# An exponential equation of state of dark energy in the light of 2018 CMB
Planck data

Mónica N. Castillo-Santos1<EMAIL_ADDRESS>A. Hernández-Almada1
<EMAIL_ADDRESS>corresponding author Miguel A. García-Aspeitia2
<EMAIL_ADDRESS>Juan Magaña3<EMAIL_ADDRESS>1 Facultad de
Ingeniería, Universidad Autónoma de Querétaro, Centro Universitario Cerro de
las Campanas, 76010, Santiago de Querétaro, México  2 Depto. de Física y
Matemáticas, Universidad Iberoamericana Ciudad de México, Prolongación Paseo
de la Reforma 880, México D. F. 01219, México  3 Escuela de Ingeniería,
Universidad Central de Chile, Avenida Francisco de Aguirre 0405, 171-0164 La
Serena, Coquimbo, Chile

###### Abstract

The dynamics of the Universe is analyzed using an exponential function for the
dark energy equation of state, known as Gong-Zhang parameterization. The phase
space of the free parameters presented in the model is constrained using
Cosmic Microwave Background radiation, Cosmic Chronometers, modulus distance
from Hydrogen II Galaxies, Type Ia Supernovae and measurements from Baryon
Acoustic Oscillations, together with a stronger bound from a Joint analysis.
The cosmological model is confronted with $\Lambda$CDM, observing there is a
strong evidence for $\Lambda$CDM in the Joint analysis although the
exponential model is preferred when the data are separated. Based on the Joint
analysis, a value of $\omega_{0}=-1.202^{+0.027}_{-0.026}$ is found for the
characteristic parameter presented in the equation of state. Additionally, the
cosmographic parameters at current times are reported, having
$q_{0}=-0.789^{+0.034}_{-0.036}$, $j_{0}=1.779^{+0.130}_{-0.119}$, and a
transition deceleration-acceleration redshift $z_{T}=0.644^{+0.011}_{-0.012}$.
Furthermore, the age of the Universe is estimated as
$t_{U}=13.788^{+0.019}_{-0.019}$ Gyrs. Finally, we open a discussion if this
model could alleviate the $H_{0}$ and $S_{8}$ tensions.

###### pacs:

Dark energy, equation of state, cosmology

##  I Introduction

Since the discovery of the accelerated expansion of the Universe at current
times through Ia Type Supernovae (SNIa) in 1998 Riess  _et al._ (1998);
Perlmutter  _et al._ (1999), its nature is one of the most elusive open
question in cosmology. This phenomenon, associated to the called Dark Energy
(DE), was later confirmed through the radiation coming from early times of the
Universe, the well-known Cosmic Microwave Background radiation (CMB) Aghanim
and et. al. (2020). There is no doubt the simplest way to model the DE is with
a cosmological constant ($\Lambda$) introduced into Einstein field equations
as a new field. By adding a matter component (baryons and Dark Matter (DM))
and a relativistic component (photons and neutrinos), has been constructed the
standard cosmological model known as $\Lambda$-Cold Dark Matter
($\Lambda$CDM). Until now, this paradigm has been tested at different scales
and data and it is successful to fit and be consistent with them. However
$\Lambda$CDM is not free of problems, apart from the coincidence and fine-
tuning problems Carroll (2001); Zeldovich (1968); Weinberg (1989) related to
$\Lambda$, recent studies Zhao  _et al._ (2017); Yang  _et al._ (2019a);
Akarsu  _et al._ (2020); Di Valentino  _et al._ (2021a); Chudaykin  _et al._
(2021); Yang  _et al._ (2021); Colgáin  _et al._ (2021); Escamilla and Vazquez
(2021); Sharma  _et al._ (2022) indicate that the DE Equation of State (EoS)
could be dynamical instead of a constant.

Furthermore, a recent problem in Cosmology is related to the Hubble constant
($H_{0}$) measurements. There is a deviation about $4\sigma$ between the local
measurements of $H_{0}$ using Type Ia Supernovae (SNIa) Riess  _et al._ (2019)
corresponding at late times of the Universe and those performed using CMB
radiation at early times Aghanim and et. al. (2020). This problem is also
known as the $H_{0}$ tension in which $\Lambda$CDM does not give an answer.
Nevertheless Ref. Efstathiou (2021) has recently suggested that this might be
related with systematics of the distance ladder measurements instead of an
“exotic new physics”. Finally a review of cosmological models that could
resolve the $H_{0}$ tension is presented in Di Valentino  _et al._ (2021b);
Motta  _et al._ (2021); Bamba  _et al._ (2012); Di Valentino  _et al._ (2021c,
d, e, f); Perivolaropoulos and Skara (2022); Abdalla  _et al._ (2022).

Additionally to the $H_{0}$ problem, there is another tension in the amplitude
of matter perturbations smoothed over $8h^{-1}\mathrm{Mpc}$, $\sigma_{8}$,
where $h$ is the dimensionless Hubble parameter. Indeed, the tension is
quantified through the quantity $S_{8}\equiv\sigma_{8}\sqrt{\Omega_{0m}/0.3}$,
where $\Omega_{0m}$ is the density parameter of matter. The $\Lambda$CDM
inferred value from CMB anisotropies is not consistent with those from weak
gravitational lensing measurements Abbott  _et al._ (2022); Asgari  _et al._
(2021); Heymans  _et al._ (2021). Notice that Planck collaboration predicts
$S_{8}=0.834\pm 0.016$ while Kilo-Degree Survey (KiDS-1000) Heymans  _et al._
(2021) predicts $S_{8}=0.766^{+0.020}_{-0.014}$, which produce a tension of
about $3\sigma$ between both results. See for instance Di Valentino  _et al._
(2021) for details.

