ournal of C osmology and A stroparticle Analyzing the Hubble tension through hidden sector dynamics in the early universe

. The recent analysis from the SH0ES collaboration has conﬁrmed the existence of a Hubble tension between measurements at high redshift ( z > 1000) and at low redshift ( z < 1) at the 5 σ level with the low redshift measurement giving a higher value. In this work we propose a particle physics model that can help alleviate the Hubble tension via an out-of-equilibrium hidden sector coupled to the visible sector. The particles that populate the dark sector consist of a dark fermion, which acts as dark matter, a dark photon, a massive scalar and a massless pseudo-scalar. Assuming no initial population of particles in the dark sector, feeble couplings between the visible and the hidden sectors via kinetic mixing populate the dark sector even though the number densities of hidden sector particles never reach their equilibrium distribution and the two sectors remain at diﬀerent temperatures. A cosmologically consistent analysis is presented where a correlated evolution of the visible and the hidden sectors with coupled Boltzmann equations involving two temperatures, one for the visible sector and the other for the hidden sector, is carried out. The relic density of the dark matter constituted of dark fermions is computed in this two-temperature formalism. As a consequence, BBN predictions are upheld with a minimal contribution to ∆ N eﬀ . However, the out-of-equilibrium decay of the massive scalar to the massless pseudo-scalar close to the recombination time causes an increase in ∆ N eﬀ that can help weaken the Hubble tension. extra relativistic degrees of freedom at CMB times while respecting BBN bounds on ∆ N eﬀ . Our analysis is based on a two-sector model, one visible and the other hidden, where we assume a negligible particle abundance of the hidden sector particles in the early universe, and where the hidden sector particles are generated by the out-of-equilibrium annihilation of SM particles into particles in the hidden sector. Because of that, the two sectors reside in heat baths at diﬀerent temperatures with the hidden sector at a lower temperature than the visible sector. Thus our analysis uses a set of coupled Boltzmann equations which depend on temperatures of both the visible and the hidden sectors and involves correlated temperature evolution of both sectors. In this set up we show that for a range of kinetic mixing and dark photon mass consistent with experimental limits, the visible and the dark sectors do not thermalize which implies that the dark sector particles never reach their equilibrium number densities. However, their phase space distributions can still be described by a Maxwell-Boltzmann distribution since they achieve chemical and kinetic equilibrium during the dark sector thermal history. The fact that the dark particles, especially the relativistic particle φ , never reach their equilibrium distribution allows to evade the constraint on ∆ N BBNeﬀ . With a speciﬁc choice of couplings, late decays of S → φφ can contribute the right amount to ∆ N CMBeﬀ thus resolving the Hubble tension. The model we present is cosmologically consistent since the formalism allows for correlated evolution of the temperatures of the hidden and the visible sectors which are not in thermal equilibrium. In fact it is the lack of thermal equilibrium between the hidden and visible sectors which forces consistency with the BBN constraint while allowing the enhancement of N eﬀ at CMB time. However, as noted earlier, while ∆ N eﬀ at recombination time would weaken the Hubble tension, a better understanding of how this aﬀects the CMB power spectrum requires a ﬁt of the cosmological parameters to the CMB data.


Introduction
Recently the SH0ES collaboration [1] has confirmed at 5σ level the tension between measurements of the Hubble parameter H 0 at high redshift (z > 1000) and at low redshift (z < 1). Based on the Lambda-cold dark matter (ΛCDM) model and an analysis of data from the times of Big Bang Nucleosynthesis (BBN), the Cosmic Microwave Background (CMB) and Baryon Acoustic Oscillations (BAO), the Hubble parameter at high z is determined to be [2] H 0 = (67.4 ± 0.5) km/s/Mpc, (1.1) while the low z value arising from the analysis of Cepheids and SNIa gives [1] H 0 = (73.04 ± 1.04) km/s/Mpc, (1.2) which leads to a 5σ difference between the Planck result and the result arising from local observation. Since the Planck analysis is based on the ΛCDM model, the discrepancy between the high z and the low z results represents an indication of physics beyond the standard ΛCDM model. There is a large number of theory models which attempt to explain this difference (for a review see ref. [3] and references therein, e.g., ref. [4]). One of the simplest avenues is the possibility of having extra relativistic degrees of freedom present during recombination, which would cause an increase in H 0 and bring it closer to its measured local value. Extensions of the SM with new particles in thermal equilibrium with the neutrinos, -1 -

