, such as [37]

$$ Q = \delta\phi - \frac{\dot{\phi}_0}{H}C, \qquad (23.27) $$

which describes the scalar field perturbations on spatially-flat ($C=0$) hypersurfaces. This is simply related to the curvature perturbation on uniform-field ($\delta\phi=0$) hypersurfaces:

$$ \mathcal{R} = C - \frac{H}{\dot{\phi}_0} \delta\phi = - \frac{H}{\dot{\phi}_0} Q, \qquad (23.28) $$

which coincides in slow-roll inflation, $\rho \simeq \rho(\phi)$, with the curvature perturbation on uniform-density hypersurfaces [16]

$$ \zeta = C - \frac{H}{\dot{\rho}_0} \delta\rho. \qquad (23.29) $$

Thus scalar field and scalar metric perturbations are coupled by the evolution of the inflaton field.

### 23.3.2 Gravitational waves from inflation

The tensor metric perturbation, $h_{ij}$ in Eq. (23.25), is gauge-invariant and decoupled from the scalar perturbations at first order. This represents the free excitations of the spacetime, i.e., gravitational waves, which are the simplest metric perturbations to study at linear order.

Each tensor mode, with wavevector $\vec{k}$, has two linearly-independent transverse and trace-free polarization states:

$$ h_{ij}(\vec{k}) = h_{\vec{k}} q_{ij} + \bar{h}_{\vec{k}} \bar{q}_{ij}. \qquad (23.30) $$

The linearised Einstein equations then yield the same evolution equation for the amplitude as that for a massless field in RW spacetime:

$$ \ddot{h}_{\vec{k}} + 3H\dot{h}_{\vec{k}} + \frac{k^2}{R^2}h_{\vec{k}} = 0, \qquad (23.31) $$

(and similarly for $\bar{h}_{\vec{k}}$). This can be re-written in terms of the conformal time, $\eta = \int dt/R$, and the conformally rescaled field:

$$ u_{\vec{k}} = \frac{M_P R h_{\vec{k}}}{\sqrt{32\pi}}. \qquad (23.32) $$
This conformal field then obeys the wave equation for a canonical scalar field in Minkowski spacetime with a time-dependent mass:

$$v_{\vec{k}}'' + \left(k^2 - \frac{R''}{R}\right) u_{\vec{k}} = 0. \qquad (23.33)$$

During slow roll

$$\frac{R''}{R} \approx (2 - \epsilon) R^2 H^2. \qquad (23.34)$$

This makes it possible to quantise the linearised metric fluctuations, $u_{\vec{k}} \to \hat{u}_{\vec{k}}$, on sub-Hubble scales, $k^2/R^2 \gg H^2$, where the background expansion can be neglected.

Crucially, in an inflationary expansion, where $\ddot{R} > 0$, the co-moving Hubble length $H^{-1}/R = 1/\dot{R}$ decreases with time. Thus, all modes start inside the Hubble horizon and it is possible to take the initial field fluctuations to be in a vacuum state at early times or on small scales:

$$\langle u_{\vec{k}_1} u_{\vec{k}_2} \rangle = \frac{i}{2} (2\pi)^3 \delta^{(3)}(\vec{k}_1 + \vec{k}_2). \qquad (23.35)$$

In terms of the amplitude of the tensor metric perturbations, this corresponds to

$$\langle h_{\vec{k}_1} h_{\vec{k}_2} \rangle = \frac{1}{2} \frac{\mathcal{P}_t(k_1)}{4\pi k_1^3} (2\pi)^3 \delta^{(3)}(\vec{k}_1 + \vec{k}_2), \qquad (23.36)$$

where the factor 1/2 appears due to the two polarization states that contribute to the total tensor power spectrum:

$$\mathcal{P}_t(k) = \frac{64\pi}{M_P^2} \left(\frac{k}{2\pi R}\right)^2. \qquad (23.37)$$

On super-Hubble scales, $k^2/R^2 \ll H^2$, we have the growing mode solution to Eq. (23.33), $u_{\vec{k}} \propto R$, corresponding to $h_{\vec{k}} \to$ constant, i.e., tensor modes are frozen-in on super-Hubble scales, both during and after inflation. Thus, connecting the initial vacuum fluctuations on sub-Hubble scales to the late-time power spectrum for tensor modes at Hubble exit during inflation, $k = R_*H_*$, we obtain

$$\mathcal{P}_t(k) \approx \frac{64\pi}{M_P^2} \left(\frac{H_*}{2\pi}\right)^2. \qquad (23.38)$$

In the de Sitter limit, $\epsilon \to 0$, the Hubble rate becomes time-independent and the tensor spectrum on super-Hubble scales becomes scale-invariant [39]. However slow-roll evolution leads to weak time dependence of $H_*$ and thus a scale-dependent spectrum on large scales, with a spectral tilt

$$n_t \equiv \frac{d \ln \mathcal{P}_t}{d \ln k} \approx -2\epsilon_* . \qquad (23.39)$$

### 23.3.3 Density Perturbations from single-field inflation

The inflaton field fluctuations on spatially-flat hypersurfaces are coupled to scalar metric perturbations at first order, but these can be eliminated using the Einstein constraint equations to yield an evolution equation

$$\dot{Q}_{\vec{k}} + 3H\dot{Q}_{\vec{k}} + \left[ \frac{k^2}{R^2} + V'' - \frac{8\pi}{M_P^2 R^3} \frac{d}{dt} \left( \frac{R^3 \dot{\phi}^2}{H} \right) \right] Q_{\vec{k}} = 0. \qquad (23.40)$$

Terms proportional to $M_P^{-2}$ represent the effect on the field fluctuations of gravity at first order. As can be seen, this vanishes in the limit of a constant background field, and hence is suppressed in the slow-roll limit, but it is of the same order as the effective mass, $V'' = 3\eta H^2$, so must be included if we wish to model deviations from exact de Sitter symmetry.

This wave equation can also be written in the canonical form for a free field in Minkowski spacetime if we define [37]

$$v_{\vec{k}} \equiv R Q_{\vec{k}}, \qquad (23.41)$$

to yield

$$v_{\vec{k}}'' + \left(k^2 - \frac{z''}{z}\right) v_{\vec{k}} = 0, \qquad (23.42)$$

where we define

$$z \equiv \frac{R \dot{\phi}}{H}, \qquad \frac{z''}{z} \approx (2 + 5\epsilon - 3\eta) R^2 H^2, \qquad (23.43)$$

where the last approximate equality holds to leading order in the slow-roll approximation.

As previously done for gravitational waves, we quantise the linearised field fluctuations $v_{\vec{k}} \to \hat{v}_{\vec{k}}$ on sub-Hubble scales, $k^2/R^2 \gg H^2$, where the background expansion can be neglected. Thus, we impose

$$\langle v_{\vec{k}_1} v_{\vec{k}_2}' \rangle = \frac{i}{2} \delta^{(3)}(\vec{k}_1 + \vec{k}_2). \qquad (23.44)$$

In terms of the field perturbations, this corresponds to

$$\langle Q_{\vec{k}_1} Q_{\vec{k}_2} \rangle = \frac{\mathcal{P}_Q(k_1)}{4\pi k_1^3} (2\pi)^3 \delta^{(3)}(\vec{k}_1 + \vec{k}_2), \qquad (23.45)$$

where the power spectrum for vacuum field fluctuations on sub-Hubble scales, $k^2/R^2 \gg H^2$, is simply

$$\mathcal{P}_Q(k) = \left(\frac{k}{2\pi R}\right)^2, \qquad (23.46)$$

yielding the classic result for the vacuum fluctuations for a massless field in de Sitter at Hubble exit, $k = R_*H_*$:

$$\mathcal{P}_Q(k) \approx \left(\frac{H}{2\pi}\right)_*^2. \qquad (23.47)$$

In practice there are slow-roll corrections due to the small but finite mass ($\eta$) and field evolution ($\epsilon$) [40].

