# Varying-constant cosmology from hyperlight, coupled scalars

Masha Baryakhtar<EMAIL_ADDRESS>Department of Physics, University of
Washington, Seattle, Washington, 98195, USA  Olivier Simon
<EMAIL_ADDRESS>Princeton Center for Theoretical Science, Princeton
University, Princeton, New Jersey, 08544, USA  Department of Physics,
Princeton University, Princeton, New Jersey, 08544, USA  Stanford Institute
for Theoretical Physics, Stanford University, Stanford, California, 94305, USA
Zachary J. Weiner<EMAIL_ADDRESS>Department of Physics, University of
Washington, Seattle, Washington, 98195, USA

(August 28, 2024)

###### Abstract

The fundamental constants at recombination can differ from their present-day
values due to degeneracies in cosmological parameters, raising the possibility
of yet-undiscovered physics coupled directly to the Standard Model. We study
the cosmology of theories in which a new, hyperlight scalar field modulates
the electron mass and fine-structure constant at early times. We find new
degeneracies in cosmologies that pair early recombination with a new
contribution to the matter density arising at late times, whose predictions
can be simultaneously consistent with CMB and low-redshift distance
measurements. Such “late dark matter” already exists in the Standard Model in
the form of massive neutrinos, but is necessarily realized by the scalar
responsible for shifting the early-time fundamental constants. After detailing
the physical effects of varying constants and hyperlight scalar fields on
cosmology, we find that variations of the electron mass and fine structure
constant are constrained at the percent and permille level, respectively, and
a hyperlight scalar in the mass range $10^{-32}\leavevmode\nobreak\
\mathrm{eV}\lesssim m_{\phi}\lesssim 10^{-28}\leavevmode\nobreak\ \mathrm{eV}$
can impact what variations are allowed while composing up to a percent of the
present dark matter density. We comment on the potential for models with a
varying electron mass to reconcile determinations of the Hubble constant from
cosmological observations and distance-ladder methods, and we show that
parameter inference varies significantly between recent baryon acoustic
oscillation and type Ia supernova datasets.

###### Contents

  1. I Introduction
  2. II Cosmology with varying constants
    1. II.1 Early-Universe signatures
      1. II.1.1 Recombination
      2. II.1.2 Acoustic oscillations and matter effects
      3. II.1.3 Small-scale damping
      4. II.1.4 Polarization
      5. II.1.5 Angular power spectra
      6. II.1.6 Degeneracies of the early Universe
    2. II.2 Late-Universe signatures
      1. II.2.1 Late-time integrated Sachs-Wolfe effect and CMB lensing
      2. II.2.2 Baryon acoustic oscillations
      3. II.2.3 Low-redshift distances from supernovae
      4. II.2.4 Matter clustering
  3. III Cosmology with coupled, hyperlight scalars
    1. III.1 Impact on the cosmological background
    2. III.2 Impact on cosmological perturbations
    3. III.3 Parameter space
  4. IV Cosmological parameter inference
    1. IV.1 Varying constants
      1. IV.1.1 Planck data alone
      2. IV.1.2 Impact of low-redshift distance measurements
    2. IV.2 Coupled, hyperlight scalars
      1. IV.2.1 Planck data alone
      2. IV.2.2 Impact of low-redshift distance measurements
      3. IV.2.3 Comparison to massive neutrinos
    3. IV.3 Implications for cosmological concordance
  5. V Conclusions
  6. A Numerical implementation of cosmological scalar fields
    1. A.1 Dynamics
    2. A.2 Effective fluid treatment
      1. A.2.1 Background
      2. A.2.2 Perturbations
    3. A.3 Initial conditions
    4. A.4 Implementation details
  7. B Parameter inference
    1. B.1 Likelihoods
    2. B.2 Sampling methods

##  I Introduction

Cosmology is an essential frontier to study the dark sector and, at the same
time, is a unique probe of the Standard Model and gravity at very early times.
The cosmic microwave background (CMB) is largely governed by the physics of
photon recombination and acoustic plasma waves, which in turn depend
sensitively on quantum electrodynamics and thus the electromagnetic coupling
strength and the electron mass in the early Universe. Some theories of physics
beyond the Standard Model motivate variations of the fundamental physical
constants over cosmological history; such microphysical models introduce new
degrees of freedom which influence cosmological dynamics through gravity. The
CMB anisotropies are also sensitive to all of the universe’s content through
gravitation, both via cosmic expansion and gravitational potentials, and both
during and after recombination.

Many previous studies of the CMB have placed limits on phenomenological
variations of the fine-structure constant $\alpha$ [1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12] and of the mass of the electron $m_{e}$ [13, 14, 15, 16, 17, 18,
19] at recombination under the assumption that the gravitational effects of
any new degrees of freedom are negligible. The possibility that fundamental
constants—in particular, the electron mass—differed from their present-day
values in the early Universe, triggering recombination earlier than standard,
is also a candidate proposal [20, 21, 22, 23] to reconcile determinations of
the present-day expansion rate $H_{0}$ as inferred by the CMB in
$\Lambda$–cold-dark-matter ($\Lambda$CDM) cosmology with those based on local,
distance-ladder measurements [24, 25, 26]. Early recombination alone is
insufficient to explain observations of large-scale structure and type Ia
supernovae in addition to the CMB, however; existing proposals therefore
further invoke spatial curvature [21, 22]. Comparatively little literature has
considered specific microphysical models responsible for the variations and
investigated whether added matter content qualitatively alters the predicted
cosmologies [27, 28, 29, 30, 31]. Implementing concrete theories is likewise
imperative to understand the field evolution and the interplay of measurements
throughout cosmic time—whether nucleosynthesis at earlier times, astrophysical
probes at later times, or laboratory experiments today.

In this work and a companion article [32], we test models of a new scalar
field that induces early-time shifts in fundamental constants against
precision cosmological datasets. We specifically consider a massive,
hyperlight field with particle mass $10^{-32}\leavevmode\nobreak\
\mathrm{eV}\lesssim m_{\phi}\lesssim 10^{-28}\leavevmode\nobreak\ \mathrm{eV}$
whose Standard Model (SM) couplings effectively modulate the strength of
electromagnetism and the mass of the electron. The scalar is produced as a
cold, spatially homogeneous field in the early universe and makes up a
subcomponent of cosmological dark matter today. In the mass range of interest,
the amplitude of the scalar field is frozen until some point after
recombination, after which it oscillates and decays; the fundamental constants
thus take values before recombination that are constant in space and time but
different from their values today. Our work complements prior study of models
akin to quintessence ($m_{\phi}\lesssim 10^{-33}\leavevmode\nobreak\
\mathrm{eV}$) in which $\alpha$ varies via a coupling to the photon [33, 34,
35].

We show on semianalytic grounds that models with contributions to the matter
density at late times—“late dark matter” scenarios—open up a novel degeneracy
in the background cosmology that, taken at face value, has the potential to
reconcile late-time observations with scenarios of early recombination. The
scalar field conjectured to shift the values of constants naturally makes such
a late contribution to dark matter, and so do massive neutrinos. The
hyperlight scalar fields we consider also suppress structure formation to an
extent controlled by their mass and abundance, yielding distinct signatures in
the CMB that further test the scenario. We assess the interplay of these
effects on quantitative constraints with a detailed analysis of cosmological
data and study the effect of various dataset combinations. That the additional
freedom of scalar field models allows for CMB observations and low-redshift
distance measurements to be simultaneously satisfied in varying-$m_{e}$
scenarios was observed in the related analysis of Ref. [29], though without
identifying its physical origin.

The outline of this paper is as follows. In Section II, we detail the physical
impact of varying fundamental constants on the CMB and other cosmological
observables. We devise a general set of parameter combinations that encode the
main physical effects relevant to the CMB anisotropies in $\Lambda$CDM,
including parameters that are typically fixed in standard analyses—not just
the electron mass and fine-structure constant, but also the present-day CMB
temperature, the helium yield, and the density of radiation beyond photons. We
then discuss the distinct effects arising from a scalar field hypothesized to
be responsible for the variations in the fundamental constants in Section III,
considering the scalar’s purely gravitational effects and their interplay with
varying constants. Section IV presents our inference of parameters within this
extension to the standard cosmological model, comparing the results of
scenarios with and without a scalar field. We also critically assess the
concordance of current cosmological datasets within the context of cosmologies
with a varying electron mass. We conclude in Section V, discussing the
potential impact of near-term and future datasets and possible model
extensions motivated by Ref. [32].

We use natural units in which $\hbar=c=1$ and define the reduced Planck mass
$M_{\mathrm{pl}}\equiv 1/\sqrt{8\pi G}=2.435\times 10^{18}\leavevmode\nobreak\
\mathrm{GeV}$. Repeated spatial indices (Latin characters) are implicitly
summed regardless of their placement. We use upright boldface to denote
spatial vectors. Unless otherwise specified, we fix a homogeneous, conformal-
time Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime with metric

$\displaystyle g_{\mu\nu}\equiv a(\tau)^{2}\eta_{\mu\nu}$ (1)

where $\eta_{\mu\nu}$ is the “mostly positive” Minkowski metric and $a(\tau)$
the scale factor. We use primes to denote derivatives with respect to
conformal time $\tau$. We occasionally work in terms of cosmic time $t$
(defined by $\mathrm{d}t=a\mathrm{d}\tau$), and use dots to denote $t$
derivatives; the Hubble rate is $H\equiv\dot{a}/a$.

##  II Cosmology with varying constants

We begin by grounding our discussion with a detailed review of the impact of
variations in fundamental constants on cosmology, seeking to quantify the
parameter dependence of physical effects measured by cosmological data in the
extended $\Lambda$CDM parameter space. We focus on the cases where the
electron mass $m_{e}$ and the fine-structure constant $\alpha$ are different
prior to recombination than at the present day; in this Section we do so
without considering additional effects from any specific microphysical model
thereof. We denote the constants’ early-time values as $\alpha_{i}$ and
$m_{e,i}$. We first discuss early-universe signatures, summarizing key
parameter combinations that are well constrained by data and the resulting
degeneracies in the physics up to and including recombination. We then
consider complementary probes of the late universe, after the recombination
epoch.

As detailed in this Section, the usual flat $\Lambda$CDM model has sufficient
parameter freedom to allow for the scale factor of recombination to differ
substantially from the standard prediction without affecting the primary CMB
anisotropies. In standard cosmology, the scale factor of recombination is
fixed by the ratio of the CMB monopole temperature at the present time to that
at photon last scattering, $a_{\star}/a_{0}=T_{0}/T_{\star}$. The former is
directly and precisely measured [36, 37], while the latter is determined by
the standard atomic and thermal physics that govern recombination. This
degeneracy direction has been explored in the context of the Hubble tension by
changing $T_{\star}$ through variations of the early-time electron mass [21].
Separately, Ref. [38] reframed the Hubble tension as a (counterfactual, if
illustrative) tension in the present-day CMB temperature $T_{0}$. Our
discussion captures the arguments of Refs. [38, 21] in terms of the common
physics from which they derive; we also comment on connections to the
rescaling invariance of the Boltzmann and Einstein equations as explored in
detail by Refs. [39, 40, 41].

Given that the CMB, baryon acoustic oscillations, and supernovae datasets we
consider here are individually fit well by the $\Lambda$CDM model compared to
any known extension thereof, our discussion centers on what degeneracies can
be realized in extended parameter spaces that identically reproduce
$\Lambda$CDM predictions. That is, we take the best-fit $\Lambda$CDM model as
a proxy for the data itself and investigate correlated changes to parameters
that leave its predictions invariant. Such a description is sufficient to
anticipate whether an extension of $\Lambda$CDM might contain directions in
parameter space that should be poorly constrained by data. However, only a
quantitative analysis can assess whether a model’s additional freedom in fact
improves upon the fit to individual datasets by explaining additional
(possibly unanticipated) features in those data. We present the results of
parameter inference in Section IV.

Throughout, we refer to the contributions of various species to the present-
day, average energy density with the abundance parameters
$\omega_{X}=\bar{\rho}_{X,0}/3H_{100}^{2}M_{\mathrm{pl}}^{2}$, where
$H_{0}=h\cdot 100\leavevmode\nobreak\
\mathrm{Mpc}^{-1}\mathrm{km}/\mathrm{s}\equiv hH_{100}$. Specifically,
$\omega_{b}$, $\omega_{c}$, $\omega_{\Lambda}$, $\omega_{\gamma}$, and
$\omega_{\nu}$ are the abundances of baryons, cold dark matter, dark energy,
photons, and neutrinos, respectively. We also use
$\omega_{m}=\omega_{b}+\omega_{c}$ and
$\omega_{r}=\omega_{\gamma}+\omega_{\nu}$ to denote the total matter and
radiation abundances, including contributions only from species that are
nonrelativistic and relativistic at early times, respectively. The early-time
energy density of neutrinos is typically quantified in terms of the effective
number of degrees of freedom $N_{\mathrm{eff}}$ as

$\displaystyle\omega_{\nu}$
$\displaystyle=\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}N_{\mathrm{eff}}\cdot\omega_{\gamma},$
(2)

which accounts for the increase in the photon temperature relative to the
neutrino temperature due to the entropy transferred from electrons and
positrons when they annihilate. In standard cosmology, the neutrinos’
temperature is slightly enhanced by their residual coupling to the plasma
during electron-positron annihilation; numerical solutions of the SM kinetic
equations yield $N_{\mathrm{eff}}=3.044$ [42, 43, 44].111Calculations of the
final digit continue to be refined [45], including finite-temperature QED
corrections and neutrino mass mixing [42, 43, 44]. If electron-positron
annihilation occurs earlier (later) due to a heavier (lighter) electron, the
neutrino temperature would be larger (smaller) than the SM prediction.
However, this effect is small: over the range of electron-mass variations
relevant to our main results (at most $20\%$ in either direction), the
relative change in $N_{\mathrm{eff}}$ is a few tenths of a percent [46, 47].
As differences of this size are smaller than the anticipated $1\sigma$
sensitivity of future CMB experiments like CMB-S4 [48, 49, 50], we neglect
this effect and fix $N_{\mathrm{eff}}=3.044$.

###  II.1 Early-Universe signatures

At linear order in perturbation theory, the effect of varying constants is
entirely captured by accounting for their time dependence in the Thomson cross
section and in the recombination equations whose solution gives the ionization
history. These dynamics (implemented in CLASS [51, 52] and HyRec [53, 54])
constitute the focus of our analysis. Subleading effects include changes to
the electrons’ and baryons’ energy densities through a varying electron mass
and fine-structure constant, respectively. As electrons make up a small
fraction $\propto m_{e}/m_{p}\approx 1/1836$ of the total energy density in
“baryonic” matter and the electromagnetic contribution to the nucleon masses
is at the permille level [32], we neglect these corrections.

In Section II.1.1, we study the effect of the fundamental constants on both
when and how quickly recombination proceeds. The earlier recombination occurs,
the less abundant dark matter and baryons are relative to radiation; Section
II.1.2 explains the resulting effect on the dynamics of the photon-baryon
plasma. Photon diffusion (Section II.1.3) is sensitive to the fundamental
constants not only implicitly via the time recombination occurs but also
explicitly via the Thomson cross section. In addition, the faster
recombination occurs, the narrower the interval over which photons last
scatter; Sections II.1.3 and II.1.4 discuss the resulting impact on small-
scale damping and the generation of polarization. Finally, the earlier
hydrogen recombines, the smaller the sound horizon is at last scattering.
Section II.1.5 shows how the distance to last scattering must compensate to
preserve its manifestation in angular correlations on the sky. Drawing from
these derivations, in Section II.1.6 we discuss numerous degeneracies in
extended $\Lambda$CDM parameter space for which the primary CMB anisotropies
are completely (or partially) unchanged.

####  II.1.1 Recombination

The CMB observed today comprises photons that last scattered off electrons
with a distribution in time determined by the physics of electrons recombining
into neutral atoms and Thomson scattering. This distribution—the so-called
visibility function—encodes both when (i.e., at what temperature) and over
what redshift interval most CMB photons last scattered; both features directly
impact the visible anisotropies in CMB temperature and polarization. The
Thomson scattering rate is

$\displaystyle\frac{\mathrm{d}{\kappa}}{\mathrm{d}{\tau}}$
$\displaystyle\equiv a\bar{n}_{e}\sigma_{T},$ (3)

where the cross section $\sigma_{T}=8\pi\alpha_{i}^{2}/3m_{e,i}^{2}$ and
$\bar{n}_{e}$ is the number density of free electrons. The visibility function
is then defined by

$\displaystyle g(\tau)$
$\displaystyle=\frac{\mathrm{d}{\kappa}}{\mathrm{d}{\tau}}\exp\left[-\int_{\tau}^{\tau_{0}}\mathrm{d}\tilde{\tau}\,\left.\frac{\mathrm{d}{\kappa}}{\mathrm{d}{\tau}}\right|_{\tau=\tilde{\tau}}\right]\equiv\kappa^{\prime}e^{-\kappa}$
(4)

[the latter equality defining $\kappa(\tau)$ as the optical depth from today,
$\tau_{0}$]. We define the moment of recombination as the time of maximum
visibility, with the scale factor of recombination $a_{\star}$ given by
$g_{\star}\equiv g(\tau(a_{\star}))=\max g(\tau(a))$. Quantities with a
subscript star generally indicate those evaluated at recombination.

