#  Observational Constraints on the Dark Energy with a Quadratic Equation of
State

Hossein Moshafi  moshafi86<EMAIL_ADDRESS>Alireza Talebian
<EMAIL_ADDRESS>Ebrahim Yusofi<EMAIL_ADDRESS>Eleonora Di Valentino
<EMAIL_ADDRESS>School of Astronomy, Institute for Research in
Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran  Innovation and
Management Research Center, Ayatollah Amoli Branch, Islamic Azad University,
Amol, Mazandaran, Iran  School of Mathematics and Statistics, University of
Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom

###### Abstract

In this study, we introduce a novel late-time effective dark energy model
characterized by a quadratic equation of state (EoS) and rigorously examine
its observational constraints. Initially, we delve into the background
dynamics of this model, tracing the evolution of fluctuations in linear order.
Our approach involves substituting the conventional cosmological constant with
a dynamically effective dark energy fluid. Leveraging a diverse array of
observational datasets encompassing the Planck 2018 Cosmic Microwave
Background (CMB), Type Ia Supernovae (SNe), Baryon Acoustic Oscillations
(BAO), and a prior on the Hubble constant $H_{0}$ (R21), we constrain the
model parameters. We establish the model’s consistency by comparing the Hubble
parameter as a function of redshift against observational Hubble data (OHD),
benchmarking its performance against the Standard $\Lambda$CDM model.
Additionally, our investigation delves into studies of the model’s dynamical
behavior by computing cosmological parameters such as the deceleration
parameter, relative Hubble parameter, and the evolution of the Hubble rate.
Furthermore, employing Bayesian analysis, we determine the Bayesian Evidence
for our proposed model compared to the reference $\Lambda$CDM model. While our
analysis unveils the favorable behavior of the model in various observational
tests, the well-known cosmological tensions persist when the full dataset
combination is explored.

###### keywords:

Dark Energy , Quadratic Equation of State , Cosmic Voids , Hubble Tension

††journal: Physics of The Dark Universe

###### Contents

  1. 1 Introduction
  2. 2 The Model with a Quadratic EoS
  3. 3 Observational data
  4. 4 Statistical methods and Analysis
    1. 4.1 Bayesian inference
  5. 5 Testing $\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip 2.5pt$\mathbb{V}$}}$CDM model with Cosmological Data
  6. 6 Summary and Discussions

##  1 Introduction

One of the most groundbreaking discoveries in recent cosmology is the
observation of cosmological acceleration in our Universe. Despite this
revelation, the true nature of this acceleration remains elusive and is
referred to as dark energy (DE). According to the widely accepted standard
model of cosmology, called $\Lambda\rm{CDM}$ (which aligns remarkably well
with current cosmological observations), DE is a cosmological constant and
accounts for $\sim 70\%$ of the energy budget of the Universe. For a more
comprehensive understanding of DE, various cosmological models have been
introduced. Constructing a cosmological model requires two essential elements:
a gravitational theory defining the Universe’s geometry and a comprehensive
picture of its matter content. In this context, DE can be conceptualized
either as an altered form of matter (such as quintessence, k-essence, phantom,
$\cdots$) or as a modification within gravitational theories (like $f(R)$
gravity, Gauss–Bonnet, $\cdots$) [10].

The models based on modified matter can be achieved through the introduction
of an exotic matter source exhibiting negative pressure. The most
straightforward instances include fluids characterized by a linear equation of
state (EoS) [36, 52, 60, 61, 62, 85], denoted as $p=w\rho$, where $p$
represents pressure and $\rho$ signifies energy density.

In diverse cosmological models, including those rooted in extra dimensions
(such as String theory, Brane-worlds, etc.), and within intricate and more
realistic systems, the relationship between pressure and density may not
adhere to a linear correlation [84, 39, 23, 25]. If we assume $p=p(\rho)$ for
any barotropic fluid to be an analytic function, we can consider a more
general form of the equation of state of the form
$p=p_{0}+A_{1}\rho+A_{2}\rho^{2}+\mathcal{O}(\rho^{3})$. In the Brane-world
[76, 19, 48] scenario, the assumption of the extra dimensions produces a
quadratic term in the energy density. In addition, it has been shown that Loop
Quantum Gravity (LQG) corrections result in a modified Friedmann equation with
the modification appearing as a negative term which is quadratic in the energy
density [81, 15, 16, 70, 73].

In this paper, we study an effective dark energy fluid with a quadratic EoS
[13]. The non-linear term in EoS can be originated from the merger process of
clusters/voids and may act as a possible source for cosmic acceleration [88,
55]. The model can explain the evolution of voids [74, 80] in an under-dense
environment within a void-dominated universe [66]. Also, it is shown that this
fluid has a possible balancing effect on the cosmological expansion [55]. In
this work, we substitute a new energy density component $\rho_{\rm v}$,
instead of the cosmological constant $\rho_{\rm\Lambda}$ in $\Lambda$CDM, and
we call it $\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[7.50002pt]{\hskip 2.5pt$\mathbb{V}$}}$CDM model. We consider the
observational data (including the deceleration parameter, relative Hubble
parameter, and the evolution of the Hubble rate) in the framework of the
proposed model.

For comprehensive evaluation against other cosmological models, examining how
the $\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[7.50002pt]{\hskip 2.5pt$\mathbb{V}$}}$CDM model performs with
respect to the cosmological tensions, including the Hubble constant $H_{0}$
tension [83, 67, 27], proves to be particularly insightful. There was another
well-studied problem called $S_{8}$ tension [14, 12, 1, 5, 53, 24, 28] which
seems to be alleviated by the recent study by [2]. Several models have been
proposed for alleviating these problems in the literature [47, 40, 29, 3, 42,
30, 79, 82, 44, 60, 59, 46, 4]. Roughly speaking, two primary categories of DE
models aimed at addressing the $H_{0}$ tension by introducing novel physics:
early-time and late-time DE models. These categories depend on the
modifications to the expansion history, whether they occur before or after the
recombination epoch, respectively. Within this framework, the proposed
$\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model in our study aligns with the latter category.

The paper is organized as follows: in Section 2 we explain the dynamics of the
model and derive the equation of motion of linear perturbations. For testing
the model, we introduce the observational data sets and the statistical
methods used in this work in Sections 3 and 4, respectively. Our results for
this model in the light of the combination of different datasets are
summarized in Section 5. Finally, we discuss the results in Section 6.

##  2 The Model with a Quadratic EoS

We consider a model consisting of a new variable energy density $\rho_{\rm v}$
instead of the constant dark energy density in the standard $\Lambda{\rm CDM}$
model. The new source is a cosmological barotropic fluid with a non-linear
EoS. In this work, we are focusing on the quadratic case [13, 45, 55],

$\displaystyle p_{\rm v}=w_{\rm v}~{}\rho_{\rm v}+\beta~{}\rho_{\rm v}^{2}$
(1)

in which $P_{\rm v}$ and $w_{\rm v}$ are the pressure and the equation of
state parameter, respectively. The new parameter $\beta$ comes from the non-
linearity effect in the over-dense regime that can originate from the merger
of clusters/voids in this area and sets the characteristic energy scale
$\rho_{\rm c}$ of the quadratic term

$\displaystyle\beta\equiv\dfrac{b}{\rho_{\rm c}}$ (2)

where $b$ is a constant non-zero parameter that must be determined with the
observational data.

As an example of application, the equation of state of (1) can describe a
cosmic fluid with two different regimes: under-dense regime (linear term
$\propto\rho_{\rm v}$) and over-dense regime (non-linear term
$\propto\rho_{\rm v}^{2}$). Our model requires the existence of some
phenomenological interactions as the possibility of merging and collapsing of
these two regimes. More precisely, starting with the linear EoS for the fluid,
the merging (collapsing) of two under-dense (over-dense) regimes leads to the
final pressure of the fluid being modified by the non-linear term in (1) [45].
In the general case, the rate of merging and collapsing could be a function of
space and time, and two under-dense and over-dense regions can coexist at the
same time [75].

