# Dynamical dark energy confronted with multiple CMB missions

Mahdi Najafi<EMAIL_ADDRESS>Department of Physics, K.N. Toosi
University of Technology, P.O. Box 15875-4416, Tehran, Iran  PDAT Laboratory,
Department of Physics, K.N. Toosi University of Technology, P.O. Box
15875-4416, Tehran, Iran  Supriya Pan<EMAIL_ADDRESS>Department of Mathematics, Presidency University, 86/1 College Street, Kolkata
700073, India  Institute of Systems Science, Durban University of Technology,
PO Box 1334, Durban 4000, Republic of South Africa  Eleonora Di Valentino
<EMAIL_ADDRESS>School of Mathematics and Statistics, University
of Sheffield, Hounsfield Road, Sheffield S3 7RH, United Kingdom  Javad T.
Firouzjaee<EMAIL_ADDRESS>Department of Physics, K.N. Toosi University
of Technology, P.O. Box 15875-4416, Tehran, Iran  PDAT Laboratory, Department
of Physics, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran,
Iran  School of Physics, Institute for Research in Fundamental Sciences (IPM),
P.O. Box 19395-5531, Tehran, Iran

###### Abstract

The measurements of the cosmic microwave background (CMB) have played a
significant role in understanding the nature of dark energy. In this article,
we investigate the dynamics of the dark energy equation of state, utilizing
high-precision CMB data from multiple experiments. We begin by examining the
Chevallier-Polarski-Linder (CPL) parametrization, a commonly used and
recognized framework for describing the dark energy equation of state. We then
explore the general Exponential parametrization, which incorporates CPL as its
first-order approximation, and extensions of this parametrization that
incorporate nonlinear terms. We constrain these scenarios using CMB data from
various missions, including the Planck 2018 legacy release, the Wilkinson
Microwave Anisotropy Probe (WMAP), the Atacama Cosmology Telescope (ACT), and
the South Pole Telescope (SPT), as well as combinations with low-redshift
cosmological probes such as Baryon Acoustic Oscillations (BAO) and the
Pantheon sample. While the $\Lambda$CDM cosmology remains consistent within
the 68% confidence level, we observe that the extensions of the CPL
parametrization are indistinguishable for Planck data. However, for ACT and
SPT data, the inclusion of additional terms begins to reveal a peak in $w_{\rm
a,DE}$ that was previously unconstrained.

##  I Introduction

One of the noteworthy discoveries of the twentieth century is the detection of
the late-time accelerating expansion of the universe, which opened a new
chapter in modern cosmology Riess  _et al._ (1998); Perlmutter  _et al._
(1999). This phenomenon requires the existence of some ’exotic’ kind of
fluid(s) with large negative pressure. Despite the lack of a clear
understanding of the origin of such exotic fluid(s) based on current
observational evidence, there are several possible explanations for this
accelerating expansion of the universe. The simplest approach is to modify the
matter sector of the universe by introducing some exotic fluid(s), referred to
as ’dark energy’ (DE), while assuming that the gravitational sector of our
universe is perfectly described by Einstein’s General Relativity (GR). In this
framework, the evolution of DE density could either remain constant or vary
with cosmic time. The cosmological constant represents the simplest DE
candidate, with a density that remains constant throughout the evolution of
the universe. However, there are numerous examples of DE models that exhibit
time-dependent behavior Copeland  _et al._ (2006); Bamba  _et al._ (2012).
Conversely, modifications of GR or new gravitational theories beyond GR can
also give rise to effective exotic types of fluids with negative pressure
(often termed as geometrical DE), which consequently serve as a source for the
late-time cosmic acceleration. Pursuing this avenue, various geometrical DE
models have been proposed Nojiri and Odintsov (2006a); Sotiriou and Faraoni
(2010); De Felice and Tsujikawa (2010); Clifton  _et al._ (2012); Capozziello
and De Laurentis (2011); Cai  _et al._ (2016); Nojiri  _et al._ (2017). In a
spatially flat, homogeneous, and isotropic Friedmann-Lemaître-Robertson-Walker
(FLRW) universe, if we consider Einstein’s GR in the background and assume a
DE fluid alongside normal matter (NM) in the matter sector, the Friedmann
equations take the form: $\rho_{\rm NM}+\rho_{\rm
DE}=\frac{3H^{2}}{\kappa^{2}}$, and $p_{\rm NM}+p_{\rm
DE}=-\frac{2\dot{H}+3H^{2}}{\kappa^{2}}$, where $H$ is the Hubble rate of the
FLRW universe, an overhead dot represents the cosmic time derivative,
$\kappa^{2}$ is Einstein’s gravitational constant, and $\rho_{i},p_{i}$ are
respectively the energy density and pressure of the $i$-th fluid ($i={\rm
NM},{\rm DE}$). Now, if the gravitational sector is described by a modified
version of GR or by a completely different gravity theory, then the Friedmann
equations in the FLRW universe can be expressed similarly to those in the case
of GR Nojiri  _et al._ (2017):

$\displaystyle\rho_{\rm NM}+\rho_{\rm eff}=\frac{3H^{2}}{\kappa^{2}},\quad
p_{\rm NM}+p_{\rm eff}=-\frac{2\dot{H}+3H^{2}}{\kappa^{2}},$ (1)

where $\rho_{\rm eff}$ and $p_{\rm eff}$ are respectively the energy density
and pressure of the effective fluid that mimics the effects of DE in GR. For
example, if one considers a modified version of GR described by $F(R)=R+f(R)$
(where $R$ is the Ricci scalar), then $\rho_{\rm eff}$ and $p_{\rm eff}$ will
contain the terms responsible for $f(R)$. In reality, one can propose any DE
(in terms of a specific energy-momentum tensor) or geometrical DE model
(assuming any modified version of GR or a completely new gravitational theory)
and investigate the underlying model with the observational data. Among these
existing cosmological models, the $\Lambda$-Cold Dark Matter ($\Lambda$CDM)
cosmological model within the framework of GR stands out. In this model,
$\Lambda>0$ serves as the source of the DE (the energy density of DE, i.e.,
$\Lambda/\kappa^{2}$, is constant over time), and DM is pressure-less.
$\Lambda$CDM represents the simplest cosmological scenario that agrees with
several, but not all, observational datasets. However, $\Lambda$CDM has faced
challenges for various reasons Abdalla  _et al._ (2022), prompting suggestions
for a revision of $\Lambda$CDM cosmology based on existing discrepancies
between early and late cosmological probes.

Along this line, a popular approach to revise the $\Lambda$CDM model is to
parametrize the equation of state of the dark component (either DE or
geometrical DE), which is defined as the ratio of its pressure to the energy
density. Parametrizing the equation of state of DE or geometrical DE can offer
a potential route to understand the inherent nature of the dark sector. By
examining the gravitational equations (1), one can reconstruct the underlying
gravitational theory through
$w_{\text{eff}}=p_{\text{eff}}/\rho_{\text{eff}}$. Over the last several
years, various forms of $w_{\text{DE}}$ ($w_{\text{eff}}$) have been explored
with observational data (see, for example, Efstathiou (1999); Chevallier and
Polarski (2001); Astier (2001); Weller and Albrecht (2002); Linder (2003);
Rubano  _et al._ (2003); Guo  _et al._ (2005); Feng  _et al._ (2006); Nesseris
and Perivolaropoulos (2005); Jassal  _et al._ (2005); Gong and Zhang (2005);
Nojiri and Odintsov (2006b); Kurek  _et al._ (2008); Barboza and Alcaniz
(2008); Kurek  _et al._ (2010); Barboza  _et al._ (2009); Li and Zhang (2011,
2012); Pace  _et al._ (2012); Feng and Lu (2011); Feng  _et al._ (2012);
Akarsu  _et al._ (2015); Chávez  _et al._ (2016); Pantazis  _et al._ (2016);
Yang  _et al._ (2017, 2018); Pan  _et al._ (2018); Rezaei  _et al._ (2017);
Mehrabi and Basilakos (2018); Yang  _et al._ (2019a, b); Panotopoulos and
Rincón (2018); Pan  _et al._ (2020a); Chudaykin  _et al._ (2021); Yang  _et
al._ (2021a, b, c); Benaoum  _et al._ (2020); Wang (2022); Yang  _et al._
(2023); Rezaei  _et al._ (2024); Escamilla  _et al._ (2024a); Adil  _et al._
(2024); Rezaei (2024); Escamilla  _et al._ (2024b); Moshafi  _et al._ (2024);
Reyhani  _et al._ (2024) and the references therein). In this article, we
consider an Exponential model for the DE equation of state Dimakis  _et al._
(2016); Pan  _et al._ (2020b), which, up to its first order of expansion,
returns the well-known Chevallier-Polarski-Linder parametrization Chevallier
and Polarski (2001); Linder (2003). The second and third terms of the
expansion add corrections to the CPL parametrization. Our aim is to constrain
the generalized Exponential form and its expansion up to the third order using
cosmic microwave background (CMB) observations from multiple missions. With
increasing sensitivity in astronomical data and the availability of various
datasets, it is crucially important to examine whether the latest cosmological
probes are offering any new information in this direction.

The article is organized as follows. In Section II, we describe the
gravitational equations of a general DE model, considering its evolution at
both the background and perturbation levels. We then introduce the DE
parametrizations that we wish to study. In Section III, we describe the
cosmological probes and the numerical methodology adopted to constrain the
present DE parametrizations.