In general, the efforts to search for alternatives to understand the DE comes
from two perspectives. One approach considers that the cosmic acceleration
comes from a modified gravity theory such as $f(R)$ theories, Einstein-Gauss-
Bonnet gravity, extra dimensions, fractional calculus Garcia-Aspeitia  _et
al._ (2018); García-Aspeitia  _et al._ (2022). Another considers there is an
extra component (fluid or field) that drives the accelerated expansion of the
Universe and some of them are Chaplygin gas Chaplygin (1904); Hernandez-Almada
_et al._ (2019), viscous fluids Hernández-Almada  _et al._ (2020a); Herrera-
Zamorano  _et al._ (2020) and emergent fluids Li and Shafieloo (2019);
Hernández-Almada  _et al._ (2020b), among others. In general, these dark
energy fluids have an evolving EoS that can be parametrized as a function of
redshift. A well-studied parametrization able to solve the $\Lambda$ problems
is a linear function of the scale factor (Chevallier-Polarski and Linder (CPL)
parametrization Chevallier and Polarski (2001); Linder (2003)), however it
present divergences at the future Ma and Zhang (2011a). Although some EoS
parameterizations as function of the redshift or scale factor have been
proposed, some of them present problems. For instance, an EoS represented by a
linear function of the redshift Huterer and Turner (2001); Weller and Albrecht
(2002) diverges at high redshift (in the past) and is in disagreement with CMB
Caldwell and Doran (2004) and Big Bang Nucleosynthesis constraints Johri
(2002). Although, there are proposals to solve these conflicts, most of them
involve two or more parameters in the EoS Upadhye  _et al._ (2005); Jassal
_et al._ (2005); Liu  _et al._ (2008); Barboza and Alcaniz (2008a); Ma and
Zhang (2011a); Li and Zhang (2011); Feng  _et al._ (2012); Magaña  _et al._
(2014); Hu  _et al._ (2016); Roman-Garza  _et al._ (2019); Singh and Nagpal
(2020); Perković and Štefančić (2020) that need to be determined by the
observations. Moreover, in Bouhmadi-Lopez  _et al._ (2015); Yang  _et al._
(2019a); Poulin  _et al._ (2019) it is possible to study models of one-
parameter extension of $\Lambda$, similarly in Acquaviva  _et al._ (2021);
Akarsu  _et al._ (2021); Poulin  _et al._ (2022); Akarsu  _et al._ (2022)
these models are able to relax $H_{0}$ or $S_{8}$ tensions.

In this work we explore a parameterization proposed by Gong and Zhang (GZ) for
DE EoS Gong and Zhang (2005), which is formulated to avoid the divergences at
the future or at the early stages of the Universe 111 Other models that does
not diverge in the past or in the future can be founded in Bassett  _et al._
(2002, 2004); Shafieloo  _et al._ (2009); De Felice  _et al._ (2012); Wei  _et
al._ (2014); Hazra  _et al._ (2015); Akarsu  _et al._ (2015); Barboza and
Alcaniz (2008b); Ma and Zhang (2011b)., only adding one extra parameter to the
standard cosmology. In Yang  _et al._ (2019a), the GZ parameterization is
constrained using CMB, SNIa (JLA sample), baryon acoustic oscillation (BAO)
and observational Hubble data (OHD, cosmic chronometers). Their constraints
for the current DE EoS ($w_{0}$) show a trend to the phantom region ($w<-1$).
Also, the $H_{0}$ bounds are in agreement with those obtained with CMB, hence
the GZ model could reduce the $H_{0}$ tension. Later on, Yang  _et al._ (2023)
provides updated GZ constraints from recent CMB data, SNIa (Pantheon and JLA),
BAO and OHD. They also obtained that the $w_{0}$ tends to the phantom regime,
a similar result as in Yang  _et al._ (2019a), whereas that point out that the
$H_{0}$ problem remains unsolvable (see Yang  _et al._ (2023)). In this line,
we confront the GZ cosmology with several cosmological datasets to provide
strong updated bounds on their parameters, in both background and perturbation
levels. We analyze the behavior of the deceleration and jerk parameters and
compare with the standard paradigm. Furthermore, we analyze if this model
could alleviate the $H_{0}$ tension through the $\mathbf{\mathbb{H}}0(z)$
diagnostic.

The organization of the paper is as follows: In Sec. II we present the
mathematical formalism for the Gong-Zhang parameterization in where it is
presented the background and perturbed cosmology. In Sec. III we show the data
sets and constraints obtained through CMB, Cosmic Chronometers, Baryon
Acoustic Oscillations, Hydrogen II Galaxies and SNIa observations. Sec. IV
presents our constraints and finally in V we summarize our results and present
a discussion about the model and its consequences. We henceforth use units in
which $c=\hbar=k_{B}=1$, unless explicitly stated otherwise.

##  II Gong-Zhang parameterization cosmology

###  II.1 Background cosmology

Under a flat Friedmann-Leamitre-Robertson-Walker (FLRW) spacetime, we consider
a cosmological model composes by DM and baryon fluid with an EoS
$\omega_{m}=0$, a radiation fluid with $\omega_{r}=1/3$, and a dynamical DE
fluid whose EoS is parameterised Gong and Zhang (2005), as

$\omega_{GZ}(z)=\frac{w_{0}}{1+z}e^{z/(1+z)},$ (1)

where $z$ is the redshift and $w_{0}$ is a free parameter which represents the
$\omega_{GZ}$ value at $z=0$. Notice that $w_{GZ}\to 0$ when $z\to\infty$ and
when $z\to-1$. Notice that this model only adds one extra parameter over
$\Lambda$CDM. Figure 1 displays the evolution of $\omega_{GZ}(z)$ for several
values of $\omega_{0}$.

Figure 1: Evolution of the GZ EoS for several values of $\omega_{0}$, all of
these EoS under the general relativity constraint of Universe acceleration
$w_{0}<-1/3$.

At the background level, the cosmological system can be expressed as

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}}{3}(\rho_{m}+\rho_{r}+\rho_{de}),$ (2)
$\displaystyle\dot{H}+H^{2}$ $\displaystyle=$
$\displaystyle-\frac{\kappa^{2}}{6}\sum_{i}(\rho_{i}+3p_{i}),$ (3)
$\displaystyle\dot{\rho}_{m}$ $\displaystyle=$ $\displaystyle-3H\rho_{m},$ (4)
$\displaystyle\dot{\rho}_{r}$ $\displaystyle=$ $\displaystyle-4H\rho_{r},$ (5)
$\displaystyle\dot{\rho}_{de}$ $\displaystyle=$
$\displaystyle-3H(1+\omega_{GZ})\rho_{de},$ (6)

where the dot symbol ($\,\cdot\,$) represents derivative with respect to
cosmic time, $\rho_{m},\rho_{r}$ and $\rho_{de}$ refer to matter, radiation
and dark energy densities respectively and the sum is also by the three
components, $H\equiv\dot{a}/a$ is the Hubble parameter, $a$ is the scale
factor related with redshift through $a=(1+z)^{-1}$, $\kappa^{2}\equiv 8\pi
G$, and $G$ is the Newton gravitational constant.

