# Cosmological observational constraints on the power law $f(Q)$ type modified
gravity theory

Sanjay Mandal 0000-0003-2570-2335<EMAIL_ADDRESS>Faculty of
Mathematics and Computer Science, Transilvania University, Iuliu Maniu Str.
50, 500091 Brasov, Romania Sneha Pradhan 0000-0002-3223-4085
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute of
Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078, India P.K.
Sahoo 0000-0003-2130-8832<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad
Campus, Hyderabad-500078, India Tiberiu Harko<EMAIL_ADDRESS>Department of Theoretical Physics, National Institute of Physics and Nuclear
Engineering (IFIN-HH),077125 Bucharest, Romania Department of Physics, Babes-
Bolyai University, 1 Mihail Kogalniceanu Street, 400084 Cluj-Napoca, Romania

###### Abstract

In modern cosmology, the curiosity of ultimately understanding the nature of
the dark energy controlling the recent acceleration of the Universe motivates
us to explore its properties by using some novel approaches. In this work, to
explore the properties of dark energy we adopt the modified $f(Q)$ gravity
theory, where the non-metricity scalar $Q$, emerging from Weyl geometry, plays
the dynamical role. For the function $f(Q)$ we adopt the functional form
$f(Q)=Q+6\gamma\,H_{0}^{2}(Q/Q_{0})^{n}$, where $n,\,\gamma,\,H_{0}$ and
$Q_{0}$ are constants. Then, we test our constructed model against the various
observational datasets, such as the Hubble, and the Pantheon+SHOES samples,
and their combined sample, through the Markov Chain Monte Carlo (MCMC)
statistical analysis. We also employ the parameter estimation technique to
constrain the free parameters of the model. In addition, we use the
constrained values of the model parameters to explore a few implications of
the cosmological model. A detailed comparison of the predictions of our model
with the $\Lambda$CDM model is also performed. In particular, we discuss in
detail some cosmographic parameters, like the deceleration, the jerk, and the
snap parameters, as well as the behavior of the dark energy and matter energy
densities to see the evolution of various energy/matter profiles. The $Om$
diagnostics is also presented to test the dark energy nature of our model, as
compared to the standard $\Lambda$CDM paradigm. Our findings show that the
considered version of the non-metric $f(Q)$ type modified gravity theory,
despite some differences with respect to the $\Lambda$CDM paradigm, can still
explain the current observational results on the cosmological parameters, and
provide a convincing and consistent account for the accelerating expansion of
the Universe.

Keywords: $f(Q)$ gravity; dark energy; parameter estimation; cosmography;
equation of state parameter.

###### Contents

  1. I Introduction
  2. II Brief review of the $f(Q)$ gravity theory
  3. III The cosmological model
    1. III.1 The generalized Friedmann equations
    2. III.2 The equation of state of the dark energy
    3. III.3 The generalized Friedmann equations in the redshift space
  4. IV Observational Data
    1. IV.1 Cosmic Chronometer (CC) Sample
    2. IV.2 Type Ia Supernovae Sample
    3. IV.3 CC + Type Ia Supernovae Sample
    4. IV.4 Information Criteria and Model Selection Analysis
    5. IV.5 Numerical results
  5. V Cosmological applications
    1. V.1 Cosmographic parameters
      1. V.1.1 The Hubble parameter
      2. V.1.2 The deceleration, jerk and snap parameters
      3. V.1.3 Dimensionless density parameters
      4. V.1.4 $Om$ Diagnostics
  6. VI Conclusion

##  I Introduction

In present day cosmology, one of the primary objective is to explain the
accelerating expansion of our Universe, an effect whose existence was
extensively proven, and investigated, over the past two decades [1, 2]. To
understand the accelerating phase of the Universe, one must either modify
Einstein’s General Relativity, or add a new exotic component, called dark
energy (DE) to the universe’s energy budget. DE is an exotic fluid type
component, having a negative pressure that causes gravity to behave in a
repulsive manner at large cosmological scales [3]. The equation-of-state
parameter $\omega(z)$, defined as the ratio of the fluid’s pressure to its
energy density, is usually employed to characterize the dynamical features of
DE. The most straightforward hypothesis to explain the cosmological
observations is to assume that dark energy is a cosmological constant, with
the parameter of the equation of state given by the redshift independent
$\omega=-1$. The cosmological constant, together with the assumption of the
existence on the Universe of a called dark matter component are the conceptual
basis of the $\Lambda$CDM cosmological paradigm. Alternative cosmological
models that depart from the conventional $\Lambda$CDM model, but still predict
an accelerating expanding Universe include braneworld models [4], K-essence,
quintessence, and non-minimally coupled scalar fields [5, 6, 7, 8, 9],
modified gravity [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], anisotropic
universes [21, 22, 23], interacting dark energy [24, 25, 26], and many others
[27, 28, 29, 30, 31, 32, 33, 34].

Based on the equivalence principle, the view of the gravitational force as a
manifestation of the curvature of the space-time became the dominant paradigm
for the understanding of the gravitational force. This assumption implies that
the gravitational interaction, and the geometry of the space-time, are
completely determined by the nature of the matter fields. The Ricci scalar
curvature plays a vital role in the curved space-time geometry. The Ricci
scalar curvature $R$ is the basic quantity from which the standard Einstein’s
general relativity has been built initially in a Riemannian geometry, where
the torsion and the non-metricity do vanish. Although it is well known that
Einstein’s general relativity provides an outstanding description of the local
gravitational phenomena, at the level of the Solar System, the theory has been
theoretically challenged by specific observational evidence coming from the
realization that the Universe is accelerating, and from the galactic
phenomenology that is usually explained by postulating the existence of dark
matter. These observations suggest that for explaining the gravitational
dynamics and galactic and extra-galactic scales one should go beyond the
standard formalism of general relativity.

The simplest way to construct extensions of general relativity is to include
either an additional component in the Einstein-Hilbert Lagrangian, or to
modify the structure of the Einstein-Hilbert gravitational Lagrangian (the
Ricci scalar) itself. These approaches have led to many important extensions
of general relativity, including $f(R)$ gravity [35], $f(G)$ gravity [36],
$f(P)$ gravity[37], Horndeski scalar-tensor theories[38] etc. However, from a
general differential geometric perspective, by taking into account the affine
properties of a manifold, the curvature is not the only geometric object that
may be used within a geometrical framework to construct gravitational
theories. Torsion and nonmetricity are two other essential geometric objects
connected to a metric space, along with the curvature. They can be used to
obtain the $f(T)$ and the $f(Q)$ gravity theories, respectively.

In the current paper, we are going to describe the current accelerated
expansion of the Universe, and the observational data, through a specific
modified gravity theory, the symmetric teleparallel gravitation theory,
alternatively called $f(Q)$ gravity. The $f(Q)$ gravity was first proposed by
Nester and Yo [39], and later extended by Jimenez et al. [40]. In $f(Q)$
gravity the non-metricity $Q$, originating from the Weyl geometric background,
describes the gravitational interaction in a flat geometry, in which the
curvature vanishes. $f(Q)$ gravity was extensively used to investigate the
cosmological evolution of the Universe. By considering the $f(Q)$ Lagrangian
of the theory as polynomial function in the redshift $z$, Lazkoz et al.[41]
obtained an important number of restrictions on $f(Q)$ gravity. This
investigation demonstrated that viable $f(Q)$ models have coefficients
comparable to those of the GR model, specifically the $\Lambda$CDM model. To
investigate if this new formalism offers any workable alternatives to explain
the Universe’s late-time acceleration, the validity of various models at the
background level was investigated. Several observational probes for the
analysis have been employed, including the expansion rates of the early-type
galaxies, Type Ia supernovae, Quasars, Gamma Ray Bursts, Baryon Acoustic
Oscillations, and Cosmic Microwave Background distance priors. It turns out
that the novel approach proposed in $f(Q)$ gravity offers a different
perspective on constructing modified, observationally reliable cosmological
models.

The exploration of stellar models in the $f(Q)$ modified gravity theory has
been performed in [42], in which observational restrictions in the context of
$f(Q)$ gravity are obtained from the study of compact general relativistic
objects. Focusing on a particular model in $f(Q)$ gravity, Frusciante [43]
found that while it is identical to the $\Lambda$CDM model at the background
level, it exhibits novel and measurably different signatures at the level of
linear perturbations. By examining the external and internal solutions for
compact stars, Lin and Zhai [44] investigated the application of $f(Q)$
gravity to the static spherically symmetric configurations and illustrated the
consequences of the $f(Q)$ gravity theory. Mandal et al.[45] explored the dark
energy parameters for the non-linear and power-law $f(Q)$ models that depict
the observable behavior of the cosmos. Jimenez et al.[46] investigated the
modified gravity theories based on nonlinear extensions of the nonmetricity
scalar, and they examined several interesting baseline cosmologies (including
accelerating solutions related to inflation and dark energy), and assessed how
cosmic perturbations behaved. Harko et al.[47] considered an extension of
$f(Q)$ gravity, by considered the effects of a non-minimal coupling between
geometry and matter. Several cosmological applications of the theory were
considered, by obtaining the generalized Friedmann equations (the cosmological
evolution equations), and by imposing specific functional forms of the
function $f(Q)$, such as power-law and exponential dependence of the
nonminimal couplings. A full theory in which nonmetricity couples to matter,
called $f(Q,T)$ gravity, where $T$ is the trace of the matter energy-momentum
tensor, was introduced and developed in [48] and [49]. Some astrophysical
implications of the $f(Q,T)$ theory were investigated in [50]. The inclusion
of the torsion in the formalism of theories with geometry-matter coupling was
considered in [51]. In addition, for studying various types of energy
restrictions for the investigation of the logarithmic and polynomial functions
in the $f(Q)$ gravity, Mandal et al.[52] used cosmographic quantities to
reconstruct the proper structure of the $f(Q)$ function. The evolution of
matter perturbations in the modified $f(Q)$ gravity was investigated by
Khyllep et al. [53], who also considered the power-law structure of the cosmic
perturbations.

