# Cosmological aspects of a hyperbolic solution in $f(R,T)$ gravity

Ritika Nagpal<EMAIL_ADDRESS>Department of Mathematics, Netaji
Subhas Institute of Technology, Faculty of Technology, University of Delhi,
New Delhi-110 078, India  J. K. Singh<EMAIL_ADDRESS>Department of Mathematics, Netaji Subhas Institute of Technology, Faculty of
Technology, University of Delhi, New Delhi-110 078, India  A. Beesham
<EMAIL_ADDRESS>Department of Mathematical Sciences, University of
Zululand, Kwa-Dlangezwa 3886, South Africa  Hamid Shabani
<EMAIL_ADDRESS>Physics Department, Faculty of Sciences, University
of Sistan and Baluchestan, Zahedan, Iran

###### Abstract

This article deals with a cosmological scenario in $f(R,T)$ gravity for a flat
FLRW model of the universe. We consider the $f(R,T)$ function as $f(R)+f(T)$
which starts with a quadratic correction of the geometric term $f(R)$ having
structure $f(R)=R+\alpha R^{2}$, and a linear matter term $f(T)=2\lambda T$.
To achieve the solution of the gravitational field equations in the $f(R,T)$
formalism, we take the form of a geometrical parameter, i.e. scale factor
$a(t)=sinh^{\frac{1}{n}}(\beta t)$ cha , where $\beta$ and $n$ are model
parameters. An eternal acceleration can be predicted by the model for $0<n<1$,
while the cosmic transition from the early decelerated phase to the present
accelerated epoch can be anticipated for $n\geq 1$. The obtained model
facilitate the formation of structure in the Universe according to the Jeans
instability condition as our model transits from radiation dominated era to
matter dominated era. We study the varying role of the equation of state
parameter $\omega$. We analyze our model by studying the behavior of the
scalar field and discuss the energy conditions on our achieved solution. We
examine the validity of our model via Jerk parameter, Om diagnostic, Velocity
of sound and Statefinder diagnostic tools. We investigate the constraints on
the model parameter $n$ and $H_{0}$ (Hubble constant) using some observational
datasets: $SNeIa$ dataset, $H(z)$ (Hubble parameter) dataset, $BAO$ (Baryon
Acoustic Oscillation data) and their combinations as joint observational
datasets $H(z)$ \+ $SNeIa$ and $H(z)$ \+ $SNeIa$ \+ $BAO$. It is testified
that the present study is well consistent with these observations. We also
perform some cosmological tests and a detailed discussion of the model.

PACS number: 98.80 cq

Keywords: $f(R,T)$ theory, FLRW metric, Parametrization, Observational
constraints.

##  I  Introduction

Einstein field equations (EFE) are

$R_{ij}+\frac{1}{2}Rg_{ij}+\Lambda g_{ij}=\frac{8\pi G}{c^{4}}T_{ij}.$ (1)

In the above equation, $R_{ij}$ is the Ricci tensor, $R$ the Ricci scalar,
$g_{ij}$ the covariant metric tensor of order 2, $\Lambda$ the cosmological
constant, $G$ the gravitational constant, $c$ indicates the speed of the
light, and $T_{ij}$ the energy-momentum-tensor (EMT). Despite the fact that
general relativity (GR) is extremely well tested, alternatives are always
present. According to observations, $95\%$ of the matter content of the
Universe is unexplored. GR has several problems, as the problem of initial
big-bang spacetime singularity car ; ein ; yil  and it is not yet quantised.
GR has to be reconciled with quantum physics to discuss quantum effects. GR
together with quantum physics forms the backbone of modern physics. The
$\Lambda$CDM model of the cosmology is quite successful, but there remain
several unresolved issues such as the fine tuning problem bull . Hence it is
worthwhile to examine alternative theories of gravity.

Amongst large range of modified theories of gravity, $f(R)$ gravity tho  is
considered as an interesting alternative. A more general function of $R$ i.e.
$f(R)$ is considered in the Einstein-Hilbert action . An intensively study on
$f(R)$ gravity seems to indicate that it is an improvement over GR suj ; noji
; cap1 . It can also explain both phases of cosmic acceleration even in the
absence of $\Lambda$ (early and late times) paul . $f(R)$ gravity behaves
extremely well on large scales, but the theory does not hold good on all
observational tests, such as on rotation of curved spiral galaxies chi1 ;
olmo1  and the solar system regime eri ; olmo2 . The generalisation of $f(R)$
gravity to $f(R,S_{m})$, where the matter Lagrangian $S_{m}$ is considered as
a general function of trace $T$ of the EMT is termed $f(R,T)$ gravity. Some
solar system tests myr1 ; mor2  have been favorably applied to modified
$f(R,T)$ theory of gravity to resolve the above-mentioned issues. To introduce
exotic imperfect fluids and quantum effects, a trace $T$ dependent term is
determined. Generally, the source term is a function of the matter Lagrangian
$S_{m}$, which yields an explicit set of field equations. A lot of remarkable
work in cosmology and astrophysics has already been done in $f(R,T)$ gravity
by several authors shab2 ; alv ; sin1 ; shab3 ; shab5 ; sin2 ; sin3 ; sin7
and also this theory can resolve the dark matter issue sin4 ; zar ; sin5 .

The above study has prompted us to compose a cosmological scenario within
$f(R,T)$ gravity. The paper is arranged as follows. In Sect. II, we give a
concise discussion on $f(R,T)$ theory. We obtain highly non-linear field
equations by considering the $f(R,T)$ function as the combination of a
quadratic R-dependent term and a linear T-dependent term. To determine the
solution of the field equations, we use an ansatz for the scale factor $a(t)$,
and find the behavior of the other geometrical parameters $H(t)$, $q(t)$. We
also present the graphical behavior of $\rho$, $p$ and $\omega$ for the
obtained model in Sect. III. Next Sect. IV is devoted to the analysis and
interpretation of the obtained solution by examining the potential of the
scalar field and energy conditions. In Sect. V, we perform some observational
tests using Jerk parameter, Om diagnostic, Velocity of sound and Statefinder
diagnostic tools to explore the validity of our model. In Sect. VI, we observe
that present study is well behaved with some observational datasets. In Sect.
VII, some cosmological tests are discussed to calculate distances in cosmology
for the accepted parametrization. Finally, we summarize our results by
providing a brief conclusion about the work in Sect. VIII.

##  II  Review of f(R,T)=f(R)+2f(T) cosmology

The general action for $f(R,T)=f(R)+2f(T)$ gravity har  coupled with the
action of a matter field with matter Lagrangian $S_{m}$ reads

$S=\int\Big{(}\frac{1}{16\pi G}f(R,T)+S_{m}\Big{)}\sqrt{-g}dx^{4},$ (2)

where $f(R,T)$ being an arbitrary function of $R$ and $T$. Here, we consider
$f(R)=R+\alpha R^{2}$ which is the first model for inflation and it was
proposed by Starobinsky star1 . This form of $f(R)$ function takes its origin
in the quantum correction to Friedmann equations. The term $R^{2}$ appears in
the functional form of $f(R)$ is the natural correction to GR and it naturally
provides an inflationary scenario in early Universe. Also Starobinsky model
shows the best compatibility according to the latest observations of the
Universe pla  and this model serves as a possible substitute to the scalar
field models describing inflation jdb . Consequently, if one extend the above
assumed $f(R)$ form with negative exponents of curvature term, then this model
is able to express the recent accelerating expansion. Therefore, with the most
general model $f(R)=R+\alpha R^{m}+\beta\frac{1}{R^{n}}$, where $\alpha$ and
$\beta$ are arbitrary constants, both the acceleration in the Universe (early
and late time acceleration) can be explained by the theories beyond GR paul .
To introduce exotic imperfect fluids and taking quantum effects in to account
with the above-defined $f(R)$ model, a trace $T$ dependent term is much
needed. This source term is a function of matter Lagrangian $S_{m}$ which
yields an explicit set of field equations. Here, in this study, we assume
$f(T)$ as a linear function of $T$ defined as $f(T)=2\lambda T$. So the
complete form of $f(R,T)$ function is $R+\alpha R^{2}+2\lambda T$.

