Transit cosmological models with observational constraints in $f(Q,T)$ gravity
Anirudh Pradhan1, Archana Dixit2

1,2Department of Mathematics, Institute of Applied Sciences & Humanities, GLA
University,

Mathura -281 406, Uttar Pradesh, India

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

This cosmological model is a study of modified $f(Q,T)$ theory of gravity
which was recently proposed by Xu et al. (Eur. Phys. J. C 79, 708 (2019)). In
this theory of gravity, the action contains an arbitrary function $f(Q,T)$
where $Q$ is non-metricity and $T$ is the trace of energy-momentum tensor for
matter fluid. In our research, we have taken the function $f(Q,T)$ quadratic
in $Q$ and linear in $T$ as $f(Q,T)=\alpha Q+\beta Q^{2}+\gamma T$ where
$\alpha$, $\beta$ and $\gamma$ are model parameters, motivated by $f(R,T)$
gravity. We have obtained the various cosmological parameters in Friedmann-
Lemaitre-Robertson-walker (FLRW) Universe viz. Hubble parameter $H$,
deceleration parameter $q$ etc. in terms of scale-factor as well as in terms
of redshift $z$ by constraining energy-conservation law. For observational
constraints on the model, we have obtained the best-fit values of model
parameters using the available data sets like Hubble data sets $H(z)$, Joint
Light Curve Analysis (JLA) data sets and union $2.1$ compilation of SNe Ia
data sets by applying $R^{2}$-test formula. We have calculated the present
values of various observational parameters viz. $H_{0}$, $q_{0}$, $t_{0}$ and
statefinder parameters $(s,r)$, these values are very close to the standard
cosmological models. Also, we have observed that the deceleration parameter
$q(z)$ shows signature-flipping (transition) point within the range $0.423\leq
z_{t}\leq 0.668$ through which it changes its phase from decelerated to
accelerated expanding universe with equation of state (EoS)
$-1.071\leq\omega-0.96$ for $0\leq z\leq 3$.

Keywords: FLRW universe, Modified $f(Q,T)$ gravity, Transit universe,
Observational constraints.

##  1 Introduction

Even if, Einstein’s general relativity (GR) proposed in $1915$ is still
considered to be one of the most successful theories, but in the wake of new
observational advances in cosmology, there appears some limitations on
standard GR in explaining those phenomena. The observations in [1]-[7] have
strong evidence that our present universe is undergoing an accelerated
expansion phase. These same observations predict the existence of two unknown
components in the Universe, called dark matter (DM) and dark energy (DE) along
with the usual baryonic matter. The heavy negative pressure due to DE is
supposed to be the possible cause of the accelerated expansion.

To adapt new findings, there are mainly two ways, one to provide alternative
theories of gravity and the other to modify the Einstein’s GR. Theories like
Brans and Dicke [8] , Nordtvedt [9], Wagoner [10], Dunn [11], Saez and
Ballester [12], Barber [13] are some of the prominent alternating theories. To
understand the dark energy problem in a better way, time-to-time various
cosmologists have proposed some well-known modified theories of gravitation
viz. $f(G)$ gravity [14], $f(R)$ gravity [15], $f(R,T)$ gravity [16],
$f(\mathcal{T})$ gravity [17], $f(G,T)$ [18], $f(R,T,R_{\mu\nu}T^{\mu\nu})$
gravity [19], $f(Q)$ theory [20] and recently proposed $f(Q,T)$ gravity [21].

In standard GR the action for formulation of Einstein field equations is given
by $S=\int(\frac{R}{2\kappa}+\mathcal{L}_{m})\sqrt{-g}d^{4}x$, here
$g=det(g_{\mu\nu})$ is defined the determinant of metric tensor $g_{\mu\nu}$,
$R$ is the Ricci scalar-curvature, $\mathcal{L}_{m}$ is the matter Lagrangian
and $\kappa=8\pi Gc^{-4}$ is Einstein’s constant. In the attempt to modify
Einstein’s GR, one of the simplest way is to replace $R$ by some more general
entity. A modified theory of gravity is formulated by replacing the Ricci
scalar-curvature $R$ by an arbitrary function $f(R)$ in the Einstein–Hilbert
action, called as $f(R)$ gravity has proposed by Nojiri and Odintsov [15]. The
different cosmological features of $f(R)$ gravity have been studied by several
researchers in different perspectives [22]-[32]. An extension of $f(R)$
gravity is proposed by Harko et al. [17] by including the trace of energy-
momentum tensor $T_{ij}$ in $f(R)$ function called as $f(R,T)$ gravity. The
theory of $f(R,T)$ gravity has been rigorously studied by several authors. The
references [33]-[37] are a small list of the study on $f(R,T)$ gravity.

