heavy relics that decouple early, for instance by entropy injection episodes [26]. Notice also that the freeze-out calculation strongly depends on the assumed background cosmology, and changes e.g. if the early universe is not radiation-dominated around DM decoupling.

The calculation of the freeze-out relic abundance hinges on a Boltzmann equation relating the Liouville operator to the collision operator acting on the phase space density. Under a variety of simplifying assumptions including homogeneity and isotropy, it is possible to reduce the relevant equation for the number density n of a single species pair-annihilating with particles in the thermal bath via 2-to-2 processes to

$$ \frac{dn}{dt} - 3Hn = -\langle\sigma v\rangle (n^2 - n_{\text{eq}}^2), \qquad (27.2) $$

where $\langle\sigma v\rangle$ is the thermally-averaged pair-annihilation cross section times relative velocity (see Ref. [27]), and $n_{\text{eq}}$ is the equilibrium number density. Relics for which the freeze-out temperature is much larger than the particle mass (and thus that freeze-out as ultra-relativistic) are called hot relics; if the opposite is true, the relic is instead considered cold.

A straightforward calculation shows that to leading order the frozen-out density of hot relics is linearly proportional to the relic particle mass. The comoving number density $Y = n/s$, where s is the entropy density, for a hot relic is approximately given by its equilibrium value,

$$ Y_{\text{f.o.}} \approx Y_{\text{eq}} \approx 0.278 \frac{g_{\text{eff}}}{g_{*s}}, \qquad (27.3) $$

where $g_{\text{eff}}$ is the relic's effective number of degrees of freedom, and $g_{*s}$ is the number of entropic relativistic degrees of freedom, both calculated at $T_{\text{f.o.}}$. The resulting relic abundance, assuming an iso-entropic expansion, is

$$ \Omega_{\text{hot}} h^2 = \frac{m Y_{\text{f.o.}} s_0 h^2}{\rho_c} \approx \frac{m}{93 \text{ eV}}, \qquad (27.4) $$

with $s_0$ the entropy density today, and with the latter equality holding for the case of SM neutrinos, with a freeze-out temperature around 1 MeV (which enters in the final relic abundance through the degrees of freedom dependence on the right-hand side of Eq. (27.3)).

For cold relics, the leading-order dependence of the relic abundance on the DM particle properties is an inverse proportionality relation to the pair-annihilation cross section,

$$ \Omega_{\text{cold}} h^2 \approx 0.1 \left( \frac{x_{\text{f.o.}}}{20} \right) \left( \frac{10^{-8} \text{ GeV}^{-2}}{\sigma_{\text{DM+DM}\leftrightarrow\text{anything}}} \right), \qquad (27.5) $$

where $x \equiv m_{\text{DM}}/T$. In turn, the freeze-out temperature is approximately given by the solution to the equation

$$ \sqrt{x} \cdot e^{-x} = (m_{\text{DM}} \cdot M_P \cdot \sigma_{\text{DM+DM}\leftrightarrow\text{anything}})^{-1}, \qquad (27.6) $$

where $M_P \approx 2.435 \times 10^{18}$ GeV is the reduced Planck mass. As a result, $T_{\text{f.o.}} \approx m_{\text{DM}}/x_{\text{f.o.}}$, with $x_{\text{f.o.}}$ a number between 10 and 50, depending on the cross section, with only a logarithmic dependence on the DM mass. Since for electroweak-scale cross sections and masses $\sigma_{\text{DM+DM}} \approx 10^{-8} \text{ GeV}^{-2}$, “weakly-interacting massive particles”, or WIMPs have gained exceptional popularity. Notice that Eq. (27.5) bears, however, no connection to the weak scale [28], despite the relation being known as “WIMP miracle”.

Numerous scenarios exist, including notably supersymmetry [29,30] and models with universal extra dimensions [31,32] where the relic abundance of the DM is controlled by processes involving a slightly heavier, unstable, co-annihilating species [33]. In this case the calculation of the abundance of the stable species proceeds similarly to what outlined above, with an effective pair-annihilation cross section that captures the effects of co-annihilation replacing the pair-annihilation cross section [30].

**Freeze-in:** Collisional processes can lead to the production of out-of-equilibrium particles that progressively accumulate over cosmic time, a process sometimes called freeze-in. The abundance

of the frozen-in particles produced at a given redshift depends on the product of the production rate times the Hubble time at that redshift. Freeze-in generally implies that the lightest observable-sector particles decay to the DM with relatively long lifetimes, giving peculiar signals at colliders (see e.g. [34]). Gravitinos are an example of DM candidates possibly produced via a freeze-in type scenario, albeit the portal coupling is in that case via a higher dimensional, Planck-suppressed operator [35].

**Cannibalization and other dark-sector number-changing processes:** Thermal processes can drive the abundance of the DM beyond simple 2-to-2 number-changing interactions. For instance, DM can “cannibalize” [36, 37] itself if $n \to 2$ processes exist. In this case, a critical aspect is whether or not the DM sector is in thermal contact with the Standard Model thermal bath. If it is, $n \to 2$ processes can drive the relic abundance, e.g. in the Strongly Interacting Massive Particles (SIMP) scenario [38]. Models exist where the kinetic decoupling (i.e. the decoupling from the thermal equilibrium velocity distribution) of the two sectors drives the abundance of the DM (elastically decoupling relics, or ELDERs [39]). When the two sectors are not in thermal contact, $n \to 2$ processes heat the DM sector dramatically, rapidly affecting the temperature ratio between the visible and dark sectors [36,38]. If the relevant cross sections are large enough, and the DM mass light enough, significant effects can arise in structure formation [36].

**Non-thermal production:** DM production can proceed via processes out of thermal equilibrium (“non-thermal” production). These include DM production via the decay of a “mother” particle [40,41] (or of topological defects [42], moduli [43] etc.) to the DM, or production via gravitational effects.

**Asymmetric DM:** An enticing alternative possibility for DM production is that of asymmetric DM [44,45]: the relic DM abundance arises from an asymmetry between anti-DM and DM. This asymmetry may or may not be related to the baryon-antibaryon asymmetry. If it is, then depending on the model and its thermal history, a relation exists between the mass of the DM and the proton mass. A variety of proposals have been put forward where alternately baryogenesis is explained from a DM sector asymmetry, or vice-versa (see e.g. Ref. [46] for a review).

**Primordial Black Holes production:** A qualitatively standalone class of DM candidates, primordial black holes (PBHs), arises from entirely different mechanisms from what reviewed above. PBHs are thought to originate from gravitational collapse of large density fluctuations in the early universe [47, 48]. The over-densities could be produced in a variety of ways, such as topological defects like cosmic strings, necklaces or domain walls, curvature fluctuations from a period of ultra-slow-roll, a sound speed “resonance”, an early phase of matter domination, or sub-horizon phenomena including a phase transition and pre-heating. Albeit the calculation depends on the details of gravitational collapse, the formation time is connected to the PBH mass via $M = \gamma M_{\text{PBH}} \approx 2 \times 10^5 \gamma (\frac{t}{18}) M_{\odot}$, with $\gamma \approx (1/\sqrt{3})^3$ during radiation domination [49].

## 27.4 Density and velocity distribution of dark matter

### 27.4.1 Local density and velocity distribution

The density and distribution of DM in the Milky Way encipher relevant dynamical information about our Galaxy, and are particularly important for direct and indirect detection experiments. The local density ($\rho_0$) is an average over a volume of a few hundred parsecs in the Solar neighbourhood.

To determine the local density from observations, two classes of methods are used [50]. So-called local measures rely on the vertical motion of tracer stars in the vicinity of the Sun, while global measures extrapolate $\rho_0$ from the measured rotation curve, with additional assumptions about the Galactic halo shape. Conversely, by comparing the extrapolated local density with the one obtained from local measures, one can constrain the local shape of the Milky Way halo. A major source of uncertainty on $\rho_0$ is the contribution of baryons (stars, gas, stellar remnants) to the local dynamical mass. For instance, the motion of tracer stars used in local measures is dictated by the total potential generated by
baryons and DM, and a robust baryonic census must be available to infer the additional contribution from DM. Recent determinations from global methods lie in the range (0.2 – 0.6) GeV/cm³, while new studies of the local DM density from Gaia satellite data yield (0.3 – 1.5) GeV/cm³, depending on the type of stars used in the study.

Other observational quantities that enter in the phase space distribution of DM, and provide constraints on mass models of the Milky Way are the local circular speed $v_c$ and the escape velocity $v_{esc}$. The local circular speed is measured by various methods, roughly divided into measurements of the Sun's velocity with respect to an object assumed to be at rest with respect to the Galactic centre or direct measurements of the local radial force. These methods yield values of $v_c$ = (218 – 246) km/s. A recent estimate of the escape velocity, defined as the speed above which objects are not gravitationally bound to our galaxy, is $v_{esc}=533_{-4}^{+54}$ km/s [51] (see also the more recent estimate in [52] of $v_{esc}=484.6_{-7.4}^{+17.8}$ km/s, which is not in tension with the value we use).

The local velocity distribution of DM particles can not be measured directly at present, and is mostly derived from simulations. In general, experiments use the simplest, so-called *Standard Halo Model (SHM)* for their data analysis. It assumes an isotropic, isothermal sphere of DM particles with a density profile of $\rho(r) \propto r^{-2}$, for which the velocity distribution is Maxwellian, with a velocity dispersion $\sigma_v = v_c/\sqrt{2}$. This distribution, which formally extends to infinity, is truncated at $v_{esc}$ [53]. Earlier high resolution, dark-matter-only simulations found velocity distributions that markedly deviated from a Maxwell-Boltzmann distribution and in addition revealed components above the dominant smooth distribution, including narrow spikes due to tidal streams. Recent hydrodynamical simulations of Milky Way-like galaxies including baryons, which have a non-negligible effect on the DM distribution in the Solar neighbourhood, find velocity distributions that are indeed close to Maxwellian, arguing that the SHM is a good approximation.

Ultimately the goal is to determine the velocity distribution from observations (for example by studying the motion of stars that share the same kinematics as the DM), and the Gaia satellite data offers a unique opportunity to study the various stellar populations. Recently it was revealed that the local stellar halo has two components: a quasi-spherical, weakly rotating structure with metal-poor stars, and a flattened, radially anisotropic structure of metal-rich stars, which arose due to accretion of a large ($10^{11-12}$ M$_\odot$) dwarf galaxy around ($8-10) \times 10^9$ y ago [54]. The expectation is that the local DM halo shows a similar bimodal structure, and first velocity distributions of the two components - using the stellar populations as tracers - were inferred in [55]. In Ref. [56], an updated halo model is introduced: it includes the anisotropic structure seen in the Gaia data and provides an analytic expression for the velocity distribution. The value of the local DM density is updated to $(0.55 \pm 0.17)$ GeV/cm³, where the 30% error accounts for the systematics. The circular rotation and the escape speeds are updated to $v_c$ = (233 ± 3) km/s and $v_{esc}=528_{-25}^{+24}$ km/s. The former is in agreement with a new Bayesian estimation of the circular velocity using Gaia DR3, of $v_c$ = (233 ± 7) km/s [57]. With Gaia DR3, new stellar streams are being discovered, some of these likely originating from the disruption and merger of dwarf galaxies. The study of the velocities of such streams will allow for increasingly improved constraints on both the DM subhalo density profiles and the velocity distributions.

### 27.4.2 *Small-scale challenges*

The ΛCDM framework is tremendously successful at explaining the observed large-scale structures of the Universe (corresponding to distances ≥1 Mpc, the typical inter-galactic distance), as well as the main properties of galaxies that form within DM haloes, see the comprehensive review in Ref. [58]. The observed large-scale structure is consistent with point-like, cold DM particles that interact purely via the gravitational force. However, in the past decades, observations at scales below ~1 Mpc, where structure formation becomes strongly nonlinear, turned out more problematic to be described within the ΛCDM model. The main small-

scale challenges are known as: the missing satellites problem, the cusp-core problem and the too-big-to-fail problem. Initially these issues, which are not all independent of one another, arose by comparing theoretical predictions from dark-matter-only simulations to observation. While their most likely solutions are in dissipative, baryonic physics (such gas cooling, star formation, supernovae feedback), see the recent reviews in Ref. [59, 60], the small-scale problems could in addition call for a modification or an extension of the ΛCDM paradigm. Most importantly, the ever increasing amount of data on the satellites of the MW and M31 are used to constrain alternative DM models.

**The missing satellites problem:** High-resolution cosmological simulations of DM haloes the size of the MW predict hundreds or thousands of subhaloes with masses that are in principle large enough to allow for galaxy formation ($> 10^7$ M$_\odot$). Yet less than ~100 satellite galaxies with masses down to ~300 M$_\odot$ are known to orbit our galaxy within 300 kpc. Galaxies in the field show a similar under-abundance. One solution could be that galaxy formation becomes increasingly inefficient as the halo mass drops, and thus the smallest DM haloes have naturally failed to form galaxies. Progress in both observations and theoretical modeling of observable properties of dwarf galaxies show that there is very scant evidence for a “missing satellite problem” [60]. Nonetheless, additional, related strong tensions remain, including the diversity of observed rotation curves [61] and the unexpected alignment of satellite planes, the latter issue possibly solved by high-precision simulations tracking inflow along filaments of the cosmic web [62].

**The cusp-core problem:** The mass density profiles of DM haloes in ΛCDM simulations rise steeply at small radii, $\rho(r) \propto r^{-\gamma}$, with $\gamma \sim 0.8 - 1.4$. This is in contrast to the observed density profiles of many low-mass galaxies (albeit not all), the rotation curves of which are best fit with constant-density cores, $\gamma \simeq 0 - 0.5$. A related issue is that simulations predict more DM than measured in the central regions of galaxies (also known as the central density problem). A likely solution is that baryonic feedback modifies the structure of DM haloes. Hydrodynamic simulations which include the effects of baryons on galaxy formation have shown that baryonic feedback (e.g., supernova-driven blowouts) can erase the central cusps and produce core-like density profiles. Surveys of satellites of hundreds of galaxies similar to the Milky Way, such as SAGA [63], will likely provide important observational information.

**The too-big-to-fail problem:** This problem is related to the fact that the dark matter halos of the largest observed Milky Way satellites are typically much smaller than those of the most massive subhalos of simulated Milky Way-like halos. DM haloes of such masses are thought to be too massive to have failed to form stars (hence the name of the problem), especially if lower-mass subhaloes are capable of doing so. A similar issue is present in Andromeda and in field galaxies outside the Local Group. Possible solutions include lowering the mass assumed for the Milky Way host halo, or baryonic effects including tidal stripping and larger-than-expected inner cores [60]. Other issues relate to the origin of ultra-faint galaxies, and the opposite problem of dwarf galaxies with very little dark matter given their stellar mass; there exist proposed solutions to both challenges, including the “bullet dwarf” scenario of Ref. [64].

The solutions that were briefly mentioned above do not require modifications to the ΛCDM framework. Other solutions involve either modifications of linear theory predictions (via the nature of the DM particle, e.g., Warm DM - WDM) or modifications of nonlinear predictions (via DM models that involve a self-interaction of DM particles - SIDM). WDM models postulate particles with masses at the keV-scale, and the observed number of dark-matter-dominated satellites is used to set a lower limit on the number of subhaloes in the MW and thus a lower limit on the particle's mass [59]. Current constraints are in the range $m_{WDM}$ > (1.6 – 2.3) keV. Cosmological simulations with SIDM find that $\sigma/m_{SIDM} \simeq (0.5-10)$ cm$^2$/g can alleviate the cusp-core and too-big-to-fail problems, giving rise to DM cores in dwarf galaxies with sizes of (0.3-1.5) kpc [13]. Galaxy clusters provide important constraints, and their large central DM densities pre-
fer models with $\sigma/m_{DM} \lesssim 0.1 \text{ cm}^2/\text{g}$ [65]. Thus, if SIDM is to solve the small-scale CDM problems and obey the constraints observed on the scales of clusters, $\sigma$ must depend on the velocity of the particle: it must increase as the rms speed of the particle decreases from the scale of clusters ($v \sim 10^3 \text{ km/s}$) to the scale of dwarf galaxies ($v \sim 10 \text{ km/s}$).

## 27.5 Dark matter models

Particle DM model building is deeply intertwined with the question of the nature of physics beyond the Standard Model (BSM) of particle physics<fnref n="1" />. Directions in this area have followed a few strategies, including, but not limited to

(1) pursuing DM candidates embedded in frameworks that include solutions to other open issues in particle physics, for example WIMPs in connection with electroweak-scale new physics that addresses the hierarchy problem, such as supersymmetry (see the Supersymmetry reviews Sec. 88 and 88); axions in connection with frameworks that address the strong CP problem (see Axions and Other Similar Particles—Sec. 90); sterile neutrinos in connection with the problem of neutrino masses and mixing (see the Neutrino Masses, Mixing, and Oscillations—Sec. 14); or (2) ad hoc, or bottom-up models built with the intent of addressing or explaining a putative experimental (e.g. particle physical anomalies) or observational (e.g. astronomical) signal.

**WIMPs:** The WIMP paradigm has been a preferred framework chiefly because it often arises in beyond the Standard Model scenarios that address the hierarchy problem whilst also providing a simple mechanism to explain the observed relic abundance via the “WIMP miracle” described above. Perhaps the most notable example of a framework containing a paradigmatic WIMP is the minimal supersymmetric extension to the Standard Model, if the lightest supersymmetric particle is a neutralino (the mass eigenstate resulting from the mixing of the supersymmetric partners to the Higgses and to the SU(2) and hypercharge gauge bosons, and, possibly, of additional singlet scalars); purely SU(2) sneutrinos have long been ruled out by direct detection, but with suitable mixing with “inert” (gauge-singlet) sneutrinos they can also play the role of WIMP candidates. For more details see the Supersymmetry—Sec. 88 and 89 in this Review. Other non-supersymmetric WIMP models include models with a Higgs or Z (or Z') portal, universal extra dimensions [32], and other models with extra (warped or flat) dimensions, little Higgs theories, technicolor and composite Higgs theories, among others (see e.g. the review in [66]).

