29. Cosmic Microwave Background1

29. Cosmic Microwave Background
Revised August 2023 by D. Scott (U. of British Columbia) and G.F. Smoot (HKUST; UC Berkeley;
LBNL; DIPC; Paris U.).

29.1 Introduction
The energy content in electromagnetic radiation from beyond our Galaxy is dominated by
the cosmic microwave background (CMB), discovered in 1965 [1]. Its spectral distribution is welldescribed by a blackbody function with T = 2.7255 K, which is a principal pillar of the hot Big
Bang model for the Universe, with the lack of any observed deviations from a Planckian spectrum
constraining physical processes over cosmic history at redshifts z . 107 (see earlier versions of thisreview).
The key information in the CMB sky is extracted from the observed angular variation of its
temperature (or intensity) correlations, and to a growing extent polarization [2]. After the firstdetection of CMB anisotropies by the Cosmic Background Explorer (COBE) satellite in 1992 [3],there has been intense activity to map the sky at increasing levels of sensitivity and angular resolution by ground-based and balloon-borne measurements. These were joined in 2003 by the first
results from NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) [4], which were improved
upon by analyses of data added every 2 years, culminating in the 9-year results [5]. In 2013 we
had the first results [6] from the third generation CMB satellite, ESA’s Planck mission [7], whichwere enhanced by results from the 2015 Planck data release [8, 9], and then the final 2018 Planck
data release [10, 11]. Additionally, CMB anisotropies have been extended to smaller angular scales
by ground-based experiments, particularly the Atacama Cosmology Telescope (ACT) [12] and theSouth Pole Telescope (SPT) [13]. Together these observations have led to a stunning confirmationof the ‘Standard Model of Cosmology.’ In combination with other astrophysical data, the CMB
anisotropy measurements place quite precise constraints on a number of cosmological parameters,
and have launched us into an era of precision cosmology. With more than half a century of study of
the CMB, the program to map temperature anisotropies is effectively wrapping up, and attention
is increasingly focussing on polarization measurements, which promise further tests of fundamental
physics.

29.2 CMB Spectrum
It is well-known that the spectrum of the microwave background is very precisely that of blackbody radiation, whose temperature evolves with redshift as T (z) = T0 (1 + z) in an expanding
Universe. As a direct confirmation of its cosmological origin, this relationship has been tested bymeasuring the strengths of emission and absorption lines in high-redshift systems (e.g., Ref. [14]).
Measurements of the spectrum are consistent with a blackbody distribution over more than
three decades in frequency (there is a claim by ARCADE [15] of a possible unexpected extragalactic emission signal at low frequency, but the interpretation is debated [16]). All viable cosmologicalmodels predict a very nearly Planckian spectrum to within the current observational limits. Because of this, measurements of deviations from a blackbody spectrum have received little attention
in recent years, with only a few exceptions. However, that situation will likely change as proposed
experiments [17] are built that have the potential to dramatically improve early Universe energy
release constraints. It now seems feasible to probe spectral distortion mechanisms that are required in the standard picture, such as those arising from the damping and dissipation of relatively
small-scale primordial perturbations, or the average effect of inverse Compton scattering. A more
ambitious goal would be to reach the precision needed to detect the residual lines from the cosmological recombination of hydrogen and helium and hence to test whether conditions at z & 1000
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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2

29. Cosmic Microwave Backgroundaccurately follow those in the standard picture [18].

29.3 Description of CMB Anisotropies
Observations show that the CMB contains temperature anisotropies at the 10−5 level andpolarization anisotropies at the 10−6 (and lower) level, over a wide range of angular scales. Theseanisotropies are usually expressed using a spherical harmonic expansion of the CMB sky:T (θ, φ) =X

a`m Y`m (θ, φ)(29.1)`m(with the linear polarization pattern written in a similar way using the so-called spin-2 sphericalharmonics). Increasing angular resolution requires that the expansion goes to higher multipoles.
Because only very weak phase correlations are observed in the CMB sky and no preferred direction
is seen, the vast majority of the cosmological information is found in the temperature 2-point
function, i.e., the variance as a function only of angular separation. Equivalently, the anisotropyP
power per unit ln ` is ` m |a`m |2 /4π.
29.3.1 The Monopole
The CMB has a mean temperature of Tγ = 2.7255 ± 0.0006 K (1σ) [19], which can be consideredas the monopole component of CMB maps, a00 . Because all mapping experiments involve difference
measurements, they are insensitive to this average level; monopole measurements can only bemade with absolute temperature devices, such as the FIRAS instrument on the COBE satellite[20]. The measured kTγ is equivalent to 0.234 meV or 4.60 × 10−10 me c2 . A blackbody of themeasured temperature has a number density nγ = (2ζ(3)/π 2 ) Tγ3 ' 411 cm−3 , energy densityργ = (π 2 /15) Tγ4 ' 4.64 × 10−34 g cm−3 ' 0.260 eV cm−3 , and a fraction of the critical densityΩγ ' 5.38 × 10−5 .
29.3.2 The Dipole
The largest anisotropy is in the ` = 1 (dipole) first spherical harmonic, with amplitude 3.3621 ±0.0010 mK [10]. The dipole is interpreted to be the result of the Doppler boosting of the monopole
caused by the Solar System motion relative to the nearly isotropic blackbody field, as broadly
confirmed by measurements of the radial velocities of local galaxies (e.g., Ref. [21]); the intrinsic(non-Doppler) part of the signal is expected to be 2 orders of magnitude smaller (and fundamentallydifficult to distinguish). The motion of an observer with velocity β ≡ v/c relative to an isotropic
Planckian radiation field of temperature T0 produces a Lorentz-boosted temperature pattern
T (θ) = T0 (1 − β 2 )1/2 /(1 − β cos θ)
h







' T0 1 + β cos θ + β 2 /2 cos 2θ + O β 3i.

(29.2)

