# Observational constraints on phenomenological emergent dark energy and
barotropic dark matter characterized by a constant equation of state parameter

Jian-Qi Liu  0009-0005-7615-0149 Yan-Hong Yao  0000-0001-5283-3635
<EMAIL_ADDRESS>Yan Su  0009-0003-1993-0769 Jun-Chao Wang
0000-0002-4099-9520 Jia-Wei Wu  School of Physics and Astronomy, Sun Yat-sen
University, 2 Daxue Road, Tangjia, Zhuhai, 519082, People’s Republic of China

###### Abstract

While cold dark matter is well-supported by a broad array of cosmological data
alongside a cosmological constant, it faces several inherent challenges at
small scales. These challenges have spurred the exploration of various
alternative dark matter candidates beyond cold dark matter, leaving the
question of ”What is DM?” relatively open. Therefore, we propose a new
cosmological model that considers dark matter as a barotropic fluid with a
constant equation of state parameter and interprets dark energy as the
phenomenological emergent dark energy rather than a cosmological constant.
This proposal is based on extensive research on the extended properties of
dark matter in the context of a cosmological constant and the intriguing
findings that have emerged from our exploration of dark matter properties
within the context of PEDE in our previous studies. We then place constraints
on this model in light of the Planck 2018 Cosmic Microwave Background (CMB)
anisotropies, baryon acoustic oscillation (BAO) measurements, the Pantheon
compilation of Type Ia supernovae, a prior on $H_{0}$ that based on the latest
local measurement by Riess et al., and the combination of KiDS and the VISTA
Kilo-Degree Infrared Galaxy Survey (KiDS+VIKING-450). The results indicate a
preference for a positive dark matter equation of state parameter at 68%
confidence level for CMB+BAO, CMB+BAO+Pantheon and CMB+BAO+Pantheon+$H_{0}$
datasets. Furthermore, the Hubble tension between all of the datasets we used
with R22 is very close to those of the PEDE, and the $S_{8}$ tension between
Planck 2018 and KiDS+VIKING-450 is reduced from 2.3$\sigma$ in the PEDE model
to 0.4$\sigma$ in the new model. However, Bayesian evidence indicates that
PEDE favors our new model with very strong evidence from all the datasets
considered in this study. Consequently, we conclude that the PEDE+$w_{\rm dm}$
model is not a viable alternative to the PEDE model.

##  I introduction

Dark matter (DM), a mysterious component of the universe that does not
interact with photons, is expected to account for one-fourth of the energy
budget of the universe today. Although the nature of DM remains uncertain, we
currently have a simple and popular DM candidate, i.e., the cold dark matter
(CDM). In the standard $\Lambda$-Cold Dark Matter ($\Lambda$CDM) model, CDM is
considered as a non-interacting perfect fluid having zero equation of state
(EoS) parameter as well as zero sound speed and zero viscosity. Accompanied by
a cosmological constant, CDM is supported by a wide range of cosmological data
(Riess et al., 1998; Perlmutter et al., 1999; Dunkley et al., 2011; Hinshaw et
al., 2013; Story et al., 2015; Alam et al., 2017; Troxel et al., 2018; Aghanim
et al., 2020a). However, CDM loses its magic at small scales, facing issues
inherent in this paradigm, such as the missing satellite (Klypin et al., 1999;
Moore et al., 1999a), too-big-to-fail (Boylan-Kolchin et al., 2012), and core-
cusp problem (Moore et al., 1999b; Springel et al., 2008). Small-scale issues
of the CDM paradigm have motivated many new DM candidates beyond CDM,
including warm DM Blumenthal et al. (1982); Bode et al. (2001), fuzzy DM Hu et
al. (2000); Marsh and Silk (2014), interacting DM Spergel and Steinhardt
(2000) and decaying DM Wang et al. (2014). These new candidates are proposed
to suppress the formation of low-mass structures and to be consistent with
large-scale data simultaneously.

Inspired by the limitations of CDM and the diversity of DM candidates, in this
article, we will investigate the extended properties of DM through the
generalized dark matter (GDM) framework. The GDM first proposed in Hu (1998)
(for follow-up studies on the GDM model, we recommend taking a look at Mueller
(2005); Kumar and Xu (2014); Armendariz-Picon and Neelakanta (2014); Kopp et
al. (2018); Kumar et al. (2019); Ilić et al. (2021); Pan et al. (2023)) and is
specified by the DM EoS, the rest-frame sound speed, and the viscosity. Since
there have already been many researches on the extended properties of DM in
the context of a cosmological constant Mueller (2005); Kopp et al. (2018);
Kumar et al. (2019); Ilić et al. (2021); Armendariz-Picon and Neelakanta
(2014), in this work we will follow the approach of Ref. (Yao et al., 2024),
i.e., considering the phenomenological emergent dark energy (PEDE) as the
invisible fuel that accelerates the expansion rate of the current universe.
The PEDE model, initially proposed by the authors of Ref.Li and Shafieloo
(2019) to address the Hubble tension, has been subsequently extended to a
generalized parameterization form capable of accommodating both the
cosmological constant and the PEDE modelLi and Shafieloo (2020) (also see Yang
et al. (2021); Hernández-Almada et al. (2020)). Notably, the PEDE model
features an identical number of free parameters as the spatially flat
$\Lambda$CDM model. This framework was further examined in Ref. Pan et al.
(2020), which explored its complete evolution encompassing both background and
perturbations, yielding results consistent with previous works regarding the
$H_{0}$ tension by using recent observational datasets. In the Ref. (Yao et
al., 2024), the authors set the DM EoS parameter as a free parameter and the
DM sound speed to zero in the context of PEDE. After placing constraints on
this model with some specific combinations of the Planck 2018 Cosmic Microwave
Background (CMB) anisotropies measurements, baryon acoustic oscillation (BAO)
measurements, and the Pantheon compilation of Type Ia supernovae, the final
results indicate a preference for a negative DM EoS parameter at $95\%$ CL for
CMB+Pantheon and CMB+BAO+Pantheon datasets, which is very interesting and
worth further investigation. In order to investigate whether introducing other
GDM parameters in the context of PEDE would lead to different interesting
results, in this paper, building upon the assumption of keeping the DM EoS
parameter as a free parameter, we further assume that the DM sound speed is a
parameter and hence not being fixed at zero. For the sake of model simplicity,
we assume that the DM non-adiabatic sound speed and viscosity are zero, i.e.,
DM is barotropic. Then we will constrain the new model using CMB, BAO, Ia
supernovae (SN Ia), and weak lensing (WL) observational data, and analyze the
fitting results of relevant free parameters to see if there are indeed some
interesting outcomes emerging.

This paper is organized as follows. Section II outlines the key equations of
the PEDE+$w_{\mathrm{dm}}$ model. In section III, we present the observational
datasets and the statistical methodology. In section IV, we present the
observational constraints and implications of the PEDE + $w_{\mathrm{dm}}$
model. In the last section, we make a brief conclusion to this paper.