It is the goal of the present paper to consider a detailed investigation, in
the framework of $f(Q)$ gravity, of a specific cosmological model, obtained by
assuming a simple power law form of the $f(Q)$ function, $f(Q)=Q+\gamma
6H_{0}^{2}(Q/Q_{0})^{n}$, where $n,\gamma$ and $Q_{0}=6H_{0}^{2}$ are
constants. After writing down the generalized Friedmann equations, an
effective dark energy model can be constructed. As for the parameter of the
equation of state of the dark energy we assume a specific, redshift dependent
form. In order to test the predictions of the model we have adopted several
numerical techniques, including the MCMC fitting, which allow us to study the
observational implications of this modified $f(Q)$ gravity model, which gives
us the possibility of constraining the cosmological model parameters, using
various observational datasets.

This manuscript is organized in the following manner. We start with the
presentation of the basic formulation of the $f(Q)$ gravity in Section II. We
present the basic assumptions and ideas of a specific $f(Q)$ type cosmological
model in Section III. Thereafter, in Section IV, we present the different
observational samples, the numerical methods, and we present the data analysis
outputs. Moreover, we discuss the obtained results in detail. In addition, in
Section V, we explore the behavior in our model of various cosmological
quantities, like the deceleration parameter, jerk and snap parameters, and the
dark energy and dark matter densities, respectively. Finally, we discuss and
conclude our results in Section VI.

##  II Brief review of the $f(Q)$ gravity theory

The basic idea of the $f(Q)$ theory is that gravitational phenomena can be
fully described in the Weyl geometry [39], in which the metric conditions is
not anymore satisfied, and the covariant divergence of the metric tensor is
given by

$\nabla_{\lambda}g_{\mu\nu}=Q_{\lambda\mu\nu},$ (1)

where $Q_{\lambda\mu\nu}$ is called the nonmetricity. The scalar non-
metricity, given by

$Q\equiv-g^{\mu\nu}\left(L^{\alpha}_{\ \beta\nu}L^{\beta}_{\
\nu\alpha}-L^{\alpha}_{\ \beta\alpha}L^{\beta}_{\ \mu\nu}\right),$ (2)

plays a fundamental role in the theory, where $L^{\lambda}_{\ \mu\nu}$ is
defined as,

$L^{\lambda}_{\
\mu\nu}=-\frac{1}{2}g^{\lambda\gamma}\left(Q_{\mu\gamma\nu}+Q_{\nu\gamma\mu}-Q_{\gamma\mu\nu}\right).$
(3)

Now, we introduce the action for the $f(Q)$ gravity theory, given by [40],

$\displaystyle
S=\int\left[\frac{1}{2}f(Q)+\mathcal{L}_{m}\right]\sqrt{-g}d^{4}x,$ (4)

where $f(Q)$ is a general function of the non-metricity scalar $Q$, $g$
represents the determinant of the metric $g_{\mu\nu}$, and $\mathcal{L}_{m}$
is the matter Lagrangian density. The non-metricity tensor is given as,

$\displaystyle
Q_{\alpha\mu\nu}=\nabla_{\alpha}g_{\mu\nu}=-L^{\rho}_{\alpha\mu}g_{\rho\nu}-L^{\rho}_{\alpha\nu}g_{\rho\mu}.$
(5)

The following two equations give the expressions of the non-metricity tensor’s
two independent traces

$\displaystyle
Q_{\alpha}=Q^{~{}\beta}_{\alpha~{}\beta},~{}~{}\tilde{Q}_{\alpha}=Q^{\beta}_{~{}~{}\alpha\beta},$
(6)

while the deformation term is given by

$\displaystyle
L^{\alpha}_{\mu\nu}=\frac{1}{2}Q^{\alpha}_{\mu\nu}-Q^{~{}~{}~{}\alpha}_{(\mu\nu)}.$
(7)

Moreover, the nonmetricity scalar $Q$ is obtained as,

$\displaystyle
Q=-g^{\mu\nu}(L^{\alpha}_{\beta\nu}L^{\beta}_{\mu\alpha}-L^{\beta}_{\alpha\beta}L^{\alpha}_{\mu\nu})=-P^{\alpha\beta\gamma}Q_{\alpha\beta\gamma}.$
(8)

Here, $P^{\alpha\beta\gamma}$ is the non-metricity conjugate, and is defined
as

$\displaystyle
P^{\alpha}_{~{}~{}\mu\nu}=\frac{1}{4}\left[-Q^{\alpha}_{~{}~{}\mu\nu}+2Q^{\alpha}_{(\mu\nu)}-Q^{\alpha}g_{\mu\nu}-\tilde{Q}^{\alpha}g_{\mu\nu}-\delta^{\alpha}_{(\mu}Q_{\nu})\right].$

The field equation of the $f(Q)$ gravity theory is obtained by varying (4)
with respect to $g_{\mu\nu}$, and it takes the following form:

$\displaystyle-\frac{2}{\sqrt{-g}}\nabla_{a}(\sqrt{-g}f_{Q}P_{\mu\nu}^{\alpha})+\frac{1}{2}g_{\mu\nu}f$
(10)
$\displaystyle+f_{Q}(P_{\nu}^{\alpha\beta}Q_{\mu\alpha\beta}-2P^{\alpha\beta}_{~{}~{}\mu}Q_{\alpha\beta\nu})=\kappa
T_{\mu\nu},$ (11)

where $f_{Q}=\frac{\partial f}{\partial Q}$, and the energy-momentum tensor
$T_{\mu\nu}$ is given by

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{2}{\sqrt{-g}}\frac{\delta\sqrt{-g}\mathcal{L}_{m}}{\delta\sqrt{g_{\mu\nu}}},$
(12)

By varying the action with respect to the affine connection, the following
equation can be obtained:

$\displaystyle\nabla_{\mu}\nabla_{\nu}(\sqrt{-g}f_{Q}P^{\mu\nu}_{~{}~{}~{}\alpha})=0.$
(13)

Within the framework of $f(Q)$ gravity, the field equations guarantee the
conservation of the energy-momentum tensor, and given the choice of $f(Q)=Q$,
the Einstein equations are retrieved.

##  III The cosmological model

The standard Friedmann-Lemaitre-Robertson-Walker line element, which describes
our flat, homogeneous, and isotropic Universe, is given by,

$ds^{2}=-dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2}).$ (14)

Here $t$ is the cosmic time, and $x,\,y,\,z$ denote the Cartesian co-
ordinates. Moreover, $a(t)$ is the cosmic scale factor. The Hubble parameter
$H(t)$ is defined by $H(t)=\frac{\dot{a}}{a}$, where $\dot{a}$ denotes the
derivative of $a$ with respect to the cosmic time $t$. Moreover, we introduce
the cosmological redshift $z$ defined as $1+z=1/a$.

###  III.1 The generalized Friedmann equations

For the FLRW geometry we get the non-metricity scalar as $Q=6H^{2}$. We
consider the matter content of the Universe as consisting of a perfect and
isotropic fluid, with energy-momentum tensor given by

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle(p+\rho)u_{\mu}u_{\nu}+pg_{\mu\nu},$ (15)

where $p$ and $\rho$ are the pressure and the energy density of the fluid,
$u_{\mu}$ is the four velocity vector normalized according to
$u^{\mu}u_{\mu}=-1$.

Now we are considering the splitting of $f(Q)$ as $f(Q)=Q+F(Q)$. By
considering the FLRW metric, we get two Friedmann equations as [54, 55]

$3H^{2}=\rho+\frac{F}{2}-QF_{Q},$ (16)
$(2QF_{QQ}+F_{Q}+1)\dot{H}+\frac{1}{4}(Q+2QF_{Q}-F)=-2p$ (17)

where $F_{Q}=\frac{dF}{dQ}$ and $F_{QQ}=\frac{d^{2}{F}}{dQ^{2}}$.