This picture is consistent with the findings on the evolution and merging of
the vast cosmic voids [74, 80, 66] if voids are considered as the under-dense
regime of our fluid while the density of the fluid on the boundary of voids
where the dark matter halos are located is high and non-linear. More
specifically, in the evolution of voids, two processes are reported: merging
of two voids into larger voids in large-scale under-dense regions and their
collapsing and disappearance in local over-dense regions [74]. For instance,
two small voids merge and form a giant cosmic void. When the adjacent sub-
voids meet up and merge, the matter between them is squeezed into thin walls
and filaments [80]. In such a merging cosmic fluid, our universe consists of
under-dense and over-dense regions in the form of a cosmic web, in which
under-density regions are located inside the voids, and over-dense regions
including strings, clusters, and shells are located at the boundary of the
voids [66].

In our model, we assume the characteristic energy scale $\rho_{\rm c}$ be the
critical energy density of the Universe, i.e. $\rho_{\rm c}\sim\rho_{0}$. This
assumption is based on the previous studies [87, 88] in which the non-linear
term in the total pressure can be related to the surface tension of the voids.

Using the continuity equation, the energy density of the fluids $\rho_{\rm v}$
as a function of the scale factor $a$ for the cases $w_{\rm v}\neq-1$ is given
by [45, 55]

$\rho_{\rm v}(a)=\frac{\rho_{\rm v_{0}}a^{-3(1+w_{\rm
v})}}{1+\frac{\beta\rho_{\rm v_{0}}}{1+w_{\rm v}}[1-a^{-3{(1+w_{\rm
v})}}]}\,,$ (3)

and for the special case ($w_{\rm v}=-1$) is given by

$\rho_{\rm v}(a)=\frac{\rho_{\rm v_{0}}}{1+3\beta\rho_{\rm v_{0}}\ln{a}},$ (4)

where $\rho_{\rm v_{0}}$ is the energy density of the fluid at present $a=1$.
In the limit $\beta\rightarrow 0$, the model (3) covers the phantom ($w_{\rm
v}<-1$) [21] and quintessence ($w_{\rm v}\geqslant-1$) models [63] while the
model (4) meets the well-known $\Lambda{\rm CDM}$ model.

As seen, the model is a dynamical dark energy in which the cosmological
constant is replaced by a quadratic barotropic fluid. The cosmic background
evolution is described through the Friedmann equation,

$H^{2}(z)=\frac{8\pi G}{3}\sum_{i}\rho_{i}(z)$ (5)

in which $G$ is the gravitational constant and $H$ is Hubble parameter and the
total energy density consists of

$\sum_{i}\rho_{i}(z)=\rho_{\rm r}(z)+\rho_{\rm b}(z)+\rho_{\rm
cdm}(z)+\rho_{\rm v}(z)$ (6)

where $\rho_{\rm r}$, $\rho_{\rm b}$ and $\rho_{\rm cdm}$ are the energy
density of radiation, baryon, and cold dark matter, respectively, while
$\rho_{\rm v}$ is given by (3) or (4) depending on $w_{\rm v}$. In the
following, we study some important cosmological parameters in the framework of
the $\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[7.50002pt]{\hskip 2.5pt$\mathbb{V}$}}$CDM model, by careful
examination of the background evolution and first-order perturbations in the
light of observational data.

To investigate the linear growth of density perturbations of both the
pressureless dark matter and dark energy fluid in our model, we use perturbed
Einstein’s equation in the conformal-Newtonian gauge. In this formalism, the
linear density contrast of non-relativistic DM, $\delta_{\rm m}$, and DE,
$\delta_{\rm v}$, satisfy the following equations in Fourier space [11]

$\displaystyle
k^{2}\Phi+3\mathcal{H}\left(\Phi^{\prime}-\mathcal{H}\Psi\right)$
$\displaystyle=4\pi Ga^{2}\left[\rho_{\rm m}\delta_{\rm m}+\rho_{\rm
v}\delta_{\rm v}+\rho_{\rm r}\delta_{\rm r}\right],$ (7) $\displaystyle
k^{2}\left(\Phi^{\prime}-\mathcal{H}\Psi\right)$ $\displaystyle=-4\pi
Ga^{2}\left[\left(1+w_{\rm m}\right)\rho_{\rm m}\theta_{\rm m}+\left(1+w_{\rm
v}+\beta\rho_{\rm v}\right)\rho_{\rm v}\theta_{\rm v}+\left(1+w_{\rm
r}\right)\rho_{\rm r}\theta_{\rm r}\right],$ (8) $\displaystyle\Psi$
$\displaystyle=-\Phi,$ (9)
$\displaystyle\Phi^{\prime\prime}+2\mathcal{H}\Phi^{\prime}-\mathcal{H}\Psi^{\prime}-\left(\mathcal{H}^{2}+2\mathcal{H}^{\prime}\right)\Psi$
$\displaystyle=-4\pi Ga^{2}\left[c^{2}_{\rm s,m}\rho_{\rm m}\delta_{\rm
m}+\left(w_{\rm v}+2\beta\rho_{\rm v}\right)\rho_{\rm v}\delta_{\rm
v}+c^{2}_{\rm s,r}\rho_{\rm r}\delta_{\rm r}\right],$ (10)
$\displaystyle\delta^{\prime}_{\rm v}+3\mathcal{H}\beta\rho_{\rm v}\delta_{\rm
v}$ $\displaystyle=-\left(1+w_{\rm v}+\beta\rho_{\rm
v}\right)\left(\theta_{\rm v}+3\Phi^{\prime}\right),$ (11)
$\displaystyle\theta^{\prime}_{\rm v}+\theta_{\rm v}\left[\left(1-3w_{\rm
v}\right)\mathcal{H}-6\beta\rho_{\rm v}\right]$
$\displaystyle=k^{2}\left(\frac{w_{\rm v}+2\beta\rho_{\rm v}}{1+w_{\rm
v}+\beta\rho_{\rm v}}\delta_{\rm v}+\Psi\right)$ (12)

where $\eta$ is the conformal time and the prime denotes the derivative to the
conformal time. $\Psi$ & $\Phi$ are Bardeen potentials and $\mathcal{H}$ is
conformal Hubble parameter. $c^{2}_{\rm s,r}$ and $c^{2}_{\rm s,m}$ are the
sound speeds and $\theta_{\rm v},\theta_{\rm m}$ and $\theta_{\rm r}$ are
divergence of velocities of corresponding components. Finally $\rho_{\rm
m}=\rho_{\rm b}+\rho_{\rm cdm}$ and $\rho_{\rm r}=\rho_{\gamma}+\rho_{\nu}$
are the density matter and density radiation, respectively. The power spectrum
of matter perturbations $P(k)$ is defined as

$\langle{\delta}_{\rm m}(\vec{k})~{}{\delta}_{\rm
m}(\vec{k^{\prime}})\rangle=\left(2\pi\right)^{3}\delta_{D}\left(\vec{k}+\vec{k^{\prime}}\right)P(k)$
(13)

where $\delta_{D}$ is a Dirac delta function and ${\delta}_{\rm m}$ is Fourier
transformed density field. The variance of the mass distribution smoothed on
$R=8h^{-1}\rm{Mpc}$ scales is defined

$\sigma_{8}^{2}=\frac{1}{2\pi^{2}}\int
P(k)\left[\frac{3j_{1}(kR)}{kR}\right]^{2}k^{2}dk$ (14)

where $j_{1}(kR)$ is a spherical Bessel function.

We’ve implemented the background and the perturbation equations with correct
modifications into the CAMB code [51] and calculated matter, and temperature
and polarization power spectra.

##  3 Observational data

In this section, we describe the cosmological data sets used in this work in
brief. The description of the data catalogs is as follows:

  1. 1.

