# A cool dark sector, concordance, and a low $\sigma_{8}$

Ellie Hughes<EMAIL_ADDRESS>Department of Physics, Bryn Mawr College, Bryn
Mawr, Pennsylvania 19010, USA  Department of Physics and Astronomy, University
of California, Davis, California 95616, USA  Departamento de Física, Facultad
de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile  Fei
Ge<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
California, Davis, California 95616, USA  Francis-Yan Cyr-Racine  Department
of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico
87106, USA  Lloyd Knox  Department of Physics and Astronomy, University of
California, Davis, California 95616, USA  Srinivasan Raghunathan  Center for
AstroPhysical Surveys, National Center for Supercomputing Applications,
Urbana, Illinois 61801, USA

(November 9, 2023)

###### Abstract

We investigate a cosmological model in which a fraction of the dark matter is
atomic dark matter (ADM). This ADM consists of dark versions of the electron
and of the proton, interacting with each other and with dark photons just as
their light sector versions do, but interacting with everything else only
gravitationally. We find constraints given current cosmic microwave background
(CMB) and baryon acoustic oscillation (BAO) data, with and without an $H_{0}$
prior, and with and without enforcing a big bang nucleosynthesis consistent
helium abundance. We find that, at low dark photon temperature, one can have
consistency with BAO and CMB data, with a fraction of dark matter that is ADM
($f_{\rm adm}$) as large as $\sim 0.1$. Such a large $f_{\rm adm}$ leads to a
suppression of density fluctuations today on scales below about 60 Mpc that
may be of relevance to the $\sigma_{8}$ tension. Our work motivates
calculation of nonlinear corrections to matter power spectrum predictions in
the ADM model. We forecast parameter constraints to come from future ground-
based CMB surveys, and find that if ADM is indeed the cause of the
$\sigma_{8}$ tension, the influence of the ADM, primarily on CMB lensing, will
likely be detectable at high significance.

##  I Introduction

The physics of dark matter remains a mystery. While weakly interacting massive
particles (WIMPs) [1, 2] and the quantum chromodynamic axion [3, 4] remain
leading dark matter candidates, the possibility that dark matter could be part
of an extended “dark sector” has gained traction in the last decade [5]. One
intriguing possibility is that this dark sector could contain a copy of the
Standard Model field content, such as in the twin Higgs model (see e.g. Refs.
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]). In such scenarios, part of the
dark matter can be charged under a new $U_{\rm d}(1)$ gauge interaction [37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56,
57] and even form atom-like bound states [58, 59, 60, 61, 62, 63, 64, 65, 66,
67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81]. In its simplest
incarnation, this “atomic” dark matter (ADM) involves two massive dark
fermions (a “dark proton” and a “dark electron”) interacting with a massless
dark photon. In the early Universe, such a dark sector forms an ionized plasma
in which the dark fermions are tightly coupled with the dark photon bath. Once
its temperature falls below the binding energy between the two dark fermions,
neutral dark atoms can form in a process similar to standard hydrogen
recombination [82].

This ADM scenario can have important impact on the way structure assembles in
our Universe. At early times, radiation pressure from the $U_{\rm d}(1)$ dark
photon bath opposes gravitational infall of dark matter into potential wells,
hence slowing down the growth of structure and modifying the anisotropies of
the cosmic microwave background (CMB). At late times, dark atoms behave as a
form of dissipative dark matter, allowing it to cool and form structures such
as dark disks [71, 72, 74, 83, 84, 85, 86], mirror stars [87, 88], and exotic
compact objects [89, 90, 91, 92, 93, 94, 95].

Given its possible impact on a broad range of scales, it is natural to ask if
ADM, whether it forms part or the entirety of the dark matter, could be
playing a role in some of the tensions facing the $\Lambda$ Cold Dark Matter
($\Lambda$CDM) cosmological model [96]. A particularly relevant one is the so-
called $\sigma_{8}$ tension, which refers to a potential discrepancy between
different measurements of the matter fluctuation amplitude on
$8h^{-1}\,\textrm{Mpc}$ scales [97]. Assuming standard $\Lambda$CDM cosmology,
CMB observations from _Planck_ find $\sigma_{8}=0.811\pm 0.006$ [98]. However,
measurements of cosmic shear from the KiDS-1000 survey find
$\sigma_{8}=0.76^{+0.025}_{-0.020}$ [99]. Although these two specific
measurements are only in $\sim 3\sigma$ disagreement, multiple observations of
the large-scale structure of the Universe using a variety of techniques have
found $\sigma_{8}$ values that are consistently lower than those inferred from
the CMB [100, 99, 101, 102, 103, 104, 105], potentially pointing to an
underlying problem with $\Lambda$CDM as the standard model of cosmology.

The cosmological impact of ADM on the large-scale structure of the Universe
has been extensively studied in the literature (see e.g. Refs. [62, 64, 68,
70, 22, 106, 107]). Much like in the visible sector where the baryon acoustic
oscillations (BAO) get imprinted on the matter power spectrum, the large dark
photon radiation pressure at early times can leave a dark acoustic oscillation
(DAO) pattern on the spectrum of matter fluctuations. On scales smaller than
this “dark” acoustic horizon, such a DAO pattern is typically damped due to
the finite mean free path of dark photons, in a process reminiscent of Silk
damping [108]. However, if the dark sector is significantly cooler than the
visible sector, the relatively weak pressure support (which scales as the
fourth power of the dark sector temperature) leads to a heavily damped DAO
spectrum. In this case, the main impact of ADM on structure formation is a
suppressed matter power spectrum on scales smaller than the dark acoustic
horizon. This suppression, which also occurs in other models involving dark
matter-dark radiation interaction (see e.g. Refs. [109, 110, 111, 112, 113,
114, 115, 116, 117, 118]), could play an important role in addressing the
$\sigma_{8}$ tension.

In this paper, we study the impact that ADM could have on the $\sigma_{8}$
tension while maintaining concordance with a broad range of cosmological data
including CMB, BAO, Cepheid-calibrated supernovae constraints on the Hubble
constant, and the primordial abundance of helium. We show that a cool dark
sector made of dark atoms could offer a possible solution to the $\sigma_{8}$
tension. We also present forecasts for how future observations could help
constrain this scenario.

This paper is organized as follows. In Sec. II, we outline the parameter space
explored and the physics of the ADM model. In Sec. III, we present the
constraints placed by current data on ADM parameters, including an allowed
region of low $\sigma_{8}$ and low dark photon temperature. In Sec. IV, we
confirm the observational viability of this allowed region and explore its
impacts on matter power spectra and CMB power spectra. In Sec. V, we forecast
the sensitivity of future CMB experiments to ADM in the low $\sigma_{8}$ and
low dark photon temperature regime. In Sec. VI, we summarize and conclude.

##  II Physical effects of changing atomic dark matter parameters

We examine in this section the impact ADM has on CMB and matter power spectra,
and how those impacts depend on ADM parameters. Because of the possibility for
resolving the $\sigma_{8}$ tension, we focus on the “cool” dark sector regime,
as mentioned in the introduction, where the dark photon temperature today
$T_{{\rm d}\gamma}^{0}$ is between 0.6 K and 1.0 K. Prior work by some of us
[119] identified the fraction of non-relativistic matter that is pressure
supported as a useful quantity for understanding changes in observables. We
will indeed find that to be the case here.

Before focusing on the cool regime, we begin with a quick overview of the ADM
model and its phenomenology. More detailed discussions can be found in Refs.
[58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76,
77, 78, 79, 80, 25, 81]. The ADM consists of massive dark protons and dark
electrons interacting with massless dark photons. At early times, dark sector
particles are highly coupled together, forming a dark plasma. Via a Thompson-
scattering-like interaction, dark electrons scatter off dark photons at a per-
particle rate higher than the cosmic expansion rate. When the dark photons
cool down, dark protons and dark electrons can form bound states, similar to
the recombination process in the visible sector. After decoupling from the
dark baryons, dark photons freely stream with a mean free path larger than the
Hubble distance. Dark baryons, after dark recombination, experience very
little pressure support and so evolve much like cold dark matter (CDM).

There are four parameters to characterize the atomic dark sector: the
temperature of the dark photons today $T_{{\rm d}\gamma}^{0}$, the binding
energy of dark hydrogen $B_{\rm d}$, the mass of dark hydrogen $m_{\rm d}$,
and the dark fine structure constant $\alpha_{\rm d}$. In this work we set
$m_{\rm d}$ and $\alpha_{\rm d}$ to have the same values as in the visible
sector.

In previous work [120, 119], the ADM model was used to mimic a scaling
transformation that preserves dimensionless cosmological observables. To do
so, both the dark hydrogen binding energy to dark photon temperature ratio
($B_{\rm d}/T_{{\rm d}\gamma}^{0}$) and the dark baryon-to-photon number ratio
($\eta_{\rm d}$) are set to their values in the visible sector. The same
$B_{\rm d}/T_{{\rm d}\gamma}^{0}$ ensures recombination happens around the
same time in both dark and visible sectors. The same baryon-to-photon ratio
ensures that the photon and baryon perturbations are in phase in both sectors.
In this work, we drop these requirements on the values of $B_{\rm d}/T_{{\rm
d}\gamma}^{0}$ and $\eta_{\rm d}$, allowing them to vary freely. As a result,
the redshift of dark recombination is free to vary, as is the dark photon
temperature. This section is aimed at elucidating the impact on power spectra
of variation of these two parameters.

