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

Krishna Naidoo<EMAIL_ADDRESS>Center for Theoretical Physics, Polish
Academy of Sciences, al. Lotników 32/46 Warsaw, Poland  Department of Physics
& Astronomy, University College London, Gower Street, London, WC1E 6BT, UK
Mariana Jaber  Center for Theoretical Physics, Polish Academy of Sciences, al.
Lotników 32/46 Warsaw, Poland  Wojciech A. Hellwing  Center for Theoretical
Physics, Polish Academy of Sciences, al. Lotników 32/46 Warsaw, Poland  Maciej
Bilicki  Center for Theoretical Physics, Polish Academy of Sciences, al.
Lotników 32/46 Warsaw, Poland

###### Abstract

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, while also
providing an explanation for the anomalously large integrated Sachs-Wolfe
(ISW) effect from cosmic voids. 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 $\sim 2$ greater in this model than what is expected from the
standard model $\rm\Lambda CDM$.

††preprint: APS/123-QED

_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], have placed intense scrutiny on
observations, methods and the assumptions of the standard model of cosmology
$\rm\Lambda CDM$ (see [5, 6, 7, 8, 9, 10, 11, 12] 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 [13], in a spatially flat universe with a cosmological constant
($\Lambda$). The equation-of-state (EoS) for dark matter ($w_{\rm dm}$) 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 [14,
15] 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 $\Lambda$ referred to as $\rm\Lambda eDM$.

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; 16] signal from cosmic voids [17, 18, 19] (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) [20] and ranging in significance from
2-4$\sigma$ [17, 18, 20], while for ‘spectroscopic’ voids (smaller and not
aligned with the LOS) the ISW is larger but to a lesser extent and remains
consistent with $\rm\Lambda CDM$ [21]. In contrast, observations of void
lensing are either consistent with $\rm\Lambda CDM$ or lower than expected
[22].

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
letter 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 letter is organised as follows: (1) we
discuss the assumptions and theoretical background of the $\rm\Lambda eDM$
model and show that this model naturally couples the $H_{0}$ and $S_{8}$
tensions; (2) we describe the implications for the ISW and lensing from cosmic
voids; (3) we constrain $\rm\Lambda eDM$ parameters using cosmological
observations; and (4) we discuss the implications for tensions and anomalies
in cosmology.

_$\rm\Lambda eDM$ and Tensions_.—The evolution of the dark matter density
$\rho_{\rm dm}$ can be described as a function of the scale factor $a$ and a
time-varying EoS $w_{\rm dm}(a)$ defined as

$\displaystyle\rho_{\rm dm}(a)=\frac{\rho_{\rm dm,0}}{a^{3}}\,W(a),$ (1)
$\displaystyle W(a)=\exp\left(3\int_{a}^{1}\frac{w_{\rm
dm}(a^{\prime})}{a^{\prime}}{\rm d}a^{\prime}\right),$ (2)

where $\rho_{\rm dm,0}$ is the dark matter density today and $a^{\prime}$ 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,

$w_{\rm dm}(a)=\begin{dcases}w_{\rm dm,0}\left(\frac{a-a_{\rm nz}}{1-a_{\rm
nz}}\right),\,\,&\text{for }a\geq a_{\rm nz},\\\
0\vphantom{\frac{0}{0}},&\text{otherwise,}\\\ \end{dcases}$ (3)

_i.e._ a linearly increasing/decreasing EoS where $w_{\rm dm,0}$ is the
current value of the dark matter EoS and $a_{\rm nz}$ the scale factor at
which dark matter’s EoS becomes non-zero. At early times (i.e. for $a<a_{\rm
nz}$) constraints from CMB measurements have shown the EoS must be very close
to zero ($|w_{\rm dm}|\lesssim 0.001$) [14, 15], however, at late times the
constraints are more relaxed [15]. For this reason we fix the EoS at early
times to zero and allow the EoS to be non-zero at late times. Generalised dark
matter models with a cosmological constant, like the model considered here,
are equivalent to interacting dark energy models [23], models where dark
matter interacts with dark energy. We can therefore think of this model
describing a dark matter-dark energy interaction [24], which could occur only
at low energies or densities, or through the decay of unstable dark matter
particles with very long half-lives (see [25] for the evanescent matter model
with similar properties). In both cases the interaction is only relevant at
very late-times and would grow over time, following the behaviour of
$\rm\Lambda eDM$. In Suppl. Mat. [26] we show the sensitivity to $w_{\rm
dm,0}$ is much stronger than $a_{\rm nz}$ and that the parameters are strongly
degenerate. We break this degeneracy by setting $a_{\rm nz}=0.5$, meaning dark
matter’s EoS only becomes non-zero at $z\leq 1$.