Einstein’ GR is basically a geometric theory based on Reimannian geometry. In
one of the classical approach to modify GR is to a apply some more general
geometric structures that could describe the gravitational field. The first
attempt in this approach is due to Weyl [38], but the main aim of the Weyl
geometry was to unify electromagnetism and gravitation. On the similar line,
Gauss-Bonnet gravity, or $f(G)$ gravity is an interesting modified theory, in
which gravitation Lagrangian is obtained by adding a generic function $f(G)$
in the Hilbert-Einstein action, where
$G=R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}-4R_{\alpha\beta}R^{\alpha\beta}+R^{2}$
is the Gauss-Bonnet invariant ($R_{\alpha\beta\mu\nu}$, $R_{\alpha\beta}$ and
$R$ are Riemann tensor, Ricci tensor and Ricci scalar respectively) [14].
$f(G)$ gravity is further studied by many researchers. See [39]-[43]. Sharif
and Ikram [18] presented an extension of $f(G)$ gravity theory named as
$f(G,T)$ gravity by including the trace of the energy-momentum tensor $T$
along with $G$ in the modified Hilbert-Einstein action. For more study on
$f(G,T)$ gravity see references [44, 45]

In another approach, the so called teleparallel equivalent to general
relativity (TEGR), the basic idea is to replace the metric tensor $g_{\mu\nu}$
of the space-time, by a set of tetrad vectors $e^{i}_{\mu}$. The curvature is
replaced by the torsion generated by the tetrad, which can be used to entirely
describe gravitational effects [17, 46, 47, 48]. The teleparallel or
$f(\mathcal{T})$ gravity are extensively used to explain the current cosmic
accelerated expansion without using the concept of dark energy [49]-[60].

In symmetric teleparallel gravity (STG), the basic geometric variable
describing the properties of the gravitational interaction is represented by
the non-metricity $Q$ of the metric. The non-metricity tensor is the covariant
derivative of the metric tensor which geometrically describes the variation of
the length of a vector in the parallel transport. This approach is first
introduced by Nester and Yo in (1999) [20]. The STG was further developed as
$f(Q)$ gravity theory or non-metric gravity and various geometrical and
physical aspects of it have been studied in past few years. In the study of
the cosmology of the $f(Q)$ theory, it has been found that the accelerated
expansion is the intrinsic property of the Universe without need of either
exotic dark energy or extra field [61]-[75].

Recently, the $f(Q)$ theory is further extended by Xu et al. [21] in the form
of $f(Q,T)$ theory by coupling the non-metricity $Q$ with the trace of the
matter energy-momentum tensor $T$. The Lagrangian of the gravitational field
is assumed to be general function of both $Q$ and $T$. The action in $f(Q,T)$
theory is taken in the form
$S=\int{\left(\frac{1}{16\pi}f(Q,T)+\mathcal{L}_{m}\right)\sqrt{-g}d^{4}x}$.
The field equations of the theory were obtained by varying the gravitational
action with respect to both metric and connection. They investigated the
cosmological implications of the theory and obtained the cosmological
evolution equations for a flat, homogeneous, isotropic geometry, by assuming
some simple functional form of $f(Q,T)$. Arora et al. have recently studied
$f(Q,T)$ gravity models with observational constraints [76].

In the present study, we have also interpreted the cosmological model with
Friedmann-Lemaitre-Robertson-walker (FLRW) Universe in $f(Q,T)$ theory with
$f(Q,T)=\alpha Q+\beta Q^{2}+\gamma T$ and this supported by $f(R,T)$-gravity
form $f(R,T)=R+\alpha R^{2}+\beta T$ in which the presence of square term of
$R$ reveals the existence of dark energy and dark matter. Using the field
equations in a different approach, applying the energy conservation condition
$\dot{\rho}+3H(\rho+p)=0$, we have obtained the various cosmological
parameters viz. Hubble parameter $H(z)$, deceleration parameter $q(z)$ etc. in
terms of scale-factor $a(t)$ as well as in terms of redshift $z$. Applying
$R^{2}$-test formula and using the available observational data sets Hubble
data sets [77]-[83], Joint Light Curve Analysis (JLA) data sets [84] and union
$2.1$ compilation of supernovae type Ia data sets [85] the best fit values of
the model parameters are determined. We have obtained the energy density
$\rho$ and isotropic pressure $p$ explicitly and then estimated the equation
of state parameter $\omega=\frac{p}{\rho}$.

The present paper is investigated in the following five sections: first
section contains a brief introduction, in Sect. $2$ some preliminary
definitions and an overview of the $f(Q,T)$ theory are given. Cosmological
solutions of the FLRW Universe in $f(Q,T)$ theory is given in Sect. $3$.
Result analysis and discussions are given in Sect. $4$. Finally, conclusions
are summarized in Sect. $5$.