Let us specify our fiducial $\Lambda$CDM and ADM models. The fiducial
$\Lambda$CDM model is the $\Lambda$CDM model that best fits the _Planck_ 2018
TT,TE,EE+lowE+lensing [121] likelihoods. We refer to it as the base model. It
has parameter values

$\displaystyle\\{H_{0},\Omega_{\rm b}h^{2},\Omega_{\rm dm}h^{2},\tau,A_{\rm
s},n_{\rm s},Y_{\rm P},N_{\rm eff},\Sigma m_{\nu}\\}$ (1) $\displaystyle=$
$\displaystyle\\{{67.37\ {\rm km/s/Mpc}},0.02233,0.1198,0.0540,{2.097\times
10^{-9}},$ $\displaystyle 0.9652,0.2454,3.046,0.06\ {\rm eV}\\}.$

For the base model, the dark matter consists only of CDM, namely $\Omega_{\rm
dm}h^{2}=\Omega_{\rm c}h^{2}$. For ADM models, we use three additional
parameters to specify the dark sector properties: $T_{{\rm d}\gamma}^{0}$,
$B_{\rm d}/T_{{\rm d}\gamma}^{0}$, and the fraction of dark matter in dark
baryons

$f_{\rm adm}\equiv\frac{\omega_{\rm db}}{\omega_{\rm dm}}=\frac{\omega_{\rm
db}}{\omega_{\rm db}+\omega_{\rm c}}.$ (2)

For the fiducial ADM model, we fix the total dark matter density to be the
same as in the base model,

$\omega_{\rm dm}^{\rm ADM}=\omega_{\rm dm}^{\rm\Lambda CDM},$ (3)

fix other cosmological parameters to the values in Eq. (1), and pick the
following ADM parameters:

$\\{f_{\rm adm},B_{\rm d}/T_{{\rm d}\gamma}^{0},T_{{\rm
d}\gamma}^{0}\\}=\\{0.1,\ 10\ \mathrm{eV/K},\ 0.6\ \mathrm{K}\\}.$ (4)

By setting $f_{\rm adm}=0.1$, we assume that 10% of the total dark matter is
dark baryons. Comparing to the base model, 10% of the base model CDM is
replaced by dark baryons in the ADM model. In the following discussion, we
will change $T_{{\rm d}\gamma}^{0}$ and $B_{\rm d}/T_{{\rm d}\gamma}^{0}$
separately to see the impacts on observables.

We also remind the reader that the matter fluctuation amplitude $\sigma_{8}$
is the rms fractional mass variance, computed in linear perturbation theory,
of a mass field convolved with a tophat sphere of radius $8$ Mpc/$h$, where
$h=H_{0}/(100\;\textrm{km}\;\textrm{s}^{-1}\;\textrm{Mpc}^{-1})$. This is
related to the matter power spectrum via

$\sigma_{8}^{2}=\frac{1}{2\pi^{2}}\int
k^{3}P(k)|\widetilde{W}(k)|^{2}\frac{dk}{k},$ (5)

where $\widetilde{W}(k)$ is the Fourier transform of the tophat window
function of radius 8 Mpc/$h$ and $P(k)$ is the linear-theory matter power
spectrum today [e.g. 122].

###  II.1 Altering the dark baryon-to-photon ratio

Figure 1: Linear matter power spectra for the $\Lambda$CDM model and the ADM
models with different dark photon temperatures today, $T_{{\rm d}\gamma}^{0}$,
as well as the evolution of perturbations with $k=0.1\ \mathrm{Mpc^{-1}}$ and
$k=2\ \mathrm{Mpc^{-1}}$. In the ADM models, we set $T_{{\rm d}\gamma}^{0}$ to
0.6 K, 0.8 K and 1.0 K and keep the other parameters the same as in the
fiducial ADM model in Eqs. (3) and (4). On the top of the middle and right
panels, we show the ratios of the dimensional total matter perturbations
between the ADM models and the base model. We show the ratio of dark baryon
overdensity ($\delta\rho$) to the total dark matter overdensity of each model
in the bottom part of the middle and the right panels. The vertical dashed
(dotted) lines show the scale factor of matter-radiation equality (ADM
decoupling).

First, we investigate the impacts of changing $T_{{\rm d}\gamma}^{0}$, with
all other parameters in Eqs. (1), (3), and (4) held fixed at their fiducial
values. Note that this means keeping $B_{\rm d}/T_{{\rm d}\gamma}^{0}$ fixed
and, therefore, adjusting the binding energy accordingly. By varying $T_{{\rm
d}\gamma}^{0}$, we also effectively change $\eta_{\rm d}$, since the dark
baryon density is fixed. To keep $N_{\rm eff}$ fixed while varying the dark
photon temperature today, we adjust $N_{\nu}$ according to

$N_{\textrm{eff}}=N_{\nu}+\left(\frac{T_{{\rm
d}\gamma}^{0}}{T_{\textrm{CMB}}^{0}}\right)^{4}\frac{8}{7}\left(\frac{11}{4}\right)^{\frac{4}{3}},$
(6)

where $T_{\textrm{CMB}}^{0}\approx 2.7255$ K is the temperature of the CMB
photons today [123]. We look into three specific cases with dark photon
temperature of 0.6 K, 0.8 K and 1.0 K, which correspond to a dark photon
energy density that is $0.34\%\ \mathrm{to}\ 2.62\%$ of the energy density in
light relics.

Figure 2: Comparisons of CMB lensing, TT, and EE spectra between the
$\Lambda$CDM model and the ADM models with three different values of $T_{{\rm
d}\gamma}^{0}$. The ADM models and the fiducial $\Lambda$CDM model are the
same as those in Fig. 1. We show from left to right in the top panels the CMB
lensing potential, TT, and EE spectra. The fractional differences of these
spectra with respect to the base $\Lambda$CDM model are shown in the bottom
panels. In the TT and EE fractional residual plots, dotted (dashed) lines are
for unlensed (lensed) spectra.

We find that the main effect, on CMB observables, of changing $T_{{\rm
d}\gamma}^{0}$ is due to CMB lensing effects. In general, most of the CMB
lensing is generated during the matter-dominated era when matter perturbations
are the main source of gravitational potentials. Therefore, we will look at
the matter power spectra first.

In Fig. 1, matter power spectra for different $T_{{\rm d}\gamma}^{0}$ are
shown in the left panel. At small scales, there is a suppression of power in
ADM predictions compared to the base model prediction. This suppression is
fundamentally due to the pressure support that the ADM receives from the dark
photons, which slows the growth of $\delta\rho_{\rm adm}$. This in turn
inhibits growth of $\delta\rho_{\rm cdm}$ as well, since the ADM is still
contributing to the expansion rate (which acts against growth), while its
contribution to gravitational potential gradients is suppressed. In the base
model, all of the dark matter freely falls into the gravitational potential
wells after horizon entry. However, in the ADM models, the freely falling CDM
is just 90% of the total dark matter (by mass), while 10% of the dark matter
mass is dark baryons, which receive at least some amount of pressure support
until dark recombination.

The dependence of $P(k)$ suppression on $T_{{\rm d}\gamma}^{0}$ arises from
the dependence of that pressure support on temperature. In the tight-coupling
limit, pressure gradients are proportional to the square of the dark sound
speed, which is given by

$c_{\rm s,d}^{2}=\frac{1}{3}{\left(1+\frac{3\rho_{\rm db}}{4\rho_{{\rm
d}\gamma}}\right)}^{-1},$ (7)

where

$\frac{\rho_{\rm db}}{\rho_{{\rm d}\gamma}}\simeq
6.5\left[\frac{10^{4}}{1+z}\right]\left[\frac{f_{\rm
adm}}{0.1}\right]\left[\frac{0.8\ \mathrm{K}}{T_{{\rm
d}\gamma}^{0}}\right]^{4}.$ (8)

The evolution of $c_{\rm s,d}^{2}$ is shown in Fig. 3. The dark sound speed is
higher when the dark photon temperature is higher, which leads to greater
pressure support of the ADM while it is tightly coupled to the dark photons.

At large scales, the ADM predicted matter power spectra are similar to those
predicted by the base model. This is because these modes enter the horizon
after dark recombination, when dark baryons behave just like CDM because they
have decoupled from the dark photons. The transition between these two regimes
of large scales, where the ADM has no impact, and small scales, where there is
suppression, is set by the wavenumber of the modes crossing the horizon during
dark recombination, $k=3.25\times 10^{-2}\ \mathrm{Mpc^{-1}}$.

Figure 3: The dark sound speed $c_{\rm s,d}^{2}$ as a function of the scale
factor for the indicated dark photon temperatures, with all other parameters
set to those of our fiducial model. The horizon entries of modes with $k=2\
\mathrm{Mpc^{-1}}$ and $k=0.1\ \mathrm{Mpc^{-1}}$ are marked with $a_{\rm
h}^{2}$ and $a_{\rm h}^{0.1}$ respectively. The scale factor at dark
recombination is indicated by $a_{*}^{\rm d}$. At $a_{*}^{\rm d}$, the
$k=3.25\times 10^{-2}\ \mathrm{Mpc^{-1}}$ mode enters the horizon. Note that
for this plot we ignore the impact of dark recombination, which guarantees the
end of tight coupling and leads to a dramatic reduction of the pressure
support felt by the dark baryons.

To understand the impact on $P(k)$ in more detail, we show in the center and
right panels of Fig. 1 the evolution of a mode in the transition region, with
$k=0.1\ \mathrm{Mpc^{-1}}$, and a mode more fully in the high-$k$ suppression
region, with $k=2\ \mathrm{Mpc^{-1}}$. In the top plots, we show the ratios of
the dimensional perturbations of the total matter between the ADM models and
the base model. For both modes, the dimensional perturbations of the total
matter in the ADM models ($\delta\rho_{\rm mtot}^{\rm ADM}$) are suppressed
compared to the base model prediction ($\delta\rho_{\rm mtot}^{\Lambda\rm
CDM}$), and the ADM models with higher $T_{{\rm d}\gamma}^{0}$ show a larger
suppression. For the different modes within the same ADM model, the deviation
of $\delta\rho_{\rm mtot}^{\rm ADM}$ relative to $\delta\rho_{\rm
mtot}^{\Lambda\rm CDM}$ at $a=1$ is larger for the $k=2\ \mathrm{Mpc^{-1}}$
mode than for the $k=0.1\ \mathrm{Mpc^{-1}}$ mode. This is caused by the
different scale factor of horizon entry for the two modes. The $k=2\
\mathrm{Mpc^{-1}}$ mode enters the horizon earlier than the $k=0.1\
\mathrm{Mpc^{-1}}$ mode. Therefore, there is a longer duration of pressure
support for the $k=2\ \mathrm{Mpc^{-1}}$ mode. As a result, the ratio
$\delta\rho_{\rm mtot}^{\rm ADM}/\delta\rho_{\rm mtot}^{\Lambda\rm CDM}$ is
smaller at $a=1$ in the $k=2\ \mathrm{Mpc^{-1}}$ mode than in the $k=0.1\
\mathrm{Mpc^{-1}}$ mode of the same ADM model. Given the same background
matter density in each model, this is consistent with the matter power spectra
shown in the left panel of the figure.

