Alleviating the $H_0$ and $σ_8$ anomalies with a decaying dark matter model
Kanhaiya L. Pandey Tanvi Karwal Subinoy Das

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    ϵ#1Eq. (<ref>)#1Eqs. (<ref>)ln#1ln(#1)#1ln^2(#1)#1Li_2(#1) TeV#1section <ref>
        [a]International Centre for Theoretical Sciences, 
        
        Survey No. 151, Shivakote, Hesaraghatta Hobli, Bengaluru, Karnataka 560089, India[b]Department of Physics and Astronomy, Johns Hopkins University, 
        
        3400 N. Charles St., Baltimore, Maryland – 21218, United States[c]Indian Institute of Astrophysics, 
        
        Sarjapura Road, 2nd Block Koramangala, Bengaluru, Karnataka – 560034, Indiaa]Kanhaiya L. Pandey,kanhaiya.pandey@icts.res.inb]Tanvi Karwaltkarwal@jhu.educ]and Subinoy Dassubinoy@iiap.res.in
        The Hubble tension between the ΛCDM-model-dependent prediction of the current expansion rate H_0 using Planck data and direct, model-independent measurements in the local universe from the SH0ES collaboration disagree at >3.5σ. 
        Moreover, there exists a milder ∼ 2σ tension between similar predictions for the amplitude S_8 of matter fluctuations and its measurement in the local universe. 
        As explanations relying on unresolved systematics have not been found, theorists have been exploring explanations for these anomalies that modify the cosmological model, altering early-universe-based predictions for these parameters. 
        However, new cosmological models that attempt to resolve one tension often worsen the other. 
        In this paper, we investigate a decaying dark matter (DDM) model as a solution to both tensions simultaneously. 
        Here, a fraction of dark matter density decays into dark radiation.
        The decay rate Γ is proportional to the Hubble rate H through the constant α_ dr, the only additional parameter of this model. 
        Then, this model deviates most from ΛCDM in the early universe, with α_ dr being positively correlated with H_0 and negatively with S_8. 
        Hence, increasing α_ dr (and allowing dark matter to decay in this way) can then diminish both tensions simultaneously. 
        When only considering Planck CMB data and the local SH0ES prior on H_0,
        ∼ 1% dark matter decays,
        decreasing the S_8 tension to 0.3σ 
        and increasing the best-fit H_0 by 1.6 km/s/Mpc.
        However, the addition of intermediate-redshift data (the JLA supernova dataset and baryon acoustic oscillation data) weakens the effectiveness of this model. 
        Only ∼ 0.5% of the dark matter decays
        bringing the S_8 tension back up to ∼ 1.5 σ and the increase in the best-fit H_0 down to 0.4 km/s/Mpc. 
        
        
        Alleviating the H_0 and σ_8  anomalies with a decaying dark matter model
            [
            
        ========================================================================
        
        
        
        
        § INTRODUCTION
        
        
        
        The simple ΛCDM concordance model has been immensely successful in describing numerous cosmological observables at different epochs <cit.>. 
        Nontheless, when fit to measurements of the early universe, the ΛCDM model finds results inconsistent with observations of the late universe <cit.>. 
        These include the persistent Hubble tension <cit.> as well as the milder S_8 tension <cit.>. 
        
        The current state-of-the-art experiment Planck which measures the cosmic microwave background (CMB) radiation, assumes a flat ΛCDM model to extract cosmological parameter values and finds the local expansion rate H_0 to be 67.37± 0.54 km/s/Mpc <cit.>. 
        On the other hand, the SH0ES collaboration finds the larger value H_0 = 73.52± 1.62 km/s/Mpc <cit.> through model-independent measurements of the local universe, at ≳ 3.5σ tension with the Planck value. 
        This tension between the early and late universe exists even without Planck CMB data or the SH0ES distance ladder <cit.>. 
        Another direct measurement of H_0 = 72.5^+2.1_-2.3 km/s/Mpc <cit.> from the H0LiCOW collaboration based on lensing time delays is in moderate tension with Planck, 
        while a constraint from Big Bang nucleosynthesis (BBN) combined with baryon acoustic oscillation (BAO) data of H_0 = 66.98 ± 1.18 km/s/Mpc <cit.> is inconsistent with SH0ES. 
        
