ARNP 2023

# The Hubble Tension and Early Dark Energy

Marc Kamionkowski1 and Adam G. Riess1,2 1William H. Miller III Department of
Physics and Astronomy, Johns Hopkins University, 3400 N. Charles St.,
Baltimore, Maryland, 21218, USA  2Space Telescope Science Institute, 3700 San
Martin Dr., Baltimore, MD 21218, USA

###### Abstract

Over the past decade, the disparity between the value of the cosmic expansion
rate directly determined from measurements of distance and redshift or instead
from the standard $\Lambda$CDM cosmological model calibrated by measurements
from the early Universe, has grown to a level of significance requiring a
solution. Proposed systematic errors are not supported by the breadth of
available data (and “unknown errors” untestable by lack of definition). Simple
theoretical explanations for this “Hubble tension” that are consistent with
the majority of the data have been surprisingly hard to come by, but in recent
years, attention has focused increasingly on models that alter the early or
pre-recombination physics of $\Lambda$CDM as the most feasible. Here, we
describe the nature of this tension, emphasizing recent developments on the
observational side. We then explain why early-Universe solutions are currently
favored and the constraints that any such model must satisfy. We discuss one
workable example, early dark energy, and describe how it can be tested with
future measurements. Given an assortment of more extended recent reviews on
specific aspects of the problem, the discussion is intended to be fairly
general and understandable to a broad audience.

###### doi:

10.1146/????

###### keywords:

cosmology; early Universe; cosmic microwave background

††journal: Ann. Rev. Nucl. Part. Sci.

###### Contents

  1. 1 INTRODUCTION
  2. 2 OBSERVATIONS AND MEASUREMENTS
    1. 2.1 Defining the Hubble Constant
    2. 2.2 The Local Distance Ladder: Geometry to Cepheids to SN Ia
    3. 2.3 Other Local measurements
      1. 2.3.1 Tip of the Red Giant Branch (TRGB)
      2. 2.3.2 Gravitational waves
      3. 2.3.3 Mira variables
      4. 2.3.4 Surface Brightness Fluctuations (SBF)
      5. 2.3.5 Masers
    4. 2.4 Model-dependent local measurements
      1. 2.4.1 Strong gravitational lensing
      2. 2.4.2 Ages and Aging
  3. 3 EARLY-UNIVERSE MEASURES
    1. 3.1 The sound horizon, the cosmic microwave background, and large-scale structure
      1. 3.1.1 The early Universe
      2. 3.1.2 The sound horizon in the baryon-photon fluid and baryon acoustic oscillations
      3. 3.1.3 $H_{0}$ from acoustic oscillations in the CMB power spectrum
      4. 3.1.4 CMB results
      5. 3.1.5 Galaxy surveys and baryon acoustic oscillations
      6. 3.1.6 Distance scale of matter-radiation equality
  4. 4 THEORY AND MODELS
    1. 4.1 Early- versus late-time solutions
      1. 4.1.1 Late-time solutions
    2. 4.2 Early dark energy
      1. 4.2.1 Oscillating-field models
      2. 4.2.2 Slow-roll models
      3. 4.2.3 Generalized EDE
      4. 4.2.4 Specific implementations
    3. 4.3 Other EDE observables
      1. 4.3.1 Cosmic birefringence
      2. 4.3.2 Nonlinear evolution of oscillating EDE field
      3. 4.3.3 The light horizon
      4. 4.3.4 Recurrent cosmological-constant-like behavior?
    4. 4.4 Other early-Universe solutions
      1. 4.4.1 Modified gravity
      2. 4.4.2 Changing $N_{\rm eff}$
      3. 4.4.3 Changing physics of the baryon-photon fluid
    5. 4.5 Recent results from CMB/LSS data
  5. 5 CONCLUSIONS

##  1 INTRODUCTION

In his “Chronology of the History of Science and Discovery” [1], Isaac Asimov
identifies Hubble’s discovery of the cosmic expansion as one of the two
defining events of 20th-century science (the other being the discovery of the
structure of DNA). Interestingly enough, though, the value Hubble inferred for
the expansion rate (the Hubble constant)—the ratio of the recessional velocity
to distance for the galaxies he observed—turned out to be too high by an order
of magnitude, a rate providing less than 2 billion years for the Universe to
have grown to its present size, far smaller than the age of the Earth! This
first “Hubble tension”, defined here as any discrepancy between the locally
measured and cosmologically inferred expansion rate, was resolved with the
discovery of two generations of stars and a consequence of prior measurements
intermingling the two.

Determination of the Hubble constant has been a central aim of cosmology ever
since, with measurements differing by almost a factor of two as late as the
early 1990s before reaching a celebrated result of $H_{0}=72\pm 8$ km sec-1
Mpc-1 [2] a 10% state-of-the-art precision by the new millennium, through use
of the Hubble Space Telescope to resolve Cepheid variables in distant galaxies
(later recalibrated to $74\pm 2$ [3]). Coupled with the theoretical
expectation of $\Omega_{m}\sim 1$, the low expansion age implied by these
measurements, still a few Gyr younger than the oldest stars, set off another
“Hubble tension” until the discovery of cosmic acceleration amended the
composition and recent expansion history and the age of the Universe grew
comfortably higher.

Cosmology has, however, blossomed over the quarter century since then,
resembling high-energy physics experiments, with huge data sets, sophisticated
analyses, careful attention to systematic errors, and a successful standard
cosmological model ($\Lambda$CDM) with precisely determined parameters. Still,
two decades into the new century, we find ourselves yet again with a third
Hubble tension, smaller in scale than prior ones but highly significant, a
$\sim 8\%$ discrepancy with $\raisebox{-3.69899pt}{~{}\shortstack{$>$ \\\
[-0.07cm] $\sim$}}~{}5\sigma$ confidence. Each past Hubble tension has taught
us something more interesting than the value of a parameter and a new tension
provides an opportunity for discovery. Will this one (again) auger new
astrophysics or fundamental physics?

The lower value $H_{0}=67.4\pm 0.5$ km sec-1 Mpc-1 [4] of the Hubble constant
is anchored by measurements of angular temperature and polarization
fluctuations in the cosmic microwave background (CMB) which calibrate free
parameters in the $\Lambda$CDM cosmological model. Similar values are obtained
from spatial fluctuations in the galaxy distribution [5, 6] whose physical
scale is calibrated by the CMB, an approach called an “inverse distance
ladder.” These data map the statistical properties of the distribution of mass
to a spectrum of physical scales set by the $\Lambda$CDM cosmological model.
Chief of these scales is the “sound horizon,” the distance a primordial
fluctuation can travel at the sound speed in an expanding Universe before its
size is frozen when the Universe becomes transparent at $z\sim 1000$. The
Hubble constant is one of six parameters that are optimized to find agreement
between this model and the data. Absent the CMB, comparing primordial
deuterium abundance to BBN predicted by the cosmological model in the early
Universe provides similar results, leading to the useful summary that the most
precise but indirect measures of the Hubble constant derived from $\Lambda$CDM
as calibrated in the pre-recombination or “early” Universe give values in the
range of 67–68 km sec-1 Mpc-1.

A higher range of 70–75 km sec-1 Mpc-1 covers essentially all precise ($\leq$
5%), recent “late” Universe measurements of the Hubble constant determined
“locally” or directly—inferred by comparing (as Hubble did) the recessional
velocities and distances of galaxies. The leading approach in terms of
community investment of Hubble Space Telescope time, and most replicated,
yields $H_{0}=73.0\pm 1.0$ km sec-1 Mpc-1 [7, 8, 9], near the middle of the
range (see Figures 1 and 2). Here, the galaxy distance is inferred from the
apparent brightness of a “standard candle,” an astronomical source of fixed
luminosity or from the angular size of a “standard rod”, a source of
calculable length. While Hubble used Cepheid variables, supergiant stars whose
pulsation period correlates with their luminosity, as standard candles,
current local measurements use a variety of standard candles and rods, but
many rely on Type Ia supernovae, a class of thermonuclear supernovae (stars
with degenerate matter which approach the Chandrasekhar mass), to measure deep
into the Hubble flow. We will discuss uncertainties in the measurements in the
next Section.

Figure 1: Extended MCMC sampling of the posterior for $H_{0}$ to measure out
to the $5\sigma$ confidence level. The upper panel shows the probability
density for the baseline from SH0ES and from the Planck Collaboration et al.
(2020) chains. The bottom panel shows the log of the probability density to
improve the ability to see the tails.

The current Hubble tension has persisted while gaining in significance for
nearly a decade, making it hard to ignore. Well-posed proposals of systematic
errors in measurements have been tested and are not supported by the data
while non-specific suggestions of measurement “unknown unknowns” are
unsatisfactory and by definition, untestable. A viable explanation is of
interest not just to cosmology, but also to physics. The standard cosmological
model, assembled in recent decades, is remarkably successful, but works only
with ingredients that involve new physics beyond general relativity and the
$SU(3)\times SU(2)\times U(1)$ Standard Model of elementary-particle physics.
The Universe is observed on the largest distance scales to be quite smooth but
with small density fluctuations well described as a realization of a Gaussian
random field with a nearly scale-invariant power spectrum. The primordial
origin of these perturbations, which have large correlation lengths, requires
new physics (which can be very well described in the context of inflation)
beyond the Standard Model. The time evolution of the perturbations in the
baryons and in the photons visible to us as the cosmic microwave background
(CMB)—as well as dynamics of galaxies and galaxy clusters in the Universe
today—requires some form of collisionless dark matter which, again, requires
new physics. Moreover, evidence for some negative-pressure dark energy (e.g.,
a cosmological constant) comes from supernova measurements [10, 11] and the
detailed characterization of cosmological perturbations and also necessitates
new physics.

Figure 2: 68% confidence-level constraints on H0 from different cosmological
probes. From Ref. [12] (based on Refs. [49, 50]).

Still, once collisionless dark matter, a cosmological constant, and a nearly
scale-invariant spectrum of primordial perturbations are postulated, all of
the data we have on the statistical properties of perturbations in the early
and late Universe can be described by a model that is parameterized by (i) an
overall amplitude for the primordial perturbation power spectrum; (ii) a
power-law index for the power spectrum; (iii) a baryon density; (iv) a dark-
matter density; (v) the Hubble constant; and (vi) a “reionization optical
depth,” which quantifies the fraction of CMB photons that are primordial.

While the Hubble tension looks prima facie to be a breakdown of $\Lambda$CDM’s
ability to connect two ends of cosmic time, it does not yield to easily
anticipated, new-physics solutions. In considering the solution space it is
important to recognize that $\Lambda$CDM plays two distinct roles in the
model-dependent calculation of $H_{0}$. First the model is calibrated in its
pre-recombination form ($z\raisebox{-3.69899pt}{~{}\shortstack{$>$ \\\
[-0.07cm] $\sim$}}~{}1000$) by comparison to primordial measurements which
fixes its six free parameters. Second, $\Lambda$CDM predicts the expansion
history between $z\sim 1000$ and $z\sim 0$ which leads to a prediction for
$H_{0}$ with or without additional refinement of the model parameters which
may come from comparison to low-redshifts measures of the expansion history. A
late-time solution affecting this extrapolation is attractive but appears less
tractable. If the discrepancy were reversed—with a higher value for $H_{0}$
coming from the CMB—it could be easily attributed to the nature of dark energy
as quintessence-like [13], where the dark-energy density is slowly decreasing.
However, such an explanation would require, given the larger $H_{0}$ from
local measurements, that dark energy violate the dominant energy condition,
something like a relativistic notion of creating energy from the vacuum. Even
if we are willing to allow for such an exotic theoretical possibility, though,
such a model is disfavored by galaxy-clustering or high-redshift supernova
measurements which prefer a nearly constant $\Lambda$-like dark energy density
independent of the primordial measurements. Neither can the simplest dials on
existing dark-matter models be turned to solve the Hubble tension. Most
analyses are done assuming a flat Universe, but allowing for some nonzero
curvature actually drives the CMB-inferred $H_{0}$ even lower. Workable
solutions, and thus the ones we will focus upon, can be obtained by modifying
$\Lambda$CDM at early times and its early cosmic-expansion history. One
promising way is to postulate some sort of “early dark energy” (EDE) [14, 15]
that behaves like a cosmological constant before matter-radiation equality but
then decays away faster than radiation afterwards. Recent CMB measurements,
since EDE was proposed, have improved sensitivity to polarization fluctuations
on small angular scales, and may even favor such models over $\Lambda$CDM.
Still, we must await future CMB measurements and galaxy surveys to disprove
either model.

A vigorous campaign to develop microphysical EDE models is now under way—at
the time of submission of this article, there were on order one new EDE model
appearing on arXiv every week. Most of them repurpose ideas explored earlier
in connection with dark energy and/or inflation, but there are some novelties
associated with the coincidence between the time that EDE becomes dynamical
and the epoch of matter-radiation equality. The machinery to produce precise
model predictions is moreover available, and so are the tools to make detailed
comparisons with data sets. As a result, model building proceeds hand in hand
with careful comparisons with ever improving data. Some of the models suggest
entirely new experimental/observational consequences of new EDE physics, which
will hopefully bring new avenues to understanding the Hubble tension. There
are also a slew of independent new techniques for local measurements that will
further test the results of supernova measurements.

Here we will review the Hubble tension and early dark energy at a colloquium
level. There have been far too many developments in measurements and theory in
this subject for us to review them in detail. We choose, therefore, in our
discussion of measurements to focus primarily on recent developments, current
questions, and future prospects. The theory discussion will emphasize
fundamental issues and model ingredients, with relevant calculations presented
schematically. We will then describe the new-physics ingredients for a handful
of EDE models.

Fortunately, there are excellent recent and broad reviews on various aspects
of the Hubble tension [16, 17, 18, 19, 20, 21] and the various types of models
that have been invoked to explain it [19, 22]. Some recent reviews on (late-
time) dark energy [23, 13, 24] also cover some overlapping issues in theory
and cosmological-parameter determination.