For the $k=2\ \mathrm{Mpc^{-1}}$ modes, horizon entry happens early when the
sound speed of the dark plasma is still large. The dark baryons oscillate due
to the dark photon pressure support, while the CDM perturbation is able to
grow. As a result, the ratio $\delta\rho_{\rm db}/\delta\rho_{\rm dm}$ quickly
decreases and oscillates around zero with decreasing amplitude. This is shown
in the bottom right panel of Fig. 1. At horizon entry, the sound speeds of the
three ADM models are similarly large. As a result, the dark photon pressure-
induced suppression on the gravitational potential is similar for all the
three ADM models. The relative differences in $\Delta P(k)/P(k)$ and
$\delta\rho_{\rm mtot}^{\rm ADM}/\delta\rho_{\rm mtot}^{\Lambda\rm CDM}$ among
the three ADM models are small.

For the $k=0.1\ \mathrm{Mpc^{-1}}$ modes, horizon entry is later, occurring
after a greater decrease in the dark sound speed than for the $k=2\
\mathrm{Mpc^{-1}}$ modes. For these modes in the transition region, the decay
is not only larger than in the $k=2\ \mathrm{Mpc^{-1}}$ case, but the amount
of decay depends significantly on $T_{{\rm d}\gamma}^{0}$. As shown in Fig. 3,
$c_{\rm s,d}^{2}$ for the $T_{{\rm d}\gamma}^{0}=0.6$ K model is below 0.1,
while it is still above 0.2 for the $T_{{\rm d}\gamma}^{0}=1.0$ K model. The
difference in dark pressure support is more significant at horizon entry among
the three models. Additionally, the late horizon entry leads to a shorter
duration of the dark acoustic oscillations, where dark recombination occurs
before the completion of the first dark oscillation period. As a result, the
fraction $\delta\rho_{\rm db}/\delta\rho_{\rm dm}$, as well as the amount of
suppression in $\delta\rho_{\rm mtot}^{\rm ADM}$ compared to the
$\delta\rho_{\rm mtot}^{\Lambda\rm CDM}$, is significantly different among the
three models. This is shown in the middle panel in Fig. 1. The $T_{{\rm
d}\gamma}^{0}=0.6$ K model has the least amount of dark photon pressure. The
dark baryons can cluster more easily with less pressure resistance. Therefore,
the ADM model with $T_{{\rm d}\gamma}^{0}=0.6$ K has the largest
$\delta\rho_{\rm adm}/\delta\rho_{\rm dm}$, as well as the smallest
suppression in $\delta\rho_{\rm mtot}^{\rm ADM}/\delta\rho_{\rm
mtot}^{\Lambda\rm CDM}$. As a result, the dispersion of $\Delta P(k)/P(k)$ is
larger among the three ADM models for modes with this wavenumber.

After understanding the differences in the matter power spectra, we now turn
to the CMB lensing spectra of the ADM models. In the left panel of Fig. 2, we
see that at small scales the lensing potential spectrum amplitudes predicted
by the ADM models are lower than the base model amplitude predictions, with a
higher $T_{\rm d\gamma}^{0}$ leading to a larger suppression. This is
consistent with the changes seen in the matter power spectra. Recall that the
lensing potential spectrum [124] is

$C_{L}^{\phi\phi}\propto\int\frac{dk}{k}\left[\int_{0}^{\chi_{*}}d\chi\Phi(k;\eta_{0}-\chi)j_{L}(k\chi)\left(\frac{\chi_{*}-\chi}{\chi_{*}\chi}\right)\right]^{2},$
(9)

where $\chi$ is the comoving distance, $\eta_{0}$ is the conformal time today,
$j_{L}$ is the spherical Bessel function, $\Phi$ is the Weyl potential, and
the subscript $*$ indicates recombination. Most of the lensing effects come
from redshift $z\lesssim 20$ [e.g. 124], deep in the matter-dominated era.
Therefore, we can use the Poisson equation to replace $\Phi$ with $\delta_{\rm
mtot}$ via

$\Phi(k,z)=-\frac{3}{2}\Omega_{\rm m}H_{0}^{2}(1+z)k^{-2}\delta_{\rm
mtot}(k,z).$ (10)

Given that the background cosmology is the same in the ADM models and in the
base model, the differences in the CMB lensing potential spectra are caused by
the matter power spectra differences. Comparing the matter power spectra and
the lensing potential spectra, this expected correlation is seen in Figs. 2
and 1.

The CMB TT and EE spectra are shown in the middle and right panels of Fig. 2.
In both panels, the fractional changes relative to the base model predictions
for the lensed and unlensed spectra are shown. We find that the main effects
on the lensed TT and EE spectra are caused by CMB lensing. The unlensed ADM
spectra deviate only mildly from base model predictions, but the lensed
spectra deviate significantly at small scales. This is expected due to the
mixing of power at different scales by CMB lensing [124]. At small scales,
where the unlensed CMB spectra are close to zero due to diffusion damping, the
lensing effect of power mixing provides the main contribution to the amplitude
of the lensed spectra. The lensing potential amplitude is smaller when the
dark photon temperature is higher, leading to lower mixing power. As a result,
the lensed CMB spectra with lower $T_{{\rm d}\gamma}^{0}$ deviate less from
the base model prediction.

###  II.2 Changing the dark binding energy-to-dark photon temperature ratio

In this subsection we investigate the physical effects on the matter and CMB
power spectra of altering the dark binding energy-to-dark photon temperature
ratio ($B_{\rm d}/T_{{\rm d}\gamma}^{0}$). We adjust $B_{\rm d}/T_{{\rm
d}\gamma}^{0}$ by varying $B_{\rm d}$ while keeping all other parameters in
Eqs. (1), (3), and (4) fixed. We show spectra for models with $B_{\rm d}$ set
to 3.6, 6, and 8.4 eV. The pressure in the dark baryon-photon plasma prior to
dark recombination is the same for all of these models since they all have the
same dark photon temperature. However, varying $B_{\rm d}/T_{{\rm
d}\gamma}^{0}$ changes the time of dark recombination and, therefore, changes
the time at which this pressure support precipitously drops.

Figure 4: Similar to Fig. 1. The dark photon temperature is fixed in all the
ADM models to $T_{{\rm d}\gamma}^{0}=0.6\ {\rm K}$, and the ratio of dark
hydrogen binding energy-to-dark photon temperature is set to either $B_{\rm
d}/T_{{\rm d}\gamma}^{0}=6,\ 8\ {\rm or}\ 10\ \mathrm{eV/K}$. The dashed
vertical lines show the scale factor of matter-radiation equality. The colored
vertical dotted lines show the scale factor of ADM decoupling for the
different ADM models. Figure 5: Similar to Fig. 2. The dark photon temperature
is fixed in all the ADM models to $T_{{\rm d}\gamma}^{0}=0.6\ {\rm K}$, and
the ratio of dark hydrogen binding energy-to-dark photon temperature is set to
either $B_{\rm d}/T_{{\rm d}\gamma}^{0}=6,\ 8\ {\rm or}\ 10\ \mathrm{eV/K}$.

As in Sec. II.1, we first discuss the matter power spectra. In the left panel
of Fig. 4, we show the matter power spectrum predictions for the ADM models
with different $B_{\rm d}/T_{{\rm d}\gamma}^{0}$, as well as the prediction
from the base model. At large scales, at which the modes enter the horizon
after dark recombination, the ADM predicted matter power spectra are identical
to those predicted by the base model. At small scales, ADM predicted matter
power spectra are suppressed relative to the base model prediction.

The key factor leading to the difference in $\Delta P(k)/P(k)$ is the duration
of the pressure support between horizon entry and dark recombination. We pick
two modes ($k=0.1\ \mathrm{Mpc^{-1}}$ and $k=2\ \mathrm{Mpc^{-1}}$ in the
middle and right panels of Fig. 4, respectively) to see the perturbation
evolution. When the dark binding energy is higher, more energy is required for
a dark photon to photoionize a bound state, leading to an earlier dark
recombination epoch. After dark recombination, dark photons begin to free
stream, and the ADM pressure support drops to near zero, stopping any further
suppression of the total matter perturbations compared to the base model
prediction. For the same mode, the perturbation evolutions before dark
recombination are quite similar among the three ADM models. This is expected
because the dark photon pressure amplitudes are the same when the dark photon
temperature is fixed. The differences only appear after dark recombination has
occurred in one of the models. For the model with dark recombination at a
later scale factor, the pressure support lasts longer, leading to a longer
suppression of the clustering of ADM despite the same background evolution as
other ADM models. Thus, $\delta\rho_{\rm mtot}$ is further suppressed, leading
to a smaller $\Delta P(k)/P(k)$.

Label | Datasets | BBN consistency?
---|---|---
BBN consistent, with SH0ES data |  _Planck_ , BAO, SH0ES | yes
BBN consistent, without SH0ES data |  _Planck_ , BAO | yes
BBN inconsistent, with SH0ES data |  _Planck_ , BAO, SH0ES | no
BBN inconsistent, without SH0ES data |  _Planck_ , BAO | no
Table 1: Summary of datasets and constraints used in this paper. CMB data are
from the _Planck_ 2018 high-$\ell$ TT+TE+EE, low-$\ell$ TT, low-$\ell$ EE, and
lensing datasets [98]; BAO data are from 6dFGS [125], SDSS MGS [126], and BOSS
DR12 [127]; and the SH0ES $H_{0}$ value is from Ref. [128]. BBN consistency
indicates constraints on helium abundance from big bang nucleosynthesis (BBN).