By integrating (4)-(6) and substituting into (2), the Hubble parameter is
given by

$\left(\frac{H(z)}{H_{0}}\right)^{2}=\Omega_{m0}(1+z)^{3}+\Omega_{r0}(1+z)^{4}+X_{de}(z),$
(7)

where

$X_{de}(z)=\Omega_{de0}(1+z)^{3}\exp\left[3w_{0}\left(e^{\frac{z}{1+z}}-1\right)\right]\,,$
(8)

and $\Omega_{i0}=\kappa^{2}\rho_{i0}/3H^{2}_{0}$ with $i=m,r$ and
$\Omega_{de0}=1-\Omega_{m0}-\Omega_{r0}$. For radiation density parameter we
use $\Omega_{r0}=2.469\times 10^{-5}h^{-2}(1+0.2271g_{*})$, where $g_{*}=3.04$
is the effective number of relativistic species Komatsu  _et al._ (2011).
Notice that the expression (7), satisfies the flatness condition $H(0)=1$.
Additionally, notice that the DE energy density in eq. (8) behaves as an
effective DM density due to the cubic term implying a similar dust evolution.
For instance, when $z\to-1$ we have
$X_{de}\sim\Omega_{de0}e^{-3\omega_{0}}(1+z)^{3}$, and when $z\gg 1$ the
equation is $X_{de}\sim\Omega_{de0}e^{-3\omega_{0}(1-e)}(1+z)^{3}$, but at
present it is a constant $X_{de}\sim\Omega_{de0}$.

On the other hand, the deceleration parameter $q(z)$ is estimated as

$\displaystyle q(z)$ $\displaystyle=$
$\displaystyle-1+\frac{H_{0}^{2}}{H^{2}(z)}\left[\frac{3}{2}\Omega_{m0}(1+z)^{3}+2\Omega_{r0}(1+z)^{4}\right.$
(9)
$\displaystyle+\left.\frac{3}{2}\left(1+\omega_{0}-\omega_{0}\frac{z}{1+z}\right)X_{de}(z)\right]\,,$

and the jerk parameter is obtained through the expression

$j(z)=q(2q+1)+(1+z)\frac{dq}{dz}\,.$ (10)

It is worth to mention that the jerk parameter for $\Lambda$CDM tends to
$j\approx 1$, and we have $j\rightarrow 1$ as $z\rightarrow-1$ in GZ model.

###  II.2 Perturbed cosmology

Within the EoS parameterization (1), the DE is dynamical and it plays an
important role in the perturbation analysis. In this sense, by perturbing the
FLRW metric under the conformal Newtonian gauge as

$ds^{2}=a^{2}(\eta)[-(1+2\psi)d\eta^{2}+(1-2\phi)\delta_{ij}dx^{i}dx^{j}]\,,$
(11)

where $\eta$ is the conformal time, $\psi(\eta,\vec{r})$ and
$\phi(\eta,\vec{r})$ are scalar potentials. Thus the perturbed equations for
the DE density and velocity are Bardeen (1980); Kodama and Sasaki (1984); Ma
and Bertschinger (1995)

$\displaystyle\delta^{\prime}$ $\displaystyle=$
$\displaystyle-(1+\omega_{GZ})(\theta-3\phi^{\prime})-3\mathcal{H}(c_{s}^{2}-\omega_{GZ})\delta\,,$
(12) $\displaystyle\theta^{\prime}$ $\displaystyle=$
$\displaystyle-\mathcal{H}(1-3\omega_{GZ})-\frac{\omega_{GZ}^{\prime}}{1+\omega_{GZ}}\theta+\frac{c_{s}^{2}}{1+\omega_{GZ}}k^{2}\delta$
(13) $\displaystyle+k^{2}\psi\,,$

respectively. In this notation,
$\delta=\delta\rho/\rho=(\rho-\bar{\rho})/\bar{\rho}$ is the density contrast
between the density field $\rho$ and the spatial average density $\bar{\rho}$,
$\theta=\nabla_{i}v^{i}$ is the velocity divergence and $k^{2}$ is the wave
vector. The prime (′) denotes the derivatives with respect to $\eta$, the
conformal Hubble parameter is $\mathcal{H}=a^{\prime}/a$ and the square sound
speed is defined in the form $c_{s}^{2}=\delta p/\delta\rho$. It is worthy to
note that for simplicity, the GZ parameterization is introduced as a fluid
into the perturbation equations instead of a modification to GR. The another
approach is that the GZ EoS is an effective EoS sourced by extra terms in the
cosmological dynamical equations due to GR modifications. In this case, Eqs.
(12)-(13), could be not valid in general and they must be modified.

##  III Datasets and constraints

The parameter phase-space of the GZ model is given by
$\Theta=\\{h,\Omega_{b0},\Omega_{m0},\omega_{0},\log(10^{10}A_{s}),n_{s},\tau{}_{reio}\\}$
where $h$ is the dimensionless Hubble constant, $A_{s}$ is the amplitude of
the initial power spectrum, $n_{s}$ is the scalar spectral index,
$\tau{}_{reio}$ is the optical depth to re-ionization,
$\Omega_{b0}=\kappa^{2}\rho_{b0}/3H^{2}_{0}$ with $\rho_{b0}$ is the current
baryon density, and $\Omega_{m0}$ and $\omega_{0}$ were defined in the
previous section. The aim is to bound them using CMB, CC, BAO, HIIG and SNIa
datasets and Markov Chain Monte Carlo (MCMC) tools. Furthermore, a stronger
bound on the phase-space is established by a joint analysis. Thus, a modified
CLASS version Blas  _et al._ (2011) is used to implement the GZ
parametrization and also a likelihood for HIIG sample is added to Montepython
Brinckmann and Lesgourgues (2018); Audren  _et al._ (2013) to perform the
mentioned analysis. The convergence of the chains is verified through the
Gelman-Rubin criterion.

Parameter | Prior
---|---
$h$ | Gauss($0.7324$,$0.0174$) Riess  _et al._ (2016)
$\Omega_{b0}$ | [$0.001,0.15$]
$\Omega_{m0}$ | [$0,1$]
$\omega_{0}$ | [$-2,0$]
$\log(10^{10}A_{s})$ | [$1.7,5.0$]
$n_{s}$ | [$0.7,1.3$]
$\tau{}_{reio}$ | [$0.004,0.08$]

Table 1: Priors considered over free parameters at background and perturbative
level.

The configuration for the priors are Uniform distributions except for $h$
which is a Gaussian distribution as shown in Table 1. We use a Gaussian prior
over $h$ to obtain a largest deviation between our result and the Planck value
instead of use a flat prior on $h$. In this sense, we would restrict the $h$
value to the SH0ES value Riess  _et al._ (2016) and leave to vary freely the
rest of parameters, in particular, we are interested to observe the behaviour
of the EoS parameter $\omega_{0}$. Hence the figure-of-merit for the joint
analysis is built through the a Gaussian log-likelihood given as
$-2\ln(\mathcal{L}_{\rm data})\varpropto\chi^{2}_{\rm data}$ and

$\chi^{2}_{\rm data}=\chi_{\rm CMB}^{2}+\chi_{\rm CC}^{2}+\chi_{\rm
BAO}^{2}+\chi_{\rm HIIG}^{2}+\chi_{\rm SNIa}^{2}\,,$ (14)

where each terms refer to the $\chi^{2}$-function for every single dataset.
Now, a description of all data is given in the rest of the section.