##  2 OBSERVATIONS AND MEASUREMENTS

###  2.1 Defining the Hubble Constant

The Hubble constant $H_{0}$ is defined as the constant of proportionality in
the relation $cz=H_{0}D$ between distance $D$ and redshift $z$ in the limit
$z\rightarrow 0$. For measurements that necessarily involve sources at $z>0$,
the linear relation is generalized to

$\displaystyle D={cz\over H_{0}}\Bigg{\\{}1-\left[1+{q_{0}\over
2}\right]{z}+\left[1+q_{0}+{q_{0}^{2}\over 2}-{j_{0}\over
6}\right]z^{2}+O(z^{3})\Bigg{\\}},$

which follows from a Taylor expansion of the scale factor

$\displaystyle a(t)=a_{0}\;\Bigg{\\{}1+H_{0}\;(t-t_{0})-{1\over
2}\;q_{0}\;H_{0}^{2}\;(t-t_{0})^{2}+{1\over
3!}\;j_{0}\;H_{0}^{3}\;(t-t_{0})^{3}+O([t-t_{0}]^{4})\Bigg{\\}}.$

with $H(t)=+(da/dt)/a$ the expansion rate at time $t$;
$q(t)=-(d^{2}a/dt^{2})\left[H(t)\right]^{-2}/a$ the deceleration parameter;
and $j(t)=(d^{3}a/dt^{3})\left[H(t)\right]^{-3}/a$ the jerk parameter, and so
on. For simplicity these relations are defined without curvature but can be
generalized with curvature. The Hubble “constant” is then the expansion rate
$H_{0}$=$H(t_{0})$ today (time $t_{0}$). We can determine $H_{0}$ (and
$q_{0}$, $j_{0}$, etc.) from measurements of distances and redshifts directly
from this definition and independent of the cosmological model.

Redshifts are easily measured from the change in wavelength of observed atomic
transitions (usually emission lines of galaxies) as compared to experimental,
laboratory values. Relative (also called uncalibrated or scale-free) distance
measurements at $D>100$ Mpc, where the $\sim 200-300\,{\rm km/sec}$ “peculiar”
velocities (random velocities relative to the Hubble flow) become negligible,
distances are readily obtained from the brightness of Type Ia supernovae which
provide for determination of $q_{0}$ and $j_{0}$, while $H_{0}$ drops out of
the calculation [25]. The parameters $q_{0}$ and $j_{0}$ can also be
constrained from other cosmological data. In either case, the uncertainties in
local determinations of $H_{0}$ from uncertainties in $q_{0}$ and $j_{0}$ are
irrelevantly small; a change $\Delta q_{0}$ changes $H_{0}$ by $O(\Delta
q_{0}$) in km sec-1 Mpc-1. Measurement of $H_{0}$ then requires absolute
distance measurements; these must ultimately be calibrated by geometry and are
harder to come by.

###  2.2 The Local Distance Ladder: Geometry to Cepheids to SN Ia

For reasons related to the homogeneity and luminosity of different classes of
astronomical objects, the most widely supported route for measuring $H_{0}$
has been to construct a 3-step distance ladder using geometry to calibrate
Cepheid variables followed by SN Ia. Here we make reference to the specific
implementation and most recent iteration of this approach by the SH0ES Team
[9].

[] MpcMegaparsec, $3\times 10^{24}$ cm, roughly, the typical spacing between
galaxies SN IaType IA supernova, a SN from thermonuclear detonation of a white
dwarf when it exceeds the Chandrasekhar mass CepheidA Yellow Supergiant star
pulsating in the fundamental (radial) mode whose light curve period strongly
correlates to its mass and luminosity SH0ESSupernovae and H0 for the Equation
of State HSTHubble Space Telescope WFC3Wide Field Camera 3 UVIS/IRRefers to
the two channels on WFC3: UV/visible (200 to 1000 nm) and infrared (800 to
1700 nm)

Cepheids have been favored as primary distance indicators for more than a
century because they are very luminous (100,000 solar luminosities), extremely
precise (3% in distance per source [26]), easy to identify due to their
periodicity (since Leavitt 1912), and well understood (since Eddington 1917).
They are massive, pulsating supergiant stars, overshooting hydrostatic
equilibrium due to the $\kappa$ temperature-dependent opacity mechanism [27].
There is a tight coupling between the period of the pulsation (weeks to
months), the mass, and the luminosity of these stars, with the latter inferred
empirically from the former from Cepheids at a common distance. They are also
the most consistently calibrated standard candle, an important issue for
reduction of errors, thanks to use of a single, stable telescope and
instrument, HST WFC3 UVIS/IR used for all measurements in local SN Ia hosts
and in three geometric calibrators of Cepheid luminosities: the megamaser host
NGC4258 [28], Milky Way parallaxes from the ESA Gaia mission [29], and the LMC
[26] (via detached eclipsing binaries) with a precision of 1.5%, 1.0% and
1.2%, respectively. Milky Way parallaxes from the ESA Gaia mission in
particular and through successive iterations, have become the best source of
geometric distance measurements, moving the calibration of Cepheids ahead of
other stellar distance indicators and even allowing for additional, self-
calibration of ESA Gaia parallax errors [30]. Near-infrared (NIR) observations
of Cepheids are used to mitigate the impact of uncertainties related to dust
which reduces systematic uncertainties relative to past, optical-only data.
Just recently, a serendipitous early James Webb Space Telescope (JWST)
observation of NGC 1365, a nearby galaxy on the Cepheid-supernova calibration
path allowed a measurement of the near-infrared Cepheid period-luminosity
relation, with JWST’s improved resolution [31]. The results are consistent
with results from HST, but show that future JWST observations will be an
important tool in the increasingly precise characterization of Cepheids.

SN Ia are rarer than Cepheids (one per galaxy per century vs hundreds per
galaxy at any time), hence none have been near enough for a parallax
measurement in 400 years, but they are far more luminous (billions of solar
luminosities). Standardizeable to 6% in distance per source, they have no
rival in their ability to witness cosmic expansion. In the past, the
uncertainty in $H_{0}$ was limited by the rarity of SN Ia whose hosts were in
range of Cepheids (at $D\leq 40$ Mpc, about one per year) but better
instruments on HST and persistence has produced a complete sample of forty-two
well-observed, prototypical SN Ia from the last four decades (the era of
digital photometry). (The redshifts of these nearby SN Ia hosts do not enter
the calculation of $H_{0}$). Great efforts have been made to calibrate and
standardize these consistently with the thousands of SN Ia in the Hubble flow
[25, 32] (typically at $0.02<z<0.15$), including the modeling of data
covariance. The result is a measurement from the SH0ES and Pantheon+ data of
$H_{0}=73.04\pm 1.04$ km sec-1 Mpc-1 including systematic uncertainties; see
Figure 3.

This result has passed a wide range of null tests and has been replicated from
the published Cepheid photometry [33, 34] and the Cepheid photometry has been
replicated independently [35]. Sixty-seven variants of the baseline analysis,
see Figure 4, demonstrate it is difficult to move the central value below
$\sim$ 72.5 or above $\sim$ 73.5 (see Ref. [30] for discussion of
uncertainties). The relation between Cepheid metallicity and luminosity, a
past source of uncertainty, has been well-calibrated [36], and due to the
breadth of anchors, has little effect on $H_{0}$ in any case. Well-posed,
experimental challenges to these measurements have been extensively studied
and are certain to continue but the present evidence does not support a
significant challenge to the conclusions of a highly significant tension. The
comparison to Planck with $\Lambda$CDM yields the strongest evidence for the
Hubble tension at 5$\sigma$, or 5.3$\sigma$ including new calibrations from
Gaia clusters [37].

[] SBFsSurface-brightness fluctuations TRGBTip of the red-giant branch
AGBAsymptotic-giant branch (like red giants, but having burned their helium
cores to carbon or heavier elements) SN-IIType II supernova, a core-collapse
SN distinguished by hydrogen in its spectrum CCHPChicago-Carnegie Hubble
Project EDDExtragalactic Distance Database

Nevertheless, it is important to test the individual rungs of this ladder
which can be done with other independent distant indicators, e.g., SBFs, TRGB,
SN II, Miras, etc. (see figure 2), which presents a composite view of a large
sample of local-measurements results). For reference, to calibrate the nearest
SN Ia, a distance indicator needs to reach galaxies at $D>10$ Mpc. To well-
calibrate the Hubble flow, a distance indicator needs to reach $D\sim 100$
Mpc, so that $O(1000\,{\rm km}~{}{\rm sec}^{-1})$ peculiar velocities are
small compared to the Hubble-flow velocity. To be calibrated by parallax the
distance indicator needs to be present in the Milky Way.

Figure 3: The cosmic distance ladder used by SH0ES to infer $H_{0}$. The
luminosities of nearby Cepheid variables are calibrated to parallaxes.
Supernova luminosities are then calibrated to Cepheid luminosities at larger
distances. The Hubble constant is then inferred from the brightnesses of more
distant supernovae. Figure 4: Values of $H_{0}$ inferred by SH0ES under
different assumptions and analyses. The $\pm 1\sigma$ vertical dotted lines
indicate the statistical error in the baseline result shown at the bottom.
$H_{0}$ is given in units of km sec-1 Mpc-1. Figure 5: This Figure, from Wu et
al 2022, illustrates the steps involved in the inference of distances from the
tip of the red giant branch and some of the metrics used to judge the quality
of the result. The tip contrast $R$ is the most important of these.

###  2.3 Other Local measurements

####  2.3.1 Tip of the Red Giant Branch (TRGB)

When a main-sequence star has burned all the hydrogen in its core to helium,
the star continues to burn hydrogen in a shell around the core, thus slowly
increasing the helium-core mass. During this time, the star becomes cooler but
brighter, a process that continues until the mass, and thus temperature, of
the helium core becomes large enough (about 0.5 solar masses) to ignite helium
burning to carbon. The luminosity of the I-band (in the infrared) is seen to
be insensitive to the star’s metallicity and mass, and so they can be used as
a standard candle. Since the TRGB luminosity is 10 times fainter than long-
period Cepheids, it is more limited in distance to $D<20$ Mpc. It has,
however, been used in lieu of Cepheid variables to calibrate nearby Type Ia
supernovae.

In practice the location of the tip in the color-magnitude diagram of stars
often appears “fuzzy” due to the presence of AGB stars which have the same
color as RGB stars but are both brighter and fainter, reducing the contrast
and blurring the location of the tip; see Figure 5 for an illustration of the
TRGB measurement process. Varying degrees of contamination of the old, metal-
poor halo tip by new star formation or crowding may be partially mitigated by
careful selection of regions to analyze [38] but leave some ambiguity about
the location of the tip which is field dependent. Techniques to measure the
tip include Sobel edge detection, parameteric luminosity function fitting, and
maximum likelihood methods. The TRGB, like Cepheids, are not generally used to
measure $H_{0}$ directly, but contribute to such a measurement by calibrating
longer range indicators.

A direct comparison of Cepheids and TRGB and distance measures can be made in
seven SN Ia hosts (both deriving calibration from the same geometric distance
measure in NGC 4258) and these yield agreement to better than 2% in the mean
[30]. However, literature differences in the determination of $H_{0}$
involving TRGB are seen to arise from the other two rungs; the (first)
calibration rung and a (third) SN Ia rung. There are several measures of
$H_{0}$ that use TRGB including the Chicago-Carnegie Hubble Project (CCHP)
[39] ($70\pm 2$), the Extragalactic Distance Database EDD [40] ($71.5\pm 2.0$)
which connect to SN Ia, and a measure that calibrates surface brightness
fluctuations with TRGB [41] ($73\pm 3$). The main source of differences in the
CCHP and EDD TRGB studies come from differences in the determination of the
TRGB in NGC 4258 which differ by 1.5 km sec-1 Mpc-1 (in $H_{0}$). These groups
use different fields in NGC 4258 and more work may elucidate how these fields
compare to the fields used to measure TRGB around SN Ia hosts. Parallaxes from
Gaia of field stars [42] or the nearby, massive globular cluster, $\Omega$
Centauri [43] may also aid TRGB calibration (see Figure 6 in [42] for a
summary of recent TRGB calibrations). The lower value of $H_{0}$ from the CCHP
study also partially sources from differences in its measurement of the Hubble
flow from SN Ia; accounting for local peculiar motions [44] and cross-
calibration of SN samples [45] would raise that $H_{0}$ by $\sim$ 1.5 km sec-1
Mpc-1 (or likewise we find removing these steps in the SH0ES Team measurement
lowers that measurement to 71.8 km sec-1 Mpc-1). These are small differences,
important to understand to reach a 1% uncertainty. However, as seen in [12]
Figure 2, all local measures at this precision level are above the 67.4 $\pm
0.5$ km sec-1 Mpc-1 expectation (Planck) while none of the local measures are
in tension with each other (at $>$ 1.5 $\sigma$). Even the lowest value of
these measures is above the 4$\sigma$ confidence level of Planck.

####  2.3.2 Gravitational waves

The gravitational-wave (GW) signal from a merging neutron-star binary can be
used to determine the Hubble parameter [46]. The spindown frequency and GW
amplitude can be used to determine the distance to the source, as long as the
inclination of the binary is known. The inclination can in principle be
constrained with knowledge of the GW polarization. Even without polarization,
the distribution of inclination angles can be averaged over in multiple
events. This measurement requires, though, a redshift obtained from an
electromagnetic counterpart.

This technique was implemented with the detection [47, 48, 49] of a
gravitational-wave signal consistent with a neutron-star–neutron-star or
neutron-star–black hole merger at a distance $\sim 40$ Mpc, as well as
detection of an associated soft gamma-ray burst and a slew of subsequent
electromagnetic observations. This event, GW170817, then provided a
measurement $H_{0}=74^{+16}_{-8}$ km sec-1 Mpc-1 ($68\%$ CL) of the Hubble
parameter [50], but the associated uncertainties—dominated by that for the
inclination angle—were too large to be of value. Subsequent detection and
modeling of a late-time radio jet from the merger then constrained the
inclination angle and thus improved the measurement to
$H_{0}=70.3^{+5.3}_{-5.0}$ km sec-1 Mpc-1. The error reduction is, however,
accompanied possibly by radio-jet modeling uncertainties. In addition, this
event at $D\sim 40$ Mpc ($z\sim 0.01$) was too close to provide a clean
measure of the Hubble flow with peculiar velocities producing an 8%
uncertainty. Given the expected increase over the next decade in detection
range, as well as improved characterization from multiple observatories, a
robust $\raisebox{-3.69899pt}{~{}\shortstack{$<$ \\\ [-0.07cm] $\sim$}}~{}2\%$
GW measurement of the Hubble parameter may be possible in the coming decade.