Slow-roll corrections to the field fluctuations are small on sub-Hubble scales, but can become significant as the field and its perturbations evolve over time on super-Hubble scales. Thus, it is helpful to work instead with the curvature perturbation, $\zeta$ defined in equation (Eq. (23.29)), which remains constant on super-Hubble scales for adiabatic density perturbations both during and after inflation [16,41]. Thus we have an expression for the primordial curvature perturbation on super-Hubble scales produced by single-field slow-roll inflation:

$$\mathcal{P}_{\zeta}(k) = \left[ \left( \frac{H}{\dot{\phi}} \right)^2 \mathcal{P}_Q(k) \right]_* \approx \frac{4\pi}{M_P^2} \left[ \frac{1}{\epsilon} \left( \frac{H}{2\pi} \right)^2 \right]_* . \qquad (23.48)$$

Comparing this with the primordial gravitational wave power spectrum (Eq. (23.38)) we obtain the tensor-to-scalar ratio

$$r \equiv \frac{\mathcal{P}_t}{\mathcal{P}_{\zeta}} \approx 16\epsilon_* . \qquad (23.49)$$

Note that the scalar amplitude is boosted by a factor $1/\epsilon_*$ during slow-roll inflation, because small scalar field fluctuations can lead to relatively large curvature perturbations on hypersurfaces defined with respect to the density if the potential energy is only weakly dependent on the scalar field, as in slow roll. Indeed, the de Sitter limit is singular, since the potential energy becomes independent of the scalar field at first order, $\epsilon \to 0$, and the curvature perturbation on uniform-density hypersurfaces becomes ill-defined.

We note that in single-field inflation the tensor-to-scalar ratio and the tensor tilt (Eq. (23.39)) at the same scale are both determined by the first slow-roll parameter at Hubble exit, $\epsilon_*$, giving rise to an important consistency test for single-field slow-roll inflation:

$$n_t \approx -\frac{r}{8}. \qquad (23.50)$$

This may be hard to verify if $r$ is small, making any tensor tilt $n_t$ difficult to measure. On the other hand, it does offer a way to rule out single-field slow-roll inflation if either $r$ or $n_t$ is large.

Given the relatively large scalar power spectrum, it has proved easier to measure the scalar tilt, conventionally defined as $n_s - 1$.
<figure><img src="image_133.png" /><figcaption>Figure 23.1: The marginalized joint 68 and 95% CL regions for the tilt in the scalar perturbation spectrum, $n_s$, and the relative magnitude of the tensor perturbations, $r$, obtained from the Planck 2018 and lensing data alone, and their combinations with BICEP2/Keck Array (BK15) and (optionally) BAO data, confronted with the predictions of some of the inflationary models discussed in this review. This figure is taken from [38].</figcaption></figure>

Slow-roll corrections lead to slow time-dependence of both $H_*$ and $\epsilon_*$, giving a weak scale-dependence of the scalar power spectrum:

$$n_s - 1 \equiv \frac{d \ln P_{\zeta}}{d \ln k} \approx -6\epsilon_* + 2\eta_*, \quad (23.51)$$

and a running of this tilt at second-order in slow roll:

$$\frac{dn_s}{d \ln k} \approx -8\epsilon_*(3\epsilon_* - 2\eta_*) - 2\xi_*^2, \quad (23.52)$$

where the running introduces a new slow-roll parameter at second-order:

$$\xi^2 = \frac{M_P^4 V'V'''}{64\pi^2 V^2}. \quad (23.53)$$

### 23.3.4 Observational Bounds

The observed scale-dependence of the power spectrum makes it necessary to specify the comoving scale, $k$, at which quantities are constrained and hence the Hubble-exit time, $k = a_* H_*$, when the corresponding theoretical quantities are calculated during inflation. This is usually expressed in terms of the number of e-folds from the end of inflation [42]:

$$N_*(k) \approx 67 - \ln \left( \frac{k}{a_0 H_0} \right) + \frac{1}{4} \ln \left( \frac{V_*^2}{M_P^4 \rho_{end}} \right) + \frac{1}{12} \ln \left( \frac{\rho_{rh}}{\rho_{end}} \right) - \frac{1}{12} \ln(g_*) \quad (23.54)$$

where $H_0^{-1}/a_0$ is the present comoving Hubble length. Different models of reheating and thus different reheat temperatures and densities, $\rho_{rh}$ in Eq. (23.54), lead to a range of possible values for $N_*$ corresponding to a fixed physical scale, and hence we have a range of observational predictions for a given inflation model, as seen in Fig. 23.1.

The Planck 2018 temperature and polarization data (see Chap. 29, “Cosmic Microwave Background” review) are consistent with a smooth featureless power spectrum over a range of comoving

wavenumbers, $0.008 h^{-1} \text{ Mpc}^{-1} \le k \le 0.1 h \text{ Mpc}^{-1}$. In the absence of running, the data measure the spectral index to be [38]

$$n_s = 0.9649 \pm 0.0042, \quad (23.55)$$

corresponding to a deviation from scale-invariance exceeding the $7\sigma$ level. If running of the spectral tilt is included in the model, this is constrained to be [38]

$$\frac{dn_s}{d \ln k} = -0.0045 \pm 0.0067 \quad (23.56)$$

at the 95% CL, assuming no running of the running. The most recent analysis [43] of the BICEP, Keck Array and Planck data places an upper bound on the tensor-to-scalar ratio

$$r < 0.036, \quad (23.57)$$

at the 95% CL.

These observational bounds can be converted into bounds on the slow-roll parameters and hence the potential during slow-roll inflation. Setting higher-order slow-roll parameters (beyond second-order in horizon-flow parameters [44]) to zero, the Planck collaboration obtain the following 95% CL bounds when lensing and BK15 data are included [38]

$$\epsilon < 0.0044, \quad (23.58)$$

$$\eta = -0.015 \pm 0.006, \quad (23.59)$$

$$\xi^2 = 0.0029^{+0.0073}_{-0.0069}, \quad (23.60)$$

which can be used to constrain models, as discussed in the next Section.

Fig. 23.1, which is taken from [38], compares observational CMB constraints on the tilt, $n_s$, in the spectrum of scalar perturbations and the ratio, $r$, between the magnitudes of tensor and scalar perturbations. Important rôles are played by data from the Planck satellite and on lensing, the BICEP2/Keck Array (BK15) and measurements of baryon acoustic oscillations (BAO). The reader is referred to [38] for technical details. These experimental constraints are compared with the predictions of some
<figure><img src="image_134.png" /><figcaption>**Figure 23.2:** The result of reconstructing a single-field inflaton potential using a cubic-spline power-spectrum mode expansion and the full *Planck*, lensing, BK15 and BAO data set. This figure is taken from [38].</figcaption></figure>

of the inflationary models discussed in this review. Generally speaking, models with a concave potential are favored over those with a convex potential, and models with power-law inflation are now excluded, as opposed to models with de Sitter-like (quasi-exponential expansion).

There is no significant evidence for local features within the range of inflaton field values probed by the data [38]. However, the data may be used to reconstruct partially the effective inflationary potential over a range of inflaton field values, assuming that it is suitably smooth. The result of one such exercise by the *Planck* collaboration [38] in the framework of a generic single-field inflaton potential is shown in Fig. 23.2. This reconstruction assumes a cubic-spline power-spectrum mode expansion and employs the full *Planck*, lensing, BK15 and BAO data set. The reader is again referred to [38] for technical details. We see that the effective inflaton potential is relatively well reconstructed over field values $\phi$ within $\pm 0.5$ of the chosen pivot value, but the potential is only very weakly constrained for larger values of $|\phi - \phi_{pivot}|$, providing wide scope for inflationary model-builders.

## 23.4 Models

### 23.4.1 Pioneering Models

The paradigm of the inflationary Universe was proposed in [2], where it was pointed out that an early period of (near-exponential expansion, in addition to resolving the horizon and flatness problems of conventional Big-Bang cosmology as discussed above (the possibility of a de Sitter phase in the early history of the Universe was also proposed in the non-minimal gravity model of [1], with the motivation of avoiding an initial singularity), would also dilute the prior abundance of any unseen heavy, (meta-)stable particles, as exemplified by monopoles

in grand unified theories (GUTs; see Chap. 93, “Grand Unified Theories” review). The original proposal was that this inflationary expansion took place while the Universe was in a metastable state (a similar suggestion was made in [45, 46], where in [45] it was also pointed out that such a mechanism could address the horizon problem) and was terminated by a first-order transition due to tunnelling though a potential barrier. However, it was recognized already in [2] that this ‘old inflation’ scenario would need modification if the transition to the post-inflationary Universe were to be completed smoothly without generating unacceptable inhomogeneities.

This ‘graceful exit’ problem was addressed in the ‘new inflation’ model of [13] (see also [14] and footnote [39] of [2]), which studied models based on an SU(5) GUT with an effective potential of the Coleman-Weinberg type (i.e., dominated by radiative corrections), in which inflation could occur during the roll-down from the local maximum of the potential towards a global minimum. However, it was realized that the Universe would evolve to a different minimum from the Standard Model [47], and it was also recognized that density fluctuations would necessarily be too large [15], since they were related to the GUT coupling strength.