        There is also evidence of ≳ 2σ tension between the constraints from Planck on the matter density Ω_m and the amplitude σ_8 of matter fluctuations in linear theory and those from local measurements <cit.>. 
        Planck derives S_8 = σ_8 (Ω_m/0.3)^0.5 = 0.832±0.013 whereas local measurements find the smaller values: 
        S_8^ SZ=σ_8(Ω_m/0.27)^0.3=0.78±0.01 from Sunyaev-Zeldovich cluster counts <cit.>, 
        S_8=0.783^+0.021_-0.025 from DES  <cit.>
        and S_8=0.745 ± 0.039 from KiDS-450 <cit.> weak-lensing surveys. 
        The CFHTLenS weak-lensing survey also finds support for disagreement with Planck CMB predictions <cit.>. 
        
        Although systematic causes for these discrepancies cannot entirely be ruled out, numerous potential systematics have been investigated and exonerated over the years while the tensions have persisted and worsened <cit.>. 
        Hence, we must consider the alternative - that the model-dependent results from the early universe are inconsistent with the model-independent measurements of the late universe because the 
        ΛCDM model of cosmology is incorrect. 
        
        There have been numerous attempts at resolving these discrepancies via non-standard cosmological models 
        <cit.>, however, most such attempts at solving the Hubble tension worsen the S_8 tension and vice-versa. 
        Solutions to the Hubble tension either reduce the size r_s of the sound horizon with an early-universe modification <cit.>, 
        or increase the angular diameter distance D_A to the CMB with new physics in the post-recombination universe. 
        Then, to keep the locations of the peaks in the CMB fixed, H_0 increases, diminishing the tension. 
        On the other hand, a solution to the S_8 tension would require either late-universe physics that leads to a suppression of the linear matter power spectrum or a decrease in the CMB-predicted value of Ω_m. 
        
        In this paper, we tackle both requirements with a decaying dark matter (DDM) model which has a decay rate proportional to the Hubble rate. 
        In this scenario, a fraction of dark matter density decays into dark radiation per Hubble time <cit.>, 
        with the effect being amplified close to the onset of matter domination. 
        This leads to an increase in the expansion rate relative to ΛCDM around recombination, resulting in a decrease in r_s. 
        Fits to the CMB then predict a higher H_0, alleviating the Hubble tension. 
        These fits also predict smaller Ω_m, leading to smaller S_8.
        This model can hence simultaneously diminish both the Hubble and the S_8 tensions. 
        Testing against various cosmological datasets, we find that this DDM model can provide a  better fit to some datasets and simultaneously alleviate the two aforementioned tensions, but not fully resolve them. 
        We also find that at most, a fraction f_ dm≲ 0.003 of dark matter can decay into dark radiation in the light of recent Planck, supernova and BAO data, and an external prior on H_0 from R18. 
        This paper is organised as follows.
        A brief description of the model is given in Section <ref>, along with its effect on observables. 
        In Section <ref>, we provide a detailed description of our analysis. 
        Our results are presented in Section <ref> and discussed in Section <ref> where we also conclude. 
        
        
        
        
        § DECAYING DARK MATTER MODEL
        
        
        Motivation for exploring a DDM resolution to the Hubble tension comes from considering the effective radiation degrees of freedom <cit.>. 
        Increasing the amount of radiation in the early universe such that Δ∼ 0.4-1 has been shown to diminish the Hubble tension  <cit.>. 
        This extra radiation must be `dark' as the presence of an extra photon-like component is strongly constrained by both BBN and the CMB <cit.>. 
        A fourth, massive, sterile neutrino could provide such extra dark radiation however, the existence of such a particle is constrained by oscillation experiments <cit.>. 
        
