1

28. Dark Energy

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 ρm and constant positive space curvature. Although Einstein
did not frame it this way, one can view the “cosmological constant” Λ as representing a constantenergy 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, andEinstein and de Sitter [6] advocated adopting an expanding, homogeneous and isotropic, spatiallyflat, matter-dominated Universe as the default cosmology until observations dictated otherwise.
Such a model has matter density equal to the critical density, Ωm ≡ ρm /ρc = 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 (H0 ), 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 Ωm ' 0.3, ΩΛ ' 0.7) as the preferred alternative to the Ωm = 1 scenario. Shortly
thereafter, CMB evidence for a spatially flat Universe [10, 11], and thus for Ωtot ' 1, cemented thecase for cosmic acceleration by firmly eliminating the free-expansion alternative with Ωm  1 andΩΛ = 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) ofa homogeneous and isotropic Universe governed by GR grows at an accelerating rate if the pressurep < − 13 ρ (in c = 1 units). A cosmological constant has ρΛ = constant and pressure pΛ = −ρΛ(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 cosmologicalconstant). 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 newform 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 literaturereferences for the discussion in Secs. 28.2 and 28.3.
Below we will use the abbreviation ΛCDM to refer to a model with cold dark matter, a cosmological constant, inflationary initial conditions, standard radiation and neutrino content, anda flat Universe with Ωtot = 1 (though we will sometimes describe this model as “flat ΛCDM” toemphasize this last restriction). We will use wCDM to denote a model with the same assumptions
but a free, constant value of w. Models with the prefix “o” (e.g., owCDM) allow non-zero spacecurvature.
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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28. Dark Energy

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 cosmological constant zero or at least negligibly small) is that theaccelerating cosmic expansion is driven by a new form of energy such as a scalar field [13] withpotential 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
1 2
2 φ̇  |V (φ)|, the scalar field acts like a cosmological constant, with pφ ' −ρφ . 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 = pφ /ρφ in scalar field models evolves with time in a way that
depends on V (φ) and on the initial conditions (φi , φ̇i ); 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 ρφ is even approximately constant, then it becomes dynamicallyinsignificant at high redshift, because the matter density scales as ρm ∝ (1 + z)3 . “Early darkenergy” models are ones in which ρφ 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 dominantcomponent before itself coming to dominate at low redshift.
Instead of introducing a new energy component, one can attempt to modify gravity in a waythat 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 valueof GN ) 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 ofcosmic structure growth to the history of cosmic expansion. Within GR, these two are linked by aconsistency 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 forgravitational lensing and the dynamics of non-relativistic tracers (such as galaxies or stars) indifferent ways (see Sec. 22.4.7 in this Review), leading to order unity mismatches between the massesof 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 themso 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
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28. Dark Energy

3ture 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 empiricallytest 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 meanthat 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 ofredshift. Within GR, the expansion rate H(z) is governed by the Friedmann equation (see thearticles 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, ΩΛ , is replaced by a redshift-dependent contribution. The evolution of the dark energy
density follows from Eq. (22.10),zρde (z)dz 0[1 + w(z 0 )]= Ωde exp 3
ρde (z = 0)
1 + z0
0

 Z

Ωde



= Ωde (1 + z)3(1+w) ,(28.1)

where the second equality holds for constant w. If Ωm , Ωr , and the present value of Ωtot are known,
then measuring H(z) pins down w(z). (Note that Ωde is the same quantity denoted Ωv 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, DA,c (z) = (1 + z)DA (z).
In GR-based linear perturbation theory, the density contrast δ(x, t) ≡ ρ(x, t)/ρ̄(t) − 1 of pressureless matter grows in proportion to the linear growth function G(t) (not to be confused withthe gravitational constant GN ), which follows the differential equation
3
G̈ + 2H(z)Ġ − Ωm H02 (1 + z)3 G = 0 .
2(28.2)

To a good approximation, the logarithmic derivative of G(z) is"d ln GH2
f (z) ≡ −' Ωm (1 + z)3 2 0
d ln(1 + z)
H (z)

#γ

,

(28.3)

where γ ' 0.55 for relevant values of cosmological parameters [21]. In an Ωm = 1 Universe,
G(z) ∝ (1 + z)−1 , but growth slows when Ωm 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, theamplitude of clustering is determined mainly by G(z), so observations of non-linear structure canbe used to infer the linear G(z), provided one has good theoretical modeling to relate the two.
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428. Dark Energy

