# A Ratio-Preserving Approach to Cosmological Concordance

Kylar Greene<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of New Mexico
Albuquerque, New Mexico 87131  Theoretical Physics Department, Fermi National
Accelerator Laboratory
Batavia, Illinois 60510  Francis-Yan Cyr-Racine<EMAIL_ADDRESS>Department of
Physics and Astronomy, University of New Mexico
Albuquerque, New Mexico 87131

###### Abstract

Cosmological observables are particularly sensitive to key ratios of energy
densities and rates, both today and at earlier epochs of the Universe. Well-
known examples include the photon-to-baryon and the matter-to-radiation
ratios. Equally important, though less publicized, are the ratios of pressure-
supported to pressureless matter and the Thomson scattering rate to the Hubble
rate around recombination, both of which observations tightly constrain.
Preserving these key ratios in theories beyond the $\Lambda$ Cold-Dark-Matter
($\Lambda$CDM) model ensures broad concordance with a large swath of datasets
when addressing cosmological tensions. We demonstrate that a mirror dark
sector, reflecting a partial $\mathbb{Z}_{2}$ symmetry with the Standard
Model, in conjunction with percent level changes to the visible fine-structure
constant and electron mass which represent a phenomenological change to the
Thomson scattering rate, maintains essential cosmological ratios.
Incorporating this ratio preserving approach into a cosmological framework
significantly improves agreement to observational data
($\Delta\chi^{2}=-35.72$) and completely eliminates the Hubble tension with a
cosmologically inferred $H_{0}=73.80\pm 1.02$ km/s/Mpc when including the
S$H_{0}$ES calibration in our analysis. While our approach is certainly
nonminimal, it emphasizes the importance of keeping key ratios constant when
exploring models beyond $\Lambda$CDM.

††preprint: APS/123-QED

##  I Introduction

The success of the $\Lambda$ cold-dark-matter ($\Lambda$CDM) model at
explaining cosmic microwave background (CMB) observations is undeniable,
earmarked by its simplicity - six degrees of freedom to explain the Universe
[1, 2, 3]. Despite its success, $\Lambda$CDM encounters notable discrepancies
in reconciling distant cosmic observations with local measurements of the
expansion rate today, $H_{0}$ [4, 5, 6, 7]. The systematic discrepancies
between local and distant measurements have been meticulously examined, yet no
resolution has been found [8, 9, 10, 11, 12, 13, 14].

Consequently, we investigate beyond the $\Lambda$CDM paradigm to produce
concordant cosmologies, drawing on several observational clues as our guide.
The initial hint originates from $\Lambda$CDM directly. The model tightly
constrains several critical ratios in the early Universe: the baryon-photon
ratio, the matter-radiation ratio, the fluid-freestreaming radiation ratio,
and the balance between pressure-supported and pressureless matter.
$\Lambda$CDM’s effectiveness at predicting CMB features hinges on its precise
predictions of these ratios while maintaining the minimal number of degrees of
freedom possible. The second clue emerges from the significant similarities
between the baryon density and the estimated dark matter density, suggesting a
potential link in the mechanisms which populate the two sectors. Given the
visible sector’s dynamic and complex nature, it is reasonable that if there is
a connection between the population of both sectors, the dark sector might be
more intricate than just a single particle species. However, this notion goes
against one of $\Lambda$CDM’s foundational assumptions. It assumes that the
majority of the Universe’s matter consists of a single type of cold, non-
interacting, and collisionless dark matter, termed cold dark matter (CDM),
detectable solely through its gravitational effects. Despite its elusive
nature and absence from the Standard Model, it is evident that a significant
portion of the dark sector must conform to CDM’s phenomenological criteria.
However, the question arises: Must the entire dark sector adhere to the CDM
paradigm? Consequently, guided by these clues, our investigation suggests the
necessity for a subdominant addition to the dark sector that: maintains the
ratios predicted by $\Lambda$CDM and has more dynamic degrees of freedom than
CDM. Separately, we require that the introduction of new physics becomes
cosmologically relevant after Big Bang Nucleosynthesis to not interfere with
light elemental abundances such as in Refs. [15, 16] where the dark sector
thermalizes with the visible sector after BBN.

These clues converge toward a singular solution: a mirror sector inspired by
$\mathbb{Z}_{2}$ symmetries [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59]. Introducing mirror sectors as
atomic dark matter (ADM), realizable through broken $\mathbb{Z}_{2}$
symmetries, offers remarkable potential for resolving tensions in both
particle physics and cosmology [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70,
71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106,
107, 108, 109, 110]. Here, a mirror sector can succinctly satisfy both
outlined phenomenological criteria. Firstly, they provide a natural link
between the two sectors, elucidating the similarity in energy densities
through connections in baryogenesis [111, 112, 113, 114]. Secondly, they
inherently justify how the mirror sector contributes the required radiation
and pressure supported matter preserving $\Lambda$CDM’s predictive accuracy
for the specified ratios [115, 116].

We utilize a symmetry of cosmological observables to guide our model building
process, the Free-Fall, Amplitude, and Thomson (FFAT) scaling [115, 117]. The
FFAT symmetry prescribes a way to increase the Hubble expansion rate of the
Universe by proportionally adjusting the length and rate scales of the
Universe. Specifically, the following rescaling of the energy density
$\rho_{i}$ of species $i$, the Thomson scattering rate $\dot{\kappa}$, and the
amplitude of scalar fluctuations $A_{\rm s}$ by a real parameter $\lambda>1$
leaves dimensionless cosmological observables invariant

$\\{\rho_{i}\xrightarrow{\lambda}\lambda^{2}\rho_{i},\dot{\kappa}\xrightarrow{\lambda}\lambda\dot{\kappa},A_{\rm
s}\xrightarrow{\lambda}A_{\rm s}/\lambda^{n_{\rm{s}}-1}\\},$ (1)

while leading to a larger Hubble expansion rate $H\xrightarrow{\lambda}\lambda
H$. In the above $n_{\rm s}$ is the scalar spectral index. Integrating the
FFAT scaling into a cosmological scenario requires addressing two, nontrivial
model building questions:

  * •

What is the form of the additional energy density needed to satisfy
$\rho_{i}\rightarrow\lambda^{2}\rho_{i}$?

  * •

How do we modify the visible Thomson scattering rate to satisfy
$\dot{\kappa}\rightarrow\lambda\dot{\kappa}$?

Notably, ADM naturally satisfies the first model building criteria of the FFAT
symmetry by allowing the inclusion of proportional additions of radiation and
pressure supported matter while preserving the critical ratios predicted by
$\Lambda$CDM. However, as a generic feature of models seeking to alleviate the
Hubble tension, the increase in $H_{0}$ will begin to cause issues with CMB
observables as the ratio of the Thomson scattering rate to background
expansion rate (the characteristic timescale of recombination) moves away from
the $\Lambda$CDM predicted value.

To protect cosmic observables from alterations to the characteristic time
scale of recombination, Thomson scattering rate must be proportionally scaled
compared to the expansion rate as part of its second model building criteria.
Scaling the Thomson scattering rate, which depends on electron number density
and the Thomson cross section, introduces theoretical challenges. In this
study, we utilize the strategy first suggested by Ref. [116], modifying the
parameter values for the electromagnetic fine-structure constant and electron
mass at the percent level. This adjustment achieves the necessary scaling of
the Thomson scattering rate while maintaining the binding energy of light
elements at the Standard Model value. As we will see, this effectively renders
the CMB insensitive to these changes at first order. However, varying
fundamental constants is notoriously difficult to realize in a physical
Lagrangian, for example see Refs. [118, 119]. We point out that this method is
not the only way to scale the Thomson scattering rate. It represents a
straightforward phenomenological model that captures interactions beyond the
Standard Model potentially affecting the Thomson scattering rate. Potential
interactions between the dark and visible sectors, currently beyond
experimental detection limits, could be incorporated within the mirror sector
framework. Such integration may provide a compelling ‘two-for-one’ realization
of the necessary phenomenology, effectively leveraging the unique capabilities
of the new, subdominant ADM component [120, 121]. The relationship between
fundamental constant variation and cosmology is a well discussed topic, for
more models and recent discussion please see Refs. [122, 123, 124, 125, 126,
127, 128, 129, 130, 131, 132, 133, 134, 135, 136].