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such as a majoron [5,6], or with an extra U(1) symmetry and Z bosons decaying to neutrinos or otherwise [7,8] can explain the tension. We note here that models that can successfully explain the Hubble tension are those where the extra degrees of freedom would affect only the dynamics in the CMB era, but not impact BBN. This is so because the precise predictions of light element synthesis during BBN can be translated into strong constraints on the effective degrees of freedom such that N BBN eff = 2.88 ± 0.27 at 68% C.L. [9,10]. This appears to be consistent with the SM prediction of N SM eff 3.046 [11]. Since the effective degrees of freedom are tied to the energy density in the universe, which affects the Hubble parameter, any deviation from N eff at BBN would have an impact on the expansion rate of the universe as well as on predictions of light element synthesis during this epoch. Thus any model looking to add extra relativistic degrees freedom at CMB time to help alleviate the Hubble tension should not violate the BBN constraints on N eff .
Further, N eff is also constrained, but to a lesser extent, during the period the CMB was generated. This is so because extra relativistic degrees of freedom affect the CMB power spectrum, which has been measured using Planck polarization data and BAO, so that [2] N eff = 2.99 +0. 34 −0.33 (95% C.L., TT,TE,EE+lowE+lensing+BAO). (1.3) Two earlier local measurements of H 0 [12,13] (dubbed R18 and R19) have shown an increasing discrepancy with high redshift values of H 0 at 3.7σ and 4.4σ deviation, respectively. The preferred N eff value, when combining the local measurements of H 0 with Planck and BAO data, becomes [2] N eff = 3.27 ± 0.15 (68% C.L., TT,TE,EE+lowE+lensing+BAO+R18). (1.4) Extra relativistic degrees of freedom, ∆N eff , increase the sound horizon r * during recombination. Since the acoustic scale θ * = r * /D M is precisely measured, 1 then D M ∝ 1/H 0 must decrease, or H 0 must increase to keep inline with experimental measurements. Based on the difference between the early and local measurements of H 0 , it was suggested that 0.2 ∆N eff 0.5 (CMB+BAO+R18), [2] (1.5) and 0.2 ∆N eff 0.4 (CMB+BAO+Pantheon [14]+R19+BBN), [15][16][17] (1.6) could relax the Hubble tension between high redshift values of H 0 and the local R18 and R19 measurements. 2 In light of the most recent measurement of H 0 by the SH0ES collaboration [1] one can expect a revised range in order to accommodate higher values of ∆N eff . In this work we present a cosmologically consistent model based on the Stueckelberg extension of the SM with a hidden sector to help alleviate the Hubble tension between the high and the low z measurements. By a cosmologically consistent model we mean that we take into account the fact that the hidden sector and the visible sector in general live in different heat baths at different temperatures and the Boltzmann equations that evolve the number density of the particle species in the visible and in the hidden sectors must use a two-temperature evolution which, however, are correlated because of the couplings between the two sectors. Thus the hidden sector possesses a U(1) X gauge symmetry and the couplings between the hidden U(1) X and the hypercharge U(1) Y can occur via kinetic mixing and Stueckelberg mass where z * is the redshift at the time of last scattering. 2 For an estimate of the Hubble parameter in different cosmologies using the Pantheon data, see refs. [18,19].
mixing. We assume that, aside from the U(1) X gauge field (which gives the dark photon γ ), the hidden sector contains a Dirac fermion D, a scalar field S and a pseudo-scalar field φ. The coupling between the visible sector and the hidden sector is assumed feeble while that between the dark fermion and the dark photon could be of normal size, i.e., of size O(1). Assuming that there was no initial particle density of the hidden sector particles, the feeble interactions due to small kinetic mixing between the hidden sector U(1) X and the visible sector U(1) Y gauge groups allow for a gradual buildup of particle density in the hidden sector. Dirac fermions D (which act as dark matter) and dark photons γ can be generated from the SM via 2 → 2 and 2 → 1 processes. The U(1) X gauge coupling g X is assumed strong enough to allow D and γ to enter into chemical equilibrium in the dark sector. It is to be noted that just like the SM thermal bath of massless photons, the dark sector can generate its own thermal bath which we assume here to consist of massless pseudoscalar particle φ. Along with φ, a light long-lived scalar particle S can be produced. Both φ and S have no direct couplings with the SM and are produced purely through dark sector interactions, which we assume to be at a different temperature than for the SM. The resolution of the Hubble tension comes from the decay S → φφ prior to recombination, which adds to the relativistic degrees of freedom at this epoch and drive the Hubble parameter to larger values. Particle physics models that attempt to explain the Hubble tension either assume the visible and the hidden sectors are initially in thermal contact and decouple later or they start as outof-equilibrium systems and eventually a hidden sector particle thermalizes with the neutrino sector before decaying close to recombination. Our analysis does not involve the neutrinos and the resolution of the Hubble tension comes entirely from the dark sector. Further, the model discussed here can explain the Hubble tension while preserving BBN predictions due to the fact that the particles S and φ do not trace their equilibrium distributions in the dark sector owing to the small coupling between them and D and γ . The model parameters are constrained by the DM relic density, limits on the dark photon mass, and by the chemical and kinetic decoupling of the species and ∆N eff at BBN.
The paper is organized as follows. In section 2 we give an overview of the model used in the analysis. In section 3 we discuss the set of coupled Boltzmann equations used to evolve the particle yields and the temperatures of the visible and hidden sectors. A discussion of kinetic decoupling of hidden sector particles is given in section 4. In section 5 we give a numerical analysis of the particle yields and the temperature evolution of the sectors taking into accounts the relevant experimental constraints. A discussion on the different contributions to ∆N eff is given in section 6 followed by an analysis of energy densities and ∆N eff up to recombination in section 7. Conclusions are given section 8. Further detail of the analysis is given in section 9.