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 involvea further integration over functions of H(z). Oscillations in w(z) over a range ∆z/(1 + z)  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) = w0 + wa (1 − a) = wp + wa (ap − a),(28.4)

where a ≡ (1 + z)−1 , w0 is the value of w at z = 0, and wp is the value of w at a “pivot”
redshift zp ≡ a−1
p − 1, where it is best constrained by a given set of experiments. For typical
data combinations, zp ' 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 projectedcombination of errors [σ(wp )σ(wa )]−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 ∆γ inthe index of Eq. (28.3), together with an overall multiplicative offset relative to the G(z) expectedfrom extrapolating the CMB-measured fluctuation amplitude to low redshift. However, these twoparameters may not accurately capture the growth predictions of all physically interesting models.
Another important parameter to constrain is the ratio of the gravitational potentials governingspace 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 thetask 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 ≡ H0 /100 km s−1 Mpc−1 determines the present day
value of the critical density and the overall scaling of distances inferred from redshifts.
• Ωm and Ωtot affect the expansion history and the distance-redshift relation.
R• The sound horizon rs = 0trec cs (t)dt/a(t), the comoving distance that pressure waves canpropagate 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 matterclustering [24].
• The amplitude of matter fluctuations, conventionally represented by the quantity σ8 (z), scales
the overall amplitude of growth measures such as weak lensing or redshift-space distortions(discussed in the next section).
Specifically, σ8 (z) refers to the rms fluctuation of the matter overdensity ρ/ρ̄ in spheres of radius
8 h−1 Mpc, computed from the linear theory matter power spectrum at redshift z, and σ8 on its
own refers to the value at z = 0 (just like our convention for Ωm ).
While discussions of dark energy are frequently phrased in terms of values and errors on quantities like wp , wa , ∆γ, and Ωtot , parameter precision is the means to an end, not an end in itself. The
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5

underlying goal of empirical studies of cosmic acceleration is to address two physically profoundquestions:• 1. Does acceleration arise from a breakdown of GR on cosmological scales or from a newenergy component that exerts repulsive gravity within GR?
• 2. If acceleration is caused by a new energy component, is its energy density constant inspace and time, as expected for a fundamental vacuum energy, or does it show variations thatindicate 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 andfundamental 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 particularSecs. 25 (Cosmological Parameters) and 29 (The Cosmic Microwave Background), and in Ref. [12],which provides extensive references to background literature. Recent observational results fromthese 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, andradiation 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 θs = rs /DA,c (zrec ), the angular size of the sound horizon at recombination, from the angular location of the acoustic peaks,
measurements of Ωm h2 and Ωb h2 from the heights of the peaks, and normalization of the amplitude of matter fluctuations at zrec from the amplitude of the CMB fluctuations themselves. The
comoparison 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 rs , which scales as (Ωm h2 )−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 anisostropies, 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 whitedwarfs, exhibit 10–15% scatter in peak luminosity after correction for light curve duration (thetime 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 DL (z) at different
redshifts. If one is comparing a high-redshift sample to a local calibrator sample measured withmuch higher precision (and distances inferred from Hubble’s law), then one essentially measures theluminosity distance in h−1 Mpc, constraining the combination hDL (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 byuncertainties associated with photometric calibration and dust extinction corrections and modelingthe 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 betested 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 thesesystematic uncertainties internally in parallel with cosmological constraints, with an overall error
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28. Dark Energy

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, whichappears in the galaxy correlation function as a localized peak at the sound horizon scale rs , or in the
power spectrum as a series of oscillations. Since observed galaxy coordinates consist of angles andredshifts, measuring this “standard ruler” scale in a galaxy redshift survey determines the angular
diameter distance DA (z) and the expansion rate H(z), which convert coordinate separations tocomoving distances. Errors on the two quantities are correlated, and in existing galaxy surveys the
2 (z)/H(z)]1/3 . As a useful rule ofbest determined combination is approximately DV (z) = [czDA,cthumb, 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 DA,c (z) with a fractional error of
0.005(V /10 Gpc3 )−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 bemeasured 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 fieldalong 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 enhanceclustering 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 underdevelopment. 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 toa well sampled spectroscopic survey of the same area, and angular BAO measurements do notdirectly constrain H(z). BAO distance measurements complement SN distance measurements byproviding absolute rather than relative distances (with precise calibration of rs from the CMB) andby 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 mattershears 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 deducethe 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 σ8 Ωαm , with α ' 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 distributionof photometric redshifts, and intrinsic alignments of galaxy orientations that could contaminate thelensing-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 thelarge-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 physicalprocesses such as star formation and feedback on the matter power spectrum. Strong gravitational
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28. Dark Energy