In Section II, we establish the theoretical framework for the model’s two key
components: the $\mathbb{Z}_{2}$ symmetry inspired model of ADM and
fundamental constant variation (FCV). Section III details our numerical
investigation of this model using Monte-Carlo-Markov-Chain methods. We present
the results in Section IV, revealing a cosmological inference of $H_{0}$ that
aligns with locally measured values. Concluding remarks follow in Section V.
Throughout this paper, we adopt natural units where $c=\hbar=1$, representing
the speed of light and reduced Planck constant, respectively. We also use the
reduced Planck mass, $M_{\rm pl}=(8\pi G)^{-1/2}$, where $G$ is Newton’s
gravitational constant.

##  II The Synchronous Mirror Recombination Model

To understand how a mirror sector containing a subdominant ADM component may
allow for concordant cosmology, we explain here how its parameters can be
chosen to exploit the FFAT symmetry direction, ensuring alignment with CMB and
LSS data. This requires satisfying the phenomenological criteria outlined
above: $\rho_{i}\rightarrow\lambda^{2}\rho_{i}$ and
$\dot{\kappa}\rightarrow\lambda\dot{\kappa}$. Subsection II.1 details how we
fulfill the energy density requirement to increase $H$, and subsection II.3
describes the scaling of the Thomson scattering rate to ensure consistent
recombination history.

###  II.1 Atomic Dark Matter

Mirror-like dark sectors can vary widely, ranging from those featuring
multiple, exact replicas of the Standard Model to others with only partial
duplications [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59]. Importantly, these frameworks often allow for
masses and coupling strengths of mirror particles to be comparable in
magnitude to the visible sector. Given the variety and scope of models
exploring ADM and $\mathbb{Z}_{2}$ symmetries, our approach in this study is
broad, aiming to delineate dark sector phenomenology that could be compatible
with various mirror ADM implementations [60, 61, 62, 63, 64, 65, 66, 67, 68,
69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,
88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105,
106, 107, 108, 109, 110].

We assume the mirror sector primarily consists of two massive stable
particles, each bearing a charge under a dark $U(1)$ symmetry reminiscent of
the visible electromagnetic sector. For the sake of simplicity in modeling,
these particles are assumed to be fundamental fermions, though we recognize
that composite particles could serve equally well in a more specific
realization. These fermions are distinguished by their masses: one mirroring
the mass of the visible electron and the other mirroring the mass of the
visible proton. We refer to the lighter of these as the dark electron and the
heavier as the dark proton, with their interactions mediated by a massless
dark photon introducing a dark radiation component to the Universe. These
additions to the dark sector interact with the visible sector exclusively
through gravitational forces and represent a subdominant portion of the
overall dark sector energy density. For the scope of this work, we do not
define the specific characteristics of the remaining dark matter, aside from
its adherence to CDM behavior. However, we note an interesting possibility and
connection to recent research: the CDM could for example be a stable ‘dark
neutron’ left over from a dark nucleosynthesis era in the early Universe which
may naturally agree with our ADM approach [29].

This schematic model presents three significant phenomenological implications.
Firstly, it enables the formation of bound states akin to visible baryons.
Secondly, the capacity to form bound states means the mirror sector will
undergo a recombination process analogous to that experienced by the visible
sector. Thirdly, ADM models allow the critical ratios established by
$\Lambda$CDM to be preserved:

  * •

The precisely measured baryon-photon ratio is preserved through the
proportional inclusion of ADM and dark radiation.

  * •

The matter-radiation ratio is maintained through the proportional inclusion of
CDM, ADM, neutrinos, and dark radiation.

  * •

Dark radiation lends pressure support to ADM, and the integration of extra
cold dark matter ensures a balance between pressure-supported and pressureless
matter.

  * •

At appropriate temperatures, radiation interacting with ADM behaves as a
fluid, maintaining this state up to ‘mirror recombination’, akin to the
recombination process in the visible sector. Following this, the dark
radiation shifts to freestreaming, thereby maintaining the fluid-freestreaming
radiation ratio when the recombination events align.

Models of atomic dark matter, especially those derived from partial
$\mathbb{Z}_{2}$ symmetries, naturally define a parameter space encompassing
mirror recombination which occur near visible recombination. The proportional
inclusion of the mirror sector densities is the same statement as the second
clue outlined above. It is natural that the ratios of dark sector energies
matches that of the visible sector if they are populated by the same criteria.
Together, this approach circumvents numerous fine-tuning challenges.

###  II.2 Thermal History of Mirror Sector

To limit the number of free parameters of the theory during our numerical
investigation, we fix the dark proton mass $m_{\rm p,d}$, dark electron mass
$m_{\rm e,d}$, and dark fine structure constant $\alpha_{\rm d}$ to be
functions of the mirror sector temperature $\xi_{\rm d}$ such that mirror
recombination occurs at the same time as the visible sector, ensuring the
fluid-freestreaming radiation ratio is preserved. The fraction of total dark
matter which is atomic, $f_{\rm ADM}$, is left as a free parameter. We call
this scenario the ‘synchronous mirror recombination’ case. Later, in section
IV.3 we will consider a model where the mirror sector is allowed to recombine
independently of the visible sector in the ‘asynchornous mirror recombination’
case. To ensure synchronous mirror recombination and enforce the fourth
condition regarding the fluid-freestreaming radiation ratio, we make three
assumptions regarding the mirror sector: (1) $\dot{\kappa}$ is similar in both
sectors, (2) the sound speeds in both the sectors are similar to first order,
and (3) the ratio of binding energy to temperature is congruent between the
two sectors. The first constraint ensures dark photons appropriately couple to
dark electrons. The second constraint guarantees that pressure waves in the
mirror plasma propagate at speeds matching those in the visible sector;
otherwise, differing speeds of dark pressure waves would gravitationally
affect visible sector pressure waves, altering their profile slightly. The
third constraint determines the conditions under which recombination in the
mirror sector becomes energetically favorable.

The first assumption is that the Thomson scattering rates for both sectors is
the same:

$\lambda n_{\rm e}a\sigma_{\rm T}=n_{\rm e,d}a\sigma_{\rm T,d}.$ (2)

where $\lambda=\tilde{H}_{0}/H_{\rm 0,\Lambda CDM}$ with $\tilde{H}_{0}$ being
the increased value and the subscript $d$ denotes dark sector values. This
formulation introduces a new variable; the dark electron number density.
Similar to the visible sector, we account for this by assuming global
neutrality under the dark $U(1)$ symmetry, leading to the relation

$n_{\rm e,d}=n_{\rm p,d}\approx\frac{\rho_{\rm ADM}}{m_{\rm p,d}}$ (3)

where $n_{\rm p,d}$ and $\rho_{\rm ADM}$ represent the dark proton number
density and ADM energy density, respectively. Therefore, the mass of the dark
proton emerges as a key determinant of the dark electron number density.
Equation (3) holds true specifically when $m_{\rm e,d}\ll m_{\rm p,d}$, a
condition inherently met in mirror-like models.

From the degeneracy direction pointed out by the FFAT symmetry, the visible
baryon density is related to the ADM density such that

$\rho_{\rm ADM}=\left(\lambda^{2}-1\right)\rho_{\rm b}.$ (4)

Therefore, equation (2) becomes

$\lambda\frac{\rho_{\rm b}}{m_{\rm p}}\sigma_{\rm
T}=\frac{\left(\lambda^{2}-1\right)\rho_{\rm b}}{m_{\rm p,d}}\sigma_{\rm
T,d}.$ (5)

For clarity, we define the ratio of a quantity $\chi$ and its dark sector
component $\chi_{\rm d}$ as $r_{\chi}=\frac{\chi_{\rm d}}{\chi}$. With this
notation, the constraint from (2) becomes

$\frac{r_{\rm mp}r_{\rm me}^{2}}{r_{\alpha}^{2}}=\frac{\xi_{\rm
d}^{4}}{\sqrt{\xi_{\rm d}^{4}+1}}$ (6)

where $\alpha$ is the visible fine-structure constant and we have made the
substitution

$\xi_{\rm
d}\equiv\frac{T_{\rm\gamma,d}}{T_{\rm\gamma}}=\left(\lambda^{2}-1\right)^{1/4}$
(7)

with $T_{\rm\gamma}$ being the visible sector temperature and
$T_{\rm\gamma,d}$ being the mirror sector temperature.