It may also be possible to constrain $H_{0}$ from GW events without optical
counterparts. The idea of this “statistical-siren” or “dark-siren” approach is
that if the origin of the GW signal can be localized on the sky, then a
catalog of candidate host galaxies can be obtained. Although $H_{0}$ cannot be
determined from this method, the redshift distribution of the candidate hosts
translates to a probability distribution for $H_{0}$ [51]. By combining
results from a large number of such events, the possible values of $H_{0}$ can
be narrowed. The LIGO/Virgo and Dark Energy Survey (DES) Collaborations
applied this technique in Ref. [52] to GW170814, a particularly bright and
reasonably well localized event, although the $H_{0}$ constraints from this
initial foray are not yet significant.

####  2.3.3 Mira variables

Mira variables are pulsating, low-mass, intermediate age (AGB) stars with
great luminosity, comparable to Cepheids in the near-infrared. The oxygen-rich
subtype has a simple, linear period-luminosity relation. However, these
objects are more challenging to find and measure than Cepheids because their
periods are far longer, hundreds to a thousand days, and there are
contaminating stars in the same period range (Carbon-rich Miras and long-
period-variables) which must be distinguished using light-curve amplitudes and
colors. First attempts to use Miras to supplant Cepheids or TRGB are promising
[53], but the sample size is small.

####  2.3.4 Surface Brightness Fluctuations (SBF)

This technique determines distances to galaxies by the surface-brightness
fluctuations that arise from the finite number of stars in the galaxy [54].
The electromagnetic energy flux $F$ from a given galaxy is related to its
distance $D$ by $F\propto D^{2}$, and the surface brightness is $\propto
D^{-2}$. If the galaxy is resolved, then there will be pixel-to-pixel
variations in the surface brightness, induced by the finite number of stars
contributing to the flux in any given pixel, with root-variance $\propto
D^{-1}$. This technique yields distances with precision approaching SN Ia but
cannot reach the same distances. Nevertheless, they offer an important
alternative to SN Ia and a recent, state-of-the-art study of SBF using HST in
the NIR from [41, 55] yields similar results as SN Ia, with $H_{0}\sim 73.3\pm
2.5$ with the same result whether calibrated by Cepheids or TRGB.

####  2.3.5 Masers

Water masers in Keplerian motion around supermassive black holes in the
centers of galaxies can be observed in the radio using VLBI. By tracking
proper motions and accelerations, a purely geometric distance can be measured
to the maser host. However, such objects are rare due to the requirement of
edge-on alignment of the inner accretion disk with our line of sight coupled
with the need for an optimal density profile of the disk. The Maser Cosmology
Project (MCP) has measured 6 such systems in the Hubble flow reported by Ref.
[28] which yields $H_{0}=74\pm 3$, independent of any previously reviewed
rungs. While this approach is limited in precision due to the small samples
and limited resolution of the galaxy nucleus, future observations of these
same maser hosts with the Event Horizon Telescope [56] could yield a dramatic
improvement and are highly anticipated.

###  2.4 Model-dependent local measurements

We now review several additional techniques to determine the Hubble constant
that we refer to as “model-dependent” because they require additional
modeling: For gravitational lensing, it is a model for the lens mass; for
ages/aging, it is a model for stellar evolution.

####  2.4.1 Strong gravitational lensing

If a given time-varying cosmological source is multiply lensed by a massive
foreground object, there will be time delays between the light curves observed
in the different images. These time delays depend on the angular-diameter
distances to the lens and source, thus providing a route to determination of
$H_{0}$ [57]. Quasars have proved to be ideal targets for this measurement as
they are bright, time-variable, and long-lived [58, 59, 60, 61]. In 2019, the
H0liCOW Collaboration reported $H_{0}=73.3^{+1.7}_{-1.8}$ km sec-1 Mpc-1from
an analysis of 6 strongly lensed quasars with time delays [62]. One issue in
the measurement is the mass-sheet degeneracy [63]: a flat mass distribution in
the plane along the line of sight can add to the time delay without affecting
the image locations and brightnesses. In practice, the mass distribution of
the lens must be constrained with dynamical constraints to the lens mass, thus
introducing new uncertain astrophysics into the measurement. As an
illustration of the possible impact, a subsequent analysis of the H0liCOW
quasars (with a seventh added) relaxing the assumption of a conventional
galaxy mass profile for the lenses (NFW or power-law) in one of two different
ways obtained values consistent with either end of the $H_{0}$ discrepancy and
with larger error bars [64]. So if lenses share the same mass profiles as
local, well-studied elliptical galaxies, lensing conforms with the other local
values, and if it does not, the way in which it does not becomes the leading
source of uncertainty. The strong-lensing determination of $H_{0}$ is now
being advanced by the TDCOSMO Collaboration [65] with new systems, new
analysis pipelines, and careful attention to identification and mitigation of
systematic effects and astrophysical uncertainties.

Figure 6: The $1\sigma$ allowed parameter regions in triads corresponding to
the age of the Universe and the Hubble constant (upper left); the sound
horizon at radiation drag and the reduced Hubble constant (bottom left); and
the total matter density and the square of the reduced Hubble constant (bottom
right). Note that all points in each panel sum to 0, while the ticks in the
axes determine the direction of equal values for each axis. From Ref. [66].

[] $\Lambda$CDMThe standard cosmological model with dark energy taken to be a
cosmological constant wCDMThe cosmological model with dark energy with an
equation-of-state parameter $w$ LIGOLaser Interferometric Gravitational-Wave
Observatory VirgoThe European gravitational-wave observatory DESDark-Energy
Survey

####  2.4.2 Ages and Aging

The expansion rate $H(z)$ at any given redshift $z$ is inversely proportional
to the ratio of the time interval $\Delta t$ associated with some given
redshift interval $\Delta z$. Ref. [67] thus proposed using differential
stellar ages to determine the expansion history, and at low redshifts, the
Hubble parameter $H_{0}$. This type of measurement currently is most precise
at a redshift $z\simeq 0.45$ [68, 69]. This measurement is subject to stellar-
astrophysics uncertainties which have only recently been well-studied by Refs.
[70, 71]. Folding these in constrains $H_{0}$ to $67.8^{+8.7}_{-7.2}$ and
$66.5\pm 5.4$ km sec-1 Mpc-1, respectively, for a generic open wCDM and for a
flat $\Lambda$CDM cosmology, results which are not very constraining in the
present landscape.

The age of the Universe also provides a constraint to $H_{0}$. In a wCDM
cosmology with matter density $\Omega_{m}$ and dark-energy equation-of-state
parameter $w$, it is

$t_{0}=\frac{1}{H_{0}}\int_{0}^{\infty}\frac{dz}{(1+z)\sqrt{\Omega_{m}(1+z)^{3}+(1-\Omega_{m})^{-3(1+w)}}}.$
(1)

The oldest (observable) stars in the Universe are found in low-metallicity
(thus formed from nearly primordial gas) globular clusters in the Milky Way
halo. The stars in these globular clusters have nearly uniform metallicities
and exhibit a clear main sequence in their color-magnitude diagrams, thus
suggesting a uniform burst of star formation. The age is determined by fitting
main-sequence isochrones from stellar-evolution models to the observed color-
magnitude diagram, as the main sequence is the best understood phase of
stellar evolution. Recent parallaxes from Gaia have reduced the uncertainties
in the stellar luminosity and thus in the inferred ages. Using ages from 22
Milky Way globular clusters [72], Ref. [73] obtains a value $H_{0}=71.0\pm
2.7$ km sec-1 Mpc-1. This provides a distance-ladder independent measurement
and thus complements other $H_{0}$ determinations. The degeneracies with other
cosmological parameters also complement those of CMB and local measurements,
as shown in Figure 6 [66].

##  3 EARLY-UNIVERSE MEASURES

###  3.1 The sound horizon, the cosmic microwave background, and large-scale
structure

####  3.1.1 The early Universe

The density of the early Universe (“early” here means within the first $\sim
400,000$ years of the Universe, before the CMB photons last scattered) was the
same to $\raisebox{-3.69899pt}{~{}\shortstack{$<$ \\\ [-0.07cm]
$\sim$}}~{}10^{-5}$ everywhere. It consisted of photons, baryons ($\sim 75\%$
protons by weight, $\sim 25\%$ alpha particles, and electrons), all three
neutrino mass eigenstates, and dark matter. The cosmological constant (or
other form of dark energy) was dynamically insignificant. Efficient electron-
photon scattering implies that the photons and baryons comprised one tightly
coupled baryon-photon fluid. The neutrinos were non-interacting (from a few
seconds after the Big Bang) but had a thermal velocity distribution with a
temperature $\sim 0.7$ times the photon temperature. The dark matter is
assumed to be entirely collisionless, an assumption verified by increasingly
constraining null searches for dark-matter interactions with baryons, photons,
neutrinos, or itself.

The primordial Universe was also populated by adiabatic density perturbations
well described as a realization of a gaussian random field with power spectrum
$P(k)\propto k^{n_{s}}$ as a function of wavenumber $k$. Here, “adiabatic”
implies that the fractional density perturbation in each species was
equivalent; i.e., the baryon:photon:DM:neutrino ratio was the same everywhere.
The scalar spectral index $n_{s}$ is determined empirically to be $n_{s}\simeq
0.96$. The characteristics of this density field—i.e., $n_{s}$ is close to,
but not precisely equal to, unity; the adiabaticity; and gaussianity—are all
consistent with the simplest single-field slow-roll models of inflation. The
details of inflation (or even whether it ever occurred) are not relevant to
the Hubble tension—we can simply take the flat Universe and nearly scale-
invariant spectrum of adiabatic perturbations as empirical facts.

####  3.1.2 The sound horizon in the baryon-photon fluid and baryon acoustic
oscillations

Consider a Dirac-delta-function adiabatic overdensity of matter at some
particular point in an otherwise perfectly homogeneous early Universe. The
pressure in the baryon-photon fluid associated with this overdensity drives a
shock wave that expands at the sound speed $c_{s}$ of the baryon-photon fluid
(see Figure 10 in Ref. [24]). Since the energy density at these early times is
dominated by the photons, this sound speed $c_{s}$ is just a bit smaller than
$c/\sqrt{3}$, where $c$ is the speed of light. When photons and baryons
decouple, at a time $t_{\rm ls}\simeq 400,000$ years after the Big Bang (when
the plasma temperature falls to $T\raisebox{-3.69899pt}{~{}\shortstack{$<$ \\\
[-0.07cm] $\sim$}}~{}$eV allowing electrons to combine with nuclei), the
shock-induced overdensity in the baryon-photon fluid has a radius $\sim
c_{s}t_{\rm ls}$.

The solution to the fluid equations with this Dirac-delta-function initial
condition provides the Green’s function for the time evolution of primordial
perturbations. When it is convolved with the primordial mass distribution, a
realization of a random field, it provides the two-point correlation function
at some later time. Thus, the two-point correlation function for the baryon
density—and thus the total-matter density, given that baryons constitute $\sim
1/5$ of the total matter density—has a bump at a comoving distance given by
the sound horizon at matter-radiation equality. This bump shows up in the
galaxy autocorrelation function at a distance $\sim 150$ Mpc. The relatively
sharp feature in configuration space then gives rise to oscillatory structures
in the Fourier domain. These are the celebrated baryon acoustic oscillations
in the matter power spectrum.

####  3.1.3 $H_{0}$ from acoustic oscillations in the CMB power spectrum

These oscillations also appear as the acoustic peaks in the angular power
spectrum $C_{l}$ of the cosmic microwave background, since the photon density
traces the baryon density at the time, $\sim 400,000$ years after the Big
Bang, the photons are released. In 1995, Ref. [74] argued that measurement of
these acoustic peaks could be used to determine the Hubble constant, along
with the values of other cosmological parameters, by comparing theoretical
calculations of $C_{l}$ with measurements. The way that the Hubble constant
comes out of this black box can be understood heuristically, however.

The multipole moment $l_{s}$ of the first acoustic peak determines the angle
subtended by the sound horizon at the surface of last scatter, given the
correspondence $\ell_{s}\simeq 2/\theta_{s}$ between the angular variation
$\theta$ of a spherical harmonic of multipole $l$. The angle subtended by the
sound horizon is $\theta_{s}=r_{s}/D_{A}$, where $D_{A}$ is the angular-
diameter distance to the CMB surface of last scatter, and $r_{s}\sim
c_{s}t_{\rm dec}$ is the sound horizon. The parameter $\theta_{s}=(1.04109\pm
0.00030)\times 10^{-2}$ is the most precisely determined parameter extracted
from CMB measurements, determined to roughly one part in $10^{4}$.

More precisely, the sound horizon is obtained by integrating the sound speed
$c_{s}(t)$ over time from the Big Bang to recombination. The comoving sound
horizon can be represented by an integral,

$r_{s}=\int_{z_{\rm
ls}}^{\infty}\frac{c_{s}(z)\,dz}{H(z)}=\frac{c}{\sqrt{3}H_{\rm
ls}}\int_{z_{\rm ls}}^{\infty}\frac{dz}{\left[\rho(z)/\rho(z_{\rm
ls})\right]^{1/2}\left(1+R\right)^{1/2}},$ (2)

over redshift $z$. Here, $z_{\rm ls}\simeq 1080$ is the redshift at which CMB
photons last scatter, $c_{s}(z)=c\left[3(1+R)\right]^{-1/2}$ is the sound
speed of the photon-baryon fluid, with
$R=(3/4)(\omega_{b}/\omega_{\gamma})/(1+z)$, and $\rho(z)$ is the total energy
density at redshift $z$. Here, $\omega_{b}=\Omega_{b}h^{2}$ is the current
physical baryon density (today), where $h\equiv H_{0}/(100\,{\rm km}\,{\rm
sec}^{-1}\,{\rm Mpc}^{-1})$ is a dimensionless Hubble constant. This
$\omega_{b}$ is determined by the higher-peak structure in the CMB power
spectrum far more precisely than $\Omega_{b}$ or $H_{0}$ separately. Planck’s
$\Lambda$CDM value is $\omega_{b}=0.0224\pm 0.0001$. At last scattering $R\sim
0.5$ and it is smaller at higher redshifts. And $\omega_{\gamma}=2.47\times
10^{-5}$ is the physical photon energy density [75]. The expansion rate at
last scattering is

$H_{\rm ls}=100\,{\rm km}\,{\rm sec}^{-1}\,{\rm
Mpc}^{-1}\,\omega_{r}^{1/2}(1+z_{\rm
ls})^{2}\sqrt{1+\frac{\omega_{m}}{\omega_{r}}\frac{1}{1+z_{\rm ls}}},$ (3)

where $\omega_{m}=\Omega_{m}h^{2}$ is the physical nonrelativistic-matter
density today—this, again, is fixed fairly precisely by the higher-peak
structure in the CMB; Planck’s $\Lambda$CDM value is $\omega_{m}=0.142\pm
0.001$. In the standard cosmological model, the early-Universe energy density
is $\rho(z)\propto\omega_{m}(1+z)^{3}+\omega_{r}(1+z)^{4}$. The physical
radiation density is

$\omega_{r}=\left[1+\frac{7}{8}N_{\rm
eff}\left(\frac{4}{11}\right)^{4/3}\right]\omega_{\gamma},$ (4)

where the second term accounts for additional nonrelativistic degrees of
freedom. In the standard cosmological model, these include the three neutrino
mass eigenstates, and $N_{\rm eff}=3.06$ differs slightly from 3 because of
the details of neutrino decoupling [76].