The total matter density is defined by

$\rho_{\rm m}(a)=\frac{\rho_{\rm m,0}}{a^{3}}\,\Gamma(a),$ (4)

with the additional term

$\Gamma(a)=f_{\rm b}+f_{\rm\nu}+f_{\rm dm}\,W(a).$ (5)

Here $f_{\rm i}$ is the fraction of the matter constituent $\rm i$ to the
total matter content today (_i.e._ $f_{\rm i}=\rho_{\rm i,0}/\rho_{\rm m,0}$);
the subscripts $\rm b$ indicate baryons, $\nu$ non-relativistic neutrinos and
$\rm dm$ dark matter. Another useful relation is the derivative of $\Gamma(a)$
with respect to $\ln a$, which is given by

$\Upsilon(a)=\frac{{\rm d}\,\Gamma(a)}{{\rm d}\ln a}=3\,f_{\rm
dm}\,\frac{w_{\rm dm}(a)\,W(a)}{\Gamma(a)}.$ (6)

In $\rm\Lambda CDM$ these functions simplify to $\Gamma(a)=1$ and
$\Upsilon(a)=0$.

We modify the background evolution and the linear perturbation equations of
dark matter in the cosmological Boltzmann solver CLASS [27], following the
above relations and equation 2 of [15]. From CLASS we output the power
spectrum and growth functions.

To understand the role that $\rm\Lambda eDM$ 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 $A_{\rm
s}=2.101\times 10^{-9}$, $n_{\rm s}=0.9649$, $\tau=0.0544$ and $\Omega_{\rm
b}h^{2}=0.02236$, respectively. For dark matter this means fixing its value at
early times to

$\Omega_{\rm dm}W_{0}h^{2}=\Omega_{\rm dm}^{\rm init}h^{2}=0.1202,$ (7)

where $\Omega_{\rm dm}^{\rm init}$ is the fractional density of dark matter if
the EoS is zero and $W_{0}=W(a_{\rm nz})$.

In Fig. 1 we fix the parameters of $\rm\Lambda eDM$ 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 $H_{0}=67.27\,{\rm
km\,s^{-1}\,Mpc^{-1}}$ 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 $\sigma_{8}$. Since
$\sigma_{8}\propto S_{8}$ this naturally couples the two tensions. If $H_{0}$
is set to the values observed by SH0ES [28] (of $H_{0}=73.3\,{\rm
km\,s^{-1}\,Mpc^{-1}}$) we obtain $\sigma_{8}$$\sim 0.5$.

Figure 1: The Hubble function $H(z)$, dark matter EoS $w_{\rm dm}(z)$ and
power spectrum $P(k)$ are displayed for models of $\rm\Lambda eDM$ as a
function of the Hubble constant $H_{0}$, with early universe physics fixed to
CMB constraints. A higher $H_{0}$ requires a negative EoS for dark matter,
which leads to a decrease in the $P(k)$ amplitude and therefore a low
$\sigma_{8}$ (shown in the inset plot on the right). Therefore, $\rm\Lambda
eDM$ can alleviate both the $H_{0}$ and $S_{8}$ tensions as a high-$H_{0}$ is
naturally coupled to a low-$\sigma_{8}$. For reference the profiles for
$\rm\Lambda CDM$ assuming early universe CMB constraints are shown with dark
blue solid lines and the profiles for $\rm\Lambda eDM$ assuming late universe
constraints on $H_{0}$ are shown with red dashed lines.  Figure 2: The ISW
(top panels) and $\kappa$ (bottom panels) angular profiles for three void
types (spec-$z$ on the left, photo-$z$ in the middle and Eridanus voids on the
right) are shown as a function of $H_{0}$ in $\rm\Lambda eDM$ with all other
parameters fixed to early universe CMB constraints. In the inset plots we show
the amplitude of the ISW and $\kappa$ profiles with respect to $\rm\Lambda
CDM$, shown with a $\sigma_{8}$ bias correction (solid purple lines) and
without a bias correction (dashed purple lines). Measured values are indicated
with horizontal grey bands. A higher $H_{0}$ resolves the void-ISW anomaly and
recovers lower $\kappa$ amplitudes. For reference the profiles for $\rm\Lambda
CDM$ assuming early universe CMB constraints are shown with dark blue solid
lines and the profiles for $\rm\Lambda eDM$ assuming late universe constraints
on $H_{0}$ are shown with red dashed lines.

_Implications for the ISW and lensing_.—The ISW $T_{\rm ISW}$ is defined as

$\frac{T_{\rm ISW}(\hat{\eta})}{T}=\frac{2}{c^{3}}\int_{0}^{\chi_{\rm
LS}}\dot{\Phi}\left[\chi\hat{\eta},t(\chi)\right]\,a(\chi)\,{\rm d}\chi,$ (8)

where $\dot{\Phi}$ is the time-derivative of the gravitational potential
$\Phi$, $c$ is the speed of light, $\chi$ is the transverse comoving distance
in a direction $\hat{\eta}$ and $\chi_{\rm LS}$ is the comoving distance to
the last scattering surface. The gravitational potential is related to the
density contrast $\delta$ by the Poisson equation which in the linear regime
simplifies to

$\Phi(x,t)=\frac{3}{2}H_{0}^{2}\,\Omega_{\rm
m}\,\Gamma(t)\,\frac{D(t)}{a(t)}\,\nabla^{-2}\delta(x,0),$ (9)

where $\Omega_{\rm m}$ is the total fractional matter density today, $D(t)$
the linear growth function and $\delta(x)$ the density contrast today (_i.e._
at $z=0$). Consequently, the time-derivative is given by

$\dot{\Phi}(x,t)=H(t)\,\Big{[}f(t)-1+\Upsilon(t)\Big{]}\Phi(x,t).$ (10)

where $f(t)$ is the linear growth rate.