On defining EMT of matter har

$T_{ij}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}S_{m})}{\delta g^{ij}},$ (3)

where its trace is given by $T=g^{ij}T_{ij}$. Also if $S_{m}$ is dependent
only on $g_{ij}$, in that case one can write

$T_{ij}=g_{ij}S_{m}-2\frac{\delta S_{m}}{\delta g^{ij}}.$ (4)

Taking a variation of action (2) w.r.t. $g_{ij}$, we have

$f^{R}(R,T)R_{ij}-\frac{1}{2}g_{ij}f(R,T)+(g_{ij}\Box-\nabla_{i}\nabla_{j})f^{R}(R,T)=8\pi
G\,T_{ij}-f^{T}(R,T)(T_{ij}+\Theta_{ij}),$ (5)

where $f^{R}(R,T)$ and $f^{T}(R,T)$ represent the derivative of $f(R,T)$
w.r.t. $R$ and $T$ respectively, $\Box$ is the d’ Alembert operator defined by
$\Box=g^{ij}\nabla_{i}\nabla_{j}$ and $\nabla_{i}$ indicates the covariant
derivative w.r.t. $g_{ij}$ associated with the symmetric Levi-Civita
connection. $\Theta_{ij}$ is of the following form

$\Theta_{ij}\equiv g^{lm}\frac{\delta T_{lm}}{\delta
g^{ij}}=-2T_{ij}+g_{ij}S_{m}-2g^{lm}\frac{\delta^{2}S_{m}}{\delta g_{ij}\delta
g^{lm}}.$ (6)

We consider perfect fluid in the thermodynamic equilibrium, so in this way, in
present study, one can simply set the matter Lagrangian $S_{m}=-p$ and we take
EMT of matter as

$T_{ij}=(\rho+p)u_{i}u_{j}-pg_{ij},$ (7)

where $\rho$ is the energy density and $p$ is the pressure of the fluid
present in the Universe. Using (6), the expression for the variation of EMT of
perfect fluid is given by

$\Theta_{ij}=-2T_{ij}-pg_{ij}.$ (8)

Using Eq. (8) in Eq. (5), we get the gravitational equation of motion as

$f^{R}(R,T)R_{ij}-\frac{1}{2}g_{ij}f(R,T)+(g_{ij}\Box-\nabla_{i}\nabla_{j})f^{R}(R,T)=8\pi
GT_{ij}+f^{T}(R,T)(T_{ij}+pg_{ij}).$ (9)

The connection between Ricci Scalar $R$ and $T$ can be seen by contracting the
Eq. (9) w.r.t $g^{ij}$,

$Rf^{R}(R,T)-2f(R,T)+3\Box f^{R}(R,T)=8\pi G\,T+(T+4p)f^{T}(R,T).$ (10)

On rearranging the terms in Eq. (8), the Ricci tensor $R_{ij}$ takes the form

$R_{ij}=\frac{1}{f^{R}(R,T)}\Big{(}8\pi
G\,T_{ij}+\frac{1}{2}g_{ij}+(\nabla_{i}\nabla_{j}-g_{ij}\Box)f^{R}(R,T)+f^{T}(R,T)(T_{ij}+pg_{ij})\Big{)}.$
(11)

Let us define a new operator $\Diamond_{ij}$ as,

$\Diamond_{ij}=\nabla_{i}\nabla_{j}-g_{ij}\Box.$ (12)

So the Eq. (11) becomes

$R_{ij}=\frac{1}{f^{R}(R,T)}\Big{(}8\pi
G\,T_{ij}+\frac{1}{2}g_{ij}+\Diamond_{ij}f^{R}(R,T)+f^{T}(R,T)(T_{ij}+pg_{ij})\Big{)}.$
(13)

The expression for Ricci scalar $R$ can be written by arranging the terms in
Eq. (10)

$R=\frac{1}{f^{R}(R,T)}\Big{(}8\pi G\,T+2f(R,T)-3\Box
f^{R}(R,T)+(T+4p)f^{T}(R,T)\Big{)}.$ (14)

By using Eqs. (13) and (14), the Eq. (9) can be represented as the field
equations with LHS as the Einstein tensor $G_{ij}$,

$\displaystyle G_{ij}=R_{ij}-\frac{1}{2}Rg_{ij}$ $\displaystyle=$
$\displaystyle\frac{8\pi
G\,T_{ij}}{f^{R}(R,T)}+\frac{1}{f^{R}(R,T)}\Big{[}\frac{1}{2}g_{ij}(f(R,T)-Rf^{R}(R,T))+\Diamond_{ij}f^{R}(R,T)+(T_{ij}+pg_{ij})f^{T}(R,T)\Big{]},$
$\displaystyle=$ $\displaystyle\frac{8\pi
G}{f^{R}(R,T)}(T_{ij}+T_{ij}^{{}^{\prime}}),$

where $T_{ij}^{{}^{\prime}}=\frac{1}{8\pi
G}\Big{(}\frac{1}{2}g_{ij}(f(R,T)-Rf^{R}(R,T))+\Diamond_{ij}f^{R}(R,T)+(T_{ij}+pg_{ij})f^{T}(R,T)\Big{)}.$
From the above field equations, EFE in GR can be resumed by fixing $\alpha=0$
and $\lambda=0$. Applying the Bianchi identity on Eq. (II) leads to111Note
that this equation has been obtained in shab4. However, because of the metric
signature in the present work, the last term in Eq. (16) has obtained the
opposite sign.

$\displaystyle\Big{(}8\pi
G+f^{T}(R,T)\Big{)}\nabla^{i}T_{ij}+\frac{1}{2}f^{T}(R,T)\nabla_{i}T+T_{ij}\nabla^{i}f^{T}(R,T)+\nabla_{j}\Big{(}pf^{T}(R,T)\Big{)}=0.$
(16)

##  III Cosmological dynamics of the Universe

We study the dynamics of the Universe by considering a homogeneous and
isotropic Universe in the form of spatially flat FLRW line element given by

$ds^{2}=dt^{2}-a^{2}(t)(dx^{2}+dy^{2}+dz^{2}),$ (17)

where $a(t)$ being the scale factor. The trace $T$ of the EMT (7) and scalar
curvature $R$ are

$T=\rho-3p,$ (18) $R=-6(2H^{2}+\dot{H}),$ (19)

where $H$ is the Hubble parameter defined as $\frac{\dot{a}}{a}$ and overhead
dot indicates the differentiation w.r.t. to $t$. Taking $f(R,T)=R+\alpha
R^{2}+2\lambda T$ and using Eqs. (7), (18), (19) in Eq. (II), we get the
following field equations

$3H^{2}=\frac{1}{1+2\alpha R}\Big{[}8\pi\rho+\lambda(3\rho-p)+2\alpha
U(a,\dot{a},\ddot{a},\dddot{a})\Big{]},$ (20)
$2\dot{H}+3H^{2}=\frac{1}{1+2\alpha R}\Big{[}-8\pi p+\lambda(\rho-3p)+2\alpha
V(a,\dot{a},\ddot{a},\dddot{a},\ddddot{a})\Big{]},$ (21)

where
$U(a,\dot{a},\ddot{a},\dddot{a})=\frac{-9}{a^{4}}(5\dot{a}^{4}+a\ddot{a}^{2}-2a^{2}\dot{a}\,\dddot{a})$
and
$V(a,\dot{a},\ddot{a},\dddot{a},\ddddot{a})=\frac{3}{a^{4}}(\dot{a}^{4}-18a\dot{a}^{2}\ddot{a}+4a^{2}\dot{a}\,\dddot{a}+a^{2}(-2\dot{a}\dddot{a}+a\ddddot{a}))$
are the functions of scale factor $a$ and its derivatives up to fourth order
respectively. Also, we have set the units so that $G=1$.

Substituting the mentioned choice for function $f(R,T)$ in Eq. (16) leads to

$\frac{8\pi+3\lambda}{8\pi+2\lambda}\dot{\rho}-\frac{\lambda}{8\pi+2\lambda}\dot{p}+3H(\rho+p)=0.$
(22)

Eq. (20) can be rewritten as

$\left(8\pi+3\lambda\right)\rho-\lambda p=\mathcal{U},$ (23)

where we have defined

$\mathcal{U}\equiv
3H^{2}+18\alpha\left(\dot{H}^{2}-4H^{2}\dot{H}-2H\ddot{H}\right).$ (24)

Solving Eqs. (22) and (23) gives

$\displaystyle\rho=\frac{3\mathcal{U}-\lambda\left(8\pi+2\lambda\right)^{-1}H^{-1}\dot{\mathcal{U}}}{3\left(8\pi+4\lambda\right)},$
(25) $\displaystyle
p=\frac{\left(8\pi+3\lambda\right)\rho-\mathcal{U}}{\lambda}.$ (26)

Solutions (25) and (26) show that to obtain the exact solutions for $\rho$ and
$p$ and to study dark energy model, we need to adopt a parametrization of
either $a(t)$ or $H(t)$. This technique is called model independent way to
explore dark energy models. This work deals with an ad hoc choice of $a(t)$,
which is the outcome of a time-dependent deceleration parameter (DP) cha  as

$a(t)=\sinh^{\frac{1}{n}}(\beta t),$ (27)

where $\beta$ and $n$, $n>0$ are arbitrary constants.

The Hubble parameter $H(t)$ and DP $q(t)$ can be found from Eq. (27) as

$H(t)=\frac{\beta\coth(\beta t)}{n},$ (28)

and

$q(t)=n\left[1-\tanh^{2}(\beta t)\right]-1.$ (29)

In the present study, we are curious to examine the different regimes of the
Universe i.e. the phase transition from decelerated to accelerated expansion
by constraining a model parameter $n$. From Eq. (29), DP depends on $t$ and
inflation in the Universe depends on the sign of $q$. A positive $q$ refers
the decelerating expansion while a negative $q$ corresponds to accelerating
phase of the model. For the above parametrization of $a(t)$, our model
entirely accelerates and decelerates according as
$t<\frac{1}{\beta}\tanh^{-1}(1-\frac{1}{n})^{\frac{1}{2}}$ and
$t>\frac{1}{\beta}\tanh^{-1}(1-\frac{1}{n})^{\frac{1}{2}}$ respectively, and
it predicts phase transitions i.e. $q=0$ when
$t=\frac{1}{\beta}\,\tanh^{-1}(1-\frac{1}{n})^{\frac{1}{2}}$. As it is well
acknowledged that the Universe experiences an accelerating phase in late time,
so it must had a slow expansion in the past rie ; per , in such case the
parametrization of the scale factor is rational.