lensing can also provide constraints on dark energy, either through time delay measurements thatprobe 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 σ8 Ωαm , where α ' 0.3–0.5. These halos can be identified as dense
concentrations of galaxies or through the signatures of hot (107 –108 K) gas in X-ray emission or
SZ distortion of the CMB. The critical challenge in cluster cosmology is calibrating the relation
P (Mhalo |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 forclusters to remain competitive with other growth probes. This method requires accurate modelingof completeness and contamination of cluster catalogs, projection effects on cluster selection andweak lensing measurements, and possible baryonic physics effects on the mass distribution withinclusters.
Redshift-Space Distortions (RSD) and the Alcock-Paczynski (AP) Effect: Redshift-space distortionsof galaxy clustering, induced by peculiar motions, probe structure growth by constraining theparameter combination f (z)σ8 (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 thegalaxy and matter distributions currently limit application of the method to large scales (comoving−1−1<
separations r >∼ 10 h Mpc or wavenumbers k ∼ 0.2h Mpc ). 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 [27]. Demanding isotropy ofclustering at redshift z constrains the parameter combination H(z)DA (z). The main challenge for
the AP method is correcting for the anisotropy induced by peculiar velocity RSD.
Low Redshift Measurement of H0 : The value of H0 sets the current value of the critical density
ρc = 3H02 /8πGN , and combination with CMB measurements provides a long lever arm for constraining the evolution of dark energy. The challenge in conventional H0 measurements is establishingdistances 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 = H0 d. This can be done by building a ladderof distance indicators tied to stellar parallax on its lowest rung, or by using gravitational-lens timedelays 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 [28], which focused on
projects in which the U.S. has either a leading role or significant participation. References andlinks to further information about these projects can be found in Ref. [28]. We have adjusted someof the dates in this Table relative to those in Ref. [28] 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 datataking. We are now at an exciting phase in which most of the “Stage III” experiments (in theparlance 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 1/8of the sky to a depth roughly 2 magnitudes deeper than the Sloan Digital Sky Survey (SDSS),
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8

Table 28.1: A selection of major dark-energy experiments, based on Ref. [28]. 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), andredshift-space distortions (RSD).
Project

Dates

Area/deg2

DataSpec-z Range

Methods

BOSS

2008–2014

10,000Opt-S

BAO/RSD

KiDS
DES

2011–20192013–2019

1350
5000

Opt-I
Opt-I

0.3–0.7 (gals)
2–3.5 (LyαF)
——
——

eBOSS

2014–2018

7500

SuMIRE

2014–2029

1500

DESI

2021–2026

14,000

VRO/LSST 2025–2035

20,000

Euclid2023–2029

15,000Roman

2027–2032

2200

WL/CL
WL/CL
SN/BAO
Opt-S
0.6–2.0 (gal/QSO) BAO/RSD
2–3.5 (LyαF)
Opt-I
WL/CL
Opt/NIR-S
0.8–2.4 (gals)
BAO/RSD
Opt-S
0–1.7 (gals)
BAO/RSD
2–3.5 (LyαF)
Opt-I
——
WL/CL
SN/BAO
Opt-I
WL/CL
NIR-S
0.7–2.2 (gals)
BAO/RSD
NIR-I
WL/CL/SN
NIR-S
1.0–3.0 (gals)
BAO/RSD

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 thousands 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 deg2 , 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 deg2 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 multi-band imaging over a widearea, 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 ofType 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
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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 fromthe final data sets of the BOSS and eBOSS programs. The Dark Energy Spectroscopic Instument(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 anorder-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 andwavelength 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 anda far lower NIR sky background. The Euclid mission and the Nancy Grace Roman Space Telescope(formerly WFIRST ) will exploit these advantages, conducting large area imaging surveys for weaklensing and cluster studies and slitless spectroscopic surveys of emission-line galaxies for BAO andRSD studies. Roman will also incorporate an imaging and spectrophotometric supernova (SN)survey, extending to redshift z ' 1.7. One can roughly characterize the difference 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 measurements, 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 Roman survey will provide
denser sampling of structure in a smaller volume. Euclid was launched successfully in July 2023 andhas released first light images; it will begin its 6.5-year main mission survey in late 2023. Romanis 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 andCMB-based measures of low-redshift structure through lensing and SZ cluster counts. Cosmologicalsupernova samples have increased in size from tens to thousands, with continuous coverage fromz = 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 increasingly 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 clusters allow a multi-faceted approach to mass calibration, improving statistical precision but also revealing sources of astrophysical 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 H0 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 expansion history, Fig. 28.1 compares
distance-redshift measurements from SN and BAO data to the predictions for a flat Universe with
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28. Dark Energy