The (comoving) angular-diameter distance to the surface of last scatter is
then an integral,

$D_{A}=\frac{c}{H_{0}}\int_{0}^{z_{\rm
ls}}\frac{dz}{\left[\rho(z)/\rho_{0}\right]^{1/2}},$ (5)

from recombination until the current time $t_{0}$ when the total energy
density is $\rho_{0}$. The denominator here is
$\rho(z)/\rho_{0}=\Omega_{m}(1+z)^{3}+(1-\Omega_{m})(1+z)^{-3(1+w)}$ in the
standard cosmological model, with a dark-energy equation-of-state parameter
$w$. The cosmological constant corresponds to $w=-1$.

From $\theta_{s}=r_{s}/D_{A}$, we infer a Hubble constant,

$H_{0}=\sqrt{3}H_{\rm ls}\theta_{s}\frac{\int_{0}^{z_{\rm
ls}}dz\,\left[\rho(z)/\rho_{0}\right]^{-1/2}}{\int_{z_{\rm
ls}}^{\infty}dz\,\left[\rho(z)/\rho(z_{\rm
ls})\right]^{-1/2}\left(1+R\right)^{-1/2}},$ (6)

from the CMB. This is a function of $\omega_{b}$ through its appearance in
$R$. There is a dependence on $\omega_{m}$ through its appearance in
$\rho(z)/\rho(z_{\rm ls})$ at early times and in $H_{\rm ls}$. There is also a
dependence on $\omega_{m}$ through the appearance of
$\Omega_{m}=\omega_{m}/h^{2}$ in $\rho(z)$ at late times. Thus, strictly
speaking, equation 6 is an implicit equation for $H_{0}$, given that
$h=H_{0}/(100{\rm km}~{}{\rm sec}^{-1}~{}{\rm Mpc}^{-1})$ appears and in the
right-hand side. There is a dependence on $\omega_{\gamma}$ in the $R$ and in
the expression for $\omega_{r}$, which also depends on $N_{\rm eff}$. The
dependence on $\omega_{r}$ comes about in $H_{\rm ls}$ and in $\rho(z)$ at
early times. The redshift $z_{\rm ls}\simeq 1080$ of the last-scattering
surface corresponds to the time when the rate for a photon to Thomson scatter
from free electrons—which are becoming scarce as they become bound into
hydrogen atoms—becomes smaller than the expansion rate. There is some
dependence of $z_{\rm ls}$ on $\omega_{b}$ and $\omega_{m}$ that is taken into
account in detailed analyses but is too small to be relevant for the Hubble
tension.

In practice, all of the unknown cosmological parameters are determined
simultaneously by fitting precise numerical calculations of CMB power spectra
to data. Still, $\omega_{m}$ and $\omega_{b}$ are determined primarily by
characteristics in the CMB power spectrum such as the Silk damping at higher
$l$ and the relative heights of the even- and odd-numbered peaks. The Hubble
constant then follows from equation 6. Through numerical differentiation of
this expression, it can be found that the Hubble constant varies as $(\Delta
H_{/}H_{0})\simeq 0.1\,(\Delta\omega_{b}/\omega_{b})$ for small changes
$\Delta\omega_{b}$ to the baryon density (holding all other parameters fixed)
and with $\omega_{m}$ as $(\Delta
H_{/}H_{0})\simeq-0.77\,(\Delta\omega_{m}/\omega_{m})$ (keeping
$\Omega_{m}=\omega_{m}/h^{2}$). This equation also illustrates how some simple
modifications to the standard assumptions might affect the results. For
example, if the number $N_{\rm eff}$ of relativistic degrees of freedom is
increased, then the radiation density $\omega_{r}$ is accordingly increased
leading to a higher $H_{\rm ls}$ and thus a higher $H_{0}$. Alternatively, if
we take $w<-1$, then the integral in the numerator of equation 6 is increased,
thus leading to a higher $H_{0}$.

####  3.1.4 CMB results

The first effort to determine $H_{0}$ from the CMB was in 2000 [77], but the
results were not constraining because these initial measurements lacked enough
information about the higher peaks in the CMB power spectrum to fix
$\omega_{m}$ and $\omega_{b}$. This was accomplished with NASA’s Wilkinson
Anisotropy Probe (WMAP), which arrived at $H_{0}=69.3\pm 0.8$ km sec-1 Mpc-1
[78] for their final mission value for the Hubble parameter (improving upon
their first-year result, $H_{0}=73\pm 5$ km sec-1 Mpc-1 [79]).111Incidentally,
WMAP’s $\sim 1\%$ measurement of $H_{0}$ improved upon the $\sim 10\%$
forecast in Ref. [74] because WMAP’s capabilities turned out to be better than
anticipated in that work, but also because the acoustic-peak amplitudes turned
out to be higher than expected with 1995 best-fit cosmological parameters.
Subsequently, the European Space Agency’s Planck satellite [4] provided power
spectra to multipole moments $\ell\sim 2500$, as opposed to $\ell\sim 800$
from WMAP, finding $H_{0}=67.4\pm 0.5$ km sec-1 Mpc-1. These measurements have
then been complemented at even smaller angular scales by the ACT and SPT
Collaborations, which arrive at similar values of $H_{0}$ with  $<$ [-0.07cm]
$\sim$ km sec-1 Mpc-1 errors [80, 81].

####  3.1.5 Galaxy surveys and baryon acoustic oscillations

The sound horizon appears as a bump in the galaxy autocorrelation function at
a distance scale $\sim 150$ Mpc. In a galaxy-redshift survey, galaxy locations
are parameterized by their position on the sky and by their redshift $z$, a
proxy for the line-of-sight distance in the limit that peculiar velocities can
be neglected. A pair of galaxies at similar redshift and some fixed angular
separation have a physical separation proportional to the angular-diameter
distance $D_{A}(z)$, which is inversely proportional to $H_{0}$ and has a
dependence on $\Omega_{m}$; cf., equation 2.1. A pair of galaxies along a
given line of sight separated in redshift by $\Delta z$ have a physical
separation inversely proportional to the expansion rate
$H(z)=H_{0}\sqrt{\Omega_{m}(1+z)^{3}+(1-\Omega)}$. To provide some indication
of the state of the art, the transverse and radial BAO scales were measured in
BOSS to $\sim 1.6\%$ and $\sim 2.7\%$ in redshift bins of width $\Delta z\sim
0.25$ [82]. The degeneracy between $H_{0}$ and $\Omega_{m}$ in $H(z)$ or
$D_{A}(z)$ is different at high and low redshifts, and so can be broken by
combining BAO measurements at different redshifts (e.g., Figure 5 in Ref.
[5]), and the BAO measurements now span the range
$0.15\raisebox{-3.69899pt}{~{}\shortstack{$<$ \\\ [-0.07cm]
$\sim$}}~{}z\raisebox{-3.69899pt}{~{}\shortstack{$<$ \\\ [-0.07cm]
$\sim$}}~{}3$. Using the sound horizon inferred either from the CMB or from
the value obtained by fixing the baryon density from big-bang nucleosynthesis
then allows a determination of the Hubble parameter with a similar error. In
practice, galaxy-survey analyses typically add to this “pure-BAO” measurement
information from the correlation-function shape and its time evolution, and
then combine with constraints to cosmological parameters from the CMB, weak
gravitational lensing of galaxies or the CMB, or other measurements (see,
e.g., Figure 20 in Ref. [6] for a comparison of the constraints derived under
various assumptions). Currently such BAO+ measurements provide (assuming a
sound horizon determined from the CMB) $H_{0}$ values consistent with the CMB
value and with errors  $<$ [-0.07cm] $\sim$ km sec-1 Mpc-1 [6, 5].

####  3.1.6 Distance scale of matter-radiation equality

The Hubble parameter can also constrained by the wavenumber $k_{\rm eq}$ of
matter-radiation equality obtained from galaxy surveys. The primordial linear-
theory matter power spectrum $P(k)$ transitions from its large-wavelength
($k\to 0$) behavior $P(k)\propto k^{n_{s}}$ (with $n_{s}\simeq 0.96$ the
scalar spectral index) to $P(k)\propto k^{n_{s}-3}$ at $k\to\infty$ at a
wavenumber $k_{\rm eq}$ corresponding to the mode that enters the horizon at
matter-radiation equality. Given the rough coincidence between this distance
scale and those corresponding to the sound horizon, the BAO wiggles in the
power spectrum must be modeled out. Once they have been subtracted, though,
the technique provides a sound-horizon–independent measurement of the Hubble
parameter. This is the idea behind the ShapeFit algorithm [83] which in a
preliminary application to BOSS data finds a low $H_{0}$. It is also the
approach in Ref. [84] which obtains a value $H_{0}=64.8^{+2.2}_{-2.5}$ km
sec-1 Mpc-1. However, this measurement assumes the standard $\Lambda$CDM power
spectrum. If, however, the early expansion history is changed in a manner
suggested by early dark energy, then, as discussed below, the Hubble parameter
inferred from this measurement is raised to a value consistent with the local
SH0ES measurement [85].

##  4 THEORY AND MODELS

###  4.1 Early- versus late-time solutions

Barring a combination of systematics that address multiple types of
observations, the Hubble tension implies some new physics beyond the
ingredients (collisionless dark matter, a cosmological constant, and Standard
Model interactions for baryons/photons) found in the standard cosmological
model. Given that local measurements of the Hubble constant are fairly
straightforward, the aim of most solutions to the Hubble tension is to
introduce new physics that increases the value of $H_{0}$ inferred from the
CMB.

These solutions are typically categorized as “late time” or “early time,” a
classification that can be understood from equation 5. Late-time solutions
postulate that the energy density in the post-recombination Universe is
smaller than in the standard model, holding the current density fixed: i.e.,
$\rho(z)/\rho_{0}(z)\leq\left[\rho(z)/\rho_{0}(z)\right]_{\rm standard}$; cf.
equation 6. This then increases the comoving distance to the surface of last
scatter and thus leads to a larger $H_{0}$. Early-time solutions postulate
that the energy density is somehow increased before recombination so that the
sound horizon at recombination is decreased. We will also discuss models that
decrease the sound horizon by changing the physics of the baryon-photon fluid.

####  4.1.1 Late-time solutions

Given the plethora of models and the continued inventiveness of theorists,
care should be taken in making blanket statements. Still, there are
theoretical reasons that make late-time solutions unpalatable and empirical
constraints that make them elusive. A late-time solution requires that the
energy density at times between decoupling and now is smaller than that in the
standard model, but keeping the energy density today fixed [86]. Given that
the scaling of the radiation and matter densities with redshift is known, this
requires some exotic matter whose energy density increases with time. This is
most easily accomplished by postulating that the cosmological constant is a
phantom field [87], a fluid with an equation-of-state parameter $w=p/\rho<-1$,
where $p$ and $\rho$ are here the dark-energy pressure and energy density.
This, however, implies a fluid that violates the strong energy condition;
i.e., it effectively creates energy out of nowhere. This seems strange, but is
this what the Hubble tension is telling us? Even if we are willing to accept a
violation of the strong energy condition, though, such models are difficult to
reconcile with the sound horizon seen in the galaxy correlation function [88,
89]. They are also difficult to reconcile with constraints to the equation-of-
state parameter $w$ inferred recently from SNe Ia at high redshifts [25].

###  4.2 Early dark energy

Figure 7: The evolution of the energy densities of radiation, nonrelativistic
matter (baryons and cold dark matter, and the cosmological constant as a
function of redshift (so time increases to the left, with the Big Bang far off
to the right and today off to the left.) Also shown is the energy density
postulated for early dark energy (EDE). The bottom panel shows the fractional
contribution of EDE to the total energy density. The EDE curves are
schematic—the key point is that it contributes $\sim 10\%$ a bit before
recombination but is otherwise dynamically unimportant. Figure courtesy T.
Karwal.

The basic idea behind early dark energy (EDE) is to postulate some exotic
fluid that contributes $\sim 10\%$ of the total energy density of the Universe
briefly before recombination and then has an energy density that decays faster
than radiation at late time, so that it leaves the late evolution of the
Universe unchanged. This increases $\rho(z)/\rho(z_{\rm ls})$ in the
denominator of equation 6, thus leading to a higher $H_{0}$. Although the
basic idea is simple, specific models are highly constrained by the very well
measured structure of the high-$\ell$ peaks in the CMB power spectra. Fourier
modes of the density field that correspond to the highest multipole moments
($\ell\sim 3000$) probed by current measurements entered the cosmological
horizon and became dynamical at a redshift $z\sim 10^{6}$, when the Universe
was only $\sim$yr old. The measured CMB power spectrum thus constrains the
expansion history to far earlier times that the time of last scattering.
Moreover, a fluid with a density that evolves with time implies, for a
relativistically-invariant theory, the possibility of spatial fluctuations in
the EDE energy density. This, along with the already complicated interplay
between baryon-photon acoustic waves, dark matter, neutrinos, and the
gravitational field, implies that any physical model for EDE will be highly
constrained.