Table 1:  Dynamics of the Universe for $a(t)=sinh^{\frac{1}{n}}(\beta t)$ Time ($t$) |  Redshift ($z$) |  ${\small a}$ |  ${\small q}$ |  ${\small H}$
---|---|---|---|---
${\small t\rightarrow 0}$ | ${\small z\rightarrow\infty}$ | ${\small 0}$ | $n-1$ | ${\small\infty}$
${\small t\rightarrow\infty}$ | ${\small z\rightarrow-1}$ | ${\small\infty}$ | ${\small-1}$ |  finite quantity $(=\frac{\beta}{n})$

From Eq. (29), the model parameters $\beta$ and $n$ are related as

$\beta t_{0}=\tanh^{-1}\Big{(}\frac{n-q_{0}-1}{n}\Big{)}^{\frac{1}{2}},$ (30)

where $t_{0}$ denotes the present time and $q_{0}$ indicates the present value
of DP. Providing different values to $n$ will give rise to different values of
$\beta$ on taking present value of $t$ and $q$. Here, we consider $t_{0}=13.8$
and $q_{0}=-0.54$ abd  and plot $a(t)$, $H(t)$ and $q(t)$ for various values
of $n=0.5,1,1.35,2$.

Using the relation

$\frac{a}{a_{0}}=\frac{1}{1+z},$ (31)

where $a_{0}$ is present value of scale factor, we evaluate $t(z)$, $H(z)$ and
$q(z)$ in terms of redshift $z$ as

$t(z)=\frac{\sinh^{-1}\sqrt{\frac{n-(1+q_{0})}{(z+1)^{2n}(q_{0}+1)}}}{\beta}.$
(32)
$H(z)=\frac{\beta\,\coth\left(\sinh^{-1}\sqrt{\frac{n-(q_{0}+1)}{(z+1)^{2n}(q_{0}+1)}}\right)}{n},$
(33)
$q(z)=n-1-n\Bigg{[}\tanh\Bigg{(}\sinh^{-1}\sqrt{\frac{n-(1+q_{0})}{(z+1)^{2n}(q_{0}+1)}}\Bigg{)}\Bigg{]}^{2}.$
(34)

The graphs of scale factor $a$, Hubble parameter $H$ and DP $q$ w.r.t $z$ are
shown as:

$\begin{array}[]{c@{\hspace{.1in}}cc}\includegraphics[width=158.99377pt,height=130.08621pt]{a-z.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=158.99377pt,height=130.08621pt]{H-z.pdf}&\includegraphics[width=158.99377pt,height=130.08621pt]{q-z.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)&\mbox{(}c)\end{array}$

Figure 1: Graphical representations of $a$, $H$ and $q$ Vs. $z$. Table 2:  Existence of various substances according to EoS parameter Substance |  EoS parameter |  Observations
---|---|---
Pressureless (Cold) matter | $\omega=0$ | 32% of the Universe
Hot matter | $\omega\in(0,\frac{1}{3})$ | Insignificant at present time
Radiation | $\omega=\frac{1}{3}$ | Influential in past
Hard Universe | $\omega\in(\frac{1}{3},1)$ | Excessive high densities
Stiff matter | $\omega=1$ |
Ekpyrotic matter | $\omega>1$ | Resist Dominant Energy Condition
Quintessence | $\omega\in(\frac{-1}{3},-1)$ | 68% of the Universe
Cosmological constant | $\omega=-1$ | Inconsistent with observations
Phantom Universe | $\omega<-1$ |  Lead to Big Rip, resist Weak Energy Condition

From Fig. 1(c), we observe that phase transitions occur when $n\geq 1$ and the
model shows eternal acceleration when $0<n<1$ i.e. the phase transitions w.r.t
redshift $z$ directly depend on the value of $n$. Also Fig. 1(c) clearly
specifies that our model is consistent with the recent observational dataset
of $SNeIa$, $BAO$ and $CMBR$ (Cosmic Microwave Background Radiation) with some
fine tuning, corresponding to the value of model parameter $N=4$ abd  when
$n=1.35$, $q=0$ at $z_{tr}=0.883752$, and is supportive with the fitting
result of Gold SNIa for $N=182$ with 1σ errors zha  when $n=2$, $q=0$ at
$z_{tr}=0.352666$. Thus we can predict that the Universe started from
decelerating phase and ended up with accelerating phase in late times in the
case when $n\geq 1$ while the model represents total eternal acceleration
right from the evolution of the Universe upto late time when $0<n<1$.

The EoS parameter $\omega$ is considered as one of the vital parameter in
cosmology, which explains the different cosmic regimes. In a more generic way,
this parameter can be defined as $\omega=p/\rho$, where in the case of
solutions (25) and (26) one obtains

$\omega=-\frac{\left(8\pi+3\lambda\right)\dot{\mathcal{U}}+3\left(8\pi+2\lambda\right)H\mathcal{U}}{3\left(8\pi+2\lambda\right)H\mathcal{U}-\lambda\dot{\mathcal{U}}}.$
(35)

In GR from the Friedmann equations, it can be observed that there is only one
approach to achieve accelerated expanding Universe by considering
$1+3\omega<0$, which can be realised for an exotic matter, which explicitly
refers negative pressure as we considered $\rho$ to be positive always. The
various substances present in the Universe lead to different eras of the
Universe which can be seen by providing particular values to $\omega$ (see
Table II).

$\begin{array}[]{c@{\hspace{.1in}}cc}\includegraphics[width=158.99377pt,height=130.08621pt]{rho.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=158.99377pt,height=130.08621pt]{pres.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 2: Graphical representations of energy density $\rho$ and isotropic
pressure $p$ Vs. $t$ for $\alpha=0.5$ and $\lambda=2$.

It is assumed that there are two major stages in the evolution of the Universe
after the big bang known as the radiation and matter eras. A radiation
dominated era is requisite to anticipate primordial nucleosynthesis.
Therefore, deviation of more than $10\%$ in expanding rate of the Universe
related to the $\Lambda$CDM at the time of nucleosynthesis epoch clashes with
the observed Helium abundance. Radiation and matter dominated stages are
defined as the key events that help to shape the Universe. The Universe has
the ability to create elements in the matter dominated era defined by the
presence and pre-dominance of matter in the Universe. It features three epochs
namely atomic, galactic and stellar epochs that span billion of years and
includes the present day. All the three epochs are required to formulate the
large structure in the Universe that we can observe today.

One can inspect the behavior $\rho$ and $p$ from solutions (25) and (26) using
the definition (24). Fig. 2(a) highlights the behavior of energy density
$\rho$ which is very high i.e. $\rho\rightarrow\infty$ in the beginning of the
Universe corresponding to $n=0.5,\,1,\,1.35,\,2$, falls off as time unfolds
and $\rho\to 0$ as $t\to\infty$. Fig. 2(b) enacts the trait of matter pressure
for all the values of $n$ mentioned earlier. For $n=0.5$, $p\to-\infty$ at
$t\to 0$, remains negative throughout the evolution and approaches to negative
constant value in late times which indicates the eternal cosmic accelerated
expansion. The isotropic pressure in the early phase of the Universe for
particularized values of $n=1,\,1.35,\,2$ reaches an extensively high value
and tends to $-\frac{3\beta^{2}}{n^{2}\left(8\pi+4\lambda\right)}$ as
$z\rightarrow-1$. Negative pressure in the universe is subjected to the
acceleration in the cosmos according to the standard cosmology. Therefore, the
present study exhibits accelerating phase at current epoch as well as in the
near future. From Fig. 2(b), we can realize that structure formation is
achievable in our model for the case $n=1,\,1.35,\,2$ because decelerated
expansion is required for the structure formation that could appear in the
presence of a kind of matter fluid which produces Jeans instability james .

It can be useful to study the matter density and pressure in the limit of
small and large times. Straightforward calculations show that both quantities
tend to infinity whose signature depend upon the model constants $\lambda$,
$\alpha$ and $n$. In the limit of large times one obtains

$\lim_{t\to\infty}\rho=-\lim_{t\to\infty}p=\frac{3\beta^{2}}{n^{2}\left(8\pi+4\lambda\right)}.$
(36)

The limit value (36) includes some interesting information. It is noted that
it does not depend on the coupling constant $\alpha$ which incorporates the
curvature correction term in the Lagrangian. On the contrary, result (36)
depends on the coupling constant of the matter part, i.e. $\lambda$. From (36)
we see that matter behaves like the DE in the late times. Also, value (36)
implies that to have a gravitational model with observationally accepted
values for the matter density and pressure in the late times, the constraint
$\lambda>-2\pi$ must hold. More precisely, by taking the limit value of the
matter density in the early times one finds that to guarantee the weak energy
condition (WEC), $\rho\geq 0$, constraints which are indicated in the Table
III must hold.