The lensing convergence, in the linear regime, is given by

$\kappa(\hat{\eta})=\frac{3H_{0}^{2}}{2c^{2}}\,\Omega_{\rm
m}\,\int_{0}^{\chi_{\rm s}}\frac{\chi}{\chi_{\rm s}}\,(\chi_{\rm
s}-\chi)\frac{D(\chi)}{a(\chi)}\,\Gamma(\chi)\,\delta(\chi)\,{\rm d}\chi,$
(11)

where $\chi_{\rm s}$ is the comoving distance to the lensing source.

Both the ISW and lensing are dependent on $\Gamma(t)$ (see Eqs. 8, 9, 10 and
11), while the ISW is also dependent on $\Upsilon(t)$ (see Eq. 8 and Eq. 10).
This means the ISW and lensing signals will be different in $\rm\Lambda eDM$
than in $\rm\Lambda CDM$ and can be constrained from studies cross-correlating
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
[29]). This makes distinguish the effect of $\rm\Lambda eDM$ 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 $\rm\Lambda eDM$
because they accumulate the departures in $\Gamma(t)$ and $\Upsilon(t)$ from
$\rm\Lambda CDM$; an effect which is more relevant for larger voids elongated
along the LOS, such as those measured by photometric surveys.

The ISW and $\kappa$ profiles for three types of voids are explored, assuming
the elliptical void profile of [30]: (1) spec-$z$ voids refers to typical
voids found from spectroscopic surveys [31, 32, 21, 33]; (2) photo-$z$ voids
from photometric surveys [34, 20, 22], _i.e._ large and preferentially
elongated along the LOS; and (3) the combined profiles of the Eridanus voids
[35] which are located in the direction of the CMB Cold Spot anomaly [36] and
for which the role of the ISW from voids has generated considerable discussion
[37, 38, 39, 40, 30, 41, 42, 43, 19, 44, 45]. The void parameters used in this
study are summarised in [26].

Measurements of the void density profiles are based on observations of
galaxies, biased tracers of the underlying density field. To correct for the
galaxy bias, a linear galaxy bias parameter is usually fitted based on
clustering analysis. However, if $\rm\Lambda CDM$ is not the correct model,
the inferred bias will be incorrect. By making the approximation that the
$P(k)$ for $\rm\Lambda eDM$ can be related to the $P(k)$ in $\rm\Lambda CDM$
by an amplitude shift, we can make a $\sigma_{8}$ bias correction to the
central void density $\delta_{0}$ (the central density contrast obtained from
$\rm\Lambda CDM$) by multiplying by the factor $\sqrt{\sigma_{8}^{\Lambda{\rm
CDM}}/\sigma_{8}^{\Lambda{\rm eDM}}}$ where $\sigma_{8}^{\Lambda{\rm CDM}}$ is
the $\sigma_{8}$ obtained from $\rm\Lambda CDM$ and $\sigma_{8}^{\Lambda{\rm
eDM}}$ from $\rm\Lambda eDM$.

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 $\Gamma\neq 1$ and $\Upsilon\neq
0$. 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.

Figure 3: Constraints on the derived parameters of $H_{0}$ and $\sigma_{8}$
are shown for $\rm\Lambda eDM$. These are constrained with Planck CMB
measurements (dashed blue contours), the addition of Pantheon type Ia
supernovae and Cepheid constraints on $M_{B}$ (SN+$M_{B}$; solid blue
contours) and BAO (purple shaded contours). The contours indicate regions of
68% and 95% confidence intervals. Constraints on $\rm\Lambda eDM$ provide
higher values of $H_{0}$ and lower values of $\sigma_{8}$ easing the $H_{0}$
and $S_{8}$ tensions.

_Constraints on $\rm\Lambda eDM$ and predictions for voids_.—We constrain
$\rm\Lambda eDM$ parameters (standard $\rm\Lambda CDM$ parameters plus the
current dark matter EoS $w_{\rm dm,0}$) using measurements from Planck CMB
(referred to as P18, [46, 47]), Type Ia supernovae from Pantheon [48]
(referred to as SN), SH0ES constraints on the absolute magnitude of Type Ia
supernovae $M_{B}$ from Cepheids [49, 50, 51] (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 $\rm\Lambda CDM$ and $\rm\Lambda eDM$ are provided in Table I
of the Suppl. Mat. [26]. In Fig. 3 we highlight the constraints on the derived
quantities of $H_{0}$ and $\sigma_{8}$, showing that $\rm\Lambda eDM$ allows
for considerably larger values of $H_{0}$, easing tensions with measurements
of SH0ES $M_{B}$ [49]. The larger $H_{0}$ constraints allow for lower values
of $\sigma_{8}$, a result which eases tensions with weak lensing measurements
[3, 4].