CMB: We use the most precise full-sky measurements of Cosmic Microwave
Background (CMB) radiation performed by _Planck_ satellite. We use both
high-$\ell$ temperature and polarization angular power spectra from the final
release of “ _Planck 2018_ ” baseline PLIK-LITE TTTEEE along with _Planck_
low-$\ell$ and low-E (SimAll) ($\ell\leq 30$) [5, 6, 7]. We mention all of
_Planck_ data (including temperature and polarization) by “CMB”.

  2. 2.

R21: To test the ability of this model to reconcile the $H_{0}$ tension, we
additionally include a Gaussian prior of the form $H_{0}=73.04\pm 1.04$
$\rm{km}/\rm{s}~{}\rm{Mpc}$, as reported by the SH0ES Collaboration [68]. We
refer to this prior as “R21”.

  3. 3.

BAO: We also consider the various measurements of the Baryon Acoustic
Oscillations (BAO) from different galaxy surveys as the _Planck_ collaboration
in their 2018 analysis [5], i.e. 6dFGS [17], SDSS-MGS [69], and BOSS DR12 [9].
We mention all these data points by “BAO”.

  4. 4.

SN: We include the measurements of the $1048$ Supernovae Type Ia luminosity
distance in the red-shift interval $z\in[0.01,2.3]$, from the Pantheon sample
[72]. We show this catalog of SuperNovae by “SN”.

In some analysis, we use a combination of “CMB+BAO+SN” data which we refer to
this combination as “CBS”.

##  4 Statistical methods and Analysis

Here, we briefly introduce the statistical methods used in our analysis. The
cosmological analysis we carry out in this work is based on Bayesian
inference. To analyze the data and extract the constraints on the cosmological
parameters for $\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[7.50002pt]{\hskip 2.5pt$\mathbb{V}$}}$CDM model, we have
modified carefully the widely used cosmological Markov Chain Monte Carlo
package CosmoMC [50, 49], which is publicly
available.111http://cosmologist.info/cosmomc This package is equipped with a
convergence diagnostic based on the Gelman and Rubin statistic [35], assuming
$|R-1|<0.01$ for all parameters, and implements an efficient sampling of the
posterior distribution using the fast/slow parameter decorrelations [49].

One measure to check the consistency of our model with observational data in
comparison to the standard model is the “Akaike Information Criterium” (AIC)
which is a tool to measure the improvement of the fit. We compute the AIC of
the extended model $\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[7.50002pt]{\hskip 2.5pt$\mathbb{V}$}}$CDM relative to that of
$\Lambda$CDM, defined as

$\Delta{\rm AIC}=\chi^{2}_{\mathrm{min},{\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[5.25pt]{\hskip
2.5pt$\mathbb{V}$}}\rm{CDM}}}-\chi^{2}_{\mathrm{min},\Lambda\rm
CDM}+2(N_{\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[5.25pt]{\hskip 2.5pt$\mathbb{V}$}}\rm{CDM}}-N_{\Lambda\rm
CDM})~{},$ (15)

where $N_{\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[5.25pt]{\hskip 2.5pt$\mathbb{V}$}}\rm{CDM}}$ and $N_{\Lambda\rm
CDM}$ stand for the number of free parameters of $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model and $\Lambda$CDM model [8], respectively.

To put the model to the test for Hubble tension we use the “rule of thumb
difference in mean” or Gaussian Tension (GT) [65], defined as

$\frac{{\bar{x}}_{\mathcal{D}}-{\bar{x}}_{\rm
SH0ES}}{({\sigma}_{\mathcal{D}}^{2}+{\sigma}_{\rm SH0ES}^{2})^{1/2}}\,,$ (16)

where $\overline{x}_{\mathcal{D}}$ and $\sigma_{\mathcal{D}}$ are the mean and
standard deviation of the $H_{0}$ for model based on observational data
$\mathcal{D}$, and $\overline{x}_{\rm SH0ES}$ and $\sigma_{\rm SH0ES}$
correspond to $H_{0}=73.04\pm 1.04$ $\rm{km}/\rm{s}~{}\rm{Mpc}$ [68].

Parameter | Priors
---|---
$w_{\rm v}$ | $[-1.5,+1.5]$
$b$ | $[0.0,1.0]$
$\Omega_{b}h^{2}$ | $[0.005,0.1]$
$\Omega_{c}h^{2}$ | $[0.001,0.99]$
$\tau_{\rm re}$ | $[0.01,0.8]$
$100\Theta_{\rm MC}$ | $[0.5,10]$
$\log\left[10^{10}A_{s}\right]$ | $[1.6,3.91]$
$n_{s}$ | $[0.8,1.2]$
Table 1: Flat priors used on various free parameters of $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model, during statistical analysis.

We do a Maximum Likelihood Analysis by using CosmoMC as a cosmological MCMC
code. To find the best fit of parameters of $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model, we assume a parameter space as:

$\displaystyle\mathcal{P}_{0}\equiv\Bigl{\\{}w_{\rm
v},b,\Omega_{b}h^{2},\Omega_{c}h^{2},100\theta_{\rm
MC},\tau,n_{s},\ln[10^{10}A_{s}]\Bigr{\\}}~{},$ (17)

where $b$ is the void merging parameter, $w_{\rm v}$ represents the equation
of state parameter, $\tau_{\rm re}$ is the reionization optical depth, $n_{s}$
is the scalar spectral index, $A_{s}$ is the amplitude of the scalar
primordial power spectrum, and $\Theta_{\rm MC}$ is an approximation of
$\theta_{*}$, which represents the angular scale of the sound horizon at
decoupling. We always consider a flat Universe ($\Omega_{\rm K}=0$). The
priors assumed for parameters are summarized in Table1.

###  4.1 Bayesian inference

A natural question that arises is how the model is efficient compared to the
standard $\Lambda$CDM cosmology. For this purpose, we need a statistical
measure for comparison between models where the base model will be fixed as
$\Lambda$CDM. This statistical comparison comes through the Bayesian Evidence.
Here, we use publicly available code MCEvidence [37, 38] to compute the
Bayesian evidence of the models.

According to Bayes’ theorem, the probability of a model $M$ with a set of
parameters $\Theta$, in light of the observed data $D$, is given by the
Posterior $\mathcal{P}$:

$\mathcal{P}(\Theta\mid D,M)=\frac{\mathcal{L}(D\mid\Theta,M)\Pi(\Theta\mid
M)}{\mathcal{E}(D\mid M)},$ (18)

where $\mathcal{L}$ is the Likelihood function, $\Pi$ represents the set of
Priors, containing the a priori information about the parameters of the model.
$\mathcal{E}$ is the so-called Evidence, to which we pay particular attention.

For a given model $M$, the Bayesian evidence $\mathcal{E}$ is the normalizing
constant in the right-hand side of Eq. (18). It normalizes the area under the
posterior $\mathcal{P}$ to unity and is given by

$\mathcal{E}(D\mid M)=\int d\Theta\mathcal{L}(D|\Theta,M)\Pi(\Theta|M)\,.$
(19)

The evidence can be neglected in model fitting, but it becomes important in
model comparison. When comparing two different models $M_{1}$ and $M_{2}$
using Bayes’ theorem (18), the ratio of posterior probabilities of the two
models $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ will be proportional to the
ratio of their evidence, this is

$\frac{\mathcal{P}_{1}(\Theta_{1}\mid D,M_{1})}{\mathcal{P}_{2}(\Theta_{2}\mid
D,M_{2})}=\frac{\Pi_{1}(\Theta_{1}|M_{1})}{\Pi_{2}(\Theta_{2}|M_{2})}\frac{\mathcal{E}_{1}(D\mid
M_{1})}{\mathcal{E}_{2}(D\mid M_{2})}\,.$ (20)

This ratio between posteriors leads to the definition of the Bayes Factor
$B_{12}$, which in logarithmic scale is written as

$\ln\mathcal{B}_{12}\equiv\log\left[\frac{\mathcal{E}_{1}(D\mid
M_{1})}{\mathcal{E}_{2}(D\mid M_{2})}\right]=\ln\left[\mathcal{E}_{1}(D\mid
M_{1})\right]-\ln\left[\mathcal{E}_{2}(D\mid M_{2})\right]\,.$ (21)

If $\ln\mathcal{B}_{12}$ is larger (smaller) than unity, the data favors model
$M_{1}$ ($M_{2}$). To assess the strength of the evidence contained in the
data, we use the empirical revised Jeffreys [41] scale, see Table 2.