Table 3:  Conditions to guarantee the WEC |  $\alpha<0$ |  $\alpha>0$
---|---|---
$-2\pi<\lambda<0$ |  $n>-(\frac{6\pi}{\lambda}+\frac{3}{2})$ |  $0<n<-(\frac{6\pi}{\lambda}+\frac{3}{2})$
$\lambda>0$ |  |  $n>0$

It is worth discussing the behavior of the EoS parameter (35). In the early
times the EoS parameter (35) goes to a constant value and in the late times it
mimics the DE, i.e.,

$\displaystyle\omega_{i}\equiv\lim_{t\to
0}\omega=1+\frac{2(\lambda+4\pi)(2n-3)}{\lambda(2n+3)+12\pi},$ (37)

where the subscript ‘$i$’ represents the values of $\omega$ in the early
times. For the late times one obtains

$\displaystyle\lim_{t\to\infty}\omega=-1,$ (38)

which is independent of the model constants. Some simple calculations show
that constraining the EoS parameter as $0<\omega_{i}<1$ (which is plausible in
cosmology) forces one to choose
$\big{(}(3\lambda+12\pi)/(6\lambda+16\pi)\textasciibreve\big{)}<n<3/2$ for
$\lambda>-2\pi$. Also, for
$-2\pi<\lambda<0,3/2<n<-\big{(}(12\pi+3\lambda)/2\lambda\big{)}$ and
$\lambda>0,~{}n>3/2$, the EoS parameter can behave like ekpyrotic matter
$(\omega>1)$ which resists DEC followed by matter dominated era where energy
density is extensively high to radiation dominated era in the early Universe.
Results (37) and (38) contains an important informations about different
epochs of cosmic evolution. From (37) we learn that the underlying
gravitational model can describe two different cosmological transitions:

  * (i)

for $n=3/2$ one obtains $\omega^{s}_{i}=1$. That is independent of other model
parameters and only for $n=3/2$ the model describe a transition from a state
in which stiff matter dominates in the early eras to a state which behaves
like the DE in the late times.

  * (ii)

it is also possible a transition from a pressure-less matter dominated era
with $\omega^{p}_{i}=0$ in the early Universe to a DE like Universe in the
late times. In this case one obtains the following relation

$\displaystyle n=\frac{3(\lambda+4\pi)}{2(3\lambda+8\pi)}.$ (39)

It has been seen that the parameters $\beta$ and $\alpha$ do not play any role
in determining the initial and final states of cosmological evolution. It
means that only modifications in the matter part of the Lagrangian are
responsible for different states of the Universe. The evolution of EoS
parameter $\omega$ w.r.t. time $t$ as well as redshift $z$ are represented
graphically in Fig. 3(a) and Fig. 3(b) when
$n=3/2,\,\frac{3(\lambda+4\pi)}{2(3\lambda+8\pi)}$.

$\begin{array}[]{c@{\hspace{.15in}}cc}\includegraphics[width=158.99377pt,height=130.08621pt]{omega-t.pdf}\hfil\hskip
10.84006pt&\includegraphics[width=158.99377pt,height=130.08621pt]{omega-z3.pdf}\\\
\mbox{(}a)\hfil\hskip 10.84006pt&\mbox{(}b)\end{array}$

Figure 3: The plots of (a) EoS parameter $\omega$ Vs. $t$ and (b) EoS
parameter $\omega$ Vs. $z$ when
$n=3/2,\,\frac{3(\lambda+4\pi)}{2(3\lambda+8\pi)}$ for a fix value of
$\lambda=2$, $\alpha=0.5$ and $\beta=0.6$.

The profound discussion on the behavior of EoS parameter which depends on the
range of matter-geometry coupling constant $\lambda$ in $f(R,T)$ gravity is
worthy and it has been investigated in Fig. 3. For a fix value of
$\alpha=0.5$, and $\lambda=2$, $\omega\in$ quintessence region for high
redshift $z$ and as time unfolds, $\omega\to-1$ in infinite future (i.e.
$z\to-1$) when $n=\frac{3(\lambda+4\pi)}{2(3\lambda+8\pi)}$, which is
consistent with the observations of temperature fluctuation in cosmic
microwave background radiations (CMBR) hin . Also, the EoS parameter indicates
another possibility for evolution of the Universe when $n=3/2$. In this case
our model evolves from a state of a stiff-matter fluid dominated era to a DE
like era in the late times.

The present value of the EoS parameter for the mentioned types of evolution
era can be obtained respectively, as follows

$\displaystyle\omega_{0}^{(n=3/2)}=\frac{\lambda\left(176\alpha\beta^{2}-9\right)+8\pi\left(46\alpha\beta^{2}-9\right)}{5\lambda\left(32\alpha\beta^{2}+9\right)+16\pi\left(19\alpha\beta^{2}+9\right)},$
(40)

for models with $n=3/2$ and

$\displaystyle\omega_{0}^{(n(\lambda))}=-\frac{(3\lambda+8\pi)\Big{[}3\lambda\left(8\alpha\beta^{2}+9\right)+4\pi\left(8\alpha\beta^{2}+27\right)\Big{]}}{9\lambda^{2}\left(88\alpha\beta^{2}+13\right)+12\pi\lambda\left(320\alpha\beta^{2}+63\right)+128\pi^{2}\left(35\alpha\beta^{2}+9\right)},$
(41)

for $n=\frac{3(\lambda+4\pi)}{2(3\lambda+8\pi)}$, where $\omega_{0}$ stands
for the current value of the EoS parameter. From Eqs. (40) and (41), it can be
seen, both coupling constants $\alpha$ and $\beta$ in the matter Lagrangian
affect the present value of the EoS parameter. Therefore, some suitable
astronomical data can be used to constrain the values of the coupling
constant. The EoS parameters $\omega\simeq-0.509$ and $\omega\simeq-0.2629$ at
present epoch $z=0$ for $n=3/2,\,\frac{3(\lambda+4\pi)}{2(3\lambda+8\pi)}$
respectively, which is in good agreement with the observation abd .

##  IV  Interpretation of the model

###  IV.1  Scalar field correspondence

In section 3, we have already discussed that the idea to predict the
acceleration in the Universe is to filled with an exotic form of matter which
satisfy $1+3\omega<0$. According as the observations, the energy which
produces the acceleration satisfies $\omega\simeq-1$. If $\omega<0$, there are
many models that can explain inflation exactly such as quintessence model,
phantom model etc. rat ; sam ; cald ; sah1 . In section 3, we have also study
above the construction of EoS parameter $\omega$ for our model, so it is
appropriate to consider a matter field which shows exotic behavior and is able
to produce anti-gravitational effects. Here, we consider dark energy as
quintessence to explain the cosmic acceleration whose action is given by

$S=\int\Big{(}\frac{1}{16\pi G}R+S_{m}\Big{)}\sqrt{-g}\,dx^{4},$ (42)

with the matter Lagrangian density

$S_{m}=\int\Big{(}-\frac{1}{2}\partial_{\mu}{\phi}\,\partial^{\mu}{\phi}-V(\phi)\Big{)}\sqrt{-g}\,dx^{4},$
(43)

where $\phi$ is the time-dependent scalar field. Therefore, we can consider
scalar field $\phi$ as a perfect fluid with energy density $\rho$ and pressure
$p$ as

$\rho=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$ (44)

$p=\frac{1}{2}\dot{\phi}^{2}-V(\phi).$ (45)

Here $\frac{1}{2}\dot{\phi}^{2}$ is the kinetic energy $(KE)$ and $V(\phi)$ is
the potential energy $(PE)$ of the scalar field. So it can be noticed that
$\omega=\omega(t)$ i.e. it can no more be treated as a constant. The
quintessence or phantom model is consistent with the observations provided
$\omega\simeq-1$. Thus, we need $\dot{\phi}^{2}<<V(\phi)$ i.e. the $KE$ of
$\phi$ is insignificant in comparison to the $PE$. In this study, we consider
that $\phi$ is the only source of DE with $V(\phi)$, so one can consider
energy density and pressure of scalar field as $\rho_{\phi}$ and $p_{{\phi}}$
respectively for flat FLRW space-time under Barrow’s scheme bar  using Eqs.
(44) and (45) as

$\begin{array}[]{c@{\hspace{.1in}}cc}\includegraphics[width=180.67499pt,height=180.67499pt]{sf2.pdf}\hfil\hskip
7.22743pt\end{array}$

Figure 4: The plot of potential energy $V(\phi)$ Vs. scalar field $\phi$.

$\rho=\frac{1}{2}\dot{\phi}^{2}+V(\phi)=\rho_{\phi},$ (46)
$p=\frac{1}{2}\dot{\phi}^{2}-V(\phi)=p_{\phi}.$ (47)

The $KE$ and $PE$ can be obtained by solving the Eqs. (46) and (47). Fig. 4
demonstrates the potential energy $V(\phi)$ plots w.r.t. scalar field $\phi$
for the same considered values of model parameters as we have taken in Fig. 1,
2. From Fig. 4, we notice that the potential $V(\phi)$ is present in the
interval $-1<\phi<0$ and $V(\phi)\simeq 0$ at $\phi\simeq 0$. Therefore, we
can predict that the scalar field $\phi$ is the only source of DE with
potential $V(\phi)$. Thus we conclude that our model is an accelerating dark
energy model.