The dependence on the fundamental constants of the Thomson scattering cross
section is manifest in its definition, but that of the electron ionization
history is slightly more complex. The visibility function peaks roughly when
the photons decouple from the plasma, which occurs when the Universe becomes
nearly neutral. Defining the ionization fraction
$X_{e}\equiv\bar{n}_{e}/n_{\mathrm{H}}$, where $n_{\mathrm{H}}$ is the number
density of hydrogen nuclei, ionized and not, and neglecting the presence of
helium (which effectively finishes recombining before hydrogen recombination
begins),

$\displaystyle\bar{n}_{e}$
$\displaystyle=X_{e}\frac{(1-Y_{\mathrm{He}})\bar{\rho}_{b}}{m_{\mathrm{H}}}.$
(5)

Here $Y_{\mathrm{He}}\equiv\bar{\rho}_{\mathrm{He}}/\bar{\rho}_{b}$ is the
helium fraction by mass and $m_{\mathrm{H}}$ the mass of hydrogen. In spite of
the complexities of the recombination process [55, 56, 57, 53, 58], its
dynamics largely depend on temperature relative to the ionization energy of
hydrogen, $E_{I}=\alpha_{i}^{2}m_{e,i}/2$. The temperature at recombination
therefore scales as $T_{\star}\propto\alpha_{i}^{2}m_{e,i}$, so

$\displaystyle a_{\star}$
$\displaystyle=\frac{T_{0}}{T_{\star}}\propto\frac{\sqrt[4]{\omega_{\gamma}}}{\alpha_{i}^{2}m_{e,i}}.$
(6)

The scale factor relative to that at recombination, $x\equiv a/a_{\star}$,
proves to be the most convenient time coordinate with which to characterize
the CMB anisotropies [21]. Moreover, writing the photon visibility as a
dimensionless distribution over $\ln x$, i.e., as
$g/aH=e^{-\kappa}\mathrm{d}\kappa/\mathrm{d}\ln x$, isolates superficial
changes to its shape that only arise in a dimensionful form [e.g., over
conformal time as in Eq. 4]. Figure 1 depicts the effect of varying constants
on the ionization history $X_{e}$ and on the dimensionless visibility
function; plotted against the scale factor relative to that of peak
visibility, both are nearly indistinguishable when varying the electron mass
by $15\%$ in either direction.

Figure 1:  Impact on the ionization fraction (Eq. 5, top panels) and the
dimensionless visibility function (LABEL:{eqn:g-star-dependence}, bottom
panels) in cosmologies varying the electron mass $m_{e,i}$ (left panels) and
the fine-structure constant $\alpha_{i}$ (right panels) by color. The colors
on the right-hand panels label values of $\alpha_{i}^{2}$ so that curves of
the same color in the left-hand panels depict the same shift in the time of
recombination. Results are plotted versus the logarithm of the scale factor
relative to that when the visibility function peaks, $a_{\star}$, to emphasize
changes in dimensionless shape; these are negligible for varying $m_{e,i}$ but
significant for varying $\alpha_{i}$. Each curve varies $\omega_{b}$ and
$\omega_{c}$ in proportion to $1/a_{\star}$ to keep baryon to photon and
matter to radiation densities fixed at recombination (Section II.1.2) and
fixes the helium mass fraction $Y_{\mathrm{He}}$ (whose effect is discussed in
Section II.1.3). All results also fix the angular size of the sound horizon
$\theta_{s}$ (Section II.1.5), a choice which has a negligible effect on these
results.

The rate of recombination increases more noticeably with the fine-structure
constant, evident in the change to $X_{e}(x)$ in Fig. 1, due to the more
sensitive dependence of the effective rates of recombination, photoionization,
etc., on $\alpha_{i}$ [59, 60]. As a consequence, the visibility function
narrows and broadens substantially more when varying the fine-structure
constant than the electron mass (i.e., at a commensurate variation in
$a_{\star}$). We discuss the physical impacts of these changes to the
visibility and ionization history and empirically determine their dependence
on $\alpha_{i}$ in Section II.1.3. In addition, $a_{\star}$ in Fig. 1
increases slightly faster than estimated from the ionization energy in Eq. 6
$\alpha_{i}$; a numerical fit estimates $a_{\star}\propto\alpha_{i}^{-2.08}$.

####  II.1.2 Acoustic oscillations and matter effects

The dynamics of the photon-baryon plasma are primarily driven by three
effects: the competition between radiation pressure and gravitational
collapse, the increasing importance of matter to the expansion rate and
gravitational potentials, and the damping of plasma oscillations on scales
below the photons’ mean free path. The semianalytic solutions of Ref. [61]
(see also Ref. [62]) suggest that the Sachs-Wolfe, Doppler, and damping
effects depend on cosmological parameters only through particular
dimensionless combinations and depend on wave number only relative to the
sound scale at recombination $1/r_{s,\star}$, to the comoving Hubble scale at
matter-radiation equality $k_{\mathrm{eq}}$, and to the effective damping
scale $k_{D,\star}$. We discuss (and define) the role of each in turn.

The importance of baryons to the dynamics of the plasma is measured by the
baryon-to-photon ratio

$\displaystyle R(x)\equiv\frac{3\bar{\rho}_{b}}{4\bar{\rho}_{\gamma}}$
$\displaystyle=R_{\star}x,$ (7)

written in terms of that at recombination, $R_{\star}\equiv
3\omega_{b}a_{\star}/4\omega_{\gamma}$. Acoustic waves in the photon-baryon
fluid travel with an effective sound speed

$\displaystyle c_{s}(x)$ $\displaystyle=\frac{1}{\sqrt{3}\sqrt{1+R(x)}},$ (8)

with a corresponding sound horizon of

$\displaystyle r_{s}(a)$
$\displaystyle\equiv\int_{0}^{a}\frac{\mathrm{d}\tilde{a}}{\tilde{a}}\,\frac{c_{s}(\tilde{a})}{\tilde{a}H(\tilde{a})}=\frac{a_{\star}}{\sqrt{3}}\int^{a/a_{\star}}_{0}\mathrm{d}x\,\frac{1}{\sqrt{1+R_{\star}x}}\frac{1}{(a_{\star}x)^{2}H(x)}.$
(9)

Independently of the physical extent of the sound horizon itself, the
measured, relative heights of the acoustic peaks tightly constrain
$R_{\star}$. At recombination, dark energy is negligible and the Universe well
described by matter and radiation alone. The latter term in the integrand of
Eq. 9 thus takes the form

$\displaystyle a^{2}H=(a_{\star}x)^{2}H(x)$
$\displaystyle=H_{100}\sqrt{\omega_{r}}\sqrt{1+x/x_{\mathrm{eq}}},$ (10)

where

$\displaystyle x_{\mathrm{eq}}$ $\displaystyle=a_{\mathrm{eq}}/a_{\star}$ (11)

is the scale factor at equality relative to that at recombination. Since
$a_{\mathrm{eq}}/a_{0}=\omega_{r}/\omega_{m}$,
$x_{\mathrm{eq}}=\bar{\rho}_{r,\star}/\bar{\rho}_{m,\star}$ measures the ratio
of the radiation and matter densities at recombination. Because Eq. 10 is
independent of $a_{\star}$, the sound horizon at recombination [Eq. 9] is
simply proportional to $a_{\star}/\sqrt{\omega_{r}}$ if $x_{\mathrm{eq}}$ and
$R_{\star}$ are held fixed; indeed, in a matter-radiation Universe Eq. 9 takes
the closed form [63]

$\displaystyle r_{s,\star}$
$\displaystyle=\frac{2a_{\star}\sqrt{x_{\mathrm{eq}}/R_{\star}}}{H_{100}\sqrt{3\omega_{r}}}\ln\left(\frac{\sqrt{R_{\star}}\sqrt{1+x_{\mathrm{eq}}}+\sqrt{1+R_{\star}}}{1+\sqrt{R_{\star}x_{\mathrm{eq}}}}\right),$
(12)

which is exact in $\Lambda$CDM up to the $\mathcal{O}(10^{-9})$ relative
contribution from dark energy at recombination.

The relative importance of matter to gravity (at the background and perturbed
level) at early times is fully characterized by when matter-radiation equality
occurs relative to recombination ($x_{\mathrm{eq}}$). The heights of the peaks
are sensitive to the onset of matter domination via the radiation driving
effect [64, 65]: oscillations of the plasma are driven by the decay of the
gravitational potentials at horizon crossing, but to a decreasing extent as
the Universe becomes increasingly dominated by matter. The early integrated
Sachs-Wolfe (early ISW) effect similarly depends on the time evolution of
metric potentials as the Universe transitions from radiation domination, i.e.,
on $x_{\mathrm{eq}}$ only [66]. Moreover, in Universes that are well described
by the matter-radiation solution, the full scale-dependent shape of these
effects is identical for cosmologies with the same $x_{\mathrm{eq}}$—a feature
Ref. [67] emphasized as important beyond simply fixing the relative size of
the sound horizon and the comoving horizon at equality. In particular, the
comoving Hubble rate in a matter-radiation Universe is

$\displaystyle aH$
$\displaystyle=\frac{\sqrt{\omega_{r}}}{a_{\star}}\frac{H_{100}}{x_{\mathrm{eq}}}\sqrt{(x/x_{\mathrm{eq}})^{-2}+(x/x_{\mathrm{eq}})^{-1}}\equiv\frac{k_{\mathrm{eq}}}{\sqrt{2}}\sqrt{(x/x_{\mathrm{eq}})^{-2}+(x/x_{\mathrm{eq}})^{-1}},$
(13)

defining $k_{\mathrm{eq}}$ as the comoving Hubble scale at equality
($x=x_{\mathrm{eq}}$). Just like the sound horizon, the comoving horizon at
equality ($1/k_{\mathrm{eq}}$) scales with $a_{\star}/\sqrt{\omega_{r}}$ if
$x_{\mathrm{eq}}$ is held fixed, situating all associated features at the same
relative scales.

In sum, the shape of the CMB power spectrum by itself measures relative energy
densities at recombination, constraining the parameter combinations
$R_{\star}\propto\omega_{b}a_{\star}/\omega_{\gamma}$ and
$x_{\mathrm{eq}}\propto\omega_{r}/\omega_{m}a_{\star}$. These combinations may
be held fixed trivially because $\omega_{b}$ and
$\omega_{c}=\omega_{m}-\omega_{b}$ are free parameters in standard cosmology,
in which case $r_{s,\star}k_{\mathrm{eq}}$ is constant. These phenomena alone
are therefore unable to discern whether the photon temperature today or at
recombination differ from the SM prediction [Eq. 6], or even if the radiation
density $\omega_{r}$ is different than that from photons and neutrinos as
predicted in the SM.

####  II.1.3 Small-scale damping

Fluctuations in the plasma are damped on small scales by two distinct effects,
one due to photon diffusion and the other due to the finite duration of
recombination. We consider each of their parameter dependence in turn. The
damping scale associated to photon diffusion is [68, 69]

$\displaystyle\frac{1}{k_{D,\star}^{2}}$
$\displaystyle\equiv\int_{0}^{1}\frac{\mathrm{d}x/x}{(a_{\star}xH)^{2}}\frac{R^{2}+16(1+R)/15}{6(1+R)^{2}}\left(\frac{\mathrm{d}{\kappa}}{\mathrm{d}{\ln
x}}\right)^{-1}.$ (14)

The baryon-to-photon ratio $R$ is invariant as a function of $x$ in
cosmologies with the same value for $R_{\star}$. Because the Thomson
scattering rate per $e$-fold, $\mathrm{d}\kappa/\mathrm{d}\ln
x=\bar{n}_{e}\sigma_{T}/H$, is dimensionless, the comoving diffusion scale
$k_{D,\star}$ is proportional to $k_{\mathrm{eq}}$ [Eq. 13] at fixed
$x_{\mathrm{eq}}$. Whether or not $k_{D,\star}/k_{\mathrm{eq}}$ varies with
parameters is therefore determined by the dimensionless Thomson scattering
rate.

Inserting the definition of free electron number density $\bar{n}_{e}$, Eq. 5,
and rewriting the baryon abundance in terms of $R_{\star}$ gives

$\displaystyle\frac{\mathrm{d}{\kappa}}{\mathrm{d}{\ln
x}}=\frac{n_{e}(x)\sigma_{T}}{H}$
$\displaystyle=\frac{R_{\star}K_{\star}}{x_{\mathrm{eq}}}\frac{X_{e}(x)}{\sqrt{(x/x_{\mathrm{eq}})^{2}+(x/x_{\mathrm{eq}})^{3}}}$
(15)

where

$\displaystyle K_{\star}$ $\displaystyle\equiv
4H_{100}M_{\mathrm{pl}}^{2}\frac{\sqrt{\omega_{\gamma}}\sigma_{T}}{a_{\star}^{2}}\sqrt{\frac{\omega_{\gamma}}{\omega_{r}}}\frac{1-Y_{\mathrm{He}}}{m_{\mathrm{H}}}$
(16)

is a rescaling of the Thomson scattering rate at recombination that encodes
its parameter dependence complementary to that of $R_{\star}$,
$x_{\mathrm{eq}}$, and the ionization history. In more detail, the acoustic
dynamics of the plasma (Section II.1.2) motivate expressing the density of
electron scatterers relative to the photon energy density at recombination,
i.e., in terms of the baryon-to-photon ratio $R_{\star}$ via $n_{e}\propto
R_{\star}x\rho_{\gamma}=R_{\star}\rho_{\gamma,\star}/x^{4}$. The impact of
matter on plasma dynamics suggests expressing the Hubble rate in terms of the
(square root of) the total radiation density and encoding the matter
contribution with $x_{\mathrm{eq}}$ as
$H\propto\sqrt{\rho_{r,\star}/x^{4}}\sqrt{1+x/x_{\mathrm{eq}}}$. Their ratio
yields an overall dependence on
$\sqrt{\rho_{\gamma,\star}}\propto\sqrt{\omega_{\gamma}}/a_{\star}^{2}\propto
T_{\star}^{2}$ times the square root of the fraction of the total radiation
density in photons. Finally, the factor of
$(1-Y_{\mathrm{He}})/m_{\mathrm{H}}$ in Eq. 16 simply translates the total
number density of baryons to that of electrons that do not recombine into
helium.

If the ionization history $X_{e}$ is unchanged as a function of $x$ and both
$R_{\star}$ and $x_{\mathrm{eq}}$ are fixed, then the full $x$ dependence of
the integrand of Eq. 14 is unchanged. The diffusion damping scale then scales
in tandem with $k_{\mathrm{eq}}$ and $1/r_{s,\star}$ if $K_{\star}$ is fixed.
Inserting Eq. 6 for $a_{\star}$,

$\displaystyle K_{\star}$
$\displaystyle\propto\alpha_{i}^{6}\sqrt{\frac{\omega_{\gamma}}{\omega_{r}}}\frac{1-Y_{\mathrm{He}}}{m_{\mathrm{H}}/M_{\mathrm{pl}}}$
(17)

At fixed $K_{\star}$, $x_{\mathrm{eq}}$, and ionization history, the impact of
diffusion damping on CMB anisotropies is independent of both the early-time
electron mass $m_{e,i}$ and the present photon temperature $T_{0}$. Namely,
the increase of the electron density relative to the Hubble rate with
$T_{\star}^{2}\propto m_{e,i}^{2}\alpha_{i}^{4}$ compensates the Thomson cross
section’s inverse dependence on $m_{e,i}^{2}$ but compounds with its
proportionality to $\alpha_{i}^{2}$. The irrelevance of $T_{0}$ merely
reflects that diffusion damping only depends on the properties of the Universe
during recombination, not on the expansion that has elapsed since then.
Equation 17 further encodes the salient dependence of the diffusion scale on
the helium yield and the effective number of neutrino species [Eq. 2].