The model
We now briefly describe the hidden sector model and its coupling to the visible sector which is used in the analysis of this work. The model is a U(1) X extension of the SM gauge group which has gauge kinetic mixing [20,21] between the two abelian gauge groups. The new gauge field obtains its mass through the Stueckelberg mechanism [22,23] for the gauge fields. Thus the extended model has the Lagrangian

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where L SM is the Standard Model Lagrangian, and contains additional terms arising from the hidden sector and its coupling to the visible sector through kinetic energy mixing and Stueckelberg mass mixing. In eq. (2.2), C µ is the gauge field of the U(1) X and B µ is the gauge field of the U(1) Y and The axionic field σ transforms dually under U(1) X and U(1) Y so that the mass term remains gauge invariant and can be discarded in the unitary gauge. The dark fermions, with a charge Q X under U(1) X , interact with the gauge field C µ via the gauge interaction with strength g X , and with the (pseudo-)scalar field (φ) S with strength (y φ ) y S . We assume a kinetic mixing between the hypercharge field B µ and the U(1) X gauge field C µ characterized by the mixing parameter δ. The mass growth for the gauge field occurs via the Stueckelberg mechanism and is characterized by the parameters M 1 and M 2 . The scalar sector of the model contains the SM Higgs boson, S and φ. Therefore, the scalar potential is given by The mixing between the gauge field B µ and C µ in the Stueckelberg mass mixing term will give the dark fermions a millicharge, which is constrained by several experiments. For simplicity here we set M 2 = 0. After electroweak symmetry breaking and in the canonical basis where the kinetic energies of the gauge fields are diagonalized and normalized, we have the following set of couplings among particles of the dark sector and among particles of the dark sector and of the visible sector so that where f runs over all SM fermions. In the limit of small kinetic mixing, g D γ g X Q X which is a normal strength coupling. The couplings g D Z and g D γ are given in appendix B. The vector and axial couplings are given by where s δ = sinh(δ), T 3f is the third component of isospin, Q f is the electric charge for the fermion and the angles θ and ψ are defined in appendix B. We also record here the couplings of the Z µ and A γ µ in the canonically diagonalized basis which are given by [24,25]

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Further details can be found in refs. [26][27][28][29]. The free parameters of the model described above are subject to a number of theoretical and experimental constraints. The theoretical constraints require that the model produce the desired amount of dark matter, which in turn necessitates a careful analysis of chemical and kinetic decoupling of different particle processes. It also requires that the model maintains the success of the BBN analysis within the SM paradigm, i.e., that the model produces only negligible extra relativistic degrees of freedom at BBN time, but makes a significant contribution ∆N eff at the time the CMB was produced. These put severe constraints on the dark photon mass, the mass of the dark fermion, and on chemical and kinetic decoupling of species in the dark sector. Additionally, the kinetic mixing and the dark photon mass are severely constrained by current experiments. Despite all the stringent constraints, we show that a considerable parameter space remains viable to construct a cosmologically consistent particle physics model that can help reduce the Hubble tension.

Evolution of hidden and visible sector temperatures
Hidden and visible sectors in general reside in heat baths at different temperatures [26,28,[30][31][32]. In models of the type we consider, the bath for the hidden sector is at a much lower temperature than the bath for the visible sector. However, because of the coupling between the visible and the hidden sectors, their temperature evolution is correlated. Specifically, the entropy in each sector is not separately conserved, but it is only the total entropy that is conserved. Thus the total entropy is S = sR 3 , where R is the scale factor and s = s v + s h , where s h is the hidden sector entropy density, s v is the entropy density of the visible sector, and Here h v eff and h h eff are the visible and the hidden effective entropy degrees of freedom and T (T h ) is the temperature of the visible (hidden) sector. The conservation of the total entropy leads to the evolution equation for the entropy density ds/dt + 3Hs = 0. (3. 2) The temperature evolution of the two sectors is also affected by the dependence of the Hubble parameter on the temperatures of the two sectors. Thus the Friedman equation gives where ρ v (T ) and ρ h (T h ) are the energy densities of the visible and of the hidden sectors and are given in terms of their respective energy density degrees of freedom as For the visible sector effective degrees of freedom, g v eff and h v eff , the recent tabulated results from lattice QCD [33] are used, while for the hidden sector, g h eff and h h eff are given in the form of integrals in appendix A. The time evolution of the energy density in the hidden sector is given by -5 -