In Table I (of the Suppl. Mat. [26]) we provide the $\chi^{2}$ and Akaike
information criterion (AIC; [55]) for model selection. Comparisons of the
$\chi^{2}$ show that combinations of P18 with SN, SN+$M_{B}$ and
SN+$M_{B}$+BAO are better fitted with a $\rm\Lambda eDM$ model however the AIC
penalises the necessity for the $w_{\rm dm,0}$ parameter, showing a marginal
preference for $\rm\Lambda CDM$. In this Table we show the significance of
tensions between the constraints on the Hubble constant (comparing to
$H_{0}=73.3\pm 1.04$ [28]), $M_{B}$ (comparing to $M_{B}=-19.244\pm 0.020$
[49]) and $S_{8}$ (comparing to $S_{8}=0.766\pm 0.017$ [56] and in brackets to
$S_{8}=0.790\pm 0.016$ [57], see [26] for a definition of $S_{8}$ in
$\rm\Lambda eDM$). 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 $H_{0}$ and $M_{B}$ tensions are
reduced from 4-5$\sigma$ to $\sim 3.5\sigma$ while the $S_{8}$ tension is
reduced from $\sim 3\sigma$ to $\sim 1\sigma$.

In Table 1 we make predictions for the void ISW and lensing amplitude for
$\rm\Lambda eDM$ with respect to $\rm\Lambda CDM$ 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 $\rm\Lambda eDM$ model.

_Discussions_.—In this letter we introduce $\rm\Lambda eDM$, a
phenomenological model of dark matter with a non-zero EoS at late times. We
show that $\rm\Lambda eDM$ 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 [58])
provide a high $H_{0}$ at the cost of greater clustering and therefore larger
$\sigma_{8}$ – putting them in conflict with the $S_{8}$ tension [59, 60].
$\rm\Lambda eDM$, 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 $\rm\Lambda eDM$ imply dark
matter’s EoS is negative at late times, strongly implying interactions between
dark matter and dark energy [23], which have been shown to alleviate both the
$H_{0}$ and $S_{8}$ tensions [61, 62, 63, 64, 65, 66]; if this is proven to be
true this will have a profound impact on fundamental physics.

Table 1: Predictions for the ISW $A_{\rm ISW}$ and lensing convergence $A_{\kappa}$ for the Spec-$z$, Photo-$z$ and Eridanus voids from constraints of P18+SN+$M_{B}$+BAO on $\rm\Lambda CDM$ and $\rm\Lambda eDM$. For $\rm\Lambda CDM$ the predictions are very close to $~{}1$, while $\rm\Lambda eDM$ allows for $A_{\rm ISW}$ to be a factor of up to $2$ times larger than $\rm\Lambda CDM$ and $A_{\kappa}$ a factor of up to $0.9$ times smaller than $\rm\Lambda CDM$.  Void | $A_{\rm ISW}$ | $A_{\kappa}$
---|---|---
| $\rm\Lambda CDM$ | $\rm\Lambda eDM$ | $\rm\Lambda CDM$ | $\rm\Lambda eDM$
Spec-$z$ | $1^{+0.0087}_{-0.0093}$ | $1.55^{+0.31}_{-0.49}$ | $1_{-0.032}^{+0.033}$ | $0.92_{-0.072}^{+0.079}$
Photo-$z$ | $1^{+0.0096}_{-0.01}$ | $1.71^{+0.44}_{-0.64}$ | $1^{+0.035}_{-0.034}$ | $0.93^{+0.069}_{-0.065}$
Eridanus | $1\pm 0.012$ | $1.97^{+0.71}_{-0.86}$ | $1\pm 0.037$ | $0.96^{+0.048}_{-0.046}$

However $\rm\Lambda eDM$ does not only alleviate the $H_{0}$ and $S_{8}$
tensions, it also provides an explanation to a long-standing anomaly – the
void-ISW anomaly, while remaining consistent with void lensing measurements.
This allows the model to be tested in the near future, including testing other
shapes and parameterisations for the EoS, from galaxy surveys such as
Euclid111https://www.euclid-ec.org/ and LSST222https://www.lsst.org/. So far,
observations appear to be consistent with constraints to $\rm\Lambda eDM$,
with the ISW of voids predicted to be roughly a factor of $\sim 2$ and lensing
to be a factor of $\sim 0.9$ with respect to $\rm\Lambda CDM$. Future
measurements of void ISW and lensing, and other probes (such as galaxy cluster
abundances, galaxy lensing and CMB temperature auto and cross-correlations)
will allow us to provide tighter constraints on $\rm\Lambda eDM$ and help
determine whether new physics in dark matter (or an interacting dark sector)
is the cause of the $H_{0}$ and $S_{8}$ tension.