At every point in the sky, one observes a blackbody spectrum, with temperature T (θ). The spectrumof the dipole has been confirmed to be the differential of a blackbody spectrum [22]. At higher
order there are additional effects arising from aberration and from modulation of the anisotropy
pattern, which have also been observed [23].
The implied velocity for the Solar System barycenter is v = 369.82 ± 0.11 km s−1 , assuming
a value T0 = Tγ , towards (l, b) = (264.021◦ ± 0.011◦ , 48.253◦ ± 0.005◦ ) [10]. This Solar System
motion implies a velocity for the Galaxy and the Local Group of galaxies relative to the CMB ofvLG = 620 ± 15 km s−1 towards (l, b) = (271.9◦ ± 2.0◦ , 29.6◦ ± 1.4◦ ) [10]; most of the error comesfrom uncertainty in the velocity of the Solar System relative to the Local Group.
The dipole is a frame-dependent quantity, and one can thus determine the ‘CMB frame’ (insome sense this is a special frame) as that in which the CMB dipole would be zero. Any velocity of
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329. Cosmic Microwave Backgroundthe receiver relative to the Earth and the Earth around the Sun is removed for the purposes of CMB
anisotropy studies, while our velocity relative to the Local Group of galaxies and the Local Group’s
motion relative to the CMB frame are normally removed for cosmological studies. The dipole is
now routinely used as a primary calibrator for mapping experiments, either via the time-varying
orbital motion of the Earth, or through the cosmological dipole measured by satellite experiments.
29.3.3 Higher-Order Multipoles
The variations in the CMB temperature maps at higher multipoles (` ≥ 2) are interpreted as
being mostly the result of density perturbations in the early Universe, manifesting themselves at theepoch of the last scattering of the CMB photons. In the hot Big Bang picture, the expansion of the
Universe cools the plasma so that by a redshift z ' 1100 (with little dependence on the details of
the model), the hydrogen and helium nuclei can bind electrons into neutral atoms, a process usuallyreferred to as ‘recombination’ [24]. Before this epoch, the CMB photons were tightly coupled tothe charged baryons, while afterwards they could freely stream towards us. By measuring the a`m s
we are thus learning directly about physical conditions in the early Universe.
A statistically-isotropic sky means that all ms are equivalent, i.e., there is no preferred axis,
so that the temperature correlation function between two positions on the sky depends only on
angular separation and not orientation. Together with the assumption of Gaussian statistics (i.e., nocorrelations between the modes), the 2-point function of the temperature field (or equivalently thepower spectrum in `) then fully characterizes the anisotropies. The power summed over all ms ateach ` is (2` + 1)C` /(4π), where C` ≡ |a`m |2 . Thus, averages of a`m s over m can be used as
estimators of the C` s to constrain their expectation values, which are the quantities predicted by atheoretical model. For an idealized full-sky observation, the variance of each measured C` (i.e., thevariance of the variance) is [2/(2` + 1)]C`2 . This sampling uncertainty (known as ‘cosmic variance’)
comes about because each C` is χ2 distributed with (2` + 1) degrees of freedom for our observable
volume of the Universe. For fractional sky coverage, fsky , this variance is increased by 1/fsky andthe modes become partially correlated.
It is important to understand that theories predict the expectation value of the power spectrum,
whereas our sky is a single realization. Hence, the cosmic variance is an unavoidable source of
uncertainty when constraining models; it dominates the scatter at lower `s, while the effects of
instrumental noise and resolution dominate at higher `s [25].
Theoretical models generally predict that the a`m modes are Gaussian random fields to highprecision, matching the empirical tests, e.g., standard slow-roll inflation’s non-Gaussian contribution is expected to be at least an order of magnitude below current observational limits [26].
Although non-Gaussianity of various forms is possible in early Universe models, tests show that
Gaussianity is an extremely good simplifying approximation [27]. The only current indications of
any non-Gaussianity or statistical anisotropy are some relatively weak signatures at large scales,
seen in both WMAP [28] and Planck data [29], but not of high enough significance to reject thesimplifying assumption. Nevertheless, models that deviate from the inflationary slow-roll conditions
can have measurable non-Gaussian signatures. So while the current observational limits make thepower spectrum the dominant probe of cosmology, it is worth noting that higher-order correlations
are tools for constraining otherwise viable theories.
29.3.4 Angular Resolution and Binning
There is no one-to-one conversion between multipole ` and the angle subtended by a particularspatial scale projected onto the sky. However, crudely speaking, a single spherical harmonic Y`mcorresponds to angular variations of θ ∼ π/`. CMB maps contain anisotropy information from thesize of the map (or in practice some fraction of that size) down to the beam-size of the instrument,σ (the standard deviation of the beam, in radians). One can think of the effect of a Gaussian beam
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429. Cosmic Microwave Background2

as rolling off the power spectrum with the function e−`(`+1)σ .
For less than full sky coverage, the ` modes become correlated. Hence, experimental results
are usually quoted as a series of ‘band powers,’ defined as estimators of `(` + 1)C` /2π over different ranges of `. Because of the strong foreground signals in the Galactic plane, even ‘all-sky’surveys, such as WMAP and Planck, involve a cut sky. The amount of binning required to obtain
uncorrelated estimates of power also depends on the map size.

29.4 Cosmological Parameters
The current ‘Standard Model’ of cosmology contains around 10 free parameters, only six ofwhich are required to have non-null values (see The Cosmological Parameters—Sec. 25 of thisReview). The basic framework is the Friedmann-Robertson-Walker (FRW) metric (i.e., a Universethat is approximately homogeneous and isotropic on large scales), with density perturbations laid
down at early times and evolving into today’s structures (see Big-Bang cosmology—Sec. 22 of
this Review). The most general possible set of density variations is a linear combination of anadiabatic density perturbation and some isocurvature perturbations. Adiabatic means that there is
no change to the entropy per particle for each species, i.e., δρ/ρ for matter is (3/4)δρ/ρ for radiation.
Isocurvature means that the set of individual density perturbations adds to zero, for example,
matter perturbations compensate radiation perturbations so that the total energy density remains
unperturbed, i.e., δρ for matter is −δρ for radiation. These different modes give rise to distinct(temporal) phases during growth, with those of the adiabatic scenario being fully consistent withthe data. Models that generate mainly isocurvature type perturbations (such as most topologicaldefect scenarios) are not viable. However, an admixture of the adiabatic mode with up to 1.7%
isocurvature contribution (depending on details of the mode) is still allowed [30].
29.4.1 Initial Condition Parameters
Within the adiabatic family of models, there is, in principle, a free function describing thevariation of comoving curvature perturbations, R(x, t). The great virtue of R is that it is constantin time on super-horizon scales for a purely adiabatic perturbation. There are physical reasons toanticipate that the variance of these perturbations will be described well by a power law in scale,i.e., in Fourier space |R|2k ∝ k ns −4 , where k is wavenumber and ns is the spectral index as usually
defined. So-called ‘scale-invariant’ initial conditions (meaning gravitational potential fluctuations
that are independent of k) correspond to ns = 1. In inflationary models [31] (see Inflation—Sec. 23of this Review), perturbations are generated by quantum fluctuations, which are set by the energy
scale of inflation together with the slope and higher derivatives of the inflationary potential. One
generally expects that the Taylor series expansion of ln Rk (ln k) has terms of steadily decreasing
size. For the simplest models, there are thus two parameters describing the initial conditions fordensity perturbations, namely the amplitude and slope of the power spectrum. These can be
explicitly defined, for example, through
D

E

2
PR
≡ k 3 |R|2k /2π 2 ' As (k/k0 )ns −1 ,(29.3)

2 (k ) and k = 0.05 Mpc−1 , say. There are other equally valid definitions of thewith As ≡ PR
00amplitude parameter (see also Secs. 22, 23, and 25 of this Review), and we caution that therelationships between some of them can be cosmology-dependent. In slow-roll inflationary models,
this normalization is proportional to the combination V 3 /(V 0 )2 , for the inflationary potential V (φ).
The slope ns also involves V 00 , and so the combination of As and ns can constrain potentials.
Inflation generates tensor (gravitational wave) modes, as well as scalar (density perturbation)
modes. This property introduces another parameter, measuring the amplitude of a possible tensor
component, or equivalently the ratio of the tensor to scalar contributions. The tensor amplitude is
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529. Cosmic Microwave BackgroundAt ∝ V , and thus one expects a larger gravitational wave contribution in models where inflation
happens at higher energies. The tensor power spectrum also has a slope, often denoted nt , butsince this seems unlikely to be measured in the near future (and there is also a consistency relation
with tensor amplitude), it is sufficient for now to focus only on the amplitude of the gravitational
wave component. It is most common to define the tensor contribution through r, the ratio of tensor
to scalar perturbation spectra at some fixed value of k (e.g., k = 0.002 Mpc−1 or k = 0.05 Mpc−1 ,although it was historically defined in terms of the ratio of contributions at ` = 2). Different
inflationary potentials will lead to different predictions, e.g., for 50 e-folds, λφ4 inflation givesr = 0.32 and m2 φ2 inflation gives r = 0.16 (both now strongly disfavored by the data), while othermodels can have arbitrarily small values of r. In any case, whatever the specific definition, andwhether they come from inflation or something else, the ‘initial conditions’ give rise to a minimumof three parameters, As , ns , and r.