**Axions and axion-like particles:** Axions are an especially compelling example of a broad category of DM candidates encompassing very light scalar or pseudoscalar fields. The QCD axion provides a solution to the strong CP problem, and is at present a viable DM candidate (see Sec. 90 for details on motivations, production mechanisms, and detection prospects for the QCD axion). Ultra-light, bosonic DM generally implies the imprint of quantum effects on macroscopic scales (hence the name of wave or fuzzy DM). Specifically, some of the small-scale issues mentioned in Sec. 27.4 can be addressed if the de Broglie wavelength of the DM, of mass $m_a$ and velocity $v_a$,

$$ \frac{\lambda}{2\pi} = \frac{\hbar}{m_a v_a} \approx 1.9 \text{ kpc} \left( \frac{10^{-22} \text{ eV}}{m_a} \right) \left( \frac{10 \text{ km/s}}{v_a} \right) \quad (27.7) $$

is comparable to the size of the smallest observed gravitationally collapsed structures, roughly, for a self-gravitating system of mass M, a scale $r \simeq GM/v^2$. The typical expectation is the formation of a soliton-like core in the DM density profile of size $\lambda$, thus inversely proportional to the DM mass, with an upper limit on the central density of around

$$ \rho_s \lesssim 7 M_{\odot}/\text{pc}^3 \left( \frac{m_a}{10^{-22} \text{ eV}} \right)^6 \left( \frac{M}{10^9 M_{\odot}} \right) \quad (27.8) $$

for a halo of virial mass M. Additionally, wave DM predicts that halos lighter than around $10^7(m_a/10^{-22} \text{ eV})^{-3/2}M_{\odot}$ should not

exist [67], and that the number of halos in the local universe with a mass at or less $10^9(m_a/10^{-22} \text{ eV})^{-4/3}M_{\odot}$ [68] be significantly depleted, addressing in part the too big to fail and missing satellite problems (see Sec. 27.4 above). Light bosonic DM is necessarily produced non-thermally [69], and the connection with the visible sector need not, but might, exist.

**Dark photons:** Light vector bosons such as a “dark photon” V with a mass below $m_V < 2m_e$, can be cosmologically stable (depending upon its kinetic mixing coupling with the visible photon) and be a viable DM candidate. Light dark photons can be produced in the early universe through scattering or annihilation via processes such as $\gamma e^{\pm} \to V e^{\pm}$ or $e^{+} e^{-} \to V \gamma$, or via resonant photon-dark photon conversion, or from a condensate seeded by inflationary perturbations [70], or from a misalignment mechanism similar to the one commonly invoked for axion production; constraints on the parameter space stem from a combination of direct detection experiments, where the dark photon is absorbed and leads to a large ionization signal, from stellar cooling constraints from the Sun, horizontal branch stars, and red giants, and from CMB and the diffuse radiation from the $V \to 3\gamma$ decay mode. More broadly, light dark (pseudo-)scalars and vectors can be best constrained with experiments that rely on their wave-like behaviour and/or on their possible “portal” with the visible sector. A broad assortment of experiments is sensitive to the range of masses between $10^{-22} \text{ eV}$ and $10^{-2} \text{ eV}$. Among these experimental efforts, the lowest masses are probed by torsion balance experiments [71, 72], atom interferometry [73], magnetometers [74, 75], and even gravitational wave detectors [76]; at increasing masses, if the light bosons couple electromagnetically, they can generate effective currents which are detectable with different apparatus depending on the relevant, mass-dependent target frequency. The experimental portfolio includes the broadband axion search ABRACADABRA [77, 78], the LC resonator DM Radio [79], lumped-element LC resonators [80], and cavity resonators such as HAYSTAC [81] and ADMX [82].

**Sterile Neutrinos:** Sterile (gauge-singlet) neutrinos, assumed to share a Dirac mass term with ordinary, $SU(2)_L$-active neutrinos, have long been considered viable DM candidates [83]. The mostly-sterile mass eigenstate participates in $SU(2)_L$ interactions via a mixing parameter $\theta \ll 1$ that controls much of the particle’s phenomenology. In particular, the sterile neutrino possesses an inverse-lifetime on the order of $\tau^{-1} \sim G_F^2 m_\nu^5 \theta^2$, forcing the mixing to not exceed

$$ \theta < 3.3 \times 10^{-4} \left( \frac{10 \text{ keV}}{m_\nu} \right)^5 \quad (27.9) $$

in order for the lifetime to exceed the age of the universe. While the main decay channel is to three active neutrinos, observationally the radiative decay mode to one neutrino plus a photon is much more relevant, giving rise to a quasi-monochromatic photon line at half the sterile neutrino mass. A recent tentative signal at 3.5 keV was reported from stacked observations of clusters of galaxies, individual clusters [84, 85], and the Galactic center [86] with both the XMM and Chandra X-ray observatories. The signal however was not detected in a large sample of galaxies and groups of galaxies [87] and dwarf galaxies [88], and especially Draco [89], shedding strong doubts on its sterile neutrino decay origin. Future observations with increased energy resolution might conclusively pinpoint the origin of the 3.5 keV emission [90].

**Models with rich dark sectors:** The absence of any conclusive signals from DM as a particle thus far motivates the hypothesis that the DM be charged under some new “hidden” dark-sector force, an idea that dates back many decades [91], including in the guise of “mirror DM” (more recently in the context of “neutral naturalness”). Top-down motivation for hidden-sector DM comes from string theory [92], although TeV-scale BSM framework such as supersymmetry and composite Higgs models can also naturally accommodate hidden sectors [93]. Although no coupling of the visible sector to the hidden sector need exist in principle, there are a few reasons to expect it [94]. The mass scale for hidden-sector DM is broader than, but overlapping with, that for WIMPs (this latter being limited to roughly between a few GeV and a few TeV). In particular, while some motivation exists for electroweak-scale

<footnote n="1">¹Notice that this includes the case of PBHs, as successful formation of the correct number density of PBHs involves new ingredients beyond standard cosmology and particle physics</footnote>
hidden sectors, light, sub-GeV hidden sectors have a strong theoretical underpinning, and offer novel detection avenues and opportunities. The phenomenology of hidden-sector DM depends primarily on the nature of the force and its force carrier. The most widely considered cases are (pseudo-)scalar and (axial-)vector mediators. Among the structures for the mediators’ coupling to the visible sector, renormalizable “portals” include the $H^\dagger H$ operator, through Lagrangian terms of the type $(\mu\phi + \lambda\phi^2)H^\dagger H$, coupling to the hypercharge field strength $B^{\mu\nu}$ via kinetic mixing, $e' B_{\mu\nu} F'^{\mu\nu}$, and the “neutrino” portal, $y_n L_{HN}$, where L is the lepton doublet of any generation, N is a right-handed neutrino, H is the SM Higgs doublet, and $y_n$ the Yukawa coupling. Other possibilities are for instance a vector mediator directly coupled to SM fermions charged under its corresponding symmetry [95], or a $Z'$ associated to $U(1)_{B-L}$. Additional possibilities, arising for instance from vector couplings to anomalous global symmetries of the SM like baryon or lepton number, also exist [94]. The accelerator program necessary to probe hidden-sector DM often involves small-scale colliders and fixed-target experiments, with experiments utilizing missing energy and momentum offering the best sensitivity. Beam-dump experiments can test large ranges of DM-mediator couplings as long as mediators decay or scatter inside the detector (see e.g. the recent review [96]). Such experiments can also probe dark sectors with light vectors coupled to visible matter besides gauge kinetic mixing: an instance are neutrino trident scattering used to place bounds on e.g. $L_\mu - L_\tau$ $Z'$ gauge bosons. Being virtually unconstrained, the phenomenology of dark sectors can be arbitrarily rich, with possibilities ranging from dark non-Abelian gauge interactions creating non-trivial self-interacting and/or particle number-changing dynamics, to models of “dynamical” DM, with multi-component, unstable DM candidates and a time-variable effective total DM abundance and equation of state [97].

## 27.6 Laboratory detection of dark matter

Laboratory searches for DM particles can be roughly classified in direct detection experiments, axion searches (see Axions and Other Similar Particles—Sec. 90), and searches at accelerators and colliders.

### 27.6.1 Searches at Accelerators and Colliders

Various searches for dark matter have been carried out by the CMS and ATLAS collaborations at the LHC in pp collisions [98–102]. In general, these assume that dark matter particles escape the detector without interacting leading to significant amounts of missing energy and momentum.

Searches for DM with the LHC and other colliders have targeted DM models that interact with the SM via Higgs or Z boson exchange, effective field theories with heavy mediators, UV-complete models such as supersymmetry, models with long-lived particles, and models with rich dark sectors. The experimental program correspondingly includes searches for invisible-particle production mediated by a SM boson, generic searches for invisible particles produced via new particle mediators, and specific searches for complete models.

There are a variety of types of signals for DM, as noted by Ref. [98]:

(a) the imbalance in the transverse momentum in an event due to the presence of DM particles, produced together with one Standard Model particle,

(b) a bump in the di-jet or di-lepton invariant mass distributions, or

(c) an excess of events in the di-jet angular distribution, produced by a dark matter mediator. No signal for DM has been observed in the LHC experiments so far. Instead, limits are set on masses, couplings, and cross-sections. The latter can be compared, generally in a model-dependent way, with direct detection experiments.

Searches strategies are designed to optimize signal-to-noise by selecting search-specific cuts: a model-independent instance is initial-state electromagnetic or strong-interaction radiation plus missing transverse energy. Collider searches for DM inform, and are informed, by DM searches through direct or indirect detection (see below), and, if possible, by the inferred thermal relic

DM abundance. The collider searches alone cannot prove that a discovery is of dark matter.

In the latter category, searches for DM with the LHC and other colliders have targeted DM models that interact with the SM via Higgs or Z boson exchange, effective field theories with heavy mediators, UV-complete models such as supersymmetry, models with long-lived particles, and models with rich dark sectors. The experimental program correspondingly includes searches for invisible-particle production mediated by a SM boson, generic searches for invisible particles produced via new particle mediators, and specific searches for complete models. Searches strategies are designed to optimize signal-to-noise by selecting specific search-specific cuts: a model-independent instance is initial-state electromagnetic or strong-interaction radiation plus missing transverse energy.

New fixed-target experiments are increasingly probing lighter, sub-GeV dark sectors; thermal, or quasi-thermal relics (such as asymmetric, SIMP, ELDER DM models) provide sharp, largely model-independent experimental targets in the case of direct annihilation to Standard Model particles; even “secluded” annihilation to dark-sector particles can be probed with fixed-target experiments, albeit with some model-dependent caveats. Accelerator searches provide unique ways to test light DM models in that they present a lower dependence on the DM particle nature, that could e.g. drive suppressed velocity-dependent direct detection cross sections, and on kinematic thresholds that can similarly suppress rates with other search strategies; additionally, accelerator searches offer opportunities to explore dark sectors in greater detail than other search strategies [94].

Experimental approaches to explore scenarios with a light DM $\chi$ and mediator $A'$ include (i) searches for missing mass in exclusive reactions such as $e^+e^- \to \gamma(A'\chi\bar{\chi})$ or $e^-p \to e^-p(A'\chi\bar{\chi})$, (ii) missing momentum/energy, e.g. in fixed-target reactions such as $eZ \to eZ(A'\chi\bar{\chi})$, (iii) electron and proton beam dump, where the DM particle is detected from scattering in a downstream detector after being produced in the beam dump, (iv) direct dark photon searches, when the mediator cannot decay to the dark matter, $m_{A'} < 2m_\chi$. Current facilities actively probing the methods listed above include APEX and HPS and JLab, MiniBooNE at FermiLab, NA64 at CERN, and TREK at J-PARC. Future experiments that will eventually contribute to covering the mentioned thermal targets include Belle-II at KEK, MAGIX at MESA, PADME at LNF, SHIP at CERN, VEPP3 at BINP, BDX at JLab, COHERENT at ORNL, DarkLight at JLab, LDMX at SLAC or JLab, MMAPS at Cornell, SBN at Fermilab, and SeaQuest [94].

### 27.6.2 Direct detection formalism

Direct detection experiments mostly aim to observe elastic or inelastic scatters of Galactic DM particles with atomic nuclei, or with electrons in the detector material. Predicted event rates assume a certain mass and scattering cross section, as well as a set of astrophysical parameters: the local density $\rho_0$, the velocity distribution $f(\vec{v})$, and the escape velocity $v_{esc}$ (see Sec. 27.4).

**Interactions with atomic nuclei:** For DM scattering off nuclei, the differential scattering rate R as a function of nuclear recoil energy $E_R$ is

$$ \frac{dR(E_R, t)}{dE_R} = N_T \frac{\rho_0}{m_{DM}} \int_{v>v_{min}} v f(\vec{v} + \vec{v}_E(t)) \frac{d\sigma(E_R, v)}{dE_R} d^3 v, \qquad (27.10) $$

where $N_T$ is the number of target nuclei, $m_{DM}$ is the mass of the DM particles, $v = |\vec{v}|$ is the speed of the particle in the experiment’s rest frame, $f(\vec{v} + \vec{v}_E(t))$ is the velocity distribution in the Earth’s frame, $v_{min}$ is the minimum speed of the DM particles that can cause a recoil energy $E_R$ and $\sigma$ is the scattering cross section on the nucleus [29, 120]. For elastic scattering, the minimum velocity is $v_{min} = (m_N E_R / 2m_r^2)^{1/2}$, with $m_N$ being the mass of the nucleus, and $m_r = (m_N m_{DM}) / (m_N + m_{DM})$ the reduced mass of the nucleus-DM system. In case of inelastic scattering, the minimum speed becomes $v_{min} = (m_N E_R / 2m_r^2)^{1/2} + E^* / (2m_N E_R)^{1/2}$, with the nuclear excitation energy $E^*$, for part of the kinetic energy of the incoming particle will be spent on exciting the nucleus. The prompt de-excitation energy, if observed
<table><caption>**Table 27.1:** Best constraints from direct detection experiments on the SI (at high >5 GeV and low < 5 GeV masses) and SD DM-nucleon couplings.</caption><thead><tr><th>Experiment</th><th>Target</th><th>Fiducial mass [kg]</th><th>Cross section [cm<sup>2</sup>]</th><th>DM mass [GeV]</th><th>Ref.</th></tr></thead><tbody><tr><td colspan="6" style="text-align:center; font-weight:bold;">Spin independent high mass (&gt;5 GeV)</td></tr><tr><td>LUX-ZEPLIN</td><td>Xe</td><td>5500</td><td>9.2 &times; 10<sup>-48</sup></td><td>36</td><td>[103]</td></tr><tr><td>PandaX-4T</td><td>Xe</td><td>2670</td><td>3.8 &times; 10<sup>-47</sup></td><td>40</td><td>[104]</td></tr><tr><td>XENONnT</td><td>Xe</td><td>4180</td><td>2.6 &times; 10<sup>-47</sup></td><td>30</td><td>[105]</td></tr><tr><td>SuperCDMS</td><td>Ge</td><td>12</td><td>1.0 &times; 10<sup>-44</sup></td><td>46</td><td>[106]</td></tr><tr><td>DarkSide-50</td><td>Ar</td><td>20</td><td>1.9 &times; 10<sup>-43</sup></td><td>10</td><td>[107]</td></tr><tr><td>DEAP-3600</td><td>Ar</td><td>2000</td><td>3.9 &times; 10<sup>-45</sup></td><td>100</td><td>[108]</td></tr><tr><td colspan="6" style="text-align:center; font-weight:bold;">Spin independent low mass (&lt;5 GeV)</td></tr><tr><td>LUX (Migdal)</td><td>Xe</td><td>118</td><td>6.9 &times; 10<sup>-38</sup></td><td>2</td><td>[109]</td></tr><tr><td>XENON1T (Migdal)</td><td>Xe</td><td>1042</td><td>3 &times; 10<sup>-40</sup></td><td>2</td><td>[110]</td></tr><tr><td>XENON1T (ionisation only)</td><td>Xe</td><td>1042</td><td>3.6 &times; 10<sup>-41</sup></td><td>3</td><td>[111]</td></tr><tr><td>DarkSide-50 (ionisation only)</td><td>Ar</td><td>20</td><td>1.4 &times; 10<sup>-42</sup></td><td>2</td><td>[107]</td></tr><tr><td>SuperCDMS (CDMSlite)</td><td>Ge</td><td>0.6</td><td>2 &times; 10<sup>-40</sup></td><td>2</td><td>[112]</td></tr><tr><td>SuperCDMS (CDMSlite, Migdal)</td><td>Ge</td><td>0.6</td><td>6 &times; 10<sup>-38</sup></td><td>2</td><td>[113]</td></tr><tr><td>CRESSST</td><td>CaWO<sub>4</sub> - O</td><td>0.024</td><td>1 &times; 10<sup>-39</sup></td><td>2</td><td>[114]</td></tr><tr><td>CRESSST</td><td>Si</td><td>0.0035</td><td>4.5 &times; 10<sup>-32</sup></td><td>0.15</td><td>[115]</td></tr><tr><td>DAMIC</td><td>Si</td><td>0.3</td><td>1 &times; 10<sup>-40</sup></td><td>4</td><td>[116]</td></tr><tr><td>NEWS-G</td><td>Ne</td><td>0.3</td><td>1 &times; 10<sup>-38</sup></td><td>2</td><td>[117]</td></tr><tr><td colspan="6" style="text-align:center; font-weight:bold;">Spin dependent proton</td></tr><tr><td>PICO60</td><td>C<sub>3</sub>F<sub>8</sub> - F</td><td>49</td><td>3.2 &times; 10<sup>-41</sup></td><td>25</td><td>[118]</td></tr><tr><td>PandaX-4T</td><td>Xe</td><td>2670</td><td>1.7 &times; 10<sup>-40</sup></td><td>40</td><td>[119]</td></tr><tr><td>LUX-ZEPLIN</td><td>Xe</td><td>5500</td><td>4.2 &times; 10<sup>-41</sup></td><td>32</td><td>[103]</td></tr><tr><td>XENONnT</td><td>Xe</td><td>4180</td><td>1.4 &times; 10<sup>-40</sup></td><td>30</td><td>[105]</td></tr><tr><td colspan="6" style="text-align:center; font-weight:bold;">Spin dependent neutron</td></tr><tr><td>PandaX-4T</td><td>Xe</td><td>2670</td><td>5.8 &times; 10<sup>-42</sup></td><td>40</td><td>[119]</td></tr><tr><td>LUX-ZEPLIN</td><td>Xe</td><td>5500</td><td>1.5 &times; 10<sup>-42</sup></td><td>30</td><td>[103]</td></tr><tr><td>XENONnT</td><td>Xe</td><td>4180</td><td>4.3 &times; 10<sup>-42</sup></td><td>30</td><td>[105]</td></tr></tbody></table>

in addition to the nuclear recoil energy, will boost the region-of-interest to higher energies [121].