_Acknowledgements_.—We thank Enrique Gaztañaga, Ofer Lahav, Constance Mahony,
Adi Nusser and Joe Silk for providing useful suggestions and insightful
discussions in preparation of this letter. Cosmological constraints on
$\rm\Lambda eDM$ were obtained using the MCMC [67, 68, 69] package Cobaya [70,
71]. 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.

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Supplemental Material

##  I Model Degeneracies

Figure 1: The angular power spectra $\mathcal{D}_{\ell}$ are shown as a
function of $w_{\rm dm,0}$ with all other parameters held fixed to P18
$\rm\Lambda CDM$ cosmology and $a_{\rm nz}=0.5$. In the subplots we show the
CMB temperature autocorrelation ($TT$, top left), temperature to E-mode cross
correlation ($TE$, top right), E-mode autocorrelation ($EE$, middle left),
B-mode autocorrelation ($BB$, middle right), lensing potential autocorrelation
($\phi\phi$, bottom left) and temperature to lensing potential cross
correlation ($T\phi$, bottom right). The text in each subplot shows the
multiplication factor used for plotting each angular power spectra.

The $\rm\Lambda eDM$ model is defined with the addition of two free
parameters, $w_{\rm dm,0}$ which defines the dark matter EoS today (i.e. at
$z=0$) and $a_{\rm nz}$ the scale factor at which the EoS becomes non-zero. In
Fig. 1 and 2 we show the angular power spectra of the cosmic microwave
background (CMB) temperature, E and B-modes, and lensing potentials for the
$\rm\Lambda eDM$ model with base parameters (i.e. those in common with
$\rm\Lambda CDM$) fixed to best-fit $\rm\Lambda CDM$ P18 constraints. In Fig.
3 we show the power spectra at redshift $z=0$ with base parameters fixed to
best-fit $\rm\Lambda CDM$. In Fig. 1 and 3 the analytical angular and 3D power
spectra are shown as a function of $w_{\rm dm,0}$ varying between $-1$ and
$0.3$, and $a_{\rm nz}=0.5$. Similarly, In Fig. 2 and 3 the analytical angular
and 3D power spectra are shown as a function of $a_{\rm nz}$ varying between
$0.2$ and $0.9$, and $w_{\rm dm,0}=-0.2$.

We see that smaller $w_{\rm dm,0}$ and $a_{\rm nz}$ 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 B-mode
and lensing auto/cross-correlations. At low-$\ell$ 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 $w_{\rm dm,0}$ than $a_{\rm nz}$. For this reason we
fix $a_{\rm nz}$ to $0.5$ in our analysis.

Figure 2: The angular power spectra $\mathcal{D}_{\ell}$ are shown as a
function of $a_{\rm nz}$ with all other parameters held fixed to P18
$\rm\Lambda CDM$ cosmology and $w_{\rm dm,0}=-0.2$. In the subplots we show
the CMB temperature autocorrelation ($TT$, top left), temperature to E-mode
cross correlation ($TE$, top right), E-mode autocorrelation ($EE$, middle
left), B-mode autocorrelation ($BB$, middle right), lensing potential
autocorrelation ($\phi\phi$, bottom left) and temperature to lensing potential
cross correlation ($T\phi$, bottom right). The text in each subplot shows the
multiplication factor used for plotting each angular power spectra.  Figure 3:
The power spectrum at redshift zero for $\rm\Lambda eDM$, with varying $w_{\rm
dm,0}$ and $a_{\rm nz}=0.5$ and on the left and varying $a_{\rm nz}$ with
$w_{\rm dm,0}=-0.2$ on the right. The plots show a strong degeneracy between
these parameters, smaller values of $a_{\rm nz}$ and $w_{\rm dm,0}$ dampen
$P(k)$ more strongly, an effect which is more sensitive to $w_{\rm dm,0}$.

##  II Parameter Inference

Constraints on cosmological parameters were obtained using the cosmological
Markov chain Monte Carlo (MCMC) software Cobaya [70, 71]. 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:

  * •

Planck CMB measurements of temperature, polarisation auto/cross correlation
for high-$\ell$ (i.e. $\ell>30$), temperature and polarisation auto
correlation for low-$\ell$ (i.e. $\ell\leq 30$) and lensing [47]. The joint
likelihood for Planck CMB measurements are referred to in the paper as P18.

  * •

Pantheon Type Ia supernovae [48] referred to as SN.

  * •

SH0ES Cepheid constraints on the Type Ia absolute magnitude [49], referred to
as SN+$M_{B}$.