$|\ln\mathcal{B}_{12}|$ |  Evidence for model ${M}_{1}$
---|---
$|\ln\mathcal{B}_{12}|<1$ | Inconclusive
$1\leq|\ln\mathcal{B}_{12}|<2.5$ | Significant
$2.5\leq|\ln\mathcal{B}_{12}|<5$ | Strong
$|\ln\mathcal{B}_{12}|\geq 5$ | Decisive
Table 2: Revised Jeffreys scale quantifying the observational viability of
cosmological model $M_{1}$ compared to some reference model $M_{2}$ [43].

##  5 Testing $\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[7.50002pt]{\hskip 2.5pt$\mathbb{V}$}}$CDM model with
Cosmological Data

In this section, we aim to find observational constraints on the cosmological
parameters of the $\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[7.50002pt]{\hskip 2.5pt$\mathbb{V}$}}$CDM model by considering
the priors assumed in Table 1. We have summarized the observational
constraints on parameters of the $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model in Table 3 where we report mean values and 68%
confidence intervals of the parameters from “CMB”, “CMB+R21”, “CMB+SN”,
“CMB+SN+R21” and “CMB+BAO+SN (CBS)” datasets. It is worth mentioning that by
using only “CMB” data, our model formally reduces the tension in the $H_{0}$
parameter, but this is due to a volume effect from the increase of the error
bars for a larger number of degrees of freedom, and the mean value is actually
shifting in the wrong direction.

In Fig. 1, we have presented relative 1D posteriors of $H_{0}$ from different
combinations of datasets. The shaded area shows the measurement of $H_{0}$
done by the SH0ES team [68]. Moreover, in Fig. 2 we see a two-dimensional
contour plot of $H_{0}$ vs. $\Omega_{m}$ for different combinations of data.
As we see considering the R21 prior for the $H_{0}$ parameter leads to higher
values for the Hubble constant but the combination of CMB, BAO, and SN data
increases the tension again. Finally, different contour plots at $1\sigma$ and
$2\sigma$ confidence levels showing the correlation between $H_{0}$ and
$\Omega_{m}$ vs. different parameters of the model are shown in Fig. 3.

Parameter | CMB | CMB+R21 | CMB+SN | CMB+SN+R21 | CBS
---|---|---|---|---|---
$w_{\rm v}$ | $-0.43\pm 0.30$ | $-0.84^{+0.21}_{-0.25}$ | $-0.71^{+0.18}_{-0.27}$ | $-0.79^{+0.14}_{-0.29}$ | $-0.69^{+0.17}_{-0.28}$
$b$ | $\rm{unconstrained}$ | $\rm{unconstrained}$ | $\rm{unconstrained}$ | $<0.554$ | $\rm{unconstrained}$
$\Omega_{m}$ | $0.409^{+0.068}_{-0.10}$ | $0.2684\pm 0.0085$ | $0.314\pm 0.012$ | $0.2836\pm 0.0079$ | $0.3080\pm 0.0079$
$H_{0}$ | $60.3^{+5.7}_{-7.0}$ | $72.9\pm 1.0$ | $67.6\pm 1.0$ | $70.59\pm 0.75$ | $67.91\pm 0.82$
$S_{8}$ | $0.898\pm 0.060$ | $0.792\pm 0.019$ | $0.833\pm 0.021$ | $0.795\pm 0.019$ | $0.820\pm 0.014$
$10^{9}A_{s}$ | $2.092\pm 0.034$ | $2.092\pm 0.034$ | $2.091\pm 0.033$ | $2.091\pm 0.035$ | $2.091\pm 0.034$
$n_{s}$ | $0.9618\pm 0.0057$ | $0.9649\pm 0.0056$ | $0.9630\pm 0.0055$ | $0.9684\pm 0.0055$ | $0.9660\pm 0.0046$
$\tau$ | $0.0518\pm 0.0080$ | $0.0532\pm 0.0081$ | $0.0520\pm 0.0079$ | $0.0544\pm 0.0082$ | $0.0535\pm 0.0080$
$\chi^{2}_{\rm R21}$ | $--$ | $1.1$ | $--$ | $7.3$ | $--$
$\chi^{2}_{\rm BAO}$ | $--$ | $--$ | $--$ | $--$ | $6.3$
$\chi^{2}_{\rm SN}$ | $--$ | $--$ | $1035.8$ | $1040.7$ | $1035.6$
$\chi^{2}_{\rm CMB}$ | $630.2$ | $633.4$ | $630.8$ | $632.8$ | $630.7$
$\Delta\rm AIC$ | $+3.9$ | $-13.1$ | $+4.69$ | $-2.06$ | $+5.07$
$\rm\ln\mathcal{B}_{12}$ | $-0.26$ | $+6.08$ | $-2.39$ | $+0.20$ | $-2.86$
Table 3:  68 $\%$ CL constraints on free and derived parameters of the
$\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[6.00002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model from _Planck_ CMB power spectra in combination
with BAO and Type Ia SuperNovae data plus a Gaussian prior on the Hubble
constant $H_{0}$ as measured by SH0ES team. Figure 1: 1D posteriors of $H_{0}$
are shown for the different dataset combinations. The shaded area shows the
measurement of $H_{0}$ done by the SH0ES team and its $1\sigma$ and $2\sigma$
errors [68].  Figure 2: 2D contour plots at $1\sigma$ and $2\sigma$ confidence
levels for $H_{0}$ vs. $\Omega_{m}$ for the $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[6.00002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model for different combination of data. The dashed
line marker is a central value of the $H_{0}$ measurement reported by the
SH0ES team (R21) and the shaded area shows the $1\sigma$ and $2\sigma$
confidence levels of this measurement.

Figure 3: 2D contour plots of $H_{0}$ and $\Omega_{m}$ vs. the parameters
$w_{\rm v}$ , $b$, and $S_{8}$ for the $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[6.00002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model for different combination of data. The vertical
dashed line in the upper panel indicates the measurement of the Hubble
parameter made by [68] and the shaded area depicts the 1$\sigma$ and 2$\sigma$
error bars.

In the last row of Table 3 we have shown the values of Bayes Factor in the
logarithmic scale $\ln\mathcal{B}_{12}$ computed for $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model against the $\Lambda$CDM one considering all the
datasets. We find that the values of $\ln\mathcal{B}_{12}$ for “CMB+SN” and a
combination of “CMB+BAO+SN” are negative indicating that $\Lambda$CDM is
preferred in these datasets significantly and strongly, respectively. However,
in the combination of CMB data with a prior on $H_{0}$ (R21) we find a
positive value for $\ln\mathcal{B}_{12}$ that indicates very strong evidence
for $\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[7.50002pt]{\hskip 2.5pt$\mathbb{V}$}}$CDM model over the
reference model $\Lambda$CDM. This is because for the $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model the Hubble tension is reduced because of a
volume effect, so it is easier to accommodate a higher $H_{0}$ value than in
the standard $\Lambda$CDM scenario. Finally, for the “CMB” and “CMB+SN+R21”
cases, we find a value of $|\ln\mathcal{B}_{12}|$ that is $<1$ indicating that
the $\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[7.50002pt]{\hskip 2.5pt$\mathbb{V}$}}$CDM model is as favored as
$\Lambda$CDM from the data point of view.