##  II Parametrized Dark Energy

As mentioned in the introduction, we adopt the spatially flat FLRW line
element to describe the geometrical configuration of our universe, with its
gravitational sector assumed to be governed by Einstein’s General Relativity.
The fluids in the matter sector, including relativistic radiation,
pressureless matter, and a DE component with an evolving equation of state
$w_{\text{DE}}(z)$, do not interact with each other. The complete set of
equations describing this underlying scenario is given by:

$\displaystyle H^{2}=\frac{\kappa^{2}}{3}(\rho_{r}+\rho_{m}+\rho_{\rm DE}),$
(2) $\displaystyle 2\dot{H}+3H^{2}=-\kappa^{2}(p_{r}+p_{m}+p_{\rm DE}),$ (3)

where $\rho_{i}$ and $p_{i}$ represent, respectively, the energy density and
pressure of the $i$-th fluid ($i$ denotes radiation ($r$), pressureless matter
($m$), and dark energy (DE)). Note that from Bianchi’s identity, one can
derive the conservation equation for the total fluid
$\sum_{i}\dot{\rho}_{i}+3H\sum_{i}(p_{i}+\rho_{i})=0$ . Since the fluids do
not interact with each other, one can write down the conservation equation for
each individual fluid as follows:

$\displaystyle\dot{\rho}_{i}+3H(p_{i}+\rho_{i})=0.$ (4)

Now, for radiation, $p_{r}=\rho_{r}/3$; for pressure-less matter, $p_{m}=0$;
and for DE with a dynamical equation of state $w_{\text{DE}}(z)$, one can use
the balance equation (4) to find:

$\displaystyle\rho_{r}=\rho_{r,0}(a/a_{0})^{-4},\quad\rho_{m}=\rho_{m,0}(a/a_{0})^{-3},$
(5) $\displaystyle\rho_{\rm DE}=\rho_{\rm
DE,0}\;\exp\left[3\;\int_{a}^{a_{0}}\frac{1+w_{\rm
DE}(a^{\prime})}{a^{\prime}}da^{\prime}\right],$ (6)

where $\rho_{i,0}$ refers to the present value of $\rho_{i}$ and $a_{0}$ is
the scale factor at the current time (hereafter we set $a_{0}=1$). Thus, given
a specific form of $w_{\text{DE}}(z)$, one can, in principle, determine the
evolution of the energy density of the DE sector either analytically or
numerically. Finally, using the Friedmann equation (2), one can obtain the
expansion scale factor. This implies that the evolution of the cosmological
scenario driven by the DE equation of state can be derived at the background
level. However, along with the evolution at the background level, it is
essential to understand how the model affects the structure formation of the
universe. In other words, it is crucial to investigate whether the prescribed
DE equation of state can lead to a stable cosmological scenario at the level
of perturbations as well. The next paragraph provides the key cosmological
equations in this context. To incorporate the evolution of the underlying
model at the level of perturbations, we consider the following metric:

$\displaystyle
ds^{2}=a^{2}(\tau)\left[-d\tau^{2}+(\delta_{ij}+h_{ij})dx^{i}dx^{j}\right],$
(7)

where $\tau$ is the conformal time, $\delta_{ij}$ and $h_{ij}$ are
respectively the unperturbed and perturbed metric tensors. For the perturbed
metric (7), utilizing the conservation equation for the $i$-th fluid, i.e.,
$T^{\mu\nu}_{;\nu}=0$, the Einstein’s equations in the synchronous gauge of
Fourier space $\kappa$ can be expressed as Mukhanov  _et al._ (1992); Ma and
Bertschinger (1995); Malik and Wands (2009):

$\displaystyle\delta^{\prime}_{i}=-(1+w_{i})\,\left(\theta_{i}+\frac{h^{\prime}}{2}\right)-3\mathcal{H}\left(\frac{\delta
p_{i}}{\delta\rho_{i}}-w_{i}\right)\delta_{i}$
$\displaystyle-9\mathcal{H}^{2}\left(\frac{\delta
p_{i}}{\delta\rho_{i}}-c^{2}_{a,i}\right)(1+w_{i})\frac{\theta_{i}}{{\kappa}^{2}},$
(8) $\displaystyle\theta^{\prime}_{i}=-\mathcal{H}\left(1-3\frac{\delta
p_{i}}{\delta\rho_{i}}\right)\theta_{i}+\frac{\delta
p_{i}/\delta\rho_{i}}{1+w_{i}}\,{\kappa}^{2}\,\delta_{i}-{\kappa}^{2}\sigma_{i},$
(9)

where a prime attached to any quantity denotes the derivative with respect to
the conformal time $\tau$; $\mathcal{H}=a^{\prime}/a$ denotes the conformal
Hubble factor; $\delta_{i}=\delta\rho_{i}/\rho_{i}$ denotes the density
perturbation; $h=h^{j}{j}$ is the trace of the metric perturbations $h{ij}$;
$\theta_{i}\equiv i\kappa^{j}v_{j}$ refers to the divergence of the $i$-th
fluid velocity; $\sigma_{i}$ stands for the anisotropic stress of the $i$-th
fluid, which we have not considered further. Additionally,
$c_{a,i}^{2}=\dot{p}_{i}/\dot{\rho}_{i}$ represents the adiabatic speed of
sound of the $i$-th fluid. For an imperfect fluid, $c^{2}_{s}=\delta
p_{i}/\delta\rho_{i}$ stands for the sound speed of the $i$-th fluid. Hence,
one can establish the following relation
$c^{2}_{a,i}=w_{i}-\frac{w_{i}^{\prime}}{3\mathcal{H}(1+w_{i})}.$ For a
constant DE equation of state, $p_{\rm DE}=w_{\rm DE}\rho_{\rm DE}$,
$c_{s}^{2}=w_{\rm DE}<0$. Consequently, the sound speed in this medium becomes
imaginary, leading to an unphysical situation. In order to avoid this, we fix
$c_{s}^{2}=1$ similar to Ref. Valiviita  _et al._ (2008).

With the complete cosmological equations at both the background and
perturbation levels, one can proceed to investigate the underlying
cosmological scenario for any given DE equation of state. In the era of
precise cosmology, it has become easier to constrain cosmological parameters
with higher accuracy through observations. However, as suggested by Dimakis
_et al._ (2016); Pan  _et al._ (2020b), one can consider the DE equation of
state to have the following dynamical Exponential form:

$\displaystyle w_{\rm DE}=w_{\rm 0,DE}-w_{\rm a,DE}+w_{\rm
a,DE}\exp\left(\frac{z}{1+z}\right),$ (10)

where $w_{\rm 0,DE}$ is the present value of the DE equation of state, i.e.,
$w_{\rm DE}(z=0)=w_{\rm 0,DE}$; $w_{\rm a,DE}$ is another free parameter; and
$1+z=a_{0}/a$. This parametrization has two benefits: first, at its first
order of approximation, it will return the well-known alternative dynamical DE
model, Chevallier-Polarski-Linder (CPL) parametrization Chevallier and
Polarski (2001); Linder (2003):

$\displaystyle w_{\rm DE}=w_{\rm 0,DE}+w_{\rm a,DE}\;\frac{z}{1+z}.$ (11)

Second, it will enable us to see the effects of higher orders of correction on
cosmological parameters. To this end, we expand (10) up to second and third
orders:

$\displaystyle w_{\rm DE}(z)$ $\displaystyle=$ $\displaystyle w_{\rm
0,DE}+w_{\rm
a,DE}\left[\frac{z}{1+z}+\frac{1}{2!}\left(\frac{z}{1+z}\right)^{2}\right],$
(12) $\displaystyle w_{\rm DE}(z)$ $\displaystyle=$ $\displaystyle w_{\rm
0,DE}+w_{\rm
a,DE}\Bigg{[}\frac{z}{1+z}+\frac{1}{2!}\left(\frac{z}{1+z}\right)^{2}+$ (13)
$\displaystyle+\frac{1}{3!}\left(\frac{z}{1+z}\right)^{3}\Bigg{]}.$

We label the equation of state (12) as CPL-A and the equation of state (13) as
CPL-B. This approach enables the investigation of both the influence of
diverse cosmological probes on parameters and the assessment of higher-order
corrections to the CPL model. In summary, expressing the DE equation of state
in exponential form facilitates the study of parameter constraints using
different observational datasets and allows observation of the effects of
higher-order corrections to the CPL model through the expansion of Equation
(10).

##  III Observational Datasets and Statistical methodology

In this section, we describe the cosmological probes obtained from various
observational surveys and the numerical methodology considered to constrain
the proposed DE scenarios. Below, we outline the cosmological probes.

  * •

Cosmic microwave background (CMB) data: One of the main components of this
article involves comparing CMB data from multiple independent experiments, as
detailed below.

    * –

Planck 2018 legacy release: The baseline likelihoods include Aghanim  _et al._
(2020a, b): Commander likelihood, providing low multipoles TT data in the
range of $2\leq\ell\leq 29$; SimAll likelihood, offering low multipoles EE
data in the range of $2\leq\ell\leq 29$; Plik TT,TE,EE likelihood, providing
high multipoles TT, TE, and EE data in the range of $30\leq\ell\leq 2508$ for
TT and $30\leq\ell\leq 1996$ for TE and EE; and lensing reconstruction
likelihood, which can provide complementary information to the Planck CMB
power spectra.

    * –

Atacama Cosmology Telescope (ACT): We consider the temperature and
polarization anisotropy DR4 likelihood from CMB measurements made by the
Atacama Cosmology Telescope from 2017 to 2021 Aiola  _et al._ (2020); Choi
_et al._ (2020).

    * –

South Pole Telescope (SPT): We analyse the temperature and polarization (TT TE
EE) likelihood from CMB measurements made from the South Pole
TelescopeBalkenhol  _et al._ (2023).