We now seek to write $r_{\rm p}$ in terms of $\xi_{\rm d}$ and can do so while
enforcing the second assumption. The sound speed of the primordial plasma is
given by

$c_{\rm s}^{2}=\frac{T_{\rm b}}{\bar{\mu}}\left(1-\frac{1}{3}\frac{d{\rm
ln}T_{\rm b}}{d{\rm ln}a}\right)$ (8)

where $T_{\rm b}$ is the temperature of baryons and $\bar{\mu}$ is the reduced
molecular weight. At first order, we can ensure $c_{\rm s}=c_{\rm s,d}$ by
enforcing

$\frac{T_{\rm b}}{\bar{\mu}}=\frac{T_{\rm ADM}}{\bar{\mu}_{\rm d}}$ (9)

where $T_{\rm ADM}$ is the temperature of the atomic dark matter. Prior to
recombination, the baryons and ADM are tightly coupled to their respective
radiation baths such that $T_{\rm b}\approx T_{\gamma}$ and $T_{\rm
ADM}\approx T_{\gamma,d}$. Additionally, $\bar{\mu}\approx m_{\rm p}$ as long
as $m_{\rm p}\gg m_{\rm e}$. Therefore, (9) implies

$\begin{split}\frac{T_{\rm ADM}}{T_{\rm
b}}\approx\frac{T_{\rm\gamma,d}}{T_{\gamma}}&=\frac{m_{\rm p,d}}{m_{\rm p}}\\\
\Rightarrow\xi_{\rm d}&=r_{\rm mp}.\end{split}$ (10)

The constraint on the dark electron mass and dark fine structure constant from
equation (6) becomes

$\frac{r_{\rm me}^{2}}{r_{\alpha}^{2}}=\frac{\xi_{\rm d}^{3}}{\sqrt{\xi_{\rm
d}^{4}+1}}.$ (11)

We now turn our attention to the last and perhaps most important constraint:
keeping the ratio of the binding energy $B_{\rm e}$ to temperature in both
sectors equal such that

$\frac{B_{\rm e}}{T_{\gamma}}=\frac{B_{\rm e,d}}{T_{\rm\gamma,d}}$ (12)

which simplifies to the constraint

$\xi_{\rm d}=r_{\rm e}r_{\alpha}^{2}$ (13)

where we have used $B_{\rm e(,d)}=\frac{1}{2}m_{\rm
e(,d)}\alpha_{\rm(d)}^{2}$. Therefore, the system of equations we must solve
for $r_{\rm e}$ and $r_{\alpha}$ is

$r_{\rm me}r_{\alpha}^{2}=\xi_{\rm d}\And\frac{r_{\rm
me}^{2}}{r_{\alpha}^{2}}=\frac{\xi_{\rm d}^{3}}{\sqrt{\xi_{\rm d}^{4}+1}}.$
(14)

We find the mirror sector parameters to enforce synchronous mirror
recombination to be solely functions of temperature such that

$\begin{split}r_{\rm me}&=\left(\frac{\xi_{\rm d}^{4}}{\sqrt{\xi_{\rm
d}^{4}+1}}\right)^{1/3}\\\ r_{\rm mp}&=\xi_{\rm d}\\\
r_{\alpha}&=\left(\frac{\sqrt{\xi_{\rm d}^{4}+1}}{\xi_{\rm
d}}\right)^{1/6}.\end{split}$ (15)

This approach ensures that the forth condition of preserving the fluid-
freestreaming radiation ratio is exactly preserved. However, we note that the
mirror sector’s subdominant nature offers flexibility in these relationships;
exact matches are not required. We explore this flexibility when we permit the
mirror sector to recombine independently from the visible sector in section
IV.3. We note that this parameterization begins to break down for very cold
mirror sectors where $\xi_{\rm d}\rightarrow 0$, as not only do the masses and
couplings diverge, but the introduced ADM will be cold and behave as CDM which
we describe in section IV.2.

###  II.3 Phenomenological Changes to $\dot{\kappa}$ using FCV

The dimensionless ratio of the Thomson scattering rate, $\dot{\kappa}$, to the
Hubble rate, $H(z)$, is crucial for setting the photon diffusion scale during
recombination. This period, vital for shaping the CMB spectra at small angular
scales, demands a balance between the expansion and Thomson scattering rates
for observational consistency. The high $\ell$ regions of the CMB spectra,
where $\dot{\kappa}$ smooths perturbations resulting in power dampening, are
particularly sensitive to this ratio. Thus, the Hubble tension can be reframed
as a ‘photon diffusion tension’ in this context [116].

The Thomson cross section dictates the mean free path of photons at this time,
and variations in $\alpha$ and $m_{\rm e}$ can enhance or hinder the level of
diffusion. Our method employs a targeted FCV approach, focusing on maintaining
constant binding energy and $\dot{\kappa}/H(z)$, the physical quantities which
drives observational constraints. In essence, CMB observations are more
attuned to changes in the binding energy and this ratio, rather than specific
alterations in $\alpha$ or $m_{\rm e}$. Notably, CMB spectral observations may
be sensitive to specific variations in $\alpha$ or $m_{\rm e}$ with possible
future telescopes but remain out of reach for current observations [137, 138,
139, 140, 141]. We remind the reader that this phenomenological approach
encapsulates any modification to the Thomson scattering rate potentially
achievable through novel interactions between the dark and visible sectors,
and is not solely limited to models investigating FCV.

Our proposed scaling for $\dot{\kappa}$ hinges on two constraints. Firstly,
the binding energy of light elements $B_{\rm e}=\frac{1}{2}m_{\rm
e}\alpha^{2}$ must remain unchanged compared to the Standard Model to ensure
visible recombination timing aligns with $\Lambda$CDM. This preserves
observables related to the distance to the CMB. Secondly, the Thomson cross
section $\sigma_{\rm T}=\frac{8\pi}{3}\alpha^{2}/m_{\rm e}^{2}$ requires a
specific scaling ($\sigma_{\rm T}\rightarrow\lambda\sigma_{\rm T}$) to
maintain the $\dot{\kappa}/H$ ratio, thereby preserving CMB temperature
spectra observational consistency. Resolving these constraints yields precise
scaling predictions for the fundamental parameters:
$\alpha\rightarrow\lambda^{1/6}\alpha\And
m_{\rm{e}}\rightarrow\lambda^{-1/3}m_{\rm{e}}$ [117, 116]. These scalings
align with the degeneracy direction indicated by the FFAT symmetry, while
preserving recombination-related cosmological observables at first order. For
more discussion regarding the observational constraints imposed on FCV, see
App. A, and for results depicting the relationship between FCV and the binding
energy see Sec. II.3.

##  III  Methodology

We integrate the ADM model into the Cosmology Linear Anisotropy Solving System
(CLASS) to probe the mirror sector parameter space, and allow the prior volume
to include the FFAT symmetry direction [142]. To incorporate the FCV necessary
to keep $\dot{\kappa}/H$ invariant, we utilize CLASS’s default functionality
for varying $\alpha$ and $m_{\rm e}$ and call upon HyREC to compute
recombination for the mirror and visible sectors [143]. Combining CLASS with
the Monte-Carlo-Markov-Chain (MCMC) package MontePython [144, 145], we compute
the one and two-dimensional posteriors for the free parameters for the
synchronous mirror recombination model. To plot our derived 1 and 2
dimensional posteriors, we utilize the python package GetDist [146].