##  2 Some Preliminary definitions and Cosmological Field Equations

The Einstein-Hilbert action for $f(Q,T)$ gravity is defined in [21] as

$I=\int{\left(\frac{1}{16\pi}f(Q,T)+L_{m}\right)d^{4}x\sqrt{-g}}$ (1)

where $f(Q,T)$ is an arbitrary function of the non-metricity $Q$ and the trace
of the energy-momentum tensor $T$, and $L_{m}$ is the Lagrangian for matter
source and $g=\text{det}(g_{ij})$ read as the determinant of the metric tensor
$g_{ij}$ and

$Q\equiv-g^{ij}(L^{\alpha}_{\beta
i}L^{\beta}_{j\alpha}-L^{\alpha}_{\beta\alpha}L^{\beta}_{ij})$ (2)

where $L^{\alpha}_{\beta\gamma}$ is known as deformation tensor defined by

$L^{\alpha}_{\beta\gamma}=-\frac{1}{2}g^{\alpha\lambda}(\nabla_{\gamma}g_{\beta\lambda}+\nabla_{\beta}g_{\lambda\gamma}-\nabla_{\lambda}g_{\beta\gamma})$
(3)

The non-metricity $Q$ and the energy-momentum tensor are defined respectively
as

$Q_{\alpha}={Q_{\alpha}^{i}}_{i},~{}~{}~{}~{}T_{ij}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{m})}{\delta
g^{ij}}$ (4)

and

$\theta_{ij}=g^{\alpha\beta}\frac{\delta T_{\alpha\beta}}{\delta g^{ij}}$ (5)

By varying the action $(1)$ with respect to the metric components, we obtain
the following field equations:

$-\frac{2}{\sqrt{-g}}\nabla_{\alpha}(f_{Q}\sqrt{-g}P^{\alpha}_{ij})-\frac{1}{2}fg_{ij}+f_{T}(T_{ij}+\theta_{ij})-f_{Q}(P_{i\alpha\beta}Q_{j}^{\alpha\beta}-2{Q^{\alpha\beta}}_{i}P_{\alpha\beta
j})=8\pi T_{ij}$ (6)

where $P^{\alpha}_{ij}$ is defined as the supper-potential of the model
mentioned as in [21].

Now, let us suppose the universe is isotropic and spatially flat and hence, it
can be described by an FLRW metric given by

$ds^{2}=-N^{2}(t)dt^{2}+a(t)^{2}(dx^{2}+dy^{2}+dz^{2})$ (7)

where $a(t)$ is called scale factor which describes evolution of the universe.
The increasing value of $a(t)$ reveals the expansion and decreasing value
indicates the collapse of the universe and its rate is measured by Hubble
function defined as $H=\frac{\dot{a}}{a}$. The non-metricity $Q$ for FLRW
metric is derived as $Q=6H^{2}$ for lapse function $N(t)=1$ [21]. Let us
suppose that our universe is filled with perfect fluid and hence, the energy
momentum tensor for perfect fluid is defined by
$T_{ij}=(\rho+p)u_{i}u_{j}+pg_{ij}$ and therefore, the $\theta_{ij}$ is
defined and obtained as

$\theta^{i}_{j}=\delta^{i}_{j}p-2T^{i}_{j}=\text{diag}(2\rho+p,-p,-p,-p)$ (8)

Let us introduce the following notations $F\equiv f_{Q}$ and
$8\pi\tilde{G}\equiv f_{T}$ for simplicity of field equations. For the FLRW
metric $(7)$, the derived field equation $(6)$ gives the following equations:

$8\pi\rho=\frac{f}{2}-6FH^{2}-\frac{2\tilde{G}}{1+\tilde{G}}(\dot{F}H+F\dot{H})$
(9) $8\pi p=-\frac{f}{2}+6FH^{2}+2(\dot{F}H+F\dot{H})$ (10)

To obtain the evolution function for Hubble parameter, we have combined the
Eqs. $(9)$ and $(10)$ as

$\dot{H}+\frac{\dot{F}}{F}H=\frac{4\pi}{F}(1+\tilde{G})(\rho+p)$ (11)

In generalized $f(Q,T)$ gravity the energy-conservation law is derived in [21]
as

$\dot{\rho}+3H(\rho+p)=\frac{\tilde{G}}{16\pi(1+\tilde{G})(1+2\tilde{G})}\left[\dot{S}-\frac{(3\tilde{G}+2)\dot{\tilde{G}}}{(1+\tilde{G})\tilde{G}}S+6HS\right]$
(12)

where $S=2(\dot{F}H+F\dot{H})$.