For more insight we have plotted three-dimensional contours (see Fig. 4) for
Hubble constant $H_{0}$ vs. $\Omega_{m}$, in one case with the merging rate of
cosmic voids $b$, as the third dimensions in color, and in the other case with
the equation of state parameter $w_{\rm v}$. Fig. 4 shows that while there is
no clear correspondence between larger values of the Hubble expansion rate and
$b$, on the contrary, they occur with $w_{\rm v}$ closer to $-1$.

Figure 4: Upper: Three-dimensional contour of $H_{0}$ vs. $\Omega_{m}$, where
the $b$ parameter is shown as a third dimension by variation of its color.
Lower: Similar plot as in the upper panel where the effect of $w_{\rm v}$
parameter is shown as the third dimension.

We calculated the power spectrum of the CMB temperature anisotropies for the
$\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model for the best fit obtained with the “CMB”
dataset. The result is shown in Fig. 5 which demonstrates a very good
consistency with observational data. The shaded area in Fig. 5 denotes the
cosmic variance of the power spectrum. According to this figure for higher
multipoles ($\ell$) the prediction of $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM and $\Lambda\rm{CDM}$ are very similar. However, for
the plateau part of the power spectrum ($\ell<30$) the tail of the
$\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model is lower than the standard model, helping in
recovering the low quadruple. It shows our model has a weaker Integrated
Sachs-Wolfe effect (ISW) which can be due to the effect of large over-
densities in the cosmic super-clusters. This feature is expected because of
the large value of $S_{8}$ parameter (according to Table 3) which means we
have more clusters in $\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[7.50002pt]{\hskip 2.5pt$\mathbb{V}$}}$CDM model than
$\Lambda$CDM.

Figure 5: Power spectrum of the temperature anisotropies for the
$\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[6.00002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model with a dashed orange line obtained with the
best-fit of the parameters for the “CMB”. The theoretical prediction of the
$\Lambda\rm{CDM}$ model is shown with a solid black line. Blue points show
observational data of _Planck_ 2018 release. The shaded area represents the
cosmic variance of the power spectrum. The lower panel shows the residual of
the two models to observational data.

In Fig. 6 we see the evolution of the Hubble parameter for the $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model based on the best-fit of the different dataset
combinations in comparison with $\Lambda\rm{CDM}$ model. The data points
include the current Observational Hubble Data (OHD) in the ranges $0\leq z\leq
2.36$ (see Table 4), which were obtained by using the “cosmic chronometric
approach (CCA)” [32, 31, 57, 77]. As we can see in Fig. 6 the $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model at low redshift shows good consistency with
observational data. However, at higher redshifts, the Hubble rate based on
best-fit parameters from the analysis of only “CMB” and also “CMB+BAO+SN
(CBS)“ data show deviation from some observational data points. Moreover, for
investigating the accelerating behavior of $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model, we have represented the evolution of the
deceleration parameter $q(z)$ in Fig. 7.

The relative difference in Hubble parameter of the $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[7.50002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model with $\Lambda$CDM model is defined as follows:

$\Delta\mathcal{H}(z)=100\times\left[\frac{\mathcal{H}_{\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[5.25pt]{\hskip
2.5pt$\mathbb{V}$}}\rm{CDM}}(z)}{\mathcal{H}_{\Lambda\rm{CDM}}(z)}-1\right].$
(22)

The above quantity is illustrated in Fig. 8 as a function of redshift.

$z$ | $H(z)$ | Ref. | $z$ | $H(z)$ | Ref.
---|---|---|---|---|---
$0.0$ | $67.77\pm 1.3$ | [54] | $0.07$ | $69.0\pm 19.6$ | [78]
$0.09$ | $69.0\pm 12.0$ |  [77] | $0.01$ | $69.0\pm 12.0$ |  [78]
$0.12$ | $68.6\pm 26.2$ |  [78] | $0.17$ | $83.0\pm 8.0$ |  [78]
$0.179$ | $75.0\pm 4.0$ |  [57] | $0.1993$ | $75.0\pm 5.0$ |  [57]
$0.2$ | $72.9\pm 29.6$ |  [78] | $0.24$ | $79.7\pm 2.7$ |  [34]
$0.27$ | $77.0\pm 14.0$ |  [78] | $0.28$ | $88.8\pm 36.6$ | [78]
$0.35$ | $82.7\pm 8.4$ |  [9] | $0.352$ | $83.0\pm 14.0$ |  [57]
$0.38$ | $81.5\pm 1.9$ |  [22] | $0.3802$ | $83.0\pm 13.5$ |  [9]
$0.4$ | $95.0\pm 17$ |  [77] | $0.4004$ | $77.0\pm 10.2$ |  [58]
$0.4247$ | $87.1\pm 11.2$ |  [58] | $0.43$ | $86.5\pm 3.7$ |  [34]
$0.44$ | $82.6\pm 7.8$ |  [18] | $0.44497$ | $92.8\pm 12.9$ |  [58]
$0.47$ | $89\pm 49.6$ |  [64] | $0.4783$ | $80.9\pm 9.0$ |  [58]
$0.48$ | $97.0\pm 60.0$ |  [78] | $0.51$ | $90.4\pm 1.9$ |  [22]
$0.57$ | $96.8\pm 3.4$ |  [71] | $0.593$ | $104\pm 13$ |  [57]
$0.60$ | $87.9\pm 6.1$ |  [18] | $0.61$ | $97.3\pm 2.1$ |  [22]
$0.68$ | $92.0\pm 8.0$ |  [57] | $0.73$ | $97.3\pm 7.0$ |  [18]
$0.781$ | $105\pm 12.0$ |  [57] | $0.875$ | $125\pm 17.0$ |  [57]
$0.88$ | $90\pm 40.0$ |  [78] | $0.9$ | $117\pm 23.0$ |  [78]
$1.037$ | $154\pm 20.0$ |  [34] | $1.3$ | $168\pm 17$ |  [78]
$1.363$ | $160\pm 33.6$ |  [56] | $1.43$ | $177\pm 18$ |  [78]
$1.53$ | $140\pm 14$ |  [78] | $1.75$ | $202\pm 40$ |  [56]
$1.965$ | $186.5\pm 50.4$ |  [34] | $2.3$ | $224\pm 8.0$ |  [20]
$2.34$ | $222\pm 7.0$ |  [26] | $2.36$ | $226\pm 8$ |  [33]
Table 4: The currently available OHD measurements. Figure 6: Evolution of
Hubble parameter for the $\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[6.00002pt]{\hskip 2.5pt$\mathbb{V}$}}$CDM model, using best-fit
values of cosmological parameters for “CMB” (dashed blue), “CMB+R21” (dash-
dotted green), “CMB+SN” (dotted red), “CMB+SN+R21” (dashed orange), and “CBS”
(dotted magenta). The $H(z)$ of $\Lambda\rm{CDM}$ model is shown with a solid
black line. OHD points in Table 4 are shown in purple. Figure 7: Evolution of
deceleration parameter $q(z)$ for the $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[6.00002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model, using best-fit cosmological parameters from
Table 3 for “CMB” (dashed blue), “CMB+R21” (dash-dotted green), “CMB+SN”
(dotted red), “CMB+SN+R21” (dashed orange), and “CBS” (dotted magenta). For
comparison, the standard $\Lambda\rm{CDM}$ prediction is also shown in a solid
black line.  Figure 8: Relative difference of the Hubble parameter as a
function of the redshift for the $\mathop{\hbox
to0.0pt{$\mathbb{V}$\hss}\mkern 2.0mu\makebox[6.00002pt]{\hskip
2.5pt$\mathbb{V}$}}$CDM model, using best-fit cosmological parameters from
Table 3 for “CMB” (dashed blue), “CMB+R21” (dash-dotted green), “CMB+SN”
(dotted red), “CMB+SN+R21” (dashed orange), and “CBS” (dotted magenta).