Figure 29.1: Theoretical CMB anisotropy power spectra, using the best-fitting ΛCDM modelfrom Planck, calculated using CAMB. The panel on the left shows the theoretical expectation forscalar perturbations, while the panel on the right is for tensor perturbations, with an amplitude set
to r = 0.01 for illustration. Note that the horizontal axis is logarithmic here. For the well-measured
scalar T T spectrum, these regions, each covering roughly a decade in `, are labeled as in the text:
the ISW rise; Sachs-Wolfe plateau; acoustic peaks; and damping tail. The T E cross-correlation
power spectrum changes sign, indicated here by plotting the absolute value, but switching color forthe negative parts.

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629. Cosmic Microwave Background29.4.2 Background Cosmology Parameters
The FRW cosmology requires an expansion parameter (the Hubble constant, H0 , often represented through H0 = 100 h km s−1 Mpc−1 ) and several parameters to describe the matter andenergy content of the Universe. These are usually given in terms of the critical density, i.e., for
species ‘x,’ Ωx ≡ ρx /ρcrit , where ρcrit ≡ 3H02 /8πG. Since physical densities ρx ∝ Ωx h2 ≡ ωx arewhat govern the physics of the CMB anisotropies, it is these ωs that are best constrained by CMB
data. In particular, CMB observations constrain Ωb h2 for baryons and Ωc h2 for cold dark matter(with ρm = ρc + ρb for the matter sum).
The contribution of a cosmological constant Λ (or other form of dark energy, see Dark Energy—Sec. 28) is usually included, together with a parameter that quantifies the curvature, ΩK ≡ 1−Ωtot ,where Ωtot = Ωm + ΩΛ . The radiation content, while in principle a free parameter, is precisely
enough determined by the measurement of Tγ that it can be considered fixed, making a < 10−4
contribution to Ωtot today.
Astrophysical processes at relatively low redshift can also affect the C` s, with a particularly
significant influence during reionization. The Universe became reionized at some redshift zi , longafter recombination, affecting the CMB when it passed through the integrated Thomson-scattering
optical depth:
Z zidt
σT ne (z) dz,
τ=(29.4)dz
0
where σT is the Thomson cross-section, ne (z) is the number density of free electrons (which depends on astrophysics), and dt/dz is fixed by the background cosmology. In principle, τ could
be determined from the small-scale matter power spectrum, together with the physics of structure formation and radiative feedback processes; however, because this is a sufficiently intractablecalculation, in practice τ needs to be considered as a free parameter.
Thus, we have eight basic cosmological parameters, namely As , ns , r, h, Ωb h2 , Ωc h2 , Ωtot , andτ . One can add additional parameters to this list, particularly when using the CMB in combination
with other data sets. The next most relevant ones might be: Ων h2 , the massive neutrino contribution; w (≡ p/ρ), the equation of state parameter for the dark energy; and dns /d ln k, measuring
deviations from a constant spectral index. To these 11 one could of course add further parametersdescribing additional physics, such as details of the reionization process, features in the initial power
spectrum, a sub-dominant contribution of isocurvature modes, etc.
As well as these underlying parameters, there are other (dependent) quantities that can beobtained from them. Such derived parameters include the actual Ωs of the various components(e.g., Ωm ), the variance of density perturbations at particular scales (e.g., σ8 ), the angular scale ofthe sound horizon (θ∗ ), the age of the Universe today (t0 ), the age of the Universe at recombination,
reionization, etc. (see The Cosmological Parameters—Sec. 25).

29.5 Physics of Anisotropies
The cosmological parameters affect the anisotropies through the well understood physics of theevolution of linear perturbations within a background FRW cosmology. There are very effective,
fast, and publicly-available software codes for computing the CMB temperature, polarization, andmatter power spectra, e.g., CMBFAST [32], CAMB [33], and CLASS [34]. These have been tested over awide range of cosmologies and are considered to be accurate to much better than the 1% level [35],so that numerical errors are less than 10% of the parameter uncertainties for Planck [6].
For pedagogical purposes, it is easiest to focus on the temperature anisotropies, before moving to
the polarization power spectra. A description of the physics underlying the C`T T s can be separated
into four main regions (the first two combined below), as shown in the top left part of Fig. 29.1.
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729. Cosmic Microwave Background29.5.1 The ISW Rise, ` . 10, and Sachs-Wolfe Plateau, 10 . ` . 100
The horizon scale (or more precisely, the angle subtended by the Hubble radius) at last scatteringcorresponds to ` ' 100. Anisotropies at larger scales have not evolved significantly, and hence
directly reflect the ‘initial conditions.’ Temperature variations are δT /T = −(1/5)R(xLSS ) '(1/3)δφ/c2 , where δφ is the perturbation to the gravitational potential, evaluated on the lastscattering surface (LSS). This is a result of the combination of gravitational redshift and intrinsic
temperature fluctuations, and is usually referred to as the Sachs-Wolfe effect [36].
Assuming that a nearly scale-invariant spectrum of curvature (and corresponding density) perturbations was laid down at early times (i.e., ns ' 1, meaning equal power per decade in k), then`(` + 1)C` ' constant at low `s. This predicted near-flatness is hard to see unless the multipole
axis is plotted logarithmically (as in Fig. 29.1, and part of Fig. 29.2).
Time variation of the potentials (i.e., time-dependent metric perturbations) at late times leads
to an upturn in the C` s in the lowest several multipoles; any deviation from a total equation of state
w = 0 has such an effect. So the dominance of the dark energy at low redshift (see Dark Energy—Sec. 28) makes the lowest `s rise above the plateau. This is usually called the integrated SachsWolfe effect (or ISW rise), since it comes from the line integral of φ̇; it has been confirmed throughcorrelations between the large-angle anisotropies and large-scale structure [37]. Specific models
can also give additional contributions at low ` (e.g., perturbations in the dark-energy component
itself [38]), but typically these are buried in the cosmic variance.
In principle, the mechanism that produces primordial perturbations could generate scalar, vector, and tensor modes. However, the vector (vorticity) modes decay with the expansion of the
Universe. The tensors (transverse trace-free perturbations to the metric) generate temperature
anisotropies through the integrated effect of the locally-anisotropic expansion of space. Since thetensor modes also redshift away after they enter the horizon, they contribute only to angularscales above about 1◦ (see Fig. 29.1). Hence, some fraction of the low-` signal could be due toa gravitational wave contribution, although small amounts of tensors are essentially impossible todiscriminate from other effects that might raise the level of the plateau. Nevertheless, the tensors
can be distinguished using polarization information (see Sec. 29.7).
29.5.2 The Acoustic Peaks, 100 . ` . 1000
On sub-degree scales, the rich structure in the anisotropy spectrum is the consequence of gravitydriven acoustic oscillations occurring before the atoms in the Universe became neutral [39]. Perturbations inside the horizon at last scattering were able to evolve causally and produce anisotropy at
the last-scattering epoch, which reflects this evolution. The frozen-in phases of these sound waves
imprint a dependence on the cosmological parameters, which gives CMB anisotropies their great
constraining power.
The underlying physics can be understood as follows. Before the Universe became neutral, theproton-electron plasma was tightly coupled to the photons, and these components behaved as asingle ‘photon-baryon fluid.’ Perturbations in the gravitational potential, dominated by the darkmatter component, were steadily evolving. They drove oscillations in the photon-baryon fluid, withphoton pressure providing most of the restoring force and baryons giving some additional inertia.
The perturbations were quite small in amplitude, O(10−5 ), and so evolved linearly. That means each
Fourier mode developed independently, and hence can be described as a driven harmonic oscillator,
with frequency determined by the sound speed in the fluid. Thus, the fluid density underwent
oscillations, giving time variations in temperature. These combine with a velocity effect, which is
π/2 out of phase and has its amplitude reduced by the sound speed.
After the Universe recombined, the radiation decoupled from the baryons and could travel freely
towards us. At that point, the (temporal) phases of the oscillations were frozen-in, and became
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829. Cosmic Microwave Backgroundprojected on the sky as a harmonic series of peaks and troughs in power. The main peak is the modethat went through 1/4 of a period, reaching maximal compression. The even peaks are maximal
under-densities, which are generally of smaller amplitude because the rebound has to fight against
the baryon inertia. The troughs, which do not extend to zero power, are partially filled by the
Doppler effect because they are at the velocity maxima.
The physical length scale associated with the peaks is the sound horizon at last scattering, which
can be straightforwardly calculated. This length is projected onto the sky, leading to an angularscale that depends on the geometry of space, as well as the distance to last scattering. Hence,
the angular position of the peaks is a sensitive probe of a particular combination of cosmologicalparameters. In fact, this characteristic angular scale, θ∗ , is the most precisely measured observable,
and hence is usually treated as an element of the cosmological parameter set.
One additional effect arises from reionization at redshift zi . A fraction of photons (τ ) will be
isotropically scattered at z < zi , partially erasing the anisotropies at angular scales smaller thanthose subtended by the Hubble radius at zi . This corresponds typically to `s above about 10,
depending on the specific reionization model. The acoustic peaks are therefore reduced by a factore−2τ relative to the plateau.
These acoustic peaks were a clear theoretical prediction going back to about 1970 [40]. One
can think of them as a snapshot of stochastic standing waves. Since the physics governing themis simple and their structure rich, one can see how they encode extractable information about thecosmological parameters. Their empirical existence started to become clear around 1994 [41], andthe emergence, over the following decade, of a coherent series of acoustic peaks and troughs is atriumph of modern cosmology. This picture has received further confirmation with the detection in
the power spectrum of galaxies (at redshifts z . 1) of the imprint of these same acoustic oscillationsin the baryon component [42], as well as through detection of the expected oscillations in CMB
polarization power spectra (see Sec. 29.7).
29.5.3 The Damping Tail, ` & 1000
The recombination process is not instantaneous, and this imparts a thickness to the LSS. This
leads to a damping of the anisotropies at the highest `s, corresponding to scales smaller than that
subtended by this thickness. One can also think of the photon-baryon fluid as having imperfectcoupling, so that there is diffusion between the two components, and hence the amplitudes ofthe oscillations decrease with time. These effects lead to a damping of the C` s, sometimes called‘Silk damping’ [43], which cuts off the anisotropies at multipoles above about 2000. So, although
in principle it is possible to measure to ever smaller scales, this becomes increasingly difficult inpractice.
29.5.4 Gravitational Lensing Effects
CMB gravitational lensing is caused by structures at lower redshift along the line of sightto the LSS. Photon paths are deflected by the lensing potential φ, such that T (n̂) → T (n̂ +∇φ). Typical deflections are around 2 arcmin, but coherent over scales of a degree or so. Lensing
preserves surface brightness, which means that a uniform temperature field would be unaffected;
however, since there are anisotropies, then several distinct effects can be identified. The C` s areconvolved with a smoothing function in a calculable way, partially flattening the peaks and troughs,
generating a power-law tail at the highest multipoles, and complicating the polarization signal [44](see Sec. 29.7.3). Additionally, the effect of lensing on the CMB can be detected through the 4-point
function, which correlates temperature gradients and small-scale anisotropies, enabling a map of
the lensing potential to be constructed [45],Lensing is important because it gives an independent estimate of As , breaking the parameter
combination As e−2τ that is largely degenerate in the temperature anisotropy power spectra. Lens31st May, 2024

