Towards a Realistic Model of Dark Atoms to Resolve the Hubble Tension

It has recently been shown that a subdominant hidden sector of atomic dark matter in the early universe provides a novel avenue towards resolving the Hubble ($H_0$) tension while maintaining good agreement with Cosmic Microwave Background era observables. However, such a mechanism requires a hidden sector whose energy density ratios are the same as in our sector and whose recombination also takes place at redshift $z \approx 1100$, which presents an apparent fine tuning. We introduce a realistic model of this scenario that dynamically enforces these coincidences without fine tuning. In our setup, the hidden sector contains an identical copy of Standard Model (SM) fields, but has a smaller Higgs vacuum expectation value (VEV) and a lower temperature. The baryon asymmetries and reheating temperatures in both sectors arise from the decays of an Affleck-Dine scalar field, whose branching ratios automatically ensure that the reheating temperature in each sector is proportional to the corresponding Higgs VEV. The same setup also naturally ensures that the Hydrogen binding energy in each sector is proportional to the corresponding VEV, so the ratios of binding energy to temperature are approximately equal in the two sectors. Furthermore, our scenario predicts a correlation between the SM/hidden temperature ratio and the atomic dark matter abundance and automatically yields values for these quantities favored by concordant early- and late-universe measurements of $H_0$.


INTRODUCTION
The longstanding tension between the early universe [1] and local [2] extractions of the Hubble constant H 0 may signal the breakdown of the standard Λ Cold Dark Matter (ΛCDM) paradigm (see [3] for reviews). Although recent local measurements using the tip of the red giant branch method suggest that late time measurements might be more compatible with early universe extractions [4], it remains to be seen whether these results will ultimately converge without the need for new physics [5]. While the tension may still be an artifact of systematic error in at least one measurement technique, many models of new physics have been proposed to resolve the discrepancy (see [6][7][8] for recent reviews).
It has recently been shown that a subdominant component of atomic dark matter (ADM) [9][10][11][12][13][14][15][16] can viably increase the early universe value of H 0 while preserving a good fit to other cosmic microwave background (CMB) and baryon acoustic oscillation (BAO) observables [17]. In this scenario, the usual ΛCDM model is supplemented with a hidden sector of dark atoms, photons and neutrinos; the former accounts for a few percent of the dark matter and the latter contribute to the radiation density during the CMB era. Unlike other models that propose modifications to the early universe, this scenario mimics the behavior of visible matter by maintaining the same matter/radiation ratio and undergoing recombination at z ≈ 1100. The addition of this sector approximately mimics the scaling symmetry ρ i → f 2 ρ i , σ T n e → f σ T n e , A s → A s f 1−ns , (1) where ρ i is the i th energy density component, n e is the electron density, σ T = 8πα 2 /(3m 2 e ) is the Thomson cross section, A s is the amplitude of scalar fluctuations, n s is the spectral tilt, and f is an arbitrary constant. The transformation in Eq. (1) preserves the form of all cosmological perturbation equations in linear theory, thereby retaining the good agreement of CMB/BAO predictions with data [17].
As noted in Ref. [17], this framework presents two main observational challenges: 1) the model favors a small CMB value of the cosmological helium fraction, Y p ≈ 0.17 to within a few percent, which is in clear tension with the consensus value from Big Bang Nucleosynthesis (BBN) Y p = 0.245 ± 0.003 [18]; and 2) the best fit hidden/visible temperature ratio satisfies T /T ≈ 0.7, corresponding to a large value of ∆N eff ≈ 1.6 during BBN, assuming identical SM field content in the hidden sector. Thus the ADM mechanism shifts the tension into parameters that affect a different era of cosmological history.
Furthermore, at face value, mimicking this approximate symmetry in Eq. (1) requires an ad-hoc coincidence to ensure that dark and visible recombination both occur at z ≈ 1100. Since the reheating temperature in each sector is an initial condition and the Hydrogen binding energy depends on strong, electromagnetic, and Higgs couplings, such a coincidence across sectors with different masses and thermal histories seems extremely unlikely at first glance; this situation calls for a dynamical explanation.
In this paper we show how such a coincidence can arise in a realistic model that accounts for the full cosmological history of the atomic hidden sector. We model the hidden sector as an identical copy of the Standard Model with arXiv:2108.11386v2 [hep-ph] 11 May 2022 lighter elementary particles and derive the baryon asymmetry and initial temperature in each sector through its coupling to an Affleck-Dine scalar field that dominates the early, post-inflationary universe.
Our model preserves all of the beneficial features identified in Ref. [17] while eliminating fine tuning needed to time hidden sector recombination. Furthermore, this scenario correlates the interacting DM fraction f adm ≡ Ω adm /Ω cdm and the T /T ratio, thereby removing one free parameter to make our scenario more predictive. However, since we recover the same hidden sector studied in Ref. [17], it inherits the tension in ∆N eff and Y p .