##  II Review of the PEDE+$w_{\mathrm{dm}}$ Model

In this work, we focus on a spatially flat, homogeneous, and isotropic
spacetime described by the spatially flat Friedmann-Robertson-Walker (FRW)
metric. Additionally, we assume that the gravitational sector of the universe
is suitably described by general relativity, where matter is minimally coupled
to gravity. Furthermore, we assume that none of the fluids interact non-
gravitationally with each other, and the universe consists of radiation,
baryons, GDM, and PEDE. Consequently, we can express the Hubble parameter as

$\displaystyle\frac{H^{2}}{H_{0}^{2}}=$ $\displaystyle\hskip
2.84544pt\Omega_{\mathrm{r0}}(1+z)^{4}+\Omega_{\mathrm{dm0}}(1+z)^{3(1+w_{\mathrm{dm}})}$
$\displaystyle+\Omega_{\mathrm{b0}}(1+z)^{3}+\Omega_{\mathrm{de}}(z),$ (1)

where $H$ is the Hubble parameter, $\Omega_{\mathrm{r0}}$,
$\Omega_{\mathrm{dm0}}$, $\Omega_{\mathrm{b0}}$, and $\Omega_{\mathrm{de}}$
are the density parameters for radiation, GDM, baryons, and PEDE respectively,
here $\Omega_{\mathrm{de}}$ is parameterized in the following form Li and
Shafieloo (2019); Pan et al. (2020):

$\displaystyle\Omega_{\mathrm{de}}(z)=\Omega_{\mathrm{de0}}[1+\tanh(\log_{10}(1+z))].$
(2)

where $\Omega_{\rm de0}=1-\Omega_{\rm r0}-\Omega_{\rm dm0}-\Omega_{\rm b0}$
and $1+z=a^{-1}$ (note that we have set the current value of the scale factor
to be unity). Since we assume that none of the fluids interact non-
gravitationally with each other, the PEDE conservation equation reads

$\displaystyle\dot{\rho_{\mathrm{de}}}(z)+3H(1+w_{\mathrm{de}}(z))\rho_{\mathrm{de}}(z)=0.$
(3)

Here, an over dot represents the derivative with respect to cosmic time. From
the equation above, one can derive a following relation between PEDE’s EoS and
density:

$\displaystyle
w_{\mathrm{de}}(z)=-1+\frac{1}{1+z}\times\frac{\mathrm{d}\ln{\Omega_{\mathrm{de}}(z)}}{\mathrm{d}z},$
(4)

substituting Eq. (2) in it, we obtain an explicitly PEDE EoS as follows Li and
Shafieloo (2019); Pan et al. (2020)

$\displaystyle w_{\mathrm{de}}(z)=-1-\frac{1}{3\ln
10}\times[1+\tanh(\log_{10}(1+z))].$ (5)

It can be inferred from the above equation that the PEDE EoS exhibits an
intriguing symmetrical feature. Specifically, for the distant past, i.e.,
$z\rightarrow\infty$, one obtains $w_{\mathrm{de}}\rightarrow-1-\frac{2}{3\ln
10}$. For the distant future, i.e., $z\rightarrow-1$, one finds
$w_{\mathrm{de}}\rightarrow-1$. At the present time when $z=0$, we observe
that $w_{\mathrm{de}}=-1-\frac{1}{3\ln 10}$, indicating a phantom dark energy
(DE) EoS. As briefly described in Ref. (Li and Shafieloo, 2019), the pivot
point of transition for the PEDE EoS can be considered to be the redshift at
which matter-dark energy densities are equal.

In the conformal Newtonian gauge, the perturbed FRW metric is expressed in the
following form:

$\displaystyle ds^{2}=a^{2}(\tau)[-(1+2\psi)d\tau^{2}+(1-2\phi)d\vec{r}^{2}],$
(6)

where $\psi$ and $\phi$ represent the metric potentials, while $\vec{r}$
denotes the three spatial coordinates. By considering the first order
perturbed part of the conserved stress-energy momentum tensor, one can derive
the following continuity and Euler equations (in Fourier space) for GDM and
PEDE (Kumar et al., 2019):

$\displaystyle\delta_{\rm ds}^{\prime}$ $\displaystyle=$
$\displaystyle-(1+w_{\rm ds})\left(\theta_{\rm
ds}-3\phi^{\prime}\right)-3\mathcal{H}\left(\frac{\delta p_{\rm
ds}}{\delta\rho_{\rm ds}}-w_{\rm ds}\right)\delta_{\rm ds}\hskip 14.22636pt$
(7) $\displaystyle\theta_{\rm ds}^{\prime}$ $\displaystyle=$
$\displaystyle-\mathcal{H}(1-3c_{\mathrm{\rm ad,ds}}^{2})\theta_{\rm
ds}+\frac{\delta p_{\rm ds}/\delta\rho_{\rm ds}}{1+w_{\rm ds}}k^{2}\delta_{\rm
ds}+k^{2}\psi$ (8)

Here, a prime stands for the conformal time derivative, $\mathcal{H}$ is the
conformal Hubble parameter, and k is the magnitude of the wavevector
$\vec{k}$. $\delta_{\rm ds}$ and $\theta_{\rm ds}$ are the relative density
and velocity divergence perturbations of the dark sector (DS:DM or DE),
$w_{\rm ds}$ and $c_{\rm ad,ds}^{2}=\frac{p_{\rm
ds}^{\prime}}{\rho_{ds}^{\prime}}=w_{\rm ds}-\frac{w_{\rm
ds}^{\prime}}{3\mathcal{H}\left(1+w_{\rm ds}\right)}$ denote the DS EoS and
the square of DS adiabatic sound speed, respectively. And $\frac{\delta p_{\rm
ds}}{\delta\rho_{\rm ds}}$ is the DS sound speed in the Newtonian gauge, it
can be expressed as:

$\frac{\delta p_{\rm ds}}{\delta\rho_{\rm ds}}=c_{\mathrm{\rm
s,ds}}^{2}+3\mathcal{H}\left(1+w_{\rm ds}\right)\left(c_{\mathrm{\rm
s,ds}}^{2}-c_{\mathrm{\rm ad,ds}}^{2}\right)\frac{\theta_{\rm ds}}{k^{2}},$
(9)

here $c_{\mathrm{\rm s,ds}}^{2}=c_{\mathrm{\rm ad,ds}}^{2}+c_{\mathrm{\rm
nad,ds}}^{2}$ is the square of DS sound speed in the rest frame. And
$c_{\mathrm{\rm nad,ds}}^{2}$ is the square of DS non-adiabatic sound speed,
it describes the micro-scale properties of DS and needs to be provided
independently. In this work, we consider $c_{\mathrm{\rm nad,dm}}^{2}=0$
(Therefore $c_{\mathrm{\rm s,dm}}^{2}=c_{\mathrm{\rm ad,dm}}^{2}=w_{\rm dm}$)
and $c_{\mathrm{\rm s,de}}^{2}=1$, then the continuity and Euler equations for
GDM and PEDE can be rewritten as:

$\displaystyle\delta_{\rm dm}^{\prime}$ $\displaystyle=$
$\displaystyle-(1+w_{\rm dm})\left(\theta_{\rm dm}-3\phi^{\prime}\right)$ (10)
$\displaystyle\theta_{\rm dm}^{\prime}$ $\displaystyle=$
$\displaystyle-\mathcal{H}(1-3w_{\rm dm})\theta_{\rm dm}+\frac{w_{\rm
dm}}{1+w_{\rm dm}}k^{2}\delta_{\rm dm}+k^{2}\psi\hskip 14.22636pt$ (11)
$\displaystyle\dot{\delta}_{\mathrm{de}}=$
$\displaystyle-(1+w_{\mathrm{de}})\left(\theta_{\mathrm{de}}-3\dot{\phi}\right)-3\mathcal{H}(1-w_{\mathrm{de}})\delta_{\mathrm{de}}$
$\displaystyle-3\mathcal{H}w_{\mathrm{de}}^{\prime}\frac{\theta_{\mathrm{de}}}{k^{2}}-9(1+w_{\mathrm{de}})(1-w_{\mathrm{de}})\mathcal{H}^{2}\frac{\theta_{\mathrm{de}}}{k^{2}},$
(12) $\displaystyle\dot{\theta}_{\mathrm{de}}=$ $\displaystyle
2\mathcal{H}\theta_{\mathrm{de}}+\frac{1}{1+w_{\mathrm{de}}}k^{2}\delta_{\mathrm{de}}+k^{2}\psi.$
(13)

Having presented the equations above, the background and perturbation dynamics
of our new model (label as PEDE+$w_{\rm dm}$) is clearly understood.

At the conclusion of this section, we provide an analysis of the effects of
the PEDE+$w_{\rm dm}$ model on the CMB TT and matter power spectra across
various values of $w_{\rm dm}$. Fig. 1 illustrates the CMB TT by setting
$w_{\rm dm}=1\times 10^{-4}$, $2\times 10^{-4}$, $3\times 10^{-4}$, $4\times
10^{-4}$ and matter power spectra by setting $w_{\rm dm}=1\times 10^{-6}$,
$2\times 10^{-6}$, $3\times 10^{-6}$, $4\times 10^{-6}$ while keeping six
other parameters fixed at their mean values derived from CMB+BAO+Pantheon data
analysis.

Figure 1: The CMB TT and matter power spectra for different values of
parameter $w_{\mathrm{dm}}$, where other relevant model parameters are fixed
to their bestfit values extracted from CMB+BAO+Pantheon data analysis.

In the context of the CMB TT power spectrum, we observe that positive values
of $w_{\rm dm}$ in the PEDE+$w_{\rm dm}$ model result in a decrease in the
amplitude of the acoustic peaks relative to the large-scale anisotropy. At
large scales, where the integrated Sachs-Wolfe effect predominates, the
primary impact of $w_{\rm dm}>0$ is an enhanced decay of the gravitational
potential post-recombination up to the present time, leading to increased
anisotropy for $l<100$. Since the direct effects of the DM EoS parameter on
the order of $10^{-4}$ are negligible compared to those caused by a positive
sound speed squared of the same order of magnitude, the effects described
above are predominantly due to a positive sound speed squared rather than just
a positive DM EoS parameter.

For the matter power spectrum, we can see that positive values of $w_{\rm dm}$
decrease the matter power spectrum. Similar to the case concerning the CMB TT
power spectrum, such effects are mostly due to a positive sound speed squared
rather than just a positive DM EoS parameter.

##  III datasets and methodology

To determine the free parameters of the PEDE+$w_{\mathrm{dm}}$ model, we
utilize the recently described observational datasets outlined below.

Cosmic Microwave Background (CMB): In this work, we employ the Planck 2018
(Aghanim et al., 2020a, b) data on CMB. Specifically, we utilize the CMB
temperature and polarization angular power spectra plikTTTEEE+low$l$+lowE.
Additionally, we incorporate the Planck 2018 CMB lensing reconstruction
likelihood (Aghanim et al., 2020c) into our analysis.

Baryon acoustic oscillation (BAO): BAO distance measurements from several
astronomical surveys including 6dFGS (Beutler et al., 2011), SDSS-MGS (Ross et
al., 2015) and BOSS DR12 (Alam et al., 2017) are considered.

Supernovae Type Ia (Pantheon): We also incorporate the Pantheon catalog of
type Ia supernovae, which consists of 1048 data points in the redshift range
$z\in[0.01,2.3]$ (Scolnic et al., 2018).

Hubble constant (R22): We also present the most recent local measurement of
the Hubble constant, which yields $H_{0}=73.04\pm 1.04$ and was obtained by
the SH0ES Collaboration (Riess et al., 2022). Hereafter, we will refer to this
measurement as R22.

KiDS+VIKING-450: We will also incorporate data from the combination of KiDS
and the VISTA Kilo-Degree Infrared Galaxy Survey (VIKING), abbreviated as
KV450 (Hildebrandt et al., 2020).

To constrain the PEDE+$w_{\mathrm{dm}}$ model, we run a Markov Chain Monte
Carlo (MCMC) using the public code MontePython-v3 Audren et al. (2013);
Brinckmann and Lesgourgues (2019) and a modified version of the CLASS code
Lesgourgues (2011); Blas et al. (2011). We perform the analysis with a
Metropolis-Hasting algorithm and consider chains to be converged using the
Gelman-Rubin Gelman and Rubin (1992) criterion $R-1<0.03$. In Table 1 we
display the priors on the free parameters of PEDE+$w_{\mathrm{dm}}$, that are,
the baryon density $\omega_{b}$, the GDM density $\omega_{\mathrm{dm}}$, the
ratio of the sound horizon to the angular diameter distance $\theta_{s}$, the
amplitude and the spectral index of the primordial scalar perturbations
$A_{s}$ and $n_{s}$, the optical depth $\tau_{\mathrm{reio}}$, and the GDM EoS
parameter $w_{\mathrm{dm}}$.

Parameters | Prior
---|---
$100\omega{}_{b}$ | [0.8,2.4]
$\omega{}_{\mathrm{dm}}$ | [0.01,0.99]
$100\theta{}_{s}$ | [0.5,2.0]
$ln[10^{10}A_{s}]$ | [2.7,4.0]
$n_{s}$ | [0.9,1.1]
$\tau{}_{\mathrm{reio}}$ | [0.01,0.8]
$10^{6}w_{\mathrm{dm}}$ | [0,100]
Table 1: Uniform priors on the free parameters of the PEDE+$w_{\mathrm{dm}}$
model.