###  IV.2  Energy conditions

Energy conditions (EC) have a great utility in classical GR which discuss the
singularity problems of space-time and explain the behavior of null, space-
like, time-like or light-like geodesics. It provides some extra freedom to
analyse certain ideas about the nature of cosmological geometries and some
relations that the stress energy momentum must satisfy to make energy
positive. In general, the EC can be classified as (i) NEC (Null energy
condition), (ii) WEC (Weak energy condition), (iii) SEC (Strong energy
condition), and (iv) DEC (Dominant energy condition). The EC can be formulated
in many ways such as geometric way (EC are well expressed in terms of Ricci
tensor or Weyl tensor), physical way (EC are expressed purely by the help of
stress energy momentum tensor), or effective way (EC are expressed in terms of
energy density $\rho$, which serves as the time-like component and pressures
$p_{i},\,i=1,2,3$, which represent the $3$-space-like component). The
formulation of these four types of EC in GR are point-wise expressed
effectively as

  * •

NEC $\Leftrightarrow$ $\rho+p_{i}\geq 0$, $\forall i$,

  * •

WEC $\Leftrightarrow$ $\rho\geq 0$, $\rho+p_{i}\geq 0$, $\forall i$,

  * •

SEC $\Leftrightarrow$ $\rho+\sum_{i=1}^{3}p_{i}\geq 0$, $\rho+p_{i}\geq 0$,
$\forall i$,

  * •

DEC $\Leftrightarrow$ $\rho\geq 0$ , $|p_{i}|\leq\rho$, $\forall i$.

The graphical representation of NEC, SEC and DEC for a fix value of
$\alpha=0.1$ and varying range of $\lambda$ are shown in Fig. 5

$\begin{array}[]{c@{\hspace{.1in}}cc}\includegraphics[width=158.99377pt,height=130.08621pt]{NEC.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=158.99377pt,height=130.08621pt]{DEC.pdf}&\includegraphics[width=158.99377pt,height=130.08621pt]{SEC.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)&\mbox{(}c)\end{array}$

Figure 5: Graphical behavior of NEC, DEC and SEC for $n=1.35$.

Using the above mentioned relations, we discuss all four energy conditions in
$f(R,T)$ theory for all different values of $n=0.5,\,1,\,1.35,\,2$ and
$\alpha=0.1$ by providing different range of coupling constant $\lambda$. We
observe the evolution of energy density and validation for all the EC for both
positive and negative range of $\lambda$. We examine that for positive
$\lambda$, NEC, WEC and DEC hold but SEC violates for $n=0.5$, which directly
implies the accelerated expansion of the Universe. Also, as it is clear that
$\lambda$ is any arbitrary coupling constant so it can also accept the
negative values, so if we extend our domain of $\lambda$ upto negative values,
then it is worth emphasizing that SEC does not hold good for all the models
$n=0.5,\,1,\,1.35,\,2$, which exactly leads to accelerating phase of the
Universe. Here as a matter of discussion, we graphically sketch the figures of
the energy conditions for $n=1.35$ only.

##  V  Validation of the model

###  V.1  Jerk parameter

As we know that the Hubble parameter $H$ measures the fractional rate of
change of scale factor $a$ i.e. the instantaneous expansion and the second
derivative of scale factor $q$ measures the cosmic acceleration. Similarly
higher derivatives of scale factors are allow us to study the cosmic expansion
history and they can potentially differentiate the various dark energy models.
Jerk parameter $j$ is an extensive kinematical quantity which measures the
rate of change of third derivative of scale factor w.r.t. time $t$. On
expanding the Taylor series for scale factor around $a_{0}$, the fourth term
of the Taylor series contains the jerk parameter $j$ sin6 . The Taylor’s
expansion around $a_{0}$ containing the jerk parameter is given by

$\frac{a}{a_{0}}=1+H_{0}t-\frac{1}{2!}q_{0}H_{0}^{2}t^{2}+\frac{1}{3!}j_{0}H_{0}^{3}t^{3}-\cdot\cdot\cdot\cdot\cdot\cdot\cdot,$
(48)

where $H_{0}$, $q_{0}$ and $j_{0}$ being the the current values of the
parameters. Therefore the jerk parameter is defined as

$j=\frac{\dddot{a}}{aH^{3}},$ (49)

and in terms of $q$, jerk parameter $j$ reads vis ; rap

$j=q+2q^{2}-\frac{\dot{q}}{H}.$ (50)

Using Eqs. (27) and (28), it can be calculated as

$j=1+n(2n-3)sech(\beta t)^{2}.$ (51)

Also it will always be suitable to express jerk parameter $j$ in terms of
redshift $z$ when $q(z)$ is given sah3 ; ala . The Jerk parameter $j$ in terms
of $z$ is expressed as

$j=1+\frac{n(2n-3)}{1+2.17391(n-0.46)(1+z)^{-2n}}.$ (52)

In Fig. 6(a), the cosmic jerk parameter highlights the dynamics of the
Universe. Universe transits from decelerated to accelerated phase in a cosmic
jerk $j$ with a positive value $j_{0}\approx 1$ and negative value $q_{0}$ in
accordance with $\Lambda$CDM. It shows the evolution of $j$ parameter for
different values of $n$ and is freely seen that $j$ remains positive in all
the cases and approaches to $1$ in late times. Jerk parameter $j$ at present
($z=0$) is positive different from $1$ in all the four cases. Therefore, we
can expect another dark energy model instead of $\Lambda$CDM.

$\begin{array}[]{c@{\hspace{.1in}}cc}\includegraphics[width=158.99377pt,height=158.99377pt]{jerk.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=158.99377pt,height=158.99377pt]{om.pdf}&\includegraphics[width=158.99377pt,height=158.99377pt]{new-
vc.pdf}\\\ \mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)&\mbox{(}c)\end{array}$

Figure 6: Graphical behavior of (a) Jerk parameter $j$ Vs. time $t$, (b)
Evolution of $Om$ Vs. redshift $z$, (c) Velocity of sound $C_{s}^{2}$ Vs. time
$t$.

###  V.2  Om diagnostic

In this section we discuss the most popular diagnostic known as Om diagnostic
denoted by $Om(z)$, used to distinguish standard $\Lambda CDM$ model from
various dark energy models sah4 ; zun . This diagnostic is related to Hubble
parameter $H$ and redshift $z$. It is noted that different trajectories of
$Om(z)$ facilitate significant differences among various DE models without
actually mentioning the current value of $\Omega_{m}$ (density parameter of
matter). Om diagnostic $Om(z)$ is defined as

$Om(z)=\frac{\Big{(}\frac{H(z)}{H_{0}}\Big{)}^{2}-1}{z(z^{2}+3z+3)}.$ (53)

This tool suggest a quintessence type behavior of dark energy $(\omega>-1)$
corresponding to its negative curvature (i.e. below the $\Lambda CDM$ line),
phantom type behavior $(\omega<-1)$ corresponding to its positive curvature
(i.e. above the $\Lambda CDM$ line) and $Om(z)=\Lambda CDM$ corresponding to
zero curvature. Fig. 6(b) explains the behavior of different dark energy
models corresponding to different values of $n$. For $n=0.5,\,1,\,1.35$ model
shows quintessence type behaviour $(\omega>-1)$ as graph of $Om(z)$ shows a
downward trend as redshift $z$ increases and for $n=2$, model represents
phantom behavior $(\omega<-1)$ as $Om(z)$ has positive slope.

###  V.3  Velocity of sound

The velocity of sound is one of the stringent attempt to investigate the
validity of a cosmic model. A model is said to be physically acceptable if
velocity of sound $C_{s}^{2}$ is less than the speed of light $c$. The
stability condition for the model is given by the relation $0\leq
C_{s}^{2}=\frac{dp}{d\rho}\leq 1$. In this study we have taken the speed of
light $c$ is 1. Therefore, the model is physically realistic provided the
condition $0\leq\frac{dp}{d\rho}\leq 1$ is satisfied.

Fig. 6(c) shows the profile of $C_{s}^{2}$ for $n=0.5,\,1,\,1.35,\,2$. The
constant $\lambda$ decides the stability of the model for a fix value of
$\alpha=0.1$. The stability of the cosmic model depends on the coupling
constant $\lambda$. By considering different values of $\lambda$, we have
different stability scenarios of the model. The model satisfies the condition
$C_{s}^{2}\leq 1$ throughout the evolution with time $t$ for all the cases
$n=0.5,\,1,\,1.35,\,2$ when $\lambda$ is taken in the range
$-17.5\leq\lambda\leq-12.57$. The condition $0<C_{s}^{2}<1$ does not hold for
other cosmic ranges. Therefore, we can say that our model is partially stable.

###  V.4  Statefinder diagnostic

The present cosmic acceleration which is rational with the recent cosmological
observations and the signature flipping behaviour of deceleration parameter
$q$ from $+ve$ to $-ve$ in accordance with high redshift $z$ to low redshift
$z$ enforce us to study beyond $q$ and $H$, and find some more cosmological
models of DE other than $\Lambda$CDM. The behavior of higher derivatives of
scale factor $a$ other than $H$ and $q$ are the essential components to
explain the dynamics of the Universe. Due to this reasons, we generalize our
domain to construct geometrical parameters which involves higher derivatives
of $a$. A technique named as Statefinder diagnostic in which a pair of
geometrical parameters $\\{r,s\\}$ proposed by sah3 ; ala  is taken in to
account to describe the dynamics of various DE models. These parameters
$\\{r,s\\}$ are defined as

$r=\frac{\dddot{a}}{aH^{3}}\text{, \ \ }s=\frac{r-1}{3(q-\frac{1}{2})},$ (54)

where $q\neq\frac{1}{2}$.

