Dark matter solution to the $H_{0}$ and $S_{8}$ tensions, and the integrated Sachs-Wolfe void anomaly

We consider a phenomenological model of dark matter with an equation-of-state $w$ that is negative and changing at late times. We show this couples the $H_{0}$ and $S_{8}$ tensions, alleviating them both simultaneously, reducing the $H_{0}$ tension from $\sim5\sigma$ to $\sim3\sigma$ and the $S_{8}$ tension from $\sim3\sigma$ to $\sim1\sigma$. Furthermore, the model provides an explanation for the anomalously large integrated Sachs-Wolfe (ISW) effect from cosmic voids, a unique consequence of the changing and negative equation-of-state. Observations of high ISW from cosmic voids may therefore be evidence that dark matter plays a significant role in both the $H_{0}$ and $S_{8}$ tensions. We predict the ISW from cosmic voids to be a factor of up to $\sim2$ greater in this model than what is expected from the standard model $\Lambda$CDM. These results extend to other degenerate models of dark matter, such as unified or interacting dark matter and dark energy models.


I. INTRODUCTION
The emergence of strong tensions in the constraints of the Hubble constant H 0 and S 8 parameters, between early [1] and late universe-based observations [2][3][4][5][6], have placed intense scrutiny on observations, methods and the assumptions of the standard model of cosmology ΛCDM (see [7][8][9][10][11][12][13][14] for reviews on the Hubble tension).One of these assumptions is the presence of cold dark matter; a yet-to-be-detected massive particle that is the dominant source of gravitation in the universe.Dark matter's interactions with standard model particles and forces is known to be very weak and the specific properties of dark matter are often assumed to manifest only on small cosmological scales.
However, with no direct observations of dark matter, there is little we can presume about its properties, other than its gravitational effects and weak interactions with known particles.Rather than consider an endless list of dark matter model extensions we can instead consider phenomenological scenarios that allow us to determine the general observational implications of a whole family of models, including their role in tensions and anomalies.With this in mind, we explore the implication of a subset of the generalised dark matter model [15], in a spatially flat universe with a cosmological constant (Λ).The equation-of-state (EoS) for dark matter (w dm ) is allowed to be non-zero and evolving at late times but with null speed of sound and viscosity.For consistency with constraints from the early universe [16,17] the EoS is assumed to be effectively zero at early times.We will refer to this model as evolving dark matter (eDM), to distinguish it from other models often abbreviated to WDM (such as warm dark matter), and with the full model with Λ referred to as ΛeDM.
While the H 0 and S 8 tensions are widely discussed in the community, a lesser known anomaly is the observation of larger than expected integrated Sachs-Wolfe [ISW; 18] signal from cosmic voids [19][20][21] (the void-ISW anomaly).This anomaly is strongest for 'photometric' voids, i.e. voids measured from photometric observations of galaxies which are preferentially elongated along the line-of-sight (LOS) [22] and ranging in significance from 2-4σ [19,20,22], while for 'spectroscopic' voids (smaller and not aligned with the LOS) the ISW is larger but to a lesser extent and remains consistent with ΛCDM [23].In contrast, observations of void lensing are either consistent with ΛCDM or lower than expected [24].
The aim of this paper is to establish the role dark matter can play in alleviating tensions in cosmology and in explaining the void-ISW anomaly.The latter has rarely been considered part of the tension discussion, but in this paper we show the ISW is very sensitive to late-time changes in dark matter's EoS and can provide a unique test for a dark matter (or an interacting dark sector) solution to tensions.This paper is organised as follows: (1) we discuss the assumptions and theoretical background of ΛeDM and the implications for the H 0 and S 8 tensions, and the ISW and lensing from cosmic voids; (2) we describe ΛeDM's parameter degeneracies and methods used for parameter inference, (3) we discuss the constraints on ΛeDM parameters using cosmological observations; and (4) we discuss the implications for tensions and anomalies in cosmology.