The diffusion parameter $K_{\star}$ captures the relevant parameter dependence
of diffusion damping, except for any incurred impact to the ionization
history; Fig. 1 shows that this effect is negligible when varying $m_{e,i}$
but not $\alpha_{i}$. The damping scale involves an integral over $X_{e}$, and
to the extent that $\alpha_{i}$ simply rescales $X_{e}$, we could estimate
$k_{D,\star}\propto\sqrt{X_{e}}$ (on top of its other parameter dependence).
However, in the short interval before recombination ($\Delta\ln a\sim 0.4$),
$X_{e}/X_{e,\text{$\Lambda$CDM}}$ varies rapidly in proportionality between
$\alpha_{i}^{0}$ and $\alpha_{i}^{-5}$ (with $R_{\star}$ and $x_{\mathrm{eq}}$
fixed); the net effect on the damping scale is an integral over this
variation. Changes to the ionization history therefore cancel some of the
direct dependence of the damping scale on $\alpha_{i}$ in $K_{\star}$. We
determine empirically that $a_{\star}k_{D,\star}\sim\alpha_{i}^{1.63}$ at
fixed $Y_{\mathrm{He}}$. Similarly, the helium mass fraction $Y_{\mathrm{He}}$
enters both directly in the Thomson scattering rate and through its effect on
the ionization history, which does shift appreciably. Leading up to
recombination, $X_{e}$ varies rapidly in proportionality between
$(1-Y_{\mathrm{He}})^{0}$ and $(1-Y_{\mathrm{He}})^{-0.8}$, reducing the
damping scale’s overall dependence on $Y_{\mathrm{He}}$ just as for
$\alpha_{i}$. We compute that
$a_{\star}k_{D,\star}\propto(1-Y_{\mathrm{He}})^{0.24}$ in total. Moreover,
the helium mass fraction is also itself sensitive to $\alpha_{i}$ and
$m_{e,i}$ through the dynamics of nucleosynthesis. We derive the dependence of
$Y_{\mathrm{He}}$ on parameters in Ref. [32], leading to
$a_{\star}k_{D,\star}\sim\alpha_{i}^{1.42}$. The helium mass fraction scales
much more weakly with the electron mass, leading to
$a_{\star}k_{D,\star}\propto m_{e,i}^{-0.05}$.

The second effect that damps small-scale CMB power is due to the nonzero width
of the visibility function, which effectively averages rapidly oscillating
modes over multiple oscillations [69, 61]. This effect is referred to as
Landau damping for its analogy to the damping of oscillations comprising a
spread of frequencies. Approximating the visibility function as a Gaussian
about its peak as
$g(\tau)\approx\exp(-[\tau-\tau_{\star}]^{2}/2\sigma_{g}^{2})/\sqrt{2\pi}\sigma_{g}$,
the comoving width of the last-scattering surface may be approximated by
$\sigma_{g}=1/\sqrt{2\pi}g_{\star}$. However, $\sigma_{g}$ (as an interval in
conformal time) also depends explicitly on the expansion rate at
recombination, which, for example, is larger for earlier recombination. The
effect solely due to the shape of the visibility function is captured by the
width of the visibility function relative to $aH$ (i.e., as a dimensionless
distribution over $\ln x$), which is approximately
$a_{\star}H_{\star}/\sqrt{2\pi}g_{\star}$. The effective Landau damping scale
is then most conveniently parameterized as [61]

$\displaystyle k_{L}$
$\displaystyle=\frac{\sqrt{6(1+R_{\star})}}{\sigma_{g}}=\frac{\sqrt{\omega_{r}}}{a_{\star}}H_{100}\frac{g_{\star}}{a_{\star}H_{\star}}\sqrt{12\pi(1+R_{\star})\left(1+x_{\mathrm{eq}}^{-1}\right)}.$
(18)

To the extent that $X_{e}$ is a fixed function of $x$ (such that
$g_{\star}/a_{\star}H_{\star}$ is unchanged) the Landau damping scale is
proportional to the aforementioned physical scales at fixed $R_{\star}$ and
$x_{\mathrm{eq}}$.

Figure 1 shows that $g_{\star}/a_{\star}H_{\star}$ is effectively independent
of $m_{e,i}$ (scaling only $\sim m_{e,i}^{-0.01}$); we empirically find that
varying $\alpha_{i}$ and $Y_{\mathrm{He}}$ independently yields

$\displaystyle\frac{g_{\star}}{a_{\star}H_{\star}}$ $\displaystyle\approx
4.45\left(\frac{\alpha_{i}}{\alpha_{0}}\right)^{0.61}\left(\frac{1-Y_{\mathrm{He}}}{1-0.245}\right)^{0.11}$
(19)

at fixed $R_{\star}$ and $x_{\mathrm{eq}}$. Therefore, $r_{s,\star}k_{L}$
scales with roughly one fewer power of $\alpha_{i}$ than does
$r_{s,\star}k_{D,\star}$. Since the Landau and diffusion damping scales are of
similar size (and combine in quadrature), measurements of the damping tail do
not cleanly translate to measurements of either of the individual parameter
combinations in Eqs. 17 and 19; however, CMB polarization is independently
sensitive to $g_{\star}/a_{\star}H_{\star}$ as discussed next.

####  II.1.4 Polarization

The amplitude of the CMB polarization spectrum relative to the temperature
spectrum is also sensitive to the duration of recombination [69]. Polarization
is generated by Thomson scattering when the temperature distribution has a
nonnegligible quadrupole. The quadrupole is sourced by the dipole of the
distribution, which is itself proportional to the time derivative of the
temperature fluctuation (the monopole) [69, 70]. Efficient Thomson scattering,
however, impedes the growth of anisotropy (like the quadrupole). The observed
CMB polarization is thus generated as the photons decouple, when scattering
still occurs and generates polarization but not so frequently that the
quadrupole is suppressed.

The average interval between scatterings around decoupling is comparable to
the width of the visibility function and sets the relative amplitude of the
polarization and temperature perturbations proportional to
$c_{s}k\sigma_{g}$—i.e., truly instantaneous recombination generates no
polarization. In other words, the longer recombination takes, the more the
quadrupole of the temperature distribution is able to grow, generating greater
polarization at last scattering. (Incidentally, the amplitude of the
polarization spectrum is also sensitive to the baryon-to-photon ratio
$R_{\star}$ via the factor of the sound speed $c_{s}$.) Thus, the polarization
amplitude uniquely probes the width of the visibility function, with parameter
dependence given by Eq. 19. Ref. [71] provides subpercent-level constraints
directly on the width of the visibility function, introduced as a
phenomenological extension of the $\Lambda$CDM model; the constraints are
driven to a comparable extent by the effects on Landau damping and on the
amplitude of the polarization spectrum. We therefore expect CMB polarization
data to provide complementary constraining power on variations in the fine-
structure constant. Because the shape of the visibility function varies
negligibly with the electron mass (per Fig. 1), we further anticipate that
polarization offers no unique information for electron-mass variations
compared to temperature data.

####  II.1.5 Angular power spectra

The angular power spectrum of the CMB measures the scale of the aforementioned
features—the sound horizon, the horizon at equality, and the diffusion and
Landau damping scales—only relative to the transverse comoving distance to
last scattering. In $\Lambda$CDM cosmologies with nonzero curvature
$\Omega_{k}$, the transverse comoving distance is [72]

$\displaystyle D_{M}(a)$
$\displaystyle\equiv\chi(a)\operatorname{sinc}\left[\sqrt{-\Omega_{k}}\frac{\chi(a)}{1/H_{0}}\right],$
(20)

where the line-of-sight comoving distance is

$\displaystyle\chi(a)$
$\displaystyle=\int_{a}^{a_{0}}\frac{\mathrm{d}\tilde{a}}{\tilde{a}}\,\frac{1}{\tilde{a}H(\tilde{a})}.$
(21)

The transverse distance reduces to the line-of-sight one [$D_{M}(a)=\chi(a)$]
for zero curvature, which we assume for the remainder of the paper. In a
Universe with only matter and dark energy, characterized by a scale factor of
matter-$\Lambda$ equality
$a_{m-\Lambda}=\sqrt[3]{\omega_{m}/\omega_{\Lambda}}$ and a Hubble constant
$h=\sqrt{\omega_{m}+\omega_{\Lambda}}$, the comoving distance Eq. 21
integrates to [73]

$\displaystyle\chi(a)$
$\displaystyle=\frac{2}{\sqrt{\omega_{m}}H_{100}}\left[F_{M}(1;a_{m-\Lambda})-F_{M}(a;a_{m-\Lambda})\right],$
(22)

where

$\displaystyle F_{M}(a;a_{m-\Lambda})$
$\displaystyle\equiv\sqrt{a}\cdot{}_{2}{F}_{1}(1/6,1/2;7/6,-[a/a_{m-\Lambda}]^{3})$
(23)

and ${}_{2}{F}_{1}$ is a hypergeometric function.

The precisely measured angular extent of the sound horizon is a ratio of two
scales, $\theta_{s}=r_{s,\star}/D_{M,\star}$, that each depend only on early-
or late-time physics. The support of the comoving distance integral in Eq. 21
is dominated at late times around matter-$\Lambda$ equality, since $aH$
decreases in the matter era but increases in dark-energy domination; the sound
horizon Eq. 9 is of course only sensitive to early times, dominated by the
period shortly before recombination. Attempts to explain the discrepancy in
$H_{0}$ inferences are thus typically categorized as early-time or late-time
solutions [74].

The sound horizon at recombination Eq. 12 may be written as

$\displaystyle r_{s,\star}$
$\displaystyle=\frac{2a_{\star}}{H_{100}\sqrt{3\omega_{r}}}F_{r_{s}}(R_{\star},x_{\mathrm{eq}}),$
(24)

where the dependence on $x_{\mathrm{eq}}$ and $R_{\star}$, which are
constrained by the shape of the CMB power spectrum (Section II.1.2), scales as
$F_{r_{s}}(R_{\star},x_{\mathrm{eq}})\propto
x_{\mathrm{eq}}^{0.247}R_{\star}^{-0.0964}$ when numerically approximated
about the best-fit $\Lambda$CDM parameters [75]. The comoving distance to last
scattering does not depend explicitly upon either $a_{\star}$ or $\omega_{r}$
but does so implicitly via the matter density when holding $x_{\mathrm{eq}}$
fixed. Replacing $\omega_{m}$ in Eq. 22 with
$\omega_{r}/a_{\star}x_{\mathrm{eq}}$ and taking the limit of a matter
Universe, for which $F_{M}(a;a_{m-\Lambda}\to\infty)\to\sqrt{a}$,
$\chi_{\star}\approx\chi(0)=2\sqrt{a_{\star}x_{\mathrm{eq}}}/\sqrt{\omega_{r}}H_{100}$.
In this limit,
$\theta_{s}=\sqrt{a_{\star}}F_{r_{s}}(R_{\star},x_{\mathrm{eq}})/\sqrt{3x_{\mathrm{eq}}}\propto\sqrt{a_{\star}}/x_{\mathrm{eq}}^{0.253}R_{\star}^{0.0964}$.
In reality, $F_{M}(1;a_{m-\Lambda})\propto(\sqrt{\omega_{m}}/h)^{0.193}$ about
the best-fit $\Lambda$CDM cosmology, which encodes the correction to the
distance to last scattering in Universes due to dark energy. Substituting for
$\omega_{m}$ as above [and accounting for the small difference between
$\chi_{\star}$ and $\chi(0)$],

$\displaystyle r_{s,\star}$
$\displaystyle\propto\frac{a_{\star}}{\sqrt{\omega_{r}}}x_{\mathrm{eq}}^{0.247}R_{\star}^{-0.0964}$
(25) and $\displaystyle\chi_{\star}$
$\displaystyle\propto\omega_{m}^{-0.403}h^{-0.193}a_{\star}^{-0.0173}=\frac{a_{\star}}{\sqrt{\omega_{r}}}\left(\frac{\sqrt{\omega_{r}}}{a_{\star}^{3.18}h}x_{\mathrm{eq}}^{2.09}\right)^{1/5.173}.$
(26)

Combining the two generalizes the well-known dependence of $\theta_{s}$ on
cosmological parameters [75, 76, 77] to

$\displaystyle\theta_{s}$
$\displaystyle\propto\left(\frac{a_{\star}^{3.18}h}{\sqrt{\omega_{r}}}x_{\mathrm{eq}}^{-0.810}R_{\star}^{-0.499}\right)^{1/5.173}\propto\left(\frac{T_{0}^{1.18}h}{\alpha_{i}^{6.36}m_{e,i}^{3.18}}\sqrt{\frac{\omega_{\gamma}}{\omega_{r}}}x_{\mathrm{eq}}^{-0.810}R_{\star}^{-0.499}\right)^{1/5.173},$
(27)

the second proportionality substituting for $a_{\star}$ with Eq. 6. Within the
$\Lambda$CDM model, $a_{\star}$ is effectively fixed and Planck [75] measures
$x_{\mathrm{eq}}$ and $R_{\star}$ with roughly $0.9\%$ and $0.7\%$
uncertainty, respectively, while $\theta_{s}$ is measured with a comparatively
negligible $0.03\%$ uncertainty. Holding $T_{0}$ and
$\omega_{r}/\omega_{\gamma}$ fixed, the uncertainty on $x_{\mathrm{eq}}$
therefore translates to a $\sim 0.7\%$ uncertainty on $h$, an estimate
reproduced in full parameter inference.

Any given parameterization of the primordial power spectrum must be rescaled
to account for the shift in correspondence between wave number (in arbitrary
units) and the scales that characterize the physical effects discussed above.
The standard power-law model,

$\displaystyle\Delta_{\mathcal{R}}^{2}(k)$
$\displaystyle=A_{s}\left(\frac{k}{k_{\mathrm{p}}}\right)^{n_{s}-1},$ (28)

is overparameterized with a normalization $A_{s}$ and pivot scale
$k_{\mathrm{p}}$, the latter of which is conventionally fixed. Thus, instead
of taking $k_{\mathrm{p}}$ to scale in tandem with other scales in the problem
(i.e. in proportion to $\sqrt{\omega_{r}}/a_{\star}$), we choose $A_{s}$ to
fix the value of $A_{s}\left(\sqrt{\omega_{r}}/a_{\star}\right)^{n_{s}-1}$.
Note that, because the Universe reionizes at late times, the observed
amplitude of CMB anisotropies is suppressed by a further factor of
$e^{-2\tau_{\mathrm{reion}}}$, where $\tau_{\mathrm{reion}}$ is the optical
depth to reionization. CMB observations on small scales therefore only
constrain the combination $A_{s}e^{-2\tau_{\mathrm{reion}}}$, but reionization
also enhances CMB polarization on large scales (multipoles $\ell\lesssim 30$),
breaking this degeneracy.

####  II.1.6 Degeneracies of the early Universe

We now summarize what parameter directions leave the primary CMB anisotropies
invariant per the preceding discussion. Namely, we identify degeneracies that
preserve the relative abundance of baryons and photons at recombination,

$\displaystyle R_{\star}$
$\displaystyle=\frac{3\rho_{b,\star}}{4\rho_{\gamma,\star}}\propto\frac{3\omega_{b}}{4\omega_{\gamma}}\frac{T_{0}}{\alpha_{i}^{2}m_{e,i}}\propto\frac{3\omega_{b}}{4\alpha_{i}^{2}m_{e,i}T_{0}^{3}}$
(29a) the relative abundance of matter and radiation at recombination,
$\displaystyle x_{\mathrm{eq}}$
$\displaystyle=\frac{\rho_{r,\star}}{\rho_{m,\star}}\propto\frac{\omega_{r}}{\omega_{m}}\frac{\alpha_{i}^{2}m_{e,i}}{T_{0}}\propto\frac{\alpha_{i}^{2}m_{e,i}T_{0}^{3}}{\omega_{m}}\frac{\omega_{r}}{\omega_{\gamma}}$
(29b) the Thomson scattering rate per Hubble time, $\displaystyle K_{\star}$
$\displaystyle\propto\alpha_{i}^{6}\sqrt{\frac{\omega_{\gamma}}{\omega_{r}}}\frac{1-Y_{\mathrm{He}}}{m_{\mathrm{H}}/M_{\mathrm{pl}}},$
(29c) the width of the dimensionless visibility function,
$\displaystyle\frac{g_{\star}}{a_{\star}H_{\star}}$
$\displaystyle\propto\alpha_{i}^{0.61}\left(1-Y_{\mathrm{He}}\right)^{0.11}$
(29d) and the angular size of the sound horizon, $\displaystyle\theta_{s}$
$\displaystyle\propto\left(\frac{T_{0}^{1.18}h}{\alpha_{i}^{6.36}m_{e,i}^{3.18}}\sqrt{\frac{\omega_{\gamma}}{\omega_{r}}}x_{\mathrm{eq}}^{-0.810}R_{\star}^{-0.499}\right)^{1/5.173}.$
(29e)

Equation 29 takes the scale factor of recombination to be proportional to
$T_{0}/\alpha_{i}^{2}m_{e,i}$, per Eq. 6. When all of these parameter
combinations and the ionization history are fixed, all the scales that encode
early-Universe physics—the inverse sound horizon $1/r_{s,\star}$, the horizon
at equality $k_{\mathrm{eq}}$, the diffusion damping scale $k_{D,\star}$, and
the Landau damping scale $k_{L}$—scale in proportion to
$\sqrt{\omega_{r}}/a_{\star}\propto\alpha_{i}^{2}m_{e,i}T_{0}\sqrt{\omega_{r}/\omega_{\gamma}}$.