929. Cosmic Microwave Backgrounding is an example of a ‘secondary effect,’ i.e., the processing of anisotropies due to relatively nearby
structures (see Sec. 29.9.2). Galaxies and clusters of galaxies give several such effects; all are expected to be of low amplitude, but are increasingly important at the highest `s. Such effects carryadditional cosmological information (about evolving gravitational potentials in the low-redshift
Universe) and are receiving more attention as experiments push to higher sensitivity and angularresolution. The lensing power spectrum (see Sec. 29.8) can potentially constrain dark-energy evolution, while future measurements at high ` are a particularly sensitive probe of the sum of theneutrino masses [46].

Figure 29.2: CMB temperature anisotropy band-power estimates from the Planck, WMAP, ACT,and SPT experiments. Note that the widths of the multipole bands vary between experiments
and have not been plotted. This figure represents only a selection of the most recent available
experimental results, and some points with large error bars have been omitted. At the higher
multipoles these band-powers involve subtraction of particular foreground models, and so proper
analysis requires simultaneous fitting of CMB and foregrounds over multiple frequencies. The
horizontal axis here is logarithmic for the lowest multipoles, to show the Sachs-Wolfe plateau, andlinear for the other multipoles. The acoustic peaks and damping region are very clearly observed,
with no need for a theoretical line to guide the eye; however, the curve plotted is the best-fit Planck
ΛCDM model.

29.6 Current Temperature Anisotropy Data
Steady improvement in CMB data quality has led to the present-day cosmological model. The
most robust constraints currently available come from Planck satellite [47, 48] data (together with
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1029. Cosmic Microwave Backgroundconstraints from non-CMB cosmological data sets), although smaller-scale results from the ACT [49]and SPT [50] experiments are beginning to add useful constraining power. We plot power spectrum
estimates from these experiments in Fig. 29.2, along with WMAP data [5] for comparison (seeprevious versions of this review for data from earlier experiments). Independent experimental data
show consistency, both in maps and in derived power spectra (up to systematic uncertainties in theoverall calibration for some experiments). This makes it clear that systematic effects are largely
under control.
The band-powers shown in Fig. 29.2 are in very good agreement with a ‘ΛCDM’ model. As
described earlier, several (at least seven) of the peaks and troughs are quite apparent. The original papers present the details on how these estimates were made, on the band-power correlation
strengths, and on the information needed for their proper interpretation.

29.7 CMB Polarization
Thomson scattering of an anisotropic radiation field also generates linear polarization and the
CMB is predicted to be polarized, at the level of roughly 5% of the temperature anisotropies [51].
Polarization is a spin-2 field on the sky, and the algebra of the modes in multipole space is strongly
analogous to spin-orbit coupling in quantum mechanics [52]. The linear polarization pattern can be
decomposed in a number of ways, with two quantities required for each pixel in a map, often given
as the Q and U Stokes parameters. However, the most intuitive and physical decomposition is ageometrical one, splitting the polarization pattern into a part that comes from a divergence (often
referred to as the ‘E mode’) and a part with a curl (called the ‘B mode’) [53]. More explicitly, themodes are defined in terms of second derivatives of the polarization amplitude, with the Hessian
for the E modes having principal axes in the same sense as the polarization, while the B-mode
pattern can be thought of as a 45◦ rotation of the E-mode pattern. Globally one sees that the E
modes have (−1)` parity (like the spherical harmonics), while the B modes have (−1)`+1 parity.
The existence of this linear polarization allows for six different cross-power spectra to be determined from data that measure the full temperature and polarization anisotropy information. Parity
considerations make two of these zero, and we are left with four potential observables, C`T T , C`T E ,C`EE , and C`BB (see Fig. 29.1). Because scalar perturbations have no handedness, the B-mode
power spectrum can only be sourced by vectors or tensors. Moreover, inflationary scalar perturbations give only E modes, while tensors generate roughly equal amounts of E and B, therefore thedetermination of a non-zero B-mode signal is a way to measure the gravitational-wave contribution(and thus potentially derive the energy scale of inflation). However, because the signal is expected
to be rather weak, one must first eliminate the foreground contributions and other systematic effects down to very low levels. In addition, CMB lensing creates B modes from E modes, further
complicating the extraction of a tensor signal.
As with temperature, the polarization C` s exhibit a series of acoustic peaks generated by theoscillating photon-baryon fluid. The main ‘EE’ power spectrum has peaks that are out of phase withthose in the ‘T T ’ spectrum because the polarization anisotropies are sourced by the fluid velocity.
The ‘T E’ part of the polarization and temperature patterns comes from correlations between densityand velocity perturbations on the last-scattering surface, which can be both positive and negative,
and is of larger amplitude than the EE signal. There is no polarization Sachs-Wolfe effect, andhence no large-angle (low-`) plateau. However, scattering during a recent period of reionization
can create a polarization ‘bump’ at large angular scales.
Because the polarization anisotropies have only a small fraction of the amplitude of the temperature anisotropies, they took longer to detect. The first measurement of a polarization signal
came in 2002 from the DASI experiment [54], which provided a convincing detection, confirming
the general paradigm, but of low enough significance that it lent no real constraint to models.
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1129. Cosmic Microwave BackgroundFigure 29.3: Cross-power spectrum band-powers of the temperature anisotropies and E-mode
polarization signal from Planck (the low multipole data have been binned here), as well as WMAP,
BICEP2/Keck, ACTPol, and SPTPol/SPT-3G. The curve is the best fit to the Planck temperature,polarization, and lensing band-powers. Note that each data point is an average over a band of
multipoles, and hence to compare in detail with a model one has to integrate the theoretical curvethrough the band.