If one assumes the standard, leading order spin-independent (SI) and spin-dependent (SD) interactions, which couple to the charge/mass and spin of the nucleus, respectively, the differential cross-section is proportional to the inverse squared speed of the DM particle, $d\sigma/dE_R \propto v^{-2}$, and the dependence on the velocity distribution can be expressed as:

$$g(v_{min}, t) = \int_{v>v_{min}} \frac{f(\vec{v} + \vec{v}_E(t))}{v} d^3 v. \quad (27.11)$$

This functions allows for the comparison of various experimental results independently of the underlying velocity distribution [122], for a given DM mass. The time-integrated differential cross section is the sum of the SI and SD contributions:

$$\frac{d\sigma(E_R, v)}{dE_R} = \frac{m_N}{2m_r^2 v^2} (\sigma_0^{SI} F_{SI}^2(E_R) + \sigma_0^{SD} F_{SD}^2(E_R)), \quad (27.12)$$

where $F^2(E_R)$ are the nuclear form factors and $\sigma_0$ the cross-sections in the limit of zero momentum transfer. Since the incoming particle velocity is $v/c \sim 10^{-3}$, the nuclear recoil energy is at most tens of keV (much smaller than typical nuclear binding energies per nucleon), and the momentum transfer $q = (2m_N E_R)^{1/2} \sim \mathcal{O}(10-100)$ MeV. This implies that $1/q$ can be of the same order as nuclear radii $R \sim A^{1/3}$ fm, and that nuclei are not point-like from the perspective of a DM particle. The cross sections will thus involve nuclear form factors. These were calculated in [123] and [124] for the SI and SD case, respectively, for specific target nuclei, while the cross sections are often expressed in terms of single-nucleon cross sections and effective couplings of the DM particle to protons and neutrons. In the SI case, all the nucleons in the nucleus contribute coherently to the cross-section (under the assumption of isospin independence in the DM couplings). Dominant sources of uncertainty are the nucleon sigma terms, especially for Higgs-dominated interactions,

where the couplings are proportional to the quark masses. An overview is presented in Ref. [125]. For SD scattering, the nuclear spin contents due to the protons and neutrons must be considered.

The interactions of DM particles with nuclei can be treated in a non-relativistic effective field theory (NR-EFT) approach, which considers more general DM scenarios based on the lowest-order, four-field operators that describe the couplings to nucleons. These operators, which correspond to different types of interactions between the DM and quark fields, can be momentum- and velocity-dependent, and might be leading when momentum-independent interactions are suppressed, or even vanish in the limit of zero momentum [126, 127]. In Ref. [127, 128] all 15 operators (arising from 20 possible bilinear combinations between the DM and nucleon fields) which obey Galilean-invariance, T-symmetry and are Hermitian are written out up to quadratic order in $q$, and the nuclear response functions evaluated in shell-model calculations for DM targets made of F, Na, Ge, I and Xe isotopes. The connection to particle physics within the context of simplified DM models is made in Ref. [129, 130], where the simplified models assume a single DM particle with one mediator which couples it to quarks. More recently the DM-nucleus scattering was also analysed in the framework of chiral effective field theory (Ch-EFT), a low-energy effective theory of QCD, which allows for a consistent derivation of the nuclear responses beyond the leading-order expressions [131, 132]. Ch-EFT preserves the QCD symmetries, and predicts DM couplings to two nucleons (e.g., when the hypothetical particle couples to a virtual pion exchanged between the nucleons). It also provides a power counting that suggests a hierarchy of the various NR-EFT operators, which is however approximate given that the couplings between the DM and the Standard Model fields are not known. The generalized SI structure factors for spin-1/2 and spin-0 DM particles and various isotopes of F, Si, Ar, Ge and Xe employed in direct detection experiments are provided in Ref. [132].

**Scattering off bound electrons and absorption:** For DM particle masses below the GeV-scale, most searches for DM-nucleus scattering rapidly lose sensitivity, due to energy thresholds around a few 100 eV - few keV. As an example, a light DM particle
with a mass of 100 MeV and $v \propto 10^{-3}c$ will induce a nuclear recoil energy of about 0.5 eV in a target made of argon. Another strategy is to search for DM scattering off bound electrons, allowing for all of the kinetic energy (50 eV in the above case) to be transferred to the material [133]. The leading possibilities are ionisation, excitation, and molecular dissociation processes, which typically require energies of (1-10) eV, and thus allow to probe scattering of DM particles with masses down to the $O(\text{MeV})$ range.

For a bound electron with binding energy $E_B$ DM particle masses of $m_{DM} \ge 250$ keV $\times E_B/1$ eV can in principle be probed. The signal depends on the material, and can consist of one or more electrons (in semiconductors, noble liquids, graphene), one or more photons (in scintillators) or phonons (in superconductors and superfluids) and quasiparticles (in superconductors). As an example, the differential event rate for ionization in atoms is given by:

$$ \frac{dR_{ion}}{d \ln E_R} = N_T \frac{\rho_0}{m_{DM}} \frac{d\langle\sigma_{ion}v\rangle}{d \ln E_R}, \qquad (27.13) $$

where $E_R$ is the recoil energy transferred to the electron, $\langle\sigma_{ion}v\rangle$ is the thermally averaged ionization cross section and $N_T$ is the number of target atoms per unit mass. The cross-section is related to the non relativistic DM-electron elastic scattering cross-section ($\sigma_e$):

$$ \frac{dR_{ion}}{d \ln E_R} = \frac{6.2}{A} \left( \frac{\rho_0}{0.4 \text{ GeV cm}^{-3}} \right) \left( \frac{\sigma_e}{10^{-40} \text{cm}^2} \right) \left( \frac{10 \text{ MeV}}{m_{DM}} \right) \times \frac{d\langle\sigma_{ion}v\rangle/d \ln E_R}{10^{-3} \sigma_e} \frac{\text{events}}{\text{kg d}} \qquad (27.14) $$

Predicted differential rates in various materials (He, Ar, Ge, Xe) and for different particle masses are shown in [133], together with cross section sensitivities as a function of mass and expected background rates from neutrinos. More recently, rates of electronic transitions induced by DM-electron scattering on Xe, Si and Ge for spin 0, spin 1/2, and spin 1 DM models were calculated in [134]. The code EXCEED-DM (extended calculation of electronic excitations for direct detection of dark matter) can compute DM-electron interaction rates with inputs from a variety of ab initio electronic structure calculations [135].

Two classes of DM candidates, axion-like-particles (ALPs) and dark (or hidden) photons (see Sec. 27.5), can be absorbed in a target material by interactions with bound electrons via the axioelectric effect, which is analogous to the photoelectric effect: a boson is absorbed by a bound electron, which is then ejected from the atom [70,136,137]. The dark photon arises in extensions of the SM by a new massive or massless U(1)' field, coupled to the SM U(1)$_Y$ via a kinetic mixing term $\kappa$, see Sec. 27.5. The absorption cross-section of a massive, NR particle $m_V$ with coupling $e' = e\kappa$ to electrons is (in natural units, and for energies $E_V \ll m_e$)

$$ \sigma_{abs} = \frac{\alpha'}{\alpha} \left( \frac{E_V}{2m_e} \right)^2 \sigma_{pe}, \qquad (27.15) $$

where $\sigma_{pe}$ is the photoelectric cross-section, and an analogue to the electromagnetic fine structure constant $\alpha$ is introduced, $\alpha' = (e\kappa)^2/4\pi$. The rate per atom is

$$ R \approx \frac{\rho_0}{m_V} \times \kappa^2 \sigma_{pe}. \qquad (27.16) $$

Since the kinetic energy of the dark photon is negligible compared to its rest energy, a mono-energetic peak at its mass is expected in the spectrum of a direct detection experiment. Dark photons with a thermally generated abundance are excluded by direct detection experiments [70], however non-thermal mechanisms (e.g., via perturbations during inflation) could create the relic abundance, see Section 27.3.

Similarly to axions, ALPs arise in the spontaneous breaking of a global symmetry, and are phenomenologically described by a mass $m_a$ and a decay constant $f_a$. Unlike for QCD axions, however, there is no strict relation between $m_a$ and $f_a$. The coupling strength to electrons with mass $m_e$ is parameterised by

$g_{ae} = 2m_e/f_a$, and the absorption cross section of a particle with incoming velocity $v_a$ is related to the cross-section for the photoelectric effect as

$$ \sigma_{abs} v_a \approx \frac{3E_a^2}{4\pi\alpha f_a^2} \sigma_{pe} = \frac{3g_{ae}^2}{4\pi\alpha} \left(\frac{E_a}{2m_e}\right)^2 \sigma_{pe}. \qquad (27.17) $$

As in the case of the dark photon, the signature is a monoenergetic peak at the mass of the particle, broadened by the energy resolution of the detector. Constraints on the couplings of ALPs and dark photons to electrons from direct detection experiments in $m_a$ and $m_V$ mass ranges from $\sim (1-10^4)$ eV were derived in Ref. [138, 139], and compared to indirect limits from anomalous energy losses in the Sun, in red-giant and horizontal-branch stars. For a detailed discussion of axion and ALP searches, we refer to the Axion review.

### 27.6.3 Current and future direct detection technologies

Direct detection experiments aim to observe the small (keV-scale and below) and rare (fewer than $\sim 1$ event/(kg y)) signals which are induced by DM particle scatters in a detector, mostly in the form of ionisation, scintillation or lattice vibrations. A majority of experiments detects more than one signal, which allows to distinguish between scattering off of electrons (electronic recoils, ER) and off of atomic nuclei (nuclear recoils, NR). A 3D position resolution is required to define central detector regions (or fiducial volumes) with low background rates from surrounding materials, and the distinction between single- versus multiple-scatters rejects a significant fraction of backgrounds, given that DM will scatter at most once. We refer to [140,141] for recent reviews of the field.

**Specific signatures:** For NRs, the shape of the differential recoil spectrum is exponentially falling with recoil energy, and depends on the mass of the particle and on the nuclear mass. Unless $m_{DM} \gg m_N$, $m_{DM}$ can in principle be determined from the measured recoil spectrum, where multiple targets will provide tighter constraints [142]. The Earth's motion through the MW induces a seasonal variation of the total event rate and a forward-backward asymmetry in a directional signal [143, 144]. The annual modulation is due to the Earth's motion in the Galactic rest frame, which is a superposition of the Earth's rotation around the Sun and the Sun's rotation around the Galactic center. Since the Earth's orbital speed is much smaller than the Sun's speed, the expected amplitude of the modulation is $\sim 5\%$. In the SHM, the period is one year, and the phase is 150 d (June 2), when both speeds add up maximally. This expectation is modified for different DM distributions, e.g. in the case of sub-structures such as clumps and streams [145, 146] and a DM disc [147]. In addition, the modulation changes phase at a specific recoil energy (known as crossing-energy) [148], which depends on the DM and nuclear mass, allowing to in principle determine $m_{DM}$ if low energy thresholds can be achieved. A powerful signature is provided by the ability to detect the axis and direction of the recoiling nucleus. Since the DM flux in the laboratory frame is peaked in the direction of motion of the Sun towards the constellation Cygnus, the recoil spectrum is peaked in the opposite direction. The observation of such a dipole feature would provide a 'smoking-gun' evidence for DM, where the forward-backward rates can differ by a factor of $\sim 10$, depending on the energy threshold. Ref. [149] provides a review of the theoretical framework and of the discovery reach of directional detectors.

**Backgrounds, including neutrinos:** Early direct detection experiments employing low-background Ge spectrometers featured background levels around 2 events/(kg d keV), while the latest generation of liquid Xe experiments reduced this noise by almost five orders of magnitude, to $4 \times 10^{-5}$ events/(kg d keV). In liquid xenon detectors, the measured ER spectra at low energies are for the first time dominated by solar pp neutrino interactions, second-order weak decays, as well as $^{214}$Pb $\beta$-decays from radon mixed with the xenon. Other backgrounds are due to the radioactivity of detector components, followed by cosmic muons and their secondaries such as fast neutrons. The cosmic and environmental radiation are suppressed by going deep underground and surrounding the experiments with appropriate shielding structures (mainly large water Cherenkov detectors for the current and
next-generation detectors). Activation of materials via cosmic-ray interactions produce long-lived radio-nuclides (e.g., $^{39}$Ar, $^{60}$Co, $^{68}$Ge, $^{32}$Si, etc), while long-lived, human-made isotopes ($^{85}$Kr, $^{137}$Cs, etc) can mix with detector materials or generate surface backgrounds. For details, we refer to Section 36.6 of this Review.

The final backgrounds are due to the irreducible neutrino flux from the Sun, the atmosphere and the diffuse supernovae background [150]. Solar pp-neutrinos start dominating the electronic recoil background due to elastic neutrino-electron scatters, at a level of ~ (10–25) events/(t y) below energies of ~100 keV, while coherent elastic neutrino-nucleus scatters (CEνNS) from $^{8}$B solar neutrinos will induce up to ~ 10$^3$ events/(t y) for high-A targets, at nuclear recoil energies below ~ few keV. Nuclear recoils from atmospheric neutrinos and the diffuse supernovae neutrino background will yield event rates in the range (1–5) events/(100 t y), depending on the detector material. In general, $^{8}$B and atmospheric neutrinos will impact light (≤ 6 GeV) and heavy (100 GeV and above) DM searches for cross sections on nucleons below ~ 10$^{-45}$ cm$^2$ and ~ 10$^{-49}$ cm$^2$, respectively. The precise cross-sections where neutrinos constitute a dominant background strongly depend on the systematic uncertainties on the neutrino flux normalisation for each source [151]. For very low energy thresholds to nuclear recoils, e.g. 10-30 eV in Ge and Si detectors, CEνNS due to the $^{7}$Be neutrino flux become relevant for exposures of ~50 kg y [152]. For DM searches with electron recoils via DM-electron scattering and dark photon or ALP absorption, solar neutrinos will also limit the sensitivity to DM masses in the range ~ (1-10$^3$) MeV and ~ (1-10$^3$) eV, respectively, for large exposures ~1 t y, as shown in Ref. [153].

**Solid-state cryogenic detectors:** Current experiments using the bolometric technique (see Section 36.5 of this Review), together with either charge or light readout, are SuperCDMS (Si, Ge) at Soudan (until 2015) and SNOLAB (in construction), EDELWEISS (Ge) at the Laboratoire Souterrain de Modane (LSM) and CRESST (CaWO$_4$, Al$_2$O$_3$, Si, LiAlO$_2$) at the Laboratori Nazionali del Gran Sasso (LNGS). These experiments are optimised for low-mass DM searches, and can probe WIMP masses down to ~0.03 GeV. CRESST operated a 0.35-g thin wafer Si detector and reached a 10-eV energy threshold, probing DM masses in the range (115–500) MeV, improving existing limits by up to a factor 20 for masses below 160 MeV. CDMSlite operated detectors at higher bias voltages to amplify the phonon signals produced by drifting charges and thus have access to light DM around 1.5 GeV. A reanalysis of the CDMSlite data using the Migdal effect allowed to exclude new parameter space for WIMP-nucleon cross-sections down to 30 MeV [113]. The goal at SNOLAB is to probe the (0.05–5) GeV region down to cross sections of ~ 10$^{-44}$ cm$^2$. Much smaller, gram-scale versions of cryogenic detectors can have single-charge resolution and thus probe low-mass DM via inelastic electron recoils. A SuperCDMS single-charge sensitive Si detector placed upper limits on DM interacting with electrons for masses between (0.5–10$^4$) MeV, as well as on dark photon kinetic mixing for dark photon masses in the range (1.5–40) eV. With Ge crystals operated at Soudan, SuperCDMS constrained dark photons and ALPs in the mass range 40 eV to 500 keV.

Germanium ionisation detectors operated at 77 K can reach sub-keV energy thresholds and low backgrounds, but lack the ability to distinguish electronic from nuclear recoils. The CDEX-10 experiment at the China Jinping Underground Laboratory (CJPL) uses p-type, point-contact Ge detectors operated in liquid nitrogen to probe DM masses down to 3 GeV. It also reported constraints on the kinetic mixing of dark photons in the mass range (0.1–4.0) keV. The 0νββ-decay experiments Majorana Demonstrator and GERDA have obtained constraints on the couplings of ALPs and dark photons to electrons, with masses between (6-100) keV and (60-1000) keV, respectively. The Majorana Demonstrator improved laboratory constraints on solar axion masses between (1.2–100) eV, setting limits on the axion-photons coupling of $g_{aγ} < 1.45 \times 10^{-9}$ GeV$^{-1}$(95% CL).

**Noble liquids:** Liquid argon (LAr) and liquid xenon (LXe) are employed as DM targets, while R&D on liquid helium is ongoing. We refer to Ref. [154] for a review of the liquid noble gas detector technology in low-energy physics, as well to Section 36.4 of this Review.

At present the best constraints on DM-nucleus interactions come from multi-tonne experiments using xenon: the LZ experiment at SURF, PandaX-4T at CJPL and XENONnT at LNGS. These experiments probe particle masses down to ~6 GeV (when using both light and charge signals) and the SI DM-nucleon cross section down to $9.2 \times 10^{-48}$ cm$^2$ (at 36 GeV). LAr experiments employ pulse shape discrimination (PSD) that allows for distinguishing between ER and NR events, at the expense of higher energy thresholds than in LXe. The DarkSide-50 TPC at LNGS (until 2019) sets a minimum upper limit on the SI, DM-nucleon cross-section of $6 \times 10^{-43}$ cm$^2$ at 10 GeV, while the single-phase experiment DEAP-3600 at SNOLAB constrains the SI cross-section to values below $3.9 \times 10^{-45}$ cm$^2$ at 100 GeV.

In noble liquids, sub-GeV DM particles can be searched for by observing inelastic, ER processes following a low-energy nuclear recoil: excitation and ionisation of the recoiling atom (the hypothetical Migdal effect) and a bremsstrahlung photon [155]. As an example, LUX and XENON1T constrained DM particle masses between (0.3–5) GeV via bremsstrahlung photons and Migdal electrons. Even lower masses are accessible when using the amplified, charge signal only, at the expense of giving up discrimination between ERs and NRs. XENON1T probed particle masses down to 60 MeV, while DarkSide-50 published constraints on WIMP masses as low as 40 MeV. DM masses at the MeV-scale can also be probed by exploiting the scattering off electrons. PandaX-4T presented light DM searches with ionisation signals, with NR and ER energy thresholds of 0.77 keV and 0.07 keV, obtaining leading constraints in the mass range 40 MeV–100 GeV and 100 MeV–10 GeV for a heavy and light mediator, respectively [156]. DarkSide-50 set limits in the mass range (16–56) MeV and above 80 MeV for a heavy and light mediator, respectively [157]. Noble liquids TPCs also search for solar axions, galactic ALPs and dark photons. XENONnT set upper limits on the axion-electron coupling of $2 \times 10^{-12}$ in the mass range (10$^{-6}$–1) eV for solar axions, and probes couplings between (1–2)×10$^{-14}$ in the mass range (1–10) keV for galactic ALPs [158]. DarkSide-50 probes ALP masses down to 0.03 keV with couplings in the range 10$^{-13}$–10$^{-14}$ [157].

The next generation of liquefied noble gas detectors are in construction (DarkSide-20k) or at the design and R&D stage (Argo, DARWIN/XLZD, PandaX-xT). These will increase the sensitivity to various DM candidates by 1-2 orders of magnitude, with the ultimate goal of exploring the experimentally accessible parameter space, until the backgrounds from neutrinos will start dominating the event rates.