Figure 2 illustrates the phenomenological impact of varying $R_{\star}$,
$x_{\mathrm{eq}}$, and $\theta_{s}$ on the CMB temperature power spectrum
$C_{\ell}^{TT}$ and in particular demonstrates the degeneracy achieved under
varying $m_{e,i}$.

Figure 2:  CMB temperature power spectrum in cosmologies varying the electron
mass $m_{e,i}$ by color and holding various combinations of $R_{\star}$ [Eq.
7], $x_{\mathrm{eq}}$ [Eq. 11], and $\theta_{s}$ [Eq. 27] fixed to their
values in the best-fit $\Lambda$CDM cosmology [75]. The top-left and top-right
panels fix all but $R_{\star}$ and $x_{\mathrm{eq}}$, respectively,
illustrating the impact of changing ratios of energy densities at
recombination. The bottom-left panel varies $\theta_{s}$, shifting the
spectrum to larger multipoles when recombination occurs earlier. The bottom-
right panel fixes them all, leaving the spectrum largely unchanged. The
angular size of the sound horizon is set by choosing $h$ via numerical
optimization; the bottom-left panel simply fixes $h$. Each panel also fixes
the helium mass fraction $Y_{\mathrm{He}}$, the combination
$A_{s}a_{\star}^{1-n_{s}}$, and the optical depth to reionization
$\tau_{\mathrm{reion}}$.

The role played by baryons in acoustic oscillations is clear in the panel
varying $R_{\star}$: the relative heights of the first peaks and troughs are
reduced at larger $m_{e,i}$, for which $R_{\star}\propto a_{\star}\propto
m_{e,i}^{-1}$, and therefore the amplitude of acoustic oscillations, is
smaller. Next, if recombination happens closer to matter-radiation equality
(as it does if a larger $m_{e,i}$ triggers recombination earlier, all else
equal), radiation has a larger effect on gravity; around recombination, the
metric potentials therefore decay relatively faster at horizon entry,
increasing the driving of acoustic oscillations. In addition, the decay of the
potentials due to radiation enhances the early ISW effect for earlier
recombination, since they are constant in a Universe with only cold dark
matter. Finally, Fig. 2 shows that holding both $R_{\star}$ and
$x_{\mathrm{eq}}$ fixed but not $\theta_{s}$ (i.e., holding $h$ constant
instead) effectively shifts the entire spectrum to larger multipoles when
recombination happens earlier.

The impact of the remaining parameters on CMB polarization and on small-scale
anisotropies is illustrated well when varying $\alpha_{i}$, shown in Fig. 3.

Figure 3:  CMB temperature (top panels) and $E$-mode (middle panels) auto
power spectra and their correlation (bottom panels) in cosmologies varying the
fine-structure constant $\alpha_{i}$ (left panels) or the electron mass
$m_{e,i}$ (right panels) by color. The impact on small-scale damping and
polarization is much more significant under variations of $\alpha_{i}$ than of
$m_{e,i}$, as encoded in Eq. 29. Each curve holds all of $R_{\star}$,
$x_{\mathrm{eq}}$, $\theta_{s}$, $Y_{\mathrm{He}}$,
$A_{s}a_{\star}^{1-n_{s}}$, and $\tau_{\mathrm{reion}}$ fixed.

The impact of small-scale damping on both temperature and polarization
anisotropies is dramatic: the Thomson scattering rate increases more rapidly
than the Hubble rate with the fine-structure constant, pushing the effects of
diffusion damping to smaller scales (and therefore larger multipoles). In
addition, recalling Fig. 1, the dimensionless visibility function also narrows
with larger $\alpha_{i}$, reducing the degree of Landau damping. A narrower
visibility function also reduces the amplitude of polarization, which is most
evident at $\ell\lesssim 800$ in the middle-left and lower-left panels of Fig.
3; at larger multipoles, the reduction in damping overcomes this suppression.
The $m_{e,i}$-independence of diffusion damping, Landau damping, and the
generation of polarization at fixed $R_{\star}$, $x_{\mathrm{eq}}$, and
$\theta_{s}$ is evident in the right panels of Fig. 3, where the changes to
the CMB spectra are imperceptible even for $15\%$ changes to the early-time
electron mass.

The preceding discussion facilitates the interpretation of degeneracies in a
variety of proposed extensions to $\Lambda$CDM. The degeneracies in scenarios
featuring early recombination via varying constants, discussed in Ref. [21],
fix $R_{\star}$ and $x_{\mathrm{eq}}$ by taking constant $a_{\star}\omega_{b}$
and $a_{\star}\omega_{m}$, as discussed in Section II.1.2. Per Eq. 27, holding
$\theta_{s}$ constant then requires $h\propto a_{\star}^{-3.18}$, as found in
Ref. [21]. As demonstrated in Ref. [21] and Section II.1.3, no additional
parameter freedom is required to preserve the diffusion damping scale only
when varying the electron mass, for which $a_{\star}\propto m_{e,i}^{-1}$.
Furthermore, the ionization history is effectively invariant (as a function of
$a/a_{\star}$) only under varying $m_{e,i}$, in which case the diffusion and
Landau damping scales imprint at the same angular scale on the sky (i.e.,
$r_{s,\star}k_{D,\star}$ and $r_{s,\star}k_{L}$ invariant) and polarization is
generated with the same amplitude, as evident in Fig. 3. We therefore expect
that early-time variations in the fine-structure constant are much more
severely constrained by primary CMB data than the electron mass.

Another class of early-recombination models postulates substantial small-scale
inhomogeneities in the baryon distribution, which enhance the recombination
rate [78, 79, 80, 81]. These changes, however, do not preserve the angular
extent of the diffusion damping feature [79] as is possible with a varying
electron mass; presumably the shape of the visibility function is also
affected, with consequences described in Sections II.1.3 and II.1.4.
Similarly, though Eq. 27 shows that $\theta_{s}$ is preserved when increasing
$N_{\mathrm{eff}}$ (i.e., $\omega_{r}/\omega_{\gamma}$) and $h$
simultaneously, the diffusion damping rate [Eq. 17] is not [82, 83, 84].
Changes to the damping tail may be partially absorbed by the primordial tilt
$n_{s}$, but not without involving the overall amplitude as well as the baryon
abundance; varying the fraction of the radiation density that is collisionless
has additional, well-studied physical effects on the CMB [85, 86, 87, 88, 40].
Equation 29c shows that the effect of a change to the total radiation density
on the damping rate may be compensated not just by adjusting the helium yield
$Y_{\mathrm{He}}$, a commonly studied degeneracy, but also by altering the
early-time values of the fine-structure constant or the hydrogen mass.

The scale factor of recombination would also be different if we instead
suppose $T_{0}$ differs from its measured value. Ref. [38] phrases the CMB’s
dependence on $\omega_{b}$, $\omega_{c}$, and $T_{0}$ as being via the
combinations $\omega_{b}/T_{0}^{3}$ and $\omega_{c}/T_{0}^{3}$. Since
$T_{0}^{3}\propto\omega_{\gamma}/T_{0}=\omega_{\gamma}/a_{\star}T_{\star}$,
holding $\omega_{b}/T_{0}^{3}$, $\omega_{c}/T_{0}^{3}$, and $T_{\star}$ fixed
also fixes $R_{\star}$ and $x_{\mathrm{eq}}$. Per Eq. 17, the diffusion
damping parameter $K_{\star}$ is invariant along this direction because both
of $\sqrt{\omega_{\gamma}}/a_{\star}^{2}\propto T_{\star}^{2}$ and
$\omega_{\gamma}/\omega_{r}$ are unchanged. Since changing $T_{0}$ does not
affect physics during recombination (but rather the amount of expansion that
has elapsed since then), the visibility function is also unchanged [as a
function of $\ln(a/a_{\star})$]. Equation 27 shows that $\theta_{s}$ is fixed
if $h\propto T_{0}^{-1.18}$, matching the degeneracy direction identified in
Ref. [38]. The effects of $T_{0}$ and $m_{e,i}$ are only distinguished by the
latter’s effect on the total radiation density at recombination, explaining
why $h$ scales differently with $T_{0}$ and $m_{e,i}$ at fixed $\theta_{s}$.

The degeneracies discussed in this section are grounded in a scaling
invariance of cosmological observables. Because the dynamics of perturbations
in the early Universe is linear, the Boltzmann and Einstein equations
governing their evolution depend only on the ratio of scales, i.e., are
invariant under a common rescaling of the length scales appearing in the
problem. Moreover, the initial conditions preferred by current data are, aside
from being nearly scale invariant, also scale free: any rescaling of length
scales may be absorbed into the over normalization of the primordial power
spectrum. Because most cosmological observables are measured as features on
the sky, i.e., as dimensionless ratios of length scales, they do not measure
the value of any individual scale. The parameters of Eq. 29 enumerate the
general requirements for a common rescaling of all length scales specifically
relevant to early-Universe dynamics, which are met by varying the scale factor
of recombination (via $m_{e,i}$ or $T_{0}$ per the above discussion).

A distinct class of models, in which the scale factor of recombination is
unchanged, was explored in detail by Refs. [39, 40, 41] by taking
$\omega_{i}\propto\lambda^{2}$ for all species $i$,
$\sigma_{T}n_{e}\propto\lambda$, and $A_{s}\propto\lambda^{1-n_{s}}$. A simple
realization thereof is to vary both $T_{0}$ and $m_{e,i}$ proportionately, in
which scaling the density of all species $\propto T_{0}^{4}$ preserves all of
the parameter combinations in Eq. 29 without requiring changes to any other
parameters. Given that direct measurements of $T_{0}$ preclude this
possibility, Refs. [39, 40, 41] invoke a “mirror” dark sector with the same
particle content and parameters as the SM to allow for arbitrarily increasing
the abundances $\omega_{i}$, meaning the abundance of the visible sector does
not scale with $\lambda$. The proportional scaling of the total matter and
radiation densities preserves $x_{\mathrm{eq}}$ as well as $\theta_{s}$ (since
$h^{2}\propto\sum_{i}\omega_{i}$), while the nonscaling of the visible photon
and baryon densities preserves $R_{\star}$. Constancy of the damping parameter
$K_{\star}$, however, requires an additional ingredient. [Note that in Eq. 16,
$\omega_{\gamma}$ and $Y_{\mathrm{He}}$ are those for the visible sector,
while $\omega_{r}$ is the radiation density of the entire Universe.] Refs.
[39, 40] invoke a differing helium yield $Y_{\mathrm{He}}$ to achieve the
requisite scaling of the Thomson scattering rate, while Ref. [41] instead
takes $m_{e,i}\propto\alpha_{i}^{-2}\propto\lambda^{-1/3}$ to simultaneously
fix $K_{\star}$ and $a_{\star}$.

The symmetry of the Einstein-Boltzmann equations is broken by nonequilibrium
effects during recombination that alter the ionization history, but only
mildly so. (The symmetry is also broken by nonzero neutrino masses, but since
they are only bounded and not measured, one is free to rescale them as well.)
However, each of the aforementioned possible realizations is constrained by
direct measurements of either $T_{0}$ or $Y_{\mathrm{He}}$, including that
varying $m_{e,i}\propto\alpha_{i}^{-2}\propto\lambda^{-1/3}$ because the BBN
prediction for the helium yield is not invariant under this transformation
[32]. Moreover, the arguments of this section demonstrate that the full
rescaling symmetry is more restrictive than necessary to keep early-time
dynamics invariant. Namely, distinct degeneracy directions are afforded by
freeing the scale factor of recombination $a_{\star}/a_{0}$, whether by
varying $T_{0}$ or $T_{\star}$. But such scenarios do not achieve the full
rescaling symmetry of the Einstein-Boltzmann system. They are distinguished by
effects arising at late times, which we turn to now.

###  II.2 Late-Universe signatures

The dynamics of the late Universe are driven by the dark sector and so do not
depend explicitly on the fundamental constants. However, given the
constraining power of the early universe observables discussed in the previous
subsection, $\Lambda$CDM parameters—specifically the relative contributions of
matter and dark energy to the total energy budget—change along the degeneracy
direction as fundamental constants are varied. The discussion thus far has
entirely neglected dark energy beyond its effect on the distance to last
scattering and the sound horizon, Eq. 27. We now focus on the late-time
effects of dark energy.

In models that are well described by flat $\Lambda$CDM cosmology at late
times, the shape of the late-time expansion history is only characterized by
the scale factor of matter–dark-energy equality,

$\displaystyle a_{m-\Lambda}$
$\displaystyle\equiv\sqrt[3]{\omega_{m}/\omega_{\Lambda}}.$ (30)

Along the degeneracy direction that preserves the primary CMB spectra,
$\omega_{m}\propto\omega_{r}/a_{\star}$ but
$h\approx\sqrt{\omega_{m}+\omega_{\Lambda}}\propto\sqrt{\omega_{r}}/a_{\star}^{3.18}$.
The dark energy density thus increases much more steeply than the matter
density along the degeneracy, drastically altering when matter–dark-energy
equality occurs and the physical effects that depend upon it. Matter
perturbations grow more slowly in a Universe dominated by dark energy than one
dominated by matter; Section II.2.1 discusses the resulting impact on
secondary signatures in the CMB and Section II.2.4 that on matter clustering.
Datasets that measure distances at low redshift directly trace out the shape
of the expansion history and provide independent constraints; these are
discussed in Sections II.2.2 and II.2.3.

####  II.2.1 Late-time integrated Sachs-Wolfe effect and CMB lensing

While dark energy is entirely negligible near recombination when most of the
CMB anisotropies are seeded, at late times its presence alters the dynamics of
metric perturbations, sourcing a nonzero integrated Sachs-Wolfe effect and
altering the degree to which the primary CMB anisotropies are gravitationally
lensed. In a Universe dominated by cold dark matter, the metric potentials (in
Newtonian gauge, for instance) are constant. As CMB photons traverse potential
wells, the amount by which they blueshift upon entering is precisely
compensated by the corresponding redshift as they exit. If large-scale metric
potentials instead decay with time, which they do to an extent controlled by
the relative abundance of dark energy [89, 90], this cancellation is
incomplete, enhancing CMB power. The earlier the dark energy era begins (i.e.,
the smaller $a_{m-\Lambda}$ is), the larger this effect. Because matter–dark-
energy equality occurs rather near the present, this late ISW effect affects
only the largest observable scales—at multipoles $\ell\lesssim 30$ where
cosmic variance strongly limits the CMB’s constraining power.

The CMB is also lensed by matter fluctuations at late times [91], which grow
more after horizon crossing in Universes where matter–dark-energy equality
occurred later. Whereas an increased absolute change in metric perturbations
over time results in an enhanced ISW signal on large scales, the lensing
deflection spectrum scales in proportion to the power in matter fluctuations.
Both effects are depicted in Fig. 4.

Figure 4:  Large-scale CMB temperature (left) and lensing (right) power
spectra in cosmologies varying the electron mass $m_{e,i}$ by color, holding
all of $R_{\star}$, $x_{\mathrm{eq}}$, $\theta_{s}$, $Y_{\mathrm{He}}$,
$A_{s}a_{\star}^{1-n_{s}}$, and $\tau_{\mathrm{reion}}$ fixed. The changes in
the power spectra arise due to the correlation of $a_{m-\Lambda}$ with
$a_{\star}\propto 1/m_{e,i}$ rather than directly from processes that depend
on the constants’ values. Therefore, these results are broadly unchanged for
commensurate variations in $\alpha_{i}^{2}$ as for $m_{e,i}$ and are
effectively identical under variations in $T_{0}$ that yield the same values
of $a_{m-\Lambda}$ when holding $\theta_{s}$ and $x_{\mathrm{eq}}$ fixed.

The disproportionate increase in the dark energy density required to decrease
the distance to last scattering in proportion to the sound horizon [Eq. 27]
makes the dark energy era begin earlier, leading to the suppression of lensing
power and enhancement of the late ISW effect evident in Fig. 4.

####  II.2.2 Baryon acoustic oscillations

Late-time changes to the universe’s energy budget may also be constrained
directly via distance measurements. First, baryon acoustic oscillations
imprint an oscillatory pattern in the matter correlation function that may be
treated as a standard ruler. The BAO scale specifically corresponds to a local
maximum of the matter density correlation function, but it can be accurately
modeled by simply evaluating the comoving sound horizon evaluated at the
baryon drag epoch, $r_{\mathrm{d}}\equiv r_{s}(a_{\mathrm{drag}})$ [92, 93].
Baryons decouple slightly after recombination, $a_{\mathrm{drag}}>a_{\star}$,
because the photon number density is so much greater than that of the baryons,
so $r_{\mathrm{d}}$ is slightly larger than $r_{s,\star}$.