Although this approach introduces a considerable expansion in the model’s
degrees of freedom, they are well justified by physical principles and guided
by underlying symmetries. For clarity, we list the free parameters
incorporated into the model, along with their physical justification. First,
we consider the six standard $\Lambda$CDM parameters $\left[h,\omega_{\rm
b},\omega_{\rm cdm},A_{\rm s},n_{\rm s},\tau_{\rm reio}\right]$. To maintain a
constant ratio of fluid-freestreaming radiation, we introduce a variable
neutrino energy density $\left[N_{\rm ur}\right]$ while assuming that
neutrinos are massless. Additionally, we incorporate variations in the
fundamental parameters $\left[\alpha,m_{\rm e}\right]$ such that the Thomson
scattering rate can vary while keeping the binding energy constant. We allow
$\left[f_{\rm ADM},\xi_{\rm d}\right]$ to vary, enabling the mirror sector to
preserve the matter-radiation, baryon-photon, and pressure supported to
pressureless matter ratios. The other mirror sector parameters ($m_{\rm
p,d},m_{\rm p,e},\alpha_{\rm d}$) are fixed functions of the temperature
according to Eq. (15), ensuring a synchronous mirror recombination with the
visible sector.

We consider the following observational data sets

  * •

CMB - Planck 2018 high $\ell$ TT, TE, and EE data, Planck 2018 low $\ell$ TT
and EE data, and Planck Lensing data [1]

  * •

Baryon Acoustic Oscillations (BAO) - 6dF Galaxy Survey [147], SDSS DR7 [148],
and BOSS DR12 [149]

  * •

Uncalibrated Pantheon+ Supernova [150]

  * •

Pantheon+ Supernova calibrated by the local S$H_{0}$ES measurement [4]

in three different combinations: (A) CMB and BAO, (B) CMB, BAO, and
uncalibrated Pantheon+, and (C) CMB, BAO, and Pantheon+ calibrated by
S$H_{0}$ES, referred to hereafter as likelihood groups A, B, and C. In all
cases, we set the helium abundance to $Y_{\rm p}=0.2454$ to ensure alignment
with BBN predictions [151]. As mentioned previously, models where the dark
sector thermalizes with the visible sector after BBN, thereby avoiding $N_{\rm
eff}$ constraints [16, 68], present intriguing connections and are certainly
possible, but are beyond the scope of this work.

The MCMC analysis is allowed to run until the chains satisfy a Gelman-Rubin
convergence criterion of $R-1<0.05$ in all free parameters. To identify the
global minimum of the fits (a significant computational challenge given the
extensive parameter space spanned by likelihood groups A and B as we will see)
we employ the newly developed Procoli package which comes equipped with an
integrated global best-fit minimizer [152]. Procoli is also utilized for
conducting profile likelihood scans, enabling us to understand the
relationship between the fit to data and increasing values of $H_{0}$ when
exploring the FFAT symmetry. Additionally, we define the visible sector
variables

$\begin{split}\delta_{\alpha}=&\frac{\tilde{\alpha}-\alpha_{0}}{\alpha_{0}}\\\
\delta_{\rm e}=&\frac{\tilde{m}_{\rm e}-m_{\rm e,0}}{m_{\rm e,0}}\end{split}$
(16)

where the tilde indicates the varied value and 0 indicates the Standard Model
value.

##  IV  Results and Discussion

We tabulate the results for the scans over the synchronous mirror
recombination model in table 1 and show the posteriors in figure 1 for
likelihood groups A, B, and C. Additionally, we perform the same analysis for
the $\Lambda$CDM model using the same likelihoods to generate comparative
data. In general, we find that all parameters naturally follow the FFAT
symmetry direction, depicted by red dashed lines, and keep the critical ratios
discussed above invariant compared to $\Lambda$CDM. The binding energy is left
invariant compared to the Standard Model while the Thomson cross section is
scaled to match the change in $H$ as expected.

| Free Parameters | CMB $\And$ BAO | CMB, BAO, Pantheon+ | CMB, BAO, Pantheon+, S$H_{0}$ES
---|---|---|---|---
$\Lambda$CDM | $h$ | $\hskip 14.22636pt0.7462\hskip 2.84526pt(0.6854)\pm 0.0549\hskip 14.22636pt$ | $\hskip 14.22636pt0.7273\hskip 2.84526pt(0.7370)\pm 0.0549\hskip 14.22636pt$ | $\hskip 14.22636pt0.7380\hskip 2.84526pt(0.7380)\pm 0.0102\hskip 14.22636pt$
| $\omega_{\rm b}$ | $0.0223\hskip 2.84526pt(0.0224)\pm 0.0002$ | $0.0223\hskip 2.84526pt(0.0224)\pm 0.0002$ | $0.0223\hskip 2.84526pt(0.0226)\pm 0.0002$
| $\omega_{\rm cdm}$ | $0.1495\hskip 2.84526pt(0.1221)\pm 0.0244$ | $0.1439\hskip 2.84526pt(0.1475)\pm 0.0256$ | $0.1472\hskip 2.84526pt(0.1473)\pm 0.0065$
| $n_{\rm s}$ | $0.9585\hskip 2.84526pt(0.9580)\pm 0.0095$ | $0.9574\hskip 2.84526pt(0.9568)\pm 0.0089$ | $0.9585\hskip 2.84526pt(0.9589)\pm 0.0096$
| ln($A_{\rm s}10^{9}$) | $2.0794\hskip 2.84526pt(2.0740)\pm 0.0411$ | $2.0728\hskip 2.84526pt(2.0736)\pm 0.0368$ | $2.0792\hskip 2.84526pt(2.0802)\pm 0.0417$
| $\tau_{\rm reio}$ | $0.0518\hskip 2.84526pt(0.0550)\pm 0.0077$ | $0.0520\hskip 2.84526pt(0.0512)\pm 0.0072$ | $0.0521\hskip 2.84526pt(0.0516)\pm 0.0075$
ADM | $\xi_{\rm dark}$ | $0.6242\hskip 2.84526pt(0.4989)\pm 0.1954$ | $0.5615\hskip 2.84526pt(0.6977)\pm 0.2660$ | $0.6849\hskip 2.84526pt(0.6893)\pm 0.0707$
| $f_{\rm adm}$ | $0.0317\hskip 2.84526pt(0.0072)\pm 0.0212$ | $0.0359\hskip 2.84526pt(0.0340)\pm 0.0212$ | $0.0322\hskip 2.84526pt(0.0346)\pm 0.0103$
FCV | $\delta_{\alpha}$ | $0.0142\hskip 2.84526pt(0.0021)\pm 0.0120$ | $0.0122\hskip 2.84526pt(0.0152)\pm 0.0118$ | $0.0141\hskip 2.84526pt(0.0149)\pm 0.0040$
| $\delta_{\rm me}$ | $-0.0264\hskip 2.84526pt(-0.0014)\pm 0.0264$ | $-0.0256\hskip 2.84526pt(-0.0309)\pm 0.0260$ | $-0.0293\hskip 2.84526pt(-0.0296)\pm 0.0132$
Other | $N_{\rm ur}$ | $3.3860\hskip 2.84526pt(2.8063)\pm 0.5624$ | $3.2670\hskip 2.84526pt(3.3393)\pm 0.5370$ | $3.3713\hskip 2.84526pt(3.3793)\pm 0.3481$
| Derived Parameters |  |  |
Dark | $r_{\alpha}$ | $1.1291\hskip 2.84526pt(1.1286)\pm 0.1305$ | $1.1705\hskip 2.84526pt(1.0808)\pm 0.2130$ | $1.0816\hskip 2.84526pt(1.0822)\pm 0.0183$
| $r_{\rm me}$ | $0.5291\hskip 2.84526pt(0.3917)\pm 0.2201$ | $0.4996\hskip 2.84526pt(0.5973)\pm 0.2524$ | $0.5906\hskip 2.84526pt(0.5886)\pm 0.0774$
| $r_{\rm mp}$ | $0.6242\hskip 2.84526pt(0.4989)\pm 0.0195$ | $0.5615\hskip 2.84526pt(0.6977)\pm 0.2660$ | $0.6849\hskip 2.84526pt(0.6893)\pm 0.0707$
Visible | $\delta_{\rm Be}$ | $0.0004\hskip 2.84526pt(0.0028)\pm 0.0086$ | $-0.0023\hskip 2.84526pt(-0.0012)\pm 0.0085$ | $-0.0020\hskip 2.84526pt(-0.0005)\pm 0.0087$
| $\delta_{\sigma_{T}}$ | $0.0901\hskip 2.84526pt(0.0070)\pm 0.0831$ | $0.0903\hskip 2.84526pt(0.0974)\pm 0.0839$ | $0.0929\hskip 2.84526pt(0.0938)\pm 0.0355$
| $\Omega_{\rm m}$ | $0.3070\hskip 2.84526pt(0.3075)\pm 0.0089$ | $0.3124\hskip 2.84526pt(0.3123)\pm 0.0077$ | $0.3111\hskip 2.84526pt(0.3114)\pm 0.0082$
| $S_{8}$ | $0.8178\hskip 2.84526pt(0.8167)\pm 0.0127$ | $0.8113\hskip 2.84526pt(0.8146)\pm 0.0125$ | $0.8164\hskip 2.84526pt(0.8159)\pm 0.0128$
| $\Delta\chi^{2}$ | $-1.08$ | $-1.26$ | $-35.72$
Table 1: This table presents the mean (best-fit) values for the synchronous
mirror recombination model across three different likelihood groups, with all
errors representing the 68% CL. A double horizontal line divides the table
into two main sections. The upper section lists free parameters, categorized
into $\Lambda$CDM, atomic dark matter (ADM), and fundamental constant
variation (FCV). The lower section details derived parameters from both the
dark and visible sectors. At the bottom, the table features $\Delta\chi^{2}$
values, comparing the synchronous mirror recombination model’s $\chi^{2}$ to a
baseline $\Lambda$CDM model using identical likelihoods. Figure 1: This
triangle plot showcases the MCMC analysis results for the synchronous
recombination mirror model under three different likelihood groups: A (blue),
B (yellow), and C (green). Dark red dashed lines represent the degeneracy
direction of the FFAT symmetry. Note that the FFAT symmetry does not make a
prediction for $S_{8}$. In the $H_{0}$ column, dark and light grey bands mark
the 68% and 95% CL from the S$H_{0}$ES’ locally measured value of
$H_{0}=73.3\pm 1.04$ [4], while similar bands in the $\Omega_{\rm m}$ and
$S_{8}$ columns reflect the constraints from Pantheon+’s measured value of
$\Omega_{\rm m}=0.334\pm 0.018$ [150] and DES-Y3 result of $S_{8}=0.776\pm
0.017$ [153]. The MCMC sampler naturally follows the FFAT symmetry direction
as it ensures agreement with CMB and LSS observational data. The degeneracy is
broken only once the local S$H_{0}$ES calibration of $H_{0}$ is included,
effectively selecting a slice of the posterior.