  * •

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 $\rm\Lambda CDM$ and $\rm\Lambda eDM$ are
obtained by sampling the posterior by running an MCMC. In both cases, this
means varying the standard $\rm\Lambda CDM$ parameters, i.e. primordial power
spectrum amplitude $A_{\rm s}$, the spectral tilt $n_{\rm s}$, the sound
horizon $\theta_{*}$, the optical depth $\tau$, baryon density $\Omega_{\rm
b}h^{2}$ and dark matter density $\Omega_{\rm dm}h^{2}$. For $\rm\Lambda eDM$
we also vary the dark matter EoS today $w_{\rm dm,0}$. In $\rm\Lambda eDM$
there is a strong degeneracy between $\Omega_{\rm dm}$ and $w_{\rm dm,0}$,
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. $\Omega_{\rm dm}^{\rm init}$ which is
related to $\Omega_{\rm dm}$ by the relation $\Omega_{\rm dm}=\Omega_{\rm
dm}^{\rm init}/W_{0}$, for $\rm\Lambda CDM$ 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. 4
we show the posterior of the sampled $\rm\Lambda eDM$ parameters, showing that
with the exception of $w_{\rm dm,0}$ the posteriors are generally quite
Gaussian. A comparison to $\rm\Lambda CDM$ 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 [69]. The chains are assumed to
be converged once $R-1<0.01$.

Defining $S_{8}$ for $\rm\Lambda eDM$ is rather challenging since the general
definition $S_{8}=\sigma_{8}\sqrt{0.3/\Omega_{\rm m}}$ in the literature
generally assumes the contribution of cold dark matter to $\Omega_{\rm m}$.
Assuming that this functional form is really trying to capture an amplitude to
the power spectra and define this in relation to the fiducial $\rm\Lambda CDM$
constraints where $\Omega_{\rm m}\simeq 0.3$ the most sensible definition of
$S_{8}$ will use $\Omega_{\rm dm}^{\rm init}$ since this sets the shape and
amplitude of the initial power spectra and using $\sigma_{8}$ which captures
$\rm\Lambda eDM$’s dampening effect. Therefore, we define

$S_{8}=\sigma_{8}\sqrt{\Omega_{\rm m}^{\rm init}/0.3}$ (12)

where $\Omega_{\rm m}^{\rm init}=\Omega_{\rm dm}^{\rm init}+\Omega_{\rm
b}+\Omega_{\nu}^{\rm NR}$ and $\Omega_{\nu}^{\rm NR}$ is the contribution from
non-relativistic neutrinos.

In Table 1 we show the constraints on the base and derived parameters for
$\rm\Lambda CDM$ and $\rm\Lambda eDM$. For model comparison, we provide the
$\chi^{2}$, AIC and the significance of the Hubble, $M_{B}$ and $S_{8}$
tensions; comparing to $H_{0}=73.3\pm 1.04$ [28], $M_{B}=-19.244\pm 0.020$
[49] and $S_{8}=0.766\pm 0.017$ [56] ($S_{8}=0.790\pm 0.016$ in brackets;
[57]) respectively.