The angular BAO scales are extracted from various galaxy surveys and reported
as

$\displaystyle\theta_{\mathrm{BAO}}(a)$
$\displaystyle\equiv\frac{r_{\mathrm{d}}}{D_{V}(a)}$ (31a)
$\displaystyle\theta_{\mathrm{BAO},\parallel}(a)$
$\displaystyle\equiv\frac{r_{\mathrm{d}}}{cH(a)^{-1}}$ (31b)
$\displaystyle\theta_{\mathrm{BAO},\perp}(a)$
$\displaystyle\equiv\frac{r_{\mathrm{d}}}{D_{M}(a)}.$ (31c)

(We retain factors of $c$ in this subsection to clearly specify units.) In Eq.
31a,

$\displaystyle D_{V}(a)$
$\displaystyle=\sqrt[3]{\frac{D_{M}(a)^{2}}{H(a)/c}\left(1/a-1\right)}$ (32)

is an effective, comoving-volume–averaged distance to a galaxy sample and the
transverse distance $D_{M}$ in Eq. 31c is defined in Eq. 20. While these
measurements rely on template modeling in a fiducial cosmology, Ref. [94]
showed that any bias in application to modified cosmological models is
negligible for existing datasets.

Substituting Eq. 22 for the comoving distance in Eq. 31c shows that BAO
distances depend on $H_{0}$ only through an overall scaling in the combination
$r_{\mathrm{d}}H_{0}$:

$\displaystyle\theta_{\mathrm{BAO},\perp}(a)$
$\displaystyle=\frac{r_{\mathrm{d}}h}{c/H_{100}}\frac{\left[F_{M}(1;a_{m-\Lambda})-F_{M}(a;a_{m-\Lambda})\right]^{-1}}{2\sqrt{1+a_{m-\Lambda}^{-3}}}.$
(33)

The dependence on the scale factor $a$ is through a (nonlinear) function
depending on cosmological parameters only via $a_{m-\Lambda}$. A sufficient
number of BAO measurements, spanning a sufficient range of scale factors about
$a_{m-\Lambda}$, could therefore constrain $a_{m-\Lambda}$ independently of
$r_{\mathrm{d}}$ and $H_{0}$ individually. (The same conclusions apply for
$\theta_{\mathrm{BAO}}$ and $\theta_{\mathrm{BAO},\parallel}$.) Neglecting the
small difference between $r_{\mathrm{d}}$ and $r_{s,\star}\propto
a_{\star}/\sqrt{\omega_{r}}$ [Eq. 25], the overall scaling of Eq. 33 then
requires that $h\propto\sqrt{\omega_{r}}/a_{\star}$, which differs
substantially from $h\propto\sqrt{\omega_{r}}/a_{\star}^{3.18}$ as required by
the CMB [Eq. 27]. For models that only vary the scale factor of recombination
(whether via $T_{0}$ or early-time variations in fundamental constants), BAO
data therefore offer constraining power in the $h$–$a_{\star}$ plane
complementary to that of the CMB.

####  II.2.3 Low-redshift distances from supernovae

Type Ia supernovae (SNe Ia) occur at a characteristic mass that yields a
standard luminosity suitable for inferring distances. The apparent magnitude
of each SN Ia (labeled $i$) is parameterized as [95, 96, 97]

$\displaystyle m_{i}$
$\displaystyle=5\log_{10}\left[\frac{a_{i}}{a_{i,\mathrm{hel}}}\frac{D_{L}(a_{i})}{10\leavevmode\nobreak\
\mathrm{pc}}\right]+M_{B},$ (34)

where $M_{B}$ is the fiducial magnitude of an SN Ia (a parameter that itself
must be sampled over) and $D_{L}$ the luminosity distance,
$D_{L}(a)=D_{M}(a)/a$. The ratio of the CMB-frame ($a_{i}$) and heliocentric
($a_{i,\mathrm{hel}}$) redshifts inside the logarithm corrects for peculiar
velocities [98, 99, 100]; SNe distance datasets tabulate both these redshifts
and either the apparent magnitude $m_{i}$ or the distance modulus
$m_{i}-M_{B}$. Absorbing the peculiar velocity correction into the apparent
magnitude via the definition $\hat{m}_{i}\equiv
m_{i}-5\log_{10}(a_{i}/a_{i,\mathrm{hel}})$, we may rearrange Eq. 34 into the
apparent brightness, which may be interpreted as a “luminosity angle” in
analogy to the BAO angles Eq. 31:

$\displaystyle\theta_{L}(a_{i})\equiv\frac{10\leavevmode\nobreak\
\mathrm{pc}\cdot 10^{-M_{B}/5}}{D_{L}(a_{i})}$
$\displaystyle=10^{-\hat{m}_{i}/5}.$ (35)

The numerator of the middle expression acts as a standardized or calibrated
luminosity distance.

Akin to BAO distances, without external constraints on the fiducial magnitude
$M_{B}$ (from, e.g., Cepheids [24, 25, 26] or the tip of the red giant branch
[101, 102, 103, 104]), SN Ia distances only probe the shape of the expansion
history, not its absolute scale (i.e., $H_{0}$). The “luminosity angle” [Eq.
35] factorizes in the same manner as Eq. 33:

$\displaystyle\theta_{L}(a_{i})$
$\displaystyle=\frac{10^{\log_{10}h-M_{B}/5}}{c\leavevmode\nobreak\
\mathrm{s}/\mathrm{m}}\frac{a_{i}\left[F_{M}(1;a_{m-\Lambda})-F_{M}(a_{i};a_{m-\Lambda})\right]^{-1}}{2\sqrt{1+a_{m-\Lambda}^{-3}}}.$
(36)

The Hubble constant is only constrained via the combination
$\log_{10}h-M_{B}/5$, and as for BAO a sufficient number of SN Ia distance
measurements may constrain the relative amount of dark energy and matter
independent of their overall calibration (and of $H_{0}$).

####  II.2.4 Matter clustering

The clustering of matter is sensitive to the late-Universe expansion history,
in addition to nonstandard dynamics of dark matter and dark energy. The
amplitude of structure is often quantified by the present-day, root-mean-
squared overdensity in spheres of radius $R$, $\sigma(R)$, defined by

$\displaystyle\sigma(R)^{2}$ $\displaystyle=\int\mathrm{d}\ln
k\,W(k,R)^{2}\frac{k^{3}}{2\pi^{2}}P_{\mathrm{lin}}(k),$ (37)

where $P_{\mathrm{lin}}$ is the linear matter power spectrum and

$\displaystyle W(k,R)$
$\displaystyle=\frac{3}{(kR)^{3}}\left[\sin(kR)-kR\cos(kR)\right]$ (38)

is the Fourier transform of a top-hat function with radius $R$. A standard
metric is $\sigma_{8}\equiv\sigma(8\,h^{-1}\leavevmode\nobreak\
\mathrm{Mpc})$, which is most sensitive to scales that entered the horizon in
the radiation era; constraints, however, are often quoted as combinations of
$\sigma_{8}$ and $\Omega_{m}$ that are best constrained by particular probes;
for example, $S_{8}\equiv\sqrt{\Omega_{m}/0.3}\,\sigma_{8}$ for cosmic shear
[105, 106, 107].

The clustering of matter is sensitive to the expansion history via two
effects. First, the onset of the dark energy era suppresses structure growth;
the earlier matter–dark-energy equality ($a_{m-\Lambda}$) occurs, the stronger
the effect. Second, even at fixed $a_{m-\Lambda}$ increasing the total matter
density $\omega_{m}$ pushes the comoving horizon at matter-radiation equality
(which determines the turnover in the matter power spectrum from growing with
$\sim k$ to falling off as $\sim k^{-3}$) to smaller scales
$k_{\mathrm{eq}}\propto\omega_{m}$ [per Eq. 13], regardless of $a_{\star}$.
When fixing $a_{m-\Lambda}$ by taking $\omega_{\Lambda}\propto\omega_{m}$, $h$
scales with $\sqrt{\omega_{m}}$ and $k_{\mathrm{eq}}\cdot
8h^{-1}\leavevmode\nobreak\ \mathrm{Mpc}\propto\sqrt{\omega_{m}}$. In other
words, the earlier the matter era begins, the less suppressed the scales that
contribute to Eq. 37 are by their subhorizon evolution during the radiation
era, enhancing $\sigma_{8}$. As such, absent modifications to the dynamics of
dark matter or the early-Universe expansion history, $h$ and $\sigma_{8}$ are
generally positively correlated.

##  III Cosmology with coupled, hyperlight scalars

In the previous section, we discuss expanded phenomenology of cosmologies with
varying fundamental constants without considering a concrete model. A “bottom-
up” approach promotes a fundamental constant to a dynamical scalar described
by an action principle; the scalar’s value dictates that of the constants and
it carries stress-energy [108, 109, 110]. Such scalars also arise in theories
with extra dimensions as so-called dilaton or moduli fields [111, 112, 113,
114, 115, 116, 117, 118, 119, 120]. To realize the scenarios discussed in
Section II we consider one such simple model, that of a massive scalar whose
amplitude modulates the strength of electromagnetism $\alpha$ and the mass of
the electron $m_{e}$. We take the field to be so light as to have a
cosmologically frozen amplitude until some time after recombination, leading
to the fundamental constants taking on time-independent values prior to
recombination that differ from those measured today.

In Section III.1, we review the gravitational impact of the scalar on the
background (i.e., spatially homogeneous) cosmology. In particular, we identify
a new degeneracy in cosmological parameters when one allows for an increase in
the total energy density of matter after recombination relative to that which
was present at earlier times. While such a “late dark matter” scenario is
necessarily implemented by a hyperlight scalar field that modifies the value
of fundamental constants at earlier times, it could also be realized via other
mechanisms; massive neutrinos are an additional example. In Section III.2 we
expose how the necessary impact of the scalar’s spatial perturbations on the
growth of structure ultimately affects cosmological observables. Finally, in
Section III.3, we connect the phenomenological parameters of the modified
cosmology (energy densities and fundamental constants) with microphysical
model parameters (the mass and amplitude of the field). In combination with
Ref. [32], we find that varying-constant cosmologies can be implemented with
microphysical parameters that are as yet unconstrained by other probes.

Studying the cosmological impact of a scalar field requires solving the
covariant Klein-Gordon equation in the perturbed, expanding spacetime [Eq.
59]:

$\displaystyle\nabla^{\mu}\nabla_{\mu}\phi$
$\displaystyle=\frac{\mathrm{d}V(\phi)}{\mathrm{d}\phi}.$ (39)

Numerical solutions for massive scalar fields [i.e., with potential
$V(\phi)=m_{\phi}^{2}\phi^{2}$] provide a unique challenge by introducing a
fixed timescale, i.e., the scalar’s oscillation period $1/m_{\phi}$. As we
detail in full in Appendix A, we extend CLASS by implementing a scheme
recently devised in Ref. [121] that solves the Klein-Gordon equation for some
number of oscillations before switching to an effective fluid approximation
scheme. The prescription in Ref. [121] for matching onto fluid variables and
determining the fluid’s effective equation of state and sound speed provides
systematically improved precision compared to prior approaches. As described
in Appendix A, we extend Ref. [121]’s results, which were applied only to
scalars that begin oscillating in radiation domination, to a generalized
background expansion rate [122].

In addition, fundamental constant variations due to a scalar field inevitably
acquire spatial dependence, at least to some degree. The resulting fifth-force
effects on the dynamics of electron density perturbations may also be ignored
because electron density perturbations themselves, again neglecting their
suppressed contribution to the Einstein equations, only enter the Boltzmann
equation for photons at second order in inhomogeneities [123, 124]. That is,
anisotropies generated by Thomson scattering are already first order because
scattering in a homogeneous plasma generates no anisotropy; inhomogeneities in
the Thomson scattering rate (whether due to inhomogeneities in the electron
number density or in the Thomson cross section itself) therefore enter at
second order.

###  III.1 Impact on the cosmological background

To solve for the classical evolution of the scalar perturbatively, we expand
the scalar into a homogeneous component and a small perturbation as
$\phi(t,\mathbf{x})=\bar{\phi}(t)+\delta\phi(t,\mathbf{x})$; the homogeneous
component of the Klein-Gordon equation Eq. 39 is then

$\displaystyle\ddot{\bar{\phi}}+3H\dot{\bar{\phi}}+m_{\phi}^{2}\bar{\phi}$
$\displaystyle=0.$ (40)

The scalar contributes to the Friedmann equations an energy density
$\bar{\rho}_{\phi}=\dot{\bar{\phi}}^{2}/2+m_{\phi}^{2}\bar{\phi}^{2}/2$ and a
pressure $\bar{P}_{\phi}=\dot{\bar{\phi}}^{2}/2-m_{\phi}^{2}\bar{\phi}^{2}/2$.

If the field is sufficiently light compared to the Hubble rate at
recombination, $H_{\star}\approx 3\times 10^{-29}\leavevmode\nobreak\
\mathrm{eV}$, the scalar’s effect on the fundamental constants is effectively
time independent over the Universe’s entire history up to (and including)
recombination. Note that when a scalar’s couplings to matter are large enough,
its equation of motion can be significantly modified from Eq. 40. Here we
implicitly assume that these contributions are negligible; we consider the
possibility and its impact on scalar field phenomenology in Ref. [32]. Because
the late-time probes discussed in Ref. [32] provide relatively strong
constraints on fundamental-constant variations at late times and the present
day, the scalar (if it shifts the early-time constants from their present-day
values) cannot be too light—it must be rolling at some point so that its
oscillation amplitude redshifts by a sufficient amount by the present day.
Since in standard cosmology recombination occurs after matter-radiation
equality and matter–dark-energy equality was relatively recent, the relevant
parameter space for the scenarios we discuss requires the scalar to begin
oscillating in the matter-dominated era.

More concretely, at early times ($H\gg m_{\phi}$) the scalar is approximately
frozen at its initial value $\bar{\phi}_{i}$, i.e., the appropriate initial
condition for Eq. 40 is $\bar{\phi}\to\bar{\phi}_{i}$ and $\dot{\bar{\phi}}\to
0$ for $m_{\phi}t\to 0$. In a matter-dominated Universe with $H=2/3t$, the
solution to Eq. 40 with these initial conditions is
$\bar{\phi}(t)=\bar{\phi}_{i}\sin(m_{\phi}t)/m_{\phi}t$. This solution
motivates defining the time when oscillations begin by
$H(t_{\mathrm{osc}})=2/3t_{\mathrm{osc}}=2m_{\phi}/3$. We further define
$a_{\mathrm{osc}}$ as the scale factor and $z_{\mathrm{osc}}$ as the redshift
at that time. Well before $t_{\mathrm{osc}}$, the scalar has negligible
kinetic energy compared to that of its potential and therefore gravitates (at
the background level) like dark energy. However,
$\bar{\rho}_{\phi}/\bar{\rho}_{m}\propto(a/a_{\mathrm{osc}})^{3}$ in this
period, so its contribution to the Einstein equations is entirely negligible
until just before $t_{\mathrm{osc}}$. Noting that $a(t)\propto t^{2/3}$ in the
matter era, after $t_{\mathrm{osc}}$ the scalar’s oscillation-averaged energy
density and pressure are, to leading order in $1/m_{\phi}t$,

$\displaystyle\bar{\rho}_{\phi}$
$\displaystyle\approx\frac{m_{\phi}^{2}\bar{\phi}_{i}^{2}}{2(a/a_{\mathrm{osc}})^{3}}$
(41) $\displaystyle\bar{P}_{\phi}$ $\displaystyle\approx 0,$ (42)

just like nonrelativistic matter. (Section A.2 discusses the leading
corrections to these results.) We define
$\omega_{\phi}=\bar{\rho}_{\phi}(t_{0})/3H_{100}^{2}M_{\mathrm{pl}}^{2}$ as
for the other density parameters.

The most important gravitational effect of the scalar on the background
cosmology is therefore its delayed contribution to the matter density at late
times, illustrated in Fig. 5.

Figure 5:  Relative abundances as a function of time,
$\Omega_{X}(t)=\bar{\rho}(t)/3H^{2}M_{\mathrm{pl}}^{2}$, of matter (orange),
radiation (red), dark energy (green), and a hyperlight scalar field (blue).
Here the scalar has a mass $m_{\phi}=10^{-29}\leavevmode\nobreak\ \mathrm{eV}$
and an abundance
${F_{\phi}}\equiv\bar{\rho}_{\phi,0}/\bar{\rho}_{c,0}=10^{-2}$ relative to
that of CDM.

