Reconstruction of an Observationally Constrained $f(R,T)$ gravity model
Anirudh Pradhan1, Gopikant Goswami2, Aroonkumar Beesham3,4

1Centre for Cosmology, Astrophysics and Space Science (CCASS), GLA University,
Mathura-281 406, Uttar Pradesh, India

2Department of Mathematics, Netaji Subhas University of Technology, Delhi,
India

3Department of Mathematical Sciences, University of Zululand Private Bag X1001
Kwa-Dlangezwa 3886 South Africa

4Faculty of Natural Sciences, Mangosuthu University of Technology, P O Box
12363, Jacobs 4052, South Africa

1E-mail<EMAIL_ADDRESS>
2E-mail<EMAIL_ADDRESS>
3,4E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, an attempt is made to construct a Friedmann-Lemaitre-Robertson-
Walker model in $f(R,T)$ gravity with a perfect fluid that yields acceleration
at late times. We take $f(R,T)$ as $R$ \+ $8\pi\mu T$. As in the $\Lambda$CDM
model, we take the matter to consist of two components, viz., $\Omega_{m}$ and
$\Omega_{\mu}$ such that $\Omega_{m}$ \+ $\Omega_{\mu}$=1. The parameter
$\Omega_{m}$ is the matter density (baryons + dark matter), and $\Omega_{\mu}$
is the density associated with the Ricci scalar $R$ and the trace $T$ of the
energy momentum tensor, which we shall call dominant matter. We find that at
present $\Omega_{\mu}$ is dominant over $\Omega_{m}$, and that the two are in
the ratio 3:1 to 3:2 according to the three data sets: (i) 77 Hubble OHD data
set (ii) 580 SNIa supernova distance modulus data set and (iii) 66 pantheon
SNIa data which include high red shift data in the range $0\leq z\leq 2.36$.
We have also calculated the pressures and densities associated with the two
matter densities, viz., $p_{\mu}$, $\rho_{\mu}$, $p_{m}$ and $\rho_{m}$,
respectively. It is also found that at present, $\rho_{\mu}$ is greater than
$\rho_{m}$. The negative dominant matter pressure $p_{\mu}$ creates
acceleration in the universe. Our deceleration and snap parameters show a
change from negative to positive, whereas the jerk parameter is always
positive. This means that the universe is at present accelerating and in the
past it was decelerating. State finder diagnostics indicate that our model is
at present a dark energy quintessence model. The various other physical and
geometric properties of the model are also discussed.

Keywords: $f(R,T)$ theory; FLRW metric; Observational parameters; Transit
universe; Observational constraints

Mathematics Subject Classification 2020: 83D05, 83F05, 83C15

##  1 Introduction:

Cosmology is the study of the large scale structure and evolution of the
universe. It began seriously with the simple Einstein static universe [1]. It
then got a drastic change due to Hubble [2], who started the concept of an
expanding universe. This was later mathematically formulated into the FLRW
(Friedmann-Lemaitre-Robertson-Walker) spacetime [3, 4, 5, 6, 7]. The universe
since its inception, passed through very exotic events such as inflation [8,
9] at the beginning, then the discovery of the CMB (cosmic microwave
background radiation) [10], and then the late time acceleration[11]$-$[26].
Dark energy, which is believed to cause this acceleration is most often
represented by the very simple concordance $\Lambda$CDM model. This model fits
well on observational grounds [27]$-$[29], despite certain weaknesses [30]
that it suffers with fine tuning and the cosmic coincidence problems. To solve
these, scalar field dominated tracker field quintessence and phantom dark
energy models were proposed [31]$-$[34]. Later on, parameterizations for the
scalar $\phi$ field were suggested, and some interesting cosmological models
were developed [35] $-$ [42]. This work has been done under the