MODEL OVERVIEW
Inspired by Twin Higgs models [19], we postulate a mirror hidden sector which contains an identical copy of all Standard Model (SM) fields, coupling constants, and gauge interactions. As in Twin Higgs models, the two sectors here have different values for the Higgs vacuum expectation values (VEVs), but in our scenario we demand that v /v < 1 where v ( ) is the SM (hidden) Higgs VEV and we use primed symbols to refer to hidden sector quantities throughout this work. Since all other couplings are identical, the QCD confinement scales in both sectors satisfy Λ qcd ≈ Λ qcd ≈ 200 MeV [20], which also yields a similar proton mass for both sectors; all other elementary particle masses in the hidden sector are scaled down by an overall factor of v /v relative to the SM. Unlike in Twin Higgs models, our setup does not invoke any direct couplings between the two sectors and does not address the electroweak hierarchy problem, so there are no collider constraints on the VEV ratio.
To ensure viable bound state formation to explain the Hubble tension in this framework, we must satisfy the following conditions: 1. There must be a hidden baryon asymmetry to prevent hidden sector particles from completely annihilating into radiation.
2. Any interaction between sectors must be sufficiently feeble to prevent them from reaching thermal equilibrium.
3. The reheating temperature in each sector must be directly proportional to the corresponding Higgs VEV. Since we have exhausted all the freedom in choosing field content and coupling constants, the VEV dependence in the Hydrogen binding energy B ∝ v is compensated by T ∝ v, so B/T ≈ B /T and both sectors undergo recombination at the same time.
In what follows, we realize all of these requirements by coupling both sectors to an Affleck-Dine field whose Schematic illustration of our setup. A nonrelativistic condensate of Affleck-Dine scalar field φ dominates the energy density of the post-inflationary early universe and carries net baryon number. Upon decay, φ transfers its asymmetry to SM and hidden sector fields with branching ratios proportional to the Higgs VEV in each sector.
decays simultaneously yield the requisite particle asymmetries and temperature relations to reconcile local and CMB measurements of H 0 with a subdominant atomic dark sector; we assume the remaining CDM in this scenario arises from a different source.
While our model and the phenomenological scenario in Ref. [17] both approximately realize the symmetry in Eq. (1), we add one additional source of scaling violation in the hidden sector Thomson cross section σ T /σ T = (v/v ) 2 > 1. The hidden radiation is more tightly coupled to its matter content as a result. Nonetheless, we find that this additional symmetry violating detail does not spoil the good CMB/BAO fit from Ref. [17].