These early models of inflation assumed initial conditions enforced by thermal equilibrium in the early Universe. However, this assumption was questionable: indeed, it was not made in the model of [1], in which a higher-order gravitational curvature term was assumed to arise from quantum corrections, and the assumption of initial thermal equilibrium was jettisoned in the ‘chaotic’ inflationary model of [48]. These are the inspirations for much recent inflationary model building, so we now discuss them in more detail, before reviewing contemporary models.
In this section we will work in natural units where we set the reduced Planck mass to unity, i.e., $8\pi/M_P^2 = 1$. All masses are thus relative to the reduced Planck scale.

### 23.4.2 $R^2$ Inflation

The first-order Einstein-Hilbert action, $(1/2) \int d^4x \sqrt{-g}R$, where $R$ is the Ricci scalar curvature, is the minimal possible theory consistent with general coordinate invariance. However, it is possible that there might be non-minimal corrections to this action, and the unique second-order possibility is

$$S = \frac{1}{2} \int d^4x \sqrt{-g} \left( R + \frac{R^2}{6M^2} \right). \qquad (23.61)$$

It was pointed out in [1] that an $R^2$ term could be generated by quantum effects, and that (Eq. (23.61)) could lead to de Sitter-like expansion of the Universe. Scalar density perturbations in this model were calculated in [17]. Because the initial phase was (almost) de Sitter, these perturbations were (approximately) scale-invariant, with magnitude $\propto M$. It was pointed out in [17] that requiring the scalar density perturbations to lie in the range $10^{-3}$ to $10^{-5}$, consistent with upper limits at that time, would require $M \sim 10^{-3}$ to $10^{-5}$ in Planck units, and it was further suggested in that these perturbations could lead to the observed large-scale structure of the Universe, including the formation of galaxies.

Although the action (Eq. (23.61)) does not contain an explicit scalar field, [17] reduced the calculation of density perturbations to that of fluctuations in the scalar curvature $R$, which could be identified (up to a factor) with a scalar field of mass $M$. The formal equivalence of $R^2$ gravity (Eq. (23.61)) to a theory of gravity with a massive scalar $\phi$ had been shown in [18], see also [19]. The effective scalar potential for what we would nowadays call the ‘inflaton’ [49] takes the form

$$S = \frac{1}{2} \int d^4x \sqrt{-g} \left[ R + (\partial_\mu \phi)^2 - \frac{3}{2} M^2 (1 - e^{-\sqrt{2/3}\phi})^2 \right] \qquad (23.62)$$

when the action is written in the Einstein frame, and the potential is shown as the solid black line in Fig. 23.3. Using (Eq. (23.48)), one finds that the amplitude of the scalar density perturbations in this model is given by

$$\Delta_R = \frac{3M^2}{8\pi^2} \sinh^4\left(\frac{\phi}{\sqrt{6}}\right), \qquad (23.63)$$

The measured magnitude of the density fluctuations in the CMB requires $M \simeq 1.3 \times 10^{-5}$ in Planck units (assuming $N_* \simeq 55$), so one of the open questions in this model is why $M$ is so small. Obtaining $N_* \simeq 55$ also requires an initial value of $\phi \simeq 5.5$, i.e., a super-Planckian initial condition, and another issue for this and many other models is how the form of the effective potential is protected and remains valid at such large field values. Using Eq. (23.51) one finds that $n_s \simeq 0.965$ for $N_* \simeq 55$ and using (Eq. (23.49)) one finds that $r \simeq 0.0035$. These predictions are consistent with the present data from Planck and other experiments, as seen in Fig. 23.1.

### 23.4.3 Chaotic Models with Power-Law Potentials

As has already been mentioned, a key innovation in inflationary model-building was the suggestion to abandon the questionable assumption of a thermal initial state, and consider ‘chaotic’ initial conditions with very general forms of potential [48]. (Indeed, the $R^2$ model discussed above can be regarded as a prototype of this approach.) The chaotic approach was first proposed in the context of a simple power-law potential of the form $\mu^{4-\alpha}\phi^{\alpha}$, and the specific example of $\lambda\phi^4$ was studied in [48]. Such models make the following predictions for the slow-roll parameters $\epsilon$ and $\eta$:

$$\epsilon = \frac{1}{2} \left(\frac{\alpha}{\phi}\right)^2, \quad \eta = \frac{\alpha(\alpha - 1)}{\phi^2}, \qquad (23.64)$$

leading to the predictions

$$r \approx \frac{4\alpha}{N_*}, \quad n_s - 1 \approx -\frac{\alpha + 2}{2N_*}, \qquad (23.65)$$

which are shown in Fig. 23.1 for some illustrative values of $\alpha$. We note that the prediction of the original $\phi^4$ model lies out of the frame, with values of $r$ that are too large and values of $n_s$ that are too small. The $\phi^3$ model has similar problems, and would in any case require modification in order to have a well-defined minimum. The simplest possibility is $\phi^2$, but this is now also disfavored by the data, at the 95% CL if only the Planck data are considered, and more strongly if other data are included, as seen in Fig. 23.1. (For non-minimal models of quadratic inflation that avoid this problem, see, e.g., [51].)

Indeed, as can be seen in Fig. 23.1, all single-field models with a convex potential (i.e., one curving upwards) are disfavored compared to models with a concave potential.

### 23.4.4 Hilltop Models

This preference for a concave potential motivates interest in ‘hilltop’ models [52], whose starting-point is a potential of the form

$$V(\phi) = \Lambda^4 \left[ 1 - \left(\frac{\phi}{\mu}\right)^p + \dots \right], \qquad (23.66)$$

where the ... represent extra terms that yield a positive semi-definite potential. To first order in the slow-roll parameters, when $x \equiv \phi/\mu$ is small, one has

$$n_s \simeq 1 - p(p-1)\mu^{-2} \frac{x^{p-2}}{(1-x^p)^2} - \frac{3}{8}r, \quad r \simeq 8p^2\mu^{-2} \frac{x^{2p-2}}{(1-x^p)^2}. \qquad (23.67)$$

As seen in Fig. 23.1, a hilltop model with $p=4$ can be compatible with the Planck and other measurements, if $\mu \gg M_P$.

### 23.4.5 D-Brane Inflation

Many scenarios for inflation involving extra dimensions have been proposed, e.g., the possibility that observable physics resides on a three-dimensional brane, and that there is an inflationary potential that depends on the distance between our brane and an antibrane, with a potential of the form [53]

$$V(\phi) = \Lambda^4 \left[ 1 - \left(\frac{\mu}{\phi}\right)^p + \dots \right]. \qquad (23.68)$$

In this scenario the effective potential vanishes in the limit $\phi \to \infty$, corresponding to complete separation between our brane and the antibrane. The predictions for $n_s$ and $r$ in this model can be obtained from (Eq. 23.67) by exchanging $p \leftrightarrow -p$, and are also consistent with the Planck and other data.

### 23.4.6 Natural Inflation

Also seen in Fig. 23.1 are the predictions of ‘natural inflation’ [54], in which one postulates a non-perturbative shift symmetry that suppresses quantum corrections, so that a hierarchically small scale of inflation, $H \ll M_P$, is technically natural. In the simplest models, there is a periodic potential of the form

$$V(\phi) = \Lambda^4 \left[ 1 + \cos\left(\frac{\phi}{f}\right) \right], \qquad (23.69)$$

where $f$ is a dimensional parameter reminiscent of an axion decay constant (see the next subsection) [55], which must have a value $> M_P$. Natural inflation can yield predictions similar to quadratic inflation (which are no longer favored, as already discussed), but can also yield an effective convex potential. Thus, it may lead to values of $r$ that are acceptably small, but for values of $n_s$ that are in tension with the data, as seen in Fig. 23.1.

### 23.4.7 Axion Monodromy Models

The effective potentials in stringy models [56, 57] motivated by axion monodromy may be of the form

$$V(\phi) = \mu^{4-\alpha}\phi^{\alpha} + \Lambda^4 e^{-C\left(\frac{\phi}{\phi_0}\right)^{p\Lambda}} \cos\left[\gamma + \frac{\phi}{f}\left(\frac{\phi}{\phi_0}\right)^{p_f+1}\right], \qquad (23.70)$$

where $\mu, \Lambda, f$ and $\phi_0$ are parameters with the dimension of mass, and $C, p, p_\Lambda, p_f$ and $\gamma$ are dimensionless constants, generalizing the potential ([54]) in the simplest models of natural inflation. The oscillations in (Eq. 23.70) are associated with the axion field, and powers $p_\Lambda, p_f \neq 0$ may arise from $\phi$-dependent evolutions of
<figure><img src="image_135.png" /><figcaption>Figure 23.3: The inflationary potential V in the R² model (solid black line) compared with its form in various no-scale models discussed in detail in [50] (dashed coloured lines).</figcaption></figure>

string moduli. Since the exponential prefactor in (Eq. 23.70) is due to non-perturbative effects that may be strongly suppressed, the oscillations may be unobservable small. Specific string models having $\phi^\alpha$ with $\alpha = 4/3, 1$ or $2/3$ have been constructed in [56,57], providing some motivation for the low-power models mentioned above.