    * –

Wilkinson Microwave Anisotropy Probe (WMAP): For a complete analysis of
Planck-independent CMB measurements Bennett  _et al._ (2013), we utilized the
temperature and polarization WMAP nine-year data at low-$\ell$, excluding TE
data due to the possibility of dust contamination Ade  _et al._ (2014).
Consequently, we set the minimum multipole in TE at $\ell=24$, enabling us to
combine the WMAP data with ACT and SPT likelihoods, considering a Gaussian
prior $\tau=0.065\pm 0.015$, as conducted in the ACT DR4 analysis Aiola  _et
al._ (2020).

We label the Planck dataset simply as Planck, while we refer the combination
of ACT+$\tau$ prior+WMAP and SPT+$\tau$ prior+WMAP as ATW and STW
respectively.

  * •

Baryon acoustic oscillations (BAO) data: We have used both isotropic and
anisotropic distance and expansion rate measurements of the SDSS-IV eBOSS
survey Alam  _et al._ (2021).

  * •

Pantheon sample of Type Ia Supernovae (SNeIa): We have used the Pantheon
catalog, consisting 1048 B-band observation of relative magnitudes of Type Ia
Supernovae distributed in the redshift interval $z=[0.01,2.3]$ Scolnic  _et
al._ (2018).

To constrain the underlying scenarios, we utilized a modified version of the
publicly available cosmological code CAMB Lewis  _et al._ (2000); Howlett  _et
al._ (2012). For exploring the posterior distributions of a given parameter
space, we employed the publicly available Monte Carlo Markov Chain package
Cobaya Lewis and Bridle (2002); Lewis (2013), generating parameter spaces with
the speed hierarchy implementation "fast dragging" Neal (2005), which includes
a convergence diagnostic following the Gelman and Rubin statistic Gelman and
Rubin (1992). In Table 1, we present the flat priors on the following free
parameters: i) $\Omega_{\mathrm{b}}h^{2}$ representing the physical baryon
density, ii) $\Omega_{\mathrm{c}}h^{2}$ representing the physical cold dark
matter density, iii) $A_{\mathrm{s}}$ representing the amplitude of the
primordial scalar perturbations, iv) $n_{s}$ representing the scalar spectral
index, v) $\tau$ representing the optical depth, vi) $\theta_{MC}$
representing the ratio of the sound horizon to the angular diameter distance,
vii) $w_{\rm 0,DE}$ representing the present value of $w_{\rm DE}$, and viii)
$w_{\rm a,DE}$ describing the evolution with redshift of $w_{\rm DE}$.

For model comparison, in addition to the relative best-fit ${\Delta\chi^{2}}$,
we calculate the relative log-Bayesian evidence $\ln\mathcal{B}_{ij}$ using
the MCEvidence package111github.com/yabebalFantaye/MCEvidence Heavens  _et
al._ (2017a, b). A negative value indicates that $\Lambda$CDM is preferred
over the tested model. To interpret the results, we refer to the revised
Jeffreys’ scale by Trotta Kass and Raftery (1995); Trotta (2008): the evidence
is inconclusive if $0\leq|\ln\mathcal{B}_{ij}|<1$, weak if
$1\leq|\ln\mathcal{B}_{ij}|<2.5$, moderate if
$2.5\leq|\ln\mathcal{B}_{ij}|<5$, strong if $5\leq|\ln\mathcal{B}_{ij}|<10$,
and very strong if $|\ln\mathcal{B}_{ij}|\geq 10$.

Parameter | Prior
---|---
$\Omega_{\mathrm{b}}h^{2}$ | $[0.005,0.1]$
$\Omega_{\mathrm{c}}h^{2}$ | $[0.01,0.99]$
$\log(10^{10}A_{\mathrm{s}})$ | $[1.61,3.91]$
$n_{\mathrm{s}}$ | $[0.8,1.2]$
$\tau$ | $[0.01,0.8]$
$100\theta_{\mathrm{MC}}$ | $[0.5,10]$
$w_{\rm 0,DE}$ | $[-3,1]$
$w_{\rm a,DE}$ | $[-3,2]$
Table 1: Flat priors on the free parameters considered in the analyses. For
alternative CMB likelihoods (ACT and SPT), Gaussian prior $\tau=0.065\pm
0.015$ is considered.

##  IV Results

In this section, we summarize the constraints on all the dark energy (DE)
parametrizations proposed in this work. To compare the key parameters of all
the parametrizations, we have utilized some common datasets, namely: i)
Planck, ATW, STW; ii) Planck+BAO, ATW+BAO, STW+BAO; iii) Planck+BAO+SNeIa,
ATW+BAO+SNeIa, STW+BAO+SNeIa. As the article is focused on the Exponential
model and its various extensions, including the well-known CPL
parametrization, in light of multiple CMB probes and their combinations with
other external cosmological probes, our primary focus is to first understand
how different CMB probes affect the DE parameters $w_{\rm 0,DE}$ and $w_{\rm
a,DE}$, and then to understand how these parameters affect other cosmological
parameters, e.g., $H_{0}$, $S_{8}$, $\Omega_{\mathrm{m}}$. Tables 2 (for the
Exponential model), 3 (for CPL), 4 (for CPL-A), and 5 (for CPL-B) display the
constraints on the free and derived parameters of these DE parametrizations.
Regarding the graphical presentations, we have summarized them into four
combined figures:

  1. 1.

Fig. 1 indicates the behavior of the dark energy parameters ($w_{\rm 0,DE}$,
$w_{\rm a,DE}$) and $H_{0}$ for all the parametrizations considering only the
CMB datasets (i.e., Planck, ATW, STW). This figure also provides a comparison
among all these parametrizations for the CMB alone dataset.

  2. 2.

Fig. 2 shows the behavior of $w_{\rm 0,DE}$, $w_{\rm a,DE}$, $H_{0}$,
$\Omega_{\rm m}$ for each parametrization considering only the combined
dataset CMB (Planck/ATW/STW)+BAO+SNIa.

  3. 3.

Fig. 3 presents the whisker graph at 68% CL of $w_{\rm 0,DE}$, $w_{\rm a,DE}$,
and $H_{0}$, offering a quick summary of all the parametrizations in terms of
these parameters.

  4. 4.

Fig. 4 compares all the parametrizations considering three combined datasets:
Planck+BAO+SNIa, ATW+BAO+SNIa, STW+BAO+SNIa.

In what follows, we describe the results extracted from each parametrization.

Table 2: 68% CL constraints, upper, and lower limits on the free and derived
parameters of the cosmological scenario in which DE assumes the Exponential
parametrization using all the dataset combinations. Note that
$\Delta\chi^{2}=$ $\chi^{2}$ (Model) $-$ $\chi^{2}$ ($\Lambda$CDM), and hence,
$\Delta\chi^{2}<0$ means that the model is preferred over the $\Lambda$CDM.

Parameter | Planck | ATW | STW | Planck+BAO | ATW+BAO | STW+BAO | Planck+BAO+SNeIa | ATW+BAO+SNeIa | STW+BAO+SNeIa
---|---|---|---|---|---|---|---|---|---
$\Omega_{\mathrm{c}}h^{2}$ | $0.1191\pm 0.0012$ | $0.1200\pm 0.0027$ | $0.1170\pm 0.0028$ | $0.1201\pm 0.0011$ | $0.1206\pm 0.0019$ | $0.1187\pm 0.0021$ | $0.1199\pm 0.0010$ | $0.1199\pm 0.0018$ | $0.1181\pm 0.0019$
$\Omega_{\mathrm{b}}h^{2}$ | $0.02244\pm 0.00015$ | $0.02237\pm 0.00020$ | $0.02238\pm 0.00021$ | $0.02238\pm 0.00014$ | $0.02234\pm 0.00018$ | $0.02235\pm 0.00020$ | $0.02239\pm 0.00014$ | $0.02238\pm 0.00019$ | $0.02235\pm 0.00020$
$100\theta_{\mathrm{MC}}$ | $1.04100\pm 0.00031$ | $1.04169\pm 0.00065$ | $1.04010\pm 0.00069$ | $1.04090\pm 0.00029$ | $1.04156\pm 0.00062$ | $1.04001\pm 0.00065$ | $1.04091\pm 0.00030$ | $1.04165^{+0.00058}_{-0.00065}$ | $1.04004\pm 0.00066$
$\tau$ | $0.0518\pm 0.0076$ | $0.062\pm 0.014$ | $0.058\pm 0.014$ | $0.0530\pm 0.0074$ | $0.063\pm 0.013$ | $0.059\pm 0.014$ | $0.0537\pm 0.0074$ | $0.062\pm 0.014$ | $0.059\pm 0.014$
$n_{\mathrm{s}}$ | $0.9672\pm 0.0041$ | $0.9729^{+0.0058}_{-0.0066}$ | $0.9678\pm 0.0068$ | $0.9649^{+0.0041}_{-0.0037}$ | $0.9717\pm 0.0052$ | $0.9651\pm 0.0059$ | $0.9651\pm 0.0039$ | $0.9729\pm 0.0052$ | $0.9660\pm 0.0058$
$\log(10^{10}A_{\mathrm{s}})$ | $3.037\pm 0.015$ | $3.066\pm 0.027$ | $3.043\pm 0.028$ | $3.042\pm 0.014$ | $3.071^{+0.027}_{-0.024}$ | $3.050\pm 0.027$ | $3.043\pm 0.014$ | $3.067\pm 0.027$ | $3.049\pm 0.027$
$w_{\rm 0,DE}$ | $-1.15^{+0.37}_{-0.60}$ | $-0.79^{+0.61}_{-0.83}$ | $-0.70\pm 0.66$ | $-0.83\pm 0.16$ | $-0.82\pm 0.16$ | $-0.87^{+0.17}_{-0.20}$ | $-0.969\pm 0.058$ | $-0.979\pm 0.058$ | $-0.984\pm 0.058$
$w_{\rm a,DE}$ | $<-0.910$ | $-1.15^{+0.67}_{-1.7}$ | $-0.96^{+1.3}_{-1.1}$ | $-0.59^{+0.43}_{-0.35}$ | $-0.56^{+0.44}_{-0.40}$ | $-0.43^{+0.56}_{-0.39}$ | $-0.26^{+0.19}_{-0.17}$ | $-0.17^{+0.23}_{-0.19}$ | $-0.14^{+0.24}_{-0.18}$
$\Omega_{\mathrm{m}}$ | $0.223^{+0.018}_{-0.082}$ | $0.321^{+0.067}_{-0.17}$ | $0.343^{+0.084}_{-0.20}$ | $0.320^{+0.017}_{-0.019}$ | $0.325^{+0.019}_{-0.021}$ | $0.315^{+0.020}_{-0.025}$ | $0.3043\pm 0.0074$ | $0.3057\pm 0.0083$ | $0.3018\pm 0.0082$
$\sigma_{8}$ | $0.935^{+0.12}_{-0.052}$ | $0.85^{+0.13}_{-0.16}$ | $0.796^{+0.096}_{-0.17}$ | $0.812\pm 0.016$ | $0.825\pm 0.017$ | $0.809\pm 0.018$ | $0.8239\pm 0.0096$ | $0.832\pm 0.016$ | $0.814\pm 0.017$
$S_{8}$ | $0.785^{+0.026}_{-0.050}$ | $0.837\pm 0.053$ | $0.806^{+0.059}_{-0.052}$ | $0.838\pm 0.014$ | $0.858\pm 0.026$ | $0.828\pm 0.029$ | $0.830\pm 0.011$ | $0.840\pm 0.021$ | $0.817\pm 0.021$
$H_{0}$ | $>77.3$ | $71^{+10}_{-20}$ | $68^{+9}_{-20}$ | $67.0\pm 1.9$ | $66.6\pm 1.9$ | $67.2\pm 2.2$ | $68.56\pm 0.78$ | $68.39\pm 0.78$ | $68.39\pm 0.78$
$r_{\mathrm{drag}}$ | $147.26\pm 0.27$ | $147.10\pm 0.63$ | $147.88\pm 0.70$ | $147.08\pm 0.25$ | $146.97\pm 0.46$ | $147.48\pm 0.55$ | $147.10\pm 0.24$ | $147.12\pm 0.47$ | $147.63\pm 0.52$
$\chi^{2}$ | $2768.3$ | $5837.2$ | $7417.7$ | $2793.7$ | $5857.4$ | $7435.5$ | $3828.5$ | $6892.8$ | $8472.7$
$\Delta\chi^{2}$ | $-4.4$ | $0.05$ | $1.2$ | $-3.0$ | $-1.0$ | $-3.0$ | $-3.5$ | $-0.3$ | $-0.9$