COSMOLOGICAL EVOLUTION
We assume the post-inflationary early universe is dominated by a complex scalar field φ = re iθ / √ 2 that carries baryon number B φ and realizes Affleck-Dine baryogenesis [21] as depicted schematically in Fig. 1. In polar coordinates, the scalar potential is where m is the φ mass, |κ| |λ| are dimensionless couplings, and the explicit θ dependence in the last term provides a source of baryon number violation. After inflation 1 , φ starts rolling at t = t i , and the φ baryon asymmetry n φ (t) = B φ r 2θ evolves according to where a is the Friedman Robertson Walker scale factor and we assume a symmetric initial condition, n φ (t i ) = 0. Integrating Eq. (3) approximately yields [23] where H =ȧ/a is the Hubble rate and this expression gives the baryon number stored in φ until it decays through baryon conserving interactions to transfer the asymmetry to the two sectors. Since the baryon density in each sector is set by the corresponding φ branching fraction, the baryon-to-entropy ratios η where g ( ) ,s is the number entropic degrees of freedom in each sector and T ( ) RH is the visible (hidden) reheating temperature. Thus, the baryon asymmetry transferred to each sector will differ based on model parameters and initial conditions in this framework.
To calculate the branching ratios, we postulate baryon conserving interactions of the form where H ( ) is the visible (hidden) Higgs doublet,Ô ( ) is an operator with compensating baryon number −B φ , Λ is the cutoff scale of the effective interaction, and n is an integer chosen to ensure that the full expression has mass dimension 4. The form of the operator in Eq. (6) is schematic, and a non-trivial contraction of SU (2) L indices may be required to ensure the leading φ coupling is proportional to v 2 or v 2 ; for our purposes, any operator will suffice as long as the coefficient preserves this proportionality. The φ branching ratios to each sector satisfy This approximate expression neglects the contributions from loop level decays through virtual Higgs propagators that need not be proportional to the same powers of v; however, such process are suppressed by loop factors of order (16π 2 ) −2 ≈ 4 × 10 −5 and can be safely neglected.
Assuming instantaneous reheating through φ decays, the energy density of each sector is proportional to the corresponding branching fraction, so the reheating temperatures satisfy where g ( ) is the number of relativistic degrees of freedom in each sector. Thus, using Eqs. (7) and (8), Eq. (5) becomes yielding a simple relationship between the asymmetries of our sectors. Note that in the g = g limit, Eqs. (8) and (9) imply that all energy density ratios are equal in the two sectors (e.g., ρ b /ρ γ = ρ b /ρ γ ), as required to approximate the symmetry in Eq. (1).
Since the hidden sector satisfies m e m p , the binding energy of Hydrogen in both sectors obeys where µ ( ) is the electron-proton reduced mass. Therefore, for g = g in Eq. (8), we predict B /T ≈ B/T , which suffices to trigger hidden recombination around z ≈ 1100. In order to ensure that T RH /T RH < 1, we require v < v, so massive elementary particles are uniformly lighter in the hidden sector. Note that the expression in Eq. (10) is only an approximate equality because the redshift of recombination is logarithmically sensitive to η b and our scenario predicts η b = η b from Eq. (5). However, since the temperature ratio of the two sectors is only of order one [17], this mild deviation is negligible for our purposes. Finally, we note that our model is more predictive than the phenomenological study in Ref. [17] because the ADM fraction of the total dark matter density is where we have assumed ρ b /ρ cdm ≈ 1/5 and used Eq. (8) with g = g . 2 Thus f adm is correlated with the temperature ratio and lies naturally in the range favored to reconcile early-and late-time measurements of H 0 for the best fit value T /T ≈ 0.7 in Ref. [17].

Affleck-Dine Mass Scale
Assuming φ decays take place during a cold, matterdominated phase, both sectors are in the broken electroweak phase throughout reheating; if this were not the case, the branching fractions would not necessarily scale according to the relation in Eq. (7). Furthermore, in the broken vacuum, the H 2 proportionality in Eq. (6) can be expanded using H = [0, (v + h)/ √ 2] T to generate interactions of the form vhφÔ/Λ n , whose branching fraction scales as ∝ v 2 , not v 4 , as desired in Eq. (7). Such decays can be kinematically forbidden if m φ < m h .

Reheat Temperature
To ensure that the branching ratios in Eq. (7) are satisfied throughout the early universe, we demand that both sectors reheat to temperatures below the scale of electroweak symmetry breaking, T ( ) RH v ( ) . Since this requirement necessarily implies that some fields in each sector will not be produced, we must nonetheless ensure that g (T RH ) = g (T RH ). However, since the field content in the two sectors is identical, this can be achieved across a wide range of temperature ratios. For example, with v /v = 1/2 and T RH = 50 GeV, all leptons, light quarks, and massless gauge bosons are produced in both sectors, but neither thermalizes its W ± , Z 0 , h, or t particles.
Assuming a non-relativistic φ-dominated universe, instantaneous φ decays, and rapid equilibration (compared to Hubble expansion) in each sector, the reheating temperature of each sector can be approximated as where τ φ is the scalar lifetime. However, the scaling in Eq. (12) is highly model dependent and can be modified, for example, with additional decay channels for φ or by parametric resonance effects [24]. Our scenario is compatible with any of these reheating variations as long as φ decays in the broken electroweak vacuum and the relation in Eq. (7) is preserved to good approximation.