In the above equation (16), the energy density ($\rho$) can be written as
$\rho=\rho_{m}+\rho_{r}$ where $\rho_{m},\,\rho_{r}$ are the energy density
for dark matter and radiation, respectively. Similarly, we can write
$p=p_{r}+p_{m}$. The standard matter distribution satisfies the conservation
equation given by,

$\frac{d\rho}{dt}+3H(1+\omega)\rho=0.$ (18)

In Eq. (18), the equation of state parameter (EoS) for matter, $\omega$, takes
different values for different matter sources, like baryonic matter, and
radiation. As for the expression of $Q$, and its time derivative, they are
related to the Hubble parameter by the important relations

$Q=6H^{2},\dot{Q}=12H\dot{H}.$ (19)

###  III.2 The equation of state of the dark energy

On the other hand, to describe the features of dark energy, due to the lack of
precision of the current data, and our lack of theoretical understanding of
dark energy, extracting the value of EoS of dark energy from observational
data is particularly difficult. Under these circumstances, one must
parameterize $\omega_{de}$ empirically, usually using two or more free
parameters, to probe the dynamical evolution of dark energy. The Chevallier-
Polarski-Linder (CPL) model [56] is the most popular and thoroughly studied
among all the parametrization forms of dark energy EoS. The simplest form of
the CPL model can be written as,

$\omega_{de}(z)=\omega_{0}+\omega_{a}\frac{z}{1+z}.$ (20)

In the above expression, $z$ is the redshift, $\omega_{0}$ denotes the
present-day value of EoS $\omega(z)$, and $\omega_{a}$ characterizes its
dynamics. The main reason for considering such a parametrization form is to
resolve the divergence property of the linear form
$\omega(z)=\omega_{0}+\omega_{a}z$ at high redshifts.

In addition, the CPL parametrization has a number of advantages, as mentioned
by Linder [57], including a manageable two-dimensional phase space, well-
behaved and bounded behavior for high redshifts, high accuracy in
reconstructing numerous scalar field equations of state, a straightforward
physical interpretation, etc.

Though it has the above mentioned benefits, there are some drawbacks to the
CPL model. The CPL model only properly describes the past expansion history,
but cannot describe the future evolution, since
$\mathinner{\\!\left\lvert\omega_{de}(z)\right\rvert}$ increases and finally
diverges as $z$ approaches $-1$. The EoS is bound between
$\omega_{0}+\omega_{a}$ and $\omega_{0}$ from the infinite past to the
present.

###  III.3 The generalized Friedmann equations in the redshift space

In general, for isotropic and homogeneous spatially flat FLRW cosmologies in
the presence of radiation, non-relativistic matter, and an exotic fluid with
an equation of state $p_{de}=\omega_{de}\,\rho_{de}$, the Friedmann equations
(16), (17) becomes

$3H^{2}=\rho_{r}+\rho_{m}+\rho_{de},$ (21)
$2\dot{H}+3H^{2}=-\frac{\rho_{r}}{3}-p_{m}-p_{de},$ (22)

where $\rho_{r}$, $\rho_{m}$, and $p_{m}$ are the energy densities of the
radiation and matter components, $p_{m}$ is the matter pressure, while
$\rho_{de}$ and $p_{de}$ are the DE’s density and pressure contribution due to
the geometry, given by

$\rho_{de}=\frac{F}{2}-Q\,F_{Q},$ (23)
$p_{de}=2\dot{H}(2QF_{QQ}+F_{Q})-\rho_{de}.$ (24)

In the following we assume that the matter pressure, be it baryonic, or dark
matter, can be neglected. From Eqs. (21) and (22) we obtain immediately the
global conservation equation

$\frac{d}{dt}\left(\rho_{r}+\rho_{m}+\rho_{de}\right)+3H\left(\frac{4\rho_{r}}{3}+\rho_{m}+\rho_{de}+p_{de}\right)=0.$
(25)

When there are no interactions between the three fluids, the energy densities
satisfy the following differential equations

$\displaystyle\dot{\rho}_{r}+4H\rho_{r}$ $\displaystyle=$ $\displaystyle 0,$
(26) $\displaystyle\dot{\rho}_{m}+3H\rho_{m}$ $\displaystyle=$ $\displaystyle
0,$ (27) $\displaystyle\dot{\rho}_{de}+3H(1+\omega_{de})\rho_{de}$
$\displaystyle=$ $\displaystyle 0.$ (28)

The dark energy equation of state $\omega_{de}$ can be written as the function
of $F(Q)$ and its derivatives as

$\omega_{de}=\frac{p_{de}}{\rho_{de}}=-1+\frac{4\dot{H}(2QF_{QQ}+F_{Q})}{F-2QF_{Q}}.$
(29)

From Eqs. (26) and (27), one can quickly get the evolution of the pressureless
matter and of radiation, namely, $\rho_{m}\propto\frac{1}{a(t)^{3}}$ and
$\rho_{r}\propto\frac{1}{a(t)^{4}}$.

Moreover, by using the relationship between redshift ($z$) and the universe
scale factor $a(t)$ [$a(t)=\frac{1}{1+z}$], we can represent the relationship
between the redshift and the cosmic time as,

$\frac{d}{dt}=\frac{dz}{dt}\frac{d}{dz}=-(1+z)H(z)\frac{d}{dz}.$ (30)

Now, for the present cosmological study of the $f(Q)$ gravity, we are
considering one particular form of $F(Q)$, with

$F(Q)=6\gamma\,H_{0}^{2}\left(\frac{Q}{Q_{0}}\right)^{n},$ (31)

where $H_{0}$, $\gamma$, $n$ and $Q_{0}$ are constants. The motivation for
choosing this form is that the Friedmann equations represent a system of
ordinary differential equations, and we can find power-law and exponential
types of solutions for these types of equations. Therefore, we have considered
the power-law form in our study. With the adopted functional form of $f(Q)$ we
obtain first

$\displaystyle\rho_{de}$ $\displaystyle=$
$\displaystyle\frac{F}{2}-Q\,F_{Q}=\frac{\alpha Q^{n}}{2}-Qn\alpha
Q^{n-1}=\alpha\left(\frac{1}{2}-n\right)Q^{n}$ $\displaystyle=$ $\displaystyle
6\gamma\,H_{0}^{2}\left(\frac{1}{2}-n\right)\left(\frac{Q}{Q_{0}}\right)^{n}=6\gamma\,H_{0}^{2}\left(\frac{1}{2}-n\right)\left(\frac{H}{H_{0}}\right)^{2n},$

where we have denoted $\alpha=6\gamma\,H^{2}_{0}/Q_{0}^{n}$, and
$Q_{0}=6H_{0}^{2}$. Then for the derivative of the dark energy we obtain the
expression

$\displaystyle\dot{\rho}_{de}$ $\displaystyle=$ $\displaystyle
n\,\alpha\,Q^{n-1}\left(\frac{1}{2}-n\right)\dot{Q}$ (33) $\displaystyle=$
$\displaystyle
12n\gamma\,H_{0}^{2}\left(\frac{1}{2}-n\right)\left(\frac{H}{H_{0}}\right)^{2n}\frac{\dot{H}}{H}.$

We substitute now the expressions of the dark energy, and of its derivative,
into the conservation equation (28), together with the CPL parametrization of
the parameter of the dark energy equation of state. Hence, by also taking into
account the relation between $H$ and $Q$, we obtain

$2n\,\frac{\dot{H}}{H}+3H\left(1+\omega_{de}\right)=0,$ (34)

leading, in the redshift space, to the first order differential equation

$-2n\,(1+z)\frac{dH}{dz}+3H\left(1+\omega_{0}+\omega_{a}\frac{z}{1+z}\right)=0,$
(35)

or

$-n(1+z)\frac{d}{dz}H^{2}+3\left(1+\omega_{0}+\omega_{a}\frac{z}{1+z}\right)H^{2}=0,$
(36)

with the general solution given by

$H^{2}(z)=C_{1}^{2}(1+z)^{\frac{3(1+\omega_{o}+\omega_{a})}{n}}e^{\frac{3\omega_{a}}{n(1+z)}},$
(37)

where $C_{1}$ is an arbitrary constant of integration, which we determine so
that $H^{2}(0)=H_{0}^{2}$, giving $C_{1}^{2}=H_{0}^{2}e^{-3\omega_{a}/n}$.
Hence we obtain

$H^{2}(z)=H_{0}^{2}(1+z)^{\frac{3(1+\omega_{o}+\omega_{a})}{n}}e^{-\frac{3\omega_{a}z}{n(1+z)}}.$
(38)

Now using (37) in (III.3), we obtain for the dark energy density $\rho_{de}$
the expression

$\rho_{de}(z)=3\gamma\,(1-2n)H_{0}^{2}(1+z)^{3(1+\omega_{o}+\omega_{a})}e^{\frac{-3\omega_{a}z}{(1+z)}}.$
(39)

Alternatively, we can obtain the same result by using the considered equation
of state, which gives first

$\displaystyle\omega_{de}$ $\displaystyle=$
$\displaystyle\frac{P_{de}}{\rho_{de}}=\frac{2\dot{H}(2QF_{QQ}+F_{Q})-\rho_{de}}{\frac{F}{2}-QF_{Q}}$
(40) $\displaystyle=$
$\displaystyle-1+\frac{4\dot{H}(2QF_{QQ}+F_{Q})}{F-2QF_{Q}}.$

With the help of the CPL parametrization we successively obtain

$-1-\frac{4\dot{H}n}{Q}=\omega_{0}+\omega_{a}\frac{z}{1+z},$ (41)

and

$-\frac{2}{3}n(1+z)\frac{dH}{dz}\frac{1}{H}=-\left[1+\omega_{0}+\omega_{a}\frac{z}{1+z}\right],$
(42)

respectively, with the solution of the above differential equation given again
by Eq. (37).