**Room temperature scintillators:** Several large DM experiments using high-purity NaI(Tl) crystals are acquiring data in various underground laboratories. Of these, DAMA/LIBRA at LNGS has the highest mass, 250 kg, and the largest exposure: 1.33 t y with an energy threshold of 1 keV and 2.46 t y with an energy threshold of 2 keV. It is the only experiment in the field that reported an annually modulated event rate with a statistical significance of $12.9 \sigma$ C.L. (20 annual cycles), with a modulation amplitude around 0.02 events/(kg d keV) in the energy region (1-4) keV. These findings were interpreted as due to DM interactions via nuclear or electronic recoils. The ANAIS experiment at Canfranc operates 112.5 kg of NaI(Tl) scintillators with an energy threshold of 1 keV and a background rate of 3.6 events/(kg d keV) in the (1-6) keV region. A blind analysis of 3 y of data, for an exposure of 314 kg y is consistent with an absence of modulation at $3.3 \sigma$. The COSINE-100 experiment, located at the Yangyang Underground Laboratory in Korea, operates 106 kg of NaI(Tl) crystals in a liquid scintillator, with an energy threshold of keV and a background rate of 3.6 events/(kg d keV). Results from a 173 kg y exposure in the (1-6) keV energy range are consistent, at 68.3%C.L., with both the null hypothesis and DAMA/LIBRA’s best fit value in the same energy range. A larger exposure and a reduced energy threshold are required to test DAMA/LIBRA for $3 \sigma$ coverage. COSINE-200, under construction at the new, deeper underground laboratory Yemilab, will be a larger experiment with reduced backgrounds. The SABRE experiment plans to operate a total of 50 kg of NaI(Tl) crystals, focussing on reaching a background level of 0.1 events/(kg d keV), an order of magnitude below DAMA/LIBRA. Twin detectors will be installed at
<figure><img src="image_143.png" /><figcaption>**Figure 27.1:** Upper limits on the SI DM-nucleon cross section as a function of DM mass.</figcaption></figure>

LNGS and at the Stawell Underground Physics Laboratory in the Southern hemisphere. While a DM-induced signal is expected to have the same phase in both hemispheres, seasonal or site-related effects would show different amplitudes and phases in the twin detectors. The COSINUS experiment, under construction at LNGS, will use NaI crystals operated as cryogenic scintillating calorimeters to differentiate between nuclear and electronic recoils based on the observation of phonon and light signals. The first phase aims for an exposure of 100 kg d, while the second phase aims to achieve, with 1000 kg d, a fully model-independent cross-check of a NR origin of the DAMA/LIBRA signal.

**Room temperature ionisation detectors:** Silicon charged-coupled devices (CCDs) are employed for low-mass DM searches, as well as for hidden photon searches in the eV-mass range. Ionisation events induced in bulk silicon of high-resistivity, fully depleted CCDs are observed with charge resolutions around 1-2 e⁻ and extremely low leakage currents, at the level of few e⁻ mm⁻²d⁻¹. The position of an energy deposit is reconstructed in 3 dimensions and the particle type (electron, neutron, muon, α-particles, etc) is reconstructed based on the recorded track pattern.

The DAMIC experiment at SNOLAB yielded new constraints on DM-electron scattering, and on the hidden-photon kinetic mixing parameter in the mass range (1-30) eV with an exposure of 7.6 kg d and on low-mass WIMPs with an exposure of 11 kg d. The SENSEI projects employs the skipper technology to achieve single-electron sensitivity. A run with a 2 g detector in the MINOS cavern at Fermilab yielded world-leading constraints on DM-electron scattering for a large range of sub-GeV dark matter masses. The skipper technology is also employed in the next stage of the DAMIC programme, DAMIC-M at LSM, which plans for a kg-size mass. The goal is to achieve thresholds of 2-3 electrons and to probe the DM-nucleon cross-section down to few × 10⁻⁴³ cm² around 2-3 GeV and the DM-electron cross section down to 2 × 10⁻⁴¹ cm² at 10 MeV mass. DAMIC-M has operated first CCDs at LSM and achieved constraints on sub-GeV DM particles interacting with electrons in the mass range (0.53-1000) MeV, for an exposure of 85.23 g d.

The NEWS-G collaboration operates spherical proportional counters filled with a noble gas. Advantages of this technology are the low intrinsic electronic noise and a high amplification gain, allowing for low energy thresholds down to single-electron detec-

tion, and the possibility to use different light targets (He, Ne, etc). A 60 cm diameter chamber operated at LSM with a gas mixture of Ne + CH₄ (0.7%) at 3.1 bar, excluded SI, WIMP-nucleon cross sections above 4.4 × 10⁻³⁷ cm² at 0.5 GeV after an exposure of 9.6 kg d with an energy threshold ~100 eV. The next iteration, a 140-cm sphere detector made of very low radioactivity copper (few µBq/kg of ²³⁸U and ²³²Th) has been installed at SNOLAB in 2021. The detector has recently started to acquire physics data, with a predicted background level of (5 ± 2) events/(kg keV d) for sub-keV events.

**Superheated liquid detectors:** Investigation of the spin-dependent interaction channel calls for target nuclei with uneven total angular momentum. A particularly favourable candidate is ¹⁹F, the spin of which is carried mostly by the unpaired proton, yielding a cross-section which is almost a factor of ten higher than of other employed nuclei with spin (e.g., ²³Na, ⁷³Ge, ¹²⁷I, ¹²⁹Xe, ¹³¹Xe). Fluorine is part of the target of experiments using superheated liquids, such as the ones operated by the PICO collaboration. A search in the PICO-60 C₃F₆ bubble chamber at SNOLAB with an exposure of 1404 kg d and an energy threshold of 2.45 keV, yielded the most stringent constraint on the DM-proton SD cross-section at 3.2 × 10⁻⁴¹ cm² for a 25 GeV particle mass. In construction at SNOLAB is PICO-500, a detector with 250 l of superheated freon (C₃F₈) contained between two quartz jars inside a pressure vessel filled with mineral oil, itself inside a tank of ultra-pure water.

**Directional detectors:** Detectors capable of measuring the direction of the recoiling nucleus would unequivocally confirm the Galactic origin of a signal and could probe the region below the neutrino floor [159, 160]. Because nuclear recoils have a range which is about 10 times smaller than the one of Compton recoils of the same energy, gaseous detectors have an excellent intrinsic background rejection if they can measure the range of events precisely. Several directional detectors are presently in operation, where a 1 m³ detector has a typical mass of a few 100 g and can measure the sense of an incoming nuclear recoil above a few tens of keV. Cygnus is a proto-collaboration which coordinates the R&D efforts for gas based TPCs with 1 keV threshold. One of these is CYGNO, an experiment that aims to build a 1-m drift length gaseous TPC filled with a He:CF₄ 60:40 gas mixture at atmospheric pressure and room temperature at LNGS.

A new technique is based on fine-grained nuclear emulsions
(solid-state detectors with silver halide crystals uniformly dispersed in a gelatine film, where each crystal works as a sensor for charged particles), as proposed by the NEWSdm collaboration. These act as target and nanometric tracking device, and the expected NR tracks are sub-µm in size. Due to the small crystal size and larger number density, a superior spatial resolution compared to gaseous detectors is obtained. Simulations show that to reach the neutrino floor, exposures of 10 ty and 100 ty are required if a 30 nm and 50 nm threshold for detecting the track length is reached. This requires further R&D, since current emulsions allow for 100 nm tracking and target masses are around 1 kg, with 10 kg y exposures planned. A proposed approach for the directional detection of sub-GeV DM is to use two-dimensional materials such as monolayer graphene [161], from which the DM particle can eject electrons. Their energy and direction, correlated with the direction of the incoming DM, can be measured for instance with the proposed PTOLEMY experiment.

**New techniques:** To probe light (sub-GeV) DM particles, either via scatters off electrons or via couplings to phonons, new techniques beyond the ones discussed above are proposed. The DM particle mass that can be accessed in DM-electron scattering in noble liquids and semiconductors is limited by the minimum ionisation/excitation energy and the size of the band gap, respectively (at the ~eV-scale). To reach lower energy thresholds, materials with smaller band gaps for electron excitations (~meV), such as superconductors and superfluids, as well as Dirac materials were proposed [162]. These would in principle allow for the detection of keV-scale DM. Other ideas to detect keV-MeV scale DM are to observe NRs in superfluid He, via collective excitation modes in the fluid [163], as realised in HeRALD and DELight, or based on the breaking of chemical bonds between atoms [164].

Even lighter DM, with masses in the meV-eV range, could be detected via absorption on a conduction electron in a superconductor, followed by the emission of an thermal phonon [165]. Another proposed target for light DM are polar materials (for example GaAs, sapphire), which are especially sensitive for scattering through an ultralight dark photon, via excitation of single optical phonons [166]. If an anisotropic crystal such as sapphire is employed, a daily modulation interaction rate could be established [167]. A new class of detectors for bosonic DM, based on resonant absorption onto a gas of small polyatomic molecules, is proposed in [168]. The DM would effectively act as a laser that resonantly excites transitions in molecules when its mass closely matches the transition energy. While DM with SI couplings can efficiently excite phonons, it has been shown in [169] that if DM couples to the electron spin, magnon excitations (quanta of collective spin wave excitations) in materials with magnetic dipole order may also offer a promising detection avenue. Yet another approach for sub-GeV DM is to employ superconducting nanowires as both target and sensor, and first bounds on DM-electron interactions were placed from a 4.3 ng tungsten-silicide prototype with a 0.8 eV energy threshold and an exposure of 8.8 × 10⁻¹⁴ kg y [170].

The detection of light DM via collective excitations in condensed matter systems and other methods is a rapidly evolving field, and a growing area of research at the interface of DM physics, condensed matter and materials science. We refer to Ref. [171,172] for discussions of some of these new directions and models. Critical challenges are to detect these very small energy depositions, and to reliably assess the background noise.

Table 27.1 summarises the most stringent constraints on the DM-nucleon SI and SD cross-sections, and Figure 27.1 shows the best constraints for SI couplings in the cross-section versus DM mass parameter space, above masses of 0.3 GeV.

## 27.7 Astrophysical detection of dark matter

DM as a microscopic constituent can have measurable, macroscopic effects on astrophysical systems. Indirect DM detection refers to the search for the annihilation or decay debris from DM particles, resulting in detectable species, including especially gamma rays, neutrinos, and antimatter particles. The production rate of such particles depends on (i) the annihilation (or decay) rate (ii) the density of pairs (respectively, of individual particles) in the region of interest, and (iii) the number of final-state particles produced in one annihilation (decay) event. In formulae, the

rate for production of a final state particle f per unit volume from DM annihilation can be cast as

$$ \Gamma_f^A = c \frac{\rho_{\text{DM}}^2}{m_{\text{DM}}^2} \langle \sigma v \rangle N_f^A, \qquad (27.18) $$

where $\langle \sigma v \rangle$ indicates the thermally-averaged cross section for DM annihilation times relative velocity [27], calculated at the appropriate temperature, $\rho_{\text{DM}}$ is the physical density of DM, and $N_f^A$ is the number of final state particles f produced in one individual annihilation event. The constant c depends on whether the DM is its on antiparticle, in which case c = 1/2, or if there is a mixture of DM particles and antiparticles (in case there is no asymmetry, c = 1/4). The analog for decay is

$$ \Gamma_f^D = \frac{\rho_{\text{DM}}}{m_{\text{DM}} \tau_{\text{DM}}} \frac{1}{N_f^D}, \qquad (27.19) $$

with the same conventions for the symbols, and where $\tau_{\text{DM}}$ is the DM's lifetime.

**Gamma Rays:** DM annihilation to virtually any final state produces gamma rays: emission processes include the dominant two-photon decay mode of neutral pions resulting from the hadronization of strongly-interacting final states; final state radiation; and internal bremsstrahlung, the latter two including, possibly, the emission of massive gauge or Higgs bosons subsequently producing photons via their decay products. Similarly, neutrinos are produced from charged pion decay and from radiative processes. The flux of gamma rays and neutrinos is calculated integrating the rate $\Gamma_f$ per steradian (simply meaning, for isotropic emission, $\Gamma_f/(4\pi)$) along the line of sight within the appropriate angular region (the differential flux is obtained in the same way by simply replacing $N_f$ with the differential flux at production, at the appropriate redshift in the case of cosmologically distant sources),

$$ \phi_f = \int_{\Delta\Omega} d\Omega \int_{\text{l.o.s.}} dl \frac{\Gamma_f}{4\pi}. \qquad (27.20) $$

It is customary to factor out, in the expression for the rate, a particle physics factor, depending upon the DM particle mass and its annihilation or decay rate, and an astrophysical factor, which only depends on the observational target. The latter is sometimes denoted with $J_{\Delta\Omega}(\psi)$ with $\psi$ indicating the direction of the line of sight. Although different conventions are in use, a common choice is to define

$$ J_{\Delta\Omega}(\psi) = \int_{\Delta\Omega} \int_{\text{l.o.s.}(\psi)} \rho_{\text{DM}}^2(l, \Omega) \, dld\Omega. \qquad (27.21) $$

For a target with uniform density $\rho$ and radius $r$ at a distance $d \gg r$, such that the target is entirely within the solid angle $\Delta\Omega$,

$$ J \approx \frac{4\pi r^3 \rho_{\text{DM}}^2}{3d^2}. \qquad (27.22) $$

Searches for gamma-ray emission from DM annihilation have focused on targets chosen based on a variety of considerations, primarily intended to maximize signal to noise. Nearby dwarf spheroidal galaxies contain very small amounts of gas, and do not host any significant astrophysical background at gamma-ray or X-ray frequencies, and are thus an optimal target choice for DM searches. An accurate determination of the DM density profile in these objects results in somewhat large systematics when deriving constraints from the non-observation of emission from DM; future optical surveys will help pinpoint with greater accuracy stellar kinematics and thus reduce such uncertainty; a second target is the inner region of the Milky Way: while nearby and potentially hosting a large density of DM, the Galactic center region is however very bright at almost any wavelength, making the extraction of a signal highly problematic; nearby clusters of galaxies are also known to host significant astrophysical emission, but are potentially ideally suited to constrain DM decay. Finally, putative nearby DM clumps are also a possible source of a bright DM signal (albeit from an unknown direction), as is the annihilation of DM in all halos at all redshifts.
<figure><img src="image_144.png" /><figcaption>Figure 27.2: Upper limits and projected sensitivity from CTA on the pair-annihilation rate versus the DM mass from gamma-ray and CMB observations (figure courtesy of Logan Morrison).</figcaption></figure>

DM annihilation and decay can lead to striking spectral features. Since the process happens typically at very low particle velocities, if the DM pair-annihilates e.g. to two photons or two neutrinos, the final-state particles will be nearly monoenergetic, with an energy close to the DM particle mass and a width proportional to the DM velocity in units of c (if a γγ line is present electroweak symmetry also implies a Zγ line, if kinematically allowed). No astrophysical processes are known to produce lines at gamma-ray or neutrino energies in the GeV and above (with, perhaps, the possible exception of cold pulsar winds [173]) making this channel virtually background-free. At lower energy, lines are expected from radiative decay modes of candidates such as sterile neutrinos (see Sec. 27.5). DM annihilation or decay can lead to additional spectral features besides lines. These include (one or more) “boxes” [174], produced by boosted final states decaying to monochromatic photons (such as e.g. neutral pions), or combinations thereof. Processes occurring at higher redshift can distort these spectral features by smearing them to lower energies. Neglecting gamma-ray attenuation, the flux of gamma rays from all redshift can be cast as

$$ \frac{dN_{\gamma}}{dE_{\gamma}}(E_{\gamma}) = \frac{c}{8\pi} \int \frac{\langle \sigma v \rangle \rho_{\text{DM}}(z) dz}{H(z)(1+z)^3 m_{\text{DM}}^2} \left( \frac{dN_{\gamma}}{dE'} \right)_{E'=E_{\gamma}(1+z)}. \qquad (27.23) $$

While the calculation of the differential spectrum of gamma rays from a given final state f, $dN_{\gamma}^f/dE_{\gamma}$, is carried out using numerical tools, such as PYTHIA [175] that reproduce hadronization and particle decay for masses well above a few GeV, in the sub-GeV range gamma-ray production follows primarily from meson decay and radiative processes well outside the range of applicability of the Altarelli-Parisì splitting function. The MeV gamma-ray range will soon be probed with forthcoming satellites [176]. Recently a code that provides the expected gamma-ray spectrum for sub-GeV DM, Hazma, has become available [177].

Observations with the Fermi Large Area Telescope (LAT) and with ground-based facilities such as HESS, VERITAS, MAGIC, and HAWC have provided an unprecedented picture of the gamma-ray sky ideally suited to look for a signal from DM annihilation or decay for DM particles from a few GeV mass up to several

TeV. The LAT has provided some of the most stringent constraints to-date on DM pair-annihilation for a variety of annihilation final states, chiefly from stacked observations of nearby satellite dwarf spheroidal galaxies [178]. Excesses of gamma rays over the expected diffuse and point-source background have been claimed, most importantly from the direction of the inner Galaxy, where a signal from DM annihilation might be especially bright [179,180]. The nature of this excess is quite controversial: while the morphology and spectrum fall within what expected for a standard WIMP with a mass of a few tens of GeV [181], unresolved point sources, including especially an (expected) population of millisecond pulsars (MSPs) have been advocated as a possible plausible counterpart [182]. Statistical methods to discriminate between DM and MSPs have been utilized [183,184], but recent studies indicate that such results might not be conclusive [185]. Large uncertainties in the Galactic diffuse background emission model are additionally known to exist, and possible plague the morphological and spectral information [186, 187]. Other notable potential gamma-ray excesses include a diffuse emission from the Andromeda galaxy (M31) [188–190], possibly in excess of what expected from cosmic-ray models [191]; and a diffuse emission at 511 keV energy in the inner Galaxy from Integral-SPI observations [192]; such emission has known astrophysical counterparts [193], as well as several proposed DM explanations (e.g. [194]).

While DM annihilation and decay typically occurs at low velocities, the possibility of “boosted” DM has also been considered [195]. In this case, the DM particle might dominantly pair-annihilate to a lighter dark species, which does interact with Standard Model particle, and could be detected with neutrino telescopes or direct detection experiments. Future facilities that promise to widen the reach of gamma-ray searches for DM include especially the Cherenkov Telescope Array (CTA), see fig. 27.2.