Despite dramatic progress since then, it is still the case that polarization data mainly support thebasic paradigm, while reducing error bars on parameters by only around 20%. However, there
are exceptions to this, specifically in the reionization optical depth, and the potential to constrain
primordial gravitational waves. Moreover, the situation is expected to change dramatically as moreof the available polarization modes are measured.
29.7.1 T–E Power Spectrum
Because the T and E skies are correlated, one has to measure the T E power spectrum, as well
as T T and EE, in order to extract all the cosmological information. This T E signal has now been
mapped out extremely accurately by Planck [48], and these band-powers are shown in Fig. 29.3,
along with those from WMAP [55] and BICEP2/Keck [56], with ACTPol [57] and SPTPol/-SPT3G [58, 59] extending to smaller angular scales. The anti-correlation at ` ' 150 and the peak at` ' 300 were the first features to become distinct, but now a whole series of oscillations is clearly seenin this power spectrum (including six or seven peaks and troughs [10]). The measured shape of the
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1229. Cosmic Microwave Backgroundcross-correlation power spectrum provides supporting evidence for the general cosmological picture,
and also directly constrains the thickness of the last-scattering surface. Because the polarization
anisotropies are generated in this scattering surface, the existence of correlations at angles above
about a degree demonstrates that there were super-Hubble fluctuations at the recombination epoch.
The sign of this correlation also confirms the adiabatic paradigm.
The overall picture of the source of CMB polarization and its oscillations has also been confirmed
through tests that average the maps around both temperature hot spots and cold spots [60]. One
sees precisely the expected patterns of radial and tangential polarization configurations, as well as
the phase shift between polarization and temperature. This leaves no doubt that the oscillation
picture is the correct one and that the polarization is coming from Thomson scattering at z ' 1100.

Figure 29.4: Power spectrum of E-mode polarization from Planck, together with WMAP, BICEP2/Keck, ACTPol, and SPTPol/SPT-3G. Note that some band-powers with larger uncertainties
have been omitted and that the unbinned Planck low-` data have been binned here. Also plottedis the best-fit theoretical model from Planck temperature + polarization + lensing data.
29.7.2 E–E Power Spectrum
Experimental band-powers for C`EE from Planck, WMAP, BICEP2/Keck Array [56], ACTPol[57], and SPTPol/SPT-3G [58, 59] are shown in Fig. 29.4. Without the benefit of correlating withthe temperature anisotropies (i.e., measuring C`T E ), the polarization anisotropies are very weak andchallenging to measure. Nevertheless, the oscillatory pattern is now well established and the data
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29. Cosmic Microwave Backgroundclosely match the T T -derived theoretical prediction. In Fig. 29.4 one can clearly see the ‘shoulder’
expected at ` ' 140, the first main peak at ` ' 400 (corresponding to the first trough in C`T T ),and the series of oscillations that is out of phase with those of the temperature anisotropy power
spectrum (including six or seven peaks and troughs [10]).
Perhaps the most unique result from the polarization measurements is at the largest angularscales (` < 10) in C`T E and C`EE , where there is evidence for an excess signal (not visible inFig. 29.4) compared to that expected from the temperature power spectrum alone. This is precisely
the signal anticipated from an early period of reionization, arising from Doppler shifts during thepartial scattering at z < zi . The amplitude of the signal indicates that the first stars, presumably
the source of the ionizing radiation, formed around z ' 8 (although the uncertainty is still quite
large). This corresponds to a scattering optical depth τ ' 0.06, so roughly 6% of CMB photons werere-scattered at the reionization epoch, with the other 94% last scattering at z ' 1100. However,estimates of the amplitude of this reionization excess have come down since the first measurements
by WMAP (indicating that this is an extremely difficult measurement to make) and the latest
Planck results have reduced the value further [11, 61].
29.7.3 B–B Power Spectrum
The expected amplitude of C`BB is very small, so measurements of this polarization curl-mode
are extremely challenging. The first indication of the existence of the BB signal came from thedetection of the expected conversion of E modes to B modes by gravitational lensing, through acorrelation technique using the lensing potential and polarization measurements from SPT [62].
However, the real promise of B modes lies in the detection of primordial gravitational waves atlarger scales. This tensor signature could be seen either in the ‘recombination bump’ at around` = 100 (caused by an ISW effect as gravitational waves redshift away at the last-scattering epoch)or the ‘reionization bump’ at ` . 10 (from additional scattering at low redshifts).
Results from the BICEP2 experiment [63] in 2014 suggested a detection of the primordial Bmode signature around the recombination peak. BICEP2 mapped a small part of the CMB sky
with the best sensitivity level reached at that time (below 100 nK), but at a single frequency. Higher
frequency data from Planck indicated that much of the BICEP2 signal was due to dust within our
Galaxy, and a combined analysis by the BICEP2, Keck Array, and Planck teams [64] indicatedthat the data are consistent with no primordial B modes. The current constraint from Planck
data alone is r < 0.069 (95% at k = 0.05 Mpc−1 [11, 61]) using all CMB power spectra, and thislimit is reduced to r < 0.044 with the inclusion of BICEP2/Keck Array data [61, 65]. The mostconstraining limit is r < 0.036 from a combination of BICEP2, Keck Array, and BICEP3 data,using WMAP and Planck maps to help remove foregrounds [66].
Several experiments are continuing to push down the sensitivity of B-mode measurements,
motivated by the enormous importance of a future detection of this telltale signature of inflation(or other physics at the highest energies). A compilation of experimental results for C`BB is shown in
Fig. 29.5, coming from a combination of direct estimates of the B modes (BICEP2/BICEP3/Keck
Array [66], POLARBEAR [67], SPTPol [68], and ACTPol [57]) and indirect determinations of
the lensing B modes based on estimating the effect of measured lensing on measured E modes(Planck [69], SPT [62], and ACT [70]). Additional band-power estimates are expected from theseand other experiments in the near future, with the Simons Observatory [71], the so-called ‘Stage 4’
CMB project [72] and the LiteBIRD satellite [73], holding great promise for pushing down to ther ∼ 0.001 level.