So long as the scalar begins rolling well before matter–dark-energy equality,
any cosmological distance may effectively be computed in a pure $\Lambda$CDM
cosmology that counts the scalar field’s energy density toward the total CDM
density. Certainly any distance measurement at $a\ll a_{\mathrm{osc}}$ is well
described in this approximation. The integral that defines the comoving
distance, Eq. 21, has most of its support where $1/aH$ is largest; in the
matter- and dark-energy–dominated limits, $1/aH\propto a^{1/2}$ and $a^{-1}$,
respectively, placing the peak of the integrand at the epoch of equality. More
concretely, the small-$a$ limit of Eq. 23, $F_{M}(a,a_{m-\Lambda})\propto
a^{7/2}$, quantifies the (un)importance of early time deviations from a pure
$\Lambda$CDM cosmology on the full comoving distance in Eq. 22. As such, the
comoving distance to recombination may also be approximated by simply
augmenting the CDM density to include the scalar’s, ignoring its differing
dynamics before $a_{\mathrm{osc}}\ll a_{m-\Lambda}$. (The scale factor of
equality $a_{m-\Lambda}$ is also implicitly defined with reference to this
combined, late-time matter density.)

Defining the late-to-early matter density ratio $A_{m}$ and inserting
$\omega_{m,\mathrm{late}}=A_{m}\omega_{m,\mathrm{early}}$ for the matter
density in the comoving distance to recombination [Eq. 26], the parameter
dependence of $\theta_{s}$ in Eq. 27 extends to

$\displaystyle\theta_{s}$
$\displaystyle\propto\left(\frac{a_{\star}^{3.18}hA_{m}^{2.09}}{\sqrt{\omega_{r}}}x_{\mathrm{eq}}^{-0.810}R_{\star}^{-0.499}\right)^{1/5.173}.$
(43)

The additional freedom afforded by $A_{m}$ enables preserving both
$a_{\star}^{3.18}hA_{m}^{2.09}/\sqrt{\omega_{r}}$ and

$\displaystyle a_{m-\Lambda}$
$\displaystyle=\left(\frac{A_{m}\omega_{m,\mathrm{early}}}{\omega_{\Lambda}}\right)^{1/3}=\left(\frac{1}{h^{2}/A_{m}\omega_{m,\mathrm{early}}-1}\right)^{1/3}$
(44)

at any value of $a_{\star}$, i.e., $h$ and $A_{m}$ provide two parameters to
satisfy both conditions. (We take $\omega_{r}$ to be fixed here, though our
arguments generalize straightforwardly to scenarios that vary $T_{0}$ rather
than $T_{\star}$.) The latter condition is more algebraically tractable when
phrased as requiring $\Omega_{m}=A_{m}\omega_{m,\mathrm{early}}/h^{2}\propto
A_{m}/a_{\star}x_{\mathrm{eq}}h^{2}$ to be fixed. The resulting degeneracy
direction has $A_{m}\propto a_{\star}^{-1.034}$ and $h\propto
a_{\star}^{-1.017}$.

This degeneracy direction _also_ nearly exactly preserves $hr_{\mathrm{d}}$ as
constrained by BAO distance measurements when $a_{m-\Lambda}$ is also fixed
[per Eq. 33]. Measurements of the BAO scale at various redshifts—including
that by the CMB at recombination—trace out the expansion history using a
common ruler; distances probing either side of matter–dark-energy equality may
only jointly agree at an arbitrary size of the sound horizon if
$a_{m-\Lambda}$ remains the same. In other words, the absolute scale (via $h$)
and shape (via $a_{m-\Lambda}$) of the late-Universe expansion history are
both free to change to satisfy that any individual BAO measurement (or
$\theta_{s}$), but fitting multiple simultaneously requires the shape
($a_{m-\Lambda}$) to be unchanged. Furthermore, as argued in Section II.2.3 SN
Ia distance measurements (sans calibration) themselves are only sensitive to
$a_{m-\Lambda}$ (within flat the $\Lambda$CDM-like models considered here).

Prior literature studying electron mass variation as a solution to the Hubble
tension [21, 22, 23] considered allowing for spatial curvature to address the
inconsistency between the distance to recombination and late-Universe
distances. The analysis here points to additional matter at late times as a
more natural solution, even on an ad hoc basis, which in addition avoids
confrontation with the expectation in inflationary cosmologies that spatial
curvature is negligible. Whether models with scalar fields fulfilling these
cosmologically motivated requirements are theoretically and phenomenologically
tenable first requires mapping the parameterization of their cosmological
effects ($A_{m}$ and the early-time $m_{e,i}$) to their fundamental
parameters.

The background-level effect of the scalar field closely resembles that due to
massive neutrinos. Namely, a single neutrino species with mass $m_{\nu}$ and
temperature $T_{\nu}=\sqrt[3]{4/11}\sqrt[4]{N_{\mathrm{eff}}/3}T_{\gamma}$
becomes exactly semirelativistic at a redshift

$\displaystyle
z_{T_{\nu}=m_{\nu}}+1=\frac{m_{\nu}}{T_{\nu}(a_{0})}=356.56\,\frac{m_{\nu}}{0.06\leavevmode\nobreak\
\mathrm{eV}}\left(\frac{N_{\mathrm{eff}}}{3.046}\right)^{-1/4}$ (45)

and has a present-day density of

$\displaystyle\omega_{\nu,\mathrm{massive}}=\frac{3\zeta(3)}{2\pi^{2}}\frac{T_{\nu}^{3}}{3H_{100}^{2}M_{\mathrm{pl}}^{2}}$
$\displaystyle=\frac{m_{\nu}}{93\leavevmode\nobreak\
\mathrm{eV}}\left(\frac{N_{\mathrm{eff}}}{3.046}\right)^{3/4}.$ (46)

Neutrino oscillation experiments require the sum of the neutrino masses to be
greater than $0.06\leavevmode\nobreak\ \mathrm{eV}$ [125, 126, 127], while
Planck combined with BAO data places an upper limit of
$0.12\leavevmode\nobreak\ \mathrm{eV}$ (within $\Lambda$CDM) [75]; the
heaviest neutrino species thus becomes nonrelativistic no earlier than $z\sim
700$ but no later than $z\sim 100$. Massive neutrinos therefore contribute to
the matter density only well after recombination (but still well before
matter–dark-energy equality) with an abundance proportional to their mass,
achieving the same effect encoded by $A_{m}$ in Eqs. 43 and 44. The neutrino
mass therefore plays the same role as $\omega_{\phi}$ for the scalar field; if
the sum of the neutrino masses is taken as a free parameter (rather than being
fixed to a fiducial, minimal value as in most analyses) in addition to the
early-time electron mass, we would expect the two to be positively correlated
when considering CMB data in conjunction with low-redshift distance datasets.
This correlation was indeed observed in Ref. [23]; its physical origin is
explained by the arguments of this section.

However, the neutrinos’ masses determine both their abundance and the redshift
at which they become matterlike (whereas the scalar’s abundance is set by both
its mass and its initial misalignment), which means massive neutrinos can only
contribute so much extra matter density without becoming nonrelativistic
before recombination; concretely, Eqs. 45 and 46 combine into

$\displaystyle\omega_{\nu,\mathrm{massive}}$ $\displaystyle=6.45\times
10^{-4}\frac{z_{T_{\nu}=m_{\nu}}+1}{356.56}\frac{N_{\mathrm{eff}}}{3.046}.$
(47)

Thus, if all three neutrino flavors becomes semirelativistic at recombination,
their present-day density is of order $5\%$ of the CDM density. A massive
scalar has the potential to make a greater contribution to the late-time
matter density while still doing so only well after recombination.

In otherwise standard cosmological scenarios, the background-level effect of
hyperlight scalars and massive neutrinos also reduces the Hubble constant
inferred from CMB data: the increase in density in the matter era makes the
Universe smaller without changing the sound horizon, requiring a compensatory
reduction in the dark energy density to maintain the distance to last
scattering. Equation 43 shows that $h\propto A_{m}^{-2.09}$ at fixed
$a_{\star}$ (and $x_{\mathrm{eq}}$ and $R_{\star}$). Per Eq. 47, a massive
neutrino species with fiducial mass $m_{\nu}=0.06\leavevmode\nobreak\
\mathrm{eV}$ increases the total matter density by $\approx 0.5\%$ relative to
that from baryons and CDM ($\omega_{c}+\omega_{b}\approx 0.14$), explaining
why the CMB prefers values of $h$ about a percent lower in cosmologies with
versus without massive neutrinos. An anticorrelation between $\omega_{\phi}$
and $h$ is likewise evident in the results of Ref. [128] for ultralight axions
with mass below $\sim 10^{-29}\leavevmode\nobreak\ \mathrm{eV}$.

If the scalar’s potential is steeper than quadratic about its minimum, the
scalar’s energy density redshifts faster than that of matter once it begins
oscillating. In this case, its gravitational effects would be important only
in the brief interval near the beginning of its oscillations. While such
effects are substantial in, e.g., early dark energy scenarios where a scalar
starts oscillating around matter-radiation equality [129, 130, 131, 132, 133],
the impact of a massless scalar that does so after recombination should be
much less important.

###  III.2 Impact on cosmological perturbations

A hyperlight scalar’s effect on the expansion history may be well understood
as an enhancement of the late-time CDM density, as described in the previous
subsection; the dynamics of its perturbations, however, are qualitatively
distinct from CDM. A scalar’s mass $m_{\phi}$ associates a Jeans scale to its
perturbations, which is parametrically $k_{J}\sim a\sqrt{Hm_{\phi}}$ at any
given time; fluctuations on larger scales cluster like CDM, but on smaller
scales they oscillate [134, 135, 136, 121]. Perturbations in CDM then grow
more slowly on scales where not all of the matter content clusters, again in
close analogy to massive neutrinos [137, 138] (which effectively behave as
warm dark matter and have a free-streaming scale associated to their
temperature). On scales where a fraction $f$ of matter does not cluster, the
growth rate of CDM density fluctuations $\delta_{c}\propto a^{p}$ is reduced
from $p=1$ to $p\approx 1-3f/5$ [135]. Section A.1 provides more detailed
exposition on the dynamics of scalar fields, their perturbations, and their
contribution to the Einstein equations.

In the mass range of interest, $m_{\phi}\lesssim 10^{-28}\leavevmode\nobreak\
\mathrm{eV}$, the Jeans scale is cosmologically macroscopic: writing $H\sim
m_{\phi}/(a/a_{\mathrm{osc}})^{3/2}$ for scalars that begin oscillating in the
matter era, the Jeans scale evolves as $k_{J}\sim
a_{\mathrm{osc}}m_{\phi}\sqrt[4]{a/a_{\mathrm{osc}}}$. Before oscillations
begin, the scalar’s energy density is small relative to the total density
(evident in Fig. 5), so it has a negligible impact on the dynamics of CDM and
baryons. Because of the weak scaling with $a$, a hyperlight scalar suppresses
the growth of structure roughly on scales smaller than the horizon when
oscillations begin, $k_{\mathrm{osc}}\equiv
a_{\mathrm{osc}}m_{\phi}\sim(120\leavevmode\nobreak\
\mathrm{Mpc})^{-1}\sqrt[3]{m_{\phi}/10^{-28}\leavevmode\nobreak\
\mathrm{eV}}$, at all times after it begins oscillating. The modifications to
the growth of structure affect the CMB via the ISW effect and gravitational
lensing; the latter is constrained both by direct reconstruction of the CMB
lensing potential and by smearing of the acoustic peaks.

Figure 6 depicts the impact of a hyperlight scalar field on CMB lensing.

Figure 6:  CMB lensing (left) and $E$-mode (right) power spectra in
cosmologies varying the early-time electron mass $m_{e,i}$ with a scalar
field, by color. Each curve holds all of $R_{\star}$, $x_{\mathrm{eq}}$,
$\theta_{s}$, $Y_{\mathrm{He}}$, $A_{s}a_{\star}^{1-n_{s}}$, and
$\tau_{\mathrm{reion}}$ fixed. The scalar field has mass
$m_{\phi}=10^{-30}\leavevmode\nobreak\ \mathrm{eV}$ and a density chosen to
(approximately) fix the scale factor of matter–dark-energy equality via the
degeneracy derived from Eq. 44. The plot inset in the right panel displays the
$E$-mode power spectrum zoomed in to highlight the decreased smearing of one
of the acoustic peaks.

The results follow the degeneracy identified in Section III.1 with varying
$m_{e,i}$, setting the scalar field energy density to fix $a_{m-\Lambda}$ as
discussed below Eq. 44. The specific parameter dependence is made concrete in
Section III.3; the scalar’s present energy density relative to that of CDM is
approximately $\omega_{\phi}/\omega_{c}\approx 1.2(m_{e,i}/m_{e,0}-1)$ [see
Eq. 55 below]. The lensing spectrum is greatly suppressed by the presence of
the scalar on all scales, but especially the smaller scales which, at late
times, lens the primary CMB anisotropies, smearing the acoustic peaks. The
latter effect is subtle but most evident in the $E$-mode power spectra
displayed in Fig. 6, where the cases with the largest early-time electron mass
$m_{e,i}$ (and therefore the largest contribution from the scalar field) reach
the point where the peaks are essentially unaltered by structure at late
times. These findings are similar to those in Ref. [139], which studied the
purely gravitational effects of ultralight axions.222 Note that the
gravitational effects of pseudoscalars/axions and scalars are identical, up to
the choice of the potential. Constraints on axions like those in Refs. [128,
140, 141, 142, 139] therefore apply equally well to scalars without additional
couplings.  Observe also that the suppression of the lensing potential is far
more drastic (at fixed $m_{e,i}/m_{e,0}$) along the degeneracy direction with
a hyperlight scalar than for that without one (Fig. 4).

As discussed in Section II.2.1, in Universes dominated by cold matter the ISW
contribution to CMB anisotropies vanishes, a feature contingent on density
fluctuations growing exactly as $\delta\rho/\bar{\rho}\propto a$. The ISW
effect sourced by a hyperlight scalar is depicted in Fig. 7 for two different
masses.

Figure 7:  Large-scale CMB temperature power spectrum in cosmologies varying
the electron mass $m_{e,i}$ with a scalar field, by color. Each curve holds
all of $R_{\star}$, $x_{\mathrm{eq}}$, $\theta_{s}$, $Y_{\mathrm{He}}$,
$A_{s}a_{\star}^{1-n_{s}}$, and $\tau_{\mathrm{reion}}$ fixed and additionally
includes a scalar field with mass $m_{\phi}=10^{-29}\leavevmode\nobreak\
\mathrm{eV}$ (left) or $10^{-30}\leavevmode\nobreak\ \mathrm{eV}$ (right) with
a density chosen to (approximately) fix the scale factor of matter–dark-energy
equality via the degeneracy derived from Eq. 44. The inset panels zoom in on
$\ell<40$.

With $m_{\phi}=10^{-29}\leavevmode\nobreak\ \mathrm{eV}$, the scalar begins
oscillating at $z_{\mathrm{osc}}\approx 390$, which is near the earliest it
can while remaining effectively frozen during recombination (i.e., such that
any shift in fundamental constants is constant during recombination). However,
this is still early enough to noticeably enhance the temperature power
spectrum at the first acoustic peak (at $\ell\sim 200$). With a lighter
scalar, e.g., with $m_{\phi}=10^{-30}\leavevmode\nobreak\ \mathrm{eV}$ (and
$z_{\mathrm{osc}}\approx 86$), the ISW effect is irrelevant at $\ell>100$ but
is slightly more pronounced at smaller multipoles. Indeed, in both cases the
ISW effect has nontrivial scale dependence, even suppressing power on large
scales rather than enhancing it (as in smaller scales). The same features are
again evident in cosmologies with ultralight axions [128]. The scalar’s full
behavior—its homogeneous oscillations and dark-energy–like behavior
beforehand, and the growth of fluctuations on large scales—has effects beyond
the suppression of structure below the Jeans scale discussed here. The
dynamics of the scalar’s perturbations therefore provide a possible means to
constrain its mass through gravitational effects alone; see Refs. [128, 140,
141, 142, 139] for more detailed exploration.