Additionally, the matter density ($\rho_{m}$) and radiation density
($\rho_{r}$) can be written in terms of redshift function $z$ as,

$\displaystyle\rho_{m}\propto(1+z)^{3}\,\,;\quad\rho_{r}\propto(1+z)^{4}$ (43)

Consequently, the Friedmann equation (21) reads,

$\begin{gathered}3H^{2}(z)=\rho_{r0}(1+z)^{4}+\rho_{m0}(1+z)^{3}+3\gamma\,(1-2n)H_{0}^{2}(1+z)^{3(1+\omega_{o}+\omega_{a})}e^{\frac{-3\omega_{a}z}{(1+z)}},\\\
\frac{H^{2}(z)}{H_{0}^{2}}=\Omega_{r0}(1+z)^{4}+\Omega_{m0}(1+z)^{3}+\gamma\,(1-2n)(1+z)^{3(1+\omega_{o}+\omega_{a})}e^{\frac{-3\omega_{a}z}{(1+z)}}.\end{gathered}$
(44)

In the equation (44), the suffix $0$ represents the present day value of the
corresponding quantity. $H_{0}$ is the current Hubble value (at $z=0$) of our
present Universe.

Finally, we are going to introduce the energy density parameters, defined as

$\begin{gathered}\Omega_{m}=\frac{\rho_{m}}{3H^{2}},\,\,\,\Omega_{r}=\frac{\rho_{r}}{3H^{2}},\,\,\,\Omega_{de}=\frac{\rho_{de}}{3H^{2}}\end{gathered}$
(45)

##  IV Observational Data

In this Section we discuss the methodology, and the various observational
samples used to constrain the parameters
$H_{0},\,\Omega_{m0},\,\omega,\,\omega_{a},\,n,\gamma$ of the considered
cosmological model. In particular, we use a Markov Chain Monte Carlo (MCMC)
method to do the statistical analysis, and to obtain the posterior
distributions of the parameters. The data analysis part is done by using the
emcee package in Python. The best fits of the parameters are maximized by
using the probability function

$\displaystyle\mathcal{L}\propto\exp(-\chi^{2}/2),$ (46)

where $\chi^{2}$ is the pseudo chi-squared function [58]. More details about
the $\chi^{2}$ function for various date samples are discussed in the
following subsections.

###  IV.1 Cosmic Chronometer (CC) Sample

For the Cosmic Chronometer (CC) Sample, we used 31 points of Hubble samples,
collected from the differential age (DA) approach in the redshift range
$0.07<z<2.42$. The complete list of this sample is collectively presented in
[59]. The chi-square function for the Hubble sample is defined as

$\chi_{CC}^{2}=\sum_{i=1}^{31}\frac{[H_{i}^{th}(\theta_{s},z_{i})-H_{i}^{obs}(z_{i})]^{2}}{\sigma_{CC}^{2}(z_{i})}$
(47)

where $H_{i}^{obs}$ denotes the observed value, $H_{i}^{th}$ denotes the
Hubble’s theoretical value, $\sigma_{z_{i}}$ denotes the standard error in the
observed value, and
$\theta_{s}=(H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n,\gamma)$ is the
cosmological background parameter space. In addition, we use the following
priors to our analysis, which we present in Table 1.

Table 1: Priors for the parameter space $H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n,\gamma$. Parameter | prior
---|---
$H_{0}$ | (60,80)
$\Omega_{m0}$ | (0,1)
$\omega_{0}$ | (-2,2)
$\omega_{a}$ | (-2,2)
$n$ | (-1,1)
$\gamma$ | (-1,1)

In our MCMC analysis, we used $100$ walkers and $1000$ steps to find out the
fitting results. The $1-\sigma$ and $2-\sigma$ CL contour plot are presented
in Fig. 1, and the numerical results are presented, for the CC sample, in
Table 2. With the mean constrain value of the free parameters, we present the
Hubble parameter profile for the CC sample, together with the $\Lambda$CDM
behavior, in Fig. 2.

Figure 1: The marginalized constraints on the parameters
$H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n,\gamma$ of our model using the
Hubble sample. The dark orange shaded regions presents the $1-\sigma$
confidence level (CL), and the light orange shaded regions present the
$2-\sigma$ confidence level. The constraint values for the parameters are
presented at the $1-\sigma$ CL. Figure 2: The red line represents the Hubble
parameter profile of the power-law model $f(Q)$ model with the constraint
values of $H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n,\gamma$. The blue dots
with the green bars represent the CC sample, and the black dotted line
represents the Hubble parameter profile of the $\Lambda$CDM model.

###  IV.2 Type Ia Supernovae Sample

Supernovae samples are a powerful indicator for exploring the background
geometry and properties of the Universe. In this analysis, we adopt the
largest SNe Ia sample published to date, the Pantheon+SHOES sample, which
consists of 1701 light curves of 1550 spectroscopically confirmed SNe Ia
across 18 different surveys [60]. The Pantheon+SHOES sample significantly
increases the number of observations relative to the Pantheon data at low
redshifts, and covers the redshift range $z\in[0.00122,2.26137]$. It is the
successor of Pantheon sample [61]. The chi-square function is defined as,

$\chi^{2}_{SN}=\sum_{i,j=1}^{1701}\bigtriangledown\mu_{i}\left(C^{-1}_{SN}\right)_{ij}\bigtriangledown\mu_{j}.$
(48)

Here $C_{SN}$ is the covariance matrix [60], and

$\displaystyle\quad\bigtriangledown\mu_{i}=\mu^{th}(z_{i},\theta)-\mu_{i}^{obs}.$

is the difference between the observed value of distance modulus, extracted
from the cosmic observations, and its theoretical values, calculated from the
model, with the given parameter space $\theta$. $\mu_{i}^{th}$ and
$\mu_{i}^{obs}$ are the theoretical and observed distance modulus,
respectively.

The theoretical distance modulus $\mu_{i}^{th}$ is defined as

$\mu_{i}^{th}(z)=m-M=5\log D_{l}(z)+25,$ (49)

where $m$ and $M$ are apparent and the absolute magnitudes of a standard
candle, respectively. The luminosity distance $D_{l}(z)$ is defined as

$D_{l}(z)=(1+z)\int_{0}^{z}\frac{dz^{\ast}}{H(z^{\ast})}.$ (50)

To run the MCMC code, we used the same priors, number of walkers, and steps,
which have been used in the CC sample. The $1-\sigma$ and $2-\sigma$ CL
contour plot is presented in Fig. 3, and the numerical results for the
Pantheon+Shoes sample are presented in Table 2. With the mean constraint value
of the free parameters, we present the distance modulus parameter profile with
the Pantheon+SHOES sample and the $\Lambda$CDM model in Fig. 4.

Figure 3: The marginalized constraints on the parameters
$H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n,\gamma$ of our model using
Pantheon+Shoes sample. The dark blue shaded regions present the $1-\sigma$
confidence level (CL), and light blue shaded regions present the $2-\sigma$
confidence level. The constraint values for the parameters are presented at
the $1-\sigma$ CL. Figure 4: The blue line represents the distance modulus
profile of the power-law $f(Q)$ model with the constraint values of
$H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n,\gamma$. The blue dots with the
green bars represent the Pantheon+SHOES sample, and the black dotted line
represents the distance modulus profile of the $\Lambda$CDM model.

###  IV.3 CC + Type Ia Supernovae Sample

To perform both the CC and Type Ia supernovae samples together, we use the
following Chi-square function

$\chi^{2}_{CC+SN}=\chi^{2}_{CC}+\chi^{2}_{SN}.$ (51)

The marginalized constraints on the parameters included in the parameter space
$\theta$ are presented in Fig. 5. The numerical results are presented in Table
2.

Figure 5: The marginalized constraints on the parameters $H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n,\gamma$ of our model using the Hubble+Pantheon sample. The dark-shaded regions present the $1-\sigma$ confidence level (CL), and the light-shaded regions present the $2-\sigma$ confidence level. The constraint values for the parameters are presented at the $1-\sigma$ CL. Table 2: Marginalized constrained data of the parameters $H_{0}$, $\Omega_{m0}$, $\omega_{0},\,\,\,\omega_{a},\,\gamma$ and $n$ for different data samples with 68% and 95% confidence level.  Model | $H_{0}$ | $\Omega_{m0}$ | $\omega_{0}$ | $\omega_{a}$ | $n$ | $\gamma$
---|---|---|---|---|---|---
|  |  |  $68\%$ limits |  |  |
|  |  | CC sample |  |  |
$\Lambda$CDM | $68.80\pm 0.94$ | $0.318\pm 0.034$ | - | - | - | -
Power-law | $71.59\pm 0.54$ | $0.292\pm 0.020$ | $-1.005\pm 0.090$ | $-0.00996\pm 0.0010$ | $-0.3612\pm 0.0010$ | $0.369\pm 0.046$
|  |  | Pantheon+SHOES sample |  |  |
$\Lambda$CDM | $72.33\pm 0.28$ | $0.383\pm 0.022$ | - | - | - | -
Power-law | $71.733^{+0.085}_{-0.068}$ | $0.1899\pm 0.0069$ | $-1.005\pm 0.010$ | $-0.0100^{+0.0010}_{-0.0011}$ | $-0.3616\pm 0.0010$ | $0.4627\pm 0.0063$
|  |  | CC+Pantheon+SHOES sample |  |  |
$\Lambda$CDM | $72.66\pm 0.26$ | $0.342\pm 0.019$ | - | - | - | -
Power-law | $71.54^{+0.11}_{-0.093}$ | $0.1971\pm 0.0068$ | $-1.0284\pm 0.0096$ | $-0.0181^{+0.011}_{-0.0082}$ | $-0.343\pm 0.010$ | $0.4871\pm 0.0098$
|  |  |  $95\%$ limits |  |  |
|  |  | CC sample |  |  |
$\Lambda$CDM | $68.8^{+1.9}_{-1.8}$ | $0.318^{+0.068}_{-0.063}$ | - | - | - | -
Power-law | $71.6^{+1.0}_{-1.0}$ | $0.292^{+0.040}_{-0.040}$ | $-1.00^{+0.18}_{-0.18}$ | $-0.00996^{+0.0020}_{-0.0020}$ | $-0.3612^{+0.0020}_{-0.0020}$ | $0.369^{+0.094}_{-0.089}$
|  |  | Pantheon+SHOES sample |  |  |
$\Lambda$CDM | $72.33^{+0.55}_{-0.54}$ | $0.383^{+0.044}_{-0.044}$ | - | - | - | -
Power-law | $71.73^{+0.16}_{-0.19}$ | $0.190^{+0.013}_{-0.013}$ | $-1.005^{+0.020}_{-0.019}$ | $-0.0100^{+0.0022}_{-0.0021}$ | $-0.3616^{+0.0020}_{-0.0019}$ | $0.463^{+0.012}_{-0.012}$
|  |  | CC+Pantheon+SHOES sample |  |  |
$\Lambda$CDM | $72.66^{+0.50}_{-0.53}$ | $0.342^{+0.038}_{-0.036}$ | - | - | - | -
Power-law | $71.54^{+0.19}_{-0.22}$ | $0.197^{+0.014}_{-0.014}$ | $-1.028^{+0.020}_{-0.018}$ | $-0.018^{+0.017}_{-0.018}$ | $-0.343^{+0.019}_{-0.020}$ | $0.487^{+0.020}_{-0.020}$