Figure 8: Likelihood contours from Ref. [90] for $\Lambda$CDM and EDE in a
subset of the 9-dimensional parameter space. These are likelihoods for the
model given data from CMB and large-scale structure measurements as of late
2018. The 9 parameters include the 6 $\Lambda$CDM parameters
$(\Omega_{m},h,A_{s},\Omega_{b},n_{s},\tau)$ along with three EDE parameters:
the fractional contribution $f_{\rm ede}(a_{c})$ to the total cosmic density
at the scale factor $a_{c}$ (normalized to unity today) at which the this
fractional density peaks, and a third parameter which quantifies the sharpness
of the transition of EDE from its early-time constant-density behavior to its
later decay. Each panel shows the likelihood contours in a given two-
dimensional slice of the full parameter space, after marginalizing over the
other 7 parameters. At the top of each column is the likelihood for the given
parameter after marginalizing over all other parameters. The gray bands and
curves indicate the likelihoods for $H_{0}$ from local measurements The orange
contours show the standard $\Lambda$CDM results and indicate the preference of
CMB/LSS data for a low Hubble parameter. The other three sets of contours
describe EDE models in which the EDE decays with scale factor $a$ at late
times as $a^{-6n/(n+1)}$. It is seen that EDE expands the allowed parameter
space to regions that overlap the local values of $H_{0}$. Although it cannot
be inferred from the Figure, it can be checked that the regions of the EDE
parameter space that accommodate a larger $H_{0}$ do so with a
reduced-$\chi^{2}$ similar to that for the best-fit $\Lambda$CDM model.

Even so, it was found Ref. [15] that physical models of EDE could resolve the
Hubble tension. Figure 8 shows the results reported there for EDE models in
which the EDE density decays with scale factor $a$ at late times as
$a^{-6n/(n+1)}$. As can be seen from the $H_{0}$ likelihood distribution, the
addition of EDE allowed (as of late 2018) for larger values of $H_{0}$, that
overlap the range of values allowed by local measurements, to be inferred from
CMB/LSS data. Moreover, the best-fit EDE model had a similar
reduced-$\chi^{2}$ as the best-fit $\Lambda$CDM model. The results shown here
are for oscillating-field and slow-roll models for EDE, both of which can be
described within the required precisions with the generalized-dark-matter
formalism.

####  4.2.1 Oscillating-field models

Oscillating scalar fields underly many of the EDE models that have been
explored. Here, a scalar field $\phi$ is postulated with a confining potential
[91],

$V(\phi)=\Lambda_{\rm ede}^{4}\left[1-\cos(\phi/f_{\rm ede})\right]^{n},$ (7)

with an energy scale $\Lambda_{\rm ede}\sim$eV so that the energy density is
in the ballpark of that of the Universe before recombination. Although this
potential was postulated [91] in an ad hoc fashion, Ref. [92] argued that it
may arise in string theory. If the initial field value is $\phi/f\sim\pi$,
then the field is initially frozen and behaves gravitationally like a
cosmological constant. Later, though, the field begins to oscillate about its
minimum with an equation-of-state parameter $w_{\rm ede}=(n-1)/(n+1)$ and an
energy density that decays with scale factor $a$ as $\rho_{\rm ede}\propto
a^{-6n/(n+1)}$ [93, 94]. Thus, for $n>1$, the EDE energy density decays faster
than that of radiation, as needed. Another similar possibility is to simply
take the potential to be $V(\phi)\propto\phi^{2n}$ [95]. Oscillating-field
models are parameterized (for fixed $n$) by the energy density $\Lambda_{\rm
ede}$, the decay constant $f_{\rm ede}$, and the initial field value
$\phi_{i}$. The power-law index $n$ can be considered as a fourth parameter.

####  4.2.2 Slow-roll models

Another possibility [15, 96] is a scalar field with a smooth “slow-roll”
potential like those considered for inflation or quintessence. For example, a
potential that asymptotes to a power law $\phi^{n}$ for large positive $\phi$
but then asymptotes to zero for large negative $\phi$ will have an energy
density that is constant at early times and then scales as $\rho\propto
a^{-6}$ at late times. Ref. [96] discusses characteristics of EDE potentials
and also generalize to models with nontrivial kinetic terms.

####  4.2.3 Generalized EDE

EDE models can also be approached in a slightly more model-independent way
using the generalized dark-matter (GDM) approach [97]. Here, it is noted that
communication between the dark sector (dark matter, dark energy, EDE) and the
gravitational fields occurs in the Einstein equations only through the stress
tensor which can be parameterized in terms of a fluid density, pressure, sound
speed, and viscosity parameter. With this approach, the EDE can be described
as a fluid with these parameters. The approach can be beneficial as codes that
modify the standard cosmological Boltzmann codes [98, 99, 100, 101] to include
rapidly oscillating fields are numerically challenging, given the wide
separation between the oscillation and expansion time scales [102]. Ref. [15]
showed how the parameters of the oscillating-field model can be mapped to GDM
parameters, and Ref. [103] then mapped EDE-like physics in terms of a GDM-like
parameterization.

####  4.2.4 Specific implementations

One of the curious aspects of EDE models is the coincidence between the time
at which EDE becomes dynamical and the epoch of matter-radiation equality. It
may be possible to connect these in chameleon models [104] and possibly in
some of the alternative-gravity models reviewed below. This epoch also
coincides with the time at which the heavier neutrino mass eigenstate(s)
become(s) non-relativistic, a realization capitalized upon in Ref. [105].

Refs. [106, 107] show that the friction required for slow-roll models may be
induced by coupling an axion field to a thermal bath in non-abelian theories.
The idea of interacting thermal baths also plays a role in the scenario or
Ref. [108]. Here, interactions of strongly-coupled radiation are mediated by a
force carrier that becomes nonrelativistic at a temperature $T\sim$eV. The
mediator then deposits its entropy into the lighter species thus providing a
step in the effective number $N_{\rm eff}$ of relativistic degrees of freedom.
This provides a higher energy density just before recombination while avoiding
the problems at high $l$ in the CMB power spectrum associated with an
increased $N_{\rm eff}$.

###  4.3 Other EDE observables

If the Hubble tension is solved by EDE, it is natural to inquire whether the
new physics associated with EDE has any other observable consequences beyond
the impact on the expansion history. Given the disparity in models, there is
no general consequence, but several possibilities have been discussed.

####  4.3.1 Cosmic birefringence

If a scalar field $\phi(x)$ is coupled to the electromagnetic field-strength
tensor $F_{\mu\nu}(x)$ through a term $[\phi(x)/m_{\rm
pl}]\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$ in the Lagrangian,
then time evolution of the scalar field yields a difference in the propagation
speeds for right- and left-circularly polarized electromagnetic waves, and
thus a frequency-independent rotation of the linear polarization of an
electromagnetic wave [109, 110, 111]. One possibility for such a time-evolving
scalar field is quintessence, a candidate for dark energy [111, 13]. Another
is a slowly rolling scalar field associated with EDE [112, 113]. In either
case, the rotation angle is $\Delta\phi/m_{\rm{Pl}}$, where $\Delta\phi$ is
the change in $\phi$ between the emission and observation of the EM wave
(although, strictly speaking, this result is altered if some of the rotation
occurs before recombination [113]). This “cosmic birefringence” (CB) leads to
parity-breaking EB and TB (where T, E, and B refer to the temperature and
parity-even and parity-odd polarization modes [114, 115]) correlations in the
CMB power spectra [116, 117]. The association of CB with EDE can be
distinguished from late-Universe CB-inducing physics by the absence of CB
observed in late-Universe probes like those that seek CB in the CMB
reionization bump [118] or in kinetic-Sunyaev Zeldovich tomography [119, 120].

####  4.3.2 Nonlinear evolution of oscillating EDE field

If EDE is due to an oscillating field, nonlinear evolution of the scalar-field
perturbations may lead to strong instabilities that then generate nonlinear
spatially-inhomogeneous dynamics or soliton-like structures that then evolve
as a subdominant dark-matter component [121, 95]. There is also a possibility
that fluctuations in the initial field value may give rise to isocurvature
perturbations [121].

####  4.3.3 The light horizon

Changes to the early expansion history will affect the light horizon at
decoupling probed by the “acoustic” peaks in the CMB B-mode power spectrum
[122]. These B modes are induced by inflationary gravitational waves, but
their amplitude depends on the energy scale of inflation, which is still
undetermined. Observation of these peaks is conceivable with a ground-based
Stage-IV CMB experiment if the B-mode amplitude is near the current upper
limit. A space-based experiment, like PICO, may be required, though, if it
smaller by an order of magnitude, and the measurement cannot be done if the
gravitational-wave amplitude is any smaller.

####  4.3.4 Recurrent cosmological-constant-like behavior?

If the Hubble tension is due to EDE, it also suggests—in combination with
evidence for accelerated expansion today [11, 10] and for inflation in the
early Universe—the possibility of recurrent periods of cosmological-
constant–like behavior throughout the history of the Universe. This
possibility was anticipated in work on tracking-field models [123, 124],
quintessence models in which the potential is such that the energy density is
always a (roughly) fixed fraction of the energy density of the dominant matter
component (radiation or matter). If, however, there are small wiggles added to
this potential, then the energy density in the quintessence field can
occasionally jump and behave briefly like a cosmological constant before
decaying away. The possibility was also anticipated in a string-axiverse
scenario [91]. Here, in each logarithmic time interval in the Universe’s
history, there is an axion-like field, with potential as in equation 7, that
becomes dynamical. The initial field value is chosen from random, and if it is
sufficiently displaced from the minimum, it can briefly behave dynamically
like a cosmological constant. This scenario also resembles assisted
quintessence, explored in connection with EDE in Ref. [125]. The idea of
recurring cosmological constants motivates the search for other times in
cosmic history where the expansion rate can be probed. Big-bang
nucleosynthesis constrains the expansion rate a few minutes after the big
bang. Ref. [126] discussed the possibility to probe the expansion history at
redshifts $z\sim 17$ with the global 21-cm intensity as measured, for example,
by EDGES [127]. Velocity acoustic oscillations—oscillations in the 21-cm
angular power spectrum induced by spatial modulation of star formation induced
by baryon–dark-matter relative velocities—may also probe the expansion history
at similar redshifts [128, 129, 130]. However, there is probably more that can
be done along these lines.

###  4.4 Other early-Universe solutions

####  4.4.1 Modified gravity

Modifications to gravity are notoriously difficult: Since modifications to the
Einstein-Hilbert action generically make the scalar degree of freedom (at
least, and sometimes also the vector degrees of freedom) in the metric
dynamical, there is no obvious way to perturb away from general relativity.
Nevertheless, the accelerated cosmic expansion, the persistent mystery of dark
matter, and now gravitational-wave measurements that probe previously
inaccessible regions of strong-field gravity have yielded an active
marketplace of alternative-gravity theories that can be explored in connection
with the Hubble tension [131, 132, 133, 134, 135, 136, 137].

For example, Ref. [131] took a phenomenological approach to modified gravity
in which the evolution of cosmological perturbations took on a parameterized
departure from those in general relativity. In relativistic cosmological
perturbation theory, there are two scalar potentials $\Phi$ and $\Psi$ (in
conformal Newtonian gauge) that generalize the gravitational potential in
Newtonian gravity. The equations that relate these to the energy-density
perturbation, pressure, and anisotropic stress are modified in two ways.
First, Newton’s constant $G$ in the Fourier-space equations is multiplied by a
function $\mu(a,k)$ of the scale factor $a$ and wavenumber $k$. And second,
the the cosmic slip $\Phi-\Psi$ (sourced by the anisotropic stress) is
replaced by a $\Phi-\gamma(a,k)\Psi$. GR is recovered in the limits $\gamma\to
1$ and $\mu\to 1$. A departure of $\mu$ from unity at early times affects the
evolution of the gravitational fields so that the phase of the acoustic
oscillations is shifted in a way that mimics a shift in the sound horizon.

In conformally coupled gravity, the Ricci scalar $R$ in the Einstein-Hilbert
action is multiplied by a function $(1+\xi\phi^{2}/m_{\rm Pl}^{2})$ of a
scalar field $\phi$ with $\xi=-1/6$, so that the scalar is conformally
coupled. The equation of motion for the scalar field is
$\ddot{\phi}+3H\dot{\phi}-\xi R\phi=0$. At early times, during radiation
domination, $H^{2}\gg R$, and the scalar field is frozen at its initial value.
Near the transition to matter domination, when $R$ approaches $H^{2}$, the
field then starts to roll to its minimum (leaving gravity as it is today) and
then has an energy density which decays away as $\rho_{\phi}\propto a^{-9/2}$
in the matter-dominated era [137].

####  4.4.2 Changing $N_{\rm eff}$

The idea to reduce the sound horizon by increasing the early-Universe
expansion rate preceded EDE models; it was noted that an increase in the
number $N_{\rm eff}$ of relativistic degrees of freedom would allow for a
larger Hubble constant [8], but this solution did not provide a good fit to
CMB data, given that an increased $N_{\rm eff}$ affects the higher-$l$ modes
in the CMB that probe redshifts up to $z\sim 10^{6}$. A fractional increase in
$N_{\rm eff}$ in the range of 0.2–0.4 is still allowed and could alleviate the
Hubble tension. Detection of an additional, sterile neutrino would reopen this
solution space.

####  4.4.3 Changing physics of the baryon-photon fluid

Models with interacting neutrinos have also been explored [138]. However, they
modify the sound horizon through dynamics of the perturbations rather than
increasing the expansion rate and thus should probably not be classified as
EDE models. They also run up against laboratory constraints to neutrino
properties [139]. Primordial magnetic fields have also been suggested as a
solution to the Hubble tension [140], but the viability of the idea awaits a
more detailed calculation of the evolution of perturbations.

###  4.5 Recent results from CMB/LSS data

There is now a large literature devoted to testing various EDE models with the
ever-increasing products of ongoing galaxy surveys and CMB experiments. The
results of Ref. [90] shown in Figure 8 have been reproduced, updated, and
expanded upon with different data sets. We do not review this work in detail
here, as the literature is large and the situation rapidly evolving with new
data sets.