Scalars in the mass range we consider also suppress structure on scales that
contribute to $\sigma_{8}$, defined in Section II.2.4. At present, the CMB-
derived value of $\sigma_{8}$ (within $\Lambda$CDM) is mildly discrepant with
that inferred from large-scale structure probes [143]. While the discrepancy
is not nearly so severe as that of $H_{0}$ between Planck and the SH0ES-
calibrated distance ladder, it is worth noting that hyperlight scalars could
in principle reconcile it, since those with masses $m_{\phi}\lesssim
10^{-24}\leavevmode\nobreak\ \mathrm{eV}$ do not cluster on the scales to
which $\sigma_{8}$ is sensitive. Ref. [139] explored this possibility in
detail, including constraints from galaxy clustering. However, since $h$ is
negatively correlated with $\omega_{\phi}$ at fixed $a_{\star}$ (as explained
in Section III.1), $h$ and $\sigma_{8}$ are positively correlated when
increasing $\omega_{\phi}$. The exacerbation of the discrepancy between
$\sigma_{8}$ inferred from large-scale structure and CMB datasets in
cosmologies that raise the CMB’s preference for $H_{0}$ is cited as a common
feature of $\Lambda$CDM extensions. On the other hand, a hyperlight scalar
that also modifies the fundamental constants and triggers early recombination
has the potential to simultaneously reduce the CMB’s preference of
$\sigma_{8}$ and raise that of $H_{0}$. Similar conclusions were drawn for the
“axi-Higgs” model in Ref. [29], in which an ultralight axion shifts the
electron mass by shifting the Higgs VEV at early times. Ignoring the effect of
the Higgs VEV on the proton mass (which is not a valid approximation, as
argued in Ref. [32]), the phenomenologies of the “axi-Higgs” scenario and of a
hyperlight scalar coupled directly to the electron are effectively identical.

Finally, ultra- and hyperlight components of dark matter can suppress
structure on nonlinear scales in a manner that may not be well captured by
standard methods to estimate nonlinear corrections to the matter power
spectrum [144, 145]. Scalars in the mass range we consider, however, do not
cluster on observationally relevant scales or redshifts [146, 140, 147].
Moreover, nonlinear structure growth substantially impacts the CMB lensing
potential only on scales smaller than those Planck measures [91]. Since we
consider no other CMB lensing observations, we therefore follow Refs. [128,
140, 141, 142, 139] in neglecting any nonlinear effects in the matter power
spectrum.

###  III.3 Parameter space

The mass of a scalar that begins oscillating in the matter-dominated era is
related to the matter density as

$\displaystyle
m_{\phi}^{2}=\frac{9H(t_{\mathrm{osc}})^{2}}{4}\approx\frac{3}{4}\frac{\bar{\rho}_{c}(t_{0})+\bar{\rho}_{b}(t_{0})}{a_{\mathrm{osc}}^{3}M_{\mathrm{pl}}^{2}},$
(48)

neglecting the scalar’s own contribution to the total energy density at that
time. Using Eq. 41, the scalar’s energy density relative to that of CDM (at
$t>t_{\mathrm{osc}}$) is therefore

$\displaystyle{F_{\phi}}$
$\displaystyle\equiv\frac{\bar{\rho}_{\phi}}{\bar{\rho}_{c}}=\frac{m_{\phi}^{2}\bar{\phi}_{i}^{2}/2(a/a_{\mathrm{osc}}^{3})}{\bar{\rho}_{c}(t_{0})/a^{3}}=\frac{3\left(1+\omega_{b}/\omega_{c}\right)\bar{\varphi}_{i}^{2}}{4},$
(49)

where $\bar{\varphi}_{i}$ is the initial homogeneous component of the rescaled
field

$\displaystyle\varphi$
$\displaystyle\equiv\frac{\phi}{\sqrt{2}M_{\mathrm{pl}}}.$ (50)

Observe that this expression is independent of the mass $m_{\phi}$ and holds
for masses between (3/2 times) the Hubble rate at matter-radiation equality,
$3H_{\mathrm{eq}}/2\approx 3.4\times 10^{-28}\leavevmode\nobreak\ \mathrm{eV}$
and that at matter–dark-energy equality, $3H_{m-\Lambda}/2\approx 2.6\times
10^{-33}\leavevmode\nobreak\ \mathrm{eV}$. The matter-era solution is then

$\displaystyle\bar{\phi}(t)$
$\displaystyle\equiv\sqrt{2}M_{\mathrm{pl}}\mathcal{A}(t)\sin m_{\phi}t,$ (51)

where the time-dependent, squared oscillation envelope for at
$t>t_{\mathrm{osc}}$ is

$\displaystyle\mathcal{A}(t)^{2}$
$\displaystyle=\frac{4}{3}\frac{{F_{\phi}}}{1+\omega_{b}/\omega_{c}}\left(\frac{a(t)}{a_{\mathrm{osc}}}\right)^{-3}.$
(52)

In Ref. [32], the shift in a fundamental parameter $\lambda$ (i.e., $\alpha$
or $m_{e}$) is parameterized in terms of the scalar field by

$\displaystyle\lambda(\varphi)$
$\displaystyle\approx\lambda(0)\left[1+g_{\lambda}(\varphi)\right],$ (53)

where $\lambda(0)$ is the vacuum value of the parameter, i.e., when the scalar
is at the minimum of its potential. (For consistency with prior literature, we
replace the subscript $\alpha$’s with $e$’s for the photon couplings,
$g_{e}\equiv g_{\alpha}$.) A shift in the inverse scale factor of
recombination, with fundamental-constant dependence given by Eq. 6, may
therefore be written in terms of the scalar’s couplings to the SM as

$\displaystyle\frac{\Delta
a_{\star}^{-1}}{a_{\star}^{-1}}\approx\frac{m_{e,i}-m_{e,0}}{m_{e,0}}+2\frac{\alpha_{i}-\alpha_{0}}{\alpha_{0}}$
$\displaystyle=g_{m_{e}}(\bar{\varphi}_{i})+2g_{e}(\bar{\varphi}_{i}).$ (54)

One may translate the $\alpha_{i}$ dependence of any other quantity in Eq. 29
(e.g., those parameterizing the effects on small-scale damping and CMB
polarization in Sections II.1.3 and II.1.4) analogously. For a scalar field
that is frozen during recombination, these relationships, combined with Eq.
49, fully specify the correspondence between the phenomenological parameters
(${F_{\phi}}$ and the early-time values of $m_{e}$ and $\alpha$) to the
fundamental parameters of the model (those parameterizing $g_{m_{e}}$ and
$g_{e}$ in addition to the initial condition $\bar{\varphi}_{i}$).

To phrase the degeneracy direction identified in Section III.1 in terms of
${F_{\phi}}$, write
$\omega_{m,\mathrm{late}}=(1+{F_{\phi}})\omega_{c}+\omega_{b}$, such that
$A_{m}=1+{F_{\phi}}/(1+\omega_{b}/\omega_{c})$. When $A_{m}\propto
a_{\star}^{-1.034}$, which ensures that matter–dark-energy equality occurs at
the same scale factor regardless of $a_{\star}$ (along the identified
degeneracy),

$\displaystyle\frac{{F_{\phi}}}{1+\omega_{b}/\omega_{c}}\approx
1.034\frac{\Delta a_{\star}^{-1}}{a_{\star}^{-1}}$ (55)

for small shifts $\Delta a_{\star}^{-1}$. The scalar must therefore make a
relative contribution to the matter density at late times that is commensurate
with the shift in recombination. This condition is approximately independent
of the scalar’s mass, so long as it begins oscillating after recombination
(and therefore after the matter era begins) but sufficiently long before
matter–dark-energy equality. More precisely, lighter scalars begin
contributing to the matter abundance later and affect the distance to last
scattering over a shorter interval. Scenarios with lighter scalars therefore
require a slightly larger increase in response to an increase in
$a_{\star}^{-1}$ than given by the coefficient $1.034$ in Eq. 55 (but still
order unity), which is derived in Section III.1 under the assumption that the
scalar contributes to the matter density at all times after recombination.

Per Eq. 49, to realize the condition of Eq. 55 the scalar thus must have a
near-Planckian initial condition. Combining with Eq. 54,

$\displaystyle\bar{\varphi}_{i}^{2}=\frac{\bar{\phi}_{i}^{2}}{2M_{\mathrm{pl}}^{2}}$
$\displaystyle\approx\frac{4}{3}\cdot
1.034\left[g_{m_{e}}(\bar{\varphi}_{i})+2g_{\alpha}(\bar{\varphi}_{i})\right].$
(56)

Ref. [32] motivates models in which the leading contributions to $g_{\lambda}$
are quadratic: $g_{\lambda}(\varphi)\approx d_{\lambda}^{(2)}\varphi^{2}/2$.
Because the scalar’s energy contribution is _also_ quadratic in the field’s
value, the degeneracy direction Eq. 56 lies at a fixed value of the
dimensionless coupling coefficient. In this case, the early-time variation in
fundamental constants [$\lambda_{i}\equiv\lambda(\bar{\varphi}_{i})$] may be
written as

$\displaystyle\frac{\lambda_{i}}{\lambda(0)}-1$
$\displaystyle=g_{\lambda}(\bar{\varphi}_{i})=d_{\lambda}^{(2)}\bar{\varphi}_{i}^{2}/2=\frac{2d_{\lambda}^{(2)}{F_{\phi}}}{3\left(1+\omega_{b}/\omega_{c}\right)}.$
(57)

For instance, when $g_{e}(\bar{\varphi}_{i})$ is negligible, we expect the
quadratic coupling $d_{m_{e}}^{(2)}\approx 1.45$ to quantify the degeneracy
direction in cosmological data. Interestingly, we show in Ref. [32] that such
$\mathcal{O}(1)$ quadratic scalar couplings to electrons and photons are
allowed by all other experimental probes.

The cosmological data we consider depend only on the early-time values of
$m_{e}$ and $\alpha$, i.e., on $g_{m_{e}}(\bar{\varphi}_{i})$ and
$g_{e}(\bar{\varphi}_{i})$ regardless of their functional form (again, when
the field remains frozen until after recombination). The scalar’s cosmological
abundance is conveniently parameterized like other species via ${F_{\phi}}$,
since $\omega_{\phi}\equiv{F_{\phi}}\omega_{c}$. The simplest parameterization
of the model for parameter inference using cosmological data is therefore
simply that of the early-time constants, $m_{e,i}$ and $\alpha_{i}$, and
${F_{\phi}}$. However, physically motivated priors may only be (directly)
phrased in terms of the fundamental parameters. In particular, while
${F_{\phi}}$ depends only on $\bar{\varphi}_{i}^{2}$, variations in $m_{e,i}$
and $\alpha_{i}$ are given by the product of (some power of) the field
amplitude $\bar{\varphi}_{i}$ and dimensionless coupling coefficients. Any
fixed shift in a fundamental constant may be accommodated by arbitrarily small
$\bar{\varphi}_{i}$ with a large enough coupling. Within this class of models
and taking quadratic couplings as an example, the limit of a gravitationally
negligible scalar, $\bar{\varphi}_{i}^{2}\ll 1$ and $d_{\lambda}^{(2)}\gg
1$—the limit of strong coupling, in other words—corresponds to that of prior
work that considered variations in fundamental constants only at a
phenomenological level.

Aside from neglecting the potential cosmological importance of the late dark
matter contribution of a hyperlight scalar motivated in Section III.1, the
strong-coupling limit runs afoul of the considerations of microphysical models
in Ref. [32]. The effective scalar potential sourced by matter becomes
substantial and can affect the scalar’s dynamics at early times, even before
recombination. Shifts in fundamental constants that are time-independent
through the end of recombination, as studied by prior work on a
phenomenological basis, thus require that the responsible scalar contributes
to gravity to some extent. We exclude regimes in which the constants evolve
before recombination from our analysis here but discuss the possibility in
more detail in Ref. [32].

The opposite, weakly coupled limit is specified by $\bar{\varphi}_{i}\gtrsim
1$. One might exclude this regime on the grounds that the scalar then
traverses a super-Planckian field distance [148], but such a regime is surely
entirely excluded observationally, given that a scalar with $m_{\phi}\gtrsim
H_{0}$ would have come to dominate the energy density of the Universe before
the start of coherent oscillations and cause a period of inflation when
$\bar{\varphi}_{i}\gtrsim 1$. One might then wonder whether some or all of the
Universe’s present-day dark energy can in fact made up of the scalar (e.g.
$m_{\phi}\lesssim H_{0}$). By its very nature, a species that behaves like
dark energy or a slowly varying cosmological “constant” (e.g., a quintessence
field) must have a nearly flat bare potential (which is true of the
$m_{\phi}^{2}\phi^{2}$ potential when $m_{\phi}\ll H_{0}$). Such a requirement
would preclude any significant variation of the field between the present-day
and earlier times. However, the effective potential sourced by the
cosmological bath of SM matter and radiation (or even dark matter [149, 150])
can easily dominate the bare potential at early times. We therefore cannot
outright exclude the possibility of significant variations in fundamental
constants at early times from a field that is today dark-energy–like without
considering concrete models of its interactions and their constraints from
other probes. We restrict our analysis to scalars as subcomponents of the
present-day dark matter for simplicity.

##  IV Cosmological parameter inference

We now test the models and hypothesized parameter degeneracies discussed in
Sections II and III using cosmological data. Insofar as the degeneracy
directions identified in Sections II and III (which are calculated about the
best-fit $\Lambda$CDM cosmology) hold throughout parameter space, we expect
$\Lambda$CDM extended with a hyperlight scalar coupled to the electron to
satisfy CMB, BAO, and SN Ia data at any $m_{e}$ with $h\propto m_{e}$ and
${F_{\phi}}\equiv\omega_{\phi}/\omega_{c}\approx m_{e,i}/m_{e,0}-1$. Moreover,
beyond simply reproducing the best-fit $\Lambda$CDM prediction, the additional
freedom afforded by the electron-coupled scalar can in principle enable
precise changes in predictions that better fit the data but are inaccessible
within the $\Lambda$CDM parameter space. The scalar’s matter contribution
could also enhance the consistency of these datasets when $\alpha$ is varied,
but the impact on small-scale damping and CMB polarization presumably remains
as the strongest limitation on the scenario. Quantitatively testing these
questions requires an analysis of those data using full numerical solutions to
the Einstein-Boltzmann equations (and the Klein-Gordon equation).

We first establish baseline expectations for phenomenological varying-
constants scenarios (i.e., without a scalar field) in Section IV.1, analyzing
in detail the individual and joint constraining power of CMB and low-redshift
distance measurements. Then Section IV.2 studies the impact of consistently
including a coupled scalar field as microphysical realization of time-varying
fundamental constants. Before proceeding, we describe the methods and datasets
we employ in our analysis.

In parameter inference, we sample over the standard set of $\Lambda$CDM
parameters with broad, uninformative priors: the present baryon and CDM
densities, $\omega_{b}\sim\mathcal{U}(0.005,0.035)$ and
$\omega_{c}\sim\mathcal{U}(0.01,0.25)$; the Hubble rate
$h\sim\mathcal{U}(0.25,1.1)$; the tilt and amplitude of the scalar power
spectrum, $n_{s}\sim\mathcal{U}(1.61,3.91)$ and $A_{s}$ [the latter via the
combination as $\ln(10^{10}A_{s})\sim\mathcal{U}(1.61,3.91)$]; and the
redshift of reionization $z_{\mathrm{reion}}\sim\mathcal{U}(4,12)$. Here we
use $\mathcal{U}(a,b)$ to denote a uniform prior between $a$ and $b$. In some
circumstances it proves more computationally efficient to sample over the
angular scale of the sound horizon $\theta_{s}\sim\mathcal{U}(0.9,1.1)$ in
place of $h$; we comment on the impact of this choice where it is used. We
exclude the possibility of a negative dark energy density (which is otherwise
accessible in allowed parameter space in some situations). We sample over
$m_{e,i}/m_{e,0}$ and $\alpha_{i}/\alpha_{0}\sim\mathcal{U}(0.7,1.3)$ whenever
considering variations of each parameter. We fix a single massive neutrino
with the standard, minimal mass $m_{\nu}=0.06\leavevmode\nobreak\
\mathrm{eV}$; representing the other two by a relativistic fluid with an
effective number of degrees of freedom of $2.0308$ yields a total early-time
$N_{\mathrm{eff}}$ [Eq. 2] of $3.044$. Section B.2 details our methods for
sampling via Markov Chain Monte Carlo (MCMC).