As for dependence on scale, smaller scales enter the horizon earlier, leading
to a longer duration of pressure support between horizon entry and dark
recombination. Thus, the total matter perturbation is more suppressed and the
matter power is lower as the scale becomes smaller. This scale dependence
becomes quite weak for modes that enter the horizon deep in the radiation-
dominated era since matter is a small fraction of the total density.

In Fig. 5, we show the CMB TT, EE, and lensing potential spectra with three
different dark binding energies, $B_{\rm d}=3.6\ \mathrm{eV},\ 6\ \mathrm{eV}\
{\rm and}\ 8.4\ \mathrm{eV}$, and compare them to the fiducial $\Lambda$CDM
model. For similar reasons to those discussed in Sec. II.1, the CMB lensing
potential spectra are closely related to the matter power spectra. Comparing
the lensed and unlensed TT and EE spectra, we find that CMB lensing is the
main driver of the differences in the lensed CMB spectra at small scales.

##  III Constraints placed by current data

(a) BBN consistent, with SH0ES data

(b) BBN consistent, without SH0ES data

(c) BBN inconsistent, with SH0ES data

(d) BBN inconsistent, without SH0ES data

Figure 6: 1- and 2-dimensional marginal posterior probability densities for
$f_{\textrm{adm}}$, $\sigma_{8}$, $T_{{\rm d}\gamma}^{0}$, and $H_{0}$ for all
four model spaces. The diagonal shows the (arbitrarily-normalized)
1-dimensional posteriors. The inner and outer contours on the 2-dimensional
posteriors enclose the 68% and 95% credible regions, respectively.

In this section, we explore the parameter constraints placed by current
observational data on the ADM model described in Sec. II. Note that this
parameter space is not just $\Lambda$CDM plus three ADM parameters, but
$\Lambda$CDM + $N_{\rm eff}$ plus three ADM parameters. We let $N_{\rm eff}$
vary freely in addition to the dark photon temperature $T_{{\rm
d}\gamma}^{0}$. In practice we implement this freedom by allowing the number
of light neutrino species $N_{\nu}$ to adjust away from its $\Lambda$CDM value
of 3.044 [129]. We included this freedom so that our model space would contain
the scaling transformation direction highlighted by Refs. [120] and [119], and
its associated large posterior $H_{0}$ uncertainty. In the explored models, we
consider both cases where the primordial helium abundance is predicted by big
bang nucleosynthesis (BBN) and where it is a free parameter.

We use CMB data from _Planck_ , namely the _Planck_ 2018 high-$\ell$ TT+TE+EE,
low-$\ell$ TT, low-$\ell$ EE, and lensing datasets [98], as well as BAO data
from 6dFGS [125], SDSS MGS [126], and BOSS DR12 [127]. Additionally, we
consider parameter constraints both with and without the inclusion of the
independent 2022 measurement of $H_{0}$ from SH0ES [128]. While Ref. [128] is
no longer the latest SH0ES result, it was the most recent data available at
the time we began this work. Hereafter, SH0ES data will refer to Ref. [128].
The datasets and constraints used in the following analyses to obtain joint
constraints on parameter values are summarized in Table 1.

We use a modified version of CAMB [130, 68, 70], a publicly available
Einstein-Boltzmann solver, to make model predictions and CosmoMC [131] to
calculate the posterior probability distributions of the parameters using
Markov chain Monte Carlo (MCMC) methods. We applied a flat prior on the ADM
parameters as

$\displaystyle T_{{\rm d}\gamma}^{0}\ [{\rm K}]\in[0.5,2.7],\ \Omega_{{\rm
db}}h^{2}\in[0.0001,0.02],$ $\displaystyle{\rm and}\ B_{\rm d}/T_{{\rm
d}\gamma}^{0}\ [{\rm eV/K}]\in[2.5,15].$ (11)

This choice of prior on $B_{\rm d}/T_{{\rm d}\gamma}^{0}$ allows dark
recombination to occur at a wide range of redshifts approximately between 475
and 2700. We set the energy densities of the dark photons and dark baryons to
not exceed those of the photons and baryons in the visible sector defined in
the base model in Sec. II. The lower bound on $T_{{\rm d}\gamma}^{0}$ is due
to a computational limitation, where a lower $T_{{\rm d}\gamma}^{0}$
occasionally results in a failure of the ODE solver for dark ionization
history. However, at the lowest allowed $T_{{\rm d}\gamma}^{0}$, pressure
support is significantly reduced (relative to the maximal high-temperature
amount) at sufficiently early times to allow for the emergence of the low
$\sigma_{8}$ solution that is a major focus of this paper. We leave
improvements to address this numerical difficulty for future work.

In Sec. III.1 we discuss the emergence, in our constraint contours for the
BBN-inconsistent model without SH0ES data, of the scaling transformation of
Ref. [120]. In Sec. III.2, we examine the ADM parameter space at low $T_{{\rm
d}\gamma}^{0}$ and find that lower values of $\sigma_{8}$ are allowed in this
regime. In Sec. III.3 we explore the possibility of easing the $H_{0}$ tension
with ADM.

###  III.1 Emergence of the scaling symmetry solution

The authors of Ref. [120] introduced a scaling of rates in the Einstein-
Boltzmann equations, and associated scaling of the amplitude of the primordial
power spectrum, that leaves dimensionless cosmological observables invariant.
This FFAT (for Free-Fall rate, Amplitude, and Thomson rate) scaling
transformation is given by

$\displaystyle\sqrt{G\rho_{i}(a)}$
$\displaystyle\rightarrow\lambda\sqrt{G\rho_{i}(a)},$
$\displaystyle\sigma_{\textrm{T}}n_{e}(a)$
$\displaystyle\rightarrow\lambda\sigma_{\textrm{T}}n_{e}(a),\ {\rm and}$
$\displaystyle A_{\rm s}$ $\displaystyle\rightarrow A_{\rm s}/\lambda^{(n_{\rm
s}-1)},$ (12)

where $a=1/(1+z)$ is the scale factor and $\lambda$ is the scaling
transformation parameter [120].

As explained in Ref. [120], the FFAT scaling transformation can be mimicked in
a model with ADM, additional free-streaming light relics, and free helium.
Thus, when we allow for BBN inconsistency (so that helium is free) and do not
restrict the value of $H_{0}$ with use of SH0ES (so as not to pin down
$\lambda$ too tightly), parameters are free to move along a subspace of the
expanded parameter space we use here, corresponding to this FFAT-mimicking
scaling transformation, with little to no impact on the dimensionless
observables in our likelihood.

We therefore expect to see this parameter-space direction emerge in the
constant probability density contours of our BBN-inconsistent no-SH0ES
posterior shown in Fig. 6(d). The elongated features in Figs. 6(d)(4),
6(d)(7), and 6(d)(9) indicate a positive correlation between
$f_{\textrm{adm}}$ and $T_{{\rm d}\gamma}^{0}$, between $f_{\textrm{adm}}$ and
$H_{0}$, and between $T_{{\rm d}\gamma}^{0}$ and $H_{0}$, respectively,
consistent with the expectations due to the scaling transformation symmetry.

As pointed out in Ref. [120], the scaling transformation, at least as they
implement it, requires BBN-inconsistent helium abundances. Also noteworthy,
the helium abundance required for consistency with SH0ES is significantly
below determinations from observations [132, 133, 134, 135, 136]. These are
formidable challenges to the building of a model that can successfully exploit
the FFAT scaling to solve the Hubble tension (see however Ref. [137]).
Nevertheless, we see the explanatory power of the FFAT symmetry, as it
explains these features of the posterior.

###  III.2 Constraints at low $T_{{\rm d}\gamma}^{0}$

As expected from Sec. II.1, with low $T_{{\rm d}\gamma}^{0}$, dark photons
provide only very mild pressure support to dark baryons (and almost none at
all after dark recombination). Thus, at low $T_{{\rm d}\gamma}^{0}$, ADM is
semi-degenerate with CDM, and the ADM parameter space opens up significantly
as a result, as can be seen with the higher values of $f_{\rm adm}$ allowed in
Figs. 6(a)(4), 6(b)(4), and 6(d)(4).

However, there is not a complete degeneracy between ADM and CDM at low
$T_{{\rm d}\gamma}^{0}$, and distinct observational signatures between the two
models remain. As discussed in Sec. II.1, even with low $T_{{\rm
d}\gamma}^{0}$, the matter power spectrum in the presence of ADM is damped
relative to the $\Lambda$CDM matter power spectrum. Therefore, lower values of
$\sigma_{8}$ in line with the KiDS-1000 result [99] are allowed with lower
$T_{{\rm d}\gamma}^{0}$, as can be seen in Figs. 6(a)(5), 6(b)(5), and
6(d)(5).

Furthermore, a higher fraction of pressure-supported dark matter also yields a
dampened matter power spectrum, and therefore a reduced $\sigma_{8}$, relative
to that of $\Lambda$CDM. Thus, the above-mentioned tolerance for higher
$f_{\rm adm}$ at low $T_{{\rm d}\gamma}^{0}$ also allows for lower
$\sigma_{8}$, as can be seen in Figs. 6(a)(2), 6(b)(2), and 6(c)(2). This low
$\sigma_{8}$ and low $T_{{\rm d}\gamma}^{0}$ region of parameter space is
interesting because it could provide a solution to the $\sigma_{8}$ tension.
We discuss this potential solution more below in Sec. IV. As we will see in
Sec. V.4, this mechanism for reducing $\sigma_{8}$ appears to be testable with
future CMB measurements.

We now compare with another recent exploration of the ADM model parameter
space [107]. In addition to $f_{\rm adm}$, $T_{{\rm d}\gamma}^{0}$, and
$B_{\rm d}/T_{{\rm d}\gamma}^{0}$, they also freed up the dark proton mass and
dark fine structure constant. Somewhat surprisingly, since our model space is
a subspace of theirs, their marginal posteriors, when using the data sets we
use, do not indicate the presence of a low $\sigma_{8}$ solution. It is
possible this is due to differential volume effects as they have two
additional dimensions over which to marginalize. For a recent discussion of
volume effects and how they can cause interesting solutions to get hidden (if
one is “looking” with the posterior probability distribution), see Ref. [138].
Some support for this hypothesis comes from the fact that when they
effectively adopt a strong prior on a low $\sigma_{8}$, they find their model
can accommodate it.