Since the level at which EDE models differ from $\Lambda$CDM are at the $\sim
3\sigma$-ish level, small changes in analyses or model assumptions that one
might guess were “below the radar” can actually affect the conclusions. For
example, the galaxy-clustering constraint to $H_{0}$ changes by about 0.7 km
sec-1 Mpc-1 if the dark energy is assumed to be a cosmological constant or
described by a more general equation-of-state parameter [6]. Similar shifts
arise from different assumptions about still-undetermined neutrino masses.
Although these shifts are small, at the $1\sigma$ level, they can change a
result from the $>3\sigma$ threshold to a $<3\sigma$ result. At this level,
conclusions can also depend on the interpretation of the statistics. For
example, Refs. [141, 142, 143] argued that current data favored $\Lambda$CDM
over EDE at the $\raisebox{-3.69899pt}{~{}\shortstack{$>$ \\\ [-0.07cm]
$\sim$}}~{}3\sigma$ level, while others [144, 145] warn that the conclusions
may reflect the choice of priors.

Perhaps the most intriguing results at the time of writing are reported in
Refs. [146, 147]. They find that new measurements of small-scale polarization
from ACT Data Release 4 [80] favor EDE over $\Lambda$CDM at the
$\raisebox{-3.69899pt}{~{}\shortstack{$>$ \\\ [-0.07cm] $\sim$}}~{}3\sigma$
level. The results should be considered as provisional given some possible
inconsistencies between Planck and ACT measurements. Still, the results, based
on a fraction of current polarization data suggest that forthcoming
experiments, like the Simons Observatory and CMB-S4 (if not ACT/SPT data
obtained even sooner), should distinguish EDE from $\Lambda$CDM with high
statistical significance.

Perhaps the most important takeaway is that four years later, with several new
data sets (especially for small-scale CMB fluctuations, which were anticipated
[90] to provide the most stringent tests of EDE), EDE remains consistent and
even possibly favored over $\Lambda$CDM. We refer readers interested in a more
detailed discussion of the statistical techniques, tests, and data sets to
Ref. [22], and to Ref. [19] for a comprehensive tour of EDE models.

##  5 CONCLUSIONS

The discrepancy between local determinations of the cosmic expansion rate
based on distance and redshift measurements and the expansion rate inferred
from CMB data and galaxy clustering has over time become statistically more
significant with new data and simultaneously survived careful scrutiny of the
relevant measurements and analyses. This Hubble tension is not solved by any
quick fix to the standard cosmological model. If it were the other way (a
lower local $H_{0}$), it could be accommodated with the type of models for
(late-time) dark energy that have been considered for 20 years. Analogous
cosmological tensions in the past have yielded to new insights on stellar
populations (to explain Hubble’s anomalously large initial expansion rate) and
fundamental physics (the 1990s discrepancy between the Hubble constant and age
of the Universe, explained ultimately by the discovery of the cosmological
constant). Although it remains to be seen how the current Hubble tension will
be resolved, it is likely to provide profound new insights into astrophysics
or physics.

The most promising new-physics explanation for the Hubble tension is some new
early-Universe physics that decreases the sound horizon. The most popular
playground for such ideas has been early dark energy, which reduces the sound
horizon by increasing the expansion rate, but there are other models that
involve, for example, modified gravity or changes to the primordial-plasma
physics. Any such model, though, is highly constrained by the data—the model
must preserve the excellent agreement between disparate and precise data sets
and the canonical $\Lambda$CDM model. Still, there are models that work and,
moreover, remain viable even after comparison with several precise new data
sets. Unlike the last (1990s) Hubble tension, which was resolved ultimately by
one number—the cosmological constant—the specification of EDE models is more
complicated and thus not as easily digested by theorists. Still, it is up to
data to decide, not our prejudices.

Fortunately, the next steps in exploring the Hubble tension are clear.
Moreover, the required observational infrastructure is already in place, as it
coincides largely with that assembled to study (late-Universe) dark energy and
inflation. Ultimately, we must continue to explore astrophysical and
measurement uncertainties. As we have learned over and over in cosmology,
there is no single bullet—robust conclusions are only reached with multiple
observational avenues and a tightly knit web of calibrations, cross-
calibrations, and consistency checks.

[SUMMARY POINTS]

  1. 1.

The values of the Hubble constant inferred from CMB measurements and galaxy
surveys disagree at the $\raisebox{-3.69899pt}{~{}\shortstack{$>$ \\\
[-0.07cm] $\sim$}}~{}5\sigma$ level with those obtained from measurements of
distances and redshifts in the local Universe.

  2. 2.

The discrepancy has not yielded to any simple explanations in terms of
systematic effects, despite considerable scrutiny of the CMB and local
measurements.

  3. 3.

Of the many models with new physics explored to explain this Hubble tension,
those that involve modifications to early-Universe dynamics seem best able to
satisfy the panoply of cosmological constraints.

  4. 4.

Although there have been some hints of EDE in recent analyses, current CMB
data are not yet precise and robust enough to distinguish EDE models from the
standard $\Lambda$CDM model.

[FUTURE ISSUES]

  1. 1.

Improving local measurements are needed to refine the Hubble constant while
maintaining control of systematic errors with $\sim$1% a target goal and
independent tests at $\sim$ 3% precision providing valuable crosschecks.
Greater specificity is needed to describe any systematic errors that would
evade present detection and impact multiple, independent measures.

  2. 2.

Forthcoming experiments that map more precisely the CMB polarization on
smaller angular scales will be required to test EDE and other new-physics
models for the Hubble tension.

  3. 3.

If future measurements favor EDE models over the standard $\Lambda$CDM model,
it will be important to understand more deeply the nature of the new physics
that provides EDE-like dynamics in the early Universe and to explore other
times when DE may have affected dynamics of the Universe.

  4. 4.

It will also be important to think about laboratory, or other non-
cosmological, tests of any such new physics.

## DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or
financial holdings that might be perceived as affecting the objectivity of
this review.

## ACKNOWLEDGMENTS

We thank Foteini with help producing Figure 2 and Jiaxi Wu for help with
Figure 4. We thank Licia Verde with comments on an earlier draft. MK was
supported by NSF Grant No. 2112699 and the Simons Foundation.