Choosing The Decay Operator
To realize the branching ratio relation in Eq. (7), the operators in Eq. (6) must be chosen with care. Since the early universe is always in the broken electroweak phase, φ decays must directly generate a net baryon asymmetry; a purely lepton number asymmetry would not yield a baryon asymmetry here since sphalerons are always out of equilibrium in our scenario.
Furthermore, since SM operators with net baryon number involve many insertions of quark fields (e.g.,Ô = u c d c d c where u c and d c are respectively up-and downtype SU (2) L singlet quarks), the exponent n in Eq. (6) is a large integer. This suppression makes it generically difficult to reheat the universe above the MeV scale while keeping m φ < m h , required to maintain the relation in Eq. (7). However, in the presence of additional baryoncharged fields in each sector this problem can be avoided.
As a toy example, we can add to each sector a gauge singlet Weyl fermion χ ( ) and its Dirac partner χ c( ) with baryon number ∓B φ /2, respectively. This enables us to posit φ decay interactions of the form φ|H| 2 χχ/Λ 2 , which can ensure the relations in Eq. (7) while giving φ sufficiently prompt φ → χ ( ) χ ( ) decay channels. Once the universe is populated with χ particles at T RH > MeV, prompt χ → 3q decays can proceed through a χ c u c d c d c /M 2 operator 3 to transfer the baryon asymmetry to SM particles, where M is the mass of a heavy particle that has been integrated out. While this realization satisfies all of our requirements, the gauge and baryon charge assignments for χ also allow direct φχχ couplings, which would induce decays that violate the VEV scaling in Eq. (7). This issue can be avoided if φ carries baryon minus lepton number, which allows the operator φ(LH) 2 χχ/Λ 5 and forbids the renormalizable φχχ interaction 4 ; as noted below, this operator dimension can still yield a sufficiently high reheating temperature for BBN.
For all candidate decay operators, it is important to forbid mixed terms that involve fields from both sectors. From the example above, a complete ultraviolet model must forbid operator variations of the form φ(LH)χ(L H )χ /Λ 5 , which would spoil the required relation in Eq. (7). Such a sequestration can be achieved, for example, in extra dimensional models in which the two sectors live on different four dimensional branes, but φ lives in a higher dimensional bulk, so hybrid operators involving both sectors are forbidden by locality (see Refs. [25,26] for examples). The mixed operators can also be suppressed if χ ( ) is endowed with a parity symmetry under which χχ transforms non-trivially; this symmetry is then spontaneously broken to allow for χ ( ) decays to baryons.

Avoiding Thermalization
Since the two sectors in our setup must not thermalize with each other, we conservatively demand that φ never thermalize with the SM whose energy density is always greater. For an interaction rate based on Eq. (6) in the broken electroweak phase, Γ φ−SM ∼ v 4 T 2n−3 RH /Λ 2n < H at reheating, so the suppression scale must satisfy shows the relative shift of the TT power spectrum for two mirror sector models: one with the dark Thomson cross-section σ T equal to the SM one, σT (as assumed in Ref. [17]), and one where σ T ≈ 2σT as predicted in our set-up. The differences are negligible and both models are compatible with the observed spectra, whose binned residuals relative to ΛCDM are shown by the points with error bars [27]. The mirror sector models have H0 ≈ 73.4 km/s/Mpc, f adm = 0.027, T /T = 0.68, and Yp = 0.17 following Ref. [17] (the other cosmological parameters are given in the text). In the lower panel we show H(z)r d relative to its ΛCDM value, where r d is the sound horizon at the drag epoch, along with BAO measurements of this quantity from BOSS [28]. Note that ΛCDM and the mirror sector models are nearly indistinguishable in this observable because of the scaling symmetry H → f H, r → r/f that the mirror sector implements [17].
that T RH MeV as required by the success of standard BBN [29][30][31][32] without thermalizing φ with either sector. We expect that a careful treatment of the phase-space in the above thermalization rate would only change the bound on Λ by O(1) because of the 2n'th root. In addition to avoiding thermalizing the two sectors through the φ interactions, we must also ensure that mixed operators of the form λ|H| 2 |H | 2 are either forbidden or sufficiently suppressed such that hh ↔ h h reactions are always slower than Hubble expansion. As noted above, such hybrid interactions can be naturally suppressed by locality in extra dimensional theories that require the two sectors to live on different branes.