Figure 4: Constraints on $\rm\Lambda eDM$ parameters for P18 (dashed blue lines), P18+SN+$M_{B}$ (solid blue lines) and P18+SN+$M_{B}$+BAO (filled purple). The contours represent the 68% and 95% confidence regions. P18 does not strongly constrain $w_{\rm dm,0}$, but the addition of SN+$M_{B}$ and BAO pulls it away from $-1$ and towards $-0.3$, although these constraints remain consistent with zero. Table 1: Constraints on $\rm\Lambda CDM$ and $\rm\Lambda eDM$ parameters and statistics are shown for P18, P18+SN, P18+SN+$M_{B}$ and P18+SN+$M_{B}$+BAO. In the top section we show the constraints on the base $\rm\Lambda CDM$ parameters $A_{\rm s}$, $n_{\rm s}$, $\theta_{*}$, $\tau$, $\Omega_{\rm b}h^{2}$ and $\Omega_{\rm dm}^{\rm init}h^{2}$. For $\rm\Lambda eDM$ we provide constraints on $w_{\rm dm,0}$ and $M_{B}$ when including constraints from SH0ES on the SN Type Ia absolute magnitude [49]. In the second section we provide constraints on the derived parameters $H_{0}$, $\sigma_{8}$ and $S_{8}$. In the third section we provide estimations of the significance with respect to local measurements of $H_{0}$ [28] and $M_{B}$ [49] and weak lensing measurements of $S_{8}$ from [56] and [57] in brackets. In the last section we provide comparisons of the $\chi^{2}$ and AIC for model selection, showing that although $\rm\Lambda eDM$ provides a better fit to the data, there remains a marginal preference for $\rm\Lambda CDM$ (based on the AIC).  | P18 |  …$\rm+SN$ |  …$+M_{B}$ |  …$\rm+BAO$
---|---|---|---|---
| $\rm\Lambda CDM$ | $\rm\Lambda eDM$ | $\rm\Lambda CDM$ | $\rm\Lambda eDM$ | $\rm\Lambda CDM$ | $\rm\Lambda eDM$ | $\rm\Lambda CDM$ | $\rm\Lambda eDM$
$\ln(10^{10}A_{\rm s})$ | $3.045^{+0.029}_{-0.028}$ | $3.045^{+0.030}_{-0.028}$ | $3.045\pm 0.028$ | $3.046^{+0.029}_{-0.028}$ | $3.048^{+0.030}_{-0.028}$ | $3.050^{+0.031}_{-0.028}$ | $3.048^{+0.029}_{-0.027}$ | $3.051^{+0.030}_{-0.028}$
$n_{\rm s}$ | $0.9659$ | $0.9660$ | $0.9658$ | $0.9665$ | $0.9672$ | $0.9688$ | $0.9675$ | $0.9692$
| ${}^{+0.0080}_{-0.0079}$ | $\pm 0.0082$ | ${}^{+0.0078}_{-0.0079}$ | ${}^{+0.0082}_{-0.0080}$ | ${}^{+0.0082}_{-0.0081}$ | ${}^{+0.0080}_{-0.0081}$ | $\pm 0.0075$ | ${}^{+0.0071}_{-0.0074}$
$100\,\theta_{\rm*}$ | $1.04196$ | $1.04196$ | $1.04196$ | $1.04198$ | $1.04203$ | $1.04210$ | $1.04204$ | $1.04212$
| $\pm 0.00056$ | ${}^{+0.00057}_{-0.00060}$ | $\pm 0.00055$ | $\pm 0.00056$ | ${}^{+0.00058}_{-0.00057}$ | ${}^{+0.00057}_{-0.00058}$ | ${}^{+0.00057}_{-0.00056}$ | ${}^{+0.00055}_{-0.00056}$
$\tau$ | $0.055^{+0.015}_{-0.014}$ | $0.055^{+0.016}_{-0.014}$ | $0.055^{+0.015}_{-0.014}$ | $0.055^{+0.015}_{-0.014}$ | $0.056^{+0.016}_{-0.014}$ | $0.058^{+0.016}_{-0.015}$ | $0.057^{+0.015}_{-0.014}$ | $0.059^{+0.015}_{-0.014}$
$\Omega_{\rm b}h^{2}$ | $0.02239$ | $0.02240$ | $0.02239$ | $0.02241$ | $0.02245$ | $0.02250$ | $0.02245$ | $0.02251$
$\pm 0.00029$ | $\pm 0.00029$ | $\pm 0.00028$ | ${}^{+0.00029}_{-0.00028}$ | ${}^{+0.00029}_{-0.00028}$ | $\pm 0.00028$ | ${}^{+0.00028}_{-0.00027}$ | ${}^{+0.00027}_{-0.00026}$
$\Omega_{\rm dm}^{\rm init}h^{2}$ | $0.1199$ | $0.1198$ | $0.1199$ | $0.1196$ | $0.1194$ | $0.1187$ | $0.1193$ | $0.1185$
| ${}^{+0.0023}_{-0.0024}$ | ${}^{+0.0024}_{-0.0025}$ | $\pm 0.0023$ | ${}^{+0.0022}_{-0.0023}$ | ${}^{+0.0023}_{-0.0024}$ | ${}^{+0.0023}_{-0.0022}$ | $\pm 0.0020$ | ${}^{+0.0018}_{-0.0017}$
$w_{\rm dm,0}$ | n/a | — | n/a | $>-0.439$ | n/a | $-0.31^{+0.30}_{-0.25}$ | n/a | $>-0.462$
$M_{B}$ | n/a | n/a | n/a | n/a | $-19.411\pm 0.028$ | $-19.386^{+0.039}_{-0.038}$ | $-19.409^{+0.023}_{-0.022}$ | $-19.393^{+0.030}_{-0.028}$
Derived |  |  |  |  |  |  |  |
$H_{0}$ | $67.4\pm 1.1$ | $69.7^{+4.3}_{-3.0}$ | $67.5^{+1.0}_{-0.99}$ | $68.3^{+1.6}_{-1.5}$ | $67.9\pm 1.0$ | $69.2^{+1.8}_{-1.7}$ | $67.99^{+0.81}_{-0.80}$ | $68.8^{+1.3}_{-1.2}$
$\sigma_{8}$ | $0.811\pm 0.012$ | $0.59^{+0.21}_{-0.22}$ | $0.810\pm 0.012$ | $0.69^{+0.12}_{-0.13}$ | $0.809\pm 0.012$ | $0.64^{+0.15}_{-0.14}$ | $0.809\pm 0.012$ | $0.67\pm 0.13$
$S_{8}$ | $0.829^{+0.025}_{-0.026}$ | $0.59^{+0.23}_{-0.25}$ | $0.827\pm 0.024$ | $0.70^{+0.13}_{-0.14}$ | $0.818\pm 0.024$ | $0.63^{+0.16}_{-0.15}$ | $0.816\pm 0.020$ | $0.67^{+0.14}_{-0.13}$
Tensions |  |  |  |  |  |  |  |
$H_{0}$ | $5.04\sigma$ | $1.73\sigma$ | $5.02\sigma$ | $3.86\sigma$ | $4.64\sigma$ | $3.04\sigma$ | $4.75\sigma$ | $3.67\sigma$
$M_{B}$ | n/a | n/a | n/a | n/a | $4.14\sigma$ | $3.4\sigma$ | $4.27\sigma$ | $3.73\sigma$
$S_{8}$ | $2.94\sigma$ | $1.13\sigma$ | $2.93\sigma$ | $0.83\sigma$ | $2.5\sigma$ | $1.53\sigma$ | $2.54\sigma$ | $1.21\sigma$
| $(1.89\sigma)$ | $(1.28\sigma)$ | $(1.85\sigma)$ | $(1.12\sigma)$ | $(1.4\sigma)$ | $(1.8\sigma)$ | $(1.38\sigma)$ | $(1.52\sigma)$
Model |  |  |  |  |  |  |  |
Selection |  |  |  |  |  |  |  |
$\chi^{2}$ | $1012.02$ | $1012.24$ | $2047.35$ | $2046.87$ | $2057.42$ | $2054.85$ | $2063.32$ | $2062$
$\Delta\chi^{2}$ | $0$ | $0.22$ | $0$ | $-0.48$ | $0$ | $-2.57$ | $0$ | $-1.32$
AIC | $1013.15$ | $1015.34$ | $2048.35$ | $2049.95$ | $2061.88$ | $2062.18$ | $2067.69$ | $2068.42$
$\Delta$AIC | $0$ | $2.19$ | $0$ | $1.6$ | $0$ | $0.31$ | $0$ | $0.72$