        The scenario explored here follows the model proposed by Ref. <cit.>. 
        It involves dark radiation interacting within the dark sector, in particular, 
        a particle (beyond the framework of the standard model) decaying into an extra dark  radiation component. 
        All the dark radiation in this scenario is a product of dark matter decay and forms a small fraction of the total dark matter density. 
        The decay rate Γ determines this fraction f_ dm of dark matter energy density that decays into dark radiation. 
        This fraction remains nearly constant over time after matter-radiation equality. 
        If f_ dm is large, it can alter the expansion rate as shown in Fig. <ref>, which we demonstrate leads to a higher predicted H_0.
        Moreover, the decay naturally reduces the amount of dark matter in galaxies and clusters leading to smaller predicted values of S_8.
        A brief description of the background dynamics of the model, contribution of dark radiation to N_ eff and effect on observables is discussed in the following subsections. 
        
        
        
         §.§ Background dynamics
        
        
        A general coupling between dark matter and dark radiation can be described by the energy balance equations <cit.>
            ρ̇_ dm + 3 Hρ_ dm   =    -Q 
            ρ̇_ dr + 3 H (1+w_ dr) ρ_ dr   =    Q
        
        where ρ_ dm and ρ_ dr are the dark matter and dark radiation energy densities and H = ̇ȧ/a is the Hubble rate, where a is the scale factor and overdots denote derivatives with respect to conformal time. 
        We also assume dark radiation has an equation of state w_ dr=P_ dr/ρ_ dr = 1/3. 
        A positive rate of energy transfer Q denotes the direction of energy transfer from dark matter to dark radiation. Non-zero values of Q imply that dark matter no longer redshifts exactly as a^-3 nor dark radiation as a^-4. 
        We adopt the covariant form of the energy-momentum transfer 4-vector introduced in <cit.>
            Q = Γρ_ dm,
        
        where the exact form of the interaction rate Γ depends on the details of the particle physics of the decay process. 
        
        Many forms of Γ have been studied in the literature <cit.>. 
        Here, we explore the simple case where Γ = α H, where α is a constant and H is the Hubble rate. 
        Although we do not model the particle physics resulting in Γ∝ H, we refer the reader to two fundamental particle physics motivations for such an interaction. 
        As discussed in Section 5 of Ref. <cit.>, if dark matter is a coherently oscillating scalar field and decays into light fermions similar to the reheating mechanism, it can give rise to our DDM set up.
        It may also arise in the model proposed by Ref. <cit.>, where a fraction of dark matter converts to dark radiation through late kinetic decoupling and Sommerfeld-enhanced dark matter annihilation. 
        The mass ranges for dark matter particles in each of these models differ greatly. 
        As our analysis here is phenomenological, our constraints are independent of the mass of the dark matter particle undergoing decay. 
        Interpreting these results in the framework of a particular fundamental model can translate our constraints to particle mass and interaction cross-section. 
        
        For Γ = α H, the background evolution is readily solved  
        
            ρ_ dm   =   ρ_ dm,0 a^-(3+α_ dr)
            ρ_ dr   =   ρ_ dr,0 a^-3(1+w_ dr) + α_ dr/α_ dr-3w_ drρ_ dm,0 a^-3 (a^-3w_ dr-a^-α_ dr) 
                ,
        
        where the subscript 0 denotes values today. 
        For w_ dr = 1/3, eq:rho_dr_bg can be further simplified to
        
            ρ_ dr = β a^-4 + α_ dr/1-α_ drρ_ dm,0 a^-(3+α_ dr) ,
        
        where β is a constant. 
        The first term in eq:rho_dr_bg_simple behaves like a standard radiation density while the second behaves like a fluid with an equation of state α_ dr/3. 
        For weak couplings between dark matter and dark radiation, α_ dr≪ 1, which leads to β∼ρ_ dr,0.
        Assuming no dark radiation exists initially, we set β = 0 and only retain the second term in our analysis. 
        With this assumption, we obtain the fraction f_ dm of dark matter that decays into dark radiation 
        
            f_ dm = 
                ρ_ dr/ρ_ dm→α_ dr/3w_ dr-α_ dr = α_ dr/1-α_ dr≃α_ dr
                .
        
        Therefore, f_ dm is constant over time and the density of dark radiation ρ_ dr≃α_ drρ_ dm. Our model is then parameterised by a single parameter α_ dr. 
        