###  III.1 Cosmic Microwave Background

The latest CMB anisotropy data is provided by Planck 2018 legacy dataset
release Aghanim and et. al. (2020). We use likelihoods of the high-$\ell$ for
Temperature power spectrum (TT mode) covering a multipole interval $30\leq\
\ell<2508$, and for high TE multipole and polarization spectra EE modes in the
range $30\leq\ell\leq 1996$. Additionally, we consider the low-$\ell$ TT-only
likelihood which covers the multipole range $2\leq\ell\leq 29$.

###  III.2 Cosmic Chronometers

Cosmic chronometers consist of 31 Hubble parameter measurements which are
considered as cosmological model independent because are obtained using
differential age tools Moresco  _et al._ (2016). Furthermore, due these points
are considered uncorrelated the $\chi^{2}$-function ca be written as

$\chi^{2}_{{\rm
CC}}=\sum_{i=1}^{31}\left(\frac{H_{th}(z_{i},\Theta)-H_{obs}(z_{i})}{\sigma^{i}_{obs}}\right)^{2},$
(15)

where the sum runs over the whole sample, and $H_{th}-H_{obs}$ is the
difference between the theoretical and observational Hubble parameter at the
redshift $z_{i}$ and $\sigma_{obs}$ is the uncertainty of $H_{obs}$.

###  III.3 Baryon Acoustic Oscillations

Baryon Acoustic Oscillations give an standard ruler evolving through the
Universe since recombination epoch produced from the interactions between
baryons and photons. As BAO is a powerful tool for constraining cosmological
models, we use several measurements of the BAO data coming from different
tracers as galaxies, Ly$\alpha\times$ Ly$\alpha$ and Ly$\alpha\times$ QSO.
Table 2 summaries the BAO dataset used. BAO measurements in the transverse
direction give an estimate of $D_{M}(z)/r_{d}$ and the BAO feature through the
line of sight gives $D_{H}(z)/r_{d}=c/H(z)r_{d}$. Here,

$D_{M}(z)=c\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})}\,,$ (16)

is the comoving angular diameter for a flat cosmology at the redshift $z$,
$r_{d}=r_{s}(z_{d})$ is the size of the sound horizon at the drag epoch
redshift $z_{d}$, and $c$ is the speed of light. The datasets 6dFGS and SDSS
DR7 report the measurement $D_{V}(z)r_{d}$ where the distance scale $D_{V}$ is
defined as

$D_{V}(z)=[z\,D_{H}(z)\,D_{M}^{2}(z)]^{1/3}\,.$ (17)

The redshift $z_{d}$ is given by Eisenstein and Hu (1998)

$z_{d}=\frac{1291(\Omega_{m0}h^{2})^{0.251}}{1+0.659(\Omega_{m0}h^{2})^{0.828}}[1+b_{1}(\Omega_{b0}h^{2})^{b_{2}}],$
(18)

and

$\displaystyle b_{1}$ $\displaystyle=$ $\displaystyle
0.313(\Omega_{m0}h^{2})^{-0.419}[1+0.607(\Omega_{m0}h^{2})^{0.674}],$ (19)
$\displaystyle b_{2}$ $\displaystyle=$ $\displaystyle
0.238(\Omega_{m0}h^{2})^{0.223}\,.$ (20)

Thus the $\chi^{2}$-function is built as

$\displaystyle\chi^{2}_{{\rm BAO}}$ $\displaystyle=$
$\displaystyle\chi^{2}_{{\rm 6dFGS}}+\chi^{2}_{{\rm SDSSDR7}}+\chi^{2}_{{\rm
SDSSDR12}}$ (21) $\displaystyle+\chi^{2}_{{\rm SDSSDR14}}\,.$

Due the 6dFGS Beutler  _et al._ (2011), SDSS DR7 Ross  _et al._ (2015), SDSS
DR14 de Sainte Agathe  _et al._ (2019) datasets give uncorrelated points, each
$\chi^{2}$ corresponds to the square of the difference between the observable
measurement and the theoretical estimate divided by the uncertainty of the
measurement. SDSS DR12 Alam  _et al._ (2017) dataset reports correlated
measurements of the $D_{M}$ and $H$, thus the $\chi^{2}$ function is built
using the covariance matrix, as $\chi^{2}=\vec{X}^{T}{\rm
C}_{DR12}^{-1}\vec{X}$.

BAO measurement | Dataset | Tracer | Parameter | $z_{\rm eff}$ | Reference
---|---|---|---|---|---
6dFGS | 6dFGS | galaxies | $r_{d}/D_{V}(z_{\rm eff})$ | 0.106 | Beutler  _et al._ (2011)
SDSS MGS | SDSS DR7 | galaxies | $D_{V}(z_{\rm eff})/r_{d}$ | 0.15 | Ross  _et al._ (2015)
BOSS Gal | SDSS DR12 | galaxies |  $D_{M}(z_{\rm eff})/r_{d}$, $D_{H}(z_{\rm eff})/r_{d}$ | 0.38, 0.51, 0.61 | Alam  _et al._ (2017)
eBOSS Ly$\alpha\times$ Ly$\alpha$ | SDSS DR14 | Ly$\alpha\times$ Ly$\alpha$ |  $D_{M}(z_{\rm eff})/r_{d}$, $D_{H}(z_{\rm eff})/r_{d}$ | 2.34 | de Sainte Agathe  _et al._ (2019)
eBOSS Ly$\alpha\times$ QSO | SDSS DR14 | Ly$\alpha\times$ QSO |  $D_{M}(z_{\rm eff})/r_{d}$, $D_{H}(z_{\rm eff})/r_{d}$ | 2.35 | Blomqvist  _et al._ (2019)

Table 2: BAO measurements used in the cosmological parameter estimation.

###  III.4 Hydrogen II Galaxies

Hydrogen II galaxies are compact low mass galaxies which their luminosity is
almost dominated by a young massive burst of star formation and are useful to
constrain cosmological parameters Chávez  _et al._ (2014) because there is a
correlation between the measured luminosity, $L$, and the inferred velocity
dispersion, $\sigma$, of the ionized gas. The largest sample reported by
González-Morán  _et al._ (2019, 2021) consists of a full sample of 181 HIIG
measurements in the redshift range $0.01<z<2.6$. The $\chi^{2}$-function is
written as

$\chi^{2}_{{\rm
HIIG}}=\sum_{i}^{181}\frac{[\mu_{th}(z_{i},{\Theta})-\mu_{obs}(z_{i})]^{2}}{\epsilon_{i}^{2}},$
(22)

where $\mu_{obs}$ is the observational distance modulus given by

$\mu_{obs}=2.5(\alpha+\beta\log\sigma-\log f-40.08)\,,$ (23)

where $\alpha$ and $\beta$ correspond to the intercept and slope of the
$L$-$\sigma$ relation and $f$ is the measured flux. The quantity
$\epsilon_{i}$ is the uncertainty of $\mu_{obs}$ at $z_{i}$. It is worth to
mention that González-Morán  _et al._ (2021) reports measurements and
uncertainties of $\log\alpha$, $\log\beta$ and $\log f$, hence it is necessary
to propagate such errors to estimate $\epsilon_{i}$ (for more details see
González-Morán  _et al._ (2021)).