A. ΛeDM and Tensions
The evolution of the dark matter density ρ dm can be described as a function of the scale factor a and a time-varying EoS w dm (a) defined as where ρ dm,0 is the dark matter density today and a ′ a dummy variable for the integration by the scale factor.This general solution is derived from the assumption of energy conservation and the continuity equation.We consider the following function for the EoS, i.e. a linearly increasing/decreasing EoS where w dm,0 is the current value of the dark matter EoS and a nz the scale factor at which dark matter's EoS becomes nonzero.At early times (i.e. for a < a nz ) constraints from CMB measurements have shown the EoS must be very close to zero (|w dm | ≲ 0.001) [16,17], however, at late times the constraints are more relaxed [17].For this reason we fix the EoS at early times to zero and allow the EoS to be non-zero at late times.However, this means the EoS is non-differentiable at a nz , a detail that should be revisited in future studies of the model.Generalised dark matter models with a cosmological constant, like the model considered here, are equivalent to interacting dark energy models [25], models where dark matter interacts with dark energy.We can therefore think of this model describing a dark matter-dark energy interaction [26], which could occur only at low energies or densities, or through the decay of unstable dark matter particles with very long half-lives (see [27] for the evanescent matter model with similar properties).In both cases the interaction is only relevant at very latetimes and would grow over time, following the behaviour of ΛeDM.In section III A we will show the sensitivity to w dm,0 is much stronger than a nz and that the parameters are strongly degenerate.We break this degeneracy by setting a nz = 0.5, meaning dark matter's EoS only becomes non-zero at z ≤ 1.The total matter density is defined by with the additional term Here f i is the fraction of the matter constituent i to the total matter content today (i.e.f i = ρ i,0 /ρ m,0 ); the subscripts b indicate baryons, ν non-relativistic neutrinos and dm dark matter.Another useful relation is the derivative of Γ(a) with respect to ln a, which is given by In ΛCDM these functions simplify to Γ(a) = 1 and Υ(a) = 0. We modify the background evolution and the linear perturbation equations of dark matter in the cosmological Boltzmann solver CLASS [28], following the above relations and equation 2 of [17].From CLASS we output the power spectrum and growth functions.
To understand the role that ΛeDM may play in the H 0 tension we fix the conditions of the early universe to those inferred by Planck CMB measurements [1].This is carried out by fixing the primordial power spectrum, optical depth and the physical densities of baryons and dark matter to A s = 2.101 × 10 −9 , n s = 0.9649, τ = 0.0544 and Ω b h 2 = 0.02236, respectively.For dark matter this means fixing its value at early times to where Ω init dm is the fractional density of dark matter if the EoS is zero and W 0 = W (a nz ).
In Fig. 1 we fix the parameters of ΛeDM as described above while varying a single parameter -the Hubble constant H 0 .In this plot we show that changing H 0 to values above CMB constraints (of H 0 = 67.27km s −1 Mpc −1 indicated with a solid blue line) requires dark matter's EoS to be negative.A consequence of a high H 0 is a decrease in the power spectrum amplitude and therefore low σ 8 .Since σ 8 ∝ S 8 this naturally couples the two tensions.If H 0 is set to the values observed by SH0ES [29] (of H 0 = 73.3km s −1 Mpc −1 ) we obtain σ 8 ∼ 0.5.