This effective adoption of a strong $\sigma_{8}$ prior is their use of the
_Planck_ 2013 Sunyaev-Zel’dovich (SZ) data [139] that acts as a tight
constraint centered on a lower value of $\sigma_{8}$. Note that the _Planck_
2013 SZ likelihood used by [107], unlike other versions in _Planck_ 2015
[140], assumes zero uncertainty in the mass bias parameter which controls the
relationship between cluster mass and SZ observables. This assumption leads to
a significant underestimate of uncertainty in $\sigma_{8}$. Therefore, we
chose not to include the _Planck_ 2013 SZ data in our analysis.

###  III.3 Constraints on $H_{0}$

As discussed above in Sec. III.1, for the BBN-inconsistent model without the
SH0ES data in Fig. 6(d), a broad range of $H_{0}$ values is allowed by current
CMB and BAO data as a result of the FFAT scaling transformation symmetry in
Ref. [120]. Of course, for the cases where we include SH0ES data, the $H_{0}$
uncertainty decreases dramatically. Even without including SH0ES data,
enforcing BBN consistency prevents the Thomson rate from following FFAT
scaling, also leading to a dramatic reduction of the $H_{0}$ posterior width.

While higher values of $H_{0}$ are not preferred by current CMB and BAO data
for the BBN-consistent model without the SH0ES data in Fig. 6(b)(10),
tolerance for higher values of $H_{0}$, in line with the latest SH0ES
measurement [141], is well above what we have in the case of $\Lambda$CDM.
Thus, the partial opening of the ADM parameter space in this work still could
provide a possible, albeit partial, resolution to the $H_{0}$ tension, as seen
in Ref. [120].

It is worth noting that, for the BBN-consistent model without the SH0ES data,
higher $H_{0}$ and lower $\sigma_{8}$ are not simultaneously allowed by
current data, as seen in Fig. 6(b)(8). Similarly, the addition of the SH0ES
$H_{0}$ data in Fig. 6(a)(3) tightens the 1-dimensional constraints on
$\sigma_{8}$, as compared to the constraints without the SH0ES data in Fig.
6(b)(3), thereby excluding the lowest values of $\sigma_{8}$. This effect of
partially excluding low $\sigma_{8}$ with the inclusion of the SH0ES data can
also be seen in the 2-dimensional constraints on $\sigma_{8}$ and $T_{{\rm
d}\gamma}^{0}$ in Fig. 6(a)(5). Therefore, the low $\sigma_{8}$ and low
$T_{{\rm d}\gamma}^{0}$ region of parameter space discussed above in Sec.
III.2 does not provide a simultaneous solution to both the $H_{0}$ and
$\sigma_{8}$ tensions.

Figure 7: Primary CMB, CMB lensing, and linear matter power spectra for the
best-fit BBN-consistent model with $T_{{\rm d}\gamma}^{0}<0.75$,
$f_{\textrm{adm}}>0.09$, and fixed $N_{\nu}=3.044$. In this model, plotted in
the blue dashed lines, $H_{0}=67.52$
$\textrm{km}\;\textrm{s}^{-1}\;\textrm{Mpc}^{-1}$, $\Omega_{\rm
b}h^{2}=0.02241$, $\Omega_{\rm c}h^{2}=0.1076$, $\tau=0.05768$, $A_{\rm
s}=2.113\times 10^{-9}$, $n_{\rm s}=0.9667$, $f_{\textrm{adm}}=0.1024$,
$B_{\rm d}=7.478$ eV, $T_{{\rm d}\gamma}^{0}=0.5001$ K, $N_{\nu}=3.044$, and
$\sigma_{8}=0.7869$. The solid black lines show the fiducial $\Lambda$CDM
power spectra generated using the best fits from Ref. [121]. Top row: TT, TE,
and EE spectra from the left to right, where
$\mathcal{D}_{\ell}=\ell(\ell+1)C_{\ell}(T_{\rm CMB}^{0})^{2}/(2\pi)$. The
green data points on the TT, TE, and EE power spectra show the _Planck_ PR3
2018 data [98]. Bottom left: Lensing power spectra with _Planck_ PR4 data
[142] shown in green, SPTpol data [143] shown in orange, and ACT DR6 data
[144, 145] shown in magenta. Bottom right: Linear matter power spectra with
$|W(k)|^{2}$ appropriate for $\sigma_{8}$ shown in brown; see Eq. (5).

In order to confirm the effects of lower values of $\sigma_{8}$ on the CMB and
verify the observational viability of the possible solution to the
$\sigma_{8}$ tension discussed in Sec. III.2, we will next explore the best
fit with low $\sigma_{8}$ and low $T_{{\rm d}\gamma}^{0}$ from the BBN-
consistent model without inclusion of the SH0ES data to determine whether it
does indeed provide a good fit to the _Planck_ CMB data.

##  IV Observational viability of region of low $\sigma_{8}$ and low $T_{{\rm
d}\gamma}^{0}$

As an assessment of the observational viability of this region of low
$\sigma_{8}$ and low $T_{{\rm d}\gamma}^{0}$ and as an exploration of the
quality of the fit to current data provided by models in this region, we
assess the best-fit BBN-consistent model with $T_{{\rm d}\gamma}^{0}<0.75$,
$f_{\textrm{adm}}>0.09$, and fixed $N_{\nu}=3.044$. We chose these bounds to
ensure that we selected a model with low $\sigma_{8}$ and to ensure that the
chosen $f_{\textrm{adm}}$ is both comparable to the value of
$f_{\textrm{adm}}$ analyzed in Sec. II and is high enough to have detectable
effects. We chose to fix $N_{\nu}$ to its standard model value of 3.044 [129]
in order to reduce the amount of beyond-$\Lambda$CDM physics introduced in
this assessment. The model parameters for this best-fit model are described in
the Fig. 7 caption.

Fig. 7 compares the $\Lambda$CDM TT, TE, EE, lensing, and linear matter power
spectra with the corresponding power spectra for this best-fit model, as well
as with current CMB and lensing data. We see that the ADM and $\Lambda$CDM TT,
TE, and EE spectra are extremely similar. The lensing power spectra are quite
similar as well. In all four cases, both models appear to provide good fits to
current data. The $\Lambda$CDM and ADM matter power spectra are very similar
at low $k$, with the expected differences (see Sec. II) appearing at high $k$.

We do not show a comparison with inferences of $P(k)$ from observations. An
accurate comparison here would require calculation of nonlinear corrections to
the ADM model, a calculation that we have kept beyond the scope of this work.
Recently, the authors of Refs. [146, 147] found that the mild tension between
$S_{8}\equiv\sigma_{8}\sqrt{\Omega_{\rm m}/0.3}$ values inferred from weak
lensing shear measurements and those from _Planck_ $\Lambda$CDM cosmology can
be reconciled if the nonlinear matter power is more suppressed than
hydrodynamical simulations predict. The scale at which this suppression
emerges is similar to the scale where we start to see deviations from
$\Lambda$CDM in the linear ADM $P(k)$. ADM could provide a possible mechanism
for this additional suppression.

We now present a more quantitative comparison with the calculation of the
difference in $\chi^{2}$ values between the best-fit $\Lambda$CDM model and
our best-fit ADM model with low $\sigma_{8}$ and low $T_{{\rm d}\gamma}^{0}$.
The $\chi^{2}$ for the ADM model is $\sim 1.71$ higher than that of the best-
fit $\Lambda$CDM model. A slightly better fit with $\Delta\chi^{2}\approx
1.49$ can be found if $N_{\nu}$ is allowed to vary, with a slightly lower
$N_{\nu}$. However, in this case, the $\chi^{2}$ reduction is less than the
decrease in the number of degrees of freedom, so freeing $N_{\nu}$ does not
improve the quality of fit by more than one would expect from noise fitting.
Note that, while the $\sigma_{8}$ value of $0.7869$ for the best-fit model
with fixed $N_{\nu}$ is higher than the KiDS-1000 central value, it is less
than their 68% confidence upper limit of 0.79 [99]. Since this region of high
$f_{\rm adm}$ and low $T_{{\rm d}\gamma}^{0}$ provides an acceptable fit to
current CMB data, with a value of $\sigma_{8}$ in agreement with low-redshift
measurements, we see that it contains possible solutions to the $\sigma_{8}$
tension.

Our claim that the ADM model can improve concordance with cosmic shear
measurements is substantiated by the fact that it can accommodate a lower
$\sigma_{8}$ consistent with the values inferred from these measurements.
However, this claim should be verified via an analysis that compares with the
data at a lower-level of reduction; e.g., with the shear-shear correlation
functions they determine. We believe such a study is highly motivated by the
analysis we present here.

##  V Forecasting the sensitivity of future experiments to atomic dark matter

The low $\sigma_{8}$, low $T_{{\rm d}\gamma}^{0}$, high $f_{\rm adm}$ region
of parameter space that is consistent with current data raises the question of
its detectability with future measurements. We investigate this question for
the ongoing surveys of the South Pole Telescope with its 3rd focal plane
detector array (SPT-3G), a survey to start in 2024 (the Simons Observatory or
SO), and one planned to start near the end of this decade, CMB-S4 [148, 149].

###  V.1 Experimental setups

We consider a few current and future CMB surveys: SPT-3G [150], SO [151], and
CMB-S4 [152]. For SPT-3G, we perform forecasts for two setups: (A) SPT-3G
Ext-4k covering 4000 deg2 and (B) SPT-3G Ext-10k covering 10,000 deg2. SPT-3G
Ext-4k is the ongoing SPT survey, a combination of two fields (‘Main-1500’ and
‘Summer’) with different noise levels with a combined coverage slightly
greater than 4000 deg2. SPT-3G Ext-10k is the combination of SPT-3G Ext-4k
with an additional 6000 deg2 survey expected to be carried out in the 2024
Austral Winter.