The theoretical counterpart is given as

$\mu_{th}(z,\Theta)=5\log_{10}\left[\frac{d_{L}(z,\Theta)}{1\,{\rm
Mpc}}\right]+25,$ (24)

where $d_{L}$ is the luminosity distance measured in Mpc expressed as

$d_{L}(z,\Theta)=(1+z)D_{M}(z),$ (25)

being $D_{M}(z)$ defined by Eq. (16).

###  III.5 Type Ia Supernovae

The Pantheon sample Scolnic  _et al._ (2018) contains 1048 luminosity modulus
measurements coming from Supernovae Ia. With this sample a redshift region
$0.01<z<2.3$ is covered. In this case the measurements are correlated which is
appropriate to use a $\chi^{2}$-function as

$\chi_{\rm SNIa}^{2}=\vec{X}^{T}{\rm C}_{P}^{-1}\vec{X},$ (26)

where $\vec{X}$ is the vector of residuals between the theoretical distance
modulus and the observed one and the upper index $T$ represents the transpose
of the vector, ${\rm C}_{P}$ is the covariance matrix formed by adding the
systematic and statistic uncertainties, ${\rm C}_{P}={\rm C}_{P,sys}+{\rm
C}_{P,stat}$.

The theoretical distance modulus is estimated by

$m_{th}=\mathcal{M}+5\log_{10}\left[\frac{d_{L}(z)}{10\,pc}\right],$ (27)

where $\mathcal{M}$ is a nuisance parameter and $d_{L}(z)$ is luminosity
distance defined in Eq. (25).

##  IV Results

After confronting the cosmological model with CMB, CMB+CC, CMB+BAO, CMB+HIIG,
CMB+SNIa data and CMB+CC+BAO+HIIG+SNIa (we will refer as Joint to this data
combination), the main values of the parameter space are summarized in the
Table 3. Additionally, the 2D probability spaces at $68\%$ ($1\sigma$) and
$95\%$ ($2\sigma$) and 1D posterior distributions are displayed in Figure 2.
Furthermore, we also estimate the cosmological parameter $\sigma_{8}$, defined
as the r.m.s. density variation when smoothed with a tophat-filter of radius
of $8h^{-1}\mathrm{Mpc}$ Peacock (1998). Regarding this parameter, we find
consistent results as those found using $\Lambda$CDM. For the Joint analysis
we find our results in agreement within $2\sigma$ with those values reported
in Yang  _et al._ (2019b) (using Planck 2015+CC+BAO+JLA), and is deviated
$1.2\sigma$ from the values reported in Gong and Zhang (2005) (using
WMAP+SNIa+SDSS) for the GZ cosmology; in particular, our yield
$\omega_{0}=-1.202^{+0.027}_{-0.026}$ is within $1\sigma$ but about $50\%$
less of uncertainty. Based on this $\omega_{0}$ value, the DE behaves as
phantom today and is deviated about $7.5\sigma$ of the quintessence behaviour
instead of about $4\sigma$ reported in Yang  _et al._ (2019b). On the other
hand, to compare statistically both GZ and $\Lambda$CDM models, we estimate
their Bayesian evidences. In this test, a model $M_{1}$ is compared with
respect to a model $M_{2}$ through the ratio of their posterior probabilities
as

$\frac{P(M_{1}|D)}{P(M_{2}|D)}=B_{12}\frac{P(M_{1})}{P(M_{2})}\,,$ (28)

where the factor $B_{12}$ is known as the Bayes’ factor defined as the ratio
of the evidence of the models, $B_{12}=P(D|M_{1})/P(D|M_{2})$. In the
Jeffreys’ scale Kass and Raftery (1995), the strength of evidence is as
follow. When $|\log B_{12}|<1$ there is a weak or inconclusive evidence, it is
definite or positive evidence for $1\leq|\log B_{12}|<3$, a strong evidence
for $3\leq|\log B_{12}|<5$, and a very strong evidence or decisive for $|\log
B_{12}|>5$. Table 4 shows the difference of the $\log B_{12}=\log B_{\rm
GZ}-\log B_{\Lambda\rm CDM}$ when GZ model is confronted with $\Lambda$CDM for
the different samples used by using the publicly package MCEvidence Heavens
_et al._ (2017a, b). A negative (positive) value of $\log B_{ij}$ indicates a
preference of the data over $\Lambda$CDM (GZ cosmology). We find an
inconclusive evidence for CMB+CC and CMB+HIIG, a strong evidence of GZ model
for CMB, CMB+BAO and CMB+SNIa, and a strong evidence of $\Lambda$CDM for the
Joint analysis.