##  6 Summary and Discussions

In this paper, we have investigated the observational predictions of a
homogeneous and isotropic cosmological fluid with a non-linear EoS given by
Eq. (1) for DE. The model introduces two additional free parameters beyond the
$\Lambda$CDM model. In the standard cosmological model, the dark fluid is
attributed to the accelerating expansion of the Universe, characterized by
$w_{\rm v}\simeq-1$ and $\Omega_{\rm v}\simeq 0.7$.

We examined an effective fluid with a non-linear quadratic EoS that drives the
accelerating expansion. Such non-linear terms in the EoS may arise from the
merger process of clusters/voids. The outcome of such mergers could be the
observed accelerating expansion at cosmic scales.

In this study, we considered Einstein’s field equation at the background level
in the presence of the dark fluid to investigate the Hubble parameter tension.
Additionally, we calculated the amplitude of matter density fluctuations at
the perturbation level. We assessed the model’s consistency with observational
data by examining it with various combinations of cosmological data. Testing
the model solely with “CMB” observational data (Planck 2018), we obtained a
constraint for the Hubble constant that reads $H_{0}=60.3^{+5.7}_{-7.0}\
\rm{km/sMpc}$. This result suggests a reduction in the Hubble tension to
approximately a $2\sigma$ level. However, due to significant large error bars
in the $H_{0}$ parameter caused by a volume effect, the Hubble parameter lacks
precise constraints.

Furthermore, we demonstrated the consistency of the Hubble rate as a function
of redshift in our model compared to OHD from Table 4, contrasting it with the
Standard $\Lambda\rm{CDM}$ model in Fig. 6. Additionally, we explored the
dynamical behavior of the model by calculating other observational functions,
such as the deceleration parameter (Fig. 7), and relative Hubble parameter
(Fig. 8).
We calculated the Bayesian Evidence for our model in comparison with the
reference model $\Lambda$CDM (last row of Table 3). Our findings indicate a
strong preference for the $\mathop{\hbox to0.0pt{$\mathbb{V}$\hss}\mkern
2.0mu\makebox[7.50002pt]{\hskip 2.5pt$\mathbb{V}$}}$CDM model over the
$\Lambda$CDM model when utilizing CMB (_Planck_) data along with a prior on
$H_{0}$ (R21). However, this preference is attributed to a reduced Hubble
constant tension in this specific scenario and diminishes when R21 is not
taken into account.

It is important to note that our study focused on a specific case of a non-
linear EoS. While our analysis reveals favorable behavior in various
observational tests, it is noteworthy that the well-known cosmological
tensions persist when the full dataset combination is explored. Future work
will be directed towards considering more general non-linear EoS for dark
energy. In subsequent studies, we plan to explore these cases for the dark
energy model and also investigate the non-linear structure formation of such
models to derive statistics for dark matter halos and cosmic voids.

## Acknowledgments

The authors are grateful to Hassan Firouzjahi for his valuable remarks and
comments. Also, we thank Sunny Vagnozzi for his helpful comments. We
acknowledge Haidar Sheikhahmadi for his revision and fruitful comments. A. T.
would like to thank the Yukawa Institute for Theoretical Physics at Kyoto
University, ICTP, University of Rwanda, and ICTP-EAIFR for their kind
hospitalities when some parts of the project were carried out. E.D.V. is
supported by a Royal Society Dorothy Hodgkin Research Fellowship. E.Y. is
supported by the Islamic Azad University, Ayatollah Amoli branch, Amol, Iran.
This article is based upon work from COST Action CA21136, Addressing
observational tensions in cosmology with systematics and fundamental physics
(CosmoVerse) supported by COST (European Cooperation in Science and
Technology).