**Neutrinos:** DM can be captured in celestial bodies in significant amounts, depending on the DM scattering cross section off of nucleons, the DM mass, and the DM flux incident on the celestial body of interest. For DM masses at or around the GeV scale, evaporation from the celestial body plays an important role [196]. If enough DM accumulates, DM annihilation inside the celestial body can then lead to the production of Standard Model parti-
cles. Such particles can heat up the body, if they lose most of their energy before escaping. Utilizing models for heat production in planets, or stellar interior models in the case of stars, constraints can be put on DM particle properties. Of note are constraints from anomalous warming of cold planets such as Uranus [197], alterations to the stellar structure or the Sun's seismic activity [198], and anomalous Earth heat flow [199]. Alternately, DM annihilation in celestial bodies can result in the production of particles that can escape the body. Within the Standard Model, the only such instance is annihilation to neutrinos, but, similarly to the boosted DM case the DM can annihilate to a (stable or unstable) dark-sector particles, whose decay or interactions can be detected on Earth [200,201]. For direct annihilation to neutrinos, given the lower limit on the DM mass from evaporation, the typical neutrino energies usually exceed the energy of neutrinos from the Sun, the best target for this type of searches, making this a virtually background-free DM search. Significant neutrino fluxes can only be achieved, however, if near-equilibration is reached between the capture and annihilation rates. In turn, this requires large-enough DM-nucleon scattering cross sections, to-date very close to the limits from direct detection (see sec. 27.6). Only spin-dependent cross section, and capture in the Sun usually provide large-enough neutrino fluxes. IceCube and ANTARES searches for an anomalous flux of high-energy neutrinos from the Sun yielded null results, which can be interpreted as constraints on spin-dependent nucleon-DM interactions under the assumption of equilibration and of specific annihilation final states [202,203]. Lower-threshold detectors, including DeepCore [204] and PINGU [205], can produce interesting limits on lower-mass DM candidates above the evaporation threshold.

**Cosmic-ray Antimatter:** Stable charged particles produced by decays of products of, or directly from DM annihilation or decay, populate the cosmic radiation and are a prime target for indirect DM searches. To maximize signal to noise, searches focus on relatively rare particle species, such as positrons, antiprotons, and antinuclei. While in certain models the production of particles and antiparticles is not symmetric [206], generally DM annihilation or decay produces as many particles as antiparticles in the final state. Charged particles produced by DM propagate and lose energy prior to reaching detectors. The transport of charged particles is customarily carried out in the context of diffusion models such as the Galactic “leaky box” setup, see e.g. [207]. While progress in constraining the uncertain propagation and energy-loss processes has been steady with improved data from new detectors such as PAMELA [208] and AMS-02 [209], the calculation of the particle flux at Earth from that at production suffers from significant uncertainties [207].

An excess of high-energy positrons over the standard secondary production from inelastic cosmic-ray interactions has been firmly established by several experiments, most recently and with the highest statistics by AMS-02 [210]. The excess has been ascribed to DM annihilation, although strong constraints from the non-observation of corresponding anomalies in other channels, such as antiprotons and gamma rays, and the peculiar spectral shape, force DM models to be quite convoluted (see e.g. [211]). Alternatively, excess *primary* positrons can be produced in the magnetosphere of nearby pulsars [212, 213]. This latter explanation was questioned in [214] in connection with the detection of a TeV halo around two candidate pulsars, leading to the determination of a highly suppressed diffusion coefficient within the pulsar nebula; with such low diffusion coefficient, positrons from the pulsars would not contribute significantly to the flux at Earth; Ref. [215], however, showed that likely the diffusion coefficient is not constant, and that if it increases outside the nebula, as expected from other cosmic-ray measurements, pulsars can still be considered as the counterpart to the positron excess. Other explanations for the excess, albeit somewhat controversial, and increasingly constrained by data, also exist [216, 217].

The antiproton spectrum in the cosmic radiation as measured by AMS-02 [209] also exhibits features that might be considered as excess flux between 10 and 20 GeV, and energies above 100 GeV [218], which have been interpreted as possible signal of DM annihilation; more realistically, however, systematic uncertainties

in the antiproton production cross section, in cosmic-ray transport, reacceleration at high energy and, at low energy, solar modulation make it extremely difficult to assess the robustness of the excesses [219, 220].

Antinuclei such as anti-deuterium and anti-helium could also form as a result of DM annihilation or decay. While baryon number conservation forces the typical kinetic energy of antinuclei produced in inelastic cosmic-ray processes to large values, antinuclei arising from hadronization of DM-initiated jets have low energy, offering optimal signal-to-noise when a low-energy cut on the antinucleus kinetic energy is used [221–225]. Specific detector designs have been developed to single out low-energy cosmic-ray antinuclei (e.g. $E < 0.25$ GeV/nucleon in the case of the General Antiparticle Spectrometer, or GAPS [226]). The detection of even a single antinucleus would have considerable importance as a possible sign of new physics [227].

**Multi-wavelength studies:** Electrons and positrons from DM lose energy quite efficiently, radiating at a variety of wavelengths. This *secondary* radiative emission presents spectral and morphological features that might provide an important additional indirect detection channel. The most efficient energy loss mechanisms for high-energy electrons and positrons are inverse Compton (IC) up-scattering of background photons, and synchrotron radiation in the presence of magnetic fields. Depending on the DM mass, the emitted light from synchrotron peaks at MHz-GHz frequencies, while the IC emission at X-ray to gamma-ray frequencies, depending on the energy of the background radiation, typically ranging from CMB photons up to starlight photons [228–230]. The calculation of the secondary emission from DM entails both solving the transport of the electrons and positrons from DM, and for the radiative emission; codes exist that perform these calculations for certain astrophysical environments [231–233], with diffusion playing an increasingly critical role in smaller and smaller structures such as dwarf galaxies [229]. It has been demonstrated that for large magnetic fields and for certain final states, the synchrotron emission is more constraining than the gamma-ray emission [234].

**Stellar Physics:** Microscopic properties of the DM can meaningfully alter, and thus be constrained by, several astrophysical environments, from planets and stars, up to the universe as a whole. DM particles light enough to be produced in collisional processes inside stars, with typically temperatures in the keV, or supernovae, with energy scales in the MeV, lead to an additional energy-loss mechanism, if capable of escaping the system. If this is the case, the increased needed energy output would also result in an increased neutrino flux, leading to constraints on the masses and couplings of the DM, see e.g. [235] for a comprehensive review. DM annihilation can even have fueled early stages of stellar evolution, perhaps with measurable consequences [236]. DM capture in neutron stars could lead to the collapse of the star into black holes; the existence of neutron stars in DM-rich environments can thus be used to constrain the mass and interaction cross section of DM with nucleons [237] and even relatively light, but stable, primordial black holes [238].

**Cosmology:** Energy injection from DM processes in the early universe, as well as the contribution of DM to the effective relativistic degrees of freedom, are severely constrained from data on Big Bang Nucleosynthesis (see Sec. 24 in this *Review*) and on the energy and anisotropy power spectrum of the CMB (see Sec. 29). Codes exist that perform the calculation of the constraints on particle DM models [177, 239].

**PBH Detection:** Macroscopic DM candidates can gravitationally perturb structure and compromise the stability of, for instance, globular clusters such as Eridanus II [24], and/or disrupt wide binaries [240]. This constrains the maximal mass of a macroscopic DM candidate to not much more than $5 M_{\odot}$ [240] (see also [24]). In the specific case or primordial black holes, strong constraints also stem from the acceleration of charged particles around and after recombination, with significant effects on the CMB [241], albeit a lively debate exists over the accretion efficiency around such objects at high redshift [242, 243]. Whether or not PBH in the solar-mass range can be 100% of the DM is therefore disputed at present. Other effects produced by macroscopic,
massive DM candidates are the microlensing of stars [244, 245] and quasars [246], femtolensing of gamma-ray bursts [247], and neutron star capture [238]. It is important to note that recently constraints from both microlensing and femtolensing have been corrected after the realization of important finite-size source and wave effects [245, 248, 249], leaving a substantial window, at PBH masses $10^{17} \lesssim M_{BH}/g \lesssim 10^{21}$ where PBH can be 100% of the DM. Light black holes, $M_{BH} \lesssim 10^{17}$ g, are also constrained by the non-detection of products of evaporation [250]; future MeV gamma-ray detectors are slated to significantly improve such constraints and to enable the direct detection of Hawking radiation [251]. If evaporation shuts down at or near the Planck scale, Planck-scale relic PBH are generally expected to carry electric charge and to be detectable with direct detection and neutrino experiments, or with paleo-detectors [252].

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## 28. Dark Energy

Revised August 2023 by D.H. Weinberg (Ohio State U.) and M. White (UC Berkeley; LBNL).

### 28.1 Repulsive Gravity and Cosmic Acceleration

In the first modern cosmological model, Einstein [1] modified his field equation of General Relativity (GR), introducing a “cosmological term” that enabled a solution with time-independent, spatially homogeneous matter density $ρ_m$ and constant positive space curvature. Although Einstein did not frame it this way, one can view the “cosmological constant” $Λ$ as representing a constant energy density of the vacuum [2], whose repulsive gravitational effect balances the attractive gravity of matter and thereby allows a static solution. After the development of dynamic cosmological models [3, 4] and the discovery of cosmic expansion [5], the cosmological term appeared unnecessary, and Einstein and de Sitter [6] advocated adopting an expanding, homogeneous and isotropic, spatially flat, matter-dominated Universe as the default cosmology until observations dictated otherwise. Such a model has matter density equal to the critical density, $Ω_m ≡ ρ_m/ρ_c = 1$, and negligible contribution from other energy components [7].

By the mid-1990s, the Einstein-de Sitter model was showing numerous cracks, under the combined onslaught of data from the cosmic microwave background (CMB), large-scale galaxy clustering, and estimates of the matter density, the expansion rate ($H_0$), and the age of the Universe. As noted in a number of papers from this time, introducing a cosmological constant offered a potential resolution of many of these tensions, yielding the most empirically successful version of the inflationary cold dark matter scenario. In the late 1990s, supernova surveys by two independent teams provided direct evidence for accelerating cosmic expansion [8, 9], establishing the cosmological constant model (with $Ω_m \simeq 0.3$, $Ω_Λ \simeq 0.7$) as the preferred alternative to the $Ω_m = 1$ scenario. Shortly thereafter, CMB evidence for a spatially flat Universe [10, 11], and thus for $Ω_{tot} \simeq 1$, cemented the case for cosmic acceleration by firmly eliminating the free-expansion alternative with $Ω_m \ll 1$ and $Ω_Λ = 0$. Today, the accelerating Universe is well established by multiple lines of independent evidence from a tight web of precise cosmological measurements.

As discussed in the Big Bang Cosmology article of this Review (Sec. 22), the scale factor $R(t)$ of a homogeneous and isotropic Universe governed by GR grows at an accelerating rate if the pressure $p < -\frac{1}{3}ρ$ (in $c = 1$ units). A cosmological constant has $ρ_Λ = \text{constant}$ and pressure $p_Λ = -ρ_Λ$ (see Eq. 22.10), so it will drive acceleration if it dominates the total energy density. However, acceleration could arise from a more general form of “dark energy” that has negative pressure, typically specified in terms of the equation-of-state-parameter $w = p/ρ$ ($=-1$ for a cosmological constant). Furthermore, the conclusion that acceleration requires a new energy component beyond matter and radiation relies on the assumption that GR is the correct description of gravity on cosmological scales. The title of this article follows the common but inexact usage of “dark energy” as a catch-all term for the origin of cosmic acceleration, regardless of whether it arises from a new form of energy or a modification of GR. Our account here draws on the much longer review of cosmic acceleration in Ref. [12], which provides background explanation and extensive literature references for the discussion in Secs. 28.2 and 28.3.

Below we will use the abbreviation $\Lambda$CDM to refer to a model with cold dark matter, a cosmological constant, inflationary initial conditions, standard radiation and neutrino content, and a flat Universe with $Ω_{tot} = 1$ (though we will sometimes describe this model as “flat $\Lambda$CDM” to emphasize this last restriction). We will use $w$CDM to denote a model with the same assumptions but a free, constant value of $w$. Models with the prefix “o” (e.g., owwCDM) allow non-zero space curvature.

### 28.2 Theories of Cosmic Acceleration

#### 28.2.1 Dark Energy or Modified Gravity?

A cosmological constant is the mathematically simplest, and perhaps the physically simplest, theoretical explanation for the accelerating Universe. The problem is explaining its unnaturally small magnitude, as discussed in Sec. 22.4.7 of this Review. An alternative (which still requires finding a way to make the cos-

mological constant zero or at least negligibly small) is that the accelerating cosmic expansion is driven by a new form of energy such as a scalar field [13] with potential $V(φ)$. The energy density and pressure of the field $φ(x)$ take the same forms as for inflationary scalar fields, given in Eq. (22.52) of the Big Bang Cosmology article. In the limit that $\frac{1}{2}φ^2 \ll |V(φ)|$, the scalar field acts like a cosmological constant, with $p_φ \simeq -ρ_φ$. In this scenario, today's cosmic acceleration is closely akin to the epoch of inflation, but with radically different energy and timescale.

More generally, the value of $w = p_φ/ρ_φ$ in scalar field models evolves with time in a way that depends on $V(φ)$ and on the initial conditions ($φ_i, φ_i$); some forms of $V(φ)$ have attractor solutions in which the late-time behavior is insensitive to initial values. Many forms of time evolution are possible, including ones where $w$ is approximately constant and broad classes where $w$ “freezes” towards or “thaws” away from $w = -1$, with the transition occurring when the field comes to dominate the total energy budget. If $ρ_φ$ is even approximately constant, then it becomes dynamically insignificant at high redshift, because the matter density scales as $ρ_m \propto (1+z)^3$. “Early dark energy” models are ones in which $ρ_φ$ is a small but not negligible fraction (e.g., a few percent) of the total energy throughout the matter- and radiation-dominated eras, tracking the dominant component before itself coming to dominate at low redshift.

Instead of introducing a new energy component, one can attempt to modify gravity in a way that leads to accelerated expansion [14]. One option is to replace the Ricci scalar $R$ with a function $R + f(R)$ in the gravitational action [15]. Other changes can be more radical, such as introducing extra dimensions and allowing gravitons to “leak” off the brane that represents the observable Universe (the “DGP” model [16]). The DGP example has inspired a more general class of “galileon” and massive gravity models. Constructing viable modified gravity models is challenging, in part because it is easy to introduce theoretical inconsistencies (such as “ghost” fields with negative kinetic energy), but above all because GR is a theory with many high-precision empirical successes on solar system scales [17]. Modified gravity models typically invoke screening mechanisms that force model predictions to approach those of GR in regions of high density or strong gravitational potential. Screening offers potentially distinctive signatures, as the strength of gravity (i.e., the effective value of $G_N$) can vary by order unity in environments with different gravitational potentials.

More generally, one can search for signatures of modified gravity by comparing the history of cosmic structure growth to the history of cosmic expansion. Within GR, these two are linked by a consistency relation, as described below (Eq. (28.2)). Modifying gravity can change the predicted rate of structure growth, and it can make the growth rate dependent on scale or environment. In some circumstances, modifying gravity alters the combinations of potentials responsible for gravitational lensing and the dynamics of non-relativistic tracers (such as galaxies or stars) in different ways (see Sec. 22.4.7 in this Review), leading to order unity mismatches between the masses of objects inferred from lensing and those inferred from dynamics in unscreened environments.

At present there are no fully realized and empirically viable modified gravity theories that explain the observed level of cosmic acceleration. The constraints on $f(R)$ models now force them so close to GR that they cannot produce acceleration without introducing a separate dark energy component [18]. The DGP model is empirically ruled out by several tests, including the expansion history, the integrated Sachs-Wolfe effect, and redshift-space distortion measurements of the structure growth rate [19]. The near-simultaneous arrival of gravitational waves and electromagnetic signals from the neutron star merger event GW170817, which shows that gravitational waves travel at almost exactly the speed of light, is a further strong constraint on modified gravity theories [20]. The elimination of models should be considered an important success of the program to empirically test theories of cosmic acceleration. However, it is worth recalling that there was no fully realized gravitational explanation for the precession of Mercury's orbit prior to the completion of GR in 1915, and the fact that no complete and viable modified gravity theory exists
today does not mean that one will not arise in the future. In the meantime, we can continue empirical investigations that can tighten restrictions on such theories or perhaps point towards the gravitational sector as the origin of accelerating expansion.

### 28.2.2 Expansion History and Growth of Structure

The main line of empirical attack on dark energy is to measure the history of cosmic expansion and the history of matter clustering with the greatest achievable precision over a wide range of redshift. Within GR, the expansion rate $H(z)$ is governed by the Friedmann equation (see the articles on Big Bang Cosmology and Cosmological Parameters—Secs. 22 and 25 in this Review). For dark energy with an equation of state $w(z)$, the cosmological constant contribution to the expansion, $\Omega_{\Lambda}$, is replaced by a redshift-dependent contribution. The evolution of the dark energy density follows from Eq. (22.10),

$$ \Omega_{\text{de}} \frac{\rho_{\text{de}}(z)}{\rho_{\text{de}}(z=0)} = \Omega_{\text{de}} \exp \left[ 3 \int_{0}^{z} [1+w(z')] \frac{dz'}{1+z'} \right] \quad (28.1) \\ = \Omega_{\text{de}}(1+z)^{3(1+w)}, $$

where the second equality holds for constant $w$. If $\Omega_{\text{m}}$, $\Omega_{\text{r}}$, and the present value of $\Omega_{\text{tot}}$ are known, then measuring $H(z)$ pins down $w(z)$. (Note that $\Omega_{\text{de}}$ is the same quantity denoted $\Omega_{\text{v}}$ in Sec. 22, but we have adopted the ‘de’ subscript to avoid implying that dark energy is necessarily a vacuum effect.)

While some observations can probe $H(z)$ directly, others measure the distance-redshift relation. The basic relations between angular diameter distance or luminosity distance and $H(z)$ are given in Ch. 22 —and these are generally unaltered in time-dependent dark energy or modified gravity models. For convenience, in later sections, we will sometimes refer to the comoving angular diameter distance, $D_{\text{A,c}}(z) = (1+z)D_{\text{A}}(z)$.

In GR-based linear perturbation theory, the density contrast $\delta(\mathbf{x}, t) \equiv \rho(\mathbf{x}, t)/\bar{\rho}(t)-1$ of pressureless matter grows in proportion to the linear growth function $G(t)$ (not to be confused with the gravitational constant $G_{\text{N}}$), which follows the differential equation

$$ \ddot{G} + 2H(z)\dot{G} - \frac{3}{2}\Omega_{\text{m}}H_0^2(1+z)^3G = 0. \quad (28.2) $$

To a good approximation, the logarithmic derivative of $G(z)$ is

$$ f(z) \equiv -\frac{d \ln G}{d \ln(1+z)} \approx \left[ \Omega_{\text{m}}(1+z)^3 \frac{H_0^2}{H^2(z)} \right]^{\gamma}, \quad (28.3) $$

where $\gamma \simeq 0.55$ for relevant values of cosmological parameters [21]. In an $\Omega_{\text{m}} = 1$ Universe, $G(z) \propto (1+z)^{-1}$, but growth slows when $\Omega_{\text{m}}$ drops significantly below unity. One can integrate Eq. (28.3) to get an approximate integral relation between $G(z)$ and $H(z)$, but the full (numerical) solution to Eq. (28.2) should be used for precision calculations. Even in the non-linear regime, the amplitude of clustering is determined mainly by $G(z)$, so observations of non-linear structure can be used to infer the linear $G(z)$, provided one has good theoretical modeling to relate the two.