Sample | $\chi^{2}_{\rm min}$ | $h$ | $\Omega_{m0}$ | $\omega_{0}$ | $\log 10^{10}A_{s}$ | $n_{s}$ | $\tau{}_{reio}$ | $\sigma_{8}$ | $S_{8}$
---|---|---|---|---|---|---|---|---|---
GZ cosmology
CMB | $2841$ | $0.742^{+0.017}_{-0.017}$ | $0.262^{+0.012}_{-0.014}$ | $-1.333^{+0.057}_{-0.054}$ | $3.048^{+0.015}_{-0.016}$ | $0.965^{+0.004}_{-0.004}$ | $0.055^{+0.007}_{-0.008}$ | $0.867^{+0.020}_{-0.020}$ | $0.810^{+0.015}_{-0.016}$
CMB+CC | $2856$ | $0.739^{+0.016}_{-0.016}$ | $0.264^{+0.011}_{-0.013}$ | $-1.324^{+0.054}_{-0.053}$ | $3.049^{+0.015}_{-0.016}$ | $0.965^{+0.004}_{-0.004}$ | $0.055^{+0.007}_{-0.008}$ | $0.865^{+0.019}_{-0.019}$ | $0.811^{+0.015}_{-0.016}$
CMB+BAO | $2861$ | $0.725^{+0.011}_{-0.012}$ | $0.274^{+0.008}_{-0.009}$ | $-1.275^{+0.045}_{-0.042}$ | $3.049^{+0.015}_{-0.016}$ | $0.966^{+0.003}_{-0.003}$ | $0.056^{+0.007}_{-0.008}$ | $0.848^{+0.017}_{-0.018}$ | $0.810^{+0.013}_{-0.014}$
CMB+HIIG | $3278$ | $0.733^{+0.013}_{-0.013}$ | $0.268^{+0.009}_{-0.010}$ | $-1.309^{+0.047}_{-0.044}$ | $3.049^{+0.015}_{-0.016}$ | $0.965^{+0.004}_{-0.004}$ | $0.055^{+0.008}_{-0.008}$ | $0.860^{+0.017}_{-0.017}$ | $0.812^{+0.015}_{-0.016}$
CMB+SNIa | $3883$ | $0.690^{+0.009}_{-0.009}$ | $0.303^{+0.009}_{-0.009}$ | $-1.168^{+0.032}_{-0.031}$ | $3.050^{+0.015}_{-0.016}$ | $0.965^{+0.004}_{-0.004}$ | $0.056^{+0.007}_{-0.008}$ | $0.815^{+0.014}_{-0.014}$ | $0.819^{+0.016}_{-0.016}$
JOINT | $4367$ | $0.709^{+0.007}_{-0.007}$ | $0.284^{+0.006}_{-0.006}$ | $-1.202^{+0.027}_{-0.026}$ | $3.051^{+0.015}_{-0.017}$ | $0.968^{+0.003}_{-0.003}$ | $0.058^{+0.007}_{-0.009}$ | $0.821^{+0.013}_{-0.013}$ | $0.799^{+0.013}_{-0.013}$
$\Lambda$CDM
CMB | $2849$ | $0.680^{+0.006}_{-0.006}$ | $0.307^{+0.008}_{-0.008}$ | — | $3.048^{+0.015}_{-0.016}$ | $0.969^{+0.004}_{-0.004}$ | $0.057^{+0.007}_{-0.008}$ | $0.808^{+0.007}_{-0.008}$ | $0.817^{+0.016}_{-0.016}$
CMB+CC | $2864$ | $0.680^{+0.006}_{-0.006}$ | $0.306^{+0.007}_{-0.008}$ | — | $3.048^{+0.015}_{-0.016}$ | $0.969^{+0.004}_{-0.004}$ | $0.057^{+0.007}_{-0.009}$ | $0.808^{+0.007}_{-0.008}$ | $0.817^{+0.015}_{-0.016}$
CMB+BAO | $2860$ | $0.681^{+0.004}_{-0.004}$ | $0.305^{+0.006}_{-0.006}$ | — | $3.048^{+0.014}_{-0.016}$ | $0.970^{+0.003}_{-0.003}$ | $0.057^{+0.007}_{-0.008}$ | $0.808^{+0.007}_{-0.007}$ | $0.814^{+0.012}_{-0.013}$
CMB+HIIG | $3289$ | $0.684^{+0.006}_{-0.005}$ | $0.302^{+0.007}_{-0.007}$ | — | $3.049^{+0.015}_{-0.016}$ | $0.971^{+0.004}_{-0.004}$ | $0.058^{+0.007}_{-0.009}$ | $0.806^{+0.007}_{-0.008}$ | $0.809^{+0.014}_{-0.015}$
CMB+SNIa | $3893$ | $0.688^{+0.005}_{-0.005}$ | $0.296^{+0.006}_{-0.007}$ | — | $3.049^{+0.015}_{-0.017}$ | $0.973^{+0.003}_{-0.004}$ | $0.059^{+0.008}_{-0.009}$ | $0.803^{+0.007}_{-0.008}$ | $0.798^{+0.014}_{-0.014}$
JOINT | $4358$ | $0.688^{+0.004}_{-0.004}$ | $0.296^{+0.005}_{-0.005}$ | — | $3.049^{+0.015}_{-0.017}$ | $0.973^{+0.003}_{-0.003}$ | $0.059^{+0.008}_{-0.009}$ | $0.804^{+0.007}_{-0.007}$ | $0.799^{+0.011}_{-0.011}$

Table 3: First column is the data sample, second column is the minimum value
of the $\chi^{2}$, and the rest of the columns are the bestfit values of the
free parameters and their $1\sigma$ CL uncertainties for both GZ and
$\Lambda$CDM cosmologies.

Sample | $\log B_{12}$
---|---
CMB | 18.99
CMB+CC | -0.29
CMB+BAO | 7.95
CMB+HIIG | -1.69
CMB+SNIa | 6.20
JOINT | -10.15

Table 4: Bayes factor difference, $\log B_{12}=\log B_{\rm GZ}-\log
B_{\Lambda\rm CDM}$. Figure 2: 2D contours at $1\sigma$ (inner region) and
$2\sigma$ (outermost region) CL for the GZ cosmology.

Figure 3 displays the $C_{\ell}^{TT}$ CMB Temperature power spectrum for both
GZ and $\Lambda$CDM cosmologies at the top panel and the relative error of GZ
with respect to $\Lambda$CDM at the bottom panel. We find agreement between
both models less than $5\%$. As it is expected, the largest contribution of
cosmological models with variable EoS is at low $\ell$ as shown in the bottom
panel Feng  _et al._ (2012).

Figure 3: $C^{TT}_{\ell}$ CMB Temperature reconstruction for both GZ and
$\Lambda$CDM cosmologies at top panel. Relative difference between both
models, $\varepsilon=C^{TT}_{\ell}({\rm GZ})/C^{TT}_{\ell}(\Lambda{\rm
CDM})-1$, at bottom panel.

Figure 5 shows the reconstruction of the Hubble (left panel), the deceleration
(middle panel) and jerk (right panel) parameters for the GZ model as function
of redshift. The blue shadow regions show the $3\sigma$ error for the joint
constraints. For comparison, the $\Lambda$CDM prediction is also shown.
Although the $H(z)$ reconstruction for GZ model is consistent with
$\Lambda$CDM as shown in Fig. 5, the cosmographic parameters present different
behaviours. Firstly, an interesting feature is the slowing down of the cosmic
acceleration observed in the $q(z)$ reconstruction at $z_{sd}\approx-0.3$ as
shown in Fig. 5, where $z_{sd}$ is the slowing down redshift, being the
accelerated epoch only a transitory effect ($-0.7\lesssim z\lesssim 0.7$),
ending the evolution of the Universe similar as the matter dominated epoch.
The slowing down of cosmic acceleration is also found in some dynamical dark
energy models with parametric EoS Cardenas and Rivera (2012); Cardenas  _et
al._ (2013); Magaña  _et al._ (2014); Hu  _et al._ (2016); Magana  _et al._
(2017); Zhang and Xia (2018); Roman-Garza  _et al._ (2019); Bolotin  _et al._
(2022) but at current times. Nevertheless, this trend might not be present in
these models if their constraints are updated from the recent data samples
Zhang and Xia (2018). In general, not all the parametric models predict a
future deceleration Yang  _et al._ (2019b) and sometimes the results can be
misleading when it is used old observational data to constrain them Zhang and
Xia (2018). For the GZ model, a future deceleration is predicted instead of a
slowing down of the acceleration at current times, this trend differs from the
standard cosmological model. This phenomenon (i.e. the future deceleration) is
also found in other models (see for instance Escobal  _et al._ (2023); Mamon
(2018)). On the other hand, the $j(z)$ presents also an oscillatory behaviour
in the range $-1<z\lesssim 1$ and goes to $j\to 1$ for $z\gg 1$ instead of a
constant as the jerk for $\Lambda$CDM.