Table 2 lists the experimental beam and the white noise levels for all the
surveys. Other than the instrumental white noise, the maps also contain
atmospheric noise parameterised using $\ell_{\text{knee}}$ and
$\alpha_{\text{knee}}$ with values given in Table 3 for SPT-3G and CMB-S4. For
SO Baseline, we model the atmospheric noise using the procedure described in
Ref. [153] and adopt the values of the parameters of this noise model as given
in Ref. [151].

The total noise power spectrum in each frequency band is modelled as

$N_{\ell}=\Delta_{T}^{2}\left[1+\left(\frac{\ell}{\ell_{\text{knee}}}\right)^{\alpha_{\text{knee}}}\right]B_{\ell}^{-2},$
(13)

where $\Delta_{T}$ represents the white noise level for a given band in
$\mu\mathrm{K}-{\rm arcmin}$; $\ell_{\text{knee}}$ and $\alpha_{\text{knee}}$
are used to parameterize the atmospheric noise; and $B_{\ell}$ is the
experimental beam window function $B_{\ell}=e^{-\ell^{2}\theta_{\rm
FWHM}^{2}/16{\rm ln2}}$ with $\theta_{\rm FWHM}$ listed in Table 2.

Figure 8: The total noise and extragalactic foreground residual power $N_{\ell}^{\rm MV-ILC}$ expected after the optimal internal linear combination of data from different frequency bands for different experiments considered in this work. Survey |  $f_{\mathrm{sky}}$ |  Survey period | Beam $\theta_{\rm FWHM}$ in arcminutes ($\Delta_{T}$ in $\mu\mathrm{K}-{\rm arcmin}$)
---|---|---|---
30 GHz | 40 GHz | 90 GHz | 150 GHz | 220 GHz | 285 GHz
SPT-3G Main-1500 (5Y) | 0.036 | 2019-23 | - | - | 1.7 (3) | 1.2 (2.2) | 1 (8.8) | -
SPT-3G Main-1500 (7Y) | 0.036 | 2019-23, 2025-26 | 1.7 (2.5) | 1.2 (1.85) | 1 (7.4)
SPT-3G Summer | 0.064 | 2019-23 | 1.7 (8.5) | 1.2 (9.0) | 1 (31)
SPT-3G Wide | 0.145 | 2024 | 1.7 (14) | 1.2 (12) | 1 (42)
SO Baseline | 0.4 | 2024-29 | 7.4 (71) | 5.1 (36) | 2.2 (8) | 1.4 (10) | 1 (22) | 0.9 (54)
CMB-S4 | 0.57 | 2029-36 | 7.4 (21.3) | 5.1 (11.7) | 2.2 (1.9) | 1.4 (2.1) | 1 (6.9) | 0.9 (16.9)
Table 2: Band-dependent beam and white noise levels for different experiments considered in this work. The three SPT-3G surveys: SPT-3G Main-1500, SPT-3G Summer, and SPT-3G Wide are surveys of non-overlapping sky patches. SPT-3G Ext-4k is the combination of SPT-3G Main-1500 (5Y) and SPT-3G Summer; and SPT-3G Ext-10k in the combination of three SPT-3G surveys: SPT-3G Main-1500 (7Y), SPT-3G Summer, and SPT-3G Wide. The noise in the polarization maps are assumed to be $\sqrt{2}\Delta_{T}$. Band [GHz] | SPT-3G | CMB-S4
---|---|---
T | P | T | P
30 | - | - | 415, 3.5 | 700, 1.4
40 | 391, 3.5
90 | 1200, 3.0 | 300, -1 | 1200, 4.2
150 | 2200, 4.0 | 1900, 4.1
220 | 2100, 3.9 | 2100, 4.1
285 | - | 2100, 3.9
Table 3: Atmospheric $1/f$ noise specifications ($\ell_{\rm knee},\alpha_{\rm
knee}$ in Eq. (13)) for SPT-3G and CMB-S4 experiments. For SO Baseline, we
simply adopt the procedure outlined in Refs. [151, 153].

###  V.2 Foregrounds and Internal Linear Combination

Other than the experimental noise described above, the total variance in the
maps also receives contributions from astrophysical and galactic foreground
emission.

Extragalactic sources include radio galaxies (RG), dusty star-forming galaxies
(DG), and kinematic and thermal Sunyaev-Zel’dovich (kSZ and tSZ) signals. We
model these based on the SPT measurements [154]. For more details of the
modelling of these temperature foregrounds, we refer the reader to Ref. [153,
155]. For polarization, we assume the 2% polarization fraction for RG and
simply scale the temperature power spectrum of $C_{\ell}^{\rm RG}$ accordingly
[156, 157]. Other extragalactic foregrounds are assumed to be unpolarized.

Given that surveys like CMB-S4 plan to scan larger regions of sky, it is
crucial to consider the impact of galactic foreground signals as well.
Important galactic foregrounds include galactic dust and synchrotron signals.
While the original CMB-S4 footprint is $\sim 67\%$, we ignore the regions
($-10^{\circ}\leq b_{\rm gal}\leq 10^{\circ}$) that are strongly contaminated
by our galaxy and only consider 57% for the forecasting purposes here. The
SPT-3G Ext-4k, SPT-3G Ext-10k, and SO Baseline footprints, with sky fractions
of $\sim 10\%,\ \sim 25\%$ and $\sim 40\%$ respectively, avoid regions with
strong galactic foregrounds. We therefore ignore galactic foregrounds for the
baseline SPT-3G and SO forecasts, as well as for the CMB-S4 forecasts after
the galactic plane cut. However, as a sanity check, we assess the impact of
galactic foregrounds using pySM [158] simulations. This is described in Sec.
V.4.

To reduce the overall impact of noise and foregrounds, we use the internal
linear combination (ILC) technique [159, 160] to optimally combine data from
multiple frequency bands to produce the minimum variance (MV-ILC) CMB map as

$S_{\ell m}=\sum_{i=1}^{N_{\mathrm{bands}}}w_{\ell}^{i}M_{\ell m}^{i}\,,$ (14)

where $M_{\ell m}^{i}$ is the spherical harmonic transform of the map from the
$i$th frequency band and the multipole-dependent weights $w_{\ell}^{i}$ are
tuned in order to minimize the overall variance from noise and foregrounds.
The weights are derived as

$w_{\ell}=\frac{\mathbf{C}_{\mathrm{\ell}}^{-1}A_{\rm s}}{A_{\rm
s}^{\dagger}\mathbf{C}_{\mathrm{\ell}}^{-1}A_{\rm s}}\,,$ (15)

where $\mathbf{C}_{\mathrm{\ell}}$ is a $N_{\mathrm{bands}}\times
N_{\mathrm{bands}}$ matrix containing the covariance of foregrounds and beam-
deconvolved noise across different frequencies at a given multipole $\ell$;
$A_{\rm s}=[1,1,...,1]$ is the frequency response of the CMB in different
bands, and it is a $N_{\mathrm{bands}}\times 1$ vector. The sum of residual
noise and foregrounds power in the MV-ILC map is given as

$N_{\ell}^{\rm MV-ILC}=\dfrac{1}{{A_{\rm
s}}^{T}\mathbf{C}_{\mathrm{\ell}}^{-1}{A_{\rm s}}}\,,$ (16)

which we use for forecasting in the subsequent sections. In Fig. 8, we present
$N_{\ell}^{\rm MV-ILC}$ for all the experiments considered in this work.

To include CMB lensing information, we feed the MV-ILC residuals
$N_{\ell}^{\rm MV-ILC}$ into a lensing quadratic estimator (QE) [161, 162] and
compute the resultant lensing noise curves $N_{\ell}^{\phi\phi}$. We set
$\ell_{\rm max}=3500$ for temperature and $\ell_{\rm max}=4000$ for
polarization during lensing reconstruction. The former choice is to mitigate
the impact of mis-modelling the small-scale foregrounds in the temperature
maps. For lensing reconstruction from a more optimistic version of CMB-S4, we
increase to $\ell_{\rm max}=5000$ for both temperature and polarization.

###  V.3 Forecasting methods

We forecast expected errors on ADM parameters from measurements of CMB
intensity and linear polarization maps as follows. First, we calculate the
Fisher information matrix via

$F_{ij}=\displaystyle\sum_{\ell}\left(\frac{\partial
C_{\ell}^{\nu}}{\partial\alpha_{i}}\right)^{T}\cdot\left(\Sigma_{\ell}^{-1}\right)_{\nu\nu^{\prime}}\cdot\frac{\partial
C_{\ell}^{\nu^{\prime}}}{\partial\alpha_{j}},$ (17)

where $\nu=\\{TT,EE,TE,\phi\phi\\}$; the $\alpha_{i}$ are the model
parameters; and

$\Sigma_{\ell}=\tfrac{2}{\left(2\ell+1\right)f_{\rm sky}}\times\\\
{\tiny\begin{pmatrix}(\tilde{C}_{\ell}^{TT})^{2}&(C_{\ell}^{TE})^{2}&\tilde{C}_{\ell}^{TT}C_{\ell}^{TE}&(C_{\ell}^{T\phi})^{2}\\\
(C_{\ell}^{TE})^{2}&(\tilde{C}_{\ell}^{EE})^{2}&\tilde{C}_{\ell}^{EE}C_{\ell}^{TE}&(C_{\ell}^{E\phi})^{2}\\\
\tilde{C}_{\ell}^{TT}C_{\ell}^{TE}&\tilde{C}_{\ell}^{EE}C_{\ell}^{TE}&\frac{1}{2}\left[(C_{\ell}^{TE})^{2}+\tilde{C}_{\ell}^{TT}\tilde{C}_{\ell}^{EE}\right]&C_{\ell}^{T\phi}C_{\ell}^{E\phi}\\\
(C_{\ell}^{T\phi})^{2}&(C_{\ell}^{E\phi})^{2}&C_{\ell}^{T\phi}C_{\ell}^{E\phi}&(\tilde{C}_{\ell}^{\phi\phi})^{2}\end{pmatrix}},$
(18)

where

$\tilde{C}_{\ell}^{XX}=C_{\ell}^{XX}+N_{\ell}^{XX},$ (19)

$XX=\\{TT,EE,\phi\phi\\}$, $N_{\ell}^{XX}$ is the residual foregrounds and
noise spectrum for the MV-ILC map $N_{\ell}^{\rm MV-ILC}$ described in Sec.
V.2, and $f_{\textrm{sky}}$ is the fraction of the sky observed. We then
calculate the parameter covariance matrix as $C=F^{-1}$. Our Eq. (18) can be
derived from Eqs. 8.20 and 8.21 of Ref. [148]. For all experiments, we include
$100\leq\ell\leq 3500$ for TT, $100\leq\ell\leq 4000$ for EE/TE, and $30\leq
L\leq 3500$ for $\phi\phi$ to match the $\ell$ ranges used for lensing
reconstruction. As mentioned earlier in Sec. V.2, we ignore small-scale
($\ell>3500$) temperature information due to the difficulties in modelling the
foregrounds. Given that CMB-S4 is a expected to operate in the next decade, we
also consider an optimistic case by setting $\ell_{\rm max}=5000$, assuming we
will have better foreground mitigation strategies by then. To be specific, we
use $100\leq\ell\leq 5000$ for TT/EE/TE and $30\leq L\leq 5000$ for
$\phi\phi$.