##  3 Cosmological Solutions of the Field Equations

Let us choose such a solution of the field equations $(9)$ & $(10)$ so that
$\tilde{G}\neq 0$ and

$\dot{\rho}+3H(\rho+p)=0$ (13)

Therefore, the Eq. $(12)$ becomes

$\dot{S}-\frac{(3\tilde{G}+2)\dot{\tilde{G}}}{(1+\tilde{G})\tilde{G}}S+6HS=0$
(14)

Here, we have taken $f(Q,T)=\alpha Q+\beta Q^{2}+\gamma T$ which is obtained
by replacing $R$ by $Q$ in $f(R,T)$-gravity in which some authors have studied
the form $f(R,T)=R+\alpha R^{2}+\beta T$ with $R$ as Ricci scalar curvature
and $T$ is the trace of energy momentum tensor $T_{ij}$ and here, factor
$R^{2}$ is added to explain the late time acceleration in expanding universe
and this study motivated us to investigate the above form of $f(Q,T)$. Now, we
obtain $F=f_{Q}=\alpha+2\beta Q$ and $8\pi\tilde{G}=f_{T}=\gamma$ and also,
$\dot{\tilde{G}}=0$ and using this in Eq. $(14)$, we get

$\dot{S}+6HS=0$ (15)

Integrating Eq. $(15)$, we obtain the Hubble parameter $H$ as

$H=\frac{1}{6}\left[\frac{1}{\beta}\sqrt{\alpha^{2}+\frac{12\beta
k}{a^{6}}}-\frac{\alpha}{\beta}\right]^{\frac{1}{2}}$ (16)

where $\alpha$, $\beta$, $\gamma$ are model parameters and $k$ is an
integrating constant.

The various observational data sets are available in terms of redshift $z$ and
it can be easily compared with theoretical studies in terms of redshift.
Hence, the various parameters need to derive in terms of redshift $z$ and
hence, we are used the relationship between the scale-factor and redshift give
as

$\frac{a_{0}}{a}=1+z$ (17)

where $a_{0}$ is the present value of scale factor and as a standard
convention, we will take $a_{0}=1$ throughout the study. Using Eq. (17) in
(16), we get

$H(z)=\frac{1}{6}\left[\frac{1}{\beta}\sqrt{\alpha^{2}+12\beta
k(1+z)^{6}}-\frac{\alpha}{\beta}\right]^{\frac{1}{2}}$ (18)

Now, the deceleration parameter $q(t)$ is defined as
$q(t)=-1-\frac{\dot{H}}{H^{2}}$ and calculated as

$q(t)=-1+\frac{18\beta k}{a^{6}[\alpha^{2}+\frac{12\beta
k}{a^{6}}-\alpha\sqrt{\alpha^{2}+\frac{12\beta k}{a^{6}}}]}$ (19)

This in terms of redshift obtained as

$q(z)=-1+\frac{18\beta k(1+z)^{6}}{[\alpha^{2}+12\beta
k(1+z)^{6}-\alpha\sqrt{\alpha^{2}+12\beta k(1+z)^{6}}]}$ (20)

Now, from Eqs. $(9)$ & $(10)$ we have calculated the energy density $\rho$ and
isotropic pressure $p$ respectively as

$\rho=-\frac{3\alpha}{2(4\pi+\gamma)}H^{2}-\frac{27\beta}{(4\pi+\gamma)}H^{4}-\frac{\gamma
k}{4(4\pi+\gamma)(8\pi+\gamma)a^{6}}$ (21)

$p=\frac{\alpha}{2(4\pi+\gamma)}H^{2}+\frac{27\beta}{4\pi+\gamma}H^{4}-\frac{k(16\pi+\gamma)}{4(4\pi+\gamma)(8\pi+\gamma)a^{6}}$
(22)

From Eq. $(16)$ one can see that for $\alpha=0$, we can find
$H\propto\frac{1}{a^{3/2}}$, $H^{2}\propto\frac{1}{a^{3}}$ and
$H^{4}\propto\frac{1}{a^{6}}$ and hence, $\rho\propto\frac{1}{a^{6}}$ which
shows the existence of stiff matter in the universe. And in late time universe
it converted into ordinary matter and dark energy.

##  4 Result Analysis and Discussions

In the above section, we have obtained the various parameters viz. Hubble
function $H(z)$, deceleration parameter $q(z)$, energy density $\rho$ and
isotropic pressure $p$ in terms of $\alpha$, $\beta$, $\gamma$, $k$, $z$ and
$H$ where $\alpha$, $\beta$, $\gamma$, $k$, are called as model parameters. To
explain the various observable properties of universe, we are to find the best
fit values of the model parameters $\alpha$, $\beta$, $\gamma$, $k$ with
various observational data sets. Therefore, we have found the best fit values
of these model parameters for the best fit curve of Hubble function as well as
apparent magnitude $m(z)$ with observational data sets.