Table 1 shows that without Pantheon+ supernova data and S$H_{0}$ES
calibration, likelihood group A best-fit values favour $\Lambda$CDM model
values. This matches our clues discussed in Sec. I: without a need to
introduce an additional signal into CMB data to increase $H_{0}$, $\Lambda$CDM
out preforms other models as it accurately predicts the ratios dictating CMB
spectra with minimal parameters. Yet, posteriors are significantly expanded,
allowing for $H_{0}$ values well over 80 km/s/Mpc within the 68% confidence
level - a notable departure from traditional $\Lambda$CDM CMB and LSS
analysis. Figure 1 demonstrates an expanded $H_{0}$ posterior which opens the
parameter space to explore cosmologically concordant models. Indeed, as we
will see in our discussion of the profile likelihood for $H_{0}$, CMB and LSS
observations have little constraining power in the FFAT symmetry direction.
The minimizer finds best-fit values consistent with $\Lambda$CDM underscoring
$\Lambda$CDM’s efficacy at matching observations, while also highlighting the
need for cosmological models to adhere to its foundational ratios (photon-
baryon number, matter-radiation ratio, pressure supported to pressureless
matter ratio, fluid-freestreaming radiation ratio) for stronger statistical
agreement. Adding five parameters to the model marginally improves the fit to
CMB and LSS data alone with a $\Delta\chi^{2}=-1.0799$.

(a) CMB TT Correlation Spectra

(b) CMB EE Correlation Spectra

Figure 2: These two figures compare the TT and EE spectra from the
$\Lambda$CDM model best-fit values obtained from Ref. [1] to the synchronous
mirror recombination model best-fit values obtained when considering
likelihood group C. In both figures, $H_{0}$ = 73.8 km/s/Mpc. The top panels
of the figures demonstrate the spectra’s power correlation. The bottom panels
demonstrate the fractional difference between the $\Lambda$CDM and synchronous
model. The grey points with associated error bars are binned Planck 2018
observations of the TT and EE spectra. The plots are logarithmic on the x axis
for $2\leq\ell<30$ for the low $\ell$ data, and then linear for $\ell\geq 30$
for the high $\ell$ data.

When incorporating the uncalibrated Pantheon+ supernovae data into the
analysis, the best-fit preferences for the model shift dramatically, naturally
leading to a best-fit $H_{0}=73.70$ km/s/Mpc. The inclusion of uncalibrated
supernova data makes the model sensitive to the shape of the redshift-
luminosity relationship and probes the eras of matter domination and matter-
dark energy equality, but in this case is unaware of the S$H_{0}$ES
calibration. Notably, the best-fits agrees nicely with direct local
measurements of the expansion rate without notion of their existence. For both
likelihood groups A and B, the error bars are broad because the sampler is
traversing the FFAT symmetry direction without a calibration for $H_{0}$.
Although the best-fits now align with local measurements of $H_{0}$, the
enhancement to the overall fit remains minimal, with a
$\Delta\chi^{2}=-1.2602$. The two-dimensional posteriors for likelihood groups
A and B in figure 1 align well with our predictions from the FFAT symmetry,
indicated by the red dashed lines.

The inclusion of the S$H_{0}$ES’ local $H_{0}$ measurement into likelihood
group C calibrates the FFAT symmetry, significantly reducing parameter
uncertainties and anchoring the results. With S$H_{0}$ES calibration, we find
$H_{0}=73.80\pm 1.02$ km/s/Mpc, closely matching the local measurement of Ref.
[4] with an overall $\Delta\chi^{2}=-35.72$ compared to $\Lambda$CDM.
Furthermore, we detect variations in $\alpha$ and $m_{\rm e}$, yet the binding
energy remains consistent with the Standard Model, affirming the precise
timing of visible recombination and the insensitivity of observables to
specific variations in fundamental parameters at first order. Additionally,
the FCV represents the phenomenological need to scale the Thomson scattering
rate accordingly when exploring models with higher $H_{0}$, regardless of the
mechanism which amplifies it. The incorporation of dark radiation and ADM
upholds the ratios of matter to radiation, pressure-supported to pressureless
matter, and fluid to free-streaming radiation, in alignment with $\Lambda$CDM
predictions. The required ADM fraction is $f_{\rm adm}=0.0322\pm 0.0103$, and
the mirror sector is found to be slightly cooler than the visible sector with
$\xi_{\rm dark}=0.6849\pm 0.07$. The variation in neutrino energy density,
$N_{\rm ur}$, yields best-fit values about 10% higher than the Standard Model
predictions. However, across all likelihood groups, the results consistently
align with $N_{\rm ur}=3.044$ within the 68% confidence level (CL).

Figure 2 demonstrates the CMB temperature-temperature (TT) and polarization-
polarization (EE) spectra for the best-fit values obtained for the mirror
model when considering likelihood group C. We find that even with
significantly enhanced values of $H_{0}=73.80$ km/s/Mpc compared to
$\Lambda$CDM, we do not sacrifice agreement to CMB spectra observations both
at low $\ell$ and high $\ell$ multipoles as seen in Table 2 and Fig. 2,
although there is a slight trade for agreement for high $\ell$ from the low
$\ell$ data.

###  IV.1 Profile Likelihood with Procoli

Figure 3: This plot demonstrates the results from a profile likelihood scan
over $H_{0}$ utilizing the Procoli package for the synchronous mirror
recombination model when considering likelihood group C. The results are
normalized to each datasets minimum $\chi^{2}$ value, allowing comparison
between all data sets. We point out the $\Delta\chi^{2}=1$ line which
represents the 68% CL.