This procedure is appropriate for homogeneous coverage, when correlations
across multipole moments and types of spectra due to gravitational lensing can
be neglected, and when the posterior probability distribution of the model
parameters is close to Gaussian. The impacts of lensing-induced correlations
increase with decreasing noise levels. Even for CMB-S4, these effects tend to
cause at most 20% changes to parameter standard errors [163]. Non-Gaussianity
of the parameter posterior distribution is a concern we address below.

###  V.4 Forecasted sensitivity of future experiments

|  | Forecasted standard error ($\sigma$)
---|---|---
Parameter |  Fiducial value |  SPT-3G Ext-4k |  SPT-3G Ext-10k |  SO Baseline |  CMB-S4
$f_{\textrm{adm}}$ | $0.102$ | 0.049 | 0.038 | 0.056 | 0.014
$B_{\rm d}/T_{{\rm d}\gamma}^{0}$ (eV/K) | $15$ | 21 | 14 | 12 | 8
$T_{{\rm d}\gamma}^{0}$ (K) | $0.50$ | 0.15 | 0.11 | 0.12 | 0.05
$N_{\textrm{eff}}$ | $3.05$ | 0.13 | 0.10 | 0.09 | 0.05
$f_{\textrm{adm}}$ | $0.030$ | 0.025 | 0.018 | 0.017 | 0.009
$B_{\rm d}/T_{{\rm d}\gamma}^{0}$ (eV/K) | $13.0$ | 5.8 | 3.9 | 3.6 | 2.1
$T_{{\rm d}\gamma}^{0}$ (K) | $1.39$ | 0.41 | 0.29 | 0.26 | 0.14
$N_{\textrm{eff}}$ | $3.25$ | 0.18 | 0.13 | 0.13 | 0.06
Table 4: Forecasted constraints on SPT-3G Ext-4k, SPT-3G Ext-10k, the Simons
Observatory (SO) Baseline, and CMB-S4. Top: Forecasted constraints on the
best-fit BBN-consistent model with $T_{{\rm d}\gamma}^{0}<0.75$,
$f_{\textrm{adm}}>0.09$, and fixed $N_{\nu}=3.044$, as explored in Sec. IV and
Fig. 7. In this model, $H_{0}=67.52$
$\textrm{km}\;\textrm{s}^{-1}\;\textrm{Mpc}^{-1}$, $\Omega_{\rm
b}h^{2}=0.02241$, $\Omega_{\rm c}h^{2}=0.1076$, $\tau=0.05768$, $A_{\rm
s}=2.113\times 10^{-9}$, $n_{\rm s}=0.9667$, and $\sigma_{8}=0.7869$. Bottom:
Forecasted constraints on the best-fit BBN-consistent model with fixed
$f_{\textrm{adm}}=0.03$. In this model, $H_{0}=68.58$
$\textrm{km}\;\textrm{s}^{-1}\;\textrm{Mpc}^{-1}$, $\Omega_{\rm
b}h^{2}=0.02266$, $\Omega_{\rm c}h^{2}=0.1213$, $\tau=0.05721$, $A_{\rm
s}=2.1175\times 10^{-9}$, $n_{\rm s}=0.9714$, and $\sigma_{8}=0.7974$.

Figure 9: 1- and 2-dimensional marginal posterior probability densities for
$f_{\textrm{adm}}$ and $T_{{\rm d}\gamma}^{0}$ for the BBN-consistent model
without the inclusion of the SH0ES data, as well as the forecasted 1- and
2-dimensional posteriors for $f_{\textrm{adm}}$ and $T_{{\rm d}\gamma}^{0}$
for SPT-3G Ext-4k, SPT-3G Ext-10k, SO Baseline, and CMB-S4. The (arbitrarily-
normalized) vertical axes of the diagonal indicate probability. The contours
on the 2-dimensional posteriors enclose the 68% credible region. Left:
Forecasted constraints on the low $\sigma_{8}$ model outlined in the top part
of Table 4. Right: Forecasted constraints on the low $f_{\rm adm}$ model
outlined in the bottom part of Table 4.

In order to forecast the sensitivity of upcoming CMB experiments to the low
$\sigma_{8}$ and low $T_{{\rm d}\gamma}^{0}$ region of parameter space
discussed in Sec. III.2 and Sec. IV, we will forecast the sensitivity of
SPT-3G, SO, and CMB-S4 to the best-fit BBN-consistent model with $T_{{\rm
d}\gamma}^{0}<0.75$, $f_{\textrm{adm}}>0.09$, and fixed $N_{\nu}=3.044$ that
we explored in Sec. IV, hereafter referred to as the low $\sigma_{8}$ model.
As a second point of reference, we will also forecast the sensitivity of
SPT-3G, SO, and CMB-S4 to the best-fit BBN-consistent model with fixed
$f_{\textrm{adm}}=0.03$, hereafter referred to as the low $f_{\rm adm}$ model.
The fiducial models used for forecasts, as well as one-dimensional marginal
errors in ADM parameters and $N_{\textrm{eff}}$, are outlined in Table 4.
Forecast results are also shown in Fig. 9.

Before looking at the ADM parameter constraints, we point out that the
forecasted constraints on $N_{\rm eff}$ degrade some from their $\Lambda$CDM +
$N_{\rm eff}$ expected values due to the additional degrees of freedom. For
example, the CMB-S4 forecasted $\sigma(N_{\textrm{eff}})=0.05$ for the low
$\sigma_{8}$ fiducial model is somewhat larger than the forecasted
$\sigma(N_{\textrm{eff}})=0.0327$ for a fiducial model with no ADM [164]. If
we include up to $\ell=5000$ for TT/EE/TE/$\phi\phi$ and for lensing
reconstruction, as done in [155], we find $\sigma(N_{\textrm{eff}})=0.04$.
Presumably this mild degradation to 0.04 from 0.0327 is due to some degree of
degeneracy that emerges between $N_{\rm eff}$ and the other cosmological
parameters, with the addition of the three ADM parameters. We have checked
that when we instead eliminate the ADM, using an unmodified version of CAMB,
we recover $\sigma(N_{\rm eff})=0.033$.

The key question we wish to address here is whether CMB-S4 will be able to
detect the influence of ADM if the cause of the $\sigma_{8}$ tension is indeed
due to a low-temperature ADM. We are thus most interested in the constraints
on the low $\sigma_{8}$ fiducial model – the one with $f_{\rm adm}=0.102$ used
for the left panel of Fig. 9. As can be seen in Table 4, the expected error in
this case is 0.014, allowing for a greater than $7\sigma$ detection. If we
include up to $\ell=5000$ for TT/EE/TE/$\phi\phi$ and lensing reconstruction,
which would require successfully cleaning foregrounds out to higher $\ell$,
the expected error halves to 0.007, which would allow for a greater than
$14\sigma$ detection.

The posteriors given current data, as one can see in Figs. 6 and 9, are highly
non-Gaussian. The forecasting methods used assume Gaussianity, so the non-
Gaussian posterior is a source of concern. However, we expect that as the data
become more constraining, the posterior will become more Gaussian. We see
evidence of this from the forecasts for our second fiducial model, which has a
much lower value of $f_{\rm adm}$. The error on $f_{\rm adm}$ assuming this
fiducial model is 0.009, only slightly different from the higher $f_{\rm adm}$
(low $\sigma_{8}$) case. The marginal posterior for $f_{\rm adm}$ given CMB-S4
data thus appears to be fairly Gaussian, enough so that a $7\sigma$ detection
would almost certainly be a detection of very high significance.

Experiments prior to CMB-S4 also have a chance of seeing some, perhaps only
weak, evidence in favor of non-zero $f_{\rm adm}$. SPT-3G Ext-10k, given the
low $\sigma_{8}$ fiducial model, can nominally be expected to make a
$2.7\sigma$ detection of non-zero $f_{\rm adm}$. However, one can see greater
fiducial model sensitivity to the error on $f_{\rm adm}$ for SPT-3G Ext-4k,
SPT-3G Ext-10k, and SO Baseline, so these constraints are both weaker than
those from CMB-S4 and less certain. A clearer picture of expectations in these
cases would require calculation of posterior distributions from simulated
data.

The relative performance of SO Baseline to the SPT-3G surveys changes
significantly between our two fiducial model cases. In the low $\sigma_{8}$
case (with high $f_{\rm adm}$) the main observable impact of varying $f_{\rm
adm}$ is in lensing, where the deeper SPT-3G Main-1500 survey leads to
stronger lensing constraints at small scales than one expects from SO
Baseline. The result is tighter constraints on $f_{\rm adm}$ from the SPT-3G
surveys. In the low $f_{\rm adm}$ case, the signals show up more in the field
(TT/TE/EE) power spectra where SO’s wider sky coverage is more of an
advantage, and so SO’s performance relative to that of the SPT-3G surveys is
stronger.