B. Implications for the ISW and lensing
The ISW T ISW is defined as where Φ is the time-derivative of the gravitational potential Φ, c is the speed of light, χ is the transverse comoving distance in a direction η and χ LS is the comoving distance to the last scattering surface.The gravitational potential is related to the density contrast δ by the Poisson equation which in the linear regime simplifies to where Ω m is the total fractional matter density today, D(t) the linear growth function and δ(x) the density contrast today (i.e. at z = 0).Consequently, the timederivative is given by where f (t) is the linear growth rate.The lensing convergence is given by where χ s is the comoving distance to the lensing source.
Both the ISW and lensing are dependent on Γ(t) (see Eqs. 8, 9, 10 and 11), while the ISW is also dependent on Υ(t) (see Eq. 8 and Eq.10).This means the ISW and lensing signals will be different in ΛeDM than in ΛCDM and can be constrained from studies crosscorrelating galaxy surveys with weak lensing and CMB temperature maps.However, cross-correlation studies of the ISW typically have very large errors since the signal is highest on large angular scales and subject to cosmic variance (see [30]).This makes distinguishing the effect of ΛeDM this way rather challenging.Voids on the other hand allow us to improve the signal-to-noise by stacking void-ISW signals.They are extremely sensitive to ΛeDM because they accumulate the departures in Γ(t) and Υ(t) from ΛCDM; an effect which is more relevant for larger voids elongated along the LOS, such as those measured by photometric surveys.
Measurements of the void density profiles are based on observations of galaxies, biased tracers of the underly-TABLE I.We list the void parameters for spectroscopic voids (spec-z) [23,[32][33][34], photometric voids (photo-z) [22,24,35] and the Eridanus voids (combining profiles from voids E-1, E-2, E-3 and E-4) [36].The void parameters are the central redshift zc, the central density contrast δ0, the effective radius R and the measured amplitude of the ISW AISW and lensing Aκ signals with respect to ΛCDM.Note the measured amplitudes are omitted for the Eridanus voids as this corresponds to a singular location on the sky.Furthermore all voids are assumed to be spherical, with the exception of the photo-z voids which are preferentially aligned towards the LOS with a LOS and perpendicular ratio of R ∥ /R ⊥ = 2.6 [35].ing density field.To correct for the galaxy bias, a linear galaxy bias parameter is usually fitted based on clustering analysis.However, if ΛCDM is not the correct model, the inferred bias will be incorrect.By making the approximation that the P (k) for ΛeDM can be related to the P (k) in ΛCDM by an amplitude shift, we can make a σ 8 bias correction to the central void density δ 0 (the central density contrast obtained from ΛCDM) by multiplying In Fig. 2 we show the void ISW and lensing profiles as a function of H 0 .As was shown in Fig 1, an increase in H 0 requires a negative and decreasing EoS for dark matter, resulting in Γ ̸ = 1 and Υ ̸ = 0.This results in a significant increase in the ISW profile and a slight decrease in lensing, fitting well with observational measurements (indicated in the inset plots).For spec-z and photo-z voids the ISW and lensing measurements were made by stacking voids, something which cannot be performed for the Eridanus voids which lies on a singular patch of sky.For this reason and because these voids more closely resemble photo-z voids we will assume that their ISW and lensing profiles follow the relation for photo-z voids.

A. Model Degeneracies
The ΛeDM model is defined with the addition of two free parameters, w dm,0 which defines the dark matter EoS today (i.e. at z = 0) and a nz the scale factor at which the EoS becomes non-zero.In Fig. 3 we show the angular power spectra of the CMB temperature, E and B-modes, and lensing potentials for the ΛeDM model with base parameters (i.e.those in common with ΛCDM) fixed to best-fit ΛCDM P18 constraints.In Fig. 4 we show the power spectra at redshift z = 0 with base parameters fixed to best-fit ΛCDM.In Fig. 3 and 4 the analytical angular and 3D power spectra are shown as a function of w dm,0 varying between −1 and 0.3 with a nz = 0.5 and separately as a function of a nz varying between 0.2 and 0.9 with w dm,0 = −0.2.
We see that smaller w dm,0 and a nz lead to a very similar dampening effect on the power spectra, a shift to the right in the temperature and E-mode angular auto/cross correlation, and an amplification to the Bmode and lensing auto/cross-correlations.At low-ℓ we see a modest increase to the temperature autocorrelation (highlighted in the inset plots), this is caused by larger ISW in these models.These figures illustrate the strong degeneracy between these two parameters and that cosmological probes will be much more sensitive to w dm,0 than a nz .For this reason we fix a nz to 0.5 in our analysis.