As seen in Fig. 23.1, the simple axion monodromy models with the power $\alpha = 4/3$ or 1 are no longer compatible with the current CMB data at the 95% CL, while $\alpha = 2/3$ is only marginally compatible at 95% CL. The Planck Collaboration has also searched for characteristic effects associated with the second term in (Eq. (23.70)), such as a possible drift in the modulation amplitude (setting $p_A = C = 0$), and a possible drifting frequency generated by $p_f \neq 0$, without finding any compelling evidence [38].

### 23.4.8 Higgs Inflation

Since the energy scale during inflation is commonly expected to lie between the Planck and TeV scales, it may serve as a useful bridge with contacts both to string theory or some other quantum theory of gravity, on the one side, and particle physics on the other side. However, as the above discussion shows, much of the activity in building models of inflation has been largely independent of specific connections with these subjects, though some examples of string-motivated models of inflation were mentioned above.

The most economical scenario for inflation might be to use as inflaton the only established scalar field, namely the Higgs field (see Chap.11, “Status of Higgs boson physics” review). A specific model assuming a non-minimal coupling of the Higgs field $h$ to gravity was constructed in [58]. Its starting-point is the action

$$S = \int d^4x \sqrt{-g} \left[ \frac{M^2 + \xi h^2}{2} R + \frac{1}{2} \partial_{\mu} h \partial^{\mu} h - \frac{\lambda}{4} (h^2 - v^2)^2 \right], \quad (23.71)$$

where $v$ is the Higgs vacuum expectation value. The model requires $\xi \gg 1$, in which case it can be rewritten in the Einstein

frame as

$$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2} R + \frac{1}{2} \partial_{\mu} \chi \partial^{\mu} \chi - U(\chi) \right], \quad (23.72)$$

where the effective potential for the canonically-normalized inflaton field $\chi$ has the form

$$U(\chi) = \frac{\lambda}{4\xi^2} \left[ 1 + \exp \left( -\frac{2\chi}{\sqrt{6}M_P} \right) \right]^{-2}, \quad (23.73)$$

which is similar to the effective potential of the R² model at large field values. As such, the model inflates successfully if $\xi \simeq 5 \times 10^4 m_h/(\sqrt{2}v)$, with predictions for $n_s$ and $r$ that are indistinguishable from the predictions of the R² model shown in Fig. 23.1.

This model is very appealing, but must confront several issues. One is to understand the value of $\xi$, and another is the possibility of unitarity violation. However, a more fundamental issue is whether the effective quartic Higgs coupling is positive at the scale of the Higgs field during inflation. Extrapolations of the effective potential in the Standard Model using the measured values of the masses of the Higgs boson and the top quark indicate that probably $\lambda < 0$ at this scale [59], though there are still significant uncertainties associated with the appropriate input value of the top mass and the extrapolation to high renormalization scales.

### 23.4.9 Supersymmetric Models of Inflation

Supersymmetry [60] is widely considered to be a well-motivated possible extension of the Standard Model that might become apparent at the TeV scale. It is therefore natural to consider supersymmetric models of inflation. These were originally proposed because of the problems of the new inflationary theory [13, 14] based on the one-loop (Coleman-Weinberg) potential for breaking SU(5). Several of these problems are related to the magnitude of the effective potential parameters: in any model of inflation based on an elementary scalar field, some parameter in the effective potential must be small in natural units, e.g., the quartic
coupling $\lambda$ in a chaotic model with a quartic potential, or the mass parameter $\mu$ in a model of chaotic quadratic inflation. These parameters are renormalized multiplicatively in a supersymmetric theory, so that the quantum corrections to small values would be

under control. Hence, it was suggested that inflation cries out for supersymmetry [61], though non-supersymmetric resolutions of the problems of Coleman-Weinberg inflation are also possible: see, e.g., Ref. [62].

<figure><img src="image_136.png" /><figcaption>**Figure 23.4:** The Bayes factors calculated in [63] for a large sample of inflationary models using Planck 2015 data [64]. Those highlighted in yellow are featured in this review, according to the numbers listed in the text.</figcaption></figure>

In the Standard Model there is only one scalar field that could be a candidate for the inflaton, namely the Higgs field discussed above, but even the minimal supersymmetric extension of the Standard Model (MSSM) contains many scalar fields. However, none of these is a promising candidate for the inflaton. The minimal extension of the MSSM that may contain a suitable candidate is the supersymmetric version of the minimal seesaw model of neutrino masses, which contains the three supersymmetric partners of the heavy singlet (right-handed) neutrinos. One of these singlet sneutrinos $\tilde{\nu}$ could be the inflaton [65]: it would have a quadratic potential, the mass coefficient required would be $\sim 10^{13}$ GeV, very much in the expected ball-park for singlet (right-handed) neutrino masses, and sneutrino inflaton decays also could give rise to the cosmological baryon asymmetry via leptogenesis. However, as seen in Fig. 23.1 and already discussed, a purely quadratic inflationary potential is no longer favored by the data. This difficulty could in principle be resolved in models with multiple sneutrinos [66], or by postulating a trilinear sneutrino coupling and hence a superpotential of Wess-Zumino type [67], which can yield successful inflation with predictions intermediate between those of natural inflation and hilltop inflation in Fig. 23.1.

Finally, we note that it is also possible to obtain inflation via supersymmetry breaking, as in the model [68] whose predictions are illustrated in Fig. 23.1.

### 23.4.10 Supergravity Models

Any model of early-Universe cosmology, and specifically inflation, must necessarily incorporate gravity. In the context of supersymmetry this requires an embedding in some supergravity theory [69, 70]. An $N = 1$ supergravity theory is specified by three functions: a Hermitian function of the matter scalar fields $\phi^i$, called the Kähler potential $K$, that describes its geometry, a holomorphic function of the superfields, called the superpotential $W$, which describes their interactions, and another holomorphic function $f_{\alpha\beta}$, which describes their couplings to gauge fields $V_\alpha$ [71].

The simplest possibility is that the Kähler metric is flat:

$$K = \phi^i \phi_i^*, \qquad (23.74)$$

where the sum is over all scalar fields in the theory, and the simplest inflationary model in minimal supergravity had the superpotential [72]

$$W = m^2(1 - \phi)^2, \qquad (23.75)$$

Where $\phi$ is the inflaton. However, this model predicts a tilted scalar perturbation spectrum, $n_s = 0.933$, which is now in serious disagreement with the data from Planck and other experiments shown in Fig. 23.1.

Moreover, there is a general problem that arises in any supergravity theory coupled to matter, namely that, since its ef-
fective scalar potential contains a factor of $e^K$, scalars typically receive squared masses $\propto H^2 \sim V$, where H is the Hubble parameter [73], an issue called the 'η problem'. The theory given by (Eq. (23.75)) avoids this η problem, but a generic supergravity inflationary model encounters this problem of a large inflaton mass. Moreover, there are additional challenges for supergravity inflation associated with the spontaneous breaking of local super-symmetry [74–76].

Various approaches to the η problem in supergravity have been proposed, including the possibility of a shift symmetry [77], and one possibility that has attracted renewed attention recently is no-scale supergravity [78, 79]. This is a form of supergravity with a Kähler potential that can be written in the form [80]

$$K = -3 \ln \left( T + T^* - \frac{\sum_i |\phi^i|^2}{3} \right), \quad (23.76)$$

which has the special property that it naturally has a flat potential, at the classical level and before specifying a non-trivial superpotential. As such, no-scale supergravity is well-suited for constructing models of inflation. Adding to its attraction is the feature that compactifications of string theory to supersymmetric four-dimensional models yield effective supergravity theories of the no-scale type [81]. There are many examples of superpotentials that yield effective inflationary potentials for either the T field (which is akin to a modulus field in some string compactification) or a φ field (generically representing matter) that are of the same form as the effective potential of the $R^2$ model (Eq. (23.62)) when the magnitude of the inflaton field $\gg 1$ in Planck units, as required to obtain sufficiently many e-folds of inflation, $N_*$ [82, 83]. This framework also offers the possibility of using a suitable superpotential to construct models with effective potentials that are similar, but not identical, to the $R^2$ model, as shown by the dashed coloured lines in Fig. 23.3.