Figure 1: In this figure we show how different parametrizations respond to CMB
alone datasets, i.e. Planck alone, ATW, STW, in terms of the parameters
$w_{\rm 0,DE},w_{\rm a,DE}$ and $H_{0}$. This figure also shows a comparison
between these parametrizations.

Figure 2: The plots display the behaviour of the parameters $w_{\rm
0,DE},w_{\rm a,DE}$, $H_{0}$ and $\Omega_{\rm m}$ within the proposed DE
parametrizations for Planck+BAO+SNeIa, ATW+BAO+SNeIa and STW+BAO+SNeIa.
Figure 3: Whisker plot showing the 68% CL constraints or upper limits on
$H_{0}$ (left panel), $w_{\rm a,DE}$ (middle panel), and $w_{\rm 0,DE}$ (right
panel) from Exponential, CPL, CPL-A, and CPL-B parametrizations considering
various observational datasets. In the left panel, the cyan vertical band
corresponds to the $H_{0}$ value from the SH0ES collaboration ($H_{0}=73.04\pm
1.04$ km$s^{-1}$Mpc-1 at 68% CL) Riess  _et al._ (2021), and the reddish
vertical band corresponds to the $H_{0}$ value (within the $\Lambda$CDM
paradigm) from the Planck collaboration ($H_{0}=67.36\pm 0.54$ km$s^{-1}$Mpc-1
at 68% CL) Aghanim  _et al._ (2020c). For the middle panel, the dotted
vertical line corresponds to $w_{\rm a,DE}=0$, and the reddish vertical band
corresponds to the 68% CL constraints for Planck+BAO+SNeIa dataset combination
assuming CPL. Finally, for the right panel, the dotted vertical line
corresponds to $w_{\rm 0,DE}=-1$, and the cyan vertical band corresponds to
the 68% CL constraints for Planck+BAO+SNeIa dataset combination assuming CPL.

Figure 4: The plots offer a comparison between the proposed DE
parametrizations considering the combined datasets, namely, Planck+BAO+SNeIa
(upper plot), ATW+BAO+SNeIa (middle plot) and STW+BAO+SNeIa (lower plot).

###  IV.1 Exponential Parametrization

This parametrization is the generalized version of all the parametrizations
explored in this work, and hence, it is of special importance in this article.
We also note that for the first time constraints on this parametrization using
multiple CMB probes are given in this work. Table 2 summarizes the constraints
on this parametric DE model considering various CMB datasets and their
combinations with other astronomical probes, described above. Figures 1, 2, 3,
and 4 summarize all the findings graphically.

If we focus solely on the CMB datasets, we observe a moderate shift among
them, around $1-2\sigma$, on most of the parameters reported in Table 2, such
as, $100\theta_{\mathrm{MC}}$, $\log(10^{10}A_{\mathrm{s}})$, or $S_{8}$. Due
to the correlations among these parameters, Planck alone yields a very low
value for the matter density compared to the ground-based telescopes explored
here, resulting in a lower limit for $H_{0}$. This contrasts with the findings
from ATW and STW, where a higher value for $\Omega_{\mathrm{m}}$ is observed,
compensated by a lower value for $H_{0}$, which is now constrained.

Regarding the DE parameters, from Planck alone, a phantom behavior of $w_{\rm
0,DE}$ ($=-1.15^{+0.37}_{-0.60}$ at 68% CL) for the present epoch is found,
which is not supported by any of the ground-based experiments. The mean value
from ground-based experiments lies in the quintessence regime, although all
experiments are consistent with a cosmological constant (i.e., $w_{\rm
0,DE}=-1$). As for $w_{\rm a,DE}$, only an upper limit is obtained from Planck
alone, but this becomes constrained at 68% CL for ATW and STW. For ATW, we get
evidence of $w_{\rm a,DE}\neq 0$ at more than 68% CL, meaning that evidence of
a dynamical DE exists; however, for STW, $w_{\rm a,DE}=0$ is allowed within
68% CL. Figure 1 gives a comprehensive understanding of the behavior of these
DE parameters and their dependence on $H_{0}$ for different CMB datasets
explored here.

When the BAO data are included, while we can still see a shift for the same
parameters that move in the CMB only case, we find that the DE parameters, as
well as the matter density $\Omega_{\mathrm{m}}$ and the Hubble constant
$H_{0}$, are very well constrained and in complete agreement between
Planck+BAO, ATW+BAO, and STW+BAO. Focusing on the DE parameters, we notice
that the mean values of $w_{\rm 0,DE}$ are quintessential for all the cases;
however, specifically for Planck+BAO and ATW+BAO, $w_{\rm 0,DE}$ remains
strictly in the quintessence regime at 68% CL, while STW+BAO allows $w_{\rm
0,DE}<-1$ at 68% CL. Looking at the constraints on $w_{\rm a,DE}$ for all
three cases, we see that $w_{\rm a,DE}\neq 0$ (in fact, $w_{\rm a,DE}<0$) at
more than 68% CL for both Planck+BAO and ATW+BAO. However, for STW+BAO,
$w_{\rm a,DE}=0$ is allowed within 68% CL. Hence, a dynamical phantom nature
of $w_{\rm DE}$ at slightly more than $1\sigma$ is indicated in the past for
Planck+BAO and ATW+BAO. Finally, when SNeIa data are added, all the parameters
shift back in agreement with the $\Lambda$CDM case (except for the case with
Planck+BAO+SNeIa which recommends a mild non-zero value of $w_{\rm a,DE}$
($w_{\rm a,DE}=-0.26^{+0.19}_{-0.17}$ at 68% CL for Planck+BAO+SNeIa) and
hence a dynamical DE is endorsed), and they are very much consistent among the
different CMB datasets, as we can see in Fig. 2.

In order to have a smooth and quick understanding of the nature of the DE
parameters $w_{\rm 0,DE}$, $w_{\rm a,DE}$, as well as the Hubble constant
$H_{0}$ within this parametrized framework, in Fig. 3 we show the whisker plot
at 68% CL on these parameters. As seen from Fig. 3, when BAO and BAO+SNeIa
datasets are added to CMB (Planck/ATW/STW), the resulting scenario becomes
close and consistent with the $\Lambda$CDM model.

As expected, since this model has two additional degrees of freedom, it is
statistically disfavored compared to the $\Lambda$CDM scenario, as shown in
Table 6, if we compute the Bayesian evidence. However, we have a modest
improvement in the $\chi^{2}$ as shown in Table 2.

Table 3: Like Table 2 but for the CPL parametrization.