###  4.1 Hubble Parameter

The Hubble parameter shows the rate of geometrical evolution of the universe
(expansion rate) and it can be estimated from observational data sets as in
Table-$1$ (see [77]-[83]). The expression for the Hubble parameter $H(z)$ is
obtained as in Eq. (18). To obtain the best fit values of model parameters
$\alpha$, $\beta$ and $k$, we have found the best fit curve of Hubble function
$H(z)$ with the $29$ observed values of Hubble constant as shown in Table-$1$,
using the $R^{2}$-test formula as given below:

$R^{2}_{SN}=1-\frac{\sum_{i=1}^{29}[(H_{i})_{ob}-(H_{i})_{th}]^{2}}{\sum_{i=1}^{29}[(H_{i})_{ob}-(H_{i})_{mean}]^{2}}$

The case $R^{2}=1$ shows the exact fit the value of model parameters $\alpha$,
$\beta$ and $k$ with observational data sets. Hence, we obtain the curve-
fitting with greatest value of $R^{2}$. Thus, we have obtained the best fit
curve of the Hubble function $(18)$ shown as in figure $1$. We have found the
best fit values of model parameters $\alpha$, $\beta$ and $k$ as
$\alpha=-3.041$, $\beta=3.686\times 10^{-5}$ and $k=1.0\times 10^{4}$ (see
Table-$2$) for maximum $R^{2}=0.9098$ with root mean square error (RMSE)
$13.1300$ i.e. $H(z)\pm 13.13$ and their $R^{2}$ values only $9.02\%$ far from
the best one.

$z$ | $H(z)$ | $\sigma_{H}$ | Reference | $z$ | $H(z)$ | $\sigma_{H}$ | Reference
---|---|---|---|---|---|---|---
$0.070$ | $69$ | $19.6$ | [77] | $0.600$ | $87.9$ | $6.1$ | [81]
$0.100$ | $69$ | $12$ | [78] | $0.680$ | $92$ | $8$ | [79]
$0.120$ | $68.6$ | $26.2$ | [77] | $0.730$ | $97.3$ | $7$ | [81]
$0.170$ | $83$ | $8$ | [78] | $0.781$ | $105$ | $12$ | [79]
$0.179$ | $75$ | $4$ | [79] | $0.875$ | $125$ | $17$ | [79]
$0.199$ | $75$ | $5$ | [79] | $0.880$ | $90$ | $40$ | [82]
$0.200$ | $72.9$ | $29.6$ | [77] | $0.900$ | $117$ | $23$ | [78]
$0.270$ | $77$ | $14$ | [78] | $1.037$ | $154$ | $20$ | [79]
$0.280$ | $88.8$ | $36.6$ | [77] | $1.300$ | $168$ | $17$ | [78]
$0.350$ | $76.3$ | $5.6$ | [80] | $1.363$ | $160$ | $33.6$ | [78]
$0.352$ | $83$ | $14$ | [79] | $1.430$ | $177$ | $18$ | [78]
$0.400$ | $95$ | $17$ | [78] | $1.530$ | $140$ | $14$ | [78]
$0.440$ | $82.6$ | $7.8$ | [81] | $1.750$ | $202$ | $40$ | [78]
$0.480$ | $97$ | $62$ | [82] | $2.300$ | $224$ | $8$ | [83]
$0.593$ | $104$ | $13$ | [79] |  |  |  |
Table 1: Hubble’s constant Table. Figure 1: The best fit curve of Hubble
function $H(z)$ as in Eq. $(18)$ with observed values of $H(z)$ as shown in
Table-$1$. The best fit values of model parameters $\alpha$, $\beta$ and $k$
are obtained with $95\%$ confidence level of bounds.

The Figure $1$ represents the best fit curve of Hubble parameter $H(z)$ for
the obtained values of the constants $\alpha$, $\beta$ and $k$. For these
values, it is observed that the curve is quiet consistent. The present value
of Hubble parameter is calculated as $H_{0}=71.26\pm 13.13$, which are very
close to the recent observations [86].

Parameters | $H(z)$ | JLA | SNe Ia
---|---|---|---
$\alpha$ | $-3.041$ | $-2.814$ | $-2.940$
$\beta$ | $3.686\times 10^{-5}$ | $2.432\times 10^{-5}$ | $3.686\times 10^{-5}$
$k$ | $1.0\times 10^{4}$ | $1.0063\times 10^{4}$ | $1.0575\times 10^{4}$
$H_{0}$ | $71.26$ | $54.62$ | $83.53$
$R^{2}$ | $0.9098$ | $0.9258$ | $0.9938$
RMSE | $13.1300$ | $0.2681$ | $0.2494$
Table 2: The best fit values of model parameters $\alpha$, $\beta$ and $k$ for
the best fit curve of Hubble function $H(z)$ and apparent magnitude $m(z)$
with different observational data sets ($H(z)$, JLA, SNe Ia) with $95\%$
confidence level of bounds.