## References

  * [1] T. M. C. Abbott et al. [DES], Phys. Rev. D 107, no.8, 083504 (2023) doi:10.1103/PhysRevD.107.083504 [arXiv:2207.05766 [astro-ph.CO]].
  * [2] Abbott, T. M. C., et al., Open J. Astrophys. 6, 2305.17173 (2023) doi:10.21105/astro.2305.17173
  * [3] E. Abdalla, G. Franco Abellán, A. Aboubrahim, A. Agnello, O. Akarsu, Y. Akrami, G. Alestas, D. Aloni, L. Amendola and L. A. Anchordoqui, et al. JHEAp 34, 49-211 (2022) doi:10.1016/j.jheap.2022.04.002 [arXiv:2203.06142 [astro-ph.CO]].
  * [4] Banihashemi, A., Khosravi, N. and Shafieloo, A. JCAP 06, 003 (2021) doi:10.1088/1475-7516/2021/06/003
  * [5] N. Aghanim et al. [Planck], Astron. Astrophys. 641, A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)] doi:10.1051/0004-6361/201833910 [arXiv:1807.06209 [astro-ph.CO]].
  * [6] N. Aghanim et al. [Planck], Astron. Astrophys. 641, A8 (2020) doi:10.1051/0004-6361/201833886 [arXiv:1807.06210 [astro-ph.CO]].
  * [7] N. Aghanim et al. [Planck], Astron. Astrophys. 641, A5 (2020) doi:10.1051/0004-6361/201936386 [arXiv:1907.12875 [astro-ph.CO]].
  * Akaike [1974] Akaike, H. 1974, IEEE Transactions on Automatic Control, 19, 716 doi:10.1109/TAC.1974.1100705
  * [9] S. Alam et al. [BOSS], Mon. Not. Roy. Astron. Soc. 470, no.3, 2617-2652 (2017) doi:10.1093/mnras/stx721 [arXiv:1607.03155 [astro-ph.CO]].
  * Amendola & Tsujikawa [2010] Amendola, L., & Tsujikawa, S. 2010, Dark Energy: Theory and Observations (Cambridge University Press).  https://books.google.co.uk/books?id=Xge0hg_AIIYC
  * Amendola & Tsujikawa [2015] —. 2015, Dark Energy: Theory and Observations (Cambridge University Press)
  * [12] A. Amon et al. [DES], Phys. Rev. D 105, no.2, 023514 (2022) doi:10.1103/PhysRevD.105.023514 [arXiv:2105.13543 [astro-ph.CO]].
  * [13] K. N. Ananda and M. Bruni, Phys. Rev. D 74, 023523 (2006) doi:10.1103/PhysRevD.74.023523 [arXiv:astro-ph/0512224 [astro-ph]].
  * [14] M. Asgari et al. [KiDS], Astron. Astrophys. 645, A104 (2021) doi:10.1051/0004-6361/202039070 [arXiv:2007.15633 [astro-ph.CO]].
  * [15] A. Ashtekar, T. Pawlowski and P. Singh, Phys. Rev. D 73, 124038 (2006) doi:10.1103/PhysRevD.73.124038 [arXiv:gr-qc/0604013 [gr-qc]].
  * [16] A. Ashtekar, T. Pawlowski and P. Singh, Phys. Rev. D 74, 084003 (2006) doi:10.1103/PhysRevD.74.084003 [arXiv:gr-qc/0607039 [gr-qc]].
  * [17] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell, Q. Parker, W. Saunders and F. Watson, Mon. Not. Roy. Astron. Soc. 416, 3017-3032 (2011) doi:10.1111/j.1365-2966.2011.19250.x [arXiv:1106.3366 [astro-ph.CO]].
  * [18] Blake, C., Brough, S., Colless, M., et al. 2012, Mon. Not. Roy. Astron. Soc. 405 doi:10.1111/j.1365-2966.2012.21473.x
  * [19] H. A. Bridgman, K. A. Malik and D. Wands, Phys. Rev. D 65, 043502 (2002) doi:10.1103/PhysRevD.65.043502 [arXiv:astro-ph/0107245 [astro-ph]].
  * Busca et al. [2013] Busca, D., et al. 2013, A & A, 552, A96, doi:10.1051/0004-6361/201220724
  * [21] R. R. Caldwell, Phys. Lett. B 545, 23-29 (2002) doi:10.1016/S0370-2693(02)02589-3 [arXiv:astro-ph/9908168 [astro-ph]].
  * Chuang & Wang [2013] Chuang, C.-H., & Wang, Y. 2013, Mon. Not. Roy. Astron. Soc., 435, 255
  * [23] M. Cicoli, J. P. Conlon, A. Maharana, S. Parameswaran, F. Quevedo and I. Zavala, Phys. Rept. 1059, 1-155 (2024) doi:10.1016/j.physrep.2024.01.002 [arXiv:2303.04819 [hep-th]].
  * [24] R. Dalal, X. Li, A. Nicola, J. Zuntz, M. A. Strauss, S. Sugiyama, T. Zhang, M. M. Rau, R. Mandelbaum and M. Takada, et al. Phys. Rev. D 108, no.12, 123519 (2023) doi:10.1103/PhysRevD.108.123519 [arXiv:2304.00701 [astro-ph.CO]].
  * [25] U. Danielsson, O. Henriksson and D. Panizo, Phys. Rev. D 107, no.2, 026020 (2023) doi:10.1103/PhysRevD.107.026020 [arXiv:2211.10191 [hep-th]].
  * [26] T. Delubac et al. [BOSS], Astron. Astrophys. 574, A59 (2015) doi:10.1051/0004-6361/201423969 [arXiv:1404.1801 [astro-ph.CO]].
  * [27] E. Di Valentino, L. A. Anchordoqui, O. Akarsu, Y. Ali-Haimoud, L. Amendola, N. Arendse, M. Asgari, M. Ballardini, S. Basilakos and E. Battistelli, et al. Astropart. Phys. 131, 102605 (2021) doi:10.1016/j.astropartphys.2021.102605 [arXiv:2008.11284 [astro-ph.CO]].
  * [28] E. Di Valentino, L. A. Anchordoqui, Ö. Akarsu, Y. Ali-Haimoud, L. Amendola, N. Arendse, M. Asgari, M. Ballardini, S. Basilakos and E. Battistelli, et al. Astropart. Phys. 131, 102604 (2021) doi:10.1016/j.astropartphys.2021.102604 [arXiv:2008.11285 [astro-ph.CO]].
  * [29] E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess and J. Silk, Class. Quant. Grav. 38, no.15, 153001 (2021) doi:10.1088/1361-6382/ac086d [arXiv:2103.01183 [astro-ph.CO]].
  * [30] H. G. Escudero, J. L. Kuo, R. E. Keeley and K. N. Abazajian, Phys. Rev. D 106, no.10, 103517 (2022) doi:10.1103/PhysRevD.106.103517 [arXiv:2208.14435 [astro-ph.CO]].
  * [31] M. O. Farooq, [arXiv:1309.3710 [astro-ph.CO]].
  * [32] O. Farooq, F. R. Madiyar, S. Crandall and B. Ratra, Astrophys. J. 835, no.1, 26 (2017) doi:10.3847/1538-4357/835/1/26 [arXiv:1607.03537 [astro-ph.CO]].
  * [33] A. Font-Ribera et al. [BOSS], JCAP 05, 027 (2014) doi:10.1088/1475-7516/2014/05/027 [arXiv:1311.1767 [astro-ph.CO]].
  * [34] E. Gaztanaga, A. Cabre and L. Hui, Mon. Not. Roy. Astron. Soc. 399, 1663-1680 (2009) doi:10.1111/j.1365-2966.2009.15405.x [arXiv:0807.3551 [astro-ph]].
  * [35] A. Gelman and D. B. Rubin, Statist. Sci. 7, 457-472 (1992) doi:10.1214/ss/1177011136
  * [36] Harko, T., Asadi, K.,Moshafi, H., and Sheikhahmadi, H. Phys. Dark Univ. 38, 101131 (2022) doi:10.1016/j.dark.2022.101131
  * [37] A. Heavens, Y. Fantaye, A. Mootoovaloo, H. Eggers, Z. Hosenie, S. Kroon and E. Sellentin, [arXiv:1704.03472 [stat.CO]].
  * [38] A. Heavens, Y. Fantaye, E. Sellentin, H. Eggers, Z. Hosenie, S. Kroon and A. Mootoovaloo, Phys. Rev. Lett. 119, no.10, 101301 (2017) doi:10.1103/PhysRevLett.119.101301 [arXiv:1704.03467 [astro-ph.CO]].
  * [39] R. Jalalzadeh, S. Jalalzadeh and B. Malekolkalami, Phys. Dark Univ. 41, 101235 (2023) doi:10.1016/j.dark.2023.101235 [arXiv:2304.07229 [gr-qc]].
  * [40] K. Jedamzik, L. Pogosian and G. B. Zhao, Commun. in Phys. 4, 123 (2021) doi:10.1038/s42005-021-00628-x [arXiv:2010.04158 [astro-ph.CO]].
  * Jeffreys [1961] Jeffreys, H. 1961, Theory of probability (Oxford: Clarendon Press)
  * [42] M. Kamionkowski and A. G. Riess, Ann. Rev. Nucl. Part. Sci. 73, 153-180 (2023) [arXiv:2211.04492 [astro-ph.CO]].
  * [43] R. E. Kass and A. E. Raftery, J. Am. Statist. Assoc. 90, no.430, 773-795 (1995) doi:10.1080/01621459.1995.10476572
  * [44] A. R. Khalife, M. B. Zanjani, S. Galli, S. Günther, J. Lesgourgues and K. Benabed, [arXiv:2312.09814 [astro-ph.CO]].
  * [45] M. Khanpour, E. Yusofi and B. Khanpour, [arXiv:1709.08612 [astro-ph.CO]].