Parameter | Planck | ATW | STW | Planck+BAO | ATW+BAO | STW+BAO | Planck+BAO+SNeIa | ATW+BAO+SNeIa | STW+BAO+SNeIa
---|---|---|---|---|---|---|---|---|---
$\Omega_{\mathrm{c}}h^{2}$ | $0.1192\pm 0.0012$ | $0.1201\pm 0.0027$ | $0.1173\pm 0.0029$ | $0.1200\pm 0.0010$ | $0.1208\pm 0.0020$ | $0.1186\pm 0.0021$ | $0.1199\pm 0.0010$ | $0.1199^{+0.0020}_{-0.0017}$ | $0.1179\pm 0.0019$
$\Omega_{\mathrm{b}}h^{2}$ | $0.02244\pm 0.00015$ | $0.02236\pm 0.00021$ | $0.02236\pm 0.00021$ | $0.02237\pm 0.00014$ | $0.02235\pm 0.00018$ | $0.02234\pm 0.00020$ | $0.02238\pm 0.00014$ | $0.02237\pm 0.00019$ | $0.02238\pm 0.00020$
$100\theta_{\mathrm{MC}}$ | $1.04099\pm 0.00031$ | $1.04168\pm 0.00065$ | $1.04010\pm 0.00070$ | $1.04090\pm 0.00030$ | $1.04163\pm 0.00061$ | $1.03998\pm 0.00066$ | $1.04090\pm 0.00030$ | $1.04167\pm 0.00064$ | $1.04015\pm 0.00068$
$\tau$ | $0.0523\pm 0.0073$ | $0.061\pm 0.014$ | $0.058\pm 0.014$ | $0.0530\pm 0.0076$ | $0.064\pm 0.014$ | $0.059\pm 0.014$ | $0.0535\pm 0.0074$ | $0.063\pm 0.014$ | $0.060\pm 0.013$
$n_{\mathrm{s}}$ | $0.9668\pm 0.0042$ | $0.9728\pm 0.0063$ | $0.9674\pm 0.0070$ | $0.9650\pm 0.0038$ | $0.9713^{+0.0047}_{-0.0054}$ | $0.9651\pm 0.0062$ | $0.9653\pm 0.0038$ | $0.9731\pm 0.0051$ | $0.9664\pm 0.0058$
$\log(10^{10}A_{\mathrm{s}})$ | $3.038\pm 0.014$ | $3.065\pm 0.027$ | $3.045\pm 0.028$ | $3.041\pm 0.015$ | $3.072\pm 0.026$ | $3.050\pm 0.027$ | $3.042\pm 0.015$ | $3.069\pm 0.026$ | $3.051\pm 0.026$
$w_{\rm 0,DE}$ | $-1.23^{+0.43}_{-0.58}$ | $-0.85^{+0.65}_{-0.86}$ | $-0.74\pm 0.70$ | $-0.78^{+0.19}_{-0.21}$ | $-0.73^{+0.23}_{-0.25}$ | $-0.84^{+0.23}_{-0.26}$ | $-0.943\pm 0.077$ | $-0.963\pm 0.080$ | $-0.987^{+0.065}_{-0.080}$
$w_{\rm a,DE}$ | $<-0.635$ | $<-0.168$ | $-0.7^{+1.3}_{-1.7}$ | $-0.84^{+0.64}_{-0.53}$ | $-0.92^{+0.84}_{-0.67}$ | $-0.61^{+0.87}_{-0.63}$ | $-0.38^{+0.29}_{-0.26}$ | $-0.25^{+0.35}_{-0.31}$ | $-0.13^{+0.32}_{-0.23}$
$\Omega_{\mathrm{m}}$ | $0.215^{+0.016}_{-0.073}$ | $0.322^{+0.062}_{-0.18}$ | $0.351^{+0.081}_{-0.20}$ | $0.318^{+0.018}_{-0.020}$ | $0.327\pm 0.023$ | $0.313\pm 0.023$ | $0.3042\pm 0.0073$ | $0.3054^{+0.0076}_{-0.0088}$ | $0.3009^{+0.0076}_{-0.0090}$
$\sigma_{8}$ | $0.943^{+0.11}_{-0.047}$ | $0.85^{+0.12}_{-0.16}$ | $0.785^{+0.091}_{-0.16}$ | $0.813\pm 0.017$ | $0.823\pm 0.018$ | $0.809\pm 0.018$ | $0.823\pm 0.010$ | $0.834\pm 0.016$ | $0.813\pm 0.017$
$S_{8}$ | $0.780^{+0.023}_{-0.046}$ | $0.833\pm 0.051$ | $0.807^{+0.056}_{-0.049}$ | $0.836\pm 0.014$ | $0.859\pm 0.027$ | $0.826\pm 0.028$ | $0.829\pm 0.011$ | $0.841^{+0.021}_{-0.019}$ | $0.814^{+0.020}_{-0.023}$
$H_{0}$ | $>79.0$ | $71^{+10}_{-20}$ | $67^{+9}_{-20}$ | $67.1\pm 1.9$ | $66.4\pm 2.1$ | $67.4^{+2.1}_{-2.3}$ | $68.56\pm 0.78$ | $68.43\pm 0.78$ | $68.44^{+0.86}_{-0.76}$
$r_{\mathrm{drag}}$ | $147.23\pm 0.27$ | $147.08\pm 0.66$ | $147.83\pm 0.73$ | $147.09\pm 0.24$ | $146.92\pm 0.51$ | $147.52\pm 0.55$ | $147.12\pm 0.24$ | $147.13^{+0.42}_{-0.48}$ | $147.67\pm 0.49$
$\chi^{2}$ | $2768.9$ | $5837.4$ | $7417.8$ | $2793.8$ | $5857.5$ | $7438.6$ | $3828.2$ | $6892.6$ | $8473.6$
$\Delta\chi^{2}$ | $-3.8$ | $0.2$ | $1.3$ | $-2.9$ | $-0.9$ | $0.1$ | $-3.7$ | $-0.5$ | $0.041$

Table 4: Like Table 2 but for the CPL-A parametrization.

Parameter | Planck | ATW | STW | Planck+BAO | ATW+BAO | STW+BAO | Planck+BAO+SNeIa | ATW+BAO+SNeIa | STW+BAO+SNeIa
---|---|---|---|---|---|---|---|---|---
$\Omega_{\mathrm{c}}h^{2}$ | $0.1192\pm 0.0012$ | $0.1201\pm 0.0027$ | $0.1172\pm 0.0028$ | $0.1200\pm 0.0011$ | $0.1207\pm 0.0020$ | $0.1187^{+0.0021}_{-0.0020}$ | $0.1199\pm 0.0010$ | $0.1199\pm 0.0018$ | $0.1181\pm 0.0019$
$\Omega_{\mathrm{b}}h^{2}$ | $0.02244\pm 0.00015$ | $0.02238\pm 0.00020$ | $0.02237\pm 0.00021$ | $0.02238\pm 0.00014$ | $0.02235\pm 0.00019$ | $0.02234\pm 0.00021$ | $.02238\pm 0.00014$ | $0.02238\pm 0.00018$ | $0.02236\pm 0.00020$
$100\theta_{\mathrm{MC}}$ | $1.04100\pm 0.00030$ | $1.04169\pm 0.00065$ | $1.04009\pm 0.00069$ | $1.04091\pm 0.00030$ | $1.04154\pm 0.00064$ | $1.03997\pm 0.00065$ | $1.04091\pm 0.00028$ | $1.04169\pm 0.00062$ | $1.04001\pm 0.00065$
$\tau$ | $0.0518\pm 0.0074$ | $0.062\pm 0.014$ | $0.058\pm 0.014$ | $0.0530\pm 0.0073$ | $0.063\pm 0.014$ | $0.059\pm 0.014$ | $0.0533\pm 0.0073$ | $00.063\pm 0.014$ | $0.059\pm 0.014$
$n_{\mathrm{s}}$ | $0.9669\pm 0.0042$ | $0.9729\pm 0.0063$ | $0.9675\pm 0.0069$ | $0.9651\pm 0.0038$ | $0.9716\pm 0.0053$ | $0.9652\pm 0.0060$ | $0.9652\pm 0.0038$ | $0.9730^{+0.0050}_{-0.0055}$ | $0.9662\pm 0.0058$
$\log(10^{10}A_{\mathrm{s}})$ | $3.037\pm 0.015$ | $3.068\pm 0.027$ | $3.043\pm 0.027$ | $3.041\pm 0.014$ | $3.071\pm 0.028$ | $3.050\pm 0.027$ | $3.042\pm 0.014$ | $3.069\pm 0.027$ | $3.050\pm 0.027$
$w_{\rm 0,DE}$ | $-1.14^{+0.43}_{-0.59}$ | $-0.88^{+0.59}_{-0.82}$ | $-0.64\pm 0.70$ | $-0.79^{+0.17}_{-0.19}$ | $-0.77\pm 0.20$ | $-0.85^{+0.19}_{-0.23}$ | $-0.948\pm 0.068$ | $-0.967^{+0.069}_{-0.077}$ | $-0.969\pm 0.071$
$w_{\rm a,DE}$ | $<-0.888$ | $-0.8^{+1.3}_{-1.2}$ | $-0.9\pm 1.1$ | $-0.65^{+0.47}_{-0.39}$ | $-0.69\pm 0.51$ | $-0.48^{+0.60}_{-0.45}$ | $-0.31\pm 0.21$ | $-0.20^{+0.27}_{-0.21}$ | $-0.18^{+0.26}_{-0.22}$
$\Omega_{\mathrm{m}}$ | $0.217^{+0.016}_{-0.075}$ | $0.311^{+0.056}_{-0.17}$ | $0.353^{+0.083}_{-0.21}$ | $0.318^{+0.016}_{-0.019}$ | $0.324^{+0.019}_{-0.022}$ | $0.314^{+0.019}_{-0.023}$ | $0.3038^{+0.0069}_{-0.0077}$ | $0.3055\pm 0.0082$ | $0.3016\pm 0.0082$
$\sigma_{8}$ | $0.941^{+0.11}_{-0.049}$ | $0.86\pm 0.13$ | $0.786^{+0.085}_{-0.17}$ | $0.812\pm 0.016$ | $0.825\pm 0.019$ | $0.810\pm 0.018$ | $0.8240\pm 0.0097$ | $0.833\pm 0.016$ | $0.815\pm 0.016$
$S_{8}$ | $0.781^{+0.023}_{-0.046}$ | $0.831\pm 0.052$ | $0.808^{+0.060}_{-0.048}$ | $0.836\pm 0.014$ | $0.857\pm 0.026$ | $0.827\pm 0.027$ | $0.829\pm 0.011$ | $0.841\pm 0.020$ | $0.817\pm 0.020$
$H_{0}$ | $>78.9$ | $72^{+10}_{-20}$ | $67^{+8}_{-20}$ | $67.1\pm 1.8$ | $66.7\pm 1.9$ | $67.3\pm 2.1$ | $68.61\pm 0.76$ | $68.42\pm 0.79$ | $68.42\pm 0.80$
$r_{\mathrm{drag}}$ | $147.24\pm 0.26$ | $147.08\pm 0.63$ | $147.86\pm 0.71$ | $147.09\pm 0.25$ | $146.94\pm 0.49$ | $147.48^{+0.50}_{-0.56}$ | $147.11\pm 0.24$ | $147.11\pm 0.46$ | $147.62\pm 0.50$
$\chi^{2}$ | $2770.5$ | $5836.4$ | $7417.4$ | $2794.3$ | $5857.2$ | $7436.8$ | $3828.2$ | $6892.8$ | $8473.8$
$\Delta\chi^{2}$ | $-2.2$ | $-0.7$ | $0.9$ | $-2.4$ | $-1.2$ | $-1.7$ | $-3.8$ | $-0.3$ | $0.2$