To delve deeper into the connection between preserving ratios and cosmological
observables, we perform a profile likelihood scan over $H_{0}$ across all
three likelihood groups utilizing the newly developed Procoli package [152].
Figure 3 displays the outcomes of one of these scans, for likelihood group C,
demonstrating that CMB and LSS data are insensitive to changes in $H_{0}$
while exploring along the FFAT symmetry direction. The degeneracy is only
resolved when a calibrating measurement of $H_{0}$ is incorporated. In table
2, we present a comparison of the $\chi^{2}$ values across each dataset within
likelihood group C for three distinct cases: $\Lambda$CDM, the synchronous
mirror recombination model, and the asynchronous mirror recombination model
(see Sec.IV.3). We observe that both mirror models result in a reduced
$\chi^{2}$ for the Planck high-$\ell$ measurements and for the Pantheon+ data,
once calibrated by the S$H_{0}$ES measurement. The model’s effectiveness at
high $\ell$ stems from maintaining the $\dot{\kappa}/H$ ratio. This
preservation allows for higher values of $H$, driven by the increased ADM and
dark radiation density, ensuring alignment with the Pantheon+ with S$H_{0}$ES
dataset. The fit to the low $\ell$ TT spectra and the BOSS DR12 BAO dataset
shows a slight deterioration. However, measurements from low redshift BAO
largely compensate for this change.

|  $\Lambda$CDM | Synchronous | Asynchronous
---|---|---|---
|  | Recombination | Recombination
Planck High $\ell$ |  2347.79 |  2343.97 |  2342.18
Planck Low $\ell$ TT | 22.63 | 24.16 | 23.89
Planck Low $\ell$ EE | 396.95 | 396.09 | 396.05
Planck Lensing | 8.85 | 8.62 | 8.70
BOSS DR12 | 3.57 | 4.28 | 4.41
BOA low z | 2.32 | 1.23 | 1.198
Pantheon+S$H_{0}$ES | 1319.44 | 1287.49 | 1287.45
$\Delta\chi^{2}$ | 0 | -35.72 | -37.65
Table 2: This table provides a comparison of the fit to individual datasets
within likelihood group C across three models: the baseline $\Lambda$CDM, the
synchronous mirror recombination model, and the asynchronous mirror
recombination model. The $\Delta\chi^{2}$ value quantifies the change in
$\chi^{2}$ relative to the baseline $\Lambda$CDM model.

###  IV.2 Cold Mirror Sector Volume Effects

The one and two dimensional posteriors for likelihood groups A and B are
influenced by a potentially large volume effect in the prior space in which
$f_{\rm adm}$ becomes unconstrained when the mirror sector is very cold. This
result was first pointed out by ref. [154] and is most clearly seen when
considering likelihood group B.

In scenarios where the mirror sector temperature is cold ($\xi_{\rm d}\lesssim
0.25$), $f_{\rm adm}$ becomes unconstrained as the ADM begins to simply
resemble the background cold dark matter. This similarity introduces a
significant volume to the parameter space, echoing $\Lambda$CDM
characteristics and resulting in ‘mitten’ shaped two-dimensional contours with
‘cold thumbs’. Figure 4 demonstrates that the synchronous mirror recombination
model follows the expected FFAT symmetry direction for $f_{\rm adm}$ in the
‘mitten’ when considering likelihood group B. However, as $\xi_{\rm
d}\rightarrow 0$, $f_{\rm adm}\rightarrow 1$ and the volume effect is
introduced creating ‘cold thumbs’ of the mitten. This volume effect is removed
when calibrating the Pantheon+ supernovae with S$H_{0}$ES, as the S$H_{0}$ES
anchor strongly disfavours cold mirror sectors. Ultimately, this increased
volume influences the shape of the one dimensional posteriors when considering
likelihoods groups A and B, but do not influence the best-fit parameters. For
instance, in figure 4, we point out the best-fit values and note they do not
align with the peaks of the distribution.

Figure 4: This triangle plot showcases a volume effect shaping the posteriors
for likelihood groups A and B in the synchronous mirror recombination model.
The one-dimensional posteriors, prominently influenced by this volume effect,
display peaks that are shaped accordingly. A black dashed line in the plot
indicates the FFAT symmetry direction. The color scale indicates the
temperature of the mirror sector, with purple colors being very cold mirror
sectors. The grey lines indicate best fit values, and because of the volume
effect introduced by cold dark sectors, do not necessarily align with the peak
of the one dimensional distribution.

###  IV.3 Additional Mirror Sector Freedom

As a final test of $\mathbb{Z}_{2}$ inspired ADM models, we now consider a
scenario where the mirror sector’s recombination is asynchronous with the
visible sector in an asynchronous mirror recombination model. By varying the
mirror binding energy to mirror sector temperature ratio through changes in
the dark fine structure constant, we determine the recombination epoch for the
mirror sector: ratios greater (less) than unity indicate earlier (later)
mirror sector recombination compared to the visible sector.

Our findings reveal a statistical preference in $\Delta\chi^{2}$ for an
earlier mirror sector recombination, finishing just before the onset of
visible hydrogen recombination. Recall that in this model, the mirror sector
interacts with the visible sector only gravitationally. Prior to mirror sector
recombination, dark photons act as a fluid. Once the majority of atomic dark
matter recombines, the dark photons start to freestream. Therefore, CMB data
seems to favor an injection of freestreaming radiation before hydrogen
recombination but after helium recombination which would have the effect of
gravitationally pulling on sound horizon wavefronts, and efficiently
transporting energy out of the system [155, 156]. The asynchronous mirror
recombination model with likelihood group C finds a $\Delta\chi^{2}=-37.65$
compared to the $\Lambda$CDM model, and $H_{0}$ values which agree with local
measurements. Figure 6 illustrates the correlation between mirror sector
parameters and cosmological tensions for the asymmetric recombination model,
with parameter values given in table 3. A bimodal feature related to the
timing of mirror recombination is found, where it is disfavoured to recombine
during visible HeII to HeI recombination. As a general feature of this model
and the synchronous mirror model, the ADM experiences pressure support from
the dark radiation, slowing the infall of dark matter into initial seeds of
structure. This allows the one and two dimensional posteriors in figure 6 to
penetrate into higher values of $\Omega_{\rm m}$ and lower values of $S_{8}$.
While outside the scope of this study, it may be of great interest to
investigate how subdominant components of the mirror sector adhering to
similar ADM phenomenology influence the growth of structure on scales relevant
for the $S_{8}$ tension.