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1429. Cosmic Microwave BackgroundFigure 29.5: Power spectrum of B-mode polarization, including results from the BICEP2/BICEP3/Keck Array combined analysis, Planck, POLARBEAR, SPT, and ACT. Note thatsome of the measurements are direct estimates of B modes on the sky, while others are only sensitive to the lensing signal and come from combining E-mode and lensing potential measurements.
Several earlier experiments reported upper limits, which are all off the top of this plot.
√ A logarithmic horizontal axis is adopted here and the y-axis has been divided by a factor of ` in orderto show all three theoretically expected contributions: the low-` reionization bump; the ` ' 100
recombination peak; and the high-` lensing signature. The dotted line is for a tensor (primordial
gravitational wave) fraction r = 0.05, simply as an example, with all other cosmological parameters
set at the best Planck-derived values, for which model the expected lensing B modes have also beenshown with a dashed line.

29.8 CMB Lensing Power Spectrum
One further CMB observable is the gravitational lensing deflection, leading to the constructionof a map of the lensing potential. The latest Planck results [74] give a map that is detected atthe > 40σ level using a minimum-variance procedure from the 4-point function of temperature andpolarization data. From these maps, estimates can be constructed of C`φφ , the lensing-potential
power spectrum, which is found to be consistent with predictions from the best-fit temperatureand polarization model. Recent results from ACT give a power spectrum that has a similar overallsignal-to-noise ratio [75] and there are also interesting measurements from SPT [76]. Figure 29.6
plots the Planck, ACT, and SPT estimates of C`κκ , the lensing convergence power spectrum, which
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1529. Cosmic Microwave Backgroundis proportional to `2 (` + 1)2 times the potential power spectrum C`φφ .

Figure 29.6: Power spectrum measurements for CMB lensing, including selected results from
Planck, ACT, and SPT. The quantity plotted is the (dimensionless) potential power spectrum,
scaled by `2 (` + 1)2 and multiplied by a factor of 108 to make the numbers more manageable. Someless reliable measurements have not been plotted. The best-fit Planck ΛCDM spectrum is plotted
as a gray line.
We can think of each sky pixel as possessing three independent quantities that can be measured,
namely T , E, and φ (and potentially B, if that becomes detectable). Determining the constraining
power comes down to counting Y`m modes [77], as well as appreciating that some modes help
to break particular parameter degeneracies. We have only scratched the surface of CMB lensing
so far, and it is expected that future small-scale experiments will be able to extract more of thecosmological information. Further constraints can also be derived on the lower-redshift Universe bycross-correlating CMB lensing with other cosmological tracers of large-scale structure. Additionally,small-scale lensing, combined with E-mode measurements, can be used to ‘delens’ CMB B-mode
data, which will be important for pushing down into the r . 0.01 regime [78].

29.9 Complications
There are a number of issues that complicate the interpretation of CMB anisotropy data (andare considered to be signal by many astrophysicists), some of which we sketch out below.
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29. Cosmic Microwave Background29.9.1 Foregrounds
The microwave sky contains significant emission from our Galaxy and from extragalactic sources[79]. Fortunately, the frequency dependence of these various sources is in general substantiallydifferent from that of the CMB anisotropy signals. The combination of Galactic synchrotron,bremsstrahlung, and dust emission reaches a minimum at a frequency of roughly 100 GHz (orwavelength of about 3 mm). As one moves to greater angular resolution, the minimum moves toslightly higher frequencies, but becomes more sensitive to unresolved (point-like) sources.
At frequencies around 100 GHz, and for portions of the sky away from the Galactic plane, theforegrounds are typically 1 to 10% of the CMB anisotropies. By making observations at multiple
frequencies, it is relatively straightforward to separate the various components and determine the
CMB signal to the few per cent level. For greater sensitivity, it is necessary to use the spatial information and statistical properties of the foregrounds to separate them from the CMB. Furthermore,
at higher `s it is essential to carefully model extragalactic foregrounds, particularly the clusteringof infrared-emitting galaxies and scattering due to galaxy clusters, which dominate the measured
power spectrum as we move into the damping tail.
The foregrounds for CMB polarization follow a similar pattern to those for temperature, but
are intrinsically brighter relative to CMB anisotropies. WMAP showed that the polarized foregrounds dominate at large angular scales, and that they must be well characterized in order to bediscriminated [80]. Planck has shown that it is possible to characterize the foreground polarization
signals, with synchrotron dominating at low frequencies and dust at high frequencies [81]. On
smaller scales there are no strongly-polarized foregrounds, and hence at high multipoles it is inprinciple easier to measure foreground-free modes in polarization than in temperature. Although
foreground contamination will no doubt become more complicated as we push down in sensitivity,
making analysis more difficult, for the time being, foreground contamination is not a fundamental
limit for CMB experiments.
29.9.2 Secondary Anisotropies
With increasingly precise measurements of the primary anisotropies, there is growing theoretical and observational interest in ‘secondary anisotropies,’ pushing experiments to higher angularresolution and sensitivity. These secondary effects arise from the processing of the CMB due toionization history and the evolution of structure, including gravitational lensing (which was alreadydiscussed) and patchy reionization effects [82]. Additional information can thus be extracted about
the Universe at z  1000. This tends to be most effectively done through correlating CMB maps
with other cosmological probes of structure. Secondary signals are also typically non-Gaussian,unlike the primary CMB anisotropies.
A secondary signal of great current interest is the Sunyaev-Zeldovich (SZ) effect [83], which
is Compton scattering (γe → γ 0 e0 ) of the CMB photons by hot electrons in intergalactic plasma.
This creates spectral distortions by transferring energy from the electrons to the photons. It isparticularly important for clusters of galaxies, through which one observes a partially Comptonized
spectrum, resulting in a decrement at radio wavelengths and an increment in the submillimeter.
The imprint on the CMB sky is of the form ∆T /T = y f (x), with the y parameter being
the integral of Thomson optical depth times kTe /me c2 through the cluster, and f (x) describing
the frequency dependence. This is simply x coth(x/2) − 4 for a non-relativistic gas (the electron
temperature in a cluster is typically a few keV), where the dimensionless frequency x ≡ hν/kTγ .
As well as this ‘thermal’ SZ effect, there is also a smaller ‘kinetic’ effect due to the bulk motionof the cluster gas, giving ∆T /T ∼ τ (v/c), with either sign, but having the same spectrum as theprimary CMB anisotropies.
A significant advantage in finding galaxy clusters via the SZ effect is that the signal is largely
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29. Cosmic Microwave Background