Figure 28.1: Distance-redshift relation measured from Type Ia SNe and BAO compared to thepredictions (black curve) of a flat ΛCDM model with Ωm = 0.315 and h = 0.674, the best-fit
parameters inferred from Planck CMB data [38]. Circles show binned luminosity distances fromthe 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 ascompiled in Ref. [40]. The lower panel plots residuals from the ΛCDM prediction, with dashedand dotted curves that show the effect of changing w by ±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 soundhorizon 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 toavoid visual confusion. The two Lyman-α forest data points are slightly offset from their effective
redshift of z = 2.33 for clarity.

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a cosmological constant. SN cosmology relies on compilation analyses that try to bring data fromdifferent surveys probing 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 3year Supernova Legacy Survey (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 elementsof the covariance matrix as error bars, and we have converted the SN luminosity distances to an
equivalent comoving angular diameter 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 factor; cosmological information resides in the relative distances as afunction of redshift. The normalization used here corresponds to a Hubble parameter h = 0.674.

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 owCDM model, with constant equation-of-state parameter w andnon-zero space curvature ΩK ≡ 1 − Ωtot . Green and gray contours show the combination of Planck
CMB data with SN or BAO data, respectively. Red contours combine CMB, BAO, and the fullshape (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 w0 wa CDM model,
which assumes a flat Universe and an evolving equation of state with w(a) = w0 + wa (1 − a).
For the BAO data points in Fig. 28.1 we have used the compilation in Table 3 (the “BAOonly” row) from the summary cosmology paper by the eBOSS collaboration [40], where one canfind references for the original data sources. The individual BAO measurements come (in order ofincreasing redshift) from the SDSS-II main galaxy sample and from luminous red galaxies, emission
line galaxies, and quasars mapped by BOSS and eBOSS. The two highest redshift points, both with
an effective z ≈ 2.3, come from the auto-correlation of the Lyman-α forest in high-redshift (z > 2)
quasars from BOSS and eBOSS and from the cross-correlation of the Lyman-α forest with thequasars themselves. The BAO measurements are converted to absolute distances using the soundhorizon scale rs = 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
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we have converted values of DV to DA,c , while for other redshifts we use the DA,c determinations
measured directly by the transverse BAO scale. The galaxy and Lyman-α forest analyses alsomeasure 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 Ωm = 0.315 and h = 0.674, the best-fit values from Planck(TT+TE+EE+lowE+lensing) assuming w = −1 and Ωtot = 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 ΛCDM cosmology. This consistency has held
steady or improved as the measurements themselves have improved over the past decade. Dottedand dashed curves in the lower panel of Fig. 28.1 show the effect of changing w by ±0.1 with allother parameters held fixed, which leads to significantly worse agreement with the data. However,such a single-parameter comparison does not capture the impact of parameter degeneracies or theability of complementary data sets to break them, and if one instead forced a match to CMB
data by changing h and Ωm when changing w then the predicted BAO distances would diverge
at z = 0 rather than converging there. The recent Pantheon+ compilation [43] comprises morethan 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 fitof Pantheon+ with CMB and BAO data remains fully compatible with a flat ΛCDM 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 constantvalue of w with non-zero space curvature (owCDM, left panel) or the evolving equation of state ofEq. (28.4) in a flat Universe (w0 wa CDM, 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 theconstraining power of the data, primarily because measurement of the Alcock-Paczynski effect on
sub-BAO scales helps to break the degeneracy between DA,c (z) and H(z). Blue contours show
constraints from the full combination of CMB, BAO+FS, and SN data. Supernovae provide finegrained 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 ΛCDM parameters (w = w0 = −1, ΩK = wa = 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
ΛCDM.
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 ± 0.033 using
Planck CMB data, BAO measurements from SDSS-II and BOSS/eBOSS, and SN measurementsfrom the Pantheon compilation. With the addition of eBOSS RSD data and DES weak lensing
and galaxy clustering measurements, they constrain a more flexible model (ow0 wa CDM) that allows
non-zero curvature and an evolving equation-of-state (Eq. 28.4), finding
wp = −1.020 ± 0.032(28.5)

at a pivot redshift zp = 0.29, a tight constraint on curvature,
1 − Ωtot = −0.0023 ± 0.0022 ,31st May, 2024

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28. Dark Energy

but only a loose constraint on the evolution parameter,
wa = −0.48+0.36−0.30 .