An estimated value of $q_{0}=-0.789^{+0.034}_{-0.036}$ is found which is
deviated more than $3\sigma$ from the $\Lambda$CDM value but the value of the
deceleration-acceleration transition redshift of
$z_{T}=0.644^{+0.011}_{-0.012}$ obtained is consistent within $1\sigma$ from
the $\Lambda$CDM value. Additionally, a current value of the jerk parameter of
$j_{0}=1.779^{+0.130}_{-0.119}$ is reported and is deviated more than
$5\sigma$ from the $\Lambda$CDM value. The values of $q_{0}$ and $j_{0}$ are
also in agreement with the regions $-1.4<q_{0}<-0.3$ and $-0.1<j_{0}<6.4$
reported in Riess  _et al._ (2004) and discussed by Visser (2004).

We estimate the age of the Universe as $13.743^{+0.030}_{-0.030}$ Gyrs (CMB),
$13.747^{+0.028}_{-0.027}$ Gyrs (CMB+CC), $13.754^{+0.025}_{-0.025}$ Gyrs
(CMB+HIIG), $13.767^{+0.024}_{-0.023}$ Gyrs (CMB+BAO),
$13.834^{+0.024}_{-0.023}$ Gyrs (CMB+SNIa) $13.788^{+0.019}_{-0.019}$ Gyrs
(Joint). These values are consistent within $1\sigma$ with the value reported
by Planck Aghanim and et. al. (2020). Additionally they are in agreement
within $1.02\sigma$ with the cosmological model-independent value of the age
of the Universe ($t_{U}=13.5^{+0.16}_{-0.14}\mathrm{(stat.)}\pm
0.23\mathrm{(sys.)}$) and deviated from the age of the oldest globular
clusters (GC) ($t_{GC}=13.32\pm 0.1\mathrm{(stat.)}\pm 0.23\mathrm{(sys.)}$)
in $2\sigma$ Valcin  _et al._ (2021) (see also Valcin  _et al._ (2020); Bernal
_et al._ (2021)). Figure 4 shows the constrained contours at $1\sigma$ and
$3\sigma$ in the plane $H_{0}$-$t_{U}$ for GZ cosmology using the joint
analysis. Additionally, vertical lines with bands at $2\sigma$ represent the
corresponding $H_{0}$ values for Planck 2018 and SH0ES. Horizontal lines is
the age of GC (solid line) Valcin  _et al._ (2021), age of Universe by Valcin
2021 Valcin  _et al._ (2021) (dashed dot green line) and Planck 2018 (dashed
line) Aghanim and et. al. (2020).

Figure 4:  Constraints at $1\sigma$ (inner region) and $3\sigma$ (outer
region) in the $H_{0}$-$t_{U}$ plane for the GZ cosmology. Vertical bands are
$2\sigma$ CL region centered at best fit values (dashed lines) for Planck 2018
and SH0ES respectively. Horizontal dashed line is the estimated value for the
age of Universe by Planck 2018. Horizontal bands represent $1\sigma$ CL region
centered at the bestfit values for the globular cluster age (solid line) and
the age of Universe (dashed dot line).

Figure 5: Left to right: reconstruction of the $H(z)$, $q(z)$ and $j(z)$ in GZ
cosmology and $\Lambda$CDM for each dataset. The bands correspond to $3\sigma$
uncertainty for GZ model.

On the other hand, GZ model can relax the $H_{0}$ tension presented between
the SH0ES measurements of $H_{0}$ and those by Planck collaboration. Our $h$
constraints (Table 3), are deviated from the Planck value in $3.7\sigma$
(CMB), $3.8\sigma$ (CMB+CC), $3.8\sigma$ (CMB+BAO), $4.1\sigma$ (CMB+HIIG),
$1.4\sigma$ (CMB+SNIa) and $4.0\sigma$ (joint). Additionally, our $h$ values
are deviated within $1\sigma$ for CMB, CMB+CC, CMB+BAO and CMB+HIIG
combinations whereas it is $4.45\sigma$ for CMB+SNIa and $3.02\sigma$ joint
from SH0ES value. This means that reduce the $H_{0}$ tension from $\sim
4.8\sigma$ obtained between the Planck value and SH0ES value to $\sim 4\sigma$
for the joint analysis.

We find that GZ model could alleviate the $S_{8}$ tension. Our estimates
report deviations of $1.72\sigma$ (CMB), $1.76\sigma$ (CMB+CC), $1.80\sigma$
(CMB+BAO), $1.80\sigma$ (CMB+HIIG), $2.07\sigma$ (CMB+SNIa) and $1.38\sigma$
(Joint) from the value reported by KiDS-1000 ($S_{8}=0.766^{+0.020}_{-0.014}$)
Heymans  _et al._ (2021). This means a distance reduction of $2.65\sigma$
obtained between the Planck value and the KiDS-1000 value to $1.38\sigma$ for
the joint analysis.

Finally, we analyze the $\mathbf{\mathbb{H}}0(z)$ diagnostic Krishnan  _et
al._ (2021) in which it is able to compare the Hubble constant from two
different models, named model A and model B with an effective EoS
$\omega_{eff}^{(A)}(z)$ and $\omega_{eff}^{(B)}(z)$ respectively through

$\frac{H_{0}^{(A)}}{H_{0}^{(B)}}=\exp\left(\frac{3}{2}\int_{0}^{z}\frac{\Delta
w_{eff}(z^{\prime})}{1+z^{\prime}}dz^{\prime}\right),$ (29)

where $\Delta w_{eff}(z)=\omega_{eff}^{(B)}(z)-\omega_{eff}^{(A)}(z)$. If
$\Delta w_{eff}(z)=0$ for all redshift, the right hand side is a constant. In
particular, if we confront GZ model and $\Lambda$CDM, the diagnostic
$\mathbf{\mathbb{H}}0(z)$ is written as

$\mathbf{\mathbb{H}}0(z)=\frac{H_{GZ}(z)}{\sqrt{\Omega_{\Lambda}+\Omega_{m0}^{\mathrm{\Lambda
CDM}}(1+z)^{3}+\Omega_{r0}^{\mathrm{\Lambda CDM}}(1+z)^{4}}},$ (30)

where $\Omega_{\Lambda}=1-\Omega_{m0}^{\mathrm{\Lambda
CDM}}-\Omega_{r0}^{\mathrm{\Lambda CDM}}$, and $\Omega_{m0}^{\mathrm{\Lambda
CDM}}$, $\Omega_{r0}^{\mathrm{\Lambda CDM}}$ correspond to matter and
radiation density parameter values respectively for $\Lambda$CDM. We use the
corresponding values reported in Aghanim and et. al. (2020). In this
diagnostic, if $\mathbf{\mathbb{H}}0(z)$ is not a constant, then it suggests
that the null model ($\Lambda$CDM) needs modification.