In the following, we consider a variety of datasets of the types described in
Section II and discuss their implementations in more detail in Section B.1. We
use the 2018 Planck likelihoods via the Plik_lite variants that are
marginalized over the parameters of the foreground models [75, 151]. These
likelihoods require an additional parameter $y_{\mathrm{cal}}$ characterizing
the overall map calibration, sampled with the standard Gaussian prior with
mean $1$ and standard deviation $0.0025$ [151]. We use BAO measurements from a
common combination of surveys: the Six-degree Field Galaxy Survey (6dFGS)
[152] (with an effective sample redshift $z=0.106$), the SDSS Main Galaxy
Sample (MGS) DR7 [153] ($z=0.15$), the Baryon Oscillation Spectroscopic Survey
(BOSS) DR12 galaxies ($z=0.38$ and $0.51$), and the Extended Baryon
Oscillation Spectroscopic Survey (eBOSS) DR16 luminous red galaxies (LRG)
[154, 155] ($z=0.70$). Note that the constraining power of this combination is
dominated by eBOSS measurements, at least for the models we consider; we refer
to this commonly used combination as the SDSS+ BAO dataset. We separately use
recent BAO measurements from Dark Energy Spectroscopic Instrument (DESI) DR1
[156, 157, 158], using data from all tracers; we note that these results are
yet unpublished at the time of writing. Finally, we use supernova distance
measurements from the Pantheon [95], Pantheon+ [96, 97], DES 5YR [159], and
Union3 [160] results; we sample over the overall normalization via the
degenerate combination $\log_{10}h-M_{B}/5\sim\mathcal{U}(3.7,3.73)$ that is
well constrained independently of external measurements of the fiducial
magnitude $M_{B}$ (or of $h$, for that matter). That is, we do not include
calibration from, e.g., Cepheids [24, 25, 26] or the tip of the red giant
branch [101, 102, 103, 104].

###  IV.1 Varying constants

####  IV.1.1 Planck data alone

We first seek to test whether the parameter degeneracies argued for in Section
II bear out in a complete analysis of CMB data. Figure 8 displays the
posterior distribution for $\Lambda$CDM with a varying early-time electron
mass, evaluated with Planck 2018 CMB and lensing likelihoods.

Figure 8:  Posterior distribution over a subset of the parameters of the
varying electron mass model without a scalar field (grey) which exhibits
extended degeneracies compared to standard $\Lambda$CDM (blue), using the
likelihood for the full set of Planck 2018 data including lensing (see Section
B.1). The panels along the diagonal depict kernel density estimates of
marginalized, one-dimensional posteriors in each parameter, with vertical
lines marking the median and $16$th and $84$th percentiles. These one-
dimensional posteriors are normalized relative to their peak value to
facilitate comparison. The median and corresponding $\pm 1\sigma$
uncertainties for each parameter are reported above the diagonal panels. The
lower panels display the marginalized, two-dimensional joint posterior density
for pairs of parameters as a histogram and the 1, 2, and $3\sigma$ contours
thereof (i.e., the $39.3\%$, $86.5\%$, and $98.9\%$ mass levels). Contours are
slightly smoothed to mitigate noise due to the binning of such narrow
distributions, but not to an extent that artificially broadens their extent.
Red lines in each panel depict the expected degeneracy direction derived in
Section II.1.5 which is largely born out in the data: $\omega_{b}$ and
$\omega_{c}\propto m_{e,i}$ and $h\propto m_{e,i}^{3.18}$.

The results indeed display a strong degeneracy, which the red curves show
correspond closely to the direction $\omega_{b}\propto m_{e,i}$,
$\omega_{c}\propto m_{e,i}$, and $h\propto m_{e,i}^{3.18}$. The posterior is
cut off at low values of these parameters almost exclusively by requiring
$h^{2}\geq\omega_{b}+\omega_{c}$ (i.e., requiring a nonnegative dark energy
density). The posteriors are therefore asymmetric, falling off at values of
$m_{e,i}/m_{e,0}\gtrsim 0.92$. Namely, arbitrarily large values of the
electron mass are _not_ as favored as those with $m_{e,i}/m_{e,0}<1$. As a
consequence, $97.5\%$ of the posterior lies where $h<0.72$—not the resounding
solution to the Hubble tension the results of, e.g., Fig. 3 might promise.
Moreover, the median of the marginalized posterior over the Hubble constant is
$h=0.47$, and that of the early-time electron mass is $m_{e,i}/m_{e,0}=0.90$.

We identify several effects that disfavor larger $m_{e,i}$ (and therefore
larger $h$). As argued in Section II, the degeneracy in the varying electron
mass model is effectively perfect so far as early-time effects are concerned,
but the CMB is sensitive to the late Universe via the ISW effect and lensing
(Section II.2.1). In Fig. 9 we display the low-$\ell$ temperature power
spectrum and the lensing power spectrum evaluated over a representative sample
of the posterior.

Figure 9:  Large-scale CMB temperature (left) and lensing (right) power
spectra for $1000$ cosmologies sampled from the posterior in Fig. 8, i.e.,
that for the varying electron mass model (without a scalar field) subject to
Planck 2018 data. Each curve is colored according to its value of
$m_{e,i}/m_{e}$ as indicated by the color bar. The binned Planck 2018 spectra
and standard errors are superimposed in grey, and the result for the
$\Lambda$CDM best-fit cosmology is depicted in dashed black. These results may
be compared with those in Fig. 4 (though the vertical scales are not
identical).

The early-time electron mass correlates strongly to the ISW enhancement of the
temperature power spectrum at low multipoles because $h$ is strongly
constrained to increase with $m_{e,i}$ in order to correctly locate the
acoustic peaks. As explained in Section II.2.1, $h$ must decrease so rapidly
with decreasing $m_{e,i}$ that matter–dark-energy equality is pushed to much
later times (even to the future, $a_{m-\Lambda}>1$). As illustrated as well by
Fig. 4, matter overdensities therefore decay less, enhancing the lensing
spectrum relative to $\Lambda$CDM and reducing the ISW enhancement of large-
scale temperature power. The data in the left panel of Fig. 9 suggest that the
ISW effect by itself prefers late recombination to dark energy, given that
most of their central values skew low, even compared to curves at the lower
boundary of the posterior (which corresponds to negligible dark energy
density). Indeed, a lack of power on large angular scales is a persistent and
unexplained feature of CMB observations [161, 162, 163, 164, 165, 166].

In addition, lensing data also appear to favor the enhanced peak evident in
Fig. 9, but it is challenging to compare the relative constraining power of
the first few data points that skew high to those at higher multipoles that
skew low (and have smaller error bars). More likely, the enhanced lensing
power is favored by the “lensing anomaly” of Planck data [167, 168, 75, 169]
in which the acoustic peaks exhibit greater lensing-induced smoothing than
predicted in $\Lambda$CDM. Such a conclusion was drawn in Ref. [38], which
studied CMB power spectrum constraints on its present-day temperature $T_{0}$;
as argued in Section II.1.6, varying $T_{0}$ and $m_{e,i}$ should exhibit
nearly identical phenomenology.

Finally, the shape of a chosen prior is not necessarily negligible over
posteriors so broad as those in Fig. 8. For the results in Fig. 8 (and any
results using CMB data alone), we sample over $\theta_{s}$ instead of $h$
(using numerical optimization to solve for $h$ in this case), as is
conventional in MCMC analyses of CMB data. In scenarios with varying electron
mass, uniformly sampling over $h$ and $m_{e,i}$ covers a broad range of values
for $\theta_{s}$, most of which lie far from the narrow range of $\theta_{s}$
values allowed by Planck. The angular size of the sound horizon $\theta_{s}$,
suitably calculated, provides a measure of a broadly model-independent feature
in the data that is well constrained (the angular locations of the acoustic
peaks) and whose posterior allows for more efficient MCMC sampling (in terms
of effective sample size per MCMC sample). But $\theta_{s}$ and $h$ are
nonlinearly related, meaning a uniform prior for one results in a nonuniform
prior for the other. Equation 27 (which is evaluated near the $\Lambda$CDM
best-fit parameters) allows us to estimate that the effective prior on $h$
from sampling uniformly in $\theta_{s}$ is $\partial\theta_{s}/\partial h\sim
h^{-0.81}$, a trend that is roughly reproduced by direct samples from our
prior. This scaling does not explain all (nor even most) of the trend
displayed in Fig. 8, however. Such a penalty on larger $h$ is entirely
negligible for $\Lambda$CDM with contemporary datasets, where $h$ is
constrained at the $1\%$ level or better, but it is of at least marginal
consequence for attempts to address the Hubble tension with models that
achieve broader posteriors on $h$ (like that in Fig. 8).333 As both $h$ and
$\theta_{s}$ are derived parameters, a uniform prior in one is not necessarily
any more motivated than in the other; one could equally well choose the dark
energy density $\omega_{\Lambda}$. A more theoretically grounded prior (and
choice of parameterization) would require a concrete model of dark energy.

Turning to variations in the fine-structure constant, Fig. 10 presents
parameter constraints from Planck 2018 data alone in analogy to Fig. 8.

Figure 10:  Posterior distribution over a subset of the parameters of the
varying fine-structure constant model (without a scalar field), using the
likelihood for the full set of Planck 2018 data including lensing (see Section
B.1). Results are depicted as described in the caption of Fig. 8, additionally
plotting the scalar spectral tilt $n_{s}$ due to its correlation with
$\alpha_{i}/\alpha_{0}$.

Due to the strong dependence of small-scale damping (Section II.1.3) and of
the amplitude of the polarization spectrum (Section II.1.4) on $\alpha_{i}$,
illustrated in Fig. 3, no extended degeneracy of $\Lambda$CDM parameters with
$\alpha_{i}$ appears. Because $h$ is even more sensitive to $\alpha_{i}$ than
to $m_{e,i}$ at fixed $\theta_{s}$ [Eq. 27], the posterior distribution over
$h$ broadens slightly compared to $\Lambda$CDM, with a standard deviation
roughly twice as large. In contrast to $m_{e,i}$, the fine-structure constant
is also correlated with the tilt of the scalar power spectrum $n_{s}$ [defined
in Eq. 28]. Though diffusion and Landau damping effectively have exponential
dependence on scale, Planck’s observations cover only a limited range of
scales where they are relevant, enabling a partial degeneracy with the power-
law tilt of the initial power spectrum. Because the pivot scale
$k_{\mathrm{p}}$ that parameterizes the primordial power spectrum [Eq. 28] is
larger than the inverse sound horizon at recombination, a bluer spectrum
(larger $n_{s}$) reduces the amplitude of the first peak and requires a larger
amplitude $A_{s}$ to compensate. The degeneracy between $A_{s}$ and
$\alpha_{i}$ is evident in our results, but weaker than that between $n_{s}$
and $\alpha_{i}$ and also does not affect the marginalized posterior over
$A_{s}$. Finally, we note that neither the lensing nor low-$\ell$ anomalies
discussed around Fig. 9 for varying $m_{e,i}$ are relevant in the parameter
space available when varying $\alpha_{i}$.

Figure 11 displays residuals of the CMB power spectrum relative to the
$\Lambda$CDM best fit, evaluated for a representative sample of the posterior
for the varying-$m_{e}$ and varying-$\alpha$ scenarios using only the Planck
2018 likelihoods.

Figure 11:  Residuals of the CMB power spectrum relative to the best-fit
$\Lambda$CDM result, $\Delta C_{\ell}^{XY}=C_{\ell}^{XY}-C_{\ell}^{XY,\star}$,
evaluated for $1000$ cosmologies sampled from the posterior of the
varying-$\alpha$ (left panels) and -$m_{e}$ (right panels) models with the
Planck 2018 likelihoods alone. In order to illustrate the correlation between
parameters and features in the residuals, each curve is colored by its value
of the fundamental constant that is varied as indicated by the respective
color bar. The top and middle panels depict relative difference of the
temperature and $E$-mode auto power spectra, respectively, while the bottom
panels depict the residual of the $TE$ cross correlation normalized by
$\sqrt{C_{\ell}^{TT,\star}C_{\ell}^{EE,\star}}$. The binned Planck 2018
spectra and standard errors are superimposed in grey, and the shaded grey
regions show the extent of cosmic variance for each multipole $\ell$ (i.e.,
relative errors of $\sqrt{2/(2\ell+1)}$). (Planck’s error bars can be smaller
because they are binned over a range of multipoles.) Note that the color
scales are symmetric about one even when the posterior is not (as is the case
for $m_{e,i}/m_{e,0}$ in the right panels).

Beyond the differences exhibited at low $\ell$ in Fig. 9, the residuals in the
varying-$m_{e}$ scenario exhibit no clear trend with $m_{e,i}$, nor any easily
identifiable systematic effect. Aside from noticeable oscillations in the $TE$
cross correlation residuals, the lack of particularly distinguishable features
is consistent with the expectation from Section II that no important physical
effects correlate uniquely with the electron mass (within the full extended
$\Lambda$CDM parameter space).

The residuals for the varying-$\alpha$ scenario display several features in
line with the expectations from the effect of the fine structure constant on
the damping tail and polarization discussed in Section II and Fig. 3. The
correlation of damping with $\alpha_{i}$ is particularly evident at high
multipoles, with larger (red) and smaller (blue) $\alpha_{i}/\alpha_{0}$
displaying increased and decreased small-scale power, respectively. In
addition, the enhancement of moderate-scale ($\ell\sim 500-1200$) polarization
power for _smaller_ $\alpha_{i}/\alpha_{0}$ (in contrast to the trend at
larger $\ell$) due to the broadened visibility function is also evident in
Fig. 11, as anticipated in Fig. 3. More recent CMB polarization observations,
like the Atacama Cosmology Telescope [170] and the South Pole Telescope [171,
172] would likely marginally improve upon Planck’s ability to constrain
$\alpha_{i}$ via the effect with cosmic-variance–limited measurements at
$\ell\gtrsim 1000$. However, the ultimate constraining power of features on
moderate scales is limited by cosmic variance; current and future high-
resolution CMB experiments clearly have great potential to improve upon the
fine-structure constant constraints of Planck. In particular, Fig. 3 suggests
that high-resolution observations can break the degeneracy of $\alpha_{i}$ and
$n_{s}$ evident in Fig. 10. While the residuals in the varying-$m_{e}$
scenario also exceed cosmic variance on scales beyond Planck’s resolution,
they do not appear to correlate with $m_{e,i}$ to any noticeable degree; this
suggests that high-resolution CMB observations may not substantially improve
bounds on electron mass variation.

####  IV.1.2 Impact of low-redshift distance measurements

We now consider the impact of constraints on the late-Universe expansion
history on the findings of the previous section. In particular, the CMB itself
provides very weak constraints on the early-time electron mass that are mostly
driven by Planck’s mild (and possibly anomalous) preference for late
recombination in a Universe with a much lower dark energy abundance than in
$\Lambda$CDM. However, Section II.2.2 shows that the BAO scale at late times
exhibits a distinct degeneracy between $h$ and $a_{\star}$ (at fixed
$a_{m-\Lambda}$). To provide effective constraints using BAO data alone, we
impose an additional prior on $R_{\star}$ and $x_{\mathrm{eq}}$ derived from
the posteriors in Fig. 8. In particular, a bivariate normal distribution with
means $(0.620,0.321)$, autocovariances $(1.7,0.66)\times 10^{-5}$, and cross-
covariance $4.1\times 10^{-6}$ effectively reproduces posteriors for
$R_{\star}$ and $x_{\mathrm{eq}}$ using Planck 2018 data for _both_
$\Lambda$CDM and varying-$m_{e}$ cosmologies (as Section II argued should be
the case). This prior effectively includes the information from the shape of
the CMB as measured by Planck that drives constraints on $\omega_{b}$ and
$\omega_{c}$ marginalized over the angle subtended by the sound horizon,
$\theta_{s}$. Without this information, BAO data alone would be unable to pick
out a unique degeneracy between $h$ and $a_{\star}$.

Figure 12 compares the posteriors for $h$ and $m_{e,i}$ or $\alpha_{i}$ when
using Planck 2018 data and the SDSS+ BAO dataset (including the above-
described prior) separately and jointly.

Figure 12:  Posterior distribution over $h$ and $m_{e,i}/m_{e,0}$ (left) or
$\alpha_{i}/\alpha_{0}$ (right) in cosmologies with varying constants without
a scalar field. The same quantities are depicted in Figs. 8 and 10, comparing
results using the Planck 2018 likelihoods (grey); the BAO dataset from eBOSS,
SDSS, and 6dFGS described in the text (blue); and both simultaneously (red).
The diagonal panels display kernel density estimates of the one-dimensional,
marginalized posterior relative to their peak value, and the lower panels
depict 1, 2, and $3\sigma$ mass levels of the two-dimensional joint posterior,
again slightly smoothed to facilitate comparison.

Without any additional mechanism, varying constant scenarios may only
simultaneously fit both Planck and BAO data where their individual posteriors
have overlapping support. Their combined constraints lie modestly far from the
peak of either posterior (and in fact is localized at larger values of $h$ and
$m_{e,i}$ than either individual posterior). The $2\sigma$ mass levels for
both cases in fact hardly overlap, further highlighting that Planck data
prefer quite a different late-time Universe via the shape of the CMB than
preferred by by BAO measurements or by measurements of the distance to last
scattering in $\Lambda$CDM. Nevertheless, the posterior is localized in the
$h$-$m_{e,i}$ plane about $m_{e,i}\approx m_{e,0}$ and $h$ near its
$\Lambda$CDM region. Under variations in the fine-structure constant, a broad