Finally, we will analyze the performance of the improved
PEDE+$w_{\mathrm{dm}}$ model in comparison to the $\Lambda$CDM model and the
PEDE model. We utilized the cosmological code MCEvidence, developed by the
authors of Ref. (Heavens et al., 2017a, b), to calculate Bayesian evidence for
all datasets and referred to Ref. (Pan et al., 2018; Yang et al., 2019) for
further discussion on this topic. The performance of a cosmological model
(denoted as $B_{i}$) with respect to a reference cosmological model is
quantified by the Bayes factor $B_{ij}$ (or its logarithm $B_{ij}$) of the
model $M_{i}$ with respect to the reference model $M_{j}$. In Table 2, we
employ the modified Jeffrey’s scale which quantifies observational support for
underlying cosmological models and compare our model with both $\Lambda$CDM
and PEDE models respectively. In this paper, $j$ is $\Lambda$CDM, $i$ is PEDE
or PEDE+$w_{\mathrm{dm}}$.

$\ln B_{ij}$ | Strength of evidence for model $M_{i}$
---|---
$0\leq\ln B_{ij}<1$ | Weak
$1\leq\ln B_{ij}<3$ | Definite/Positive
$3\leq\ln B_{ij}<5$ | Strong
$\ln B_{ij}\geq 5$ | Very strong
Table 2: Revised Jeffreys’ scale quantifies the strength of evidence for model
$M_{i}$ compared to model $M_{j}$.

##  IV results and discussion

In Tab. 3 and Fig. 2, we present the constraints on the PEDE+$w_{\mathrm{dm}}$
model for CMB, CMB+BAO, CMB+BAO+Pantheon, and CMB+BAO+Pantheon+$H_{0}$
datasets. Additionally, we provide the best-fit values and 1$\sigma$ and
2$\sigma$ errors of the parameters of the PEDE model for the same datasets in
Table. 4 for comparison.

We start by examining the fitting results of the PEDE+$w_{\rm dm}$ model using
only CMB data. Subsequently, we explore the impact of incorporating additional
probes by progressively adding them to the CMB analysis. Using only CMB data,
we find no evidence for a non-zero DM parameter $w_{\rm dm}$ at the 68%
confidence level. Specifically, at the 68% confidence level, $w_{\rm dm}$
ranges from $0$ to $0.539\times 10^{-6}$. Nevertheless, differences in some of
the model parameters between the PEDE model and the PEDE+$w_{\rm dm}$ model
are presented due to the small but not negligible positive mean value of
$w_{\rm dm}$. The parameters with significant changes in fitting results are
$\sigma_{8}$ and $S_{8}$. More specifically, the small but non-vanish positive
mean value of parameter $w_{\rm dm}$ reduces the parameter $\sigma_{8}$ and
$S_{8}$ from $\sigma_{8}=0.856\pm 0.006$ (at the 68% confidence level) and
$S_{8}=0.813\pm 0.013$ (at the 68% confidence level) in the PEDE model to
$\sigma_{8}=0.771^{+0.083}_{-0.028}$ (at the 68% confidence level) and
$S_{8}=0.735^{+0.077}_{-0.031}$ (at the 68% confidence level) in the
PEDE+$w_{\rm dm}$ model. We point out that these changes mainly originate from
the effects of a non-zero mean value of DM sound speed squared, which is on
the order of $10^{-6}$, rather than directly from the effects of a DM EoS
parameter of the same magnitude, because DM EoS parameter being of this
magnitude has a negligible direct effect on the fitting results of other
parameters. As for the other parameters, introducing a DM parameter $w_{\rm
dm}$ does not cause them to undergo significant changes. Specifically, the
introduction of $w_{\rm dm}$ only changes $H_{0}$ from $H_{0}=72.5\pm 0.70$
(at the 68% confidence level) in the PEDE model to $H_{0}=72.33\pm 0.75$ (at
the 68% confidence level) in the PEDE+$w_{\rm dm}$ model. As a result, the
Hubble tension with R22 still lies within 1$\sigma$.

With the addition of BAO data to CMB, we observe some changes in the fitting
results. In particular, we find an indication for a positive DM parameter
$w_{\rm dm}$ at the 68% confidence level ($w_{\rm
dm}=0.63^{+0.19}_{-0.44}\times 10^{-6}$ at the 68% confidence level).
Therefore, due to the positive correlation between $w_{\rm dm}$ and
$\Omega_{m}$ along with the negative correlation between $w_{\rm dm}$ and
$H_{0}$, $\sigma_{8}$, and $S_{8}$, we obtain a slightly higher value of
$\Omega_{m}$ and slightly lower values of $H_{0}$, $\sigma_{8}$, and $S_{8}$
compared to those of the PEDE model. As a result, the $H_{0}$ tension with R22
is raised from 1.2$\sigma$ in the PEDE model to 1.4$\sigma$.

When Pantheon are added to CMB+BAO, we still find an indication for a positive
DM parameter $w_{\rm dm}$ at the 68% confidence level ($w_{\rm
dm}=0.67^{+0.19}_{-0.45}\times 10^{-6}$ at the 68% confidence level). And
because of the same reason we discussed in the previous paragraph, a slightly
higher value of $\Omega_{m}$ and slightly lower values of $H_{0}$,
$\sigma_{8}$, and $S_{8}$ compared to those of the PEDE model are obtained,
raising the $H_{0}$ tension with R22 from 1.6$\sigma$ in the PEDE model to
1.8$\sigma$. We can also see that, compared to the case using CMB+BAO, the fit
results for each parameter do not change significantly after adding Pantheon.

For the CMB+BAO+Pantheon+$H_{0}$ dataset, we continue to observe evidence for
a positive DM parameter $w_{\rm dm}$ at the 68% confidence level ($w_{\rm
dm}=0.61^{+0.22}_{-0.41}\times 10^{-6}$ at the 68% confidence level).
Therefore, compared to the PEDE model, there is a slight rise in $\Omega_{m}$
and slight declines in $H_{0}$, $\sigma_{8}$, and $S_{8}$, which elevates the
Hubble tension with R22 from 1.3$\sigma$ in the PEDE model to 1.5$\sigma$ in
our new model. It is worth mentioning that the fit results for each parameter
derived from the three data sets, CMB+BAO, CMB+BAO+Pantheon, and
CMB+BAO+Pantheon+$H_{0}$, are very similar.