###  4.2 Luminosity Distance, Apparent Magnitude

The luminosity distance is measured the total flux of the source of light and
is defined as $D_{L}=c(1+z)\int_{0}^{z}\frac{dz}{H(z)}$. Let us define the
apparent magnitude in terms of $D_{L}$ to obtain the best fit values of model
parameters $\alpha$, $\beta$ and $k$ for the best fit curve of $H(z)$ with SNe
Ia data sets as $m(z)=16.08+5log_{10}(\frac{H_{0}D_{L}}{0.026\text{Mpc}})$. To
find the best fit curve of apparent magnitude $m(z)$ with SNe Ia data sets we
have considered $51$ observed data sets of apparent magnitude $m(z)$ from
Joint Light curve Analysis (JLA) as in [84] and $580$ observed data sets of
apparent magnitude from union $2.1$ compilation [85]. To perform the best
curve-fitting of theoretical and observed results we used the following
$R^{2}$-test formula:

$R^{2}_{SN}=1-\frac{\sum_{i=1}^{N}[(m_{i})_{ob}-(m_{i})_{th}]^{2}}{\sum_{i=1}^{N}[(m_{i})_{ob}-(m_{i})_{mean}]^{2}}$

where sums are run over $1$ to $N$ and $N$ has values $51$ for JLA data and
$580$ for union 2.1 compilation data sets.

The ideal case $R^{2}=1$ occurs when the observed data and theoretical
function $m(z)$ agree exactly. On the basis of maximum value of $R^{2}$, we
get the best fit values of $\alpha$, $\beta$ and $k$ for the apparent
magnitude $m(z)$ function which is given in Table $2$ for both JLA and SNe Ia
data sets. The best fit curve of $m(z)$ with observed values of $m(z)$ are
shown in figures $2a$ & $2b$ respectively.

From Eq. (16) we can find the scale-factor $a(t)$ in terms of Hubble constant
as

$a(t)=\left[\frac{12\beta k}{(\alpha+36\beta
H^{2})^{2}-\alpha^{2}}\right]^{\frac{1}{6}}$ (23)

One can calculate the value of scale-factor for any value of Hubble constant
$H$ using Eq. (23) and it can see that as $H\to\infty$ then $a\to 0$ which
shows singularity of the model. Also, from Eq. $(23)$ we can derive the scale-
factor in terms of cosmic time as well as in terms of redshift $z$ using
curve-fitting or cosmographic study. Eq. (23) depicts that for $\beta\to 0$,
$a(t)\to 0$ and model shows a point universe and for $\alpha=0$ the Hubble
parameter $H\propto a^{-\frac{3}{2}}$ which gives the power-law and
exponential cosmology.

a. b.

Figure 2: a. The best fit curve of $m(z)$ with observational JLA data sets
[84] and b. best fit curve of $m(z)$ with union 2.1 compilation of supernovae
data sets [85]. The best fit values of model parameters $\alpha$, $\beta$ and
$k$ are obtained with $95\%$ confidence level of bounds.

### Deceleration Parameter

The deceleration parameter $q(z)$ is the second of the observational parameter
that mentions the nature of evolution of geometrical of the universe. The
expressions for the deceleration parameter (DP) $q(z)$ in terms of scale-
factor $a(t)$ and redshift $z$ are shown in Eqs. (19) & (20) respectively and
Figure $3$ shows the variation of DP for the best fit values of model
parameters $\alpha$, $\beta$ and $k$ with three data sets. One can see that
$q(z)$ is an increasing function of redshift $z$ and it shows signature-
flipping (transition) point at $z_{t}$ within the range $0.423\leq z_{t}\leq
0.668$ (as shown in Table-$3$), where $q(z_{t})=0$ and $q(z)<0$ for $z<z_{t}$
and $q(z)>0$ for $z>z_{t}$, also, $q(z)\to-1$ as $z\to-1$ and $q(z)$ tends to
a finite positive value as $z\to\infty$. The present value DP is calculated as
$-1\leq q_{0}<0$ which is mentioned in Table-$3$. The behaviour of DP $q(z)$
over redshift $z$ reveals that our universe is undergoing an accelerating
phase at present and decelerating in early stages which are very close to the
recent observations. Thus, our derived model depicts a transit phase model
from decelerating to accelerating universe as predicted by the recent
observational studies [1, 2, 3, 4].

From Eq. (20) we can find the general expression for the transit value of
redshift $z=z_{t}$ by taking $q(z)=0$ and solving for $z$ as:

$z_{t}=\left(\frac{2\alpha^{2}}{3\beta k}\right)^{\frac{1}{6}}-1$ (24)

From Eq. (24) one can see that as $\beta\to 0$ give $z_{t}\to\infty$ i.e. the
model of an ever accelerating universe. Also, we can see that for the finite
value of model parameters $\alpha$, $\beta$ and $k$ we obtain either an
transit phase model or decelerating universe by choosing suitable values of
the model parameters. From the Table-$3$ we can see that the transition point
lies within the range $0.423\leq z_{t}\leq 0.668$ for the best fit values of
model parameters with observational data sets.

Figure 3: The plot of deceleration parameter $q(z)$ over redshift $z$ for the
best fit values $\alpha$, $\beta$ and $k$ which is mentioned in Table-$2$.