        For a detailed description of the perturbations in our model, we refer the reader to Ref. <cit.>. 
        
        
        
        
         §.§ Calculation of Δ N_ eff
        
        
        In standard cosmology, the energy density ρ_ rad of relativistic species in terms of the photon energy density ρ_γ is
        
            ρ_ r = [ 1+ 7/8 N_ eff( 4/11)^4/3] ρ_γ
                .
        
        This includes standard model (SM) neutrinos (for which N_ν,  eff = 3.046) 
        <cit.>, and characterizes  any  free-streaming  radiation  beyond  the SM  expectation. 
        Then, any departure from the SM can be accounted for through Δ N_ eff, where N_ eff = N_ν, eff+Δ N_ eff. 
        In our case, Δ N_ eff,dr can be expressed in terms of α_ dr<cit.> as  
        
            7/8Δ N_ eff,dr( 4/11)^4/3ρ_γ,0/a^4 = 1/1-α_ drρ_ dm,0 a^-(3+α_ dr) - ρ_ dm,0 a^-3
                ,
        
        making it a derived parameter in our analysis. 
        
        
        
        
         §.§ Effects on observables
        
        
        
        To understand the effect of our model on observables, 
        Fig. <ref> shows how decaying dark matter affects the CMB TT power spectrum, the matter power spectrum and the expansion rate. 
        These plots were produced by fixing all  parameters except Ω_ dm,0 to their best-fit  values, with Ω_Λ,0 = 1 - Ω_ m,0 preserving flatness. 
        The early universe fixes Ω_ dm at early times,
        that is, if dark matter decays, Ω_ dm,0 should decrease as its decay rate increases. 
        This is also reflected in our posteriors, most evident in Fig. <ref> from the negative correlation of Ω_ dm,0 and α_ dr.
        Therefore, to better represent the cosmology in our posteriors, we fix matter-radiation equality at a_ eq 
        and hence Ω_ dm at early times and determine Ω_ dm,0 as a function of α_ dr. 
        Then, at a_ eq, equating the matter and radiation contents of the Universe, 
        
            Ω_ dm,0 a_ eq^-(3+α_ dr) +
                Ω_ b,0 a_ eq^-3 = 
                Ω_ r,0 a_ eq^-4 +
                α_ dr/1-α_ drΩ_ dm,0 a_ eq^-(3+α_ dr) 
                ,
        
        where we have substituted ρ_ dr using Eq. (<ref>). 
        Then for the dark sector today, we have 
        
            Ω_ dm,0    =    a_ eq^-1Ω_ r - Ω_ b/a_ eq^-α_ dr( 1-α_ dr/1-2α_ dr) 
                , 
            Ω_Λ,0    =    
                1 - Ω_ dm,0 - Ω_ b.
        
        
        
        The main effect of the DDM model is an alteration of the expansion history of the Universe, as seen from the bottom-right panel of Fig. <ref>. 
        At very early times, the Universe is radiation-dominated and the DDM universe resembles the  universe as dark radiation is always subdominant.  
        As we fix Ω_ dm at a_ eq, the pre-equality DDM universe has more dark matter than , increasing its expansion rate with H_ DDM/H_Λ CDM peaking at a_ eq. 
        After a_ eq, the DDM universe has less dark matter than  
        as dark matter decays into dark radiation which quickly redshifts away
        and H_ DDM decreases. 
        The two expansion rates are equal when Ω_ dm + Ω_ dr = Ω_ cdm which occurs at
        
            a_ cross = 1/(1- 2 α_ dr)^(1/α_ dr) a_ eq
                  ,
        
        ignoring minor variations due to Ω_Λ, 0. 
        The DDM expansion rate continues to decrease past this point until Λ becomes important and begins to dominate. 
        The Universe again resembles a  universe and H_ DDM = H_Λ CDM at a=1 as set by our choice of Ω_Λ,0. 
        Hence, the early, pre-recombination universe expands faster than  and the post-recombination universe slower. 
        This effect is more pronounced as α_ dr increases. 
        