Considering that when using data including CMB, both $\sigma_{8}$ and $S_{8}$
are reduced compared to those of the PEDE model. Therefore, it is necessary
for us to verify whether this result is consistent with the WL data, thus
resolving the $\sigma_{8}$ and $S_{8}$ tension. In order to do that, we the
KiDS+VIKING-450 dataset to fit the PEDE+$w_{\rm dm}$ model as well as the PEDE
model, and the constraints on these two models are shown in Fig. 3 and Tab. 5.
From the Tab. 5, we find that, for the PEDE model, the fitting results for
$\sigma_{8}$ and $S_{8}$ from the KiDS+VIKING-450 dataset are
$\sigma_{8}=0.84^{+0.13}_{-0.22}$ and $S_{8}=0.726\pm 0.035$ at the 68%
confidence level, respectively. Consequently, the tensions in $\sigma_{8}$ and
$S_{8}$ between Planck 2018 and KiDS+VIKING-450 within the PEDE model are
0.1$\sigma$ and 2.3$\sigma$, respectively. We can see that the $S_{8}$ tension
is on the verge of being resolved. While for the PEDE+$w_{\rm dm}$ model, the
KiDS+VIKING-450 dataset yields fitting results of
$\sigma_{8}=0.86^{+0.15}_{-0.20}$ and $S_{8}=0.714\pm 0.038$ at the 68%
confidence level. Accordingly, the tension between Planck 2018 and
KiDS+VIKING-450 within the PEDE+$w_{\rm dm}$ model are 0.4$\sigma$ for both
$\sigma_{8}$ and $S_{8}$. In summary, the new model can at least resolve the
$\sigma_{8}$ and $S_{8}$ tension between Planck 2018 and the KiDS+VIKING-450
dataset.

Finally, we present the $\ln B_{ij}$ values quantifying the evidence of fit of
the PEDE model and the PEDE+$w_{\rm dm}$ model with respect to the
$\Lambda$CDM model under the data sets considered in this work in Tab. 6.
Recall the Revised Jeffreys’ scale shown in Tab. 2, we find that although
PEDE+$w_{\rm dm}$ performs better than $\Lambda$CDM when it comes to
alleviating tensions in $H_{0}$, $\sigma_{8}$ and $S_{8}$, the Bayesian
evidence shows that all the data sets considered in this work favor
$\Lambda$CDM more. More specifically, except for the CMB dataset favoring
$\Lambda$CDM over PEDE+$w_{\rm dm}$ with strong evidence, all other datasets
favor $\Lambda$CDM over PEDE+$w_{\rm dm}$ with very strong evidence. In
addition, while PEDE+$w_{\rm dm}$ outperforms PEDE in resolving tensions in
$S_{8}$, the Bayesian evidence shows that all datasets favor PEDE over
PEDE+$w_{\rm dm}$ with very strong evidence.