In modified gravity models the growth rate of gravitational clustering may differ from the GR prediction. A general strategy to test modified gravity, therefore, is to measure both the expansion history and the growth history to see whether they yield consistent results for $H(z)$ or $w(z)$.

### 28.2.3 Parameters

Constraining a general history of $w(z)$ is nearly impossible, because the dark energy density, which affects $H(z)$, is given by an integral over $w(z)$, and distances and the growth factor involve a further integration over functions of $H(z)$. Oscillations in $w(z)$ over a range $\Delta z/(1+z) \ll 1$ are therefore extremely difficult to constrain. It has become conventional to phrase constraints or projected constraints on $w(z)$ in terms of a linear evolution model,

$$ w(a) = w_0 + w_a(1-a) = w_p + w_a(a_p - a), \quad (28.4) $$

where $a \equiv (1+z)^{-1}$, $w_0$ is the value of $w$ at $z=0$, and $w_p$ is the value of $w$ at a “pivot” redshift $z_p \equiv a_p^{-1} - 1$, where it

is best constrained by a given set of experiments. For typical data combinations, $z_p \simeq 0.5$. This simple parameterization can provide a good approximation to the predictions of many physically motivated models for observables measured with percent-level precision. A widely used “Figure of Merit” (FoM) for dark energy experiments [22] is the projected combination of errors $[\sigma(w_p)\sigma(w_a)]^{-1}$. Ambitious future experiments with 0.1–0.3% precision on observables can constrain richer descriptions of $w(z)$, which can be characterized by principal components.

There has been less convergence on a standard parameterization for describing modified gravity theories. Deviations from the GR-predicted growth rate can be described by a deviation $\Delta\gamma$ in the index of Eq. (28.3), together with an overall multiplicative offset relative to the $G(z)$ expected from extrapolating the CMB-measured fluctuation amplitude to low redshift. However, these two parameters may not accurately capture the growth predictions of all physically interesting models. Another important parameter to constrain is the ratio of the gravitational potentials governing space curvature and the acceleration of non-relativistic test particles. The possible phenomenology of modified gravity models is rich [23], which enables many consistency tests but complicates the task of constructing parameterized descriptions.

The more general set of cosmological parameters is discussed elsewhere in this Review (Sec. 25), but here we highlight a few that are particularly important to the dark energy discussion.

* The dimensionless Hubble parameter $h \equiv H_0/100 \text{ km s}^{-1} \text{ Mpc}^{-1}$ determines the present day value of the critical density and the overall scaling of distances inferred from redshifts.

* $\Omega_{\text{m}}$ and $\Omega_{\text{tot}}$ affect the expansion history and the distance-redshift relation.

* The sound horizon $r_s = \int_0^{t_{\text{rec}}} c_s(t) dt/a(t)$, the comoving distance that pressure waves can propagate between $t=0$ and recombination, determines the physical scale of the acoustic peaks in the CMB and the baryon acoustic oscillation (BAO) feature in low-redshift matter clustering [24].

* The amplitude of matter fluctuations, conventionally represented by the quantity $\sigma_8(z)$, scales the overall amplitude of growth measures such as weak lensing or redshift-space distortions (discussed in the next section).

Specifically, $\sigma_8(z)$ refers to the rms fluctuation of the matter overdensity $\rho/\bar{\rho}$ in spheres of radius $8h^{-1}\text{Mpc}$, computed from the linear theory matter power spectrum at redshift $z$, and $\sigma_8$ on its own refers to the value at $z=0$ (just like our convention for $\Omega_{\text{m}}$).

While discussions of dark energy are frequently phrased in terms of values and errors on quantities like $w_p$, $w_a$, $\Delta\gamma$, and $\Omega_{\text{tot}}$, parameter precision is the means to an end, not an end in itself. The underlying goal of empirical studies of cosmic acceleration is to address two physically profound questions:

* 1. Does acceleration arise from a breakdown of GR on cosmological scales or from a new energy component that exerts repulsive gravity within GR?

* 2. If acceleration is caused by a new energy component, is its energy density constant in space and time, as expected for a fundamental vacuum energy, or does it show variations that indicate a dynamical field?

Substantial progress towards answering these questions, in particular any definitive rejection of the cosmological constant “null hypothesis,” would be a major breakthrough in cosmology and fundamental physics.

## 28.3 Observational Probes

We briefly summarize the observational probes that play the greatest role in current constraints on dark energy. Further discussion can be found in other articles of this Review, in particular Secs. 25 (Cosmological Parameters) and 29 (The Cosmic Microwave Background), and in Ref. [12], which provides extensive references to background literature. Recent observational results from these methods are discussed in Sec. 28.4.
### 28.3.1 Methods, Sensitivity, Systematics

**Cosmic Microwave Background Anisotropies:** Although CMB anisotropies provide limited information about dark energy on their own, CMB constraints on the geometry, matter content, and radiation content of the Universe play a critical role in dark energy studies when combined with low-redshift probes. In particular, CMB data supply measurements of $\theta_s = r_s/D_{A,c}(z_{\text{rec}})$, the angular size of the sound horizon at recombination, from the angular location of the acoustic peaks, measurements of $\Omega_m h^2$ and $\Omega_b h^2$ from the heights of the peaks, and normalization of the amplitude of matter fluctuations at $z_{\text{rec}}$ from the amplitude of the CMB fluctuations themselves. The comparison of the acoustic scale to the diffusion damping scale helps to constrain the expansion rate near recombination. Planck data yield a 0.18% determination of $r_s$, which scales as $(\Omega_m h^2)^{-0.25}$ for cosmologies with standard matter and radiation content. The uncertainty in the matter fluctuation amplitude at the epoch of recombination is 0.5%. Secondary anisotropies, including the integrated Sachs-Wolfe effect, the Sunyaev-Zeldovich (SZ, [25]) effect, and weak lensing of primary anisotropies, provide additional information about dark energy by constraining low-redshift structure growth.

**Type Ia Supernovae (SN):** Type Ia supernovae, produced by the thermonuclear explosions of white dwarfs, exhibit 10–15% scatter in peak luminosity after correction for light curve duration (the time to rise and fall) and color (which is a diagnostic of dust extinction). Since the peak luminosity is not known a priori, supernova surveys constrain ratios of luminosity distances $D_L(z)$ at different redshifts. If one is comparing a high-redshift sample to a local calibrator sample measured with much higher precision (and distances inferred from Hubble's law), then one essentially measures the luminosity distance in $h^{-1}$Mpc, constraining the combination $hD_L(z)$. With distance uncertainties of 5–8% per well observed supernova, a sample of around 100 SNe is sufficient to achieve sub-percent statistical precision. The percent-level systematic uncertainties in current samples are dominated by uncertainties associated with photometric calibration and dust extinction corrections and modeling the correlations among supernova luminosity, color, light-curve shape, and host galaxy properties. Another potential systematic is redshift evolution of the supernova population itself, which can be tested by analyzing subsamples grouped by spectral properties or host galaxy properties to confirm that they yield consistent results. Recent analyses of large supernova compilations calibrate these systematic uncertainties internally in parallel with cosmological constraints, with an overall error budget that has similar contributions from systematic and statistical uncertainties (e.g., [26]).

**Baryon Acoustic Oscillations (BAO):** Pressure waves that propagate in the pre-recombination photon-baryon fluid imprint a characteristic scale in the clustering of matter and galaxies, which appears in the galaxy correlation function as a localized peak at the sound horizon scale $r_s$, or in the power spectrum as a series of oscillations. Since observed galaxy coordinates consist of angles and redshifts, measuring this “standard ruler” scale in a galaxy redshift survey determines the angular diameter distance $D_A(z)$ and the expansion rate $H(z)$, which convert coordinate separations to comoving distances. Errors on the two quantities are correlated, and in existing galaxy surveys the best determined combination is approximately $D_V(z) = [czD_{A,c}^2(z)/H(z)]^{1/3}$. As a useful rule of thumb, a survey that fully samples structures at redshift $z$ over a comoving volume $V$, and is therefore limited by cosmic variance rather than shot noise, measures $D_{A,c}(z)$ with a fractional error of $0.005(V/10\,\text{Gpc}^3)^{-1/2}$ and $H(z)$ with a fractional error 1.6–1.8 times higher. The most precise BAO measurements to date come from large galaxy redshift surveys probing $z < 0.8$, and these will be extended to higher redshifts by ongoing and future projects. At redshifts $z > 2$, BAO can also be measured in the Lyman-α forest of intergalactic hydrogen absorption towards background quasars, where the fluctuating absorption pattern provides tens or hundreds of samples of the density field along each quasar sightline. For Lyman-α forest BAO, the best measured parameter combination is more heavily weighted towards $H(z)$ because of strong redshift-space distortions that enhance clustering in the line-of-sight direction. Radio intensity mapping, which maps large-scale

structure in redshifted 21-cm hydrogen emission without resolving individual galaxies, offers a potential route to measuring BAO over large volumes at relatively low cost, but the technique is still under development. Photometric redshifts in optical imaging surveys can be used to measure BAO in the angular direction, though the typical distance precision is a factor of 3–4 lower compared to a well sampled spectroscopic survey of the same area, and angular BAO measurements do not directly constrain $H(z)$. BAO distance measurements complement SN distance measurements by providing absolute rather than relative distances (with precise calibration of $r_s$ from the CMB) and by having greater achievable precision at high redshift thanks to the increasing comoving volume available. Theoretical modeling suggests that BAO measurements from even the largest feasible redshift surveys will be limited by statistical rather than systematic uncertainties.

**Weak Gravitational Lensing:** Gravitational light bending by a clustered distribution of matter shears the shapes of higher redshift background galaxies in a spatially coherent manner, producing a correlated pattern of apparent ellipticities. By studying the weak lensing signal for source galaxies binned by photometric redshift (estimated from broad-band colors), one can probe the history of structure growth. “Cosmic shear” weak lensing uses the correlation of source ellipticities to deduce the clustering of intervening matter. “Galaxy-galaxy lensing” (GGL) uses the correlation between a shear map and a foreground galaxy sample to measure the average mass profile around the foreground galaxies, which can be combined with galaxy clustering to constrain total matter clustering. For a specified expansion history, the predicted signals scale approximately as $\sigma_8\Omega_m^\alpha$, with $\alpha \simeq 0.3-0.5$. The predicted signals also depend on the distance-redshift relation, so weak lensing becomes more powerful in concert with SN or BAO measurements that can pin this relation down independently. The most challenging systematics are shape measurement biases, biases in the distribution of photometric redshifts, and intrinsic alignments of galaxy orientations that could contaminate the lensing-induced signal. Weak lensing of CMB anisotropies is an increasingly powerful tool, in part because it circumvents many of these observational and astrophysical systematics. Predicting the large-scale weak lensing signal is straightforward in principle, but the number of independent modes on large scales is small, and the inferences are therefore dominated by sample variance. Exploiting small-scale measurements, for tighter constraints, requires modeling the effects of complex physical processes such as star formation and feedback on the matter power spectrum. Strong gravitational lensing can also provide constraints on dark energy, either through time delay measurements that probe the absolute distance scale, or through measurements of multiple-redshift lenses that constrain distance ratios. The primary uncertainty for strong lensing constraints is modeling the mass distribution of the lens systems.

**Clusters of Galaxies:** Like weak lensing, the abundance of massive dark-matter halos probes structure growth by constraining $\sigma_8\Omega_m^\alpha$, where $\alpha \simeq 0.3-0.5$. These halos can be identified as dense concentrations of galaxies or through the signatures of hot ($10^7-10^8$ K) gas in X-ray emission or SZ distortion of the CMB. The critical challenge in cluster cosmology is calibrating the relation $P(M_{\text{halo}}|O)$ between the halo mass as predicted from theory and the observable $O$ used for cluster identification. Measuring the stacked weak lensing signal from clusters has emerged as a promising approach to achieve percent-level accuracy in calibration of the mean relation, which is required for clusters to remain competitive with other growth probes. This method requires accurate modeling of completeness and contamination of cluster catalogs, projection effects on cluster selection and weak lensing measurements, and possible baryonic physics effects on the mass distribution within clusters.

**Redshift-Space Distortions (RSD) and the Alcock-Paczynski (AP) Effect:** Redshift-space distortions of galaxy clustering, induced by peculiar motions, probe structure growth by constraining the parameter combination $f(z)\sigma_8(z)$, where $f(z)$ is the growth rate defined by Eq. (28.3). Uncertainties in theoretical modeling of non-linear gravitational evolution and the non-linear bias between the galaxy and matter distributions currently limit application of the
<table><caption>**Table 28.1:** A selection of major dark-energy experiments, based on Ref. [27]. Abbreviations in the "Data" column refer to optical (Opt) or near-infrared (NIR) imaging (I) or spectroscopy (S). For spectroscopic experiments, the "Spec-z" column lists the primary redshift range for galaxies (gals), quasars (QSOs), or the Lyman-α forest (LyαF). Abbreviations in the "Methods" column are weak lensing (WL), clusters (CL), supernovae (SN), baryon acoustic oscillations (BAO), and redshift-space distortions (RSD).</caption><thead><tr><th>Project</th><th>Dates</th><th>Area/deg<sup>2</sup></th><th>Data</th><th>Spec-z Range</th><th>Methods</th></tr></thead><tbody><tr><td>BOSS</td><td>2008–2014</td><td>10,000</td><td>Opt-S</td><td>0.3–0.7 (gals)<br>2–3.5 (LyαF)</td><td>BAO/RSD</td></tr><tr><td>KiDS</td><td>2011–2019</td><td>1350</td><td>Opt-I</td><td>—</td><td>WL/CL</td></tr><tr><td>DES</td><td>2013–2019</td><td>5000</td><td>Opt-I</td><td>—</td><td>WL/CL<br>SN/BAO</td></tr><tr><td>eBOSS</td><td>2014–2018</td><td>7500</td><td>Opt-S</td><td>0.6–2.0 (gal/QSO)<br>2–3.5 (LyαF)</td><td>BAO/RSD</td></tr><tr><td>SuMIRE</td><td>2014–2029</td><td>1500</td><td>Opt-I</td><td>—</td><td>WL/CL</td></tr><tr><td>DESI</td><td>2021–2026</td><td>14,000</td><td>Opt/NIR-S</td><td>0.8–2.4 (gals)<br>0–1.7 (gals)<br>2–3.5 (LyαF)</td><td>BAO/RSD<br>BAO/RSD</td></tr><tr><td>VRO/LSST</td><td>2025–2035</td><td>20,000</td><td>Opt-I</td><td>—</td><td>WL/CL<br>SN/BAO</td></tr><tr><td>Euclid</td><td>2023–2029</td><td>15,000</td><td>Opt-I</td><td>—</td><td>WL/CL</td></tr><tr><td>Roman</td><td>2027–2032</td><td>2200</td><td>NIR-S</td><td>0.7–2.2 (gals)</td><td>BAO/RSD</td></tr><tr><td></td><td></td><td></td><td>NIR-I</td><td>—</td><td>WL/CL/SN</td></tr><tr><td></td><td></td><td></td><td>NIR-S</td><td>1.0–3.0 (gals)</td><td>BAO/RSD</td></tr></tbody></table>

method to large scales (comoving separations $r \gtrsim 10 h^{-1} Mpc$ or wavenumbers $k \lesssim 0.2 h Mpc^{-1}$). A second source of anisotropy arises if one adopts the wrong cosmological metric to convert angles and redshifts into comoving separations, a phenomenon known as the Alcock-Paczynski effect [28]. Demanding isotropy of clustering at redshift $z$ constrains the parameter combination $H(z)D_A(z)$. The main challenge for the AP method is correcting for the anisotropy induced by peculiar velocity RSD.

**Low Redshift Measurement of $H_0$:** The value of $H_0$ sets the current value of the critical density $\rho_c = 3H_0^2/8\pi G_N$, and combination with CMB measurements provides a long lever arm for constraining the evolution of dark energy. The challenge in conventional $H_0$ measurements is establishing distances to galaxies that are “in the Hubble flow,” i.e., far enough away that their peculiar velocities are small compared to the expansion velocity $v = H_0d$. This can be done by building a ladder of distance indicators tied to stellar parallax on its lowest rung, or by using gravitational-lens time delays or geometrical measurements of maser data to circumvent this ladder.

### 28.3.2 Dark Energy Experiments

Most observational applications of the above methods now take place in the context of large cosmological surveys, for which constraining dark energy and modified gravity theories is a central objective. Table 28.1 lists a selection of recent, ongoing, and planned dark-energy experiments, taken originally from the Snowmass 2013 Dark Energy Facilities review [27], which focused on projects in which the U.S. has either a leading role or significant participation. References and links to further information about these projects can be found in Ref. [27]. We have adjusted some of the dates in this Table relative to those in Ref. [27] and added the European-led KiloDegree Survey (KiDS). Dates in the Table correspond to the duration of survey observations, and the final cosmological results frequently require 1–3 years of analysis and modeling beyond the end of data taking. We are now at an exciting phase in which most of the “Stage III” experiments (in the parlance of Ref. [29]) have completed observations and in some cases final analysis, and the more powerful “Stage IV” experiments are underway or beginning operations in the near future.

Beginning our discussion with imaging surveys, the Dark Energy Survey (DES) observed 1/8 of the sky to a depth roughly 2 magnitudes deeper than the Sloan Digital Sky Survey (SDSS), enabling weak lensing measurements with much greater statistical precision, cluster measurements calibrated by weak lensing, and angular BAO measurements based on photometric redshifts. With repeated imaging of selected fields, DES identified thou-

sands of Type Ia SNe, which together with spectroscopic follow-up data enabled significant improvements for supernova (SN) cosmology. Cosmological results from weak lensing and galaxy clustering analyses of the 3-year (Y3) DES data set are presented in Ref. [30]. The first cosmological results from the DES supernova survey are presented in Ref. [31], and DES supernovae are incorporated into the Pantheon+ compilation analyzed in Ref. [26]. KiDS carried out an optical imaging survey of about 1000 deg$^2$, with cosmology results from weak lensing and galaxy clustering reported in Refs. [32, 33]. The Subaru Hyper-Suprime Camera (HSC) collaboration is surveying about 1500 deg$^2$ with greater depth and sharper image quality than DES; cosmology results from the 3-year HSC data set are presented in Refs. [34, 35]. Beginning in the mid-2020s, the Legacy Survey of Space and Time (LSST) of the Vera Rubin Observatory (VRO) will scan the southern sky to SDSS-like depth every four nights. The co-added data from LSST’s decade-long primary survey will provide deep multiband imaging over a wide area, enabling weak lensing, cluster, and photometric BAO studies from billions of galaxies. Additionally, LSST time-domain monitoring will identify and measure light curves for thousands of Type Ia SNe per year.