For our parametrization of $a$ in Eq. (27), the expressions of $r$ and $s$ are
given as follows:

$r=1+n(2n-3)sech(\beta t)^{2},$ (55)

where the parameter $r$ is same as the jerk parameter $j$, which is defined in
subsection $5.1$.

$s=\frac{4n(2n-3)}{-9+12n-9cosh(2t\beta)}.$ (56)

This technique facilitate us that how one can differentiate various DE models
easily by plotting the different trajectories of $r$ and $s$ (see Fig. 7). For
a brief and recent review on statefinder diagnostic, see sin2 ; rit2 ; srt .

$\begin{array}[]{c@{\hspace{.1in}}cc}\includegraphics[width=180.67499pt,height=180.67499pt]{s-r.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=180.67499pt,height=180.67499pt]{q-r.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 7: The behavior of $s-r$ and $q-r$ trajectories for different values of
$n$ and $\beta$.

Fig. 7(a) highlight the evolution of four trajectories with time for $n$ and
$\beta$ in $s-r$ plane. Each trajectory for $n=1,\beta=0.068$;
$n=1.35,\beta=0.082$ and $n=2,\beta=0.098$ exhibit the same pattern as all
begin in the region $r<1$ and $s>0$ evolving with time, approaches to
$\Lambda$CDM model i.e. the point $r=1,s=0$ . The time evolution of the
trajectory corresponding to $n=0.5,\beta=0.021$ starts from the region $r>1$
and $s<0$ and eventually approaches to $\Lambda$CDM model. From the plot 7(a),
we observe that all the trajectories deviate from SCDM which is resemble to
matter dominated universe, exhibit different dark energy candidates as
Chaplygin gas for $n=0.5,\beta=0.021$, quintessence for $n=1,\beta=0.068$;
$n=1.35,\beta=0.082$ and $n=2,\beta=0.098$, $\Lambda$CDM for $r=1,s=0$ and
SCDM for $r=1,s=1$. Thus the various DE scenarios can be observed by these
evolutionary trajectories which are the remarkable features of statefinder
diagnostic.

Fig. 7(b) states the evolution of the four trajectories with time for $n$ and
$\beta$ in $q-r$ plane. Each trajectory begins in the neighbourhood of $SCDM$
at the time of evolution of the Universe without passing through $\Lambda$CDM
and $SCDM$ converge to $SS$, the steady state model of the Universe. The
downward pattern of a trajectory corresponding to $n=0.5$, $\beta=0.021$ and
upward trend of the trajectories corresponding to $n=2$, $\beta=0.098$; $n=1$,
$\beta=0.068$ and $n=1.35$, $\beta=0.082$ converge to the point $r=1$, $q=-1$
denoted by SS i.e. the steady state model of the Universe which suggest the
steady state behavior of dark energy model in late times.

##  VI Observational constraints on the model parameters

An impressive feature of astronomy is associated with its recent progress in
observational cosmology. Study of the origin, evolution, structure formation,
properties of dark matter and dark energy in the Universe with the help of
cosmic instruments and ray detectors is called observational cosmology. There
are several types of observational data available today for different
measurements in the field of cosmology. Some of them are Sloan Digital Sky
Survey ($SDSS$) which provide the map of the galaxy distribution and encode
the current fluctuations in the Universe,$CMBR$ that serves as the evidence of
the big bang theory, Quasi Stellar Radio Sources ($QUASARS$) which are
considered as the most metal thing in the Universe and extract the matter
between observer and quasars, Baryon Acoustic Oscillations ($BAO$) that
measures the large scale structures in the Universe in order to understand
dark energy better, observations from type Ia Supernova are the tools for
measuring the cosmic distances usually known as standard candles. In the
subsequent sections, we have presented a statistical analysis by using some
observational datasets of $SNeIa$, $H(z)$ and $BAO$ to constrain Hubble
parameter $H_{0}$ and model parameter $n$ involved in our model. To constraint
model parameter $n$, we restrict the inverse hyperbolic of sine series and
hyperbolic of cotangent series in Eq. (33) upto first term and then integrate
the approximate series to calculate the Chi-square value i.e. $\chi^{2}_{min}$
using each observational data set.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=180.67499pt,height=166.2212pt]{zhz.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=180.67499pt,height=166.2212pt]{zmuz.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 8: Figures (a) and (b) are error bar plots comparing our model with
standard $\Lambda$CDM model using $H(z)$ and $SNeIa$ datasets respectively.
Black line signifies $\Lambda CDM$ model and red line displays our model in
both figures (a) and (b).

###  VI.1 Hubble observation H(z)

The Hubble parameter $H$ can be observed in terms of some physical observable
quantity such as length, time and redshift $z$. In terms of redshift $z$, $H$
reads

$H(z)=-\frac{1}{(1+z)}\frac{dz}{dt}.$ (57)

Here, the best fit value of model parameter $n$ is obtained by using $28$
points of $H(z)$ datasets Hz  in the range $(0.1,2.5)$. We take the recent
value of Hubble constant $H_{0}=67.8$ $Km/s/Mpc$ from Planck $2014$ results
Hz-Plank  to compare our model with standard $\Lambda$CDM. The constrains on
model parameter $n$ can be obtained by minimizing the Chi-square value i.e.
$\chi^{2}_{min}$, which is identical as the maximum likelihood analysis and is
expressed as

$\chi_{OH}^{2}=\sum\limits_{i=1}^{28}\frac{[H^{obs}(z_{i})-H^{th}(z_{i})]^{2}}{\sigma{(z_{i})}^{2}},$
(58)

where, $OH$ indicates the observational Hubble dataset. $H^{obs}$ and $H^{th}$
serve the observed and theoretical value of $H$. $\sigma{(z_{i})}$ denotes the
standard error in the measured value of $H$.

###  VI.2  Type Ia Supernova

Here, we fit the present model with latest union $2.1$ compilation
observational dataset of $580$ point SNeIa  and compare the results with
$\Lambda$CDM.

$\chi_{OSN}^{2}(\mu_{0})=\sum\limits_{i=1}^{580}\frac{[\mu_{th}(\mu_{0},z_{i})-\mu_{obs}(z_{i})]^{2}}{\sigma_{\mu(z_{i})}^{2}},$
(59)

where $OSN$ denotes the observational $SNeIa$ dataset. $\sigma_{\mu(z_{i})}$,
$\mu_{obs}$ and $\mu_{th}$ denote the standard error in the measurement of
$\mu(z)$, the observed and theoretical distance modulus of the model
respectively. The distance model $\mu(z)$ is given by

$\mu(z)=m-m^{\prime}=5LogD_{l}(z)+\mu_{0},$ (60)

where $D_{l}(z)$ and $\mu_{0}$ are the luminosity distance and nuisance
parameter respectively. $m$ and $m^{\prime}$ represent the apparent and
absolute magnitudes of standard candle respectively.

The left panel of Fig. 8 is not good fitted so much but right panel is better
fitted when we compare our model with $\Lambda$CDM using $H(z)$ and SNeIa data
respectively.

<EMAIL_ADDRESS>in}}cc}\includegraphics[width=158.99377pt,height=144.54pt]{cont-
hz.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=158.99377pt,height=144.54pt]{cont-
sn.pdf}&\includegraphics[width=158.99377pt,height=144.54pt]{cont-bao.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)&\mbox{(}c)\end{array}$

Figure 9: Plots (a), (b) and (c) represent the contours in $(n$-$H_{0})$ plane
from the statistical analysis of $H(z)$, $SNeIa$ and $BAO$ datasets
respectively with 1-$\sigma$ error (red shaded region ), 2-$\sigma$ error
(light green shaded region shows) and 3-$\sigma$ error (light yellow shaded
region). Black dot signifies the the well fitted constrained value of model
parameter $n$ and the values of $H_{0}$ for each observation.

###  VI.3  Baryon Acoustic Oscillations

The $BAO$ measures the structures of the universe on a large scale taking the
statistical property of matter as sound waves which is useful to study DE in a
better way. Here, we consider a sample from various surveys, namely $6dF$
Galaxy survey 6df , $SDSS$ galaxy sample padn , results from $WiggleZ$ survey
wig  and $BOSSCMASS$ boss .

$\begin{array}[]{c@{\hspace{0.1in}}c}\includegraphics[width=180.67499pt,height=166.2212pt]{cont-
hz+sn.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=180.67499pt,height=166.2212pt]{cont-
hz+sn+bao.pdf}\\\ \mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 10: Plots (a), (b) and (c) represent the contours in $(n$-$H_{0})$
plane from the statistical analysis of $H(z)$ \+ $SNeIa$ and $H(z)$ \+ $SNeIa$
\+ $BAO$ respectively with 1-$\sigma$ error (red shaded region ), 2-$\sigma$
error (light green shaded region shows) and 3-$\sigma$ error (light yellow
shaded region). Black dot signifies the well fitted constrained value of model
parameter $n$ and the values of $H_{0}$ for each observation.