Figure 2: One-dimensional posterior distributions and two-dimensional joint contours at the 68% and 95% confidence levels for the most relevant parameters of the PEDE+$w_{\rm dm}$ model are shown using the CMB, CMB+BAO, CMB+BAO+Pantheon, and CMB+BAO+Pantheon+$H_{0}$ observational datasets. The R22 results are highlighted in grey. Parameters | CMB | CMB+BAO | CMB+BAO+Pantheon | CMB+BAO+Pantheon+$H_{0}$
---|---|---|---|---
$100\omega{}_{b}$ | ${2.236\pm 0.016}^{+0.031}_{-0.030}$ | ${2.223\pm 0.013}^{+0.027}_{-0.025}$ | ${2.216\pm 0.014}^{+0.029}_{-0.028}$ | ${2.221\pm 0.013}^{+0.026}_{-0.026}$
$\omega{}_{\mathrm{dm}}$ | ${0.1201\pm 0.0013}^{+0.0026}_{-0.0026}$ | ${0.12174\pm 0.00093}^{+0.0018}_{-0.0018}$ | ${0.12256\pm 0.00094}^{+0.0018}_{-0.0018}$ | ${0.12200\pm 0.00094}^{+0.0018}_{-0.0019}$
$100\theta{}_{s}$ | ${1.04188\pm 0.00030}^{+0.00058}_{-0.00059}$ | ${1.04173\pm 0.00030}^{+0.00056}_{-0.00058}$ | ${1.04167\pm 0.00030}^{+0.00056}_{-0.00057}$ | ${1.04175\pm 0.00028}^{+0.00056}_{-0.00053}$
$\ln(10^{10}A_{s})$ | ${3.047\pm 0.015}^{+0.029}_{-0.028}$ | ${3.041\pm 0.015}^{+0.028}_{-0.028}$ | ${3.038\pm 0.017}^{+0.031}_{-0.030}$ | ${3.039\pm 0.015}^{+0.028}_{-0.028}$
$n_{s}$ | ${0.9646\pm 0.0044}^{+0.0087}_{-0.0083}$ | ${0.9607\pm 0.0037}^{+0.0072}_{-0.0072}$ | ${0.9589\pm 0.0038}^{+0.0073}_{-0.0074}$ | ${0.9602^{+0.0039}_{-0.0033}}^{+0.0068}_{-0.0076}$
$\tau{}_{reio}$ | ${0.0551\pm 0.0075}^{+0.015}_{-0.015}$ | ${0.0512\pm 0.0072}^{+0.014}_{-0.014}$ | ${0.0487^{+0.0082}_{-0.0068}}^{+0.016}_{-0.015}$ | ${0.0500\pm 0.0075}^{+0.015}_{-0.015}$
$10^{6}w_{\mathrm{dm}}$ | $<0.539<1.40$ | $0.63^{+0.19}_{-0.44}<1.17$ | $0.67^{+0.19}_{-0.45}<1.20$ | $0.61^{+0.22}_{-0.41}<1.19$
$\Omega_{m}$ | ${0.2725\pm 0.0080}^{+0.016}_{-0.015}$ | ${0.2824\pm 0.0057}^{+0.011}_{-0.011}$ | ${0.2875\pm 0.0057}^{+0.011}_{-0.011}$ | ${0.2839\pm 0.0056}^{+0.011}_{-0.011}$
$H_{0}$ | ${72.33\pm 0.75}^{+1.5}_{-1.5}$ | ${71.40\pm 0.51}^{+1.0}_{-0.98}$ | ${70.95\pm 0.51}^{+1.0}_{-0.96}$ | ${71.28\pm 0.50}^{+1.0}_{-0.96}$
$\sigma_{8}$ | ${0.771^{+0.083}_{-0.028}}^{+0.094}_{-0.13}$ | ${0.751\pm 0.051}^{+0.089}_{-0.093}$ | ${0.748\pm 0.052}^{+0.091}_{-0.089}$ | ${0.750\pm 0.050}^{+0.093}_{-0.092}$
$S_{8}$ | ${0.735^{+0.077}_{-0.031}}^{+0.096}_{-0.12}$ | ${0.729\pm 0.050}^{+0.089}_{-0.086}$ | ${0.732\pm 0.051}^{+0.088}_{-0.090}$ | ${0.730\pm 0.049}^{+0.091}_{-0.089}$
Table 3: The mean values and 1, 2$\sigma$ of the PEDE+$w_{\mathrm{dm}}$ model for CMB, CMB+BAO, CMB+BAO+Pantheon and CMB+BAO+Pantheon+$H_{0}$ datasets. Parameters | CMB | CMB+BAO | CMB+BAO+Pantheon | CMB+BAO+Pantheon+$H_{0}$
---|---|---|---|---
$100\omega{}_{b}$ | ${2.239^{+0.015}_{-0.015}}^{+0.029}_{-0.030}$ | ${2.227^{+0.013}_{-0.013}}^{+0.026}_{-0.026}$ | ${2.221^{+0.013}_{-0.013}}^{+0.026}_{-0.026}$ | ${2.226^{+0.012}_{-0.012}}^{+0.024}_{-0.025}$
$\omega{}_{\mathrm{dm}}$ | ${0.1198^{+0.0012}_{-0.0012}}^{+0.0024}_{-0.0024}$ | ${0.1213^{+0.0009}_{-0.0009}}^{+0.0018}_{-0.0018}$ | ${0.1221^{+0.0009}_{-0.0009}}^{+0.0016}_{-0.0016}$ | ${0.12157^{+0.00082}_{-0.00082}}^{+0.0016}_{-0.0016}$
$100\theta{}_{s}$ | ${1.04189^{+0.00030}_{-0.00030}}^{+0.00059}_{-0.00059}$ | ${1.04175^{+0.00030}_{-0.00030}}^{+0.00058}_{-0.00060}$ | ${1.04172^{+0.00028}_{-0.00028}}^{+0.00057}_{-0.00053}$ | ${1.04175^{+0.00028}_{-0.00028}}^{+0.00056}_{-0.00056}$
$\ln(10^{10}A_{s})$ | ${3.043^{+0.015}_{-0.015}}^{+0.029}_{-0.028}$ | ${3.039^{+0.014}_{-0.014}}^{+0.027}_{-0.028}$ | ${3.036^{+0.014}_{-0.014}}^{+0.026}_{-0.027}$ | ${3.037^{+0.014}_{-0.014}}^{+0.028}_{-0.029}$
$n_{s}$ | ${0.9655^{+0.0042}_{-0.0042}}^{+0.0080}_{-0.0084}$ | ${0.9619^{+0.0036}_{-0.0036}}^{+0.0071}_{-0.0068}$ | ${0.9603^{+0.0035}_{-0.0035}}^{+0.0068}_{-0.0069}$ | ${0.9614^{+0.0035}_{-0.0035}}^{+0.0069}_{-0.0072}$
$\tau{}_{reio}$ | ${0.054^{+0.008}_{-0.008}}^{+0.015}_{-0.014}$ | ${0.050^{+0.007}_{-0.007}}^{+0.014}_{-0.014}$ | ${0.048^{+0.007}_{-0.007}}^{+0.013}_{-0.014}$ | ${0.0494^{+0.0071}_{-0.0071}}^{+0.014}_{-0.015}$
$\Omega_{m}$ | ${0.270^{+0.007}_{-0.007}}^{+0.015}_{-0.014}$ | ${0.280^{+0.006}_{-0.006}}^{+0.011}_{-0.011}$ | ${0.285^{+0.005}_{-0.005}}^{+0.010}_{-0.010}$ | ${0.2812^{+0.0049}_{-0.0049}}^{+0.0096}_{-0.0092}$
$H_{0}$ | ${72.5^{+0.70}_{-0.70}}^{+1.4}_{-1.4}$ | ${71.7^{+0.51}_{-0.51}}^{+1.0}_{-1.0}$ | ${71.21^{+0.47}_{-0.47}}^{+0.91}_{-0.91}$ | ${71.52^{+0.44}_{-0.44}}^{+0.84}_{-0.85}$
$\sigma_{8}$ | ${0.856^{+0.006}_{-0.006}}^{+0.012}_{-0.012}$ | ${0.858^{+0.006}_{-0.006}}^{+0.012}_{-0.012}$ | ${0.859^{+0.006}_{-0.006}}^{+0.012}_{-0.012}$ | ${0.8577^{+0.0063}_{-0.0063}}^{+0.012}_{-0.012}$
$S_{8}$ | ${0.813^{+0.013}_{-0.013}}^{+0.026}_{-0.025}$ | ${0.828^{+0.011}_{-0.011}}^{+0.020}_{-0.021}$ | ${0.836^{+0.010}_{-0.010}}^{+0.020}_{-0.019}$ | ${0.830^{+0.010}_{-0.010}}^{+0.019}_{-0.020}$
Table 4: The mean values and 1, 2$\sigma$ of the PEDE model for CMB, CMB+BAO, CMB+BAO+Pantheon and CMB+BAO+Pantheon+$H_{0}$ datasets. Figure 3: One-dimensional posterior distributions and two-dimensional joint contours at the 68% and 95% confidence levels for the most relevant parameters of the PEDE model and the PEDE+$w_{\rm dm}$ model are shown using the KiDS+VIKING-450 dataset. The R22 results are highlighted in grey. Parameters | PEDE | PEDE+$w_{\mathrm{dm}}$
---|---|---
$100\omega{}_{b}$ | ${2.24^{+0.22}_{-0.33}}^{+0.36}_{-0.35}$ |
$\omega{}_{\mathrm{dm}}$ | ${0.115^{+0.034}_{-0.063}}^{+0.088}_{-0.073}$ | ${0.102^{+0.025}_{-0.054}}^{+0.077}_{-0.061}$
$ln[10^{10}A_{s}]$ | ${3.10^{+0.61}_{-1.3}}^{+1.7}_{-1.4}$ | ${3.25^{+0.79}_{-1.2}}^{+1.6}_{-1.5}$
$n_{s}$ | ${1.04^{+0.16}_{-0.12}}^{+0.24}_{-0.26}$ | ${1.14^{+0.15}_{-0.052}}^{+0.17}_{-0.25}$
$10^{8}w_{\mathrm{dm}}$ | - | $<4.27<8.69$
$\Omega_{m}$ | ${0.250^{+0.064}_{-0.12}}^{+0.17}_{-0.14}$ | ${0.225^{+0.048}_{-0.10}}^{+0.15}_{-0.12}$
$H_{0}$ | ${74.8^{+7.0}_{-2.8}}^{+7.3}_{-9.1}$ | ${74.9^{+6.9}_{-2.8}}^{+7.2}_{-9.7}$
$\sigma_{8}$ | ${0.84^{+0.13}_{-0.22}}^{+0.32}_{-0.27}$ | ${0.86^{+0.15}_{-0.20}}^{+0.30}_{-0.28}$
$S_{8}$ | ${0.726\pm 0.035}^{+0.067}_{-0.070}$ | ${0.714\pm 0.038}^{+0.072}_{-0.078}$
Table 5: The mean values and 1, 2$\sigma$ of the PEDE+$w_{\mathrm{dm}}$ model for the KiDS+VIKING-450 dataset. Model | Datasets | $\ln B_{ij}$
---|---|---
PEDE | CMB | 0.93
PEDE | CMB+BAO | -2.32
PEDE | CMB+BAO+Pantheon | -8.02
PEDE | CMB+BAO+Pantheon+$H_{0}$ | 2.00
PEDE | KiDS+VIKING-450 | -0.002
PEDE+$w_{\mathrm{dm}}$ | CMB | -4.37
PEDE+$w_{\mathrm{dm}}$ | CMB+BAO | -10.80
PEDE+$w_{\mathrm{dm}}$ | CMB+BAO+Pantheon | -17.00
PEDE+$w_{\mathrm{dm}}$ | CMB+BAO+Pantheon+$H_{0}$ | -8.06
PEDE+$w_{\mathrm{dm}}$ | KiDS+VIKING-450 | -6.94
Table 6: Summary of the $\ln B_{ij}$ values, which quantify the evidence of
fit for the PEDE model and the PEDE+$w_{\rm dm}$ model relative to the
$\Lambda$CDM model, using the CMB, CMB+BAO, CMB+BAO+Pantheon,
CMB+BAO+Pantheon+$H_{0}$, and KiDS+VIKING-450 datasets. It should be noted
that a negative $\ln B_{ij}$ value indicates that the PEDE model or the
PEDE+$w_{\rm dm}$ model is less supported compared to the base $\Lambda$CDM
model.