B. Parameter Inference
Constraints on cosmological parameters were obtained using the cosmological Markov chain Monte Carlo  (MCMC) software Cobaya [47,48].This was done using a combination of Planck CMB measurements, Pantheon Type Ia supernovae, constraints on the absolute magnitude of Type Ia supernovae, baryonic acoustic oscillations and redshift space distortion measurements.Specifically: (i) Planck CMB measurements of temperature, polarisation auto/cross correlation for high-ℓ (i.e.ℓ > 30), temperature and polarisation auto correlation for low-ℓ (i.e.ℓ ≤ 30) and lensing [49].The joint likelihood for Planck CMB measurements are re-ferred to in the paper as P18.(ii) Pantheon Type Ia supernovae [50] referred to as SN.(iii) SH0ES Cepheid constraints on the Type Ia absolute magnitude [51], referred to as SN+M B .(iv) Baryonic acoustic oscillations and redshift space distortion measurements from 6dF, SDSS DR7 and BOSS [52][53][54], which are referred to as BAO.
The constraints for parameters from ΛCDM and ΛeDM are obtained by sampling the posterior by running an MCMC.In both cases, this means varying the standard ΛCDM parameters, i.e. primordial power spectrum amplitude A s , the spectral tilt n s , the sound horizon θ * , the optical depth τ , baryon density Ω b h 2 and dark matter density Ω dm h 2 .For ΛeDM we also vary the dark matter EoS today w dm,0 .In ΛeDM there is a strong degeneracy between Ω dm and w dm,0 , which makes sampling the posterior extremely difficult.To remove this degeneracy and improve the efficiency of the MCMC sampling, we instead sample the initial dark matter density, i.e.Ω init dm which is related to Ω dm by the relation Ω dm = Ω init dm /W 0 , for ΛCDM the two are equivalent since W 0 = 1.Note, the addition of the Type Ia absolute magnitude from SH0ES means we have to additionally sample M B as a sampled parameter.In Fig. 5 we show the posterior of the sampled ΛeDM parameters, showing that with the exception of w dm,0 the posteriors are generally quite Gaussian.A comparison to ΛCDM shows the constraints on these parameters are close to identical for P18, since the high redshift CMB measurements are unable to constrain the late evolution of dark matter's EoS.To ensure the MCMC chains have converged we make use of Cobaya's inbuilt measure of the R − 1 Gelman-Rubin statistics [55].The chains are assumed to be converged once R − 1 < 0.01.
Defining S 8 for ΛeDM is rather challenging since the general definition S 8 = σ 8 0.3/Ω m in the literature generally assumes the contribution of cold dark matter to Ω m .Assuming that this functional form is really trying to capture an amplitude to the power spectra and define this in relation to the fiducial ΛCDM constraints where Ω m ≃ 0.3 the most sensible definition of S 8 will use Ω init dm since this sets the shape and amplitude of the initial power spectra and using σ 8 which captures ΛeDM's dampening effect.Therefore, we define where and Ω NR ν is the contribution from non-relativistic neutrinos.

A. Constraints on ΛeDM
We constrain ΛeDM parameters (standard ΛCDM parameters plus the current dark matter EoS w dm,0 ) using measurements from Planck CMB (referred to as P18, [49,58]), Type Ia supernovae from Pantheon [50] (referred to as SN), SH0ES constraints on the absolute magnitude of Type Ia supernovae M B from Cepheids [51,59,60] (referred to as SN+M B ), baryonic acoustic oscillations and redshift space distortions from 6dF, SDSS DR7 and BOSS [52][53][54] (referred to as BAO).Constraints on cosmological parameters for ΛCDM and ΛeDM are provided in Table II.In Fig. 6 we highlight the constraints on the derived quantities of H 0 and σ 8 , showing that ΛeDM allows for slightly larger values of H 0 (∼ 68.3 for ΛeDM compared to ∼ 67.5 for ΛCDM from P18+SN constraints), easing tensions with measurements of SH0ES M B [51] from ∼ 5σ to ∼ 3σ.The larger H 0 constraints allow for lower values of σ 8 (∼ 0.69 for ΛeDM compared to ∼ 0.81 for ΛCDM from constraints P18+SN constraints), a result which eases tensions with weak lensing measurements [3][4][5][6] from ∼ 3σ to ∼ 1σ.
In Table II we provide the χ 2 and AIC for model selection.Comparisons of the χ 2 show that combinations of P18 with SN, SN+M B and SN+M B +BAO are better fitted with a ΛeDM model however the AIC penalises the necessity for the w dm,0 parameter, showing a marginal preference for ΛCDM.In this Table we show the significance of tensions between the constraints on the Hubble constant (comparing to H 0 = 73.3± 1.04 [29]) and S 8 (comparing to S 8 = 0.766 ± 0.017 [57] and in brackets to S 8 = 0.790 ± 0.016 [5]).Here the significance is obtained by computing the difference between the best-fit values and dividing by the joint errors added in quadrature.The significance of the H 0 tension are reduced from 4-5σ to ∼ 3.5σ while the S 8 tension is reduced from ∼ 3σ to ∼ 1σ.The constraints on the dark matter EoS remain consistent with ΛCDM w dm,0 > −0.462 (for P18+SN+M B +BAO), although the maximum likelihood peaks at w dm,0 ∼ −0.2.