Table 5: Like Table 2 but for the CPL-B parametrization.

Parameter | Planck | ATW | STW | Planck+BAO | ATW+BAO | STW+BAO | Planck+BAO+SNeIa | ATW+BAO+SNeIa | STW+BAO+SNeIa
---|---|---|---|---|---|---|---|---|---
$\Omega_{\mathrm{c}}h^{2}$ | $0.1191\pm 0.0012$ | $0.1201\pm 0.0028$ | $0.1172\pm 0.0029$ | $0.1201\pm 0.0010$ | $0.1207\pm 0.0019$ | $0.1186\pm 0.0020$ | $0.1199\pm 0.0010$ | $0.1199\pm 0.0018$ | $0.1181\pm 0.0019$
$\Omega_{\mathrm{b}}h^{2}$ | $0.02245\pm 0.00015$ | $0.02238\pm 0.00020$ | $0.02237\pm 0.00022$ | $0.02238\pm 0.00014$ | $0.02234\pm 0.00018$ | $0.02234\pm 0.00020$ | $0.02238\pm 0.00014$ | $0.02239\pm 0.00018$ | $0.02236\pm 0.00020$
$100\theta_{\mathrm{MC}}$ | $1.04099\pm 0.00031$ | $1.04172^{+0.00068}_{-0.00061}$ | $1.04008\pm 0.00068$ | $1.04089\pm 0.00029$ | $1.04159\pm 0.00060$ | $1.03996\pm 0.00065$ | $1.04091\pm 0.00030$ | $1.04169\pm 0.00060$ | $1.04004\pm 0.00065$
$\tau$ | $0.0523\pm 0.0075$ | $0.061\pm 0.014$ | $0.057\pm 0.015$ | $0.0529\pm 0.0073$ | $0.063\pm 0.014$ | $0.059\pm 0.014$ | $0.0535\pm 0.0074$ | $0.061\pm 0.014$ | $0.060\pm 0.014$
$n_{\mathrm{s}}$ | $0.9672\pm 0.0041$ | $0.9729\pm 0.0064$ | $0.9672\pm 0.0068$ | $0.9648\pm 0.0038$ | $0.9716\pm 0.0052$ | $0.9653\pm 0.0061$ | $0.9652\pm 0.0038$ | $0.9731\pm 0.0052$ | $0.9661\pm 0.0058$
$\log(10^{10}A_{\mathrm{s}})$ | $3.038\pm 0.014$ | $3.066\pm 0.026$ | $3.043\pm 0.028$ | $3.041\pm 0.014$ | $3.072\pm 0.027$ | $3.049\pm 0.026$ | $3.042\pm 0.014$ | $3.065\pm 0.027$ | $3.050\pm 0.026$
$w_{\rm 0,DE}$ | $-1.08^{+0.45}_{-0.62}$ | $-0.86^{+0.60}_{-0.82}$ | $-0.63\pm 0.69$ | $-0.79^{+0.16}_{-0.19}$ | $-0.77^{+0.18}_{-0.22}$ | $-0.86^{+0.18}_{-0.21}$ | $-0.949\pm 0.068$ | $-0.972\pm 0.066$ | $-0.974\pm 0.070$
$w_{\rm a,DE}$ | $<-0.960$ | $-0.8^{+1.3}_{-1.1}$ | $-1.0\pm 1.1$ | $-0.64^{+0.47}_{-0.35}$ | $-0.66^{+0.58}_{-0.42}$ | $-0.45^{+0.54}_{-0.42}$ | $-0.29^{+0.22}_{-0.19}$ | $-0.19^{+0.26}_{-0.20}$ | $-0.16^{+0.25}_{-0.20}$
$\Omega_{\mathrm{m}}$ | $0.224^{+0.020}_{-0.083}$ | $0.314^{+0.060}_{-0.17}$ | $0.348^{+0.087}_{-0.20}$ | $0.319^{+0.016}_{-0.019}$ | $0.325\pm 0.021$ | $0.313\pm 0.021$ | $0.3043\pm 0.0073$ | $0.3045^{+0.0075}_{-0.0086}$ | $0.3016\pm 0.0080$
$\sigma_{8}$ | $0.933^{+0.12}_{-0.057}$ | $0.86^{+0.13}_{-0.15}$ | $0.792^{+0.098}_{-0.17}$ | $0.812\pm 0.016$ | $0.825\pm 0.018$ | $0.809\pm 0.018$ | $0.8236\pm 0.0098$ | $0.832\pm 0.015$ | $0.814\pm 0.016$
$S_{8}$ | $0.784^{+0.025}_{-0.050}$ | $0.833\pm 0.049$ | $0.808^{+0.059}_{-0.054}$ | $0.837\pm 0.014$ | $0.858\pm 0.026$ | $0.825\pm 0.026$ | $0.829\pm 0.011$ | $0.838\pm 0.020$ | $0.816\pm 0.020$
$H_{0}$ | $>76.8$ | $72^{+10}_{-20}$ | $68^{+10}_{-20}$ | $67.0\pm 1.8$ | $66.6\pm 1.9$ | $67.4^{+1.9}_{-2.1}$ | $68.56\pm 0.77$ | $68.53\pm 0.77$ | $68.41\pm 0.77$
$r_{\mathrm{drag}}$ | $147.26\pm 0.27$ | $147.05\pm 0.69$ | $147.83\pm 0.73$ | $147.07\pm 0.25$ | $146.95^{+0.44}_{-0.49}$ | $147.52\pm 0.52$ | $147.11\pm 0.24$ | $147.11\pm 0.47$ | $147.63\pm 0.49$
$\chi^{2}$ | $2770.9$ | $5837.2$ | $7417.8$ | $2792.0$ | $5857.3$ | $7438.9$ | $3828.3$ | $6892.9$ | $8473.9$
$\Delta\chi^{2}$ | $-1.8$ | $0.0053$ | $1.3$ | $-4.7$ | $-1.0$ | $0.4$ | $-3.6$ | $-0.2$ | $0.3$

###  IV.2 CPL

In Table 3, we present the results of this parametric DE model considering
different datasets as mentioned above, and Figs. 1, 2, 3, 4 summarize all the
findings graphically.

We start with the constraints from CMB alone (i.e., Planck, ATW, and STW).
Looking at the constraints on $w_{\rm 0,DE}$ from all the three datasets, we
notice that for Planck alone, the mean value of $w_{\rm 0,DE}$ indicates a
highly phantom nature of the DE, while within 68% CL, the quintessence
scenario is also allowed ($w_{\rm 0,DE}=-1.23^{+0.43}_{-0.58}$ at 68% CL for
Planck alone). Because of this phantom nature of $w_{\rm 0,DE}$, the expansion
rate of the universe at present time quantified by $H_{0}$ takes a very high
value, and consequently, due to the existing geometric degeneracy between
$H_{0}$ and $\Omega_{\mathrm{m}}$, the model leads to a lower value of the
matter density parameter ($\Omega_{\mathrm{m}}=0.215^{+0.016}_{-0.073}$). In
contrast, for both ATW and STW, DE exhibits its quintessential character
($w_{\rm 0,DE}=-0.85^{+0.65}_{-0.86}$ for ATW, and $w_{\rm 0,DE}=-0.74\pm
0.70$ for STW) although $w_{\rm 0,DE}<-1$ is allowed in both cases. For the
other parameter, $w_{\rm a,DE}$, we notice that for both Planck alone and ATW,
it has an upper limit but STW constrains it at 68% CL ($w_{\rm
a,DE}=-0.7^{+1.3}_{-1.7}$). It is interesting to note that while neither
Planck nor ATW can constrain $w_{\rm a,DE}$, STW can. In addition, our
impression of $H_{0}$ remains similar to what we observed in the Exponential
case for the same datasets.

When the BAO data are added to all variants of the CMB data (i.e., Planck,
ATW, STW), referring to Table 3, we find that the mean values of $w_{\rm
0,DE}$ remain in the quintessence regime for all cases. However, $w_{\rm
0,DE}>-1$ strictly at 68% CL for Planck+BAO and ATW+BAO, while STW+BAO allows
$w_{\rm 0,DE}<-1$ at 68% CL. Looking at the constraints on $w_{\rm a,DE}$ for
all three cases, we see that $w_{\rm a,DE}\neq 0$ (in fact, $w_{\rm a,DE}<0$)
at more than 68% CL for both Planck+BAO and ATW+BAO. However, for STW+BAO,
$w_{\rm a,DE}=0$ is allowed within 68% CL. Hence, a phantom nature of $w_{\rm
DE}$ is indicated in the past for Planck+BAO and ATW+BAO, while STW+BAO is
consistent with a cosmological constant.