##  V concluding remarks

Although CDM is supported by a wide range of cosmological data when it is
accompanied by a cosmological constant, it encounters many issues inherent to
the CDM paradigm at small scales, such as the missing satellite, too-big-to-
fail, and core-cusp problem. These issues have motivated many new DM
candidates beyond CDM, including warm DM, fuzzy DM, interacting DM and
decaying DM, which are all proposed to suppress the formation of low-mass
structures. The current limitations of CDM leave the question of ”What is DM?”
relatively open. In light of this, we have proposed a new cosmological model
by considering DM as a barotropic fluid characterized by a constant EoS
parameter and DE as PEDE, considering that there have already been many
researches on the extended properties of DM in the context of a cosmological
constant and there are some interesting results emerged as we investigated the
extended properties of DM in the context of PEDE. (Yao et al., 2024).

Given the basic equations of the PEDE+$w_{\rm dm}$ model both at the
background and perturbation levels, we present some analysis concerning the
impacts of the PEDE+$w_{\rm dm}$ model on the CMB TT and matter power spectra
for different values of $w_{\rm dm}$, and find that positive values of $w_{\rm
dm}$ result in a reduction in the amplitude of the acoustic peaks compared to
the large-scale anisotropy occurred, and an increase in the power at large
scale where $l<100$ due to integrated Sachs-Wolfe effect. In addition to that,
positive values of $w_{\rm dm}$ decrease the matter power spectrum. We pointed
out here that the effects on both CMB TT and matter power spectra are mostly
due to a positive sound speed squared rather than just a positive DM EoS
parameter. We have also fitted the model using data sets consisting of CMB,
CMB+BAO, CMB+BAO+Pantheon, CMB+BAO+Pantheon+$H_{0}$, and KiDS+VIKING-450. We
find that the value of parameter $w_{\rm dm}$ is positive at 68% confidence
level for CMB+BAO, CMB+BAO+Pantheon and CMB+BAO+Pantheon+$H_{0}$ datasets,
which leads to a slight rise in $\Omega_{m}$ and slight declines in $H_{0}$,
$\sigma_{8}$, and $S_{8}$. Consequently, the $H_{0}$ tension with R22 is
raised from 1.2$\sigma$ (CMB+BAO), 1.6$\sigma$ (CMB+BAO+Pantheon), and
1.3$\sigma$ (CMB+BAO+Pantheon+$H_{0}$) in the PEDE model to 1.4$\sigma$
(CMB+BAO), 1.8$\sigma$ (CMB+BAO+Pantheon), and 1.5$\sigma$
(CMB+BAO+Pantheon+$H_{0}$) in the PEDE+$w_{\rm dm}$ model. Of course, these
levels of tensions still can be attributed to the statistical fluctuations. We
also find that the $S_{8}$ tension between the Planck 2018 data and the
KiDS+VIKING-450 data are reduced from 2.3$\sigma$ in the PEDE model to
0.4$\sigma$ in the new model. Finally, it is also noteworthy that, although
PEDE+$w_{\rm dm}$ outperforms PEDE in addressing $S_{8}$ tension between CMB
and KiDS+VIKING-450, Bayesian evidence suggests that PEDE favors our new model
with very strong evidence by all the datasets considered in this study.
Therefore, we conclude in the end that the PEDE+$w_{\rm dm}$ model is not a
good alternative to the PEDE model.

###### Acknowledgements.

This work is supported by the National key R&D Program of China (Grant
No.2020YFC2201600), Guangdong Basic and Applied Basic Research Foundation
(Grant No.2024A1515012573), National Natural Science Foundation of China
(Grant No.12073088), and National SKA Program of China (Grant No.
2020SKA0110402).