independent of redshift, so in principle clusters can be found to arbitrarily large distances. The
SZ effect can be used to find and study individual clusters, and to obtain estimates of the Hubble
constant. There is also the potential to constrain other cosmological parameters, such as theclustering amplitude σ8 and the equation of state of the dark energy, through counts of detectedclusters as a function of redshift. The promise of the method has been realized through detectionsof clusters purely through the SZ effect by SPT [84], ACT [85], and Planck [86]. Results from
Planck clusters [87] suggest a somewhat lower value of σ8 than inferred from CMB anisotropies,
but there are still systematic uncertainties that might encompass the difference, and a more recent
analysis of SPT-detected clusters shows better agreement [88]. Further analysis of scaling relations
among cluster properties should enable more robust cosmological constraints to be placed in future,so that we can understand whether this ‘tension’ might be a sign of new physics.
29.9.3 Higher-order Statistics
Although most of the CMB anisotropy information is contained in the power spectra, there
will also be weak signals present in higher-order statistics. These can measure any primordial nonGaussianity in the perturbations, as well as non-linear growth of the fluctuations on small scales andother secondary effects (plus residual foreground contamination of course). There are an infinite
variety of ways in which the CMB could be non-Gaussian [26]; however, there is a generic form toconsider for the initial conditions, where a quadratic contribution to the curvature perturbations
is parameterized through a dimensionless number fNL . This weakly non-linear component can beconstrained in several ways, the most popular being through measurements of the bispectrum (or3-point function).
The constraints depend on the shape of the triangles in harmonic space, and it has becomecommon to distinguish the ‘local’ or ‘squeezed’ configuration (in which one side is much smaller thanthe other two) from the ‘equilateral’ configuration. Other configurations are also relevant for specifictheories, such as ‘orthogonal’ non-Gaussianity, which has positive correlations for k1 ' 2k2 ' 2k3 ,and negative correlations for the equilateral configuration. The constraints from the Planck teamlocal = 1 ± 5, f equil = −26 ± 47, and f ortho = −38 ± 24.
[89] are fNL
NL
NL
These results are consistent with zero, but are at a level that is now interesting for model
predictions. The amplitude of fNL expected is small, so that a detection of fNL  1 would ruleout all single-field, slow-roll inflationary models. It is still possible to improve upon these Planck
results, and it certainly seems feasible that a measurement of primordial non-Gaussianity may
yet be within reach. Non-primordial detections of non-Gaussianity from expected signatures havealready been made. For example, the bispectrum and trispectrum contain evidence of gravitational
lensing, the ISW effect, and Doppler boosting. For now the primordial signal is elusive, but should
it be detected, then detailed measurements of non-Gaussianity will become a unique probe of
inflationary-era physics. Because of that, much effort continues to be devoted to honing predictions
and measurement techniques, with the expectation that we will need to go beyond the CMB(e.g., 3D galaxy surveys) to dramatically improve the constraints.
29.9.4 Anomalies
Several features seen in the Planck data [29, 60, 90] confirm those found earlier with WMAP
[28], showing mild deviations from a simple description of the sky; these are often referred toas ‘anomalies.’ One such feature is the lack of power in the multipole range ` ' 20–30 [11, 48].
Other examples involve the apparent breaking of statistical anisotropy, caused by alignment of
the lowest multipoles, as well as a somewhat excessive cold spot and a power asymmetry between
hemispheres. No such feature is significant at more than the roughly 3σ level, and the importanceof ‘a posteriori’ statistics here has been emphasized by many authors. Since these effects are atlarge angular scales, where cosmic variance dominates, the results will not increase in significance
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29. Cosmic Microwave Backgroundwith more data, although there is the potential for more sensitive polarization measurements toprovide independent tests.

29.10 Constraints on Cosmological Parameters
The most striking outcome of the last couple of decades of experimental results is that thestandard cosmological paradigm continues to be in very good shape. A large amount of highprecision data on the CMB power spectrum is adequately fit with fewer than 10 free parameters(and only six need non-trivial values). The framework is that of FRW models, which have nearlyflat geometry, containing dark matter and dark energy, and with adiabatic perturbations having
close to scale-invariant initial conditions.
Within this basic picture, the values of the cosmological parameters can be constrained. Of
course, more stringent bounds can be placed on models that cover a restricted parameter space,e.g., assuming that Ωtot = 1 or r = 0. More generally, the constraints depend upon the adopted
prior probability distributions, even if they are implicit, for example by restricting the parameter
freedom or their ranges (particularly where likelihoods peak near the boundaries), or by using
different choices of other data in combination with the CMB. As the data become even more
precise, these considerations will be less important, but for now we caution that restrictions onmodel space and choice of non-CMB data sets and priors need to be kept in mind when adopting
specific parameter values and uncertainties.
There are some combinations of parameters that fit the CMB anisotropies almost equivalently.
For example, there is a nearly exact geometric degeneracy, where any combination of Ωm and ΩΛthat provides the same angular-diameter distance to last scattering will give nearly identical C` s.
There are also other less exact degeneracies among the parameters. Such degeneracies can be broken
when using the CMB results in combination with other cosmological data sets. Particularly usefulare complementary constraints from baryon acoustic oscillations, galaxy clustering, the abundanceof galaxy clusters, weak gravitational lensing measurements, and Type Ia supernova distances. For
an overview of some of these other cosmological constraints, see The Cosmological Parameters—
Sec. 25 of this Review.
Within the context of a 6-parameter family of models (which fixes Ωtot = 1, dns /d ln k = 0,
r = 0, and w = −1) the Planck results for T T , together with T E, EE, and CMB lensing, yield [11]:ln(1010 As ) = 3.044 ± 0.014; ns = 0.965 ± 0.004; Ωb h2 = 0.02237 ± 0.00015; Ωc h2 = 0.1200 ± 0.0012;
100θ∗ = 1.04092 ± 0.00031; and τ = 0.054 ± 0.007. Other parameters can be derived from this basic
set, including h = 0.674 ± 0.005, ΩΛ = 0.685 ± 0.007 (= 1 − Ωm ) and σ8 = 0.811 ± 0.006 (see also
Astrophysical Constants and Parameters—Sec. 2). Somewhat different (although consistent) values
are obtained using other data combinations, such as including BAO, supernova, H0 , or weak-lensing
constraints (see Sec. 25 of this Review). However, the Planck results quoted above are currently
the best available from CMB data alone. Results from other CMB experiments (e.g., SPT-3G [59])are consistent and becoming competitive.
The standard cosmological model still fits the data well, with the error bars on the parameters
continuing to shrink. Improved measurement of higher acoustic peaks has dramatically reduced theuncertainty in the θ∗ parameter, which is now detected at > 3000σ. The evidence for ns < 1 is now
at the 8σ level from Planck data alone. The value of the reionization optical depth has decreased
compared with earlier estimates; it is convincingly detected, but still not at very high significance.
The inferred value of H0 is smaller than the most precise values derived from the cosmic distanceladder. This parameter tension is discussed more fully in other sections of this Review. The CMB
anisotropies also provide the most precise estimate of the age of the Universe, with Planck giving
the value t0 = 13.797 ± 0.023 Gyr.
Constraints can also be placed on parameters beyond the basic six, particularly when including
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1929. Cosmic Microwave Backgroundother astrophysical data sets. Relaxing the flatness assumption, the constraint on Ωtot is 1.011 ±0.006. Note that for h, the CMB data alone provide only a very weak constraint if spatial flatness is
not assumed. However, with the addition of other data (particularly powerful in this context being
a compilation of BAO measurements; see Sec. 25 of this Review), the constraints on the Hubble
constant and curvature improve considerably, leading to Ωtot = 0.9993 ± 0.0019 [11].
For Ωb h2 the CMB-derived value is generally consistent with completely independent constraints
from Big Bang nucleosynthesis (see Sec. 24 of this Review). Related are constraints on additional
neutrino-like relativistic degrees of freedom, which lead to Neff = 2.99 ± 0.17 (including BAO),
i.e., no evidence for extra neutrino species.
The tightest published limit on the tensor-to-scalar ratio is r < 0.036 (measured at k =0.05 Mpc−1 ) from BICEP/Keck Array [66]. The detailed limit depends on how other parameters, especially As , nt , and dns /d ln k 6= 0 are restricted. The joint constraints on ns and r allow
specific inflationary models to be tested [30,91]. Looking at the (ns , r) plane, this means that m2 φ2(mass-term quadratic) inflation is disfavored by the data, as well as λφ4 (self-coupled) inflation.
The addition of the dark-energy equation of state w adds the partial degeneracy of being able
to fit a ridge in (w, h) space, extending to low values of both parameters. This degeneracy is broken
when the CMB is used in combination with other data sets, e.g., adding a compilation of BAO andsupernova data gives w = −1.028 ± 0.031 [11]. Constraints can also be placed on more general darkenergy and modified-gravity models [92]. However, when extending the search space, one needs tobe careful not to over-interpret some tensions between data sets as evidence for new physics.
For the reionization optical depth, a reanalysis of Planck data in 2016 resulted in a reductionin the value of τ , with the tightest result giving τ = 0.055 ± 0.009, and the newest analyses giving similar numbers. This corresponds to zi = 7.8–8.8 (depending on the functional form of thereionization history), with an uncertainty of ±0.9 [93]. This redshift is only slightly higher thatthat suggested from studies of absorption lines in high-z quasar spectra [94] and Ly α-emitting
galaxies [95], perhaps hinting that the process of reionization was not as complex as previously
suspected. The important constraint provided by CMB polarization, in combination with astrophysical measurements, thus allows us to investigate how the first stars formed and brought about
the end of the cosmic dark ages.