###  4.3 Age of the present universe

The third observational parameter is the age of the present universe $t_{0}$
which is mentioned in Gyrs. The age of the present Universe is calculated as

$\int_{t_{0}}^{t}dt=\int_{0}^{z}\frac{dt}{dz}dz$ (25)

where $t_{0}$ is the present age of the Universe and $t\leq t_{0}$. On
integration of left hand side of Eq. $(25)$ and using the relation
$\frac{dz}{dt}=-(1+z)H(z)$ in right hand side of $(25)$, we obtain

$(t_{0}-t)=\int_{0}^{z}\frac{dz}{(1+z)H(z)}$ (26)

where $H(z)$ is the Hubble parameter $H(z)$ (see Eq. $(18)$) with three
observational data sets. The plot of cosmic time $(t_{0}-t)H_{0}$ over
redshift is shown in Figure $4$. The parallel graph of time with $z$-axis
shows age of the present universe within the range $11.28\leq t_{0}\leq 16.63$
Gyrs which is mentioned in Table-$3$. One can conclude that it is very close
to age of the present universe calculated in [86].

Parameters | $H_{0}$ | JLA | SNe Ia
---|---|---|---
$q_{0}$ | $-0.7337$ | $-0.7811$ | $-0.5699$
$z_{t}$ | $0.599$ | $0.668$ | $0.423$
$H_{0}$ | $71.26$ | $54.62$ | $83.53$
$t_{0}H_{0}$ | $0.9024$ | $0.9286$ | $0.9636$
$t_{0}$ | $12.38$ | $16.63$ | $11.28$
Table 3: The present values of various parameters obtained for different data
sets. Figure 4: The plot of cosmic time $(t_{0}-t)H_{0}$ over redshift $z$ for
the best fit values $\alpha$, $\beta$ and $k$ as given in Table-$2$

### Energy density $(\rho)$, Isotropic pressure $(p)$ and EoS parameter
$(\omega)$

The expression for energy density $\rho$ is represented by the Eq. $(21)$ and
one can see that it has two singularities at $\gamma\in\\{-4\pi,-8\pi\\}$ and
$\rho\geq 0$ for all values of $\gamma<-8\pi$ and $\rho\leq 0$ for
$\gamma>-8\pi$. Also, we can find that $\rho$ is an increasing function of
redshift $z$ with $\gamma<-8\pi$ which shows that as $t\to t_{0}$ ($z\to 0$),
$\rho$ is decreasing function of redshift which is consistent with
cosmological studies in observational as well as in theoretical. Eq. $(22)$
represents the expression for isotropic pressure $p$ and we can see that the
expression of $p$ has two singularities at $\gamma\in\\{-4\pi,-8\pi\\}$ and
$p\leq 0$ for all values of $\gamma$ except $\gamma=-4\pi,-8\pi$. We can find
that the energy density $\rho\geq 0$ and the isotropic pressure $p\leq 0$ for
all values of $\gamma<-8\pi$. Also, one can see that the derived model tends
to Einstein’s GR model for $k=0$, $\beta=0$ and $\gamma=-8\pi$.

a. b. c.

Figure 5: a. The plot of energy density $\rho$ versus $z$, b. the plot of
isotropic pressure $p$ versus $z$, and c. the plot of Equation of state
parameter (EoS) $\omega$ versus $z$ for the best fit values of $\alpha$,
$\beta$ and $k$ mentioned in Table-$2$.

Figure $5a$ represents the evolution of energy density $\rho$ for
$\gamma=-8.1\pi$ and it depicts that for $z\to\infty$, $\rho\to\infty$ which
shows the big-bang singularity and at late time it tends to zero, while figure
$5b$ reveals the behaviour of isotropic pressure $p$ over redshift $z$ for
$\gamma=-8.1\pi$ and we can see that it is a decreasing function of redshift
and $p\to-\infty$ as $z\to\infty$. And figure $5c$ depicts the variation of
$\omega$ for $\gamma=-8.1\pi$ over redshift $z$ and one can see that it shows
$\omega$ is an increasing function of redshift $z$ but $\omega\leq 0$. From
figure $5c$ we can obtain the present value of $\omega$ lies within the range
$-1.071\leq\omega\leq-0.96$ for $0\leq z\leq 3$ which is agreed with
observational results in [87, 88, 89, 90]. Our model evolves from a
quintessence universe to phantom dominated universe and the range of $\omega$
is supported to an accelerated universe.