(28.7)

Obtaining a tight constraint on wp in such a flexible cosmological model requires measurementsthat 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 ΛCDM model fit to Planck CMB data alone predicts H0 = (67.4 ± 0.5) km s−1 Mpc−1(see Chapter 29 of this Review). This prediction and its error bar are sensitive to the assumptionsof constant dark energy and a flat Universe. However, by adding BAO and supernova data one
can construct an “inverse distance ladder” to measure H0 precisely, even with a general darkenergy model and free curvature [46]. Ref. [40] applies this approach to obtain H0 = (68.2 ±0.8) km s−1 Mpc−1 . As discussed in Sec. 25.3.1 of this Review, some recent measurements fromlow-redshift data yield higher values of H0 . For example, Ref. [47] uses a distance ladder based on
Cepheid calibration of Type Ia supernova luminosities to find H0 = (73.0 ± 1.0) km s−1 Mpc−1 , instrong tension with cosmologically inferred values. However, calibrations based on tip-of-the-redgiant-branch (TRGB) distances give lower H0 , consistent with the cosmological values [48].
The tension in H0 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 ΛCDM andinverse-distance-ladder values upward. Models with extra relativistic degrees of freedom or darkenergy 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 significantlywith 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 H0 measurements [49],though it still prefers H0 values below those of Ref. [47]. Numerous theory papers have examinedpossible physical solutions to the H0 tension, all of which involve significant modifications to theΛCDM 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 σ8 . Probes of low-redshift
structure yield constraints in the (σ8 , Ωm ) plane, which can be summarized in terms of the parameter
combination S8 ≡ σ8 (Ωm /0.3)0.5 . As discussed in earlier editions of this Review, many weak-lensing
and cluster studies to date yield S8 values lower than those predicted for Planck-normalized ΛCDM.
Fig. 28.3, taken from Ref. [50], illustrates the current state-of-play on S8 . (Similar figures withdifferent selections of data can be found in many recent papers and in the previous edition of thisReview.) The topmost point represents the predicted value and 1σ uncertainty, S8 = 0.830 ± 0.014,
based on a flat ΛCDM 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 S8 measured from weak lensing of the CMB, again assuming
flat ΛCDM. Although the background being lensed is the CMB, this is still a measurement of lowredshift clustering; for multipoles ` > 100 roughly half of the expected signal comes from structure
at z < 2 and nearly all from z < 10 [51].
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14

CMB: Planck CMB aniso.
CMB: Planck CMB aniso. (+Alens marg.)
CMB: WMAP+ACT CMB aniso.
CMBL: Planck CMB lensing + BAO
CMBL: SPT CMB lensing + BAO

CMBL: ACT CMB lensing + BAO
CMBL: ACT+Planck CMB lensing + BAO
WL: DES-Y3 galaxy lensing+clustering
WL: KiDS-1000 galaxy lensing+clustering
HSC-Y3 galaxy lensing (Fourier) + BAO
HSC-Y3 galaxy lensing (Real) + BAO
GC: eBOSS BAO+RSD
CX: SPT/Planck CMB lensing x DES
CX: Planck CMB lensing x DESI LRG
CX: Planck CMB lensing x unWISE
0.70

0.75
S8

0.80

0.85

8( m/0.3)0.5

0.90

Figure 28.3: Predicted values of S8 ≡ σ8 (Ωm /0.3)0.5 from a CMB-normalized ΛCDM model (green
points), compared to observational estimates from CMB lensing (red points), from optical weaklensing surveys (first four blue points), from galaxy clustering RSD (blue GC point), and fromcross-correlation of galaxies with CMB lensing maps (blue CX points). The predictions (green)
are extrapolated forward from primary CMB anisotropy measurements that measure structure atz ≈ 1100. CMB lensing (red) is most sensitive to structure at z ≈ 1 − 2, while the galaxy-based
data (blue) probe structure at z ≈ 0.2–0.8. Taken from Ref. [50].