Finally, when the SNeIa dataset is added (see Table 3), we do not find any
strong evidence beyond the $\Lambda$CDM model, because $w_{\rm 0,DE}$ for
Planck+BAO+SNeIa, ATW+BAO+SNeIa, and STW+BAO+SNeIa is consistent with $-1$,
and $w_{\rm a,DE}=0$ is allowed for ATW+BAO+SNeIa and STW+BAO+SNeIa datasets.
However, for Planck+BAO+SNeIa, we notice $w_{\rm a,DE}=-0.38^{+0.29}_{-0.26}$
at 68% CL, and therefore the slight dynamical nature of DE is not strictly
ruled out for this combined dataset. In fact, since $w_{\rm a,DE}<0$ at
slightly more than 68% CL, a slight phantom dynamical DE is suggested in this
case.

As expected, this parametrization is also statistically disfavored compared to
the $\Lambda$CDM scenario, as shown in Table 6, from a Bayesian evidence point
of view, but slightly favored by a $\chi^{2}$ analysis, as shown in Table 3.
However, this parametrization is indistinguishable from the Exponential one,
or the other CPL extensions, from a Bayesian point of view.

###  IV.3 CPL-A

In Table 4, we show the constraints on this parametrization for various
datasets, and Figs. 1, 2, 3, 4 display the results graphically.

Focusing on the CMB constraints (Table 4), we find that the results on $w_{\rm
0,DE}$ in this parametrization are similar to what we have observed with the
CPL parametrization (see Table 3). That means, for Planck alone, the tendency
of $w_{\rm 0,DE}$ being in the phantom regime is observed ($w_{\rm
0,DE}=-1.14^{+0.43}_{-0.59}$ at 68% CL), while within 68% CL, the cosmological
constant is equally allowed. However, ATW and STW agree well with the
quintessential DE. Concerning the $w_{\rm a,DE}$ parameter, Planck alone gives
its upper limit, while for the remaining CMB experiments, it is well
constrained within 68% CL. Our observation on $H_{0}$ remains the same (see
Table 3). We also refer to Fig. 1 for a quick visualization of these
parameters for these CMB datasets.

When adding BAO measurements to the CMB, referring to Table 4, we find that
the mean values of $w_{\rm 0,DE}$ lie in the quintessence regime ($w_{\rm
0,DE}>-1$) for all three datasets. Furthermore, $w_{\rm 0,DE}$ remains
strictly in the quintessence regime for Planck+BAO and ATW+BAO at the 68% CL
constraints, while for STW+BAO, $w_{\rm 0,DE}<-1$ is allowed. Focusing on
$w_{\rm a,DE}$, we see that $w_{\rm a,DE}\neq 0$ (in fact, $w_{\rm a,DE}<0$)
at more than 68% CL for both Planck+BAO and ATW+BAO, while for STW+BAO,
$w_{\rm a,DE}=0$ is allowed within 68% CL. Hence, a mild phantom nature of
$w_{\rm DE}$ is indicated in the past for both Planck+BAO and ATW+BAO at
slightly more than 68% CL. Additionally, we notice that $H_{0}$ takes similar
values as in the $\Lambda$CDM model but with much relaxed constraints.

The inclusion of SNeIa to the former dataset combinations, referring to Table
4 and Fig. 2, does not add anything new compared to our observations with the
CPL case, except for slightly stronger error bars. However, similar to the
earlier two parametrizations, namely, Exponential and CPL, we notice here that
Planck+BAO+SNeIa suggests the possibility of a dynamical DE at slightly more
than 68% CL, while ATW+BAO+SNeIa and STW+BAO+SNeIa are close to and consistent
with the $\Lambda$CDM model expectations. We refer to Fig. 3 for a quick view
of the DE parameters and $H_{0}$ for all the datasets. This whisker plot
clearly shows that the $\Lambda$CDM model is well recovered within the 68% CL.

Similar to the previously discussed parametrizations, CPL-A is also
statistically disfavored compared to the $\Lambda$CDM scenario, as shown in
Table 6, but improves the $\chi^{2}$ analysis, as shown in Table 4.

###  IV.4 CPL-B

This is the last parametrization of this series, and in Table 5 we display the
constraints that are obtained from various datasets similar to the previous
cases. Figs. 1, 2, 3, and 4 summarize all the findings graphically.

In a similar fashion, we start by looking at the constraints from the CMB
alone dataset. Similar to the previous parametrizations, our observations on
the DE parameters, namely, $w_{\rm 0,DE}$ and $w_{\rm a,DE}$, remain
consistent. This means that a phantom nature of $w_{\rm 0,DE}$ through its
mean value is mildly supported by Planck alone, but that possibility is not
supported by any of the ground-based CMB experiments considered in this
article. However, all three CMB datasets are perfectly in agreement with a
cosmological constant. On the other hand, we find that $w_{\rm a,DE}$ attains
an upper limit by Planck alone, but for the remaining CMB experiments, $w_{\rm
a}$ is well constrained at 68% CL ($w_{\rm a,DE}=-0.8^{+1.3}_{-1.1}$ for ATW
and $w_{\rm a,DE}=-1.0\pm 1.1$ for STW). Additionally, we notice that due to
the existing geometric degeneracy between $H_{0}$ and $\Omega_{\rm m}$, Planck
alone returns a lower value of $\Omega_{\rm m}$ but a higher value of $H_{0}$,
while in the cases with ATW and STW, $H_{0}$ is lower and, as a consequence,
$\Omega_{\rm m}$ is higher. We refer to Fig. 1 for a graphical understanding
of some of the key parameters of this parametrization.

The inclusion of BAO to all three CMB datasets does not offer anything
significantly new compared to the constraints observed with the earlier
parametrizations. Concerning the DE parameters, we notice that the mean values
of $w_{\rm 0,DE}$ are quintessential, and within 68% CL, $w_{\rm 0,DE}>-1$
strictly for Planck+BAO and ATW+BAO, while within 68% CL for STW+BAO, a
phantom crossing ($w_{\rm 0,DE}<-1$) is allowed. For the remaining DE
parameter, $w_{\rm a,DE}$, we notice that $w_{\rm a,DE}\neq 0$, and in
particular, $w_{\rm a,DE}<0$ at more than 68% CL for Planck+BAO and ATW+BAO.
Hence, a dynamical nature of $w_{\rm DE}$ is indicated for both Planck+BAO and
ATW+BAO at slightly more than 68% CL, and in addition, a phantom nature of
$w_{\rm DE}$ in the past is also suggested. However, for STW+BAO, $w_{\rm
a,DE}=0$ is allowed within 68% CL.

Finally, when SNeIa data are added to CMB (Planck/ATW/STW)+BAO, we do not find
any significant changes in the constraints compared to the earlier
parametrizations. Effectively, a quintessential DE is preferred for all three
datasets as indicated by the mean values of $w_{\rm 0,DE}$; however, the
cosmological constant $w_{\rm 0,DE}=-1$ is well consistent within 68% CL. On
the other hand, Planck+BAO+SNeIa favors a dynamical DE with $w_{\rm a,DE}\neq
0$ at more than 68% CL, but this is consistent with its null value within 95%
CL. For the remaining two datasets, namely, ATW+BAO+SNeIa and STW+BAO+SNeIa,
$w_{\rm a,DE}=0$ is consistent within 68% CL.

Table 6: Summary of $\ln B_{ij}$ values calculated for all the dynamical DE
parametrizations with respect to the $\Lambda$CDM model. The negative value of
$\ln B_{ij}$ indicates that $\Lambda$CDM is preferred over the model.