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where p h is hidden sector pressure density and j h is the source in the hidden sector. It encodes the heat exchange between the visible and the hidden sectors and is given by eq. (3.7) of appendix C.
Since the temperatures of the visible and the hidden sectors are correlated, a quantitative analysis of this correlation is essential. For this reason we introduce a function η = T /T h . The evolution equation for η depends on the particle number densities, and so we need to identify the interactions that enter in the evolution. There are two such types of interactions. The first one involves the interactions of the dark particles among themselves, i.e., among D,D, γ , φ and S. Then we have the feeble interactions between the visible sector and the hidden sector, which involve processes of the type iī → DD, and iī → γ , where i refers to the Standard Model particles and the interactions are those induced by the kinetic energy mixing. We assume no initial number density for the hidden sector particles and it is only through the feeble interactions of the visible sector with the hidden sector that the hidden sector particles are produced. Here we note that only D,D and γ are produced by the portal interactions with the visible sector, while φ and S are only produced by interactions within the dark sector. The set of all interactions considered appear in the system of coupled Boltzmann equations, eqs. (C.1)-(C.4) (see appendix C). Here we have introduced the dimensionless quantity Y a = n a /s, the comoving number density or yield of a particle species a, and have written the equations relative to the hidden sector temperature T h instead of time.
The ratio η ≡ T /T h is itself dependent on T h , since we have taken T h as our reference temperature. Its evolution is driven by the energy injection from the SM into the dark sector. The energy exchange is given by the source term j h . In this case one may derive the equation for η so that where j h is given by The J terms appearing in eq. (3.7) are given in appendix C. In eq. (3.6), ζ and ζ h are 1 for the radiation dominated era and 3/4 for matter dominated era. More generally they are given by ζ = 3 . The relic density of D, which is dark matter, is related to Y ∞ D , the yield today, and the entropy density today, s 0 by where ρ c is the critical density, and h = 0.674 [2]. The total relic density of dark matter as given by the Planck collaboration [34] is (Ωh 2 ) PLANCK = 0.1198 ± 0.0012. Note that since we are modifying the early cosmology of ΛCDM by adding extra radiation, the DM relic density is bound to change. This is taken into consideration in the numerical analysis as we explain later.

Departure from kinetic equilibrium
As the temperature of the universe drops, annihilation processes (i.e., number-changing processes) become progressively inefficient. Once their rate falls below the Hubble expansion rate, the involved particle species undergo chemical decoupling. The onset of chemical decoupling between D and γ in the dark sector initiates a dark freeze-out where the dark matter yield or comoving number density remains fixed with falling temperature. Despite this decoupling, particle species can remain in local thermal equilibrium through scattering processes with the thermal bath. Focusing on D, its temperature T D still follows that of the hidden sector's thermal bath T h = T φ as it scatters off relativistic species, namely S and φ, via the processes DS → DS and Dφ → Dφ. The same holds for S: after its chemical decoupling, local thermal equilibrium can be maintained via Sφ → Sφ. Even those processes will eventually lose their competition against the Hubble rate and decouple. Note that kinetic decoupling can occur before chemical decoupling in which case one needs to solve the full Boltzmann equations for the phase space distributions [35,36]. This has an effect on the relic density and is often numerically challenging. Here we restrict ourselves to the case where kinetic decoupling happens much after chemical decoupling has occurred which is the assumption made when writing the Boltzmann equations, eqs. (C.1)-(C.4).
To determine the temperature of kinetic decoupling of a dark sector species X with mass m X , one needs to solve the second moment of the Boltzmann equation. An analytical solution is quite complicated but can be simplified by assuming that the scattering particle is non-relativistic which means that T X m X and that the typical momentum transfer in the scattering processes is small compared to m X . Keeping only leading order terms in p 2 /m 2 X , we write the Boltzmann equation as [37,38] where γ(T φ ) is the momentum transfer rate evaluated at the bath temperature T φ , T X is the temperature of particle X (used to describe departure from kinetic equilibrium) with number density n X , and is defined by where f (p) is the one particle phase space distribution and the Hubble parameter is given by The momentum transfer rate in eq. (4.1) is given in terms of an average of the matrix element squared over the Mandelstam variable t [39] where E is the energy of the bath particle φ with g φ = 1. The upper sign in f ± (E)(1 ∓ f (E)) correspond to fermions and the lower sign to bosons. The temperature of kinetic decoupling T kd is calculated as assuming instantaneous decoupling. Prior to decoupling, T X traces T φ as both temperatures drop as 1/R. However, after decoupling, the temperature of X (D and S in our model) decreases as 1/R 2 , while T φ continues as 1/R assuming that there are no further interactions or decays. This situation will change in the next section when we discuss out-of-equilibrium decay of S, which adds to the energy density of φ. An illustration of how eq. (4.5) works is given in figure 1, where we show the rate of the processes Dφ → Dφ, DS → DS (blue curve) and φS → φS (gray curve) along with the Hubble parameter (dashed line) plotted against the visible sector temperature. Kinetic decoupling occurs when the rate falls below H(T ) which for D occurs at a higher temperature than that for S for this particular choice of parameters (benchmark (b) of table 1 below). Note that our model predicts late kinetic decoupling, while there exists a lower limit on T kd from Lyman-α forest measurements [40] implying that T kd > 100 eV. This will set constraints on the couplings in the dark sector.