## References

  * Asimov [1994] Isaac Asimov.  Asimov’s Chronology of Science & Discovery: Updated and Illustrate.  Harper-Collins, New York, 1994.
  * Freedman et al. [2001] W. L. Freedman et al.  Final results from the Hubble Space Telescope key project to measure the Hubble constant.  Astrophys. J., 553:47–72, 2001, astro-ph/0012376.
  * Freedman et al. [2012] Wendy L. Freedman, Barry F. Madore, Victoria Scowcroft, Chris Burns, Andy Monson, S. Eric Persson, Mark Seibert, and Jane Rigby.  Carnegie Hubble Program: A Mid-infrared Calibration of the Hubble Constant.  Astrophys. J., 758(1):24, October 2012, 1208.3281.
  * Aghanim et al. [2020] N. Aghanim et al.  Planck 2018 results. VI. Cosmological parameters.  Astron. Astrophys., 641:A6, 2020, 1807.06209.  [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * Alam et al. [2021] Shadab Alam et al.  Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological implications from two decades of spectroscopic surveys at the Apache Point Observatory.  Phys. Rev. D, 103(8):083533, 2021, 2007.08991.
  * Abbott et al. [2022] T. M. C. Abbott et al.  Dark Energy Survey Year 3 results: Cosmological constraints from galaxy clustering and weak lensing.  Phys. Rev. D, 105(2):023520, 2022, 2105.13549.
  * Riess et al. [2011] Adam G. Riess, Lucas Macri, Stefano Casertano, Hubert Lampeitl, Henry C. Ferguson, Alexei V. Filippenko, Saurabh W. Jha, Weidong Li, and Ryan Chornock.  A 3% Solution: Determination of the Hubble Constant with the Hubble Space Telescope and Wide Field Camera 3.  Astrophys. J., 730:119, 2011, 1103.2976.  [Erratum: Astrophys.J. 732, 129 (2011)].
  * Riess et al. [2016] Adam G. Riess et al.  A 2.4% Determination of the Local Value of the Hubble Constant.  Astrophys. J., 826(1):56, 2016, 1604.01424.
  * Riess et al. [2022a] Adam G. Riess et al.  A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/sec/MpcUncertainty from the Hubble Space Telescope and the SH0ES Team.  Astrophys. J. Lett., 934(1):L7, 2022a, 2112.04510.
  * Perlmutter et al. [1999] S. Perlmutter et al.  Measurements of $\Omega$ and $\Lambda$ from 42 high redshift supernovae.  Astrophys. J., 517:565–586, 1999, astro-ph/9812133.
  * Riess et al. [1998] Adam G. Riess et al.  Observational evidence from supernovae for an accelerating universe and a cosmological constant.  Astron. J., 116:1009–1038, 1998, astro-ph/9805201.
  * Abdalla et al. [2022] Elcio Abdalla et al.  Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies.  JHEAp, 34:49–211, 2022, 2203.06142.
  * Caldwell and Kamionkowski [2009] Robert R. Caldwell and Marc Kamionkowski.  The Physics of Cosmic Acceleration.  Ann. Rev. Nucl. Part. Sci., 59:397–429, 2009, 0903.0866.
  * Karwal and Kamionkowski [2016] Tanvi Karwal and Marc Kamionkowski.  Dark energy at early times, the Hubble parameter, and the string axiverse.  Phys. Rev. D, 94(10):103523, 2016, 1608.01309.
  * Poulin et al. [2018a] Vivian Poulin, Tristan L. Smith, Daniel Grin, Tanvi Karwal, and Marc Kamionkowski.  Cosmological implications of ultralight axionlike fields.  Phys. Rev. D, 98(8):083525, 2018a, 1806.10608.
  * Bernal et al. [2016] Jose Luis Bernal, Licia Verde, and Adam G. Riess.  The trouble with $H_{0}$.  JCAP, 10:019, 2016, 1607.05617.
  * Verde et al. [2019] L. Verde, T. Treu, and A. G. Riess.  Tensions between the Early and the Late Universe.  Nature Astron., 3:891, 7 2019, 1907.10625.
  * Knox and Millea [2020] Lloyd Knox and Marius Millea.  Hubble constant hunter’s guide.  Phys. Rev. D, 101(4):043533, 2020, 1908.03663.
  * Di Valentino et al. [2021] Eleonora Di Valentino, Olga Mena, Supriya Pan, Luca Visinelli, Weiqiang Yang, Alessandro Melchiorri, David F. Mota, Adam G. Riess, and Joseph Silk.  In the realm of the Hubble tension—a review of solutions.  Class. Quant. Grav., 38(15):153001, 2021, 2103.01183.
  * Shah et al. [2021] Paul Shah, Pablo Lemos, and Ofer Lahav.  A buyer’s guide to the Hubble constant.  Astron. Astrophys. Rev., 29(1):9, December 2021, 2109.01161.
  * Efstathiou [2021] George Efstathiou.  To H0 or not to H0?  Mon. Not. R. Astron. Soc., 505(3):3866–3872, August 2021, 2103.08723.
  * Schöneberg et al. [2022] Nils Schöneberg, Guillermo Franco Abellán, Andrea Pérez Sánchez, Samuel J. Witte, Vivian Poulin, and Julien Lesgourgues.  The H0 Olympics: A fair ranking of proposed models.  Phys. Rept., 984:1–55, 2022, 2107.10291.
  * Frieman et al. [2008] Joshua Frieman, Michael Turner, and Dragan Huterer.  Dark Energy and the Accelerating Universe.  Ann. Rev. Astron. Astrophys., 46:385–432, 2008, 0803.0982.
  * Weinberg et al. [2013] David H. Weinberg, Michael J. Mortonson, Daniel J. Eisenstein, Christopher Hirata, Adam G. Riess, and Eduardo Rozo.  Observational Probes of Cosmic Acceleration.  Phys. Rept., 530:87–255, 2013, 1201.2434.
  * Brout et al. [2022] Dillon Brout et al.  The Pantheon+ Analysis: Cosmological Constraints.  Astrophys. J., 938(2):110, 2022, 2202.04077.
  * Riess et al. [2019] Adam G. Riess, Stefano Casertano, Wenlong Yuan, Lucas M. Macri, and Dan Scolnic.  Large Magellanic Cloud Cepheid Standards Provide a 1% Foundation for the Determination of the Hubble Constant and Stronger Evidence for Physics beyond LambdaCDM.  Astrophys. J., 876(1):85, 2019, 1903.07603.
  * Christy [1966] R. F. Christy.  Pulsation Theory.  Ann. Rev. Astron. Astrophys., 4:353, January 1966.
  * Pesce et al. [2020] D. W. Pesce et al.  The Megamaser Cosmology Project. XIII. Combined Hubble constant constraints.  Astrophys. J. Lett., 891(1):L1, 2020, 2001.09213.
  * Riess et al. [2018] Adam G. Riess et al.  Milky Way Cepheid Standards for Measuring Cosmic Distances and Application to Gaia DR2: Implications for the Hubble Constant.  Astrophys. J., 861(2):126, 2018, 1804.10655.
  * Riess et al. [2021] Adam G. Riess, Stefano Casertano, Wenlong Yuan, J. Bradley Bowers, Lucas Macri, Joel C. Zinn, and Dan Scolnic.  Cosmic Distances Calibrated to 1% Precision with Gaia EDR3 Parallaxes and Hubble Space Telescope Photometry of 75 Milky Way Cepheids Confirm Tension with $\Lambda$CDM.  Astrophys. J. Lett., 908(1):L6, 2021, 2012.08534.
  * Yuan et al. [2022] Wenlong Yuan, Adam G. Riess, Stefano Casertano, and Lucas M. Macri.  A First Look at Cepheids in a SN Ia Host with JWST.  9 2022, 2209.09101.
  * Scolnic et al. [2018] D. M. Scolnic et al.  The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample.  Astrophys. J., 859(2):101, 2018, 1710.00845.
  * Burns et al. [2018] Christopher R. Burns et al.  The Carnegie Supernova Project: Absolute Calibration and the Hubble Constant.  Astrophys. J., 869(1):56, 2018, 1809.06381.
  * Feeney et al. [2018] Stephen M. Feeney, Daniel J. Mortlock, and Niccolò Dalmasso.  Clarifying the Hubble constant tension with a Bayesian hierarchical model of the local distance ladder.  Mon. Not. Roy. Astron. Soc., 476(3):3861–3882, 2018, 1707.00007.
  * Javanmardi et al. [2021] Behnam Javanmardi, Antoine Merand, Pierre Kervella, Louise Breuval, Alexandre Gallenne, Nicolas Nardetto, Wolfgang Gieren, Grzegorz Pietrzynski, Vincent Hocde, and Simon Borgniet.  Inspecting the Cepheid Distance Ladder: the Hubble Space Telescope Distance to the SN Ia Host Galaxy NGC 5584.  Astrophys. J., 911(1):12, 2021, 2102.12489.
  * Breuval et al. [2022] Louise Breuval, Adam G. Riess, and Pierre Kervella.  An Improved Calibration of the Wavelength Dependence of Metallicity on the Cepheid Leavitt law.  arXiv e-prints, page arXiv:2205.06280, May 2022, 2205.06280.
  * Riess et al. [2022b] Adam G. Riess, Louise Breuval, Wenlong Yuan, Stefano Casertano, Lucas M. ~Macri, J. Bradley Bowers, Dan Scolnic, Tristan Cantat-Gaudin, Richard I. Anderson, and Mauricio Cruz Reyes.  Cluster Cepheids with High Precision Gaia Parallaxes, Low Zero-point Uncertainties, and Hubble Space Telescope Photometry.  Astrophys. J., 938(1):36, 2022b, 2208.01045.
  * Hoyt et al. [2021] Taylor J. Hoyt, Rachael L. Beaton, Wendy L. Freedman, In Sung Jang, Myung Gyoon Lee, Barry F. Madore, Andrew J. Monson, Jillian R. Neeley, Jeffrey A. Rich, and Mark Seibert.  The Carnegie Chicago Hubble Program X: Tip of the Red Giant Branch Distances to NGC 5643 and NGC 1404.  Astrophys. J., 915(1):34, July 2021, 2101.12232.
  * Freedman et al. [2019] Wendy L. Freedman et al.  The Carnegie-Chicago Hubble Program. VIII. An Independent Determination of the Hubble Constant Based on the Tip of the Red Giant Branch.  7 2019, 1907.05922.
  * Anand et al. [2021] Gagandeep S. Anand, Luca Rizzi, R. Brent Tully, Edward J. Shaya, Igor D. Karachentsev, Dmitry I. Makarov, Lidia Makarova, Po-Feng Wu, Andrew E. Dolphin, and Ehsan Kourkchi.  The Extragalactic Distance Database: The Color-Magnitude Diagrams/Tip of the Red Giant Branch Distance Catalog.  Astron. J., 162(2):80, August 2021, 2104.02649.
  * Blakeslee et al. [2021] John P. Blakeslee, Joseph B. Jensen, Chung-Pei Ma, Peter A. Milne, and Jenny E. Greene.  The Hubble Constant from Infrared Surface Brightness Fluctuation Distances.  Astrophys. J., 911(1):65, April 2021, 2101.02221.
  * Li et al. [2022] Siyang Li, Stefano Casertano, and Adam G. Riess.  A Maximum Likelihood Calibration of the Tip of the Red Giant Branch Luminosity from High Latitude Field Giants using Gaia Early Data Release 3 Parallaxes.  arXiv e-prints, page arXiv:2202.11110, February 2022, 2202.11110.
  * Soltis et al. [2021] John Soltis, Stefano Casertano, and Adam G. Riess.  The Parallax of $\omega$ Centauri Measured from Gaia EDR3 and a Direct, Geometric Calibration of the Tip of the Red Giant Branch and the Hubble Constant.  Astrophys. J. Lett., 908(1):L5, February 2021, 2012.09196.
  * Peterson et al. [2021] Erik R. Peterson et al.  The Pantheon+ Analysis: Evaluating Peculiar Velocity Corrections in Cosmological Analyses with Nearby Type Ia Supernovae.  10 2021, 2110.03487.
  * Brownsberger et al. [2021] Sasha Brownsberger, Dillon Brout, Daniel Scolnic, Christopher W. Stubbs, and Adam G. Riess.  The Pantheon+ Analysis: Dependence of Cosmological Constraints on Photometric-Zeropoint Uncertainties of Supernova Surveys.  10 2021, 2110.03486.
  * Schutz [1986] Bernard F. Schutz.  Determining the Hubble Constant from Gravitational Wave Observations.  Nature, 323:310–311, 1986.
  * Abbott et al. [2017a] B. P. Abbott et al.  GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral.  Phys. Rev. Lett., 119(16):161101, 2017a, 1710.05832.
  * Abbott et al. [2017b] B. P. Abbott et al.  Multi-messenger Observations of a Binary Neutron Star Merger.  Astrophys. J. Lett., 848(2):L12, 2017b, 1710.05833.
  * Abbott et al. [2017c] B. P. Abbott et al.  Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A.  Astrophys. J. Lett., 848(2):L13, 2017c, 1710.05834.
  * Abbott et al. [2017d] B. P. Abbott et al.  A gravitational-wave standard siren measurement of the Hubble constant.  Nature, 551(7678):85–88, 2017d, 1710.05835.
  * Chen et al. [2018] Hsin-Yu Chen, Maya Fishbach, and Daniel E. Holz.  A two per cent Hubble constant measurement from standard sirens within five years.  Nature, 562(7728):545–547, 2018, 1712.06531.
  * Soares-Santos et al. [2019] M. Soares-Santos et al.  First Measurement of the Hubble Constant from a Dark Standard Siren using the Dark Energy Survey Galaxies and the LIGO/Virgo Binary–Black-hole Merger GW170814.  Astrophys. J. Lett., 876(1):L7, 2019, 1901.01540.
  * Huang et al. [2019] Caroline D. Huang, Adam G. Riess, Wenlong Yuan, Lucas M. Macri, Nadia L. Zakamska, Stefano Casertano, Patricia A. Whitelock, Samantha L. Hoffmann, Alexei V. Filippenko, and Daniel Scolnic.  Hubble Space Telescope Observations of Mira Variables in the Type Ia Supernova Host NGC 1559: An Alternative Candle to Measure the Hubble Constant.  8 2019, 1908.10883.
  * Tonry and Schneider [1988] John Tonry and Donald P. Schneider.  A New Technique for Measuring Extragalactic Distances.  Astron. J., 96:807, September 1988.
  * Garnavich et al. [2022] Peter Garnavich, Charlotte M. Wood, Peter Milne, Joseph B. Jensen, John P. Blakeslee, Peter J. Brown, Daniel Scolnic, Benjamin Rose, and Dillon Brout.  Connecting Infrared Surface Brightness Fluctuation Distances to Type Ia Supernova Hosts: Testing the Top Rung of the Distance Ladder.  arXiv e-prints, page arXiv:2204.12060, April 2022, 2204.12060.
  * [56] https://eventhorizontelescope.org.
  * Refsdal [1964] S. Refsdal.  On the possibility of determining Hubble’s parameter and the masses of galaxies from the gravitational lens effect.  Mon. Not. Roy. Astron. Soc., 128:307, 1964.
  * Vanderriest et al. [1989] C. Vanderriest, J. Schneider, G. Herpe, M. Chevreton, M. Moles, and G. Wlerick.  The value of the time delay delta T (A,B) for the ’double’ quasar 0957+561 from optical photometric monitoring.  Astron. Astrophys., 215:1–13, May 1989.
  * Keeton and Kochanek [1997] C. R. Keeton and C. S. Kochanek.  Determining the Hubble constant from the gravitational lens PG-1115+080.  Astrophys. J., 487:42, 1997, astro-ph/9611216.
  * Schechter et al. [1997] Paul L. Schechter et al.  The Quadruple gravitational lens PG1115+080: Time delays and models.  Astrophys. J. Lett., 475:L85–L88, 1997, astro-ph/9611051.
  * Koopmans et al. [2003] L. V. E. Koopmans, T. Treu, C. D. Fassnacht, R. D. Blandford, and G. Surpi.  The Hubble Constant from the gravitational lens B1608+656.  Astrophys. J., 599:70–85, 2003, astro-ph/0306216.
  * Wong et al. [2020] Kenneth C. Wong et al.  H0LiCOW – XIII. A 2.4 per cent measurement of H0 from lensed quasars: 5.3$\sigma$ tension between early- and late-Universe probes.  Mon. Not. Roy. Astron. Soc., 498(1):1420–1439, 2020, 1907.04869.
  * Falco et al. [1985] E. E. Falco, M. V. Gorenstein, and I. I. Shapiro.  On model-dependent bounds on H 0 from gravitational images : application to Q 0957+561 A, B.  Astrophys. J. Lett., 289:L1–L4, February 1985.
  * Birrer et al. [2020] S. Birrer et al.  TDCOSMO - IV. Hierarchical time-delay cosmography – joint inference of the Hubble constant and galaxy density profiles.  Astron. Astrophys., 643:A165, 2020, 2007.02941.
  * Millon et al. [2020] M. Millon et al.  TDCOSMO. I. An exploration of systematic uncertainties in the inference of $H_{0}$ from time-delay cosmography.  Astron. Astrophys., 639:A101, 2020, 1912.08027.
  * Bernal et al. [2021] José Luis Bernal, Licia Verde, Raul Jimenez, Marc Kamionkowski, David Valcin, and Benjamin D. Wandelt.  The trouble beyond $H_{0}$ and the new cosmic triangles.  Phys. Rev. D, 103(10):103533, 2021, 2102.05066.
  * Jimenez and Loeb [2002] Raul Jimenez and Abraham Loeb.  Constraining cosmological parameters based on relative galaxy ages.  Astrophys. J., 573:37–42, 2002, astro-ph/0106145.
  * Stern et al. [2010] Daniel Stern, Raul Jimenez, Licia Verde, Marc Kamionkowski, and S. Adam Stanford.  Cosmic Chronometers: Constraining the Equation of State of Dark Energy. I: H(z) Measurements.  JCAP, 02:008, 2010, 0907.3149.
  * Moresco et al. [2012] M. Moresco et al.  Improved constraints on the expansion rate of the Universe up to z~1.1 from the spectroscopic evolution of cosmic chronometers.  JCAP, 08:006, 2012, 1201.3609.