Turning to spectroscopic surveys, the Baryon Oscillation Spectroscopic Survey (BOSS) and its successor eBOSS used fiber-fed optical spectrographs to map the redshift-space distributions of millions of galaxies and quasars. These 3-dimensional maps enable BAO and RSD measurements, and Lyman-α forest spectra of high-redshift quasars extend these measurements to redshifts $z > 2$. As discussed below, the eBOSS Collaboration has now published BAO and RSD analyses from the final data sets of the BOSS and eBOSS programs. The Dark Energy Spectroscopic Instrument (DESI) follows a strategy similar to BOSS/eBOSS but on a much grander scale, using a larger telescope (4-m versus 2.5-m) and a much higher fiber multiplex (5000 versus 1000) to survey an order-of-magnitude more galaxies. DESI began full operations in May 2021, and in its first two years it has measured well over 20 million redshifts; at least 40 million redshifts are expected by the survey end. A new Prime Focus Spectrograph (PFS) for the Subaru telescope will enable the spectroscopic component of SuMIRE, starting in 2024, with the large telescope aperture and wavelength sensitivity that extends to the near-infrared (NIR) allowing it to probe a higher redshift galaxy population than DESI, over a smaller area of sky.

Compared to ground-based observations, space observations afford higher angular resolution and a far lower NIR sky background. The Euclid mission and the Nancy Grace Roman Space Telescope (formerly WFIRST) will exploit these advantages, con-
ducting large area imaging surveys for weak lensing and cluster
studies and slitless spectroscopic surveys of emission-line galax-
ies for BAO and RSD studies. *Roman* will also incorporate an
imaging and spectrophotometric supernova (SN) survey, extend-
ing to redshift *z* ≈ 1.7. One can roughly characterize the differ-
ence between the *Euclid* and *Roman* dark-energy experiments as
“wide versus deep,” with currently planned survey areas of 15,000
deg² and 2200 deg², respectively. For weak lensing shape mea-
surements, *Euclid* uses a single wide optical filter, while *Roman*
will use three NIR filters. The *Euclid* galaxy redshift survey will
cover a large volume at relatively low space density, while the *Ro-
man* survey will provide denser sampling of structure in a smaller
volume. *Euclid* was launched successfully in July 2023 and has
released first light images; it will begin its 6.5-year main mission
survey in late 2023. *Roman* is scheduled for launch in late 2026 or
early 2027, and details of its cosmology surveys will likely evolve
before operations begin. There are numerous synergies among the
LSST, *Euclid*, and *Roman* dark energy programs, as discussed in
Ref. [36].

## 28.4 Current Constraints on Expansion, Growth, and Dark Energy

The last decade has seen dramatic progress in measurements
of the cosmic expansion history and structure growth, leading
to much tighter constraints on the parameters of dark energy
models. CMB data from the *WMAP* and *Planck* satellites and
from higher resolution ground-based experiments have provided
an exquisitely detailed picture of structure at the recombination
epoch and CMB-based measures of low-redshift structure through
lensing and SZ cluster counts. Cosmological supernova samples
have increased in size from tens to thousands, with continuous
coverage from *z* = 0 to *z* ≈ 1.4, alongside major improvements
in data quality, analysis methods, and detailed understanding of
local populations. BAO measurements have advanced from the
first detections to 1–2% precision at multiple redshifts, with in-
creasingly sophisticated methods for testing systematics, fitting
models, and evaluating statistical errors. Advances in X-ray, SZ,
and weak-lensing observations of large samples of galaxy clus-
ters allow a multi-faceted approach to mass calibration, improv-
ing statistical precision but also revealing sources of astrophys-
ical uncertainty. Cluster constraints have been joined, and for
the present superseded, by precise matter-clustering constraints
from cosmic-shear weak lensing and galaxy-galaxy lensing, and
by redshift-space distortion measurements that probe different
aspects of structure growth at (presently) lower precision. The
precision of low-redshift *H*₀ measurements has sharpened from
the roughly 10% error of the *HST* Key Project [37] to claimed
precision as high as 1–2% in recent analyses.

As an illustration of current measurements of the cosmic expan-
sion history, Fig. 28.1 compares distance-redshift measurements
from SN and BAO data to the predictions for a flat Universe
with a cosmological constant. SN cosmology relies on compila-
tion analyses that try to bring data from different surveys prob-
ing distinct redshift ranges to a common scale. Here we use the
“joint light curve analysis” (JLA) sample of Ref. [39], who carried
out a careful intercalibration of the 3-year Supernova Legacy Sur-
vey (SNLS3, [41]) and the full SDSS-II Supernova Survey [3] data
in combination with several local supernova samples and high-
redshift supernovae from *HST*. For illustration purposes, we have
binned the JLA data in redshift and plotted the diagonal elements
of the covariance matrix as error bars, and we have converted the
SN luminosity distances to an equivalent comoving angular di-
ameter distance. Because the peak luminosity of a fiducial SN
Ia is an unknown free parameter, the SN distance measurements
could all be shifted up and down by a constant multiplicative fac-
tor; cosmological information resides in the relative distances as
a function of redshift. The normalization used here corresponds
to a Hubble parameter *h* = 0.674.

For the BAO data points in Fig. 28.1 we have used the compil-
ation in Table 3 (the “BAO-only” row) from the summary cosmol-
ogy paper by the eBOSS collaboration [40], where one can find
references for the original data sources. The individual BAO mea-
surements come (in order of increasing redshift) from the SDSS-II

<figure><img src="image_145.png" /><figcaption>**Figure 28.1:** Distance-redshift relation measured from Type Ia SNe and BAO compared to the predictions (black curve) of a flat $\Lambda$CDM model with $\Omega_m = 0.315$ and $h = 0.674$, the best-fit parameters inferred from *Planck* CMB data [38]. Circles show binned luminosity distances from the JLA SN sample [39], multiplied by $(1+z)^{-1}$ to convert to comoving angular diameter distance. Red squares show BAO distance measurements from the SDSS-II, BOSS, and eBOSS surveys as compiled in Ref. [40]. The lower panel plots residuals from the $\Lambda$CDM prediction, with dashed and dotted curves that show the effect of changing $w$ by $\pm 0.1$ while all other parameters are held fixed. Note that the SN data points can be shifted up or down by a constant factor to account for freedom in the peak luminosity, while the BAO points are calibrated to 0.2% precision by the sound horizon scale computed from *Planck* data. The errors on the BAO data points are approximately independent, but not entirely so. In the upper panel, error bars are plotted only at $z > 0.7$ to avoid visual confusion. The two Lyman-$\alpha$ forest data points are slightly offset from their effective redshift of $z = 2.33$ for clarity.</figcaption></figure>

main galaxy sample and from luminous red galaxies, emission line
 galaxies, and quasars mapped by BOSS and eBOSS. The two high-
est redshift points, both with an effective $z \approx 2.3$, come from the
auto-correlation of the Lyman-$\alpha$ forest in high-redshift ($z > 2$)
quasars from BOSS and eBOSS and from the cross-correlation of
the Lyman-$\alpha$ forest with the quasars themselves. The BAO mea-
surements are converted to absolute distances using the sound
horizon scale $r_s = 147.09$ Mpc from *Planck* 2018 CMB data,
whose 0.18% uncertainty is small compared to the current BAO
measurement errors. For the $z = 0.15$ and $z = 1.5$ data points
we have converted values of $D_V$ to $D_{\Lambda,c}$, while for other redshifts
we use the $D_{\Lambda,c}$ determinations measured directly by the trans-
verse BAO scale. The galaxy and Lyman-$\alpha$ forest analyses also
measure $H(z)$ at the same redshifts, providing further leverage on
expansion history that is not captured in Fig. 28.1.

The plotted cosmological model has $\Omega_m = 0.315$ and $h = 0.674$, the best-fit values from *Planck* (TT+TE+EE+lowE+lensing) assuming $w = -1$ and $\Omega_{\text{tot}} = 1$ [38]. The SN, BAO, and CMB data sets, probing a wide range of redshifts with radically different techniques, are for the most part mutually consistent with the predictions of a flat $\Lambda$CDM cosmology. This consistency has held steady or improved as the measurements themselves have improved over the past decade. Dotted and dashed curves in the lower panel of Fig. 28.1 show the effect of changing $w$ by $\pm 0.1$ with all other parameters held fixed, which leads to significantly worse agreement with the data. However, such a single-parameter com-
<figure><img src="image_146.png" /><figcaption>**Figure 28.2:** Constraints on dark energy model parameters from combinations of CMB, BAO, galaxy clustering, and supernova (SN) data, taken from Ref. [42]. The left panel shows 68% and 95% confidence contours in the $\omega_{wCDM}$ model, with constant equation-of-state parameter $w$ and non-zero space curvature $\Omega_K \equiv 1 - \Omega_{tot}$. Green and gray contours show the combination of Planck CMB data with SN or BAO data, respectively. Red contours combine CMB, BAO, and the full shape (FS) of redshift-space galaxy clustering. Blue contours add SN data to this combination. The right panel shows confidence contours for the same data combinations in the $w_0w_aCDM$ model, which assumes a flat Universe and an evolving equation of state with $w(a) = w_0 + w_a(1-a)$.</figcaption></figure>

parison does not capture the impact of parameter degeneracies or the ability of complementary data sets to break them, and if one instead forced a match to CMB data by changing $h$ and $\Omega_m$ when changing $w$ then the predicted BAO distances would diverge at $z=0$ rather than converging there. The recent Pantheon+ compilation [43] comprises more than 1500 spectroscopically confirmed Type Ia supernovae from 18 different samples, including those in the JLA. With higher statistical precision and reduced systematic uncertainties, a joint fit of Pantheon+ with CMB and BAO data remains fully compatible with a flat $\Lambda$CDM model [26]. Other good representations of recent observational constraints on the cosmic expansion history include Fig. 4 of Ref. [26] for SNIa and Fig. 2 of Ref. [40] for BAO.

Fig. 28.2, taken from Ref. [42], presents constraints on models that allow a free but constant value of $w$ with non-zero space curvature (owCDM, left panel) or the evolving equation of state of Eq. (28.4) in a flat Universe ($w_0w_aCDM$, right panel). Green contours show constraints from the combination of Planck 2015 CMB data and the JLA supernova sample. Gray contours show the combination of Planck with BAO measurements from BOSS, 6dFGS [44], and SDSS-II [45]. Red contours adopt a more aggressive analysis of the BOSS galaxy data that uses the full shape (FS) of the redshift-space power spectrum and correlation function, modeled via perturbation theory, in addition to the measurement of the BAO scale itself. The full shape analysis improves the constraining power of the data, primarily because measurement of the Alcock-Paczynski effect on sub-BAO scales helps to break the degeneracy between $D_{A,c}(z)$ and $H(z)$. Blue contours show constraints from the full combination of CMB, BAO+FS, and SN data. Supernovae provide fine-grained relative distance measurements with good bin-by-bin precision at $z < 0.7$ (see Fig. 28.1), which is complementary to BAO for constraining redshift evolution of $w$. In both classes of model, the flat $\Lambda$CDM parameters ($w = w_0 = -1$, $\Omega_K = w_a = 0$) lie within the 68% confidence contour. Many recent papers feature constraint diagrams analogous to Fig. 28.2, and the constraints within this parameter space have tightened moderately with new data while remaining consistent with flat $\Lambda$CDM.

The precision on dark energy parameters depends, of course, on both the data being considered and the flexibility of the model being assumed. For wCDM, Ref. [40] find $w = -1.026 \pm 0.033$ using Planck CMB data, BAO measurements from SDSS-II and

BOSS/eBOSS, and SN measurements from the Pantheon compilation. With the addition of eBOSS RSD data and DES weak lensing and galaxy clustering measurements, they constrain a more flexible model (owwCDM) that allows non-zero curvature and an evolving equation-of-state (Eq. 28.4), finding

$$w_p = -1.020 \pm 0.032 \qquad (28.5)$$

at a pivot redshift $z_p = 0.29$, a tight constraint on curvature,

$$1 - \Omega_{tot} = -0.0023 \pm 0.0022, \qquad (28.6)$$

but only a loose constraint on the evolution parameter,

$$w_a = -0.48^{+0.36}_{-0.30}. \qquad (28.7)$$

Obtaining a tight constraint on $w_p$ in such a flexible cosmological model requires measurements that have complementary sensitivity to its multiple free parameters, and the precision of Eq. (28.5) is a testament to the remarkable improvements in cosmological measurements over the past decade. BAO and FS measurements from DESI will likely supersede those from BOSS/eBOSS in the near future, enabling substantially sharper parameter constraints.

A flat $\Lambda$CDM model fit to Planck CMB data alone predicts $H_0 = (67.4 \pm 0.5) \text{ km s}^{-1} \text{ Mpc}^{-1}$ (see Chapter 29 of this Review). This prediction and its error bar are sensitive to the assumptions of constant dark energy and a flat Universe. However, by adding BAO and supernova data one can construct an “inverse distance ladder” to measure $H_0$ precisely, even with a general dark energy model and free curvature [46]. Ref. [40] applies this approach to obtain $H_0 = (68.2 \pm 0.8) \text{ km s}^{-1} \text{ Mpc}^{-1}$. As discussed in Sec. 25.3.1 of this Review, some recent measurements from low-redshift data yield higher values of $H_0$. For example, Ref. [47] uses a distance ladder based on Cepheid calibration of Type Ia supernova luminosities to find $H_0 = (73.0 \pm 1.0) \text{ km s}^{-1} \text{ Mpc}^{-1}$, in strong tension with cosmologically inferred values. However, calibrations based on tip-of-the-red-giant-branch (TRGB) distances give lower $H_0$, consistent with the cosmological values [48].

The tension in $H_0$ could reflect some combination of statistical flukes and systematic errors in one or more of the data sets employed in these analyses. However, if the resolution lies in new physics rather than measurement errors, then this is probably physics that operates in the pre-recombination Universe, rescaling the BAO standard ruler in a way that shifts the $\Lambda$CDM and
inverse-distance-ladder values upward. Models with extra relativistic degrees of freedom or dark energy that is dynamically significant in the early Universe can achieve this effect by increasing the early expansion rate, but they are tightly constrained by the damping tail of CMB anisotropies and by the shape of the galaxy power spectrum. These constraints will sharpen significantly with the next generation of CMB experiments, which will improve measurements of polarization anisotropies in the damping tail. A finely tuned model in which early dark energy decays rapidly after recombination can mitigate the tension between CMB data and local $H_0$ measurements [49], though it still prefers $H_0$ values below those of Ref. [47]. Numerous theory papers have examined possible physical solutions to the $H_0$ tension, all of which involve significant modifications to the $\Lambda$CDM scenario.

The amplitude of CMB anisotropies is proportional to the amplitude of density fluctuations present at recombination, and by assuming GR and a specified dark energy model one can extrapolate the growth of structure forward to the present day to predict $\sigma_8$. Probes of low-redshift structure yield constraints in the ($\sigma_8, \Omega_m$) plane, which can be summarized in terms of the parameter combination $S_8 \equiv \sigma_8(\Omega_m/0.3)^{0.5}$. As discussed in earlier editions of this Review, many weak-lensing and cluster studies to date yield $S_8$ values lower than those predicted for Planck-normalized $\Lambda$CDM.

Fig. 28.3, taken from Ref. [50], illustrates the current state-of-play on $S_8$. (Similar figures with different selections of data can be found in many recent papers and in the previous edition of this Review.) The topmost point represents the predicted value and $1\sigma$ uncertainty, $S_8 = 0.830 \pm 0.014$, based on a flat $\Lambda$CDM model normalized to Planck CMB anisotropies. The next two points illustrate a different treatment of the Planck data or an alternative CMB data combination. The next four points, in red, illustrate values of $S_8$ measured from weak lensing of the CMB, again assuming flat $\Lambda$CDM. Although the background being lensed is the CMB, this is still a measurement of low-redshift clustering; for multipoles $\ell > 100$ roughly half of the expected signal comes from structure at $z < 2$ and nearly all from $z < 10$ [51].

The remaining points, in blue, represent galaxy-based measurements that probe structure at $z \approx 0.2-0.8$, lower than the peak of the CMB-lensing kernel. The first four come from analyses of the DES, KiDS, and HSC weak lensing surveys, using cosmic shear and galaxy-galaxy lensing combined with galaxy clustering. The three points labeled CX come from cross-correlation of galaxies with CMB lensing maps, which measures the galaxy-matter cross-correlation without using optical weak lensing data. The point labeled GC (fourth from bottom) is inferred from RSD analysis of eBOSS galaxies. Details and references can be found in Sec. 3.4 of Ref. [50].

It is evident from Fig. 28.3 that CMB lensing measurements of $S_8$ (red points) are in good agreement with predictions based on primary CMB anisotropy (green points), but $S_8$ values measured from weak lensing at lower redshifts are consistently lower than this prediction (blue points, excluding the GC point derived from RSD). The discrepancy with any individual measurement is only $1-2\sigma$, but the significance is higher if one treats the points derived from separate surveys as statistically independent. For instance, after averaging the two HSC measurements, an unweighted mean of the six low redshift lensing measurements in Fig. 28.3 yields $S_8 = 0.759 \pm 0.009$, where the $1\sigma$ error is simply the dispersion of the measurements divided by $\sqrt{N-1} = \sqrt{5}$. With this estimate, the $+3\sigma$ bound $S_8 = 0.786$ lies just below the $-3\sigma$ bound of the Planck prediction, $S_8 = 0.788$. However, several or all of these measurements could be affected by a common systematic error, such as the modeling of galaxy intrinsic alignments or the modeling of structure on non-linear scales. Furthermore, the RSD measurement, which probes similar redshifts with a physically distinct observable, gives a much higher $S_8$ value, though with an uncertainty large enough to accommodate both the CMB-based prediction and the low-$z$ weak lensing measurement.

The tension between the $\Lambda$CDM prediction of $S_8$ and low redshift weak lensing measurements has persisted through several cycles of this Review, without getting obviously stronger or obviously weaker. The sharpest conflicts arise in analyses that extend

to non-linear scales and thus have the highest statistical precision, but these analyses are also the most susceptible to modeling uncertainties such as the impact of baryonic dissipation and feedback on the mass distribution. If the tension suggested by Fig. 28.3 represents a true deviation between the clustering growth extrapolated forward from the early Universe and the clustering of matter at late times, then it is difficult to explain by changing the dark energy equation of state because the expansion history is already well constrained by BAO and SN data. Instead, such a deviation might point towards modified gravity, decaying dark matter, or coupling between dark matter and dark energy.