Numerical analysis: yield and temperature evolution prior to decoupling
In this section we discuss the results pertaining to temperature and number density evolution before the full decoupling of the dark sector species. Here we numerically solve the coupled Boltzmann equations, eqs. (C.1)-(C.4), together with the temperature evolution of the two sectors, eq. (3.6), in order to track the growth of the particle yields as a function of the temperature T h . The system of equations form a set of stiff differential equations which are solved in two steps: (1) integrating from a high temperature assuming T h /T 1 until chemical decoupling sets in the dark sector at T cd and (2) integrating dY S /dT h and dY φ /dT h from T cd down to 1 eV assuming Y D = constant and Y γ = 0. In every calculation we verified that Y γ is small enough to be discarded (since γ decays away before BBN) and that D has already undergone a dark freeze-out. Splitting the temperature range this way helps the ordinary differential equation solver tackle the stiffness of the system at hand.  We select a set of representative benchmarks which satisfy all the experimental constraints on the dark matter relic density and the dark photon mass and which can help alleviate the Hubble tension. They are given in table 1. Note that a more general investigation of the available parameter space will be given later in this section and in the next sections as well. The DM relic density for benchmarks (b)-(d) is ∼ 0.12 while that for (a) is 0.07 which can be explained by the smallness of the kinetic mixing δ. We note here that κ s and κ φs have dimension of mass and one expects these to be Planck suppressed so that the first such term is of size λm 2 EW /m Pl . For m EW ∼ 10 2 GeV, m Pl ∼ 10 19 GeV, and λ ∼ 10 −3 one gets κ ∼ κ φs ∼ 10 −18 GeV.
We show in figure 2 the evolution of the particle yields as a function of the hidden sector temperature for benchmarks (b) (left panel) and (c) (right panel). The solid lines correspond to the particle yields as obtained from solving the Boltzmann equations, while the dotted lines correspond to the equilibrium yields of φ and S. One can see from the left panel that the particle yields increase gradually with D and γ taking the lead as they are produced by SM processes while the yields for S and φ are smaller since they are produced only via their interactions with D and with each other (see eqs. (C.1) and (C.2)). The yields undergo a sudden surge at T h ∼ 0.1 GeV which happens near the QCD crossover (T = ηT h < 1 GeV) where the freeze-out of the SM degrees of freedom causes an entropy dump leading to a sudden drop in the visible sector temperature relative to the hidden sector. One can see in the left panel of figure 3 that the visible temperature drops sharply at T ∼ 1 GeV while the  hidden sector temperature drops at a slower rate. This can be seen using the conservation of entropy Not only does h v eff undergo a sudden drop, but so does η (see right panel of figure 4). As a result, one can immediately see from the first equality of eq. (5.1) that T h drops at a slower rate. On the contrary, the second equality of eq. (5.1) shows that a drop in h v eff is partially compensated by an increase in the second term proportional to ξ, where we define ξ ≡ 1/η (see left panel of figure 4). As a result, the drop in the visible sector temperature is not slowed down. Following this increase, the yield of γ starts decreasing (red curve) as the decay rate overtakes the Hubble parameter (see black curve in the right panel of figure 3). At the same time dark matter annihilation in the dark sector becomes very efficient, leading to a drop in the dark matter yield (blue curve) before freeze-out sets in as the rate of the process DD → γ γ drops below the Hubble rate. At this point, D and γ are chemically decoupled after they have been in chemical equilibrium for some time, as seen in the right panel of figure 3, leading to dark matter freeze out and the yield leveling off. Note that this happens at T h ∼ 10 −2 GeV which is in the neighborhood of m D /20, the standard freeze-out temperature.
We also notice from the right panel of figure 3 that φ and S enter into chemical equilibrium (purple curve) as well as D, γ and S (pink curve). Thus there is a period in the thermal history of the dark sector where all species are in thermal equilibrium. Chemical decoupling between φ and S happens after chemical decoupling of D and γ and so the freeze-out of S happens at a lower temperature. From the left panel of figure 2, the yield of S can be seen leveling off at T h ∼ 5 × 10 −4 GeV. The drop in the yield of S is followed by a rise in φ before the yield of the latter continues at a fixed value. Notice that neither S nor φ follow their equilibrium distribution (plotted as dotted curves) which is an essential criterion in our analysis to avoid the BBN constraint on ∆N eff as we discuss later. We note also that S does not contribute to the relic density, since it will eventually decay near recombination as the rate of S → φφ overtakes the Hubble rate (see right panel of figure 3). The energy injection from the SM into the cold dark sector gradually causes its temperature to increase. This is depicted in the left panel of figure 4 for benchmarks (a), (c) and (d) of table 1. For those benchmarks, the two sectors do not thermalize and ξ reaches at most a value ∼ 0.55 (for benchmark (d)) after BBN. The degree at which the two sectors reach thermal equilibrium depends on the size of the kinetic mixing δ which is why benchmark (a) has the smallest ξ value since it has the smallest δ. A larger value of δ can produce a thermal distribution of φ, and for ξ large enough contributions to ∆N eff at BBN can be sizable which implies that δ is constrained in order to maintain the successful BBN predictions of the SM.