We also note that these forecasts indicate that some fairly precise
measurements of $T_{{\rm d}\gamma}^{0}$ may be possible with CMB-S4 data in
either fiducial model case; formally, these are $\sim 10\sigma$ constraints.
In the low $T_{{\rm d}\gamma}^{0}$ (low $\sigma_{8}$) case, one should be able
to both bound the temperature from below, and have clear evidence that the
temperature is well below that of the CMB or the cosmic neutrino background.
The temperature constraint would be a valuable clue, in either case, about a
possible thermal origin for the dark photon relic.

We have also plotted contours from our forecasts and for current data for our
two most well-constrained ADM parameters, $f_{\rm adm}$ and $T_{{\rm
d}\gamma}^{0}$, in Fig. 9. One can visually see here the qualitative
improvement over constraints from current data expected very soon from the
SPT-3G Ext-4k survey, and a continuing progression through to the CMB-S4
constraints.

As mentioned in Sec. V.2, we also check the impact of galactic foregrounds on
our constraints. We generate galactic dust and synchrotron signals using
publicly available pySM [158] simulations, which are based on Planck sky model
code [165]. We calculate the dust and synchrotron foreground power in the SPT
and CMB-S4 footprints, and include them along with other components in the ILC
step. With the inclusion of galactic foregrounds, we find that the above
constraints weaken only marginally by $\lesssim 5\%$.

It is important to note that all forecasts in this paper are based on linear
theory predictions. However, there exist significant nonlinear corrections to
the results outlined in this section, meaning that future work on nonlinear
predictions is needed to refine these forecasts, as well as for the comparison
of theoretical matter power spectra to data mentioned in Sec. IV.

Despite a caveat regarding the varying sensitivity of future experiments to
different fiducial ADM models, ultimately, these forecasts demonstrate the
promise for the ADM model in general, and the low $\sigma_{8}$ and low
$T_{{\rm d}\gamma}^{0}$ region of parameter space in particular, to be tightly
constrained with upcoming CMB experiments. These tighter constraints include
the possibility of either confirming or excluding the low $\sigma_{8}$ and low
$T_{{\rm d}\gamma}^{0}$ region of parameter space, meaning that, if this
region is the solution to the $\sigma_{8}$ tension, its effects should be
detectable with future experiments.

##  VI Conclusions

We investigated the observational consequences of an extension to $\Lambda$CDM
to include both ADM and the possibility of a non-standard energy density in
free-streaming massless particles. The ADM consists of dark protons and dark
electrons interacting via a dark electromagnetism, and hence the model
includes a thermal background of dark photons. Like in the visible sector,
these undergo a transition from a tightly-coupled plasma to dark hydrogen plus
free-streaming dark photons. Since the dark electrons, protons, and hydrogen
are all non-relativistic during the epochs of interest we consider them as
contributions to the total density of dark matter, while the massless dark
photons contribute to the total radiation density. The free parameters of the
model we took to be the fraction of dark matter that is atomic $f_{\rm adm}$,
the temperature of dark photons today $T_{{\rm d}\gamma}^{0}$, and the binding
energy of dark hydrogen $B_{\rm d}$, in addition to the standard $\Lambda$CDM
parameters and $N_{\rm eff}$.

We examined the impact of ADM on CMB and matter power spectra in the “cool”
regime (with $T_{{\rm d}\gamma}^{0}$ in the 0.6 K to 1 K range) in Sec. II. In
this regime, the main influence on the CMB temperature and polarization
spectra is via gravitational lensing, due to the impact on the matter power
spectrum $P(k)$. We saw no change from $P(k)$ in our baseline $\Lambda$CDM
model for modes entering the horizon after dark recombination, and a nearly-
uniform suppression of power for modes that enter during radiation domination
and while $c_{\rm s,d}^{2}$ has not yet dropped much below its maximum value
of 1/3. The suppression of power is due to the pressure support received by
the fraction of the dark matter that is ADM. This pressure support slows down
the gravitationally-driven growth of both the ADM and CDM perturbations, as
the ADM does not contribute as much as it would otherwise to gravitational
potential gradients. Such suppression of power is a general feature of models
in which part of the dark matter interacts with dark radiation, which others
have also noted might address the $\sigma_{8}$ tension [e.g. 109, 110, 111,
112, 113, 114, 115, 116, 117, 118, 107].

We also looked at the dependence of these effects on variations in the dark
photon temperature in Sec. II.1, and on variations in the dark hydrogen
binding energy in Sec. II.2. Because of the primary role of CMB lensing, we
also focused here on $P(k)$. Variation in $B_{\rm d}$ (at fixed $T_{{\rm
d}\gamma}^{0}$) changes when dark recombination occurs, impacting the duration
of pressure support for the ADM and, therefore, the amount of power
suppression. Varying $T_{{\rm d}\gamma}^{0}$ also changed the amplitude of
dark photon pressure. We looked at the dependence of the square of the sound
speed $c_{\rm s,d}^{2}$ of the dark photon-ADM plasma on dark photon
temperature since pressure gradients are proportional to this quantity. No
matter the dark photon temperature, at sufficiently early times, $c_{\rm
s,d}^{2}$ reaches its maximal value of 1/3. Modes that enter the horizon after
significant reduction of this quantity do not suffer the suppression in growth
due to pressure support. Reducing $T_{{\rm d}\gamma}^{0}$ thus moves the onset
of $P(k)$ suppression to higher $k$.

We then constrained the ADM model using current CMB [98] and BAO [125, 126,
127] data in Sec. III by considering four different model spaces: both with
and without BBN constraints on helium abundance, and with and without
incorporating the $H_{0}$ measurement from SH0ES [128]. In Sec. III.1, we saw
in elongated probability density contours evidence of a quasi-degeneracy
between $f_{\textrm{adm}}$, $T_{{\rm d}\gamma}^{0}$, and $H_{0}$. We pointed
out that this is expected since this model space allows for a scaling
transformation that is nearly the FFAT-scaling transformation, with associated
symmetry, introduced in Ref. [120] and discussed extensively in Ref. [119].
Likewise, this scaling transformation cannot be followed in the three cases
where we included SH0ES data and/or enforced BBN consistency, and hence we do
not see the elongated contours in these cases. These results highlight the
explanatory power of the FFAT scaling transformation symmetry.

In Sec. III.2 we presented a key finding: at low dark photon temperature, the
ADM parameter space opens up significantly, allowing for higher fractions of
ADM and therefore a lower $\sigma_{8}$. This lower $\sigma_{8}$ is in line
with results from cosmic shear and other large-scale structure measurements
[99, 100, 101, 102, 103, 104, 105], providing a possible solution to the
$\sigma_{8}$ tension. We can understand the lower $\sigma_{8}$ physically
because, as we found in Sec. II, $P(k)$ is suppressed on small scales with the
presence of the pressure-supported ADM.

We subsequently verified the observational viability of the low $\sigma_{8}$
and low $T_{{\rm d}\gamma}^{0}$ region of parameter space by comparing CMB
power spectra for a best-fit model in the region, to current data in Sec. IV.
We found that the best-fit model with low $T_{{\rm d}\gamma}^{0}$, high
$f_{\rm adm}$, and $N_{\nu}$ fixed to its standard model value provides a good
fit to current data, roughly comparable to that of $\Lambda$CDM with
$\chi^{2}$ only $\sim 1.71$ higher. We saw no significant improvement in
quality of fit by letting $N_{\nu}$ vary. However, it is worth noting, as seen
in Sec. III.3, that this possible solution to the $\sigma_{8}$ tension likely
does not provide a simultaneous solution to the $\sigma_{8}$ and $H_{0}$
tensions.

This low $\sigma_{8}$ region of the parameter space requires a dark photon
temperature of about 0.5 K. This is below the temperature expected for a boson
with two degrees of freedom even if it froze out while all standard model
particles were still relativistic. Such a cool component would point to
different post-inflation reheating temperatures for the ADM and dark photons,
relative to standard model particles, and thus have implications for the
reheating process (see e.g. Ref. [166]). A non-thermal mechanism for dark-
sector particle production could be a potential solution (see e.g. Refs. [112,
167]).

Our investigation also motivates the development of nonlinear corrections to
the linear ADM model predictions presented in this paper, both in order to
compare ADM matter power spectra with current data and to improve forecasts,
as mentioned in Sec. IV and V.4. Additionally, beyond the analysis via
$\sigma_{8}$ presented in this paper, our work highlights the need for more
robust comparison to cosmic shear measurements (possibly via shear-shear
correlation functions) to see the extent to which the low $\sigma_{8}$ region
identified in Sec. III.2 improves concordance with these measurements, as
mentioned in Sec. IV.

In order to determine whether this possible solution to the $\sigma_{8}$
tension could be tested with future observations, we forecasted the
sensitivity of future CMB experiments to this low $\sigma_{8}$ region of
parameter space. We found in Sec. V.4 that CMB-S4 will be able to tightly
constrain both the fraction of ADM and the dark photon temperature, including
the possibility of detecting nonzero $f_{\rm adm}$ with high significance. We
also see the promise of experiments prior to CMB-S4 – SPT-3G and SO – to
provide steadily tighter constraints on the ADM model. Ultimately we find
that, if the low $\sigma_{8}$ and low $T_{{\rm d}\gamma}^{0}$ region of
parameter space is the solution to the $\sigma_{8}$ tension, its effects
should be detectable with future CMB experiments.

##  VII Acknowledgements

This material is based upon work initiated as part of the Physics Research
Experiences for Undergraduates (REU) Program at the University of California,
Davis and was supported by the National Science Foundation (NSF) under grant
PHY-2150515. LK, FG, and EH were supported in part by DOE Office of Science
award DE-SC0009999. F.-Y. C.-R. is supported by the NSF under grant
AST-2008696. F.-Y. C.-R. also thanks the Robert E. Young Origins of the
Universe Chair fund for its generous support. SR is supported by the Center
for AstroPhysical Surveys (CAPS) at the National Center for Supercomputing
Applications (NCSA), University of Illinois Urbana-Champaign. LK acknowledges
the hospitality of TTK at RWTH where some of this work was completed. We thank
B. Benson and T. Crawford for feedback on a draft and G. Lynch for useful
conversations.

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