### 23.4.11 Other Exponential Potential Models

This framework also offers the possibility [82] of constructing models in which the asymptotic constant value of the potential at large inflaton field values is approached via a different exponentially-suppressed term:

$$V(\phi) = A \left[1 - \delta e^{-B\phi} + \mathcal{O}(e^{-2B\phi})\right], \quad (23.77)$$

where the magnitude of the scalar density perturbations fixes A, but δ and B are regarded as free parameters. In the case of $R^2$ inflation $\delta = 2$ and $B = \sqrt{2/3}$. In a model such as (Eq. (23.77)), one finds at leading order in the small quantity $e^{-B\phi}$ that

$$\begin{aligned} n_s &= 1 - 2B^2 \delta e^{-B\phi}, \\ r &= 8B^2 \delta^2 e^{-2B\phi}, \\ N_* &= \frac{1}{B^2 \delta} e^{+B\phi}. \end{aligned} \quad (23.78)$$

yielding the relations

$$n_s = 1 - \frac{2}{N_*}, \quad r = \frac{8}{B^2 N_*^2}. \quad (23.79)$$

These relations correspond to the class of models labeled as 'α attractors' [84] in Fig. 23.1. CMB bounds on the tensor-to-scalar ratio translate into an upper bound on the parameter $\alpha = 2/3B^2$ in α-attractor models, which determines the curvature of the Kähler manifold in supergravity models [85]. There are generalizations of the simplest no-scale model (Eq. (23.76)) with prefactors before the $\ln(\dots)$ that are 1 or 2, leading to larger values of $B = \sqrt{2}$ or 1, respectively, and hence smaller values of r than in the $R^2$ model. CMB constraints on the scalar spectral index can also place a lower bound on the magnitude of $\alpha$ [86], due to the implicit dependence of $N_*$ upon $\alpha$ [87].

## 23.5 Model Comparison

Given a particular inflationary model, one can obtain constraints on the model parameters, informed by the likelihood, corresponding to the probability of the data given a particular

choice of parameters (see Sec. 40, “Statistics” review). In the light of the detailed constraints on the statistical distribution of primordial perturbations now inferred from high-precision observations of the cosmic microwave background, it is also possible to make quantitative comparison of the statistical evidence for or against different inflationary models. This can be done either by comparing the logarithm of the maximum likelihood that can be obtained for the data using each model, i.e., the minimum $\chi^2$ (with some correction for the number of free parameters in each model), or by a Bayesian model comparison [88] (see also Sec. 40.3.4 in “Statistics” review).

In such a Bayesian model comparison one computes [7] the evidence, $\mathcal{E}(D|M_A)$ for a model, $M_A$, given the data $D$. This corresponds to the likelihood, $\mathcal{L}(\theta_{Aj}) = p(D|\theta_{Aj}, M_A)$, integrated over the assumed prior distribution, $\pi(\theta_{Aj}|M_A)$, for all the model parameters $\theta_{Aj}$:

$$\mathcal{E}(D|M_A) = \int \mathcal{L}(\theta_{Aj}) \pi(\theta_{Aj}|M_A) d\theta_{Aj}. \quad (23.80)$$

The posterior probability of the model given the data follows from Bayes' theorem

$$p(M_A|D) = \frac{\mathcal{E}(D|M_A)\pi(M_A)}{p(D)}, \quad (23.81)$$

where the prior probability of the model is given by $\pi(M_A)$. Assuming that all models are equally likely a priori, $\pi(M_A) = \pi(M_B)$, the relative probability of model A relative to a reference model, in the light of the data, is thus given by the Bayes factor

$$B_{A,ref} = \frac{\mathcal{E}(D|M_A)}{\mathcal{E}(D|M_{ref})}. \quad (23.82)$$

Computation of the multi-dimensional integral (Eq. (23.80)) is a challenging numerical task. Even using an efficient sampling algorithm requires hundreds of thousands of likelihood computations for each model, though slow-roll approximations can be used to calculate rapidly the primordial power spectrum using the APSIC numerical library [7] for a large number of single-field, slow-roll inflation models.

The change in $\chi^2$ for selected slow-roll models relative to the Starobinsky $R^2$ inflationary model, used as a reference, is given in Table 23.1 (taken from [38]). All the other inflation models require a substantial amplitude of tensor modes, and so have an increased $\chi^2$ with respect to the Starobinsky and other models with a scalar tilt but small tensor modes. Table 23.1 also shows the Bayesian evidence for $(\ln B_{A,ref} > 0)$ or against $(\ln B_{A,ref} < 0)$ a selection of inflation models using the Planck analysis priors [38]. The Starobinsky $R^2$ inflationary model may be chosen as a reference [38] that provides a good fit to current data. Higgs inflation [58] is indistinguishable using current data, making the model comparison “inconclusive” on the Jeffrey’s scale ($|\ln B_{A,ref}| < 1$). (Recall, though, that this model is disfavored by the measured values of the Higgs and top quark masses [59].) We note that although α-attractor models can provide a good fit to the data, they are disfavored relative to the Starobinsky model due to their larger prior volume. There is now strong evidence ($|\ln B_{A,ref}| > 5$) against large-field models such as chaotic inflation with a quadratic or a quartic potential. Indeed, over 30% of the slow-roll inflation models considered in Ref. [7] are strongly disfavored by the Planck data.

The Bayes factors for a wide selection of slow-roll inflationary models are displayed in Fig. 23.4, which is adapted from Fig. 3 in [63], where more complete descriptions of the models and the calculations of the Bayes factors using Planck 2015 data [64] are given. Models discussed in this review are highlighted in yellow, and numbered as follows: (1) $R^2$ inflation (Sec. 23.4.2) and models with similar predictions, such as Higgs inflation (Sec. 23.4.8) and no-scale supergravity inflation (Sec. 23.4.10); chaotic inflation models (2) with a $\phi^2$ potential; (3) with a $\phi^4$ potential; (4) with a $\phi^{2/3}$ potential, and (5) with a $\phi^p$ potential marginalising over $p \in [0.2, 6]$ (Sec. 23.4.3); hilltop inflation models (6) with $p = 2$; (7) with $p = 4$ and (8) marginalising over $p$ (Sec. 23.4.4); (9) brane
<table><caption>**Table 23.1:** Observational evidence for and against selected inflation models: $\Delta\chi^2$ and the Bayes factors are calculated relative to the Starobinsky $R^2$ inflationary model, which is treated as a reference. Results from Planck 2018 analysis [38].</caption><thead><tr><th>Model</th><th>Δχ<sup>2</sup></th><th>ln B<sub>A,ref</sub></th></tr></thead><tbody><tr><td>R<sup>2</sup> inflation</td><td>0</td><td>0</td></tr><tr><td>Power-law potential φ<sup>2/3</sup></td><td>+4.0</td><td>-4.6</td></tr><tr><td>Power-law potential φ<sup>2</sup></td><td>+21.6</td><td>&lt; -10</td></tr><tr><td>Power-law potential φ<sup>4</sup></td><td>+75.3</td><td>&lt; -10</td></tr><tr><td>Natural inflation</td><td>+9.9</td><td>-6.6</td></tr><tr><td>Hilltop quartic model</td><td>-0.3</td><td>-1.4</td></tr></tbody></table>

inflation Sec. 23.4.5); (10) natural inflation (Sec. 23.4.6); (11) exponential potential models such as α-attractors (Sec. 23.4.11). As seen in Fig. 23.4 and discussed in the next Section, constraints on reheating are starting to provide additional information about models of inflation.

## 23.6 Constraints on Reheating

One connection between inflation and particle physics is provided by inflaton decay, whose products are expected to have thermalized subsequently. As seen in Eq. (23.54), the number of e-folds required during inflation depends on details of this reheating process, including the matter density upon reheating, denoted by $\rho_{th}$, which depends in turn on the inflaton decay rate $\Gamma_{\phi}$. We see in Fig. 23.1 that, within any specific inflationary model, both $n_s$ and particularly $r$ are sensitive to the value of $N_*$ in particular, the uncertainty in the experimental measurement of $n_s$ is comparable to the variation in many model predictions for $N_* \in [50, 60]$. This implies that the data start to constrain scenarios for inflaton decay in many models. For example, it is clear from Fig. 23.1 that $N_* = 60$ would be preferred over $N_* = 50$ in a chaotic inflationary model with a quadratic potential.