##  III Void Properties

In our analysis, the integrated Sachs-Wolfe effect (ISW) and lensing
convergence signal from three types of voids was studied. The void types
studied are chosen to be representative of voids measured from observations,
namely those measured from spectroscopic surveys (referred to as spec-$z$
voids), photometric surveys (referred to as photo-$z$ voids) and the Eridanus
voids which were measured along the line-of-sight of the CMB Cold Spot
anomaly. The parameters used and the measured ISW and lensing amplitude with
respect to $\rm\Lambda CDM$ are given in Table 2.

Table 2: We list the void parameters for spectroscopic voids (spec-$z$) [31, 32, 21, 33], photometric voids (photo-$z$) [34, 20, 22] and the Eridanus voids (combining profiles from voids E-1, E-2, E-3 and E-4) [35]. The void parameters are the central redshift $z_{\rm c}$, the central density contrast $\delta_{0}$, the effective radius $R$ and the measured amplitude of the ISW $A_{\rm ISW}$ and lensing $A_{\kappa}$ signals with respect to $\rm\Lambda CDM$. 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 $R_{\parallel}/R_{\perp}=2.6$ [34]. Void | $\quad\delta_{0}\quad$ | $\quad z_{\rm c}\quad$ |  $R$ $[h^{-1}{\rm Mpc}]$ | $A_{\rm ISW}$ | $A_{\kappa}$
---|---|---|---|---|---
Spec-$z$ | $-0.6$ | $0.6$ | $35$ | $1.64\pm 0.53$ | $0.97\pm 0.19$
Photo-$z$ | $-0.55$ | $0.5$ | $60$ | $4.1\pm 2$ | $0.79\pm 0.12$
E-1 | $-0.34$ | $0.14$ | $119$ | n/a | n/a
E-2 | $-0.87$ | $0.26$ | $50$ | n/a | n/a
E-3 | $-0.8$ | $0.3$ | $59$ | n/a | n/a
E-4 | $-0.62$ | $0.42$ | $168$ | n/a | n/a

##  IV ISW and Lensing Predictions

In Fig. 5 we sample the chains from the Cobaya MCMC for $\rm\Lambda CDM$ and
$\rm\Lambda eDM$ from P18+SN+$M_{B}$+BAO and display the amplitude of the ISW
($A_{\rm ISW}$) and lensing ($A_{\kappa}$) for the three void types with
respect to the best-fit $\rm\Lambda CDM$ model. We show here that $\rm\Lambda
CDM$ makes very tight predictions for what this should be, with little
dependence on the void types. On the other hand $\rm\Lambda eDM$ allows for
much larger $A_{\rm ISW}$ and smaller $A_{\kappa}$. The Hubble constant for
the sampled points is shown with a red-blue color bar, indicating that in
$\rm\Lambda eDM$ a higher $H_{0}$ is met with larger $A_{\rm ISW}$ and smaller
$A_{\kappa}$, in better agreement with observations. A redshift evolution can
be seen in the profiles, voids at lower redshift exhibit a larger $A_{\rm
ISW}$ but weaker $A_{\kappa}$ with respect to $\rm\Lambda CDM$ for
high-$H_{0}$.

Figure 5: The amplitude of the ISW ($A_{\rm ISW}$) and lensing convergence
($A_{\kappa}$) for three void types are sampled from parameter constraints on
$\rm\Lambda CDM$ (solid black) and $\rm\Lambda eDM$ (dashed black) from
P18+SN+$M_{B}$+BAO. These are compared to the amplitude obtained from the
best-fit constraints on $\rm\Lambda CDM$. The Hubble constant value for the
sampled points in $\rm\Lambda eDM$ are indicated with a red-blue color bar.
The grey horizontal and vertical bands show measurements from observations
(see Table 2). Constraints from $\rm\Lambda CDM$ show the amplitudes should be
very close to one. For $\rm\Lambda eDM$ a larger $H_{0}$ strongly correlates
to a much larger $A_{\rm ISW}$ and smaller $A_{\kappa}$.