B. ISW and Lensing Predictions
In Fig. 7 we sample the chains from the Cobaya MCMC for ΛCDM and ΛeDM from P18+SN+M B +BAO and display the amplitude of the ISW (A ISW ) and lensing (A κ ) for the three void types with respect to the best-fit ΛCDM model.We show here that ΛCDM makes very tight predictions for what this should be, with little dependence on the void types.On the other hand ΛeDM allows for much larger A ISW and smaller A κ .The Hubble constant for the sampled points is shown with a red-blue color bar, indicating that in ΛeDM a higher H 0 is met with larger A ISW and smaller A κ , in better agreement with observations.A redshift evolution can be seen in (filled purple).The contours represent the 68% and 95% confidence regions.P18 does not strongly constrain w dm,0 , but the addition of SN+MB and BAO pulls it away from −1 and towards −0.3, although these constraints remain consistent with zero.
the profiles, voids at lower redshift exhibit a larger A ISW but weaker A κ with respect to ΛCDM for high-H 0 .
In Table III we make predictions for the void ISW and lensing amplitude for ΛeDM with respect to ΛCDM based on constraints using P18+SN+M B +BAO.These values are in excellent agreement with current observations and suggest the S 8 tensions and void-ISW anomaly can be solved and the H 0 tension alleviated with new physics in dark matter (or a unified interacting dark sector) that is being captured by the phenomenological ΛeDM model.

V. DISCUSSIONS
In this paper we introduce ΛeDM, a phenomenological model of dark matter with a non-zero EoS at late times.We show that ΛeDM is able to simultaneously ease the H 0 and S 8 tensions while also providing an explanation for the void-ISW anomaly.This is the first model, to our knowledge, that can consistently explain all three with a single modification.
Previous solutions to the H 0 tension (such as early dark energy [61]) provide a high H 0 at the cost of greater clustering and therefore larger σ 8 -putting them in conflict with the S 8 tension [62,63].ΛeDM, on the other hand, does not require changing the physical densities of dark matter at last scattering but instead imposes a late change to the evolution of dark matter.Since dark matter has never been directly observed, it is reasonable to suspect the simple assumptions of cold dark matter may be incomplete.Best fit constraints on ΛeDM imply dark matter's EoS is negative at late times, strongly implying interactions between dark matter and dark energy [25], which have been shown to alleviate both the H 0 and S 8 tensions [64][65][66][67][68][69]; if this is proven to be true this will have a profound impact on fundamental physics.However ΛeDM does not only alleviate the H 0 and TABLE II.Constraints on ΛCDM and ΛeDM parameters and statistics are shown for P18, P18+SN, P18+SN+MB and P18+SN+MB+BAO.In the top section we show the constraints on the base ΛCDM parameters As, ns, θ * , τ , Ω b h2 and Ω init dm h 2 .For ΛeDM we provide constraints on w dm,0 and MB when including constraints from SH0ES on the SN Type Ia absolute magnitude [51].In the second section we provide constraints on the derived parameters H0, σ8 and S8.In the third section we provide estimations of the significance with respect to local measurements of H0 [29] and weak lensing measurements of S8 from [3] and [5] in brackets.In the last section we provide comparisons of the χ 2 and AIC for model selection, showing that although ΛeDM provides a better fit to the data, there remains a marginal preference for ΛCDM (based on the AIC).