        The effect of DDM on the CMB is shown in the top panel of Fig. <ref>.
        The increase in the pre-recombination expansion rate and decrease in the post-recombination expansion rate lead to a smaller r_s and larger angular distance D_A to the CMB respectively. 
        The inferred angular size of the sound horizon θ_* = r_s / D_A is then smaller and the CMB peaks are shifted to smaller scales or larger multipoles ℓ as α_ dr increases.
        These can be shifted back into agreement with data by increasing the Hubble constant, relieving the tension. 
        Other changes to the CMB include the following suppressions and enhancements of power. 
        An increase in the amount of dark matter in the pre-equality universe suppresses power in CMB peaks 
        which entered the horizon during radiation domination, 
        as the enhancement due to acoustic driving is reduced.
        On the other hand, as there is less dark matter post-equality, 
        the first CMB peak receives a boost in power, 
        having entered the horizon during matter-domination.
        Finally, Ω_Λ,0 is larger for larger α_ dr and the late universe is Λ-dominated sooner. 
        The late ISW effect hence enhances power in low multipoles. 
        Small shifts in the other  parameters can absorb these changes in the CMB. 
        
        Finally, the matter power spectrum, shown in the bottom-left panel of Fig. <ref> is effectively inflected about k_ cross≃ a_ crossH where a_ cross is given by Eq. (<ref>). 
        At scales larger than k_ cross, there is more power in a DDM universe for larger α_ dr, and less at scales smaller than k_ cross. 
        Relating to the bottom-right plot in Fig. <ref> of H_ DDM/H_Λ CDM, 
        we find that the change to the expansion rate dictates the power in the matter power spectrum. 
        At early times, when the DDM universe is expanding faster than the  universe, clustering and therefore power in the matter power spectrum is suppressed. 
        The opposite is true after a_ cross, or below k_ cross - the DDM universe expands slower than  and power in the matter power spectrum is boosted. 
        These trends are enhanced for larger values of α_ dr, as expected. 
        The BAO peaks at k > k_ cross are also slightly shifted to smaller scales, same as the trend in r_s in the CMB, and sharper as the pre-equality universe has more dark matter. 
        Altogether, as power is suppressed at scales ∼ 8Mpc, the DDM model helps alleviate the S_8 tension. 
        
        
        
        
        § METHODOLOGY
        
        
        To investigate this DDM model, we use a modified version of the publicly available Boltzmann code <cit.>. 
        The modified version is based on the dynamics described in  <cit.>; 
        We vary the 6 standard ΛCDM parameters: 
        the baryon density ω_ b today,
        the dark matter density ω_ dm today, 
        the angular size θ_ MC of the sound horizon at recombination, 
        the optical depth τ to reionisation, 
        the scalar spectral index n_ s,
        and the amplitude A_ s of the primordial power spectrum. 
        To this, we add the DDM parameter α_ dr. 
        We then use the publicly available Markov chain Monte Carlo code <cit.> to explore our 7-dimensional parameter space with the following assumptions. 
        We assume a flat universe with Ω_k = 0, 
        a constant dark energy equation of state, w_ de=-1 and fix the running of the scalar spectral index dn_ s/d lnk = 0. 
        We adopt standard values for the sum of neutrino masses Σ m_ν = 0.06 eV and the SM  N_ν, eff = 3.046. 
        The entire DDM model is described by the sole parameter α_ dr. 
        The dark radiation energy density Ω_ dr and Δ N_ eff,dr are derived parameters which can be expressed in terms of α_ dr. 
        Table <ref> shows the priors for the 7 varied parameters. 
        
        We fit to various early and late-universe data sets in certain combinations. Our data include: 
        
          *  Planck :  The CMB temperature and polarization angular power spectra  (high-ℓ TT + low-ℓ TEB) released by Planck 2015 <cit.>
          *  JLA :  Luminosity distance of supernovae Type Ia coming from `joint light-curve analysis' using SNLS (Supernova Legacy Survey) and SDSS (Sloan Digital Sky Survey) catalogs <cit.>
          *  BAO :  The `Baryonic Acoustic Oscillation' data from DR12-BAO <cit.>, SDSS-6DF <cit.> and SDSS-MGS <cit.>
          *  R18 :  An external gaussian prior on H_0 = 73.52±1.62 km/s/Mpc <cit.>.
        