Progress in understanding this tension should come from several directions. Unified analyses of multiple weak lensing data sets are now testing consistency and highlighting the influence of analysis choices on cosmological results [52,53]. Final data from DES and HSC will have better statistical power, which improves the internal calibration of systematic nuisance parameters and affords greater precision at the larger scales that are easier to model. Improved CMB lensing maps will allow better cross-correlation measurements (the CX points in Fig. 28.3), which avoid some of the main systematics affecting optical weak lensing. DESI data will soon allow the first RSD-based measurements of structure growth at the 1–2% level, which will show whether the low matter clustering amplitude suggested by weak lensing data is also apparent in galaxy peculiar velocities. In modified gravity models these two methods of constraining structure can yield inconsistent results because they respond to different combinations of metric components, but either consistency or inconsistency would provide valuable guidance. Finally, if there is truly a 5–10% discrepancy between CMB-normalized $\Lambda$CDM predictions and low-redshift clustering, it should be measured at high statistical significance in Euclid weak lensing data and, subsequently, in LSST and Roman data.

## 28.5 Summary and Outlook

Figure 28.2 focuses on model parameter constraints, but to describe the observational situation it is more useful to characterize the precision, redshift range, and systematic uncertainties of the basic expansion and growth measurements. At present, supernova surveys constrain distance ratios at the 1–2% level in redshift bins of width $\Delta z = 0.1$ out to $z \approx 0.8$, with larger but still interesting error bars out to $z \simeq 1.3$. Estimated systematic uncertainties are comparable to statistical uncertainties and include effects of photometric calibration, dust reddening, host-galaxy correlations, and possible evolution of the SN population. BAO surveys have measured the angular diameter distance $D_{\Lambda,c}(z)$ and the expansion rate $H(z)$ over the range $0 < z < 2.5$, calibrated to absolute units using the CMB-based value of the sound horizon $r_s$. For $D_{\Lambda,c}(z)$, the final analyses of BOSS/eBOSS achieve precision of $1.6-1.8\%$ in three overlapping redshift bins from luminous galaxies at $z = 0.2-1.0$, 2.6% from quasars at $z \approx 1.5$, and 2.9% from the Lyman-$\alpha$ forest at $z \approx 2.3$. For $H(z)$, the precision at the same redshifts is 2.6–3.0%, 4.1%, and 2.1%, respectively. SDSS, DES, KiDS, and HSC have used combinations of weak lensing and galaxy clustering to measure the parameter combination $S_8 = \sigma_8(\Omega_m/0.3)^{0.5}$ with estimated precision of 2–4%, and the cross-correlation of galaxy distributions with CMB lensing maps achieves comparable precision. These estimates account for identified systematic uncertainties, but the measurements and modeling are challenging, and it is possible that the systematics are underestimated. Our simple average of these measurements, treating them as independent and using their dispersion as an empirical estimate of uncertainty, yields an error on the mean of 1.1%. The statistical power of these weak lensing surveys is concentrated at $z \approx 0.2-0.8$, and they provide useful constraints at lower precision on redshift evolution over this range and on individual values of $\Omega_m$ and $\sigma_8$. RSD measurements constrain the similar parameter combination $f(z)\sigma_8(z)$, but they do not yet have precision competitive with that of weak lensing measurements. Distance-ladder estimates of $H_0$ now span a small range, using overlapping data but distinct treatments of key steps. Individual studies quote uncertainties as small as 1–2%, with similar statistical and systematic contributions.
<figure><img src="image_147.png" /><figcaption>**Figure 28.3:** Predicted values of $S_8 \equiv \sigma_8(\Omega_m/0.3)^{0.5}$ from a CMB-normalized $\Lambda$CDM model (green points), compared to observational estimates from CMB lensing (red points), from optical weak lensing surveys (first four blue points), from galaxy clustering RSD (blue GC point), and from cross-correlation of galaxies with CMB lensing maps (blue CX points). The predictions (green) are extrapolated forward from primary CMB anisotropy measurements that measure structure at $z \approx 1100$. CMB lensing (red) is most sensitive to structure at $z \approx 1-2$, while the galaxy-based data (blue) probe structure at $z \approx 0.2-0.8$. Taken from Ref. [50].</figcaption></figure>

Planck data and higher resolution ground-based experiments now measure CMB anisotropies with exquisite precision; for example, CMB measurements now constrain the physical size of the BAO sound horizon to 0.2% and the angular scale of the sound horizon to 0.01%. A flat $\Lambda$CDM model with standard radiation and neutrino content can fit the CMB data and the BAO and SN distance measurements to within their estimated uncertainties. The value of $H_0$ implied by this model disagrees with recent Cepheid+SNIa distance-ladder measurements of $H_0$ at a $\sim 5\sigma$ level. This disagreement persists in models that allow non-zero curvature and low redshift evolution of $w(z)$, provided one assumes standard pre-recombination physics to compute the sound horizon $r_s$. The discrepancy could reflect underestimated systematic uncertainties in the Cepheid-based $H_0$ estimate, or it could be a sign of new physics in the early Universe that rescales $r_s$. Improved parallax data from *HST* and *Gaia*, discovery of new SNIa in nearby galaxies, and observing programs on the *James Webb Space Telescope* (*JWST*) are all allowing improvements and systematics checks in the Cepheid distance ladder. It will be equally important to bring the TRGB calibration method to a comparable level of precision and systematics control, and to understand any differences between the Cepheid and TRGB distance scales. Polarization measurements of the CMB damping tail from ground-based experiments will provide increasingly strong constraints on resolutions of the $H_0$-tension that invoke novel pre-recombination physics.

Extrapolating forward from the CMB anisotropies measured by Planck predicts a low redshift value of $S_8$ that is higher than recent estimates from weak gravitational lensing and galaxy clustering. Comparing the Planck central value to the simple average of recent experiments discussed in Sec. 28.4 implies a difference of 9.3%: $S_8 = 0.830 \pm 0.014$ versus $S_8 = 0.759 \pm 0.009$. This disagreement could reflect a common systematic that biases several of the weak lensing analyses in the same direction, and it would be weakened if the Planck $S_8$ value were high because of an unlucky statistical fluctuation or a residual systematic. If real, this discrepancy could point towards modified gravity, decaying dark matter, or coupling between dark matter and dark energy.

There are numerous directions for progress over the next several years. Analyses of the final weak lensing data sets from KiDS, HSC, and DES will yield tighter statistical constraints and more complete understanding of systematic uncertainties. Cross-correlation of galaxy maps with higher signal-to-noise CMB lensing maps will provide independent clustering measurements that avoid many of the systematics of optical weak lensing. Collectively these efforts could achieve an unambiguous determination of the amplitude of low redshift matter clustering at the 1–2% level.

The DESI redshift survey already exceeds the size of the SDSS and BOSS/eBOSS surveys, with a factor of $\sim 10$ increase expected by the end of the 5-year program. DESI will enable the first high precision BAO measurements of expansion history at $z \approx 0.7-1.4$, as well as improved BAO measurements at $z < 0.7$ from denser galaxy samples and at $z > 2$ from much larger Lyman-$\alpha$ forest samples. DESI will also enable the first percent-level measurements of structure growth through RSD. The BAO measurements will complement increasingly precise measurements of the relative distance scale at $z < 1$ from supernova compilations that include the final data from DES. Large galaxy samples will also enable more powerful applications of the Alcock-Paczynski effect and parameter measurements based on voids or higher order clustering statistics.

*Euclid*, LSST, and *Roman* provide another major leap in observational capabilities. LSST will be the ultimate ground-based optical weak-lensing experiment, measuring several billion galaxy shapes over 20,000 deg$^2$ of the southern hemisphere sky, and it will detect and monitor many thousands of SNe per year. *Euclid* and *Roman* also have weak lensing as a primary science goal, taking advantage of the high angular resolution and extremely stable image quality achievable from space. Both missions plan large spectroscopic galaxy surveys, which will provide better sampling at high redshifts than DESI or Subaru PFS because of the lower infrared sky background above the atmosphere. *Roman* is also designed to carry out what should be the ultimate supernova cosmology experiment, with deep, high resolution, near-IR observations and the stable calibration achievable with a space platform.
The 2020s will also see dramatic advances in CMB lensing from the Simons Observatory and, potentially, CMB-S4 and/or a space-based probe; cross-correlation with galaxy surveys allows precise tomographic studies of clustering as a function of redshift. While the measurement goals of these next generation facilities overlap, the history of precision cosmology demonstrates the crucial importance of pursuing these measurements with multiple experiments that have distinct challenges and complementary strengths, and with analysis teams that develop ideas independently. The ensemble of experiments will produce cosmological constraints that are much tighter and much more robust to systematic uncertainties than any one of them could achieve individually.

If the anomalies suggested in Fig. 28.3 are real, then these experiments will map out their redshift, scale, and environment dependence in great detail, providing detailed empirical constraints on dynamical dark energy or modified gravity models. If these tensions dissipate with improved measurements, then the experiments of the 2020s will achieve much more stringent tests of the ΛCDM paradigm, with the potential to reveal deviations that are still within the statistical uncertainties of current data. The critical clue to the origin of cosmic acceleration could also come from a surprising direction, such as laboratory, solar-system, or gravitational-wave tests that challenge GR, time variation of fundamental “constants,” or anomalous behavior of gravity in some astronomical environments. Experimental advances along these multiple axes could confirm today’s relatively simple, but frustratingly incomplete, “standard model” of cosmology, or they could force yet another radical revision in our understanding of energy, or gravity, or the spacetime structure of the Universe.

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## 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 well described 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 $z \lesssim 10^7$ (see earlier versions of this review).

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 first detection 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], which were 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 the South Pole Telescope (SPT) [13]. Together these observations have led to a stunning confirmation of 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 $T(z) = T_0(1+z)$ in an expanding Universe. As a direct confirmation of its cosmological origin, this relationship has been tested by measuring 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 cosmological models 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 $z \gtrsim 1000$ accurately 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 and polarization anisotropies at the $10^{-6}$ (and lower) level, over a wide range of angular scales. These anisotropies are usually expressed using a spherical harmonic expansion

of the CMB sky:

$$T(\theta, \phi) = \sum_{\ell m} a_{\ell m} Y_{\ell m}(\theta, \phi) \qquad (29.1)$$

(with the linear polarization pattern written in a similar way using the so-called spin-2 spherical harmonics). 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 anisotropy power per unit $\ln \ell$ is $\ell \sum_m |a_{\ell m}|^2 / 4\pi$.

#### 29.3.1 The Monopole

The CMB has a mean temperature of $T_\gamma = 2.7255 \pm 0.0006$ K ($1\sigma$) [19], which can be considered as the monopole component of CMB maps, $a_{00}$. Because all mapping experiments involve difference measurements, they are insensitive to this average level; monopole measurements can only be made with absolute temperature devices, such as the FIRAS instrument on the COBE satellite [20]. The measured $kT_\gamma$ is equivalent to 0.234 meV or $4.60 \times 10^{-10} m_e c^2$. A blackbody of the measured temperature has a number density $n_\gamma = (2\zeta(3)/\pi^2) T_\gamma^3 \simeq 411 \text{ cm}^{-3}$, energy density $\rho_\gamma = (\pi^2/15) T_\gamma^4 \simeq 4.64 \times 10^{-34} \text{ g cm}^{-3} \simeq 0.260 \text{ eV cm}^{-3}$, and a fraction of the critical density $\Omega_\gamma \simeq 5.38 \times 10^{-5}$.

#### 29.3.2 The Dipole

The largest anisotropy is in the $\ell = 1$ (dipole) first spherical harmonic, with amplitude $3.3621 \pm 0.0010$ 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 fundamentally difficult to distinguish). The motion of an observer with velocity $\beta \equiv v/c$ relative to an isotropic Planckian radiation field of temperature $T_0$ produces a Lorentz-boosted temperature pattern

$$T(\theta) = T_0(1 - \beta^2)^{1/2} / (1 - \beta \cos \theta) \\ \approx T_0 \left[ 1 + \beta \cos \theta + (\beta^2/2) \cos 2\theta + O(\beta^3) \right]. \quad (29.2)$$

At every point in the sky, one observes a blackbody spectrum, with temperature $T(\theta)$. The spectrum of 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 $v = 369.82 \pm 0.11 \text{ km s}^{-1}$, assuming a value $T_0 = T_\gamma$, towards $(l, b) = (264.021^\circ \pm 0.011^\circ, 48.253^\circ \pm 0.005^\circ)$ [10]. This Solar System motion implies a velocity for the Galaxy and the Local Group of galaxies relative to the CMB of $v_{LG} = 620 \pm 15 \text{ km s}^{-1}$ towards $(l, b) = (271.9^\circ \pm 2.0^\circ, 29.6^\circ \pm 1.4^\circ)$ [10]; most of the error comes from 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' (in some sense this is a special frame) as that in which the CMB dipole would be zero. Any velocity of the 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 ($\ell \ge 2$) are interpreted as being mostly the result of density
perturbations in the early Universe, manifesting themselves at the epoch 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 $z \simeq 1100$ (with little dependence on the details of the model), the hydrogen and helium nuclei can bind electrons into neutral atoms, a process usually referred to as ‘recombination’ [24]. Before this epoch, the CMB photons were tightly coupled to the charged baryons, while afterwards they could freely stream towards us. By measuring the $a_{\ell m}$s we are thus learning directly about physical conditions in the early Universe.

A statistically-isotropic sky means that all $m$s 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., no correlations between the modes), the 2-point function of the temperature field (or equivalently the power spectrum in $\ell$) then fully characterizes the anisotropies. The power summed over all $m$s at each $\ell$ is $(2\ell+1)C_\ell/(4\pi)$, where $C_\ell \equiv \langle |a_{\ell m}|^2 \rangle$. Thus, averages of $a_{\ell m}$s over $m$ can be used as estimators of the $C_\ell$ s to constrain their expectation values, which are the quantities predicted by a theoretical model. For an idealized full-sky observation, the variance of each measured $C_\ell$ (i.e., the variance of the variance) is $[2/(2\ell+1)]C_\ell^2$. This sampling uncertainty (known as ‘cosmic variance’) comes about because each $C_\ell$ is $\chi^2$ distributed with $(2\ell+1)$ degrees of freedom for our observable volume of the Universe. For fractional sky coverage, $f_{\text{sky}}$, this variance is increased by $1/f_{\text{sky}}$ and the 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 $\ell$s, while the effects of instrumental noise and resolution dominate at higher $\ell$s [25].

Theoretical models generally predict that the $a_{\ell m}$ modes are Gaussian random fields to high precision, 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 the simplifying assumption. Nevertheless, models that deviate from the inflationary slow-roll conditions can have measurable non-Gaussian signatures. So while the current observational limits make the power 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 $\ell$ and the angle subtended by a particular spatial scale projected onto the sky. However, crudely speaking, a single spherical harmonic $Y_{\ell m}$ corresponds to angular variations of $\theta \sim \pi/\ell$. CMB maps contain anisotropy information from the size of the map (or in practice some fraction of that size) down to the beam-size of the instrument, $\sigma$ (the standard deviation of the beam, in radians). One can think of the effect of a Gaussian beam as rolling off the power spectrum with the function $e^{-\ell(\ell+1)\sigma^2}$.

For less than full sky coverage, the $\ell$ modes become correlated. Hence, experimental results are usually quoted as a series of ‘band powers,’ defined as estimators of $\ell(\ell+1)C_\ell/2\pi$ over different ranges of $\ell$. 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 of which are required to have non-null values (see The Cosmological Parameters—Sec. 25 of this Review).

The basic framework is the Friedmann-Robertson-Walker (FRW) metric (i.e., a Universe that 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 an adiabatic density perturbation and some isocurvature perturbations. Adiabatic means that there is no change to the entropy per particle for each species, i.e., $\delta\rho/\rho$ for matter is $(3/4)\delta\rho/\rho$ 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., $\delta\rho$ for matter is $-\delta\rho$ for radiation. These different modes give rise to distinct (temporal) phases during growth, with those of the adiabatic scenario being fully consistent with the data. Models that generate mainly isocurvature type perturbations (such as most topological defect 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 the variation of comoving curvature perturbations, $\mathcal{R}(x, t)$. The great virtue of $\mathcal{R}$ is that it is constant in time on super-horizon scales for a purely adiabatic perturbation. There are physical reasons to anticipate that the variance of these perturbations will be described well by a power law in scale, i.e., in Fourier space $\langle |\mathcal{R}|_k^2 \rangle \propto k^{n_s-4}$, where $k$ is wavenumber and $n_s$ is the spectral index as usually defined. So-called ‘scale-invariant’ initial conditions (meaning gravitational potential fluctuations that are independent of $k$) correspond to $n_s = 1$. In inflationary models [31] (see Inflation—Sec. 23 of 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 $\ln \mathcal{R}_k(\ln k)$ has terms of steadily decreasing size. For the simplest models, there are thus two parameters describing the initial conditions for density perturbations, namely the amplitude and slope of the power spectrum. These can be explicitly defined, for example, through

$$ \mathcal{P}_{\mathcal{R}}^2 \equiv k^3 \langle |\mathcal{R}|_k^2 \rangle / 2\pi^2 \approx A_s (k/k_0)^{n_s-1}, \quad (29.3) $$

with $A_s \equiv \mathcal{P}_{\mathcal{R}}^2(k_0)$ and $k_0 = 0.05 \text{ Mpc}^{-1}$, say. There are other equally valid definitions of the amplitude parameter (see also Secs. 22, 23, and 25 of this Review), and we caution that the relationships between some of them can be cosmology-dependent. In slow-roll inflationary models, this normalization is proportional to the combination $V^3/(V')^2$, for the inflationary potential $V(\phi)$. The slope $n_s$ also involves $V''$, and so the combination of $A_s$ and $n_s$ 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 $A_t \propto V$, 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 $n_t$, but since 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., $k = 0.002 \text{ Mpc}^{-1}$ or $k = 0.05 \text{ Mpc}^{-1}$, although it was historically defined in terms of the ratio of contributions at $\ell = 2$). Different inflationary potentials will lead to different predictions, e.g., for 50 e-folds, $\lambda\phi^4$ inflation gives $r = 0.32$ and $m^2\phi^2$ inflation gives $r = 0.16$ (both now strongly disfavored by the data), while other models can have arbitrarily small values of $r$. In any case, whatever the specific definition, and whether they come from inflation or something else, the ‘initial conditions’ give rise to a minimum of three parameters, $A_s$, $n_s$, and $r$.