P18
. H 0 FIG. 7. The amplitude of the ISW (AISW) and lensing convergence (Aκ) for three void types are sampled from parameter constraints on ΛCDM (solid black) and ΛeDM (dashed black) from P18+SN+MB+BAO.These are compared to the amplitude obtained from the best-fit constraints on ΛCDM.The Hubble constant value for the sampled points in ΛeDM are indicated with a red-blue color bar.The grey horizontal and vertical bands show measurements from observations (see Table I).Constraints from ΛCDM show the amplitudes should be very close to one.For ΛeDM a larger H0 strongly correlates to a much larger AISW and smaller Aκ.

FIG. 1 .
FIG.1.The Hubble function H(z), dark matter EoS w dm (z) and power spectrum P (k) are displayed for models of ΛeDM as a function of the Hubble constant H0, with early universe physics fixed to CMB constraints.A higher H0 requires a negative EoS for dark matter, which leads to a decrease in the P (k) amplitude and therefore a low σ8 (shown in the inset plot on the right).Therefore, ΛeDM can alleviate both the H0 and S8 tensions as a high-H0 is naturally coupled to a low-σ8.For reference the profiles for ΛCDM assuming early universe CMB constraints are shown with dark blue solid lines and the profiles for ΛeDM assuming late universe constraints on H0 are shown with red dashed lines.

FIG. 3 .FIG. 4 .
FIG.3.The angular power spectra D ℓ are shown as a function of w dm,0 on the left and anz on the right, with all other parameters held fixed to P18 ΛCDM cosmology and anz = 0.5.In the subplots we show the CMB temperature autocorrelation (T T , top left), temperature to E-mode cross correlation (T E, top right), E-mode autocorrelation (EE, middle left), B-mode autocorrelation (BB, middle right), lensing potential autocorrelation (ϕϕ, bottom left) and temperature to lensing potential cross correlation (T ϕ, bottom right).The text in each subplot shows the multiplication factor used for plotting each angular power spectra.

FIG. 5 .
FIG.5.Constraints on ΛeDM parameters for P18 blue lines), P18+SN+MB (solid blue lines) and P18+SN+MB+BAO (filled purple).The contours represent the 68% and 95% confidence regions.P18 does not strongly constrain w dm,0 , but the addition of SN+MB and BAO pulls it away from −1 and towards −0.3, although these constraints remain consistent with zero.
P18... + SN + M B ... + BAO FIG.6.Constraints on the derived parameters of H0 and σ8 are shown for ΛeDM.These are constrained with Planck CMB measurements (dashed blue contours), the addition of Pantheon type Ia supernovae and Cepheid constraints on MB (SN+MB; solid blue contours) and BAO (purple shaded contours).The contours indicate regions of 68% and 95% confidence intervals.Constraints on ΛeDM provide higher values of H0 and lower values of σ8 easing the H0 and S8 tensions.TABLE III.Predictions for the ISW AISW and lensing convergence Aκ for the Spec-z, Photo-z and Eridanus voids from constraints of P18+SN+MB+BAO on ΛCDM and ΛeDM.For ΛCDM the predictions are very close to 1, while ΛeDM allows for AISW to be a factor of up to 2 times larger than ΛCDM and Aκ a factor of up to 0.9 times smaller than ΛCDM.Eridanus 1 ± 0.012 1.97 +0.71 −0.86 1 ± 0.037 0.96 +0.048 −0.046 preparation of this paper.Cosmological constraints on ΛeDM were obtained using the MCMC [55, 70, 71] package Cobaya [47, 48].This work is supported by the National Science Center, Poland, under grant agreements no: 2018/30/E/ST9/00698 and 2018/31/G/ST9/03388.MJ is supported by the Polish Ministry of Science and Higher Education through grant DIR/WK/2018/12.MB is supported by the Polish National Science Center through grants no.2020/38/E/ST9/00395 and 2020/39/B/ST9/03494, and by the Polish Ministry of Science and Higher Education through grant DIR/WK/2018/12.