| Parameter | Asymmetric Recombination
---|---|---
$\Lambda$CDM | $h$ | $\hskip 14.22636pt0.7375\hskip 2.84526pt(0.7381)\pm 0.0100\hskip 14.22636pt$
| $\omega_{\rm b}$ | $0.0224\hskip 2.84526pt(0.0022)\pm 0.0003$
| $\omega_{\rm cdm}$ | $0.1461\hskip 2.84526pt(0.1474)\pm 0.0067$
| $n_{\rm s}$ | $0.9624\hskip 2.84526pt(0.9591)\pm 0.0106$
| ln($A_{\rm s}10^{9}$) | $2.0889\hskip 2.84526pt(2.0718)\pm 0.0460$
| $\tau_{\rm reio}$ | $0.0519\hskip 2.84526pt(0.0515)\pm 0.0074$
ADM | $\xi_{\rm dark}$ | $0.6715\hskip 2.84526pt(0.7132)\pm 0.0817$
| $f_{\rm adm}$ | $0.0362\hskip 2.84526pt(0.0347)\pm 0.0143$
| $r_{\alpha}$ | $1.1804\hskip 2.84526pt(1.2127)\pm 0.3811$
FCV | $\delta_{\alpha}$ | $0.0122\hskip 2.84526pt(0.0142)\pm 0.0044$
| $\delta_{\rm me}$ | $-0.0254\hskip 2.84526pt(-0.0312)\pm 0.0140$
Other | $N_{\rm ur}$ | $3.3757\hskip 2.84526pt(3.2859)\pm 0.3628$
Dark | $r_{\rm me}$ | $0.5721\hskip 2.84526pt(0.6132)\pm 0.0950$
| $r_{\rm mp}$ | $0.6715\hskip 2.84526pt(0.7132)\pm 0.0817$
| $r_{\rm Be}/\xi_{\rm dark}$ | $1.3077\hskip 2.84526pt(1.2645)\pm 0.7095$
Visible | $\delta_{\rm Be}$ | $-0.0015\hskip 2.84526pt(-0.0035)\pm 0.0084$
| $\delta_{\sigma_{T}}$ | $0.0796\hskip 2.84526pt(0.0959)\pm 0.0385$
| $\Omega_{\rm m}$ | $0.3098\hskip 2.84526pt(0.3117)\pm 0.0827$
| $S_{8}$ | $0.8161\hskip 2.84526pt(0.8139)\pm 0.0131$
| $z_{\rm rec}$ | $1090.1\hskip 2.84526pt(1088.1)\pm 9.3337$
| $\Delta\chi^{2}$ | $-37.65$
Table 3: This table presents the mean (best-fit) values for the asynchronous
mirror recombination model when considering likelihood group C, with all
errors representing 68% CL. The format is the same as in table 1. Figure 5:
This triangle plot demonstrates the relationship between variations in
$\alpha$ and $m_{\rm e}$ in the synchronous mirror recombination model. A
black dashed line marks the Standard Model binding energy. The samples are
plotted as points with colors corresponding to the ratio of the varied visbile
binding energy compared to the Standard Model value. Figure 6: This triangle
plot showcases the asynchronous mirror recombination model. It displays three
free mirror sector parameters alongside three key observables ($H_{0}$,
$S_{8}$, and $\Omega_{\rm m}$) linked to cosmological tensions. 68% (dark
grey) and 95% (light grey) CL limits for $H_{0}$, $S_{8}$, and $\Omega_{\rm
m}$ appear in the relevant plots. The color coding reflects the ratio of the
mirror sector binding energy to its temperature relative to visible sector
values, a determinant of the mirror sector’s recombination epoch. Ratios above
one suggest earlier recombination compared to the visible sector, while values
below one indicate a later recombination. The black dashed line indicates the
values which align with the visible sector binding energy to temperature
ratio.

Finally, an interesting coincidence has appeared in the analysis of the
synchronous and asynchronous mirror recombination models when considering
likelihood groups B and C. The best-fit temperature for the mirror sector in
the asymmetric recombination model is $\xi_{\rm dark}=0.7132$ or 1.944 K. This
temperature is strikingly similar to the 1.95 K predicted for the neutrino
background. Note that this is not simply adding an additional neutrino species
and naming it something else, as this radiation behaves as a fluid until
mirror recombination. This coincidence bears intriguing implications,
connecting back to the second clue mentioned in Section I. This observation
could imply that, at some point in cosmic history, the mirror sector radiation
might have coupled to the neutrino or visible sector, allowing the two sectors
to achieve thermal equilibrium with one another and explaining this
temperature coincidence [157, 158]. A similar phenomenology to this was
realized in [15].

###  IV.4 Constant Binding Energy

A particular scaling of $\alpha$ and $m_{\rm e}$ has been determined to
maintain constant binding energy while enabling increased Thomson scattering
rates [116]. Without prior knowledge of this specific strategy, the MCMC
sampler naturally navigates this path, revealing the insensitivity of cosmic
observables to individual changes in either $\alpha$ or $m_{\rm e}$, focusing
instead on maintaining the invariance of the binding energy within the visible
sector. This degeneracy direction is depicted in Figure 5, where the plot’s
two-dimensional posterior closely follows lines of constant binding energy,
and guarantees that recombination in the visible sector occurs at a time
consistent with $\Lambda$CDM predictions, thereby preserving distance
measurements. Simultaneously, the Thomson scattering rate not left invariant
in the direction of constant binding energy, allowing the sampler to explore
regimes with constant $\frac{\dot{\kappa}}{H}$ ratios. We show likelihood
group B for its capacity to explore the FFAT symmetry’s degeneracy direction
without the S$H_{0}$ES calibration which anchors the result.

##  V  Conclusion

In this study, we leverage insights into the dark sector’s connection to the
visible sector, and the predictions of ratios by $\Lambda$CDM. We introduce a
streamlined, universal model of ADM, inspired by recent works investigating
mirror sectors. In this model, a subdominant component of the dark sector
($\sim 3\%$) mirrors the visible sector’s $U(1)$ symmetry for electromagnetic
interactions. Notably, with parameters analogous to those in the visible
sector, the ADM is expected to experience a recombination event within the
mirror sector at epochs parallel to those observed in the visible sector. To
assess the influence of adding a dynamic, subdominant ADM component to the
mirror sector, we employ the FFAT symmetry to guide our modeling approach.
This strategy safeguards cosmological observables, guaranteeing alignment with
CMB and LSS data. ADM inherently fulfills one of the FFAT symmetry’s
phenomenological requirements by contributing proportional amounts of matter,
radiation, and cold dark matter densities to the Universe. The second
criterion of the FFAT symmetry involves maintaining the ratio of the Thomson
scattering rate to the background expansion rate. To achieve this, we suggest
a variation of fundamental constants in a way that scales the Thomson
scattering rate appropriately, while keeping the binding energy of hydrogen
and helium constant. We discover that the CMB observational agreement remains
unaffected by particular variations in $\alpha$ and $m_{\rm e}$, provided the
binding energy remains unchanged. While integrating fundamental constant
variation into a realistic Lagrangian presents challenges, the underlying
phenomenology remains essential. A model capable of adjusting the Thomson
scattering rate similarly, without relying on FCV, would offer a compelling
route to explore the concordant cosmological models.

To investigate this phenomenology and its relationship to cosmic tensions, we
conduct a MCMC analysis utilizing a modified version of CLASS which
incorporates ADM. We explore two distinct cases of mirror sector
recombination: i) the synchronous mirror recombination model depicting the
mirror sector recombining concurrently with the visible sector and ii) the
asynchronous mirror recombination model allowing the mirror sector to undergo
recombination independently from the visible sector. Our findings indicate
that under both recombination scenarios, the CMB and LSS observables show
almost no sensitivity in response to changes in $H_{0}$. Incorporating data
from the local measurement of $H_{0}$, we determine a cosmologically inferred
value of $H_{0}=73.8\pm 1.0$ km/s/Mpc for both models. This results in
improved fits, especially for the high $\ell$ Planck data and Pantheon+
dataset, corresponding to a $\Delta\chi^{2}=-35.72$ for the synchronous model
and $\Delta\chi^{2}=-37.65$ for the asynchronous model. The asynchronous
mirror recombination model reveals a preference for the mirror sector to
undergo recombination slightly earlier than the visible sector. This timing
places mirror sector recombination between the conclusion of helium
recombination and the onset of hydrogen recombination.

This dual approach to investigating the Hubble tension has achieved fruitful
results, summarized as follows:

  * •

The inclusion of a subdominant (3%) atomic dark matter component in the mirror
sector with a temperature of $\xi_{\rm d}\sim 0.7$, inspired by
$\mathbb{Z}_{2}$ symmetries, alongside a mechanism that maintains a consistent
$\dot{\kappa}/H$, eliminates the tension in $H_{0}$.

  * •

The inclusion of extra mirror sector species, with mass and coupling strengths
mirroring those in the visible sector, results in mirror sector recombination
occurring at times similar to that of the visible sector. Given the freedom to
recombine independently, the mirror sector exhibits a preference for
undergoing recombination just before the visible sector hydrogen recombines.

  * •

By scaling the Thomson scattering rate in accordance with the background
expansion rate, we achieve cosmological concordance. Our approach employs a
phenomenological variation of fundamental constants, demonstrating that
targeted adjustments to $\alpha$ and $m_{\rm e}$ facilitate the necessary
scaling while maintaining compatibility with CMB observations

Our work has highlighted several intriguing potential links between mirror
models and cosmological tensions while offering avenues for further
investigation into possible dark sector interactions. Significant efforts are
required from both particle physics and cosmology disciplines to further
verify the efficacy of this or similar models at addressing cosmological
tensions. However, the phenomenology is clear: $3\%-4\%$ of the dark sector
behaving as ADM while ensuring the ratio of $\dot{\kappa}/H$ is left invariant
compared to $\Lambda$CDM produces concordant cosmology.