As a specific example, let us consider α-attractor models that predict small values of $r$. As seen in Fig. 23.1, within these models the combination of Planck, BICEP2/Keck Array and BAO data would require a limited range of $n_s$, corresponding to a limited range of $N_*$, as seen by comparing the left and right vertical axes in Fig. 23.5:

$$N_* \gtrsim 52 \quad (68\% \text{ CL}), \quad N_* \gtrsim 44 \quad (95\% \text{ CL}). \quad (23.83)$$

Within any specific model for inflaton decay, these bounds can be translated into constraints on the effective decay coupling. For example, if one postulates a two-body inflaton decay (Eq. (23.21)), the bounds (Eq. (23.83)) can be translated into bounds on $y^2 \equiv 8\pi\Gamma/m$. This is illustrated in Fig. 23.5 [85], where any value of $N_{0.05}$ (the number of e-folds for a scale 0.05 Mpc$^{-1}$, on the left vertical axis), corresponding to the scalar tilt $n_s$ shown on the right vertical axis, when projected onto the diagonal line representing one of the three illustrated choices of α-attractor models, corresponds to a specific value of $y$ (lower horizontal axis) and $T_{reh}$ (upper horizontal axis). For example, for $\alpha = 1$ one has

$$y \gtrsim 10^{-10} \quad (68\% \text{ CL}), \quad y \gtrsim 10^{-17} \quad (95\% \text{ CL}). \quad (23.84)$$

These bounds are not very constraining, but they can be expected to improve significantly in the coming years and thereby provide significant information on the connections between inflation and particle physics.

## 23.7 Beyond Single-Field Slow-Roll Inflation

There are numerous possible scenarios beyond the simplest single-field models of slow-roll inflation. These include models with features in the potential leading to deviations from slow roll, as well as theories in which non-canonical fields are considered, such as k-inflation [89] or DBI inflation [90], and multiple-field models, such as the curvaton scenario [91]. As well as altering the single-field slow-roll predictions for the primordial curvature power spectrum (Eq. (23.48)) and the tensor-scalar ratio (Eq. (23.49)), they may introduce new quantities that vanish in

single-field slow-roll models, such as isocurvature matter perturbations, corresponding to entropy fluctuations in the photon-to-matter ratio, at first order:

$$S_m = \frac{\delta n_m}{n_m} - \frac{\delta n_\gamma}{n_\gamma} = \frac{\delta \rho_m}{\rho_m} - \frac{3}{4} \frac{\delta \rho_\gamma}{\rho_\gamma}. \quad (23.85)$$

Another possibility is non-Gaussianity in the distribution of the primordial curvature perturbation (see Chap. 29, “Cosmic Microwave Background” review), encoded in higher-order correlators such as the primordial bispectrum [92]

$$\langle \zeta(\mathbf{k}) \zeta(\mathbf{k}') \zeta(\mathbf{k}'') \rangle \equiv (2\pi)^3 \delta(\mathbf{k} + \mathbf{k}' + \mathbf{k}'') B_{\zeta}(k, k', k''), \quad (23.86)$$

which is often expressed in terms of a dimensionless non-linearity parameter

$$f_{NL} \propto B_{\zeta}(k, k', k'') / P_{\zeta}(k) P_{\zeta}(k').$$

The three-point function (Eq. (23.86)) can be thought of as defined on a triangle whose sides are $\mathbf{k}, \mathbf{k}', \mathbf{k}''$, of which only two are independent, since they sum to zero. Further assuming statistical isotropy ensures that the bispectrum depends only on the magnitudes of the three vectors, $\mathbf{k}, \mathbf{k}'$ and $\mathbf{k}''$. The search for $f_{NL}$ and other non-Gaussian effects was a prime objective of the Planck data analysis [93, 94].

### 23.7.1 Enhanced perturbations on small scales

Although the observed primordial power spectrum on CMB scales is constrained to be almost scale invariant, consistent with the simplest slow-roll inflation models, the form of the power spectrum is largely unconstrained on much smaller scales, corresponding to scales that would have left the horizon closer to the end of inflation. Although free-streaming and other damping of perturbations in the primordial plasma are expected to have washed out small-scale fluctuations in the matter sector [95], gravitational relics such as primordial black holes [96, 97] or induced gravitational waves [98–101] produced by sufficiently large primordial density perturbations could have survived and might be detectable in the present day. Rapid growth of the power spectrum on small scales requires violation of slow roll in single-field models [102]. This could arise from features in the scalar field potential, such as an inflection point [103], leading to a transient phase of ultra-slow-roll evolution [104, 105] which can give a $k^4$-rise in the scalar power spectrum on small scales [106, 107].

The LIGO-Virgo-KAGRA collaboration has placed an upper limit on such a stochastic gravitational wave background (SGWB), conventionally expressed in terms of the effective density of gravitational waves per logarithmic frequency interval, of $\Omega_{GW} \le 5.8 \times 10^{-9}$ for a scale-invariant spectrum in the frequency range 20–77 Hz [108]. On the other hand, pulsar timing arrays have found growing evidence for a low-frequency SGWB at nano-Hertz frequencies [109–112]. While this could be associated with an astrophysical background due to supermassive black hole binary systems, the NANOGrav collaboration report that a cosmological background provides a better fit to their 15-year dataset [113]. This could be generated by large primordial density fluctuations on the corresponding comoving scales, $k \sim 10^6$ Mpc$^{-1}$, but these are constrained by the production of primordial black holes of approximately solar mass [114] and possible one-loop or higher-order corrections to the primordial power spectrum on CMB scales.

### 23.7.2 Effective Field Theory of Inflation

Since slow-roll inflation is a phase of accelerated expansion with an almost constant Hubble parameter, one may think of inflation in terms of an effective theory where the de Sitter spacetime symmetry is spontaneously broken down to RW symmetry by the time-evolution of the Hubble rate, $\dot{H} \ne 0$. There is then a Goldstone boson, $\pi$, associated with the spontaneous breaking of time-translation invariance, which can be used to study model-independent properties of inflation. The Goldstone boson describes a spacetime-dependent shift of the time coordinate, corresponding to an adiabatic perturbation of the matter fields:

$$\delta\phi_i(t, \vec{x}) = \phi_i(t + \pi(t, \vec{x})) - \phi_i(t). \quad (23.87)$$
<figure><img src="image_137.png" /><figcaption>**Figure 23.5:** Illustration of the impact of the BICEP/Keck [43] and other constraints on the inflaton decay coupling, $y$, and the number of e-folds at a scale $0.05 \text{ Mpc}^{-1}$, $N_{0.05}$, in $\alpha$-attractor models of inflation. The horizontal blue lines are 68% and 95% C.L. lower limits on $n_s$. The left and right axes show the relation between $N_{0.05}$ and $n_s$ and the top and bottom scales show the relation between the inflaton coupling, $y$, and the reheating temperature, $T_{reh}$. The diagonal dashed, solid and dotted black lines illustrate the correlations between these quantities for $\alpha$-attractor models with $\alpha = 0.1, 1$, and $5$. We also include the lower limit on $y$ from Big Bang Nucleosynthesis (red line), the constraint that $T_{reh}$ be no smaller than the electroweak scale (gray line), and a constraint from gravitino production (purple line) for $\alpha = 1$, which strengthens for smaller $\alpha$. Plot taken from Ref. [85].</figcaption></figure>

Thus adiabatic field fluctuations can be absorbed into the spatial metric perturbation, $\mathcal{R}$ in Eq. (23.28) at first order, in the comoving gauge:

$$ \mathcal{R} = -H\pi, \qquad (23.88) $$

where we define $\pi$ on spatially-flat hypersurfaces. In terms of inflaton field fluctuations, we can identify $\pi \equiv \delta\phi/\dot{\phi}$, but in principle this analysis is not restricted to inflation driven by scalar fields.

The low-energy effective action for $\pi$ can be obtained by writing down the most general Lorentz-invariant action and expanding in terms of $\pi$. The second-order effective action for the free-field wave modes, $\pi_k$, to leading order in slow roll is then

$$ S_{\pi}^{(2)} = - \int d^4 x \sqrt{-g} \frac{M_P^2 \dot{H}}{c_s^2} \left[ \dot{\pi}_k^2 - \frac{c_s^2}{R^2} (\nabla \pi)^2 \right], \qquad (23.89) $$

where $\epsilon_H$ is the Hubble slow-roll parameter (Eq. (23.11)). We identify $c_s^2$ with an effective sound speed, generalising canonical slow-roll inflation, which is recovered in the limit $c_s^2 \to 1$.

The scalar power spectrum on super-Hubble scales (Eq. (23.48)) is enhanced for a reduced sound speed, leading to a reduced tensor-scalar ratio (Eq. (23.49))

$$ \mathcal{P}_{\zeta}(k) \approx \frac{4\pi}{M_P^2} \frac{1}{c_s^2 \epsilon} \left(\frac{H}{2\pi}\right)_*^2, \qquad r \approx 16(c_s^2 \epsilon)_*. \qquad (23.90) $$

At third perturbative order and to lowest order in derivatives, one

obtains [115]

$$ S_{\pi}^{(3)} = \int d^4 x \sqrt{-g} \frac{M_P^2 (1 - c_s^2) \dot{H}}{c_s^2} \left[ \frac{\dot{\pi} (\nabla \pi)^2}{R^2} - \left(1 + \frac{2}{3} \frac{\tilde{c}_3}{c_s^2}\right) \dot{\pi}^3 \right]. \qquad (23.91) $$

Note that this expression vanishes for canonical fields with $c_s^2 = 1$. For $c_s^2 \neq 1$ the cubic action is determined by the sound speed and an additional parameter $\tilde{c}_3$. Both terms in the cubic action give rise to primordial bispectra that are well approximated by equilateral bispectra. However, the shapes are not identical, so one can find a linear combination for which the equilateral bispectra of each term cancel, giving rise to a distinctive orthogonal-type bispectrum [115].