        We fit to the combinations Planck, Planck+R18
        and Planck+JLA+BAO+R18.
        We adhere to the Gelman-Rubin convergence criteria of R-1<0.01 and discard the first 30% of our chains as burn-in. 
        
        
        
        
        § RESULTS
        
        
        Figures <ref>-<ref> compare our posteriors for the ΛCDM (blue) and DDM (red) models for various data sets. 
        Along with posteriors for Ω_b h^2, Ω_dm h^2 and α_ dr, we also show posteriors for the derived parameters 
        H_0 and S_8. 
        The green bands represent local measurements of H_0 and S_8.
        From these figures, the correlation of H_0 with α_ dr and the anticorrelation of S_8 with α_ dr are apparent. 
        An increase in α_ dr results in a greater H_0 and a smaller S_8. 
        This is the exact effect required to solve the H_0 and S_8 tensions simultaneously.  
        These correlations are most evident in the posteriors of Planck+R18 in Fig. <ref>.
        The inclusion of JLA and BAO data diminishes these correlations as seen in Fig. <ref>. 
        
        As seen from Fig. <ref>, Planck data places an upper bound on α_ dr (≤0.003). 
        However, the addition of an external prior on H_0 from R18 leads to a small preference for non-zero α_ dr (≈ 0.005±0.003) at the ∼ 1.5σ level. 
        With Planck+R18, the Hubble tension is reduced to  ∼ 1.5 σ and the S_8 tension to ∼ 0.3 σ. 
        The addition of JLA and BAO data weakens these resolutions, as seen form Table <ref>. 
        For Planck+JLA+BAO+R18, the H_0 and S_8 tensions remain at ∼ 2.5 σ and ∼ 1.5 σ levels respectively.
        For all dataset combinations explored, we remain consistent with ΛCDM within 1σ for all ΛCDM parameters. 
        
        Table <ref> shows the best-fit χ^2 values for the ΛCDM and DDM models. 
        The DDM model leads to a slight improvement in fit, largely due to fitting the R18 measurement better than ΛCDM. 
        
        
        
        
        In Fig. <ref>, we show how the decay parameter α_ dr improves the H_0 and S_8 tensions. 
        In the DDM scenario (red contours), α_ dr increases towards the bottom right. 
        Considering just Planck+R18, the external prior on H_0 pushes the decay rate of dark matter to be ∼ 1% of the Hubble rate. 
        These large values of α_ dr alter expansion history enough for the model to predict a larger H_0. 
        As the early universe fixes the density of dark matter, the DDM model also leads to smaller Ω_dm and therefore Ω_m, lowering the predicted S_8. 
        Without data at intermediate redshifts, such large changes in cosmology are permitted. 
        As seen from the left panel of Fig. <ref>, within the scope allowed by Planck+R18, the DDM contours intersect the 1σ local-measurement square (green). 
        For these datasets, while the 1σΛCDM contour (blue) intersects the 1σ local measurement of S_8, the 2σΛCDM contour is beyond the 1σ local H_0 measurement. 
        Therefore, the DDM model diminishes the H_0 and S_8 tensions. 
        
        Including data at intermediate redshifts, primarily supernova data from JLA, the tensions remain unresolved. 
        As seen from the right panel of Fig. <ref>, while DDM nearly resolves the S_8 tension, the Hubble tension still exists. 
        The combined datasets Planck+R18+JLA+BAO do not permit large deviations from ΛCDM cosmology, 
        which is in agreement with  other studies where it has been found that supernovae and BAO measurements generically prefer standard ΛCDM <cit.>.
        Moreover, smaller values of α_ dr are permitted, with the decay rate of dark matter constrained to be ∼ 0.5% of the Hubble rate. 
        