Figure 6 shows the reconstruction of the $\mathbf{\mathbb{H}}0(z)$ diagnostic
for the GZ cosmology and its error band at $1\sigma$ CL. Horizontal square
points represents the CMB Planck value of $H_{0}$ for $\Lambda$CDM Aghanim and
et. al. (2020) and horizontal band is its uncertainty at $1\sigma$. It is
interesting to observe that in the past (high redshift),
$\mathbf{\mathbb{H}}0(z)$ goes to Planck value and it presents a trend to go
to our prior value of $H_{0}$ obtained by SH0ES collaboration Riess  _et al._
(2019) at late times. This result is supported by the Bayesian evidence for
CMB data in which the GZ model presents a strong evidence when a value of
$h=0.742^{+0.017}_{-0.017}$ is obtained for CMB.

Figure 6: $\mathbf{\mathbb{H}}0(z)$ diagnostic for GZ cosmology in terms of
the redshift $z$. Here we present the behavior of $\mathbb{H}0(z)$ under the
joint analysis. The red squares represent the results for $\Lambda$CDM
cosmology assuming $h=0.6766$ and $\Omega_{0m}=0.3111$ according to Aghanim
and et. al. (2020). Notice that at high redshift, $\mathbf{\mathbb{H}}0(z)$
goes to Planck value whereas that at late times it presents a trend to go to
our prior value of $H_{0}$ obtained by SH0ES collaboration Riess  _et al._
(2019).

##  V Summary and Discussions

We analyzed a model of the Universe which include a DE component characterized
with an exponential EoS added to DM, baryons and radiation, namely GZ
cosmology. This model was confronted with several cosmological datasets as
CMB, CC, BAO, HIIG and SNIa, and also a joint analysis. Our results are
consistent within $2\sigma$ with those reported in Gong and Zhang (2005); Yang
_et al._ (2019b). Furthermore, the Bayes factor is estimated to compare
statistically both GZ model and $\Lambda$CDM and shows that the latter was
preferred by the Joint analysis but GZ cosmology is preferred when the data is
separated, in particular, for CMB and the combinations CMB+BAO and CMB+SNIa.
Based on the Joint constraint results, we reconstruct the deceleration and
jerk parameters as shown in Fig. 5 and we obtain current values
$q_{0}=-0.789^{+0.034}_{-0.036}$ and $j_{0}=1.779^{+0.130}_{-0.119}$
respectively. Although both quantities are deviated more than $3\sigma$ from
the $\Lambda$CDM values, the deceleration acceleration transition redshifts
for both models are in agreement within $1\sigma$. Additionally, an
interesting difference between both models is also presented in these
cosmographic parameters $q(z)$ and $j(z)$, observing oscillations at the
future as shown in Fig. 5 as happens in other models with parametric EoS
presented in the literature Magaña  _et al._ (2014). In contrast to these
models, GZ cosmology presents only an accelerated phase around $-0.7\lesssim
z\lesssim 0.7$, going to a state similar to matter dominated era as $z\to-1$.
Our constraints show that $\omega_{0}=-1.202^{+0.027}_{-0.026}$ (joint)
pointing a phantom DE behavior, however, in this case the final state of the
Universe ($z\to-1$) is a decelerated state instead of an accelerated phase and
even without a future singularity (Big Rip) commonly produced in the phantom
DE scenarios. This kind of behavior is also presented in models as Parker
Vacuum Metamorphosis Di Valentino  _et al._ (2018).

On the other hand, we report the age of the Universe as
$t_{U}=13.788^{+0.019}_{-0.019}$ Gyrs which is consistent with the standard
cosmology within $1\sigma$ and is $2\sigma$ greater than the age of the oldest
globular clusters Valcin  _et al._ (2021). Although we find slightly lower
values of the Universe age with respect to the Planck 2018 value, they are in
agreement within $1\sigma$.

We also confront GZ model and $\Lambda$CDM using the $\mathbf{\mathbb{H}}0(z)$
diagnostic and also based on results of the Bayesian statistical test for CMB
data, we find GZ cosmology is an interesting candidate to alleviate the
$H_{0}$ tension between early and late time estimation. Based on joint
analysis, we find a $h$ value deviated $3.02\sigma$ from SH0ES and a distance
of $4\sigma$ from the Planck 2018 value. However, when CMB and SNIa data are
combined, we find a $1.4\sigma$ from Planck value and about $4.5\sigma$ from
SH0ES. For the other analyzed data, we find a distance up to $4.1\sigma$ from
Planck value but $1\sigma$ from SH0ES value. The $H_{0}$ tension is reduced
from $\sim 4.8\sigma$ obtained between the Planck value (based on
$\Lambda$CDM) and SH0ES value to around $3\sigma-4\sigma$ for the joint
analysis.

Moreover, we also estimate the quantities $\sigma_{8}$ and $S_{8}$. We find
that our results for $\sigma_{8}$ are consistent within $2\sigma$ with values
reported by Yang  _et al._ (2019b) when a combination of CMB Planck 2015, CC,
BAO, JLA data are used and are within $1.2\sigma$ from the values reported in
a previous analysis for GZ model Gong and Zhang (2005) using WMAP+SNIa+SDSS.
Finally, we find $S_{8}$ values deviated up to $1.80\sigma$ with the value
reported by KiDS-1000 Heymans  _et al._ (2021). In particular we find that GZ
model could alleviate the $S_{8}$ tension reducing from $2.65\sigma$ to
$1.38\sigma$ for our joint analysis.

###### Acknowledgements.

We thank anonymous referees for thoughtful remarks and suggestions. A.H.A.
thank to the support from Luis Aguilar, Alejandro de León, Carlos Flores, and
Jair García of the Laboratorio Nacional de Visualización Científica Avanzada.
M.A.G.-A. acknowledges support from cátedra Marcos Moshinsky (MM), Universidad
Iberoamericana for support with the SNI grant and the numerical analysis was
also carried out by Numerical Integration for Cosmological Theory and
Experiments in High-energy Astrophysics (Nicte Ha) cluster at IBERO
University, acquired through cátedra MM support. J.M. acknowledges the support
from ANID REDES 190147. A.H.A, M.A.G.-A. and J.M. acknowledge partial support
from project ANID Vinculación Internacional FOVI220144.

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