Dataset | Model | $\ln B_{ij}$ | Evidence for $\Lambda$CDM | $\ln B_{ij}$ | Evidence for Exp | $\ln B_{ij}$ | Evidence for CPL
---|---|---|---|---|---|---|---
Planck | Exponential | $-0.2$ | Inconclusive | — | — | $-0.2$ | inconclusive
Planck+BAO | Exponential | $-3.7$ | Moderate | — | — | $-0.4$ | inconclusive
Planck+BAO+SNeIa | Exponential | $-5.0$ | Strong | — | — | $-0.1$ | inconclusive
ATW | Exponential | $-1.6$ | Weak | — | — | $-0.3$ | inconclusive
ATW+BAO | Exponential | $-4.2$ | Moderate | — | — | $-0.1$ | inconclusive
ATW+BAO+SNeIa | Exponential | $-7.1$ | Strong | — | — | $-1.4$ | Weak
STW | Exponential | $-1.9$ | Weak | — | — | $-0.5$ | inconclusive
STW+BAO | Exponential | $-4.8$ | Moderate | — | — | $-0.1$ | inconclusive
STW+BAO+SNeIa | Exponential | $-6.1$ | Strong | — | — | $-0.4$ | inconclusive
Planck | CPL | $-0.02$ | inconclusive | $0.2$ | inconclusive | — | —
Planck+BAO | CPL | $-3.2$ | Moderate | $0.4$ | inconclusive | — | —
Planck+BAO+SNeIa | CPL | $-4.9$ | Moderate | $0.1$ | inconclusive | — | —
ATW | CPL | $-1.3$ | Weak | $0.3$ | inconclusive | — | —
ATW+BAO | CPL | $-4.1$ | Moderate | $0.1$ | inconclusive | — | —
ATW+BAO+SNeIa | CPL | $-5.7$ | Strong | $1.4$ | Weak | — | —
STW | CPL | $-1.4$ | Weak | $0.5$ | inconclusive | — | —
STW+BAO | CPL | $-4.7$ | Moderate | $0.1$ | inconclusive | — | —
STW+BAO+SNeIa | CPL | $-5.8$ | Strong | $0.4$ | inconclusive | — | —
Planck | CPL-A | $-0.5$ | inconclusive | $-0.3$ | inconclusive | $-0.4$ | inconclusive
Planck+BAO | CPL-A | $-3.3$ | Moderate | $0.4$ | inconclusive | $-0.07$ | inconclusive
Planck+BAO+SNeIa | CPL-A | $-5.0$ | Strong | $0.04$ | inconclusive | $-0.08$ | inconclusive
ATW | CPL-A | $-1.8$ | Weak | $-0.2$ | inconclusive | $-0.5$ | inconclusive
ATW+BAO | CPL-A | $-4.0$ | Moderate | $0.2$ | inconclusive | $0.04$ | inconclusive
ATW+BAO+SNeIa | CPL-A | $-6.0$ | Strong | $1.2$ | Weak | $-0.2$ | inconclusive
STW | CPL-A | $-1.6$ | Weak | $0.3$ | inconclusive | $-0.3$ | inconclusive
STW+BAO | CPL-A | $-4.5$ | Moderate | $0.2$ | inconclusive | $0.1$ | inconclusive
STW+BAO+SNeIa | CPL-A | $-6.0$ | Strong | $0.2$ | inconclusive | $-0.1$ | inconclusive
Planck | CPL-B | $-0.1$ | inconclusive | $0.1$ | inconclusive | $-0.1$ | inconclusive
Planck+BAO | CPL-B | $-3.3$ | Moderate | $0.3$ | inconclusive | $-0.1$ | inconclusive
Planck+BAO+SNeIa | CPL-B | $-4.7$ | Moderate | $0.3$ | inconclusive | $0.2$ | inconclusive
ATW | CPL-B | $-1.8$ | Weak | $-0.2$ | inconclusive | $-0.5$ | inconclusive
ATW+BAO | CPL-B | $-4.1$ | Moderate | $0.1$ | inconclusive | $-0.05$ | inconclusive
ATW+BAO+SNeIa | CPL-B | $-7.1$ | Strong | $0.03$ | inconclusive | $-1.3$ | Weak
STW | CPL-B | $-1.7$ | Weak | $0.2$ | inconclusive | $-0.3$ | inconclusive
STW+BAO | CPL-B | $-4.5$ | Moderate | $0.3$ | inconclusive | $0.1$ | inconclusive
STW+BAO+SNeIa | CPL-B | $-6.1$ | Strong | $-0.009$ | inconclusive | $-0.4$ | inconclusive

###  IV.5 Comparison of the parametrizations

Having all the results in a nutshell, in this section we compare the proposed
DE parametrizations. We refer to Figs. 1, 2, 4, and 3 for a quick comparison
between the parametrizations, which clearly show that it is difficult to
distinguish between them for all the datasets explored here. However, it is
interesting to note that while for the Planck and STW datasets (top and bottom
panels in Fig. 1), all the parametrizations give the same constraints on the
DE parameters, for ATW (middle panel in Fig. 1), the additional terms in the
expansion (namely CPL-A and CPL-B) help in constraining $w_{\rm a,DE}$, which
is otherwise unconstrained for the CPL case. As expected, with the addition of
low-redshift data (BAO and SNeIa), all the differences disappear and the DE
parameters are very well constrained.

Additionally, we have performed the Bayesian evidence analysis Heavens  _et
al._ (2017a, b) to understand the goodness of fit of the models with respect
to some reference model. In Table 6, we have presented the values of $\ln
B_{ij}$ considering $\Lambda$CDM, Exponential, and CPL as reference models.
According to these results, we find that $\Lambda$CDM is preferred over all
the DE parametrizations, while both the Exponential and the CPL
parametrizations are indistinguishable (inconclusive evidence) among
themselves and with the extended cases CPL-A and CPL-B. The only exception is
for ATW+BAO+SNeIa, where the CPL and CPL-A parametrizations are weakly
preferred over the Exponential one, but the CPL parametrization is weakly
preferred over the CPL-A case.

##  V Summary and Conclusions

The nature of dark energy has remained elusive despite extensive astronomical
surveys. While the standard $\Lambda$CDM cosmology has effectively described
the late-time accelerating expansion and the distribution of dark matter in
the universe, emerging cosmological tensions between early (based on
$\Lambda$CDM) and late-time (model-independent) cosmological probes have
called for revisions. Therefore, any modification to the $\Lambda$CDM model is
welcome as long as observational data does not definitively rule out such
modifications.

In this article, we explore a well-established way of modification: the
parametrization of the DE equation of state. Parametrizing the DE sector
offers numerous possibilities, with one of the simplest approaches being to
vary the equation of state of DE, $w_{\rm DE}$, in different ways and then
assess the viability of the proposed parametrized models. Along this line, the
widely recognized parametrization is the Chevallier-Polarski-Linder (CPL)
model, which involves a Taylor expansion of $w_{\rm DE}$ around the present
epoch. While CPL has been extensively used and often serves as the primary
choice for many astronomical surveys when investigating the properties of DE,
in our current study, we opt for a generalized parametrization: the
Exponential parametrization. This choice is motivated by its ability to
encompass the CPL parametrization within its first-order approximation.
Additionally, the higher-order approximations of this general parametrization,
namely, CPL-A and CPL-B, have only two free parameters, which are economically
less expensive compared to DE parametrizations with more than two parameters.
Overall, we have considered four parametrizations of the DE equation of state:
Exponential, CPL, CPL-A, and CPL-B, and constrained them using CMB data from
Planck, ACT, SPT, and their combinations with BAO, as well as the Pantheon
sample from SNeIa. Tables 2, 3, 4, and 5, and Figs. 1, 2, 3, and 4 summarize
the constraints on all these parametrizations.

The main conclusion of this article regarding the key cosmological parameters
is as follows:

  * •

$\mathbf{(w_{\rm 0,DE})}$ $-$ Planck alone indicates a slight (but not
statistically significant) phantom inclination of DE (through the mean value
of $w_{\rm 0,DE}$) for all the parametrizations, which goes away for the other
two ground-based CMB experiments. However, within 68% CL, Planck, ATW, and STW
are in agreement with a cosmological constant. The inclusion of BAO makes a
difference. For all the parametrizations, $w_{\rm 0,DE}$ remains strictly in
the quintessence regime ($w_{\rm 0,DE}>-1$) at slightly more than 68% CL for
both Planck+BAO and ATW+BAO, but STW+BAO allows a phantom crossing ($w_{\rm
0,DE}<-1$) at slightly more than 68% CL. When SNeIa data are included, for all
three combined analyses, the mean values of $w_{\rm 0,DE}$ for all the
parametrizations get very close to $-1$ but they all remain quintessential,
allowing $w_{\rm 0,DE}<-1$ within 68% CL.

  * •

$\mathbf{(w_{\rm a,DE})}$ $-$ The parameter $w_{\rm a,DE}$ remains
unconstrained for Planck alone, while it becomes constrained for ATW (except
in the CPL parametrization, where $w_{\rm a,DE}$ remains unconstrained), STW,
and other combined datasets in the presence of BAO and SNeIa. Evidence of
$w_{\rm a,DE}\neq 0$ is found for several datasets in different
parametrizations. We noticed that for the Exponential parametrization, $w_{\rm
a,DE}\neq 0$ at slightly more than 68% CL for ATW, Planck+BAO, ATW+BAO, and
Planck+BAO+SNeIa. For the CPL parametrization, the same is true for Planck+BAO
and ATW+BAO. For both CPL-A and CPL-B, Planck+BAO, ATW+BAO, and
Planck+BAO+SNeIa favor a mild preference of $w_{\rm a,DE}\neq 0$ at slightly
more than 68% CL. Thus, the preference for dynamical DE within the present DE
parametrizations is supported by several cosmological probes.

  * •

$\mathbf{H_{0}}$ $-$ The most stringent constraints on $H_{0}$ are obtained by
Planck+BAO+SNeIa, ATW+BAO+SNeIa, and STW+BAO+SNeIa, where a mild reduction in
its tension ($H_{0}\gtrsim 68.4$ km/s/Mpc) is found, even if still above
$3\sigma$.

  * •

$\mathbf{S_{8}}$ $-$ With the exception of Planck alone, which reports
$S_{8}\sim 0.78$ for all the parametrizations, we do not find anything
noteworthy for the remaining datasets.

The contrast of multipole CMB probes across various dark energy equation of
state parameters, as explored in this article, presents a compelling
narrative. Particularly intriguing is the scrutiny on the current state of
$w_{\rm a,DE}$ in light of the findings from two ground-based CMB experiments.

In summary, the findings of this study show an interesting perspective: while
$\Lambda$CDM serves as the boundary of present dark energy parametrizations,
the potential dynamical nature of dark energy remains a viable consideration
in interpreting the results. This conclusion gains further support from recent
developments, such as those arising from the new DESI data Adame  _et al._
(2024).

## Acknowledgments

We thank William Giarè for the help and useful discussions. MN gratefully
thanks Saeed Fakhry and Mina Ghodsi Yengejeh for the useful discussion and
constructive comments. SP acknowledges the financial support from the
Department of Science and Technology (DST), Govt. of India under the Scheme
“Fund for Improvement of S&T Infrastructure (FIST)” (File No.
SR/FST/MS-I/2019/41). EDV acknowledges support from the Royal Society through
a Royal Society Dorothy Hodgkin Research Fellowship. This article is based
upon work from the COST Action CA21136 “Addressing observational tensions in
cosmology with systematics and fundamental physics (CosmoVerse), supported by
COST (European Cooperation in Science and Technology).

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