###  IV.4 Information Criteria and Model Selection Analysis

This subsection will discuss the various statistical information criteria and
the model selection procedures. For this purpose, we use the Akaike
information criterion (AIC) [62], and the Bayesian information criterion (BIC)
[63] to compare a set of models with their observational prediction given by
dataset(s).

On the basis of information theory, the AIC addresses the problem of model
adequacy. It is a Kullback-Leibler information estimator with the property of
asymptotic unbiasedness. The AIC estimator is given under the standard
assumption of Gaussian errors, by [64, 65]

$AIC=-2\ln{(\mathcal{L}_{max})}+2k+\frac{2k\,(k+1)}{N_{tot}-k-1},$ (52)

where $k$ is the number of free parameters in the proposed model,
$\mathcal{L}_{max}$ is the maximum likelihood value of the dataset(s)
considered for analysis, and $N_{tot}$ is the number of data points. For a
large number of data points, the above formula reduces to
$AIC\equiv-2\mathcal{L}_{max}+2k$, which is a modified form of AIC. Therefore,
the modified AIC criteria is convenient for all the cases [66].

The BIC is a Bayesian evidence estimator, given by [65, 66, 67],

$BIC=-2\ln{(\mathcal{L}_{max})}+k\log(N_{tot}).$ (53)

For a given set of comparable models, we aim to rank them according to their
fitting qualities with respect to the observational dataset. We use the
previously studied method, in particular, the relative difference between the
IC value of the given models,

$\Delta IC_{model}=IC_{model}-IC_{min},$ (54)

where $IC_{min}$ is the minimum value of IC of the set of competing models.
The $\Delta IC$ value measures the compatibility and tension between the
models. According to Jeffrey’s scale [68], the condition $\Delta IC\leq 2$
confirms the statistical compatibility of the two models, and the model most
favored by the data. The condition $2<\Delta IC<6$ indicates a mild tension
between the two models, while the condition $\Delta IC\geq 10$ suggests a
strong tension. The outputs of these tests are presented in Table 3.

Table 3: The corresponding $\chi^{2}_{min}$ of the models for each sample and the information criteria AIC, BIC for the examined cosmological models, along with the corresponding differences $\Delta IC_{model}=IC_{model}-IC_{min}$.  Model | $\chi^{2}_{min}$ |  red. $\chi^{2}$ | AIC |  $\Delta$ AIC | BIC |  $\Delta$ BIC
---|---|---|---|---|---|---
|  |  | CC |  |  |
$\Lambda$CDM | 16.07 | 0.64 | 20.07 | 0 | 22.93 | 0
Power-law | 16.06 | 0.64 | 28.06 | 7.98 | 36.66 | 13.72
|  |  | Pantheon+SHOES |  |  |
$\Lambda$CDM | 1696.84 | 1.0 | 1700.84 | 0 | 1719.15 | 0
Power-law | 1683.20 | 0.99 | 1695.20 | 5.63 | 1727.83 | 8.6
|  |  | CC+Pantheon+SHOES |  |  |
$\Lambda$CDM | 1712.9 | 1.0 | 1716.90 | 0 | 1735.28 | 0
Power-law | 1699.33 | 0.99 | 1711.33 | 5.5 | 1744.07 | 8.79

###  IV.5 Numerical results

In Tables 2 and 3, we have presented the numerical limits of the parameters
$H_{0}$, $\Omega_{m0}$, $\omega_{0},\,\,\,\omega_{a}$, $n$, and of some
cosmological parameters with the 68% and 95% confidence levels. The constraint
values on the present Hubble parameter are $71.59\pm
0.54,\,71.733^{+0.085}_{-0.068},\,71.54^{+0.11}_{-0.093}$ with 68% CL for CC,
Pantheon+SHOES, CC+Pantheon+SHOES sample respectively.

These results are consistent with recent studies (one can see the detailed
discussion on $H_{0}$ in the reference herein [69]). Furthermore, the
parameters $\omega_{0},\,\,\,\omega_{a}$ play an important role in identifying
the nature of the CPL equation of state parameter/dark energy equation of
state (EoS). This EoS reduces to $\omega_{0}$ at $z=0$, and the constraint
values on it are $-1.005^{+0.090}_{-0.090}$, $-1.005^{+0.010}_{-0.010}$,
$-1.0284^{+0.0096}_{-0.0096}$ for the respective date samples. These values
are very close to the $\Lambda$CDM model.

On the other hand $\omega_{CPL}(z)$ shows the phantom type behaviour with the
constraint values on $\omega_{0},\,\,\,\omega_{a}$ for all datasets, i.e.,
$\omega_{CPL}(z)<1$ always. From all these outputs, one can see that our
findings confirm the existence of the present accelerated expansion of the
Universe. In addition to this, we have presented the $\chi^{2}_{mim}$, the
reduced $\chi^{2}_{mim}$, the AIC, BIC, $\Delta$AIC and $\Delta$BIC values in
Table 3. From these results, we can estimate that the power law $f(Q)$ type
model is a good fit to the observational datasets, as compared with the
$\Lambda$CDM model. However, it shows a mild tension between models as per the
information criteria analysis. Our model shows mild tension compared to
$\Lambda$CDM because the modified gravity model has more degrees of freedom in
the parameter spaces than $\Lambda$CDM. To explore more about our model, we
discuss some cosmological applications in the following Section.

##  V Cosmological applications

In this Section, we shall discuss some cosmological applications of our
theoretical $f(Q)$ model, and we examine its current dynamical status. In this
respect, we investigate the basic Cosmographic Parameters, the matter
distribution profiles, and the dark energy types profiles, respectively.

###  V.1 Cosmographic parameters

The Cosmographic parameters are simply a Mathematical tool that considers the
cosmic scale factor, and its derivatives. Using these parameters’ behavior,
one can investigate the present, low redshift behavior, and predict the future
of the cosmological models. Therefore, we consider the profiles of the Hubble,
deceleration, jerk and snap parameters to present the dynamic status of our
model. Furthermore, we can write down the mathematical expressions for those
parameters as follows;

$\displaystyle
q(z)=\Omega_{r}+\frac{1}{2}\Omega_{m}(z)+\frac{1+3\omega_{de}}{2}\Omega_{de}(z),$
(55) $j(z)=q(z)(2q(z)+1)+(1+z)q^{\prime}(z),$ (56)
$s(z)=-(1+z)j^{\prime}(z)-2j(z)-3j(z)q(z).$ (57)

Here, (′) represents one time derivative with respect to $z$.

####  V.1.1 The Hubble parameter

In the previous Section, we have presented the evolution profile of the Hubble
parameter with the constraint values of the free parameters. Here, we consider
the ratio of $H_{Q}(z)/H_{\Lambda CDM}(z)$ in order to check the difference
between both models. In Fig. 6 we plot the redshift dependence of this ratio.
For low redshifts, like, for example, for $z=0.2$, the difference between the
two models is of the order of $0.0003$%, $7.06$%, and $5.58$%, respectively,
for the CC, Pantheon+SHOES, and CC+Pantheon+SHOES samples.

The differences between the models increase for high redshift, so that for
$z=2.0$, the differences are of the order of $0.003$%, $27.21$%, and $22.98$%,
respectively, for the CC, Pantheon+SHOES, and CC+Pantheon+SHOES samples,
respectively.

####  V.1.2 The deceleration, jerk and snap parameters

Furthermore, we have depicted the profiles of the deceleration, jerk, and snap
parameters with the constraint values of the free parameters for the various
observational datasets in Figs. 7, 8, and 9, respectively.