  * [46] N. Khosravi, S. Baghram, N. Afshordi and N. Altamirano, Phys. Rev. D 99, no.10, 103526 (2019) doi:10.1103/PhysRevD.99.103526 [arXiv:1710.09366 [astro-ph.CO]].
  * [47] L. Knox and M. Millea, Phys. Rev. D 101, no.4, 043533 (2020) doi:10.1103/PhysRevD.101.043533 [arXiv:1908.03663 [astro-ph.CO]].
  * [48] D. Langlois, Astrophys. Space Sci. 283, 469-479 (2003) doi:10.1023/A:1022552617831 [arXiv:astro-ph/0301022 [astro-ph]].
  * [49] A. Lewis, Phys. Rev. D 87, no.10, 103529 (2013) doi:10.1103/PhysRevD.87.103529 [arXiv:1304.4473 [astro-ph.CO]].
  * [50] A. Lewis and S. Bridle, Phys. Rev. D 66, 103511 (2002) doi:10.1103/PhysRevD.66.103511 [arXiv:astro-ph/0205436 [astro-ph]].
  * [51] A. Lewis, A. Challinor and A. Lasenby, Astrophys. J. 538, 473-476 (2000) doi:10.1086/309179 [arXiv:astro-ph/9911177 [astro-ph]].
  * [52] X. Li and A. Shafieloo, Astrophys. J. Lett. 883, no.1, L3 (2019) doi:10.3847/2041-8213/ab3e09 [arXiv:1906.08275 [astro-ph.CO]].
  * [53] X. Li, T. Zhang, S. Sugiyama, R. Dalal, R. Terasawa, M. M. Rau, R. Mandelbaum, M. Takada, S. More and M. A. Strauss, et al. Phys. Rev. D 108, no.12, 123518 (2023) doi:10.1103/PhysRevD.108.123518 [arXiv:2304.00702 [astro-ph.CO]].
  * [54] E. Macaulay et al. [DES], Mon. Not. Roy. Astron. Soc. 486, no.2, 2184-2196 (2019) doi:10.1093/mnras/stz978 [arXiv:1811.02376 [astro-ph.CO]].
  * [55] S. Mohammadi, E. Yusofi, M. Mohsenzadeh and M. K. Salem, Mon. Not. Roy. Astron. Soc. 525, no.3, 3274-3280 (2023) doi:10.1093/mnras/stad2514 [arXiv:2309.07826 [astro-ph.CO]].
  * [56] M. Moresco, Mon. Not. Roy. Astron. Soc. 450, no.1, L16-L20 (2015) doi:10.1093/mnrasl/slv037 [arXiv:1503.01116 [astro-ph.CO]].
  * [57] M. Moresco, A. Cimatti, R. Jimenez, L. Pozzetti, G. Zamorani, M. Bolzonella, J. Dunlop, F. Lamareille, M. Mignoli and H. Pearce, et al. JCAP 08, 006 (2012) doi:10.1088/1475-7516/2012/08/006 [arXiv:1201.3609 [astro-ph.CO]].
  * [58] M. Moresco, L. Pozzetti, A. Cimatti, R. Jimenez, C. Maraston, L. Verde, D. Thomas, A. Citro, R. Tojeiro and D. Wilkinson, JCAP 05, 014 (2016) doi:10.1088/1475-7516/2016/05/014 [arXiv:1601.01701 [astro-ph.CO]].
  * [59] Moshafi, H., Baghram, S. and Khosravi, N. Phys. Rev. D 104, no.6, 063506 (2021) doi:10.1103/PhysRevD.104.063506 [arXiv:2012.14377 [astro-ph.CO]].
  * [60] Moshafi, H., Firouzjahi, H. and Talebian, A. Astrophys. J. 940, no.2, 121 (2022) doi:10.3847/1538-4357/ac9c58
  * [61] B. Mostaghel, H. Moshafi and S. M. S. Movahed, Eur. Phys. J. C 77, no.8, 541 (2017) doi:10.1140/epjc/s10052-017-5085-1 [arXiv:1611.08196 [astro-ph.CO]].
  * [62] Mostaghel, B., Moshafi, H., and Movahed,S. M. S Mon. Not. Roy. Astron. Soc. 481, no.2, 1799-1808 (2018) doi:10.1093/mnras/sty2384
  * Papantonopoulos [2005] Papantonopoulos, E. 2005, The Physics of the Early Universe, Lecture Notes in Physics (Springer Berlin Heidelberg).  https://books.google.co.jp/books?id=2fkFPh2jNIkC
  * [64] A. L. Ratsimbazafy, S. I. Loubser, S. M. Crawford, C. M. Cress, B. A. Bassett, R. C. Nichol and P. Väisänen, Mon. Not. Roy. Astron. Soc. 467, no.3, 3239-3254 (2017) doi:10.1093/mnras/stx301 [arXiv:1702.00418 [astro-ph.CO]].
  * [65] M. Raveri and W. Hu, Phys. Rev. D 99, no.4, 043506 (2019) doi:10.1103/PhysRevD.99.043506 [arXiv:1806.04649 [astro-ph.CO]].
  * Redmount [1988] Redmount, I. H. 1988, Monthly Notices of the Royal Astronomical Society, 235, 1301, doi:10.1093/mnras/235.4.1301
  * [67] A. G. Riess, Nature Rev. Phys. 2, no.1, 10-12 (2019) doi:10.1038/s42254-019-0137-0 [arXiv:2001.03624 [astro-ph.CO]].
  * [68] A. G. Riess, W. Yuan, L. M. Macri, D. Scolnic, D. Brout, S. Casertano, D. O. Jones, Y. Murakami, L. Breuval and T. G. Brink, et al. Astrophys. J. Lett. 934, no.1, L7 (2022) doi:10.3847/2041-8213/ac5c5b [arXiv:2112.04510 [astro-ph.CO]].
  * [69] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden and M. Manera, Mon. Not. Roy. Astron. Soc. 449, no.1, 835-847 (2015) doi:10.1093/mnras/stv154 [arXiv:1409.3242 [astro-ph.CO]].
  * [70] K. Saaidi, H. Sheikhahmadi and A. H. Mohammadi, Astrophys. Space Sci. 338, 355-361 (2012) doi:10.1007/s10509-011-0944-y [arXiv:1201.0275 [gr-qc]].
  * [71] P. Sarmah and U. D. Goswami, Mod. Phys. Lett. A 37, no.21, 2250134 (2022) doi:10.1142/S0217732322501346 [arXiv:2203.00385 [gr-qc]].
  * [72] D. M. Scolnic et al. [Pan-STARRS1], Astrophys. J. 859, no.2, 101 (2018) doi:10.3847/1538-4357/aab9bb [arXiv:1710.00845 [astro-ph.CO]].
  * [73] R. Sengupta, B. C. Paul, M. Kalam, P. Paul and A. Aich, Eur. Phys. J. Plus 138, no.10, 929 (2023) doi:10.1140/epjp/s13360-023-04541-w [arXiv:2302.09062 [gr-qc]].
  * Sheth & Van De Weygaert [2004] Sheth, R. K., & Van De Weygaert, R. 2004, Monthly Notices of the Royal Astronomical Society, 350, 517, doi:10.1111/j.1365-2966.2004.07661.x
  * [75] J. Shim, C. Park, J. Kim and H. S. Hwang, Astrophys. J. 908, no.2, 211 (2021) doi:10.3847/1538-4357/abd0f6 [arXiv:2012.03511 [astro-ph.CO]].
  * [76] T. Shiromizu, K. i. Maeda and M. Sasaki, Phys. Rev. D 62, 024012 (2000) doi:10.1103/PhysRevD.62.024012 [arXiv:gr-qc/9910076 [gr-qc]].
  * [77] J. Simon, L. Verde and R. Jimenez, Phys. Rev. D 71, 123001 (2005) doi:10.1103/PhysRevD.71.123001 [arXiv:astro-ph/0412269 [astro-ph]].
  * [78] D. Stern, R. Jimenez, L. Verde, M. Kamionkowski and S. A. Stanford, JCAP 02, 008 (2010) doi:10.1088/1475-7516/2010/02/008 [arXiv:0907.3149 [astro-ph.CO]].
  * [79] S. Vagnozzi, Universe 9, no.9, 393 (2023) doi:10.3390/universe9090393 [arXiv:2308.16628 [astro-ph.CO]].
  * Van De Weygaert & Platen [2011] Van De Weygaert, R., & Platen, E. 2011, International Journal of Modern Physics: Conference Series, 01, 41–66, doi:10.1142/s2010194511000092
  * [81] K. Vandersloot, Phys. Rev. D 71, 103506 (2005) doi:10.1103/PhysRevD.71.103506 [arXiv:gr-qc/0502082 [gr-qc]].
  * [82] L. Verde, N. Schöneberg and H. Gil-Marín, [arXiv:2311.13305 [astro-ph.CO]].
  * [83] L. Verde, T. Treu and A. G. Riess, Nature Astron. 3, 891 doi:10.1038/s41550-019-0902-0 [arXiv:1907.10625 [astro-ph.CO]].
  * [84] K. Wood, P. M. Saffin and A. Avgoustidis, JCAP 07, 062 (2023) doi:10.1088/1475-7516/2023/07/062 [arXiv:2304.09205 [hep-th]].
  * [85] W. Yang, E. Di Valentino, S. Pan, S. Basilakos and A. Paliathanasis, Phys. Rev. D 102, no.6, 063503 (2020) doi:10.1103/PhysRevD.102.063503 [arXiv:2001.04307 [astro-ph.CO]].
  * [86] E. Yusofi, M. Khanpour, B. Khanpour, M. A. Ramzanpour and M. Mohsenzadeh, Mon. Not. Roy. Astron. Soc. 511, no.1, L82-L86 (2022) doi:10.1093/mnrasl/slac006 [arXiv:1907.12418 [astro-ph.CO]].
  * [87] E. Yusofi and M. A. Ramzanpour, [arXiv:2204.12180 [astro-ph.CO]].
  * [88] E. Yusofi, M. Khanpour, B. Khanpour, M. A. Ramzanpour and M. Mohsenzadeh, Mon. Not. Roy. Astron. Soc. 511, no.1, L82-L86 (2022) doi:10.1093/mnrasl/slac006 [arXiv:1907.12418 [astro-ph.CO]].