DISCUSSION
We have introduced a realistic model of atomic dark matter that brings local and CMB measurements of H 0 into concordance following the phenomenological study in Ref. [17]. Our approach is inspired by Twin Higgs models in which a hidden sector contains an identical copy of the SM, but with a slightly different Higgs VEV, v , and temperature, T . The baryon asymmetry and reheating temperature in each sector is set by the VEVdependent branching ratio of an Affleck-Dine scalar field φ, and all differences between sectors are governed by v /v. Since the φ branching ratio to the visible and hidden sectors scales as ∝ v 4 and v 4 , respectively, the hydrogen binding energy-to-temperature ratio, B/T , is the same in both sectors and recombination occurs for all atomic species at z ≈ 1100. As a result, there is no fine tuning required to ensure this coincidence.
Our model features several notable differences with respect to the phenomenological treatment of Ref. [17]: 1. Because our hidden sector has a smaller electron mass, the Thomson cross section satisfies σ T = (v/v ) 2 σ T = 2.1σ T . So, even if recombination still occurs at z ≈ 1100 on account of B/T = B /T , this larger σ T can change the acoustic oscillations of the hidden sector baryons and thus modify the CMB/BAO observables indirectly. To test the importance of this effect, we implemented a mirror sector model in the Boltzmann code CLASS v 3.1.1 [33] by modifying the existing interacting DM module [34]: we simply rescaled the DM-dark radiation interaction terms by the free-electron fraction of the SM (this is a reasonable approximation since fast variations in this quantity, i.e. recombination, occur simultaneously in the visible and dark sectors by construction) and scaled the interaction coefficient, a idm dr, to match σ T . The redshift-dependence of this interaction matches that of Thomson scat-tering for n = 2 in the notation of Ref. [34]. 5 In Fig. 2 we show that a different scattering crosssection has very little impact on the quality of the fit to CMB and BAO observables. There we take cosmological parameters motivated by the detailed MCMC analysis in Ref. [17]: v /v = 0.68, Y p = 0.17, f adm = 0.027; note that other parameters were not explicitly provided in that paper so we picked values that give a reasonable fit by eye: Ω cdm h 2 = 0.142, Ω b h 2 = 0.0224, 100θ s = 1.0425, ln 10 10 A s = 3.03, n s = 0.958, τ reio = 0.0539, and N fs = 3.30, where N fs is the number of freestreaming degrees of freedom (i.e. SM and mirror neutrinos). The total number of relativistic degrees of freedom, N eff ≈ 4.24, includes the contribution of the non-free-streaming dark photons. A similar Boltzmann code implementation of a twin mirror sector was recently studied in Ref. [36], but their focus was on models with v /v > 1. It would be interesting to perform a detailed cosmological analysis for our specific realization of the mirror sector in a future work -we expect this would give rise to only small shifts in the preferred values of the cosmological parameters.
2. From Eq. (11) our scenario features a one-toone correspondence between f adm and the hidden/visible temperature ratio, so our model has fewer free parameters than [17]. It is also notable that the relation in Eq. (11) is automatically consistent with the approximate best fit values f adm ≈ 0.05 and T /T ≈ 0.7 (see also footnote 2).
As noted in [17], this framework is in generic tension with both the direct measurement of the primordial helium fraction and the theoretical prediction of Big Bang nucleosynthesis accounting for the large value of ∆N eff ≈ 1.6 (assuming the full SM-like field content in the hidden sector). BBN with such a large enhancement to the expansion rate predicts Y p ≈ 0.266±0.0053, higher than in the standard cosmology with Y p ≈ 0.247 ± 0.0046 (with n b /n γ = 6.13 × 10 −10 [1]); 6 thus the theoretical prediction of Y p within this model appears to be even more inconsistent with the CMB inference of the same quantity. Even if the hidden sector is populated after BBN, through, e.g., the decay of a non-relativistic particle, the CMB-preferred value still disagrees both with the direct late-universe measurement and the (now) standard Y p yield from BBN. These points of tension persist in our scenario, but because we approximately realize the scaling relation in Eq. (1), we also inherit a good fit to CMB observables and a larger value of H 0 ≈ 73 km s −1 Mpc −1 , provided that Y p and ∆N eff are allowed to float.
It would be interesting to study whether hidden sector model variations can overcome these observational limitations. For example, it may be possible to realize our scenario with only a subset of SM generations in the hidden sector, in analogy with Fraternal Twin Higgs models [43], but such studies are beyond the scope of this work.
Finally, we note that our scenario may imply several interesting consequences that are worth exploring in detail. For example, an atomic dark sector with identical field content may yield dark nuclei [37][38][39], galactic disks [44,45], and stars [46][47][48][49][50][51] but exploring the observational implications of these structures is beyond the scope of this work. Moreover, UV completions of the various effective operators required to realize our proposal might lead to additional signatures. For example, it may be possible for neutral hidden-visible oscillations between each sector's neutrinos, neutrons, and photons, as long as the mixing interactions that enable these processes do not thermalize the two sectors at early times.