##### The deceleration parameter.

From the redshift profile of the deceleration parameter one can see clearly
that our model’s evolution started from the decelerated phase, and it is
currently in an accelerating stage, after going through the matter-dominated
era. In addition, we have found that the present values of the deceleration
parameter $q_{0}=-0.532,\,\,-0.717,\,\,-0.744$ for CC, Pantheon+SHOES,
CC+Pantheon+SHOES, respectively, are aligned with the recent observational
results [70, 71, 72].

##### Jerk and snap parameters.

The evolution of the jerk and snap parameters are presented for the present
model in Figs. 8 and 9, respectively. We have also obtained the parametric
plot $q-j$ for the redshift range $z\in[-1,2.5]$ in Fig. 10. In addition, we
have presented $1-\sigma$ CL values of the deceleration, jerk, and snap
parameters in Table 4. The present-day value of the jerk parameter for all the
observational samples is close to the $\Lambda$CDM value.

Figure 6: Evolution of the ratio $H_{Q}(z)/H_{\Lambda CDM}(z)$ as a function
of the redshift variable $z$ for the constraint values of
$H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n,\,\gamma$ for the CC,
Pantheon+SHOES, and the CC+Pantheon+SHOES samples. Figure 7: Evolution of the
deceleration parameter as functions of the redshift variable $z$ for the
constraint values of $H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n\,\gamma$ for
CC, Pantheon+SHOES, CC+Pantheon+SHOES samples. Figure 8: Evolution of jerk
parameter $j$ as a function of the redshift variable $z$ for the constraint
values of $H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n\,\gamma$ for the CC,
Pantheon+SHOES, and CC+Pantheon+SHOES samples.(Here the profile of the jerk
parameter for CC and Pantheon+SHOES samples overlaps each other.) Figure 9:
Evolution of the snap parameter $s$ as a function of the redshift variable $z$
for the constraint values of
$H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n\,\gamma$ for the CC,
Pantheon+SHOES, and CC+Pantheon+SHOES samples. Figure 10: Parametric plot of
$q=q(j)$ in the redshift range $z\in[-1,2.5]$ with the constraint values of
$H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n\,\gamma$ for the CC,
Pantheon+SHOES, and the CC+Pantheon+SHOES samples. The orange, blue, and cyan
color points represents the present value of the pair $(j_{0},q_{0})$ for
respective samples.

####  V.1.3 Dimensionless density parameters

The energy density sources of our universe evolve in time, and play a major
role in characterizing its past, present, and future. Here, we have presented
the evolution profiles of the dark energy density and of the matter density in
Figs. 11 and 12, respectively. From those Figures, one can observe that the
matter energy dominated our Universe in the early time, whereas the dark-
energy density dominates in the current phase. Dark energy is also responsible
for the present acceleration of the Universe. The present-day values of the
dark energy density are $0.685^{+0.010}_{-0.013}$,
$0.8076^{+0.0037}_{-0.0036}$, and $0.8064^{+0.0024}_{-0.0023}$ with $1-\sigma$
error for the CC, Pantheon+SHOES, and CC+Pantheon+SHOES, respectively. We also
present the constraint values of the matter density and of the dark energy
density in Tables 2 and 4, for the 68% and 95% confidence levels. In addition,
the energy densities satisfy the relation $\Omega_{m}+\Omega_{de}\simeq 1$ for
the entire period of their evolution. The dynamical profiles of the two fluids
also suggests that dark energy will continue to dominate our Universe in the
near future.

Figure 11: Profiles of the parameter of the dark energy density $\Omega_{de}$ as functions the redshift variable $z$ for the constraint values of $H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n,\gamma$ for the CC, Pantheon+SHOES, and CC+Pantheon+SHOES samples. Figure 12: Profiles of the matter-energy density parameter $\Omega_{m}$ as a function of the redshift variable $z$ for the constraint values of $H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n,\gamma$ for the CC, Pantheon+SHOES, and CC+Pantheon+SHOES samples. Table 4: Present-day values of the cosmological parameters $q_{0}$, $j_{0}$, $s_{0}$ and $\Omega_{de0}$ as predicted by the power law $f(Q)$ model for different data samples with 68% confidence level.  Model | $q_{0}$ | $j_{0}$ | $s_{0}$ | $\Omega_{de0}$
---|---|---|---|---
|  | CC sample |  |
$\Lambda$CDM | $-0.523\pm 0.0345$ | $1\pm(<\mathcal{O}(10^{-16}))$ | $-0.431\pm 0.1035$ | $0.682\pm 0.034$
Power-law | $-0.532^{+0.077}_{-0.070}$ | $1.001^{+0.298}_{-0.258}$ | $-0.439^{+0.469}_{-0.278}$ | $0.685^{+0.010}_{-0.013}$
|  | Pantheon+SHOES sample |  |
$\Lambda$CDM | $-0.4255\pm 0.033$ | $1\pm(<\mathcal{O}(10^{-16}))$ | $-0.7235\pm 0.099$ | $0.617\pm 0.022$
Power-law | $-0.717^{+0.017}_{-0.017}$ | $1.006^{+0.035}_{-0.035}$ | $0.108^{+0.075}_{-0.071}$ | $0.8076^{+0.0037}_{-0.0036}$
|  | CC+Pantheon+SHOES sample |  |
$\Lambda$CDM | $-0.487\pm 0.0285$ | $1\pm(<\mathcal{O}(10^{-16}))$ | $-0.539\pm 0.0855$ | $0.658\pm 0.019$
Power-law | $-0.744^{+0.015}_{-0.015}$ | $1.06^{+0.023}_{-0.038}$ | $0.198^{+0.011}_{-0.413}$ | $0.8064^{+0.0024}_{-0.0023}$

####  V.1.4 $Om$ Diagnostics

The $Om$ diagnostic is used to analyze the difference between standard
$\Lambda CDM$ and other dark energy models. $Om$ is more convenient than the
state-finder diagnosis [73] as it uses only the first-order temporal
derivative of the cosmic scale factor. This is because it only involves the
Hubble parameter, and the Hubble parameter depends on a single time derivative
of $a(t)$. For the spatially flat Universe, it is defined as

$Om(x)=\frac{\mathcal{H}(x)^{2}-1}{(1+z)^{3}-1},\,\,x=1+z,\mathcal{H}(x)=H(x)/H_{0},$
(58)

where $z$ is the redshift, and $H_{0}$ is the present-day value of the Hubble
parameter. For the dark energy model with the constant equation of state
$\omega$,

$\mathcal{H}(x)=\Omega_{m0}x^{3}+(1-\Omega_{m0})x^{\delta},\,\,\delta=3(1+\omega).$
(59)

Now, we can rewrite $Om(x)$ as

$Om(x)=\Omega_{m0}+(1-\Omega_{m0})\frac{x^{\delta}-1}{x^{3}-1}.$ (60)

For the $\Lambda$CDM model, we find

$Om(x)=\Omega_{m0},$ (61)

whereas $Om(x)<\Omega_{m0}$ in phantom cosmology with $\delta<0$, while
$Om(x)>\Omega_{m0}$ in the quintessence models with $\delta>0$. These results
show that: $Om(x)-\Omega_{m0}=0$, if dark energy is a cosmological constant
[73].

In another way, we can say that the $Om$ diagnostic gives us a null test of
the cosmological constant. As a consequence, $\mathcal{H}(x)^{2}$ provides a
straight line against $x^{3}$ with a constant slope $\Omega_{m0}$ for
$\Lambda$CDM, a result which can be verified by using equation (59). For other
dark energy models $Om(x)$ is curved, because

$\frac{d\mathcal{H}^{2}(x)}{dx}={\rm constant}.$ (62)

Furthermore, for $x_{1}<x_{2}$, $Om(x_{1},x_{2})\equiv Om(x_{1})-Om(x_{2})=0$
in $\Lambda$CDM, whereas $Om(x_{1},x_{2})\equiv Om(x_{1})-Om(x_{2})<0$ in
phantom models, and $Om(x_{1},x_{2})\equiv Om(x_{1})-Om(x_{2})>0$ in
quintessence cosmology. This test helps us with the interpretation of the
observational measurements, and also, provides us a null test for the
$\Lambda$CDM model. In addition to this, one can check that $Om(x)\rightarrow
0$ as $z\rightarrow-1$ for quintessence, $Om(x)$ diverges at $z<0$, suggesting
the ‘big rip’ future singularity for phantom cosmology, and $\Lambda$CDM
approached towards the de Sitter spacetime at the late times.

We have examined the $Om$ diagnostic profiles for our $f(Q)$ model with the
constraint values of the parameters. We have presented our results in Fig. 13.
One can observe that at $z=0$, $Om(x_{1},x_{2})<0$, which means that the dark
energy candidate of our model shows phantom-type behavior. But, in the late
time, $Om(x)\rightarrow 0$ when $z\rightarrow-1$ the model has quintessence-
like properties.

Figure 13: Profiles of the $Om$ diagnostic parameter as a function of $1+z$
for the constraint values of
$H_{0},\Omega_{m0},\,\omega_{0},\omega_{a},n,\gamma$ for the CC,
Pantheon+SHOES, and CC+Pantheon+SHOES samples.