## References

  * Riess et al. (1998) A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, et al., The Astronomical Journal 116, 1009 (1998).
  * Perlmutter et al. (1999) S. Perlmutter, G. Aldering, G. Goldhaber, R. Knop, P. Nugent, P. Castro, S. Deustua, S. Fabbro, A. Goobar, D. Groom, et al., The Astrophysical Journal 517, 565 (1999).
  * Dunkley et al. (2011) J. Dunkley, R. Hlozek, J. Sievers, V. Acquaviva, P. A. Ade, P. Aguirre, M. Amiri, J. Appel, L. Barrientos, E. Battistelli, et al., The Astrophysical Journal 739, 52 (2011).
  * Hinshaw et al. (2013) G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. Bennett, J. Dunkley, M. Nolta, M. Halpern, R. Hill, N. Odegard, et al., The Astrophysical Journal Supplement Series 208, 19 (2013).
  * Story et al. (2015) K. Story, D. Hanson, P. Ade, K. Aird, J. Austermann, J. Beall, A. Bender, B. Benson, L. Bleem, J. Carlstrom, et al., The Astrophysical Journal 810, 50 (2015).
  * Alam et al. (2017) S. Alam, M. Ata, S. Bailey, F. Beutler, D. Bizyaev, J. A. Blazek, A. S. Bolton, J. R. Brownstein, A. Burden, C.-H. Chuang, et al., Monthly Notices of the Royal Astronomical Society 470, 2617 (2017).
  * Troxel et al. (2018) M. A. Troxel, N. MacCrann, J. Zuntz, T. Eifler, E. Krause, S. Dodelson, D. Gruen, J. Blazek, O. Friedrich, S. Samuroff, et al., Physical Review D 98, 043528 (2018).
  * Aghanim et al. (2020a) N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. Barreiro, N. Bartolo, S. Basak, et al., Astronomy & Astrophysics 641, A6 (2020a).
  * Klypin et al. (1999) A. Klypin, A. V. Kravtsov, O. Valenzuela, and F. Prada, The Astrophysical Journal 522, 82 (1999).
  * Moore et al. (1999a) B. Moore, S. Ghigna, F. Governato, G. Lake, T. Quinn, J. Stadel, and P. Tozzi, The Astrophysical Journal 524, L19 (1999a).
  * Boylan-Kolchin et al. (2012) M. Boylan-Kolchin, J. S. Bullock, and M. Kaplinghat, Monthly Notices of the Royal Astronomical Society 422, 1203 (2012).
  * Moore et al. (1999b) B. Moore, T. Quinn, F. Governato, J. Stadel, and G. Lake, Monthly Notices of the Royal Astronomical Society 310, 1147 (1999b).
  * Springel et al. (2008) V. Springel, J. Wang, M. Vogelsberger, A. Ludlow, A. Jenkins, A. Helmi, J. F. Navarro, C. S. Frenk, and S. D. White, Monthly Notices of the Royal Astronomical Society 391, 1685 (2008).
  * Blumenthal et al. (1982) G. R. Blumenthal, H. Pagels, and J. R. Primack, Nature 299, 37 (1982).
  * Bode et al. (2001) P. Bode, J. P. Ostriker, and N. Turok, The Astrophysical Journal 556, 93 (2001).
  * Hu et al. (2000) W. Hu, R. Barkana, and A. Gruzinov, Physical Review Letters 85, 1158 (2000).
  * Marsh and Silk (2014) D. J. Marsh and J. Silk, Monthly Notices of the Royal Astronomical Society 437, 2652 (2014).
  * Spergel and Steinhardt (2000) D. N. Spergel and P. J. Steinhardt, Physical review letters 84, 3760 (2000).
  * Wang et al. (2014) M.-Y. Wang, A. H. Peter, L. E. Strigari, A. R. Zentner, B. Arant, S. Garrison-Kimmel, and M. Rocha, Monthly Notices of the Royal Astronomical Society 445, 614 (2014).
  * Hu (1998) W. Hu, The Astrophysical Journal 506, 485 (1998).
  * Mueller (2005) C. M. Mueller, Physical Review D 71, 047302 (2005).
  * Kumar and Xu (2014) S. Kumar and L. Xu, Physics Letters B 737, 244 (2014).
  * Armendariz-Picon and Neelakanta (2014) C. Armendariz-Picon and J. T. Neelakanta, Journal of Cosmology and Astroparticle Physics 2014, 049 (2014).
  * Kopp et al. (2018) M. Kopp, C. Skordis, D. B. Thomas, and S. Ilić, Physical Review Letters 120, 221102 (2018).
  * Kumar et al. (2019) S. Kumar, R. C. Nunes, and S. K. Yadav, Monthly Notices of the Royal Astronomical Society 490, 1406 (2019).
  * Ilić et al. (2021) S. Ilić, M. Kopp, C. Skordis, and D. B. Thomas, Physical Review D 104, 043520 (2021).
  * Pan et al. (2023) S. Pan, W. Yang, E. Di Valentino, D. F. Mota, and J. Silk, Journal of Cosmology and Astroparticle Physics 2023, 064 (2023).
  * Yao et al. (2024) Y.-H. Yao, J.-C. Wang, and X.-H. Meng, Physical Review D 109, 063502 (2024).
  * Li and Shafieloo (2019) X. Li and A. Shafieloo, The Astrophysical Journal Letters 883, L3 (2019).
  * Li and Shafieloo (2020) X. Li and A. Shafieloo, The Astrophysical Journal 902, 58 (2020).
  * Yang et al. (2021) W. Yang, E. Di Valentino, S. Pan, A. Shafieloo, and X. Li, Physical Review D 104, 063521 (2021).
  * Hernández-Almada et al. (2020) A. Hernández-Almada, G. Leon, J. Magaña, M. A. García-Aspeitia, and V. Motta, Monthly Notices of the Royal Astronomical Society 497, 1590 (2020).
  * Pan et al. (2020) S. Pan, W. Yang, E. Di Valentino, A. Shafieloo, and S. Chakraborty, Journal of Cosmology and Astroparticle Physics 2020, 062 (2020).
  * Aghanim et al. (2020b) N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. Barreiro, N. Bartolo, S. Basak, et al., Astronomy & Astrophysics 641, A5 (2020b).
  * Aghanim et al. (2020c) N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. Barreiro, N. Bartolo, S. Basak, et al., Astronomy & Astrophysics 641, A8 (2020c).
  * Beutler et al. (2011) F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell, Q. Parker, W. Saunders, and F. Watson, Monthly Notices of the Royal Astronomical Society 416, 3017 (2011).
  * Ross et al. (2015) A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden, and M. Manera, Monthly Notices of the Royal Astronomical Society 449, 835 (2015).
  * Scolnic et al. (2018) D. M. Scolnic, D. Jones, A. Rest, Y. Pan, R. Chornock, R. Foley, M. Huber, R. Kessler, G. Narayan, A. Riess, et al., The Astrophysical Journal 859, 101 (2018).
  * Riess et al. (2022) A. G. Riess, W. Yuan, L. M. Macri, D. Scolnic, D. Brout, S. Casertano, D. O. Jones, Y. Murakami, G. S. Anand, L. Breuval, et al., The Astrophysical journal letters 934, L7 (2022).
  * Hildebrandt et al. (2020) H. Hildebrandt, F. Köhlinger, J. Van den Busch, B. Joachimi, C. Heymans, A. Kannawadi, A. H. Wright, M. Asgari, C. Blake, H. Hoekstra, et al., Astronomy & Astrophysics 633, A69 (2020).
  * Audren et al. (2013) B. Audren, J. Lesgourgues, K. Benabed, and S. Prunet, Journal of Cosmology and Astroparticle Physics 2013, 001 (2013).
  * Brinckmann and Lesgourgues (2019) T. Brinckmann and J. Lesgourgues, Physics of the Dark Universe 24, 100260 (2019).
  * Lesgourgues (2011) J. Lesgourgues, arXiv preprint arXiv:1104.2932 (2011).
  * Blas et al. (2011) D. Blas, J. Lesgourgues, and T. Tram, Journal of Cosmology and Astroparticle Physics 2011, 034 (2011).
  * Gelman and Rubin (1992) A. Gelman and D. B. Rubin, Statistical science 7, 457 (1992).
  * Heavens et al. (2017a) A. Heavens, Y. Fantaye, A. Mootoovaloo, H. Eggers, Z. Hosenie, S. Kroon, and E. Sellentin, arXiv preprint arXiv:1704.03472 (2017a).
  * Heavens et al. (2017b) A. Heavens, Y. Fantaye, E. Sellentin, H. Eggers, Z. Hosenie, S. Kroon, and A. Mootoovaloo, Physical review letters 119, 101301 (2017b).
  * Pan et al. (2018) S. Pan, E. N. Saridakis, and W. Yang, Physical Review D 98, 063510 (2018).
  * Yang et al. (2019) W. Yang, M. Shahalam, B. Pal, S. Pan, and A. Wang, Physical Review D 100, 023522 (2019).