We define $d_{z}$ (Distance redshift ratio) for $BAO$ measurement

$d_{z}=\frac{r_{s}(z_{*})}{D_{v}(z)},$ (61)

where $z_{*}$ denotes the redshift at the time of photons decoupling redshift
which is $z_{*}=1090$ given by Planck 2015 results adep . Also $r_{s}(z_{*})$
is co-moving sound horizon during decoupling of photons waga . Further the
dilation scale
$D_{v}(z)=\big{[}\frac{d^{2}_{A}(z)z}{H(z)}\big{]}^{\frac{1}{3}}$, where
$d_{A}(z)$ indicates the angular diameter distance.

The Chi square value for $BAO$ measurement $(\chi_{BAO}^{2})$ is given by gio

$\chi_{BAO}^{2}=Y^{T}C^{-1}Y,$ (62)

where $Y$ is a matrix given by

$Y=\left[{\begin{array}[]{cc}\frac{d_{A}(z_{*})}{D_{v}(0.106)}-30.84\\\
\frac{d_{A}(z_{*})}{D_{v}(0.35)}-10.33\\\
\frac{d_{A}(z_{*})}{D_{v}(0.57)}-6.72\\\
\frac{d_{A}(z_{*})}{D_{v}(0.44)}-8.41\\\
\frac{d_{A}(z_{*})}{D_{v}(0.6)}-6.66\\\
\frac{d_{A}(z_{*})}{D_{v}(0.73)}-5.43\\\ \end{array}}\right]$

The inverse covariance matrix $C^{-1}$ gio  by adopting the correlation
coefficients existed in hing  becomes

$C^{-1}=\left[{\begin{array}[]{cccccc}0.52552&-0.03548&-0.07733&-0.00167&-0.00532&-0.00590\\\
-0.03548&24.97066&-1.25461&-0.02704&-0.08633&-0.09579\\\
-0.07733&-1.25461&82.92948&-0.05895&-0.18819&-0.20881\\\
-0.00167&-0.02704&-0.05895&2.91150&-2.98873&1.43206\\\
-0.00532&-0.08633&-0.18819&-2.98873&15.96834&-7.70636\\\
-0.00590&-0.09579&-0.20881&1.43206&-7.70636&15.28135\\\ \end{array}}\right]$

Table 4: Outcomes of data analysis for our model Data |  $\chi^{2}_{min}$ |  $H_{0}$ |  $n$
---|---|---|---
$H(z)$ |  $19.5872$ |  $68.0000$ |  $1.5176$
$SNeIa$ |  $565.5074$ |  $66.3402$ |  $1.5907$
$BAO$ |  $2.1622$ |  $66.4138$ |  $1.5009$
$H(z)$ \+ $SNeIa$ |  $589.7361$ |  $65.6148$ |  $1.5396$
$H(z)$ \+ $SNeIa$ \+ $BAO$ |  $597.2995$ |  $63.0579$ |  $1.5060$

Figures $9,10$ represent the contours with error $1-\sigma$, $2-\sigma$ and
$3-\sigma$ in $(n$-$H_{0})$ plane. The well fitted constrained value of the
model parameter $n=1.5176$ and Hubble constant $H_{0}=68.0000$ $Km/s/Mpc$ with
$1\,\sigma$, $2\,\sigma$ and $3\,\sigma$ errors are found in the ranges
$1.488<n<1.5461$, $1.469<n<1.563$ and $1.449<n<1.58$ respectively according to
the Hubble dataset $H(z)$. The well fitted constrained value of the model
parameter $n=1.5907$ and Hubble constant $H_{0}=66.3402$ $Km/s/Mpc$ with
$1\,\sigma$, $2\,\sigma$ and $3\,\sigma$ errors are found in the ranges
$1.547<n<1.634$, $1.521<n<1.662$ and $1.493<n<1.69$ respectively according to
the $SNeIa$ (union 2.1 compilation data set). The well fitted constrained
value of the model parameter $n=1.5009$ and Hubble constant $H_{0}=66.4138$
$Km/s/Mpc$ with $1\,\sigma$, $2\,\sigma$ and $3\,\sigma$ errors are found in
the ranges $1.492<n<1.510$, $1.486<n<1.516$ and $1.481<n<1.522$ respectively
according to the $BAO$. The well fitted constrained value of the model
parameter $n=1.5396$ and Hubble constant $H_{0}=65.6148$ $Km/s/Mpc$ with
$1\,\sigma$, $2\,\sigma$ and $3\,\sigma$ errors are found in the ranges
$1.516<n<1.563$, $1.501<n<1.577$ and $1.486<n<1.591$ respectively according to
the combined dataset $H(z)$ \+ $SNeIa$. The well fitted constrained value of
the model parameter $n=1.5060$ and Hubble constant $H_{0}=63.0579$ $Km/s/Mpc$
with $1\,\sigma$, $2\,\sigma$ and $3\,\sigma$ errors are found in the ranges
$1.497<n<1.515$, $1.492<n<1.521$ and $1.487<n<1.526$ respectively according to
the combined dataset $H(z)$ \+ $SNeIa$ \+ $BAO$ (see Table IV).

##  VII  Some other cosmological tests

###  VII.1  Lookback time

The time interval between the detection of light on the Earth and the emission
from the source is called Lookback time denoted by $t_{L}$. Therefore, the
total time $t_{L}$ elapsed between the light ray which emits from a galaxy at
time $t_{z}$ at particular redshift $z$ and reaches us at a time $t_{0}$ at
redshift $z=0$ is given by

$t_{L}=t_{0}-t_{z}=\int_{a}^{a_{0}}\frac{dt}{\dot{a}}.$ (63)

From (32), we have

$t_{L}=t_{0}-\frac{sinh^{-1}\sqrt{\frac{n-(1+q_{0})}{(z+1)^{2n}(q_{0}+1)}}}{\beta}.$
(64)

###  VII.2  Proper distance

The proper distance is the distance travelled by the photons from the source
to us which is also known as Instantaneous distance. Proper distance is
denoted by $d(z)$ and it can be calculated as $d(z)=a_{0}r(z)$, where $r(z)$
is the radial distance which can be obtained by

$r(z)=\int_{t}^{t_{0}}\frac{dt}{a(t)}.$ (65)

Using Eq. (32), the proper distance $d(z)$ of the model is obtained by

$d(z)=\frac{1}{\beta}\Bigg{[}-cosh(\beta t)\,S_{1}\,sinh(\beta
t)^{-\frac{1+n}{n}}(-sinh(\beta t)^{2})^{\frac{1+n}{2n}}+cosh(\beta
t_{0})\,S_{2}\,sinh(\beta t_{0})^{-\frac{1+n}{n}}(-sinh(\beta
t_{0})^{2})^{\frac{1+n}{2n}}\Bigg{]},$ (66)

where $S_{1},S_{2}$ are hypergeometric functions defined as

$S_{1}=Hypergeometric\,\,2F1[\frac{1}{2},\frac{1}{2}(1+\frac{1}{n}),\frac{3}{2},cosh(\beta
t)^{2}]$

and

$S_{2}=Hypergeometric\,\,2F1[\frac{1}{2},\frac{1}{2}(1+\frac{1}{n}),\frac{3}{2},cosh(\beta
t_{0})^{2}].$

$\begin{array}[]{c@{\hspace{.1in}}cc}\includegraphics[width=180.67499pt,height=144.54pt]{lb.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=180.67499pt,height=144.54pt]{pd.pdf}&\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)&\end{array}$

Figure 11: The plots of look back time $t_{L}$ and proper distance $d(z)$ Vs.
redshift $z$.

###  VII.3  Angular diameter

If the size of a source is known, then its angular width which is its distance
from the observer is given by

$d_{A}=\frac{\theta}{l},$ (67)

where $\theta$ is the angular size of the source and $l$ is the size of the
source. In terms of redshift $z$, the angular diameter is given by

$d_{A}=\frac{d(z)}{1+z}.$ (68)

###  VII.4  Luminosity distance

Suppose an object is located at a distance $r$ with absolute luminosity $L$
and observed luminosity $l$ of the source then we have

$L=\frac{l}{4\pi r^{2}}.$ (69)

and the luminosity distance $d_{l}$ is defined as

$d_{l}=\left(\frac{L}{4\pi l}\right)^{\frac{1}{2}}.$ (70)

In terms of $z$, the luminosity distance $d_{l}$ is given by

$d_{l}=(1+z)d(z).$ (71)

$\begin{array}[]{c@{\hspace{.1in}}cc}\includegraphics[width=180.67499pt,height=144.54pt]{add.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=180.67499pt,height=144.54pt]{ld.pdf}&\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)&\end{array}$

Figure 12: The plots of angular diameter distance $d_{A}$ and luminosity
distance $d_{l}$ Vs. redshift $z$.