##  VI Conclusion

In the present paper, we have investigated in detail the cosmological
properties of a particular $f(Q)$ gravity model, with the function $f(Q)$
given by $f(Q)=Q+6\gamma H_{0}^{2}(Q/Q_{0})^{n}$. The $f(Q)$ theory is an
interesting, and fundamental approach to the description of the gravitational
phenomena, in which the gravitational interaction is fully characterized by
the non-metricity of the space-time $Q$, defined a general functional
framework. $f(Q)$ gravity is one important component of the ”geometric trinity
of gravity”, and offers a full and convincing alternative to the curvature
description the gravitational interaction, which is used in standard general
relativity, and which was so successful in the description of the
gravitational interaction. From a geometric and mathematical point of view,
$f(Q)$ gravity uses the Weylian extension of Riemann geometry, in which one of
the fundamental prescription of this geometry, the metricity condition, is not
valid anymore. The breaking of the metricity condition is thus the source of
the gravitational phenomena, with the non-metricity scalar $Q$ playing an
analogous role to the one played by the Ricci scalar in general relativity. In
an action formulation, for $f(Q)=Q$, we exactly recover standard general
relativity. In our study we have restricted our analysis to a specific form of
the function $f(Q)$, in which the deviations from standard general relativity
are described by a power-law function in the non-metricity $Q$. After writing
down the field equations of the $f(Q)$ theory in a general form, we have
considered a specific dark energy model, in which the effective dark energy
density, and its effective pressure, which are both geometric in their origin,
are related by a linear, barotropic type equation of state, with a redshift
dependent parameter of the EOS, $\omega_{de}=\omega_{de}(z)$. For
$\omega_{de}$ we have adopted the first order CPL parameterizations, which can
be extensively used for the observational testing of cosmological models.
Moreover, we have restricted our basic model by imposing the energy
conservation of each of the considered components of the Universe, radiation,
matter, and dark energy, respectively. This procedure allows the determination
of the expression of the Hubble function in terms of the three $f(Q)$ model
parameters $H_{0}$, $\gamma$, and $n$, respectively. However, for a full
comparison with the observational data, one must extend the parameter space by
including the two parameters of the CPL equation of state of the dark energy.

To confront the power-law $f(Q)$ model with observations, several datasets
containing cosmological data have been used. In particular, we have analyzed
the model with respect to the Cosmic Chronometer (CC) dataset, as well as with
the Pantheon+SHOES database. As a firs step in our investigation we have
performed an MCMC analysis of the model, and obtained the optimal values of
the model parameters. Then, by using these values, we have considered the
general cosmological properties of this particular $f(Q)$ type theory.
Generally, the MCMC analysis of all three combinations of data sets indicate a
value of $n$ which is of the order of $n\approx-0.36$, or, approximately,
$n=-1/3$. Hence, the dependence of the function $F(Q)$ on $Q$ is of the form
$F(Q)\propto Q^{-1/3}$, that is, $F$ decreases with the increase of the
nonmetricity. This interesting result may raise the problem of the explanation
of this particular value of $n=-1/3$, obtained phenomenologically in the
present work, by a more detailed theoretical approach.

The deviations from standard general relativity are described by the parameter
$\gamma$, which turn out to be important, with $\gamma$ having values of the
order $\gamma\approx 0.45$. This indicate a large departure from the
Riemannian geometry based general relativity (in the absence of a cosmological
constant), but clearly indicates the possibility of the description of the
dark energy in this $f(Q)$ type model. The comparison with the observational
data on the Hubble parameter indicates a very good concordance between the
$f(Q)$ model, $\Lambda$CDM and observations up to a redshift of $z\approx 1$,
with some deviations appearing at higher redshifts. The AIC analysis also
confirms the existence of a mild tension between the present model and the
$\Lambda$CDM predictions, but to obtain a definite answer to this question
more observational data spreading on a larger redshift range are necessary.
The values of two free parameters $\omega_{0}$ and $\omega_{a}$ of the CPL
type equation of state parameter of the dark energy indicate that
$\omega_{0}\approx-1$, and hence at least at small redshifts the present model
mimics a cosmological constant. The correction term $\omega_{a}$, giving the
higher order redshift corrections is very small, of the order of
$\omega_{a}\approx-0.01$, indicating that an effective cosmological constant,
obtained from the Weyl geometric structure of the theory, gives the best
description of the observational data.

We have also performed a detailed investigation of several other cosmological
parameters by using the optimal values of the $f(Q)$ model parameters. Our
analysis indicate the presence of several important differences with respect
to the $\Lambda$CDM model, differences whose relevance may be addressed once
the precision and the number of observational data will significantly
increase.

The $f(Q)$ theory of gravity can also be extended to include, together with
the ordinary matter, scalar or other physical fields in the action. The
present power-law $f(Q)$ model may have some other possible applications,
like, for example, to consider inflation in the presence of both scalar fields
and nonmetricity, an approach that may lead to the formulation of a new view
on the gravitational, geometrical and cosmological processes that did shape
and influence the dynamics of the very early Universe. Another major topic of
research would be the investigation of structure formation in the power-law
$f(Q)$ theory which could be done with the use of a background cosmological
metric, obtained by solving exactly or approximately the cosmological
evolution equations. In this case the BAO, SNIa, and CMB shift parameter data
could be investigated to obtain important physical and cosmological
constraints for the power law $f(Q)$ model. This approach may lead to a
detailed investigation and analysis of the cosmic structure formation
processes, by providing a new perspective on these processes, and on the role
of Weyl non-metricity. Another direction of research would be to obtain the
Newtonian and the post-Newtonian approximations of the present power-law
$f(Q)$ gravity, and to find out what constraints the local classic Solar
System tests impose on the free parameters of the theory, and if these
constraints are consistent with the cosmological observations. The Newtonian
and the post-Newtonian limits may also prove to be extremely useful in
obtaining physical constraints from a large body of astrophysical
observations.

To conclude, in our work we have developed a particular version of the $f(Q)$
theory, with the functional form of $f$ given by a simple power law function,
and we have proven its consistency with the cosmological observations, and as
an important theoretical tool for the understanding of the accelerating
expansion of the Universe. The obtained results also suggests the necessity of
the study of further extensions and generalizations of this simple $f(Q)$ type
model. Our results have shown that the present poser-law model may represent
an interesting geometric alternatives to dark energy, going below the
Riemannian mathematical structure of general relativity, and in which the non-
metric properties of the space-time may offer the clue for a deeper
understanding of the gravitational interaction. In the present study we have
proposed some basic theoretical tools, and observational/statistical
procedures for the investigation of the basic geometric aspects of gravity,
from a different perspective than the Riemannian one, and of their
cosmological applications.

## Acknowledgements

S.M. acknowledges Transilvania University of Brasov for Transilvania
Fellowship for postdoctoral research. SP & PKS acknowledges the National Board
for Higher Mathematics (NBHM) under the Department of Atomic Energy (DAE),
Govt. of India for financial support to carry out the Research project No.:
02011/3/2022 NBHM(R.P.)/R & D II/2152 Dt.14.02.2022. The work of TH is
supported by a grant of the Romanian Ministry of Education and Research, CNCS-
UEFISCDI, project number PN-III-P4-ID-PCE-2020-2255 (PNCDI III).