  * Moresco et al. [2020] Michele Moresco, Raul Jimenez, Licia Verde, Andrea Cimatti, and Lucia Pozzetti.  Setting the Stage for Cosmic Chronometers. II. Impact of Stellar Population Synthesis Models Systematics and Full Covariance Matrix.  Astrophys. J., 898(1):82, 2020, 2003.07362.
  * Borghi et al. [2022] Nicola Borghi, Michele Moresco, Andrea Cimatti, Alexandre Huchet, Salvatore Quai, and Lucia Pozzetti.  Toward a Better Understanding of Cosmic Chronometers: Stellar Population Properties of Passive Galaxies at Intermediate Redshift.  Astrophys. J., 927(2):164, 2022, 2106.14894.
  * O’Malley et al. [2017] Erin M. O’Malley, Christina Gilligan, and Brian Chaboyer.  Absolute Ages and Distances of 22 GCs Using Monte Carlo Main-sequence Fitting.  Astrophys. J., 838(2):162, April 2017, 1703.01915.
  * Jimenez et al. [2019] Raul Jimenez, Andrea Cimatti, Licia Verde, Michele Moresco, and Benjamin Wandelt.  The local and distant Universe: stellar ages and $H_{0}$.  JCAP, 03:043, 2019, 1902.07081.
  * Jungman et al. [1996] Gerard Jungman, Marc Kamionkowski, Arthur Kosowsky, and David N. Spergel.  Cosmological parameter determination with microwave background maps.  Phys. Rev. D, 54:1332–1344, 1996, astro-ph/9512139.
  * Fixsen et al. [1996] D. J. Fixsen, E. S. Cheng, J. M. Gales, John C. Mather, R. A. Shafer, and E. L. Wright.  The Cosmic Microwave Background spectrum from the full COBE FIRAS data set.  Astrophys. J., 473:576, 1996, astro-ph/9605054.
  * Dodelson and Turner [1992] Scott Dodelson and Michael S. Turner.  Nonequilibrium neutrino statistical mechanics in the expanding universe.  Phys. Rev. D, 46:3372–3387, 1992.
  * Jaffe et al. [2001] Andrew H. Jaffe et al.  Cosmology from MAXIMA-1, BOOMERANG and COBE / DMR CMB observations.  Phys. Rev. Lett., 86:3475–3479, 2001, astro-ph/0007333.
  * Bennett et al. [2013] C. L. Bennett et al.  Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results.  Astrophys. J. Suppl., 208:20, 2013, 1212.5225.
  * Spergel et al. [2003] D. N. Spergel et al.  First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Determination of cosmological parameters.  Astrophys. J. Suppl., 148:175–194, 2003, astro-ph/0302209.
  * Aiola et al. [2020] Simone Aiola et al.  The Atacama Cosmology Telescope: DR4 Maps and Cosmological Parameters.  JCAP, 12:047, 2020, 2007.07288.
  * Dutcher et al. [2021] D. Dutcher et al.  Measurements of the E-mode polarization and temperature-E-mode correlation of the CMB from SPT-3G 2018 data.  Phys. Rev. D, 104(2):022003, 2021, 2101.01684.
  * Ross et al. [2017] Ashley J. Ross et al.  The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: Observational systematics and baryon acoustic oscillations in the correlation function.  Mon. Not. Roy. Astron. Soc., 464(1):1168–1191, 2017, 1607.03145.
  * Brieden et al. [2021] Samuel Brieden, Héctor Gil-Marín, and Licia Verde.  Model-independent versus model-dependent interpretation of the SDSS-III BOSS power spectrum: Bridging the divide.  Phys. Rev. D, 104(12):L121301, 2021, 2106.11931.
  * Philcox et al. [2022] Oliver H. E. Philcox, Gerrit S. Farren, Blake D. Sherwin, Eric J. Baxter, and Dillon J. Brout.  Determining the Hubble Constant without the Sound Horizon: A $3.6\%$ Constraint on $H_{0}$ from Galaxy Surveys, CMB Lensing and Supernovae.  4 2022, 2204.02984.
  * Smith et al. [2022] Tristan L. Smith, Vivian Poulin, and Théo Simon.  Assessing the robustness of sound horizon-free determinations of the Hubble constant.  8 2022, 2208.12992.
  * Poulin et al. [2018b] Vivian Poulin, Kimberly K. Boddy, Simeon Bird, and Marc Kamionkowski.  Implications of an extended dark energy cosmology with massive neutrinos for cosmological tensions.  Phys. Rev. D, 97(12):123504, 2018b, 1803.02474.
  * Caldwell [2002] R. R. Caldwell.  A Phantom menace?  Phys. Lett. B, 545:23–29, 2002, astro-ph/9908168.
  * Efstathiou [2021] George Efstathiou.  To H0 or not to H0?  Mon. Not. Roy. Astron. Soc., 505(3):3866–3872, 2021, 2103.08723.
  * Keeley and Shafieloo [2022] Ryan E. Keeley and Arman Shafieloo.  Ruling Out New Physics at Low Redshift as a solution to the $H_{0}$ Tension.  6 2022, 2206.08440.
  * Poulin et al. [2019] Vivian Poulin, Tristan L. Smith, Tanvi Karwal, and Marc Kamionkowski.  Early Dark Energy Can Resolve The Hubble Tension.  Phys. Rev. Lett., 122(22):221301, 2019, 1811.04083.
  * Kamionkowski et al. [2014] Marc Kamionkowski, Josef Pradler, and Devin G. E. Walker.  Dark energy from the string axiverse.  Phys. Rev. Lett., 113(25):251302, 2014, 1409.0549.
  * McDonough and Scalisi [2022] Evan McDonough and Marco Scalisi.  Towards Early Dark Energy in String Theory.  8 2022, 2209.00011.
  * Turner [1983] Michael S. Turner.  Coherent Scalar Field Oscillations in an Expanding Universe.  Phys. Rev. D, 28:1243, 1983.
  * Johnson and Kamionkowski [2008] Matthew C. Johnson and Marc Kamionkowski.  Dynamical and Gravitational Instability of Oscillating-Field Dark Energy and Dark Matter.  Phys. Rev. D, 78:063010, 2008, 0805.1748.
  * Agrawal et al. [2019] Prateek Agrawal, Francis-Yan Cyr-Racine, David Pinner, and Lisa Randall.  Rock ’n’ Roll Solutions to the Hubble Tension.  4 2019, 1904.01016.
  * Lin et al. [2019a] Meng-Xiang Lin, Giampaolo Benevento, Wayne Hu, and Marco Raveri.  Acoustic Dark Energy: Potential Conversion of the Hubble Tension.  Phys. Rev. D, 100(6):063542, 2019a, 1905.12618.
  * Hu [1998] Wayne Hu.  Structure formation with generalized dark matter.  Astrophys. J., 506:485–494, 1998, astro-ph/9801234.
  * Bertschinger [1995] Edmund Bertschinger.  COSMICS: cosmological initial conditions and microwave anisotropy codes.  6 1995, astro-ph/9506070.
  * Seljak and Zaldarriaga [1996] Uros Seljak and Matias Zaldarriaga.  A Line of sight integration approach to cosmic microwave background anisotropies.  Astrophys. J., 469:437–444, 1996, astro-ph/9603033.
  * Lewis et al. [2000] Antony Lewis, Anthony Challinor, and Anthony Lasenby.  Efficient computation of CMB anisotropies in closed FRW models.  Astrophys. J., 538:473–476, 2000, astro-ph/9911177.
  * Lesgourgues [2011] Julien Lesgourgues.  The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview.  4 2011, 1104.2932.
  * Hlozek et al. [2018] Renée Hlozek, David J. E. Marsh, and Daniel Grin.  Using the Full Power of the Cosmic Microwave Background to Probe Axion Dark Matter.  Mon. Not. Roy. Astron. Soc., 476(3):3063–3085, 2018, 1708.05681.
  * Sabla and Caldwell [2022] Vivian I. Sabla and Robert R. Caldwell.  The Microphysics of Early Dark Energy.  2 2022, 2202.08291.
  * Karwal et al. [2022] Tanvi Karwal, Marco Raveri, Bhuvnesh Jain, Justin Khoury, and Mark Trodden.  Chameleon early dark energy and the Hubble tension.  Phys. Rev. D, 105(6):063535, 2022, 2106.13290.
  * Sakstein and Trodden [2020] Jeremy Sakstein and Mark Trodden.  Early Dark Energy from Massive Neutrinos as a Natural Resolution of the Hubble Tension.  Phys. Rev. Lett., 124(16):161301, 2020, 1911.11760.
  * Berghaus and Karwal [2020] Kim V. Berghaus and Tanvi Karwal.  Thermal Friction as a Solution to the Hubble Tension.  Phys. Rev. D, 101(8):083537, 2020, 1911.06281.
  * Berghaus and Karwal [2022] Kim V. Berghaus and Tanvi Karwal.  Thermal Friction as a Solution to the Hubble and Large-Scale Structure Tensions.  4 2022, 2204.09133.
  * Aloni et al. [2022] Daniel Aloni, Asher Berlin, Melissa Joseph, Martin Schmaltz, and Neal Weiner.  A Step in understanding the Hubble tension.  Phys. Rev. D, 105(12):123516, 2022, 2111.00014.
  * Harari and Sikivie [1992] Diego Harari and Pierre Sikivie.  Effects of a Nambu-Goldstone boson on the polarization of radio galaxies and the cosmic microwave background.  Phys. Lett. B, 289:67–72, 1992.
  * Carroll et al. [1990] Sean M. Carroll, George B. Field, and Roman Jackiw.  Limits on a Lorentz and Parity Violating Modification of Electrodynamics.  Phys. Rev. D, 41:1231, 1990.
  * Carroll [1998] Sean M. Carroll.  Quintessence and the rest of the world.  Phys. Rev. Lett., 81:3067–3070, 1998, astro-ph/9806099.
  * Capparelli et al. [2020] Ludovico M. Capparelli, Robert R. Caldwell, and Alessandro Melchiorri.  Cosmic birefringence test of the Hubble tension.  Phys. Rev. D, 101(12):123529, 2020, 1909.04621.
  * Murai et al. [2022] Kai Murai, Fumihiro Naokawa, Toshiya Namikawa, and Eiichiro Komatsu.  Isotropic cosmic birefringence from early dark energy.  9 2022, 2209.07804.
  * Kamionkowski et al. [1997] Marc Kamionkowski, Arthur Kosowsky, and Albert Stebbins.  Statistics of cosmic microwave background polarization.  Phys. Rev. D, 55:7368–7388, 1997, astro-ph/9611125.
  * Zaldarriaga and Seljak [1997] Matias Zaldarriaga and Uros Seljak.  An all sky analysis of polarization in the microwave background.  Phys. Rev. D, 55:1830–1840, 1997, astro-ph/9609170.
  * Lue et al. [1999] Arthur Lue, Li-Min Wang, and Marc Kamionkowski.  Cosmological signature of new parity violating interactions.  Phys. Rev. Lett., 83:1506–1509, 1999, astro-ph/9812088.
  * Lepora [1998] Nathan F. Lepora.  Cosmological birefringence and the microwave background.  12 1998, gr-qc/9812077.
  * Liu et al. [2006] Guo-Chin Liu, Seokcheon Lee, and Kin-Wang Ng.  Effect on cosmic microwave background polarization of coupling of quintessence to pseudoscalar formed from the electromagnetic field and its dual.  Phys. Rev. Lett., 97:161303, 2006, astro-ph/0606248.
  * Hotinli et al. [2022] Selim C. Hotinli, Gilbert P. Holder, Matthew C. Johnson, and Marc Kamionkowski.  Cosmology from the kinetic polarized Sunyaev Zel’dovich effect.  4 2022, 2204.12503.
  * Lee et al. [2022] Nanoom Lee, Selim C. Hotinli, and Marc Kamionkowski.  Probing Cosmic Birefringence with Polarized Sunyaev Zel’dovich Tomography.  7 2022, 2207.05687.
  * Smith et al. [2020] Tristan L. Smith, Vivian Poulin, and Mustafa A. Amin.  Oscillating scalar fields and the Hubble tension: a resolution with novel signatures.  Phys. Rev. D, 101(6):063523, 2020, 1908.06995.
  * Jeong and Kamionkowski [2020] Donghui Jeong and Marc Kamionkowski.  Gravitational waves, CMB polarization, and the Hubble tension.  Phys. Rev. Lett., 124(4):041301, 2020, 1908.06100.
  * Griest [2002] Kim Griest.  Toward a possible solution to the cosmic coincidence problem.  Phys. Rev. D, 66:123501, 2002, astro-ph/0202052.
  * Dodelson et al. [2000] Scott Dodelson, Manoj Kaplinghat, and Ewan Stewart.  Solving the Coincidence Problem : Tracking Oscillating Energy.  Phys. Rev. Lett., 85:5276–5279, 2000, astro-ph/0002360.
  * Sabla and Caldwell [2021] Vivian I. Sabla and Robert R. Caldwell.  No $H_{0}$ assistance from assisted quintessence.  Phys. Rev. D, 103(10):103506, 2021, 2103.04999.
  * Hill and Baxter [2018] J. Colin Hill and Eric J. Baxter.  Can Early Dark Energy Explain EDGES?  JCAP, 08:037, 2018, 1803.07555.
  * Bowman et al. [2018] Judd D. Bowman, Alan E. E. Rogers, Raul A. Monsalve, Thomas J. Mozdzen, and Nivedita Mahesh.  An absorption profile centred at 78 megahertz in the sky-averaged spectrum.  Nature, 555(7694):67–70, 2018, 1810.05912.
  * Muñoz [2019a] Julian B. Muñoz.  Standard Ruler at Cosmic Dawn.  Phys. Rev. Lett., 123(13):131301, 2019a, 1904.07868.
  * Muñoz [2019b] Julian B. Muñoz.  Robust Velocity-induced Acoustic Oscillations at Cosmic Dawn.  Phys. Rev. D, 100(6):063538, 2019b, 1904.07881.
  * Sarkar and Kovetz [2022] Debanjan Sarkar and Ely D. Kovetz.  Measuring the cosmic expansion rate using 21-cm velocity acoustic oscillations.  10 2022, 2210.16853.
  * Lin et al. [2019b] Meng-Xiang Lin, Marco Raveri, and Wayne Hu.  Phenomenology of Modified Gravity at Recombination.  Phys. Rev. D, 99(4):043514, 2019b, 1810.02333.
  * Braglia et al. [2021] Matteo Braglia, Mario Ballardini, Fabio Finelli, and Kazuya Koyama.  Early modified gravity in light of the $H_{0}$ tension and LSS data.  Phys. Rev. D, 103(4):043528, 2021, 2011.12934.
  * Braglia et al. [2020] Matteo Braglia, Mario Ballardini, William T. Emond, Fabio Finelli, A. Emir Gumrukcuoglu, Kazuya Koyama, and Daniela Paoletti.  Larger value for $H_{0}$ by an evolving gravitational constant.  Phys. Rev. D, 102(2):023529, 2020, 2004.11161.
  * Ballesteros et al. [2020] Guillermo Ballesteros, Alessio Notari, and Fabrizio Rompineve.  The $H_{0}$ tension: $\Delta G_{N}$ vs. $\Delta N_{\rm eff}$.  JCAP, 11:024, 2020, 2004.05049.
  * Ballardini et al. [2020] Mario Ballardini, Matteo Braglia, Fabio Finelli, Daniela Paoletti, Alexei A. Starobinsky, and Caterina Umiltà.  Scalar-tensor theories of gravity, neutrino physics, and the $H_{0}$ tension.  JCAP, 10:044, 2020, 2004.14349.
  * Zumalacarregui [2020] Miguel Zumalacarregui.  Gravity in the Era of Equality: Towards solutions to the Hubble problem without fine-tuned initial conditions.  Phys. Rev. D, 102(2):023523, 2020, 2003.06396.
  * Abadi and Kovetz [2021] Tal Abadi and Ely D. Kovetz.  Can conformally coupled modified gravity solve the Hubble tension?  Phys. Rev. D, 103(2):023530, 2021, 2011.13853.
  * Kreisch et al. [2020] Christina D. Kreisch, Francis-Yan Cyr-Racine, and Olivier Doré.  Neutrino puzzle: Anomalies, interactions, and cosmological tensions.  Phys. Rev. D, 101(12):123505, 2020, 1902.00534.
  * Blinov et al. [2019] Nikita Blinov, Kevin James Kelly, Gordan Z Krnjaic, and Samuel D McDermott.  Constraining the Self-Interacting Neutrino Interpretation of the Hubble Tension.  Phys. Rev. Lett., 123(19):191102, 2019, 1905.02727.
  * Jedamzik and Pogosian [2020] Karsten Jedamzik and Levon Pogosian.  Relieving the Hubble tension with primordial magnetic fields.  Phys. Rev. Lett., 125(18):181302, 2020, 2004.09487.
  * Ivanov et al. [2020] Mikhail M. Ivanov, Evan McDonough, J. Colin Hill, Marko Simonović, Michael W. Toomey, Stephon Alexander, and Matias Zaldarriaga.  Constraining Early Dark Energy with Large-Scale Structure.  Phys. Rev. D, 102(10):103502, 2020, 2006.11235.
  * Hill et al. [2020] J. Colin Hill, Evan McDonough, Michael W. Toomey, and Stephon Alexander.  Early dark energy does not restore cosmological concordance.  Phys. Rev. D, 102(4):043507, 2020, 2003.07355.
  * D’Amico et al. [2021] Guido D’Amico, Leonardo Senatore, Pierre Zhang, and Henry Zheng.  The Hubble Tension in Light of the Full-Shape Analysis of Large-Scale Structure Data.  JCAP, 05:072, 2021, 2006.12420.
  * Smith et al. [2021] Tristan L. Smith, Vivian Poulin, José Luis Bernal, Kimberly K. Boddy, Marc Kamionkowski, and Riccardo Murgia.  Early dark energy is not excluded by current large-scale structure data.  Phys. Rev. D, 103(12):123542, 2021, 2009.10740.
  * Herold and Ferreira [2022] Laura Herold and Elisa G. M. Ferreira.  Resolving the Hubble tension with Early Dark Energy.  10 2022, 2210.16296.
  * Hill et al. [2022] J. Colin Hill et al.  Atacama Cosmology Telescope: Constraints on prerecombination early dark energy.  Phys. Rev. D, 105(12):123536, 2022, 2109.04451.
  * Poulin et al. [2021] Vivian Poulin, Tristan L. Smith, and Alexa Bartlett.  Dark energy at early times and ACT data: A larger Hubble constant without late-time priors.  Phys. Rev. D, 104(12):123550, 2021, 2109.06229.