###### Acknowledgements.

We would like to thank David Camarena for his support and useful comments on
the draft, as well as Ryan Janish for useful discussions. This work was
supported by the National Science Foundation (NSF) under grant AST-2008696.
F.-Y. C.-R. would like to thank the Robert E. Young Origins of the Universe
Chair fund for its generous support. We would like to thank the UNM Center for
Advanced Research Computing, supported in part by the NSF, for providing the
research computing resources used in this work. This material is based upon
work supported by the U.S. Department of Energy, Office of Science, Office of
Workforce Development for Teachers and Scientists, Office of Science Graduate
Student Research (SCGSR) program. The SCGSR program is administered by the Oak
Ridge Institute for Science and Education for the DOE under contract number
DE-SC0014664.

##  Appendix A Constraints on FCV

Adjusting fundamental constants introduces challenges that span both
theoretical and observational domains. Although we have pinpointed a variation
in the CMB during recombination that evades detection, geological records,
atomic spectra, quasar observations, and potentially BBN impose much tighter
constraints. To ensure the model is not in direct disagreement with current
observables, we briefly review the main methods for constraining FCV. For an
extensive review, see refs. [159, 160].

Geological constraints, assessing periods comparable to Earth’s age (or
$z<0.5$), are based on measuring relative abundances of various heavy element
isotopes. The most stringent of these arises from the fascinating Oklo natural
reactor phenomenon, where natural nuclear fission occurred approximately 1.7
billion years ago [161]. Analysis of this event, considering the neutron mass
and decay resonance energy, has significantly constrained the variation of
$\alpha$ to $|\Delta\alpha/\alpha|<1.1\times 10^{-8}$ over this period [162].

Atomic clocks provide a highly precise method for measuring FCV today ($z=0$).
These experiments seek to detect variation in frequencies of long term, stable
oscillators and probe the nature of $\alpha$, the electron to proton mass
ratio $\mu$, and the proton gyromagnetic ratio $g_{\rm p}$. By measuring
frequencies related to the gross, fine-structure, and hyper-fine structure
atomic transitions they measure combinations of $R_{\infty}$,
$\alpha^{2}R_{\infty}$, and $\mu\alpha^{2}R_{\infty}$, respectively, where
$R_{\infty}$ is the Rydberg constant. The scaling of $\alpha$ and $m_{\rm e}$
proposed in this study specifically leaves the Rydberg constant and
$\mu\alpha^{2}$ invariant, which could leave some atomic clock measurements
insensitive to this specific variation. However, the fine structure
measurements ($\alpha R_{\infty}$) effectively break the degeneracy. Current
atomic clock data restricts the variability of $\alpha$ to about
$|\dot{\alpha}/\alpha|<10^{-19}$ annually [163].

Quasar (QSO) absorption spectra provide the earliest directly observable
constraints on FCV. As some of the Universe’s most luminous sources, QSOs
allow observations up to redshifts of $z\sim 7$, reaching back to less than 1
Gyr after the Big Bang. Astronomers analyze absorption spectra from cold gas
clouds along QSOs’ line of sight up to $z\sim 3$ (about 2 Gyr old), setting
constraints on early Universe FCV. This method, however, introduces additional
complexity due to the Universe’s expansion redshifting the spectra. FCV
constraints using QSOs rely on identifying chromatic effects in the spectra,
impacting all wavelengths and may be difficult to disentangle from redshift.
Current constraints from QSO absorption spectra separately limit $\alpha$
variation to $|\Delta\alpha/\alpha|<10^{-6}$ at $z\sim 2.4$ [164] and
$|\Delta\mu/\mu|<10^{-5}$ at $z\sim 0.9$ [165].

Big Bang nucleosynthesis offers potential constraints on FCV mere minutes
after the end of inflation at redshifts of $z\sim 10^{8}$. Standard BBN
models, which successfully predict light elemental abundances, potentially
anchor FCV constraints to this early epoch. BBN’s sensitivity extends to
various factors, including the expansion rate, neutron-proton number density
ratio and mass difference, neutron lifetime, and the Coulomb barrier strength.
This complexity opens a challenging avenue, as BBN’s outcomes are not clear
when considering variations of $\alpha$ and $m_{\rm e}$. However, a
comprehensive analysis of FCV’s impact on BBN demands not only advanced
nuclear reaction rate calculations but also detailed lattice QCD and QED
computations [166, 167, 168, 169].

We summarize the constraints on FCV as follows:

  * •

Atomic clocks confirm the stability of $\alpha$ and $m_{\rm e}$ at present
($z=0$) and that linear variations are ruled out.

  * •

Geological and quasar (QSO) observations suggest no significant variation in
these parameters individually over the past 10 billion years ($z\sim 3$).

  * •

The exact impact of the FFAT symmetry combined with FCV on Big Bang
Nucleosynthesis outcomes remains uncertain at $z\sim 10^{8}$. See App. B for
more discussion.

However, it’s interesting to highlight that most discussions herein typically
focus on varying a single parameter. Simultaneously varying both $\alpha$ and
$m_{\rm e}$ might weaken some constraints, particularly if they primarily
depend on the binding energy, which remains unchanged under this specific FCV.
Under the most conservative assumption that simultaneous variations in
$\alpha$ and $m_{\rm e}$ do not impact constraining power, a particle physics
model aiming to realize the required Thomson scattering rate increase must
vary these parameters while satisfying several criteria: it must preserve BBN
predictions, avoid rapid variation compared to the timescale set by
recombination, and align with Standard Model values before the earliest QSO
observations, maintaining stability thereafter.

##  Appendix B Comments on BBN and FCV

Big Bang Nucleosynthesis is significantly influenced by two critical
quantities that depend on the fine structure constant and the electron mass:
the neutron to proton mass difference ($Q_{\rm np}$) and the neutron lifetime
($\tau_{\rm n}$). Ref. [170] explores the impact of varying $\alpha$ and
$m_{\rm e}$ linearly, establishing two key relationships:

$\frac{\Delta Q_{\rm np}}{Q_{\rm np}}=-0.59\frac{\Delta\alpha}{\alpha},$ (17)
$\frac{\Delta\tau_{\rm n}}{\tau_{\rm
n}}=3.86\frac{\Delta\alpha}{\alpha}+1.52\frac{\Delta m_{\rm e}}{m_{\rm e}}.$
(18)

In this analysis, we omit variations in the vacuum expectation value and quark
masses as examined in [170]. Applying the best-fit values identified in
Section IV, we ascertain that following the FFAT symmetry direction results in
a decrease in $Q_{\rm np}$ by approximately 1% and an increase in $\tau_{\rm
n}$ by around 1%.

BBN unfolds in three crucial stages, with the neutron to proton mass
difference ($Q_{\rm np}$) and the neutron lifetime ($\tau_{\rm n}$) playing
pivotal roles:

  1. 1.

Initially, the temperature remains high enough for the weak interaction rate
to maintain protons and neutrons in equilibrium.

  2. 2.

Once the weak interaction rate dips below the Hubble rate, inverse reactions
cease, and the neutron to proton ratio becomes fixed at an amount sensitive to
$Q_{\rm np}$ and the background temperature. However, neutrons are unstable
and will continue to decay according to $\tau_{\rm n}$ until the onset of
light element synthesis.

  3. 3.

Finally, the temperature falls to a level where primordial nucleosynthesis
begins, facilitated by the formation of deuterons.

This simplified overview demonstrates that BBN is influenced by $Q_{\rm np}$
and $\tau_{\rm n}$, as well as the background expansion rate. Without
comprehensive calculations, pinpointing the exact impact of these factors on
the predicted helium-4 yields is challenging and very model dependent.
However, it seems that an increase in $Q_{\rm np}$ and a decrease in
$\tau_{\rm n}$ would increase the helium-4 yields. Ultimately, BBN could serve
as the decisive assessment for any proposed resolution to the Hubble tension,
as elemental abundances offer stringent constraints on the expansion rate
during that epoch. For recent discussion regarding the connection between BBN
and FCV see Refs. [171, 172, 169, 173].

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