##  VIII  Discussions and Conclusions

In present paper, we have examined the cosmological model in the background of
FLRW metric under the theory of $f(R,T)$ gravity with the form
$f(R,T)=f(R)+2f(T)$. The functional form of $f(R)$ is considered in a sense
that it provides quadratic geometric corrections in GR and it is accepted as
the most compatible model with the latest observations. The variable $T$ is
the trace of EMT, which introduce exotic imperfect fluids and quantum effects.
Also the functional form of $T$ explains the non minimal matter-geometry
coupling discussed in rap . Consequently the effect of complete form of
$f(R,T)$ function on the general action produces an explicit set of field
equations. Therefore, the behavior of the Universe depends on the field
equations and based on the associated informations and the observations using
various statistical data can be interpreted as follows.

  * •

The geometrical parametrization of scale factor $a(t)$ has been chosen in such
a way that it leads to two types of cosmic models depending upon various the
range of $n$. The time dependent DP shows different regimes of the Universe as
(i) the model shows eternal acceleration when $0<n<1$ and (ii) the model
transits from early deceleration to late time acceleration of the Universe
which can be seen in Fig 1(c). Hence the phase transition occurs when $n\geq
1$. For $n=1.35$, $q=0$ at redshift transition values $z_{tr}=0.883752$ and
for $n=2$, $q=0$ at redshift transition values $z_{tr}=0.352666$, which are
best fitted redshift transition values according as the observations abd ; zha
respectively. Also we inspect that the model exhibits point type singularity
at time $t=0$ (see Table I).

  * •

Using the parametrization of $a(t)$, we have obtained the solution of the
field equations (20)-(21) i.e. the value of energy density $\rho$ and
isotropic pressure $p$ of the matter filled in the Universe. The plots of
$\rho$ and $p$ can be seen in Fig 2. Fig. 2(a) represents the behavior of
$\rho$ corresponding to $n=0.5,\,1,\,1.35,\,2$. Initially at the point of
singularity $t=0$, the Universe starts with very high energy density
$\rho\to\infty$ and later on falls off as time unfolds $\rho\to 0$ in late
times $t\to\infty$. Fig. 2(b) highlights the profile of pressure for all the
values of $n$ as mentioned earlier. The isotropic pressure $p\to-\infty$ as
$t\to 0$, remains negative throughout the evolution and approaches to negative
constant value in late times for $n=0.5$, which indicates the eternal cosmic
accelerated expansion. For $n=1$, $p$ varies from initially positive value in
a very short interval of times to a negative value in late times and
sequentially approaches to a negative constant. For $n=1.35,2$, the isotropic
pressure $p$ decreases monotonically with time and attains a negative value in
late times. The initially small positive value of pressure resembles the
decelerating expansion of the Universe while according to standard cosmology,
the negative pressure is indeed responsible for cosmic acceleration. The model
is rational with formation of structure in the Universe for the specified
values of $n=1,\,1.35,\,2$.

  * •

The result (36) depends on the coupling constant of the matter part $\lambda$.
From (36) we see that matter behaves like the DE in late times and the value
(36) is observationally accepted for the matter density and pressure in late
times provided $\lambda>-2\pi$. More precisely, the limit value of the matter
density in the early times is the necessary condition that our model satisfies
the weak energy condition (WEC), provided the constraints in the Table III
must hold.

  * •

From Eqs. (37) and (38), we observe that the EoS parameter (35) tends to a
constant value in the early times and it shows the DE in late times. From Eq.
(37) we find two different cosmological transitions in our model: (i) a
transition from a stiff matter dominants at the early stage to a DE dominants
in late times when $n=3/2$ , and (ii) a transition from a pressure less matter
dominated era with $\omega^{p}_{i}=0$ in the early eras to a DE dominants in
late times when $n=\frac{3(\lambda+4\pi)}{2(3\lambda+8\pi)}$. Therefore, it is
interesting that the final state of cosmological evolution depends on
$\lambda$ only. In Fig. 3., for a fix value of $\alpha=0.5$, and $\lambda=2$,
$\omega\in$ quintessence region for high redshift $z$ and as time unfolds,
$\omega\to-1$ in infinite future (i.e. $z\to-1$) when $n$ equals to the
expression (39), which is presently consistent with the observations of
temperature fluctuation in $CMBR$ hin . The current value of the $\omega$ for
the mentioned types of evolution era are obtained in Eqs. (40) and (41) when
$n=3/2,\,\frac{3(\lambda+4\pi)}{2(3\lambda+8\pi)}$respectively. Therefore, the
astronomical data can constrain the coupling constant. From Fig. 3(b), the EoS
parameters $\omega\simeq-0.509$ and $\omega\simeq-0.2629$ at present epoch
$z=0$ for $n=3/2,\,\frac{3(\lambda+4\pi)}{2(3\lambda+8\pi)}$ respectively,
which is in good agreement with the observations abd .

  * •

After inspecting the EoS parameter $\omega$, it can easily be observed that
there is some exotic matter field present in the Universe which is capable of
being produced some repulsive pressure. The matter field has negligible $KE$
as compare to $PE$ in the Universe which makes $\omega$ negative. The behavior
of the quintessence scalar field correspondence has been examined in
subsection IV A. In Fig. 4, we observe that $V(\phi)$ is present in the
interval $-1<\phi<0$ and $V(\phi)\simeq 0$ at $\phi\simeq 0$. Therefore, we
can predict that the $\phi$ is the only source of DE with $V(\phi)$. Thus we
conclude that our model is an accelerating dark energy model.

  * •

In subsection IV B, we have discussed the physical acceptability of the model
by analysing EC. The derived model for $n=0.5,\,1,\,1.35,\,2$ satisfy NEC, WEC
and DEC for a wide accepted range of $\lambda$. On the other hand, SEC fails
to satisfy in the late stages of the Universe, which produces a repulsive
force and make the Universe to get jerk. Also the validation of SEC in the
early stage leads to the early decelerated phase of the Universe (See Fig. 5).

  * •

In section V, we have examined the validity of our model through the various
observational tests as: (i) Analysis of Jerk parameter, (ii) Om diagnostic
(iii) Analysis of velocity of sound.

Fig 6(a) enacts the evolution of jerk parameter $j$ w.r.t. redshift $z$ for
the given parametrization of DP for the values of $n=0.5,\,1,\,1.35,\,2$. Jerk
parameter has attained positive values for any redshift $z$ and $j\to 1$
represents $\Lambda CDM$ in the future at $z\to 1$. At present $z=0$, the
value of $j$ for $n=0.5,1,1.35$ is clearly less than $1$ and for $n=2$, the
value of jerk parameter, which is greater than $1$, favours a dark energy
model different from $\Lambda CDM$.

Fig 6(b) expresses the behavior of different dark energy model for different
values of $n$. The model exhibits quintessence-like behavior when
$n=0.5,\,1,\,1.35$ and the phantom-like behavior when $n=2$. Further by
considering different values of $\lambda$, we have different stability
scenario of model. Model agrees on $C_{s}^{2}\leq 1$ throughout the evolution
with time $t$ for all the cases $n=0.5,\,1,\,1.35,\,2$ when $\lambda$ assumes
value in the range $-17.5\leq\lambda\leq-12.57$ (See Fig. 6(c)). Otherwise,
$0<C_{s}^{2}<1$ does not hold in other interval. i.e. the model is partially
stable.

  * •

From the Fig. 7(a), we observe that all the trajectories approach to $\Lambda
CDM$ and deviate from $SCDM$ which is resemble to matter dominated universe.
The trajectories exhibit different dark energy candidates as Chaplygin gas for
$n=0.5,\beta=0.021$, quintessence for $n=1,\beta=0.068$; $n=1.35,\beta=0.082$
and $n=2,\beta=0.098$, $\Lambda CDM$ for $r=1,s=0$ and $SCDM$ for $r=1,s=1$.
Thus the various DE scenarios can be observed by these evolutionary
trajectories, which are the remarkable features of statefinder diagnostic.
Figure 7(b) states the all the trajectories begin in the neighbourhood of
$SCDM$ at the time of evolution of the Universe without passing through
$\Lambda CDM$ and $SCDM$converge to $SS$, the steady state model of the
Universe which suggest the steady state behavior of dark energy model in late
times.

  * •

In section VI, Figs. 9 and 10. represent the likelihood contours for $n$ and
$H_{0}$ with $1\sigma$, $2\sigma$ and $3\sigma$ errors in the $n$-$H_{0}$
plane. The constrained values of $n$ are obtained $1.5176$, $1.5907$,
$1.5009$, $1.5396$ and $1.5060$ according to the Hubble $H(z)$, $SNeIa$,
$BAO$, $H(z)+SNeIa$ and $H(z)+SNeIa+BAO$ data sets for which the corresponding
values of $H_{0}$ are obtained as $68.0000$, $66.3402$, $66.4138$, $65.6148$
and $63.0579$ respectively (see Table IV). Here, we observe that both
constrained values of $n$ and $H_{0}$ are best fitted values according to
these datasets.

  * •

In section VII, we have investigated some other types of the cosmological
tests such as lookback time, proper distance, angular diameter distance and
luminosity distance and it is found that the resulting outcomes of these
kinematics test are consistent with current observations.

Acknowledgements R. Nagpal and J. K. Singh express their thanks to the Dept.
of Mathematical Sciences, University of Zululand, S.A. for financial support
and hospitality to complete the work. The author JKS also express his thanks
to CTP, Jamia Millia Islamia, New Delhi, India for some fruitful discussions
with Prof. M. Sami and Prof. S. G. Ghosh.

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