Analysis based on Planck 2018 temperature and polarization data has placed bounds on several bispectrum shapes including equilateral and orthogonal shapes [94]:

$$ f_{NL}^{\text{equil}} = -26 \pm 47, \qquad f_{NL}^{\text{orthog}} = -38 \pm 24 \quad (68\% \text{ CL}). \qquad (23.92) $$

For the simplest case of a constant sound speed, and marginalising over $\tilde{c}_3$, this provides a bound on the inflaton sound speed [94]

$$ c_s \geq 0.021 \quad (95\% \text{ CL}). \qquad (23.93) $$

For a specific model such as DBI inflation [90], corresponding to $\tilde{c}_3 = 3(1 - c_s^2)/2$, one obtains a tighter bound [94]:

$$ c_s^{\text{DBI}} \geq 0.086 \quad (95\% \text{ CL}). \qquad (23.94) $$
The Planck team have analysed a wide range of non-Gaussian templates from different inflation models, including tests for deviations from an initial Bunch-Davies vacuum state, direction-dependent non-Gaussianity, and feature models with oscillatory bispectra [94]. No individual feature or resonance is above the three-$\sigma$ significance level after accounting for the look-elsewhere effect. These results are consistent with the simplest canonical, slow-roll inflation models, but do not rule out most alternative models; rather, bounds on primordial non-Gaussianity place important constraints on the parameter space for non-canonical models.

### 23.7.3 Multi-Field Fluctuations

There is a very large literature on two- and multi-field models of inflation, most of which lies beyond the scope of this review [116, 117]. However, two important general topics merit being mentioned here, namely residual isocurvature perturbations and the possibility of non-Gaussian effects in the primordial perturbations.

One might expect that other scalar fields besides the inflaton might have non-negligible values that evolve and fluctuate in parallel with the inflaton, without necessarily making the dominant contribution to the energy density during the inflationary epoch. However, the energy density in such a field might persist beyond the end of inflation before decaying, at which point it might come to dominate (or at least make a non-negligible contribution to) the total energy density. In such a case, its perturbations could end up generating the density perturbations detected in the CMB. This could occur due to a late-decaying scalar field [91] or a field fluctuation that modulates the end of inflation [118] or the inflaton decay [119].

#### 23.7.3.1 Isocurvature Perturbations

Primordial perturbations arising in single-field slow-roll inflation are necessarily adiabatic, i.e., they affect the overall density without changing the ratios of different contributions, such as the photon-matter ratio, $\delta(n_\gamma/n_m)/(n_\gamma/n_m)$. This is because inflaton perturbations represent a local shift of the time, as described in section Sec. 23.7.2:

$$ \pi = \frac{\delta n_{\gamma}}{\dot{n}_{\gamma}} = \frac{\delta n_{m}}{\dot{n}_{m}}. \qquad (23.95) $$

However, any light scalar field (i.e., one with effective mass less than the Hubble scale) acquires a spectrum of nearly scale-invariant perturbations during inflation. Fluctuations orthogonal to the inflaton in field space are decoupled from the inflaton at Hubble-exit, but can affect the subsequent evolution of the density perturbation. In particular, they can give rise to local variations in the equation of state (non-adabatic pressure perturbations) that can alter the primordial curvature perturbation $\zeta$ on super-Hubble scales. Since these fluctuations are statistically independent of the inflaton perturbations at leading order in slow roll [117], non-adiabatic field fluctuations can only increase the scalar power spectrum with respect to adiabatic perturbations at Hubble exit, while leaving the tensor modes unaffected at first perturbative order. Thus, the single-field result for the tensor-scalar ratio (Eq. (23.49)) becomes an inequality [120]

$$ r \le 16\epsilon_{*}. \qquad (23.96) $$

Hence, an observational upper bound on the tensor-scalar ratio does not bound the slow-roll parameter $\epsilon$ in multi-field models.

If all the scalar fields present during inflation eventually decay completely into fully thermalized radiation, these field fluctuations are converted fully into adiabatic perturbations in the primordial plasma [121]. On the other hand, non-adiabatic field fluctuations can also leave behind primordial isocurvature perturbations (Eq. (23.85)) after inflation. In multi-field inflation models it is thus possible for non-adiabatic field fluctuations to generate both curvature and isocurvature perturbations leading to correlated primordial perturbations [122].

The amplitudes of any primordial isocurvature perturbations (Eq. (23.85)) are strongly constrained by the current CMB data,

especially on large angular scales. Using temperature and low-$\ell$ polarization data yields the following bound on the amplitude of cold dark matter isocurvature perturbations at scale $k = 0.002h^{-1}\text{Mpc}^{-1}$ (marginalising over the correlation angle and in the absence of primordial tensor perturbations) [38]:

$$ \frac{P_{S_m}}{P_{\zeta} + P_{S_m}} < 0.025 (95\% \text{ CL}). \qquad (23.97) $$

For fully (anti-)correlated isocurvature perturbations, corresponding to a single isocurvature field providing a source for both the curvature and residual isocurvature perturbations, the bounds become significantly tighter [38]:

$$ \frac{P_{S_m}}{P_{\zeta} + P_{S_m}} < 0.0002 (95\% \text{ CL}), \text{ correlated,} \qquad (23.98) $$

$$ \frac{P_{S_m}}{P_{\zeta} + P_{S_m}} < 0.003 (95\% \text{ CL}), \text{ anti-correlated} \qquad (23.99) $$

#### 23.7.3.2 Local-Type Non-Gaussianity

Since non-adiabatic field fluctuations in multi-field inflation may lead the evolution of the primordial curvature perturbation at all orders, it becomes possible to generate significant non-Gaussianity in the primordial curvature perturbation. Non-linear evolution on super-Hubble scales leads to local-type non-Gaussianity, where the local integrated expansion is a non-linear function of the local field values during inflation, $N(\phi_i)$. While the field fluctuations at Hubble exit, $\delta\phi_{i*}$, are Gaussian in the slow-roll limit, the curvature perturbation, $\zeta = \delta N$, becomes a non-Gaussian distribution [123]:

$$ \zeta = \sum_{i} \frac{\partial N}{\partial \phi_i} \delta\phi_i + \frac{1}{2} \sum_{i,j} \frac{\partial^2 N}{\partial \phi_i \partial \phi_j} \delta\phi_i \delta\phi_j + \dots \qquad (23.100) $$

with non-vanishing bispectrum in the squeezed limit ($k_1 \approx k_2 \gg k_3$):

$$ B_{\zeta}(k_1, k_2, k_3) \approx \frac{12}{5} f_{NL}^{\text{local}} \frac{P_{\zeta}(k_1)}{4\pi k_1^3} \frac{P_{\zeta}(k_3)}{4\pi k_3^3}, \qquad (23.101) $$

where

$$ \frac{6}{5} f_{NL}^{\text{local}} = \frac{\sum_{i,j} \frac{\partial^2 N}{\partial \phi_i \partial \phi_j}}{\left(\sum_i \frac{\partial N}{\partial \phi_i}\right)^2}. \qquad (23.102) $$

Both equilateral and orthogonal bispectra, discussed above in the context of generalised single field inflation, vanish in the squeezed limit, enabling the three types of non-Gaussianity to be distinguished by observations, in principle.

Non-Gaussianity during multi-field inflation is highly model dependent, though $f_{NL}^{\text{local}}$ can often be smaller than unity in multi-field slow-roll inflation [124]. Scenarios where a second light field plays a role during or after inflation can make distinctive predictions for $f_{NL}^{\text{local}}$, such as $f_{NL}^{\text{local}} = -5/4$ in some curvaton scenarios [123, 125] or $f_{NL}^{\text{local}} = 5$ in simple modulated reheating scenarios [119, 126]. By contrast the constancy of $\zeta$ on super-Hubble scales in single-field slow-roll inflation leads to a very small non-Gaussianity [127, 128], and in the squeezed limit we have the simple result $f_{NL}^{\text{local}} = 5(1 - n_S)/12$ [129, 130].