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We now discuss the available parameter space of the model taking into account the dark matter relic density constraint and limits on the dark photon mass. For a fixed value of m D and g X , we scan the parameter space in the kinetic mixing-dark photon mass plane. We only keep points that satisfy the dark matter relic density, (Ωh 2 ) D ∼ 0.1-0.125, 3 and ones where kinetic decoupling of D happens after its chemical decoupling. We also remove points with very late kinetic decoupling of S and those with ∆N BBN eff > 0.2. Further, there is an array of experimental limits on the dark photon mass which mixes kinetically with the SM sector. These include CHARM [41,42], E141 [43] and ν-CAL [42,[44][45][46][47], E137 [48,49], electron and muon g − 2 [50], BaBar [51], NA48 [52], NA64 [53,54], E141 [43] and LHCb [55,56]. The limits are obtained from darkcast [57]. Additionally, more stringent constraints arise from Supernova SN1987A [58]

Contributions to ∆N eff
After neutrino decoupling from photons, the visible sector has two separate thermal baths: the neutrinos with temperature T ν and the photons with temperature T γ , where T γ = (11/4) 1/3 T ν based on entropy conservation. Note that in principle, the three neutrino species have different temperatures, however, neutrino oscillations tend to thermalize the three flavors and one can describe them by a single temperature. The effective number of relativistic degrees of freedom in the SM is thus given by Precision calculations of N eff give a value slightly different than three, namely N SM eff = 3.046 [11,59] after taking into account non-instantaneous decoupling of neutrinos, finite temperature corrections, neutrino oscillations and effects of e + e − annihilation to neutrinos.

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Deviations from the SM value of N eff can be realized in beyond the SM physics with particles contributing to the energy density of the universe. At BBN time, an increase in N eff (and consequently in the energy density) would increase the expansion rate of the universe leading to observable effects on the light elements' abundances which are successfully predicted by BBN (in particular the Helium mass fraction). Thus, extra relativistic degrees of freedom in a particle physics model must not spoil BBN predictions. Deviations from N SM eff are written as where ρ DS = ρ D + ρ S + ρ φ . Here ρ D and ρ S are mass densities and are given by ρ i = m i n i (i = D, S). The energy densities of the massless particles are where g γ = 2. For the benchmarks of table 1, ∆N BBN eff ranges from O(10 −4 ) to O(10 −2 ) which is far too small and thus consistent with N SM eff . We emphasize here that the reason for this is having a particle number density in the dark sector that never traces its equilibrium distribution owing to the small coupling between the two sectors. Since at BBN the only particles that are still relativistic are φ and S, one can argue that increasing the couplings between D and (φ, S) could increase the number densities of the latter causing a conflict with BBN. This, however, reduces the relic density of D. To bring it back up to its experimental value, an increase in δ is required, which is constrained by several experiments as shown in figure 5.
Near recombination, all particles have decoupled from the dark sector bath and φ is the only relativistic degree of freedom in the dark sector. Thus, the contribution to N eff following the decay S → φφ is given by We note here that an important property of φ are sufficiently large self-interactions. This is achieved via a four-point contact interaction φφ → φφ ∝ λ 2 φ and via the exchange of S (in both s and t channels), where the latter is proportional to κ 2 φS , see eq. (2.3). Selfinteractions prevent φ from free-streaming which can cause observable effects on the CMB power spectrum [60,61]. An example in the SM are neutrinos which free-stream after their decoupling from photons. Such a free-streaming leads to a phase shift in the CMB and BAO peaks [60,62], and the measurements from Planck are sensitive to any deviation in the evolution of perturbations [62][63][64]. We can estimate the size of the coupling required to achieve efficient self-interactions of φ making it an ideal fluid. We will ignore the diagrams that are proportional to κ 2 φS since they are very small (see table 1). The smallness of this coupling ensures that S is long-lived and decays just prior to recombination. Thus, the only diagram remaining is the four-point contact interaction. The interaction rate is given by where x = n φ /n eq φ . Comparing it to the Hubble rate, we can derive a lower bound on λ φ such that where g eff = g v eff + g h eff ξ 4 . For T γ ∼ 1 eV, ξ ∼ 0.5 and x ∼ 10 −2 , we obtain λ φ 10 −12 . Thus even for such small values of the self-coupling one can achieve efficient self-interactions to prevent the free-streaming of φ.