###  4.4 Statefinder Diagnostic

In cosmology, we are known two geometrical parameters the Hubble parameter
$H=\frac{\dot{a}}{a}$ and the deceleration parameter
$q=-\frac{a\ddot{a}}{\dot{a}^{2}}$ where $a(t)$ is the scale-factor and these
parameters are describe the history of universe. There are another geometrical
parameters called as statefinder diagnostic proposed in [91] which represent
the geometric evolution of various stages of dark energy models [91, 92, 93].
The statefinder parameters $r$ and $s$ are defined in terms of scale-factor
$a(t)$ respectively as

$r=\frac{\dddot{a}}{aH^{3}}\hskip 56.9055pts=\frac{r-1}{3(q-\frac{1}{2})}$
(27)

For the model we obtain statefinder parameters as

$r=1+\frac{3k[1-12\alpha^{2}\beta
k\frac{1}{a^{6}}-144\beta^{2}k^{2}\frac{1}{a^{12}}]}{2a^{6}H^{2}\sqrt{\alpha^{2}+\frac{12\beta
k}{a^{6}}}}$ (28) $s=\frac{36\beta k[1-12\alpha^{2}\beta
k\frac{1}{a^{6}}-144\beta^{2}k^{2}\frac{1}{a^{12}}]}{[12\beta
k-a^{6}(\alpha^{2}+\frac{12\beta
k}{a^{6}}-\alpha\sqrt{\alpha^{2}+\frac{12\beta k}{a^{6}}})]}$ (29)

a. b.

Figure 6: The plot of statefinder parameters $r(z)$, $s(z)$ for the best fit values of $\alpha$, $\beta$ and $k$ mentioned in Table-$2$. Parameters | $H(z)$ | JLA | SNe Ia
---|---|---|---
$r_{0}$ | $-46.5099$ | $-19.2762$ | $-180.6409$
$s_{0}$ | $38.5101$ | $15.8268$ | $169.7697$
$(s,r)_{z\to-1}$ | $(0,1)$ | $(0,1)$ | $(0,1)$
Table 4: The present values of various parameters obtained for different data
sets.

a. b.

Figure 7: The plot of statefinder parameters $r(z)$, $s(z)$ and $q(z)$, $r(z)$
for the best fit values of $\alpha$, $\beta$ and $k$ mentioned in Table-$2$.

The plot of $r$, $s$ over $z$ are represented in figure $6a$ & $6b$ and we can
calculate the present value of $r$ & $s$ as $-180.6409\leq r_{0}\leq-19.2762$,
$15.8268\leq s_{0}\leq 169.7697$. Also, one can see that as $z\to-1$ then
$r\to 1$ and $s\to 0$. Now, figure $7a$ shows the plot of $s(z)$ over $r(z)$
and the variation of $(s,r)$ represents the various cosmological models [91,
92, 93] and we can see $(s,r)\to(0,1)$ as $z\to-1$ which corresponds to the
$\Lambda$CDM and a flat FLRW universe. Figure $7b$ represents the plot of
$r(z)$ over $q(z)$ and it shows that as $z\to-1$ then $(q,r)\to(-1,1)$ and it
confirms that our present and future universe is in accelerating phase of
expansion.

##  5 Conclusions

In our research, we have taken the function $f(Q,T)$ quadratic in $Q$ and
linear in $T$ as $f(Q,T)=\alpha Q+\beta Q^{2}+\gamma T$ where $\alpha$,
$\beta$ and $\gamma$ are model parameters. We have obtained the various
cosmological parameters in Friedmann-Lemaitre-Robertson-walker (FLRW) Universe
viz. Hubble parameter $H$, deceleration parameter $q$ etc. in terms of scale-
factor as well as in terms of redshift $z$ by constraining on energy
conservation law. For observational constrains on the model, we have obtained
the best fit values of model parameters using the available data sets like
Hubble data sets $H(z)$, Joint Light Curve Analysis (JLA) data sets and union
$2.1$ compilation of SNe Ia data sets applying $R^{2}$-test formula. We have
calculated the present values of various observational parameters
$\\{t_{0},q_{0},H_{0}\\}$. These are very close to the standard cosmological
models. The main features of our model is as follows:

  * •

The derived Hubble function is constrained by observational data sets and the
present value of Hubble constant is calculated in the range $54.62\leq
H_{0}\leq 83.53$ which is compatible with [86].

  * •

The deceleration parameter $q$ shows the signature-flipping (transition) point
within the range $0.423\leq z_{t}\leq 0.668$ that depicts our universe has
been undergoing in accelerating phase from $t_{z}$ Gyrs ago where $8.85\leq
t_{z}\leq 14.16$ which is supported by [1, 2, 3, 4].

  * •

Our derived model evolves from quintessence universe $-1<\omega<-0.96$ to
phantom dominated universe $-1.071<\omega<-1$ passing through vacuum value
$\omega=-1$ for $0\leq z\leq 3$ which is agreed with the recent observational
value obtained in [87, 88, 89, 90] and this is the goodness of the derived
model in support of late time accelerating scenario of the universe

  * •

The age of the present universe is calculated in the range $11.28\leq
t_{0}\leq 16.63$ compatible with [86].

  * •

The statefinder diagnostic is also, analyzed and we obtained $(s,r)\to(0,1)$
as $z\to-1$ (future universe). It shows that our universe model tends to
$\Lambda$CDM FLRW a flat universe model in future.

Thus, our derived model is much interesting to researchers in this field for
further investigations and the compatibility of obtained results with standard
cosmological models as well as observational data sets shows the viability of
our model.

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