29.11 Particle Physics Constraints
CMB data place limits on parameters that are directly relevant for particle physics models. For
example, there is a limit on the sum of the masses of the neutrinos, Σmν < 0.12 eV (95%) [11] comingfrom Planck together with BAO measurements (although limits are weaker when considering both
Neff and Σmν as free parameters). This assumes the usual number density of fermions, which
decoupled when they were relativistic. The limit is tantalizingly only a factor of a few higher thanthe minimum value coming from neutrino mixing experiments (see Neutrino Mixings—Secs. 14 and26). As well as being an indirect probe of the neutrino background, Planck data also require thatthe neutrino fluid has perturbations, i.e., that it possesses a sound speed c2s ' 1/3, as expected [9].
The current suite of data suggests that ns < 1, with a best-fitting value about 0.035 below unity.
This is already quite constraining for inflationary models, particularly along with r limits. There isno current evidence for running of the spectral index, with dns /d ln k = −0.004 ± 0.007 from Planck
alone [11] (with a similar value when BAO data are included), although this is less of a constraint
on models. Similarly, primordial non-Gaussianity is being probed to interesting levels, althoughtests of simple inflationary models will only come with significant reductions in uncertainty.
The large-angle anomalies, such as the hemispheric modulation of power and the dip in powerat ` ' 20–30, have the potential to be hints of new physics. Such effects might be expected ina Universe that has a large-scale power cut-off, or anisotropy in the initial power spectrum, or is
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topologically non-trivial. However, cosmic variance and a posteriori statistics limit the significanceof these anomalies, absent the existence of a model that naturally yields some of these features(and ideally also predicting other phenomena that can be tested).
Constraints on ‘cosmic birefringence’ (i.e., rotation of the plane of CMB polarization thatgenerates non-zero T B and EB power) can be used to place limits on theories involving parity
violation, Lorentz violation, or axion-photon mixing [96].
It is possible to place limits on additional areas of physics [97], for example annihilating dark
matter [9, 9], primordial magnetic fields [98], and time variation of the fine-structure constant [99],as well as the neutrino chemical potential, a contribution of warm dark matter, topological defects,
or physics beyond general relativity. Further particle physics constraints will follow as the smallerscale and polarization measurements continue to improve.
The CMB anisotropy measurements precisely pin down physics at the time of last-scattering,and so any change of physics can be constrained if it affects the relevant energies or timescales.
Future, higher sensitivity measurements of the CMB frequency spectrum will push the constraints
back to cover energy injection at much earlier times (∼ 1 year). Comparison of CMB and BBN
observables extend these constraints to timescales of order seconds, and energies in the MeV range.
And to the extent that inflation provides an effective description of the generation of perturbations,the inflationary observables may constrain physics at GUT-type energy scales.
More generally, careful measurement of the CMB power spectra and non-Gaussianity can inprinciple put constraints on physics at the highest energies, including ideas of quantum gravity,
string theory, extra dimensions, colliding branes, etc. At the moment any calculation of predictions
appears to be far from definitive. However, there is a great deal of activity on implications offundamental theories for the early Universe, and hence a chance that there might be observational
implications for specific scenarios.

29.12 Fundamental Lessons
More important than the precise values of parameters is what we have learned about the generalfeatures that describe our observable Universe. Beyond the basic hot Big Bang picture, the CMB
has taught us that:• the (observable) Universe is very close to isotropic;• the Universe recombined at z ∼ 1000 and started to become ionized again at z ∼ 10;• the geometry of the Universe is close to flat;• both dark matter and dark energy are required;• gravitational instability is sufficient to grow all of the observed large structures in the Universe;• topological defects were not important for structure formation;• there were ‘synchronized’ super-Hubble modes generated in the early Universe;• the initial perturbations were predominantly adiabatic in nature;• the primordial perturbation spectrum has a slightly red tilt;• the perturbations had close to Gaussian (i.e., maximally random) initial conditions.
These features form the basis of the cosmological standard model, ΛCDM, for which it is tempting to make an analogy with the Standard Model of particle physics (see earlier Sections of thisReview). In comparison, the cosmological model is much further from any underlying ‘fundamental
theory,’ which might ultimately provide the values of the parameters from first principles. Nevertheless, any genuinely complete ‘theory of everything’ must include an explanation for the valuesof these cosmological parameters in addition to the parameters of the Standard Model of particle
physics.
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29. Cosmic Microwave Background29.13 Future Directions
Given the significant progress in measuring the CMB sky, which has been instrumental in tying down the cosmological model, what can we anticipate for the future? There will be a steady
improvement in the precision and confidence with which we can determine the appropriate cosmological parameters. Ground-based experiments operating at smaller angular scales will continue
to place tighter constraints on the damping tail, lensing, and cross-correlations. New polarization
experiments at small scales will probe further into the damping tail, without the limitation of extragalactic foregrounds. And polarization experiments at large angular scales will push down thelimits on primordial B modes.
Planck, the third generation CMB satellite mission, was launched in May 2009, and produceda large number of papers, including a set of cosmological studies based on the first two full surveysof the sky (accompanied by a public release of data products) in 2013, a further series coming fromanalysis of the full mission data release in 2015 (eight surveys for the Low Frequency Instrument
and five surveys for the High Frequency Instrument), and a third series derived from a final analysisof the 2018 data release, including full constraints from polarization data. Planck data currently
dominate constraints on models, but that situation will change soon.
A set of cosmological parameters is now known to percent-level accuracy, and that may seem
sufficient for many people. However, we should certainly demand more of measurements that
describe the entire observable Universe! Hence a lot of activity in the coming years will continue tofocus on determining those parameters with increasing precision. This necessarily includes testingfor consistency among different predictions of the cosmological Standard Model, and searching forsignals that might require additional physics.
A second area of focus will be the smaller-scale anisotropies and ‘secondary effects.’ There is agreat deal of information about structure formation at z  1000 encoded in the CMB sky. This may
involve higher-order statistics and cross-correlations with other large-scale structure tracers, as well
as spectral signatures, with many experiments targeting the galaxy cluster SZ effect, for example.
The current status of CMB lensing is similar (in terms of total signal-to-noise) to the quality of thefirst CMB anisotropy measurements by COBE, and thus we can expect that experimental probesof lensing will improve dramatically in the coming years. All of these investigations can provide
constraints on the dark-energy equation of state, for example, which is a major area of focus forseveral future cosmological surveys at optical wavelengths. CMB lensing also promises to yield ameasurement of the sum of the neutrino masses.
A third direction is increasingly sensitive searches for specific signatures of physics at the highest
energies. The most promising of these may be the primordial gravitational wave signals in C`BB ,which could be a probe of the ∼ 1016 GeV energy range. There are several experiments underway or
being developed that are designed to search for the polarization B modes, with the most ambitious
being CMB-S4 on the ground and LiteBIRD in space. Additionally, non-Gaussianity holds thepromise of constraining models beyond single-field slow-roll inflation.
Anisotropies in the CMB have proven to be the premier probe of cosmology and the early
Universe. Theoretically the CMB involves well-understood physics in the linear regime, and isunder very good calculational control. A substantial and improving set of observational data now
exists. Systematics appear to be under control and are not currently a limiting factor. And sofor the next several years we can expect an increasing amount of cosmological information to begleaned from CMB anisotropies, with the prospect also of some genuine surprises.
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