Reheating of the dark sector bath prior to recombination
Contributions to ∆N eff at CMB have been considered before in the literature, but only in two contexts: (1) the visible and hidden sectors were initially in thermal contact and decouple later at a temperature T d causing the temperature of the hidden sector to vary as [65] and (2) the sectors were not in thermal equilibrium but eventually become thermalized [66][67][68] where a hidden sector particle decays to neutrinos after it becomes non-relativistic [6,8,69].
In the model discussed here, the particles in the dark sector are produced exclusively by SM processes. Furthermore, the massive particle S and the relativistic species φ in the dark sector are produced in the right amount so as to satisfy BBN constraints while contributing effectively to ∆N CMB eff which can be obtained entirely due to contributions from the dark sector and without participation of the neutrino sector.
After decoupling of the dark sector species from the visible sector, the main coupled set of Boltzmann equations that we need to solve are where the Hubble parameter is given by Here, m Pl = 1.22 × 10 19 GeV is the Planck mass and the decay width of S → φφ is given by Adding the Boltzmann equation for neutrinos (and for electrons) does not have any significant impact on the analysis. We integrate the equations from the temperature at which the last -14 -JCAP04(2022)042 In the top panels of figure 6, we exhibit the change in the energy densities as a function of time. One can see that the energy density of φ starts to increase as S begins to decay. Eventually ρ S drops to zero and ρ φ becomes closer to ρ γ which causes an increase in ∆N eff according to eq. (6.5). In the bottom panels, we show the variation in ∆N eff for four choices of the coupling λ φS (left) and κ φS (right). Here we see that ∆N CMB eff can be achieved in the required range of 0.2-0.5 for the chosen parameter set. A smaller λ φS means the reaction SS → φφ becomes less efficient leading to more S particles available to decay to φ just before recombination which explains the increase in ∆N CMB eff for smaller λ φS . Furthermore, a smaller κ φS leads to the same result, i.e., ∆N CMB eff increases as the decay of S happens closer to recombination.
We present in figure 7 an exclusion limit in the two parameters y S and λ φS for benchmark (b) in the left panel and benchmark (c) in the right panel. Similar limits can be realized for the other benchmarks. Here one finds that the parameter space for each (m D , m γ ) is constrained by the DM relic density and the requirement on early and late kinetic decoupling  Having extra relativistic degrees of freedom from a dark sector (i.e., a dark radiation) will have observable effects on the CMB power spectrum. The cosmological background is blind to the nature of the relativistic species when it comes to its effect on H 0 . Both free-streaming and non-free-streaming dark radiation cause an increase in the expansion rate of the universe prior to matter-radiation equality. This additional radiation pushes more high-modes to enter the horizon during radiation-domination [70] and since radiation-matter equality is required to occur at a fixed z [61], this creates a degeneracy between extra radiation and the matter density. As a result, the DM relic density is expected to grow. Furthermore, changes in the comoving sound horizons at CMB last scattering (r * ) and baryon drag (r d ) are expected. Because of the stringent constraints from CMB on r * /D M (which controls the angular scale of the CMB peaks), a change in the angular diameter distance D M should occur to keep that ratio fixed. Also, extra radiation is degenerate with the free electron fraction which is correlated with the Helium abundance that is successfully predicted by BBN. All of this shows how the different cosmological parameters are tightly related and to know the extent the additional particles S and φ and their dynamics have on these parameters and especially on H 0 would require a more extensive analysis to fit the CMB data.