A combined analysis of the Planck 2018 temperature and polarization data [94] yields the following range for $f_{NL}^{\text{local}}$ defined in (Eq. (23.102)):

$$ f_{NL}^{\text{local}} = -1 \pm 5 \quad (68\% \text{ CL}). \qquad (23.103) $$

This sensitivity is sufficient to rule out parameter regimes giving rise to relatively large non-Gaussianity, but insufficient to probe $f_{NL}^{\text{local}} = \mathcal{O}(\epsilon)$, as expected in single-field models, or the range $f_{NL}^{\text{local}} = \mathcal{O}(1)$ found in the simplest two-field models.

Local-type primordial non-Gaussianity can also give rise to a striking scale-dependent bias in the distribution of collapsed dark matter halos and thus the galaxy distribution [131, 132]. Bounds from high-redshift galaxy surveys are not currently competitive
with the best CMB constraints. An analysis based on the clustering of quasars in the final data release (DR16) of the extended Baryon acoustic Oscillation Spectroscopic Survey (eBOSS) [133] yields

$$f_{\mathrm{NL}}^{\mathrm{local}} = -12 \pm 21 \quad (68\% \mathrm{CL}). \qquad (23.104)$$

A recent analysis based on the clustering of luminous red galaxies in imaging surveys from the Dark Energy Spectroscopic Instrument (DESI) [134] finds $f_{\mathrm{NL}}^{\mathrm{local}} = 47_{-11}^{+14}$, suggesting that $f_{\mathrm{NL}}^{\mathrm{local}} > 0$ at more than 99.9% confidence, although the possibility that this could be due to unforeseen systematics cannot yet be excluded.

## 23.8 Initial Conditions and Fine-tuning

This review is based on the assumption that the inflationary paradigm is valid. However, it remains the object of many criticisms (see, e.g., [135]), many of them related to the perceived unnaturalness of the required initial conditions.

Most work on inflation is done in the context of RW cosmology, which assumes a high degree of symmetry, or small inhomogeneous perturbations (usually first order) about an RW cosmology. The isotropic RW space-time is an attractor for many homogeneous but anisotropic cosmologies in the presence of a false vacuum energy density [136], or a scalar field with suitable self-interaction potential energy [137,138]. However it is much harder to establish the range of highly inhomogeneous initial conditions that yield a successful RW Universe, with only limited studies initially (see, e.g., [139,140]). A related open question is the general nature of the pre-inflationary state of the inflaton and other fields that could have provided initial conditions suitable for inflation [135]. They would need to have satisfied non-trivial homogeneity and isotropy conditions, and one may ask how these could have arisen, whether these are plausible, and whether there may be some observable signature of the pre-inflationary state. These and other criticisms of inflation were addressed in [141], which presented studies of the sensitivity of inflation to the initial conditions. Complementing the studies reported in [141], there have been numerical relativity investigations of highly inhomogeneous initial conditions [142–144]. The general conclusion is that inflation is rather robust with respect to inhomogeneities in the initial conditions in both the scalar field profile and the extrinsic curvature, including large tensor perturbations.

To quantify the fine-tuning of initial conditions requires a measure in the space of possible cosmologies [145], however it has been argued that some of the measures historically used to frame this problem are formally invalid [146]. It is sometimes also objected that inflationary models predict the existence of a multiverse, and potentially a loss of predictive power [147], if it undergoes the process termed eternal inflation [148–150]. However, whether this is actually a bug or a feature remains a topic of debate [151, 152]. The existence of the multiverse is a purely philosophical problem, unless it has observable consequences, e.g., in the CMB.

One might expect signatures of any pre-inflationary state to appear at large angular scales, i.e., low multipoles $ℓ$. Indeed, various anomalies have been noted in the large-scale CMB anisotropies, as also discussed in Chap. 29, the “Cosmic Microwave Background” review, including a possible suppression of the quadrupole and other very large-scale anisotropies, an apparent feature in the range $ℓ \approx 20$ to 30, and a possible hemispheric asymmetry. However, none of these are highly significant statistically, in view of the limitations due to cosmic variance [64]. They cannot yet be regarded as signatures of initial conditions, the multiverse or some pre-inflationary dynamics, such as might emerge from string theory.

A different kind of initial condition problem, called the Trans-Planckian problem [153], is that the perturbations now seen in the CMB would have had wavelengths shorter than the Planck length at the onset of inflation. However, under quite general and conservative assumptions the usual inflationary predictions would be quite robust [154], with the possibility of $O((H/m_P)^n)$ corrections that might have interesting signatures in the CMB [155].

When inflation was first proposed [1] [2] there was no evidence for the existence of scalar fields or the accelerated expansion of the Universe. The situation has changed dramatically in recent

years with the observational evidence that the cosmic expansion is currently accelerating and with the discovery of a scalar particle, namely the Higgs boson (see Chap. 11, “Status of Higgs boson physics” review). Combined with the lack of any widely accepted alternative model for the origin of cosmic structure, these discoveries have lent support to the idea of a primordial accelerated expansion driven by a scalar field, i.e., cosmological inflation. In parallel, successive CMB experiments have been consistent with generic predictions of inflationary models, although without yet providing irrefutable evidence. It was concluded in [141] that the inflationary paradigm is not currently in trouble. However, we note that inflation via a formally elementary scalar inflaton should probably only be regarded as an effective field theory valid at energy densities hierarchically smaller than the Planck scale. It should eventually be embedded in a suitable ultraviolet completion, on which inflationary dynamics may be our clearest window.

## 23.9 Future Probes of Inflation

Prospective future CMB experiments, both ground- and space-based are reviewed in the separate PDG “Cosmic Microwave Background” review, Chap. 29. The main emphasis in CMB experiments in the coming years will be on ground-based experiments providing improved measurements of B-mode polarization and greater sensitivity to the tensor-to-scalar ratio $r$, and more precise measurements at higher $ℓ$ that will constrain $n_s$ better. As is apparent from Fig. 23.1 and the discussion of models such as $R^2$ inflation, there is a strong incentive to reach a $5-\sigma$ sensitivity to $r \sim 3$ to $4 \times 10^{-3}$. This could be achieved with a moderately-sized space mission with large sky coverage [156], improvements in delensing and foreground measurements. The discussion in Sec. 23.3 (see also Fig. 23.5), also brought out the importance of reducing the uncertainty in $n_s$, as a way to constrain post-inflationary reheating and the connection to particle physics. CMB temperature anisotropies probe primordial density perturbations down to comoving scales of order 50 Mpc, beyond which scale secondary sources of anisotropy dominate. CMB spectral distortions could potentially constrain the amplitude and shape of primordial density perturbations on comoving scales from Mpc to kpc due to distortions caused by the Silk damping of pressure waves in the radiation dominated era, before the last scattering of the CMB photons but after the plasma can be fully thermalised [95].

Improved sensitivity to non-Gaussianities is also a priority. In addition to CMB measurements, future large-scale structure surveys will also have roles to play as probes into models of inflation, for which there are excellent prospects. Current surveys such as DESI may reach $\Delta f_{\mathrm{NL}} \sim 4$ [157] comparable with the Planck sensitivity. In the future, radio surveys such as SKA will measure large-scale structure out to redshift $z \sim 3$ [158], initially through mapping the intensity of the neutral hydrogen 21-cm line, and eventually through radio galaxy surveys which will probe local-type non-Gaussianity to $f_{\mathrm{NL}} \sim 1$.

Galaxy clustering using DESI and Euclid satellite data could also constrain the running of the scalar tilt to a precision of $\Delta\alpha_s \approx 0.0028$, a factor of 2 improvement on Planck constraints, or a precision of 0.0016 using LSST data [157].

As an example of a proposed future satellite mission, SPHEREX [159] will use measurements of the galaxy power spectrum to target a measurement of the running of the scalar spectral index with a sensitivity $\Delta\alpha_s \sim 10^{-3}$ and local-type primordial non-Gaussianity, $\Delta f_{\mathrm{NL}} \sim 1$. Including information from the galaxy bispectrum one might reduce the measurement error on non-Gaussianity to $\Delta f_{\mathrm{NL}} \sim 0.2$, making it possible to distinguish between single-field slow-roll models and alternatives such as the curvaton scenario for the origin of structure, which generate $f_{\mathrm{NL}} \sim 1$.

Finally we note that future gravitational wave experiments such as the Laser Interferometer Space Antenna (LISA) will be sensitive to a stochastic gravitational wave background, including tensor perturbations produced during inflation or induced by large primordial density perturbations on small scales after inflation [160], which could provide constraints on inflationary perturbations on smaller scales.

## Acknowledgements

The authors are grateful to Vincent Vennin for preparing
Fig. 23.4 for this review. The work of J.E. was supported in part by the UK STFC research grant ST/T000759/1. The work of D.W. was supported in part by the UK STFC research grant ST/W001225/1.

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