## References

  * [1] Supernova Cosmology Project collaboration, S. Perlmutter et al., Measurements of Omega and Lambda from 42 high redshift supernovae, Astrophys. J. 517, 565, (1999).
  * [2] A. G. Riess et al., Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, Astrophys. J. 116, 1009, (1998).
  * [3] Li, Miao et al.,, Dark Energy, Commun. Theor. Phys., 56, 525, (2011).
  * [4] R. Maartens, Brane-world gravity, Living Rev. Rel. 7, 7, (2004).
  * [5] B. Ratra, P. Peebles, Cosmological Consequences of a Rolling Homogeneous Scalar Field, Phys. Rev. D 37, 3406, (1988).
  * [6] C. Armendariz-Picon, V.F. Mukhanov, P.J. Steinhardt, A dynamical solution to the problem of a small cosmological constant and late time cosmic acceleration, Phys. Rev. Lett. 85, 4438 (2000).
  * [7] A. Vikman, Can dark energy evolve to the phantom?, Phys. Rev. D 71, 023515, (2005).
  * [8] L.A. Urena-Lopez, T. Matos, New Cosmological Tracker Solution for Quintessence, Phys. Rev. D 62, 081302, (2000).
  * [9] T. Harko, Dissipative quintessence and its cosmological implications, Phys. Rev. D 107, 123507 (2023).
  * [10] S. Capozziello, Curvature quintessence, Int. J. Mod. Phys. D 11, 483, (2002).
  * [11] S. Nojiri, S.D. Odintsov, Modified $f(R)$ gravity consistent with realistic cosmology: From a matter dominated epoch to a dark energy universe, Phys. Rev. D 74, 086005, (2006).
  * [12] S. A. Appleby, R. A. Battye, Do consistent $F(R)$ models mimic General Relativity plus?, Phys. Lett. B 654, 7, (2007).
  * [13] W. Hu, I. Sawicki, Models of $f(R)$ Cosmic Acceleration that Evade Solar-System Tests, Phys. Rev. D 76, 064004, (2007).
  * [14] A. A. Starobinsky, Disappearing cosmological constant in $f(R)$ gravity, JETP Lett. 86, 157, (2007).
  * [15] O. Bertolami et al, Extra force in $f(R)$ modified theories of gravity, Phys.Rev.D 75, 104016 (2007).
  * [16] T. Harko, F. S. N. Lobo, $f(R,L_{m})$ gravity, Eur. Phys. J.C 70, 373 (2010).
  * [17] T. Harko et al., $f(R,T)$ gravity, Phys. Rev. D 84, 024020 (2011).
  * [18] Z. Haghani et al., Weyl-Cartan-Weitzenböck gravity as a generalization of teleparallel gravity, JCAP 10, 061 (2012).
  * [19] T. Harko, S. Shahidi, Coupling matter and curvature in Weyl geometry: conformally invariant $f(R,L_{m})$ gravity, Eur. Phys. J. C 82, 219 (2022).
  * [20] T. Harko, S. Shahidi, Palatini formulation of the conformally invariant $f(R,L_{m})$ gravity theory, Eur. Phys. J. C 82, 1003 (2022).
  * [21] O. Akarsu, C. B. Kilinc, Bianchi type-III models with anisotropic dark energy, Gen. Rel. Grav. 42, 763, (2010).
  * [22] V. Marra, M. Paakkonen, W. Valkenburg, Bias on w from large-scale structure, MNRAS, 431, 1891-1902 (2013).
  * [23] W. Valkenburg, Perceiving the equation of state of Dark Energy while living in a Cold Spot, JCAP 01, 047, (2012).
  * [24] L. Amendola, G. Camargo Campos, R. Rosenfeld, Consequences of dark matter-dark energy interaction on cosmological parameters derived from SNIa data, Phys. Rev. D 75, 083506, (2007).
  * [25] T. Clemson, K. Koyama, G.-B. Zhao, R. Maartens, J. Valiviita, Interacting dark energy: Constraints and degeneracies, Phys. Rev. D 85, 043007, (2012).
  * [26] J. Lu, Y. Wu, Y. Jin, Y. Wang, Investigate the interaction between dark matter and dark energy, Results in Physics 2, 14-21, (2012).
  * [27] R. Caldwell, R. Dave, P.J. Steinhardt, Cosmological imprint of an energy component with general equation of state, Phys. Rev. Lett. 80, 1582, (1998).
  * [28] V. Sahni, S. Habib, Does Inflationary Particle Production suggest $\Omega_{m}<1$?, Phys. Rev. Lett. 81, 1766, (1998).
  * [29] I. Zlatev, L.-M. Wang, P.J. Steinhardt, Quintessence, cosmic coincidence and the cosmological constant, Phys. Rev. Lett. 82, 896, (1999).
  * [30] S. Matarrese, C. Baccigalupi, F. Perrotta, Approaching lambda without fine-tuning, Phys. Rev. D 70, 061301, (2004).
  * [31] V. Pettorino, C. Baccigalupi, F. Perrotta, Scaling solutions in scalar-tensor cosmologies, JCAP 12, 003, (2005).
  * [32] G. Gupta, E.N. Saridakis, A.A. Sen, Non-minimal quintessence and phantom with nearly at potentials, Phys. Rev. D 79, 123013, (2009).
  * [33] J.L. Cervantes-Cota, R. de Putter, E.V. Linder, Induced Gravity and the Attractor Dynamics of Dark Energy/Dark Matter, JCAP 12, 019, (2010).
  * [34] A. de la Macorra, Scalar Field Dark Energy Parametrization, Class. Quantum Grav. 33, 095001, (2016).
  * [35] A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91, 99, (1980).
  * [36] A. De Felice, S. Tsujikawa, Construction of cosmologically viable $f(G)$ dark energy models, Phys. Lett. B 675, 1, (2009).
  * [37] C. Erices, E. Papantonopoulos, E. N. Saridakis, Cosmology in cubic and $f(P)$ gravity, Phys. Rev. D 99, 123527, (2019).
  * [38] C. Deffayet, G. Esposito-Farese, A. Vikman, Covariant Galileon, Phys. Rev. D 79, 084003, (2009).
  * [39] J.M. Nester, H-J Yo, Symmetric teleparallel general relativity, Chinese Journal of Physics 37, 113 (1999).
  * [40] J. B. Jimenez et al. Coincident general relativity, Phys. Rev. D 98, 044048, (2018).
  * [41] R. Lazkoz et al. Observational constraints of $f(Q)$ gravity, Phys. Rev. D 100, 104027, (2019).
  * [42] M. Koussour et al., Observational constraints on two cosmological models of $f(Q)$ theory, Eur. Phys. J. C., 83, 400, (2023).
  * [43] N. Frusciante, Signatures of $f(Q)$ gravity in cosmology, Phys. Rev. D 103, 044021, (2021).
  * [44] R. H. Lin, X. H. Zhai, Spherically symmetric configuration in $f(Q)$ gravity, Phys. Rev. D 103, 124001, (2021)
  * [45] S. Mandal et al., Energy conditions in $f(Q)$ gravity, Phys. Rev. D 102, 024057 (2020); Phys. Rev. D 106, 048502 (2022).
  * [46] J. B. Jimenez et al., Cosmology in $f(Q)$ geometry, Phys. Rev. D 101, 103507, (2020).
  * [47] T. Harko et al., Coupling matter in modified $f(Q)$ gravity, Phys. Rev. D 98, 084043, (2018).
  * [48] Y. Xu et al., $f(Q,T)$ gravity, Eur. Phys. J. C 79, 708 (2019).
  * [49] Y. Xu et al., Weyl type f(Q, T) gravity, and its cosmological implications, Eur. Phys. J. C 80, 449 (2020).
  * [50] J.-Z. Yang et al., Geodesic deviation, Raychaudhuri equation, Newtonian limit, and tidal forces in Weyl-type f(Q, T) gravity, Eur. Phys. J. C 81 111 (2021).
  * [51] T. Harko et al., Non-minimal geometry-matter couplings in Weyl-Cartan space-times: f(R , T , Q ,Tm) gravity, Physics of the Dark Universe 34 100886 (2021).
  * [52] S. Mandal et al., Cosmography in $f(Q)$ gravity, Phys. Rev. D 102, 124029, (2020).
  * [53] W. Khyllep et al., Cosmological solutions and growth index of matter perturbations in $f(Q)$ gravity, Phys. Rev. D 103, 103521, (2021).
  * [54] J. Beltran Jimenez, L. Heisenberg, T. Koivisto, Coincident general relativity, Phys. Rev. D 98, 044048, (2018).
  * [55] J. Beltran Jimenez et al., Cosmology in $f(Q)$ geometry, Phys. Rev. D 101, 103507, (2020).
  * [56] X. D. Li et al., Dark Energy and Fate of the Universe, Science China-Physics Mechanics & Astronomy, 55, 1330, (2012).
  * [57] E.V. Linder, Exploring the expansion history of the universe, Phys. Rev. Lett. 90, 091301, (2003).
  * [58] M.P. Hobson, A.H. Jaffe, A.R. Liddle, P. Mukherjee, D. Parkison (Eds.), Bayesian Methods in Cosmology, Cambridge University Press, Cambridge, England (2009).
  * [59] S. Mandal et al., $H_{0}$ tension in torsion-based modified gravity, Nuclear Physics B 993, 116285 (2023).
  * [60] D.M. Scolnic et al,, The Pantheon+ Analysis: The Full Data Set and Light-curve Release, ApJ 938 113 (2022)
  * [61] D. M. Scolnic, et al., The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample, Astrophys. J. 859, 101. (2018).
  * [62] H. Akaike, A new look at the statistical model identification, IEEE Trans. Autom. Control, 19, 716 (1974).
  * [63] G. Schwarz, Estimating the Dimension of a Model, Ann. Stat. 6, 461 (1978).
  * [64] D. J. Spiegelhalter, N. G. Best, B. P. Carlin, and A. van der Linde, Bayesian Measures of Model Complexity and Fit, J. R. Stat. Soc. 64, 583 (2002).
  * [65] K. Anderson, Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. (Springer, New York, 2002).
  * [66] K. P. Burnham and D. R. Anderson, Multimodel Inference: Understanding AIC and BIC in Model Selection, Sociol. Methods Res. 33, 261 (2004).
  * [67] A. R. Liddle, Information criteria for astrophysical model selection, Mon. Not. R. Astron. Soc. 377, L74 (2007).
  * [68] R. E. Kass and A. E. Raftery, Bayes Factors, J. Am. Stat. Assoc. 90, 773 (1995).
  * [69] E. Di Valentino et al., In the realm of the Hubble tension-a review of solutions, Class. quantum Grav. 38, 153001 (2021)
  * [70] Ahmad Mehrabi and Mehdi Rezaei, Cosmographic Parameters in Model-independent Approaches, ApJ 923, 274 (2021).
  * [71] Xiao-Bo Zou, Hua-Kai Deng, Zhao-Yu Yin, Hao Wei, Model-independent constraints on Lorentz invariance violation via the cosmographic approach, Physics Letters B 776, 284 (2018).
  * [72] Saeed Pourojaghi, Mohammad Malekjani, A new comparison between holographic dark energy and standard $\Lambda$-cosmology in the context of cosmography method, European Physical Journal C 81, 575 (2021).
  * [73] Varun Sahni, Arman Shafieloo, and Alexei A. Starobinsky, Two new diagnostics of dark energy Phys. Rev. D 78, 103502 (2008).