The remaining points, in blue, represent galaxy-based measurements that probe structure atz ≈ 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 combinedwith 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 weaklensing 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 S8 (red points) are in good
agreement with predictions based on primary CMB anisotropy (green points), but S8 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 isonly 1–2σ, 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√ ± 0.009, where the√ S8 = 0.759
1σ error is simply the dispersion of the measurements divided by N − 1 = 5. With this estimate,the +3σ bound S8 = 0.786 lies just below the −3σ bound of the Planck prediction, S8 = 0.788.
However, several or all of these measurements could be affected by a common systematic error, suchas 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 ob31st May, 2024

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28. Dark Energy

servable, gives a much higher S8 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 ΛCDM prediction of S8 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 thehighest 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 thetension 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 alreadywell 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 analysesof multiple weak lensing data sets are now testing consistency and highlighting the influence ofanalysis 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 andaffords 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 someof the main systematics affecting optical weak lensing. DESI data will soon allow the first RSDbased measurements of structure growth at the 1–2% level, which will show whether the low matterclustering 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 ΛCDM predictions and low-redshift clustering, it should be measured
at high statistical significance in Euclid weak lensing data and, subsequently, in LSST and Romandata.

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 ∆z = 0.1 out to z ≈ 0.8, with larger but still interesting error
bars out to z ' 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 DA,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 rs . For DA,c (z), the final analyses of BOSS/eBOSS achieveprecision 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 ≈ 1.5, and 2.9% from the Lyman-α forest at z ≈ 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 usedcombinations of weak lensing and galaxy clustering to measure the parameter combination S8 =
σ8 (Ωm /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 possiblethat the systematics are underestimated. Our simple average of these measurements, treatingthem as independent and using their dispersion as an empirical estimate of uncertainty, yields anerror on the mean of 1.1%. The statistical power of these weak lensing surveys is concentrated at
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28. Dark Energyz ≈ 0.2–0.8, and they provide useful constraints at lower precision on redshift evolution over thisrange and on individual values of Ωm and σ8 . RSD measurements constrain the similar parameter
combination f (z)σ8 (z), but they do not yet have precision competitive with that of weak lensing
measurements. Distance-ladder estimates of H0 now span a small range, using overlapping data
but distinct treatments of key steps. Individual studies quote uncertainties as small as 1–2%, withsimilar statistical and systematic contributions.
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 ΛCDM
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 H0 implied by thismodel disagrees with recent Cepheid+SNIa distance-ladder measurements of H0 at a ∼ 5σ level.
This disagreement persists in models that allow non-zero curvature and low redshift evolution ofw(z), provided one assumes standard pre-recombination physics to compute the sound horizon rs .
The discrepancy could reflect underestimated systematic uncertainties in the Cepheid-based H0 estimate, or it could be a sign of new physics in the early Universe that rescales rs . 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 H0 -tension that invoke novel pre-recombination physics.
Extrapolating forward from the CMB anisotropies measured by Planck predicts a low redshift
value of S8 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%: S8 = 0.830 ± 0.014 versus S8 = 0.759 ± 0.009. Thisdisagreement 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 S8 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 andmore complete understanding of systematic uncertainties. Cross-correlation of galaxy maps withhigher signal-to-noise CMB lensing maps will provide independent clustering measurements thatavoid 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, witha factor of ∼ 10 increase expected by the end of the 5-year program. DESI will enable the firsthigh precision BAO measurements of expansion history at z ≈ 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-αforest samples. DESI will also enable the first percent-level measurements of structure growththrough RSD. The BAO measurements will complement increasingly precise measurements of therelative 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 effectand parameter measurements based on voids or higher order clustering statistics.
Euclid, LSST, and Roman provide another major leap in observational capabilities. LSST willbe the ultimate ground-based optical weak-lensing experiment, measuring several billion galaxy
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28. Dark Energy

shapes over 20,000 deg2 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 fromspace. Both missions plan large spectroscopic galaxy surveys, which will provide better samplingat 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 calibrationachievable with a space platform. The 2020s will also see dramatic advances in CMB lensing fromthe 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 experimentsthat have distinct challenges and complementary strengths, and with analysis teams that develop
ideas independently. The ensemble of experiments will produce cosmological constraints that aremuch 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 theΛCDM 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 comefrom 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 someastronomical 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 structureof the Universe.
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