Conclusion
The discrepancy between the high redshift (z > 1000) and the low redshift (z < 1) measurements of the Hubble parameter, which now stands at 5σ, is an important clue to the model underlying the expansion of the universe. At early times, the Hubble parameter is extracted based on the ΛCDM model from measurements of the acoustic scale, which is precisely determined by Planck. Local direct measurements of the Hubble parameter H 0 have shown that this value is larger than the one calculated at early times, a tension that has recently grown to 5σ. In order to help alleviate the tension, we propose a model which can deliver JCAP04(2022)042 extra relativistic degrees of freedom at CMB times while respecting BBN bounds on ∆N eff . Our analysis is based on a two-sector model, one visible and the other hidden, where we assume a negligible particle abundance of the hidden sector particles in the early universe, and where the hidden sector particles are generated by the out-of-equilibrium annihilation of SM particles into particles in the hidden sector. Because of that, the two sectors reside in heat baths at different temperatures with the hidden sector at a lower temperature than the visible sector. Thus our analysis uses a set of coupled Boltzmann equations which depend on temperatures of both the visible and the hidden sectors and involves correlated temperature evolution of both sectors. In this set up we show that for a range of kinetic mixing and dark photon mass consistent with experimental limits, the visible and the dark sectors do not thermalize which implies that the dark sector particles never reach their equilibrium number densities. However, their phase space distributions can still be described by a Maxwell-Boltzmann distribution since they achieve chemical and kinetic equilibrium during the dark sector thermal history. The fact that the dark particles, especially the relativistic particle φ, never reach their equilibrium distribution allows to evade the constraint on ∆N BBN eff . With a specific choice of couplings, late decays of S → φφ can contribute the right amount to ∆N CMB eff thus resolving the Hubble tension. The model we present is cosmologically consistent since the formalism allows for correlated evolution of the temperatures of the hidden and the visible sectors which are not in thermal equilibrium. In fact it is the lack of thermal equilibrium between the hidden and visible sectors which forces consistency with the BBN constraint while allowing the enhancement of N eff at CMB time. However, as noted earlier, while ∆N eff at recombination time would weaken the Hubble tension, a better understanding of how this affects the CMB power spectrum requires a fit of the cosmological parameters to the CMB data.

Further details of the analysis
In this appendix we give further details of the analysis presented in the main body of the paper. In appendix A we give the formulae for the effective energy and entropy degrees of freedom, in appendix B we give details of mixings of the two U(1) sectors and in appendix C we give the Boltzmann equations with visible and hidden sector temperatures.

A Degrees of freedom
Here we focus on the hidden sector energy effective energy degrees of freedom g h eff in eq. (3.4) and on the hidden sector entropy effective degrees of freedom h h eff in eq. (3.1). They include degrees of freedom for the dark photon, the dark fermion and the scalars S and φ so that

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where g φ = g S = h φ = h S = 1 and at temperature T h , g eff and h eff for the particles γ and D are given by [71] g γ eff = 45 Here x γ and x D are defined so that x γ = m γ /T h and x D = m D /T h . The limit x γ → 0 gives g γ eff = h γ eff → 3 and the limit x D → 0 gives g D eff = h D eff → 4. Further details of the model are given in appendix B.

B Model details
In this appendix we give further details of the extended model with an additional U(1) X gauge group factor beyond the SM gauge group. In addition to the electroweak SM gauge fields A µ a (a = 1-3) of SU(2) L and B µ of U(1) Y , we have a U(1) X gauge field C µ . The mixing of the gauge fields in the neutral sector leads to a 3 × 3 mass square matrix whose diagonalization leads to the photon, the Z boson and a new massive neutral field which we label as the dark photon. Here we display the form of the couplings that enter ∆L int . Thus the couplings g D Z , g D γ and g D γ that appear in ∆L int (see eq. (2.4)) are given by where the matrix R is given by eq. (23) of [25]. It involves three Euler angles (θ, φ, ψ) which diagonalize the Stueckelberg mass matrix such that R T M 2 R = diag(m 2 γ , m 2 Z , 0), where M 2 is defined by eq. (21) of [25]. The angles φ, θ, ψ are defined as where s δ = sinh δ and c δ = cosh δ.

C Boltzmann equations with visible and hidden sector temperatures
Since, in general, the visible and the hidden sector reside in different temperature baths, the Boltzmann equations for the yields Y D , Y γ , Y φ and Y S involve these two temperatures and are given by The equilibrium yield is given by where n i is the number density of a SM particle species i, g i is its number of degrees of freedom, and m i its mass. Further, K 2 is the modified Bessel function of the second kind and degree two and the J functions in eq. (3.7) are given by We note in passing that in eq. (C.3) the processes iī → γ γ, γ Z, γ γ also contribute. However, their contributions are relatively small compared to those of iī → γ .