        
        
        
        
        § DISCUSSION AND CONCLUSIONS
        
        
        While the ΛCDM model of cosmology fits numerous datasets well, its predictions based on the early and late universe disagree <cit.>. 
        The current expansion rate H_0 is underpredicted by ΛCDM when fit to the early universe <cit.>. 
        Despite this, measurements of the late universe are in agreement with a ΛCDM expansion history, but with different parameter values <cit.>. 
        This Hubble tension has persisted and worsened over the years and no systematic cause has yet been found <cit.>.
        Moreover, ΛCDM overpredicts the amplitude of matter fluctuations S_8 relative to direct measurements in the late universe <cit.>. 
        Although this is a milder tension, combined, these tensions might hint new physics beyond the standard model of cosmology. 
        
        Theories that address each tension often worsen the other. 
        In this paper, we explored a decaying dark matter model than can simultaneously improve both tensions.
        We considered dark matter that decays into dark radiation, parameterised by a single new parameter <cit.>.
        The DDM model increases the expansion rate relative to ΛCDM, with the largest effect being close to recombination. 
        This leads to a reduced sound horizon, to compensate for which H_0 increases, alleviating the Hubble tension. 
        The DDM model also reduces the dark matter density in the late universe, suppressing structure formation and lowering the predicted value of S_8. Hence, it offers solutions to both tensions simultaneously. 
        
        Considering just data from the early and current universes, that is the Planck+R18 combination, we find that the Hubble tension is reduced below the 1.5σ level and the S_8 tension below 0.5σ. 
        DDM not only significantly diminishes both tensions, but also provides a slightly better fit to these datasets with Δχ^2_ tot = -1.9, as seen from Table <ref>. 
        However, including measurements of the Universe at intermediate redshifts with Planck+R18+JLA+BAO, we find that the DDM model is strongly constrained and the H_0 and S_8 tensions persist at the ∼ 2.5 σ and ∼ 1.5 σ levels respectively. 
        The DDM model alters expansion history relative to ΛCDM all through matter domination, as shown in Fig. <ref>. 
        As found by numerous models that aim to resolve the Hubble tension through modifications of the late universe, late-universe datasets such as JLA and BAO strongly constrain expansion history and keep such models from fully resolving the Hubble tension <cit.>. 
        In this case, the “new physics” we add is present not only in the early universe where it has maximal effect, but throughout cosmic history. 
        Its presence in the late universe would spoil the fits to JLA and BAO, keeping it from diminishing the H_0 and S_8 tensions further. 
        This can also be seen from the tilt of the H_0 and S_8 vs α_ dr contours in Figs. <ref> and <ref>. 
        Without JLA and BAO data, α_ dr has a stronger correlation with H_0 and anticorrelation with S_8 in Fig. <ref>. 
        This relaxes when intermediate-redshift datasets are added as in Fig. <ref>, implying that the addition of JLA and BAO data weakens the effectiveness of DDM at resolving both tensions. 
        
        Numerous models of dark matter interacting within the dark sector have been explored <cit.>. 
        In these models, the interaction is effective only up to a certain scale and negligible at larger scales. 
        This produces a cut-off-like feature in the matter power spectrum at small scales, keeping the power in scales ∼ 8Mpc the same as in ΛCDM. 
        For decaying dark matter with a constant time-independent decay rate <cit.>, the constraints are driven by the change to the late integrated Sachs-Wolfe (ISW) effect on the large-scale CMB data. 
        To be consistent with it, the dark matter must decay very slowly which only allows a slight improvement in the S_8 tension. 
        The DDM model considered here circumvents this by having a smaller decay rate in the early universe around decoupling which then increases with time. 
        Models which introduce a time-dependent dark-matter drag force due to dark radiation which also shut-off at late times <cit.> have similar effects.
         
        
        The S_8 and Hubble tensions are intriguing results in cosmology. 
        They require careful investigation whether from a systematic or a new-physics perspective. 
        Future data may shed further light on whether these anomalies are hints of physics beyond the standard model of cosmology after all. 
        
        
        
        
        
        
        
        We thank  Marc Kamionkowski and Vivian Poulin for their comments on the initial results of this work. 
        SD and TK acknowledge the IUSSTF-JC-009-2016 award from the Indo-US Science & Technology Forum which supported the project and facilitated the authors visiting each other. 
        TK also acknowledges support from the 2018 Johns Hopkins Discovery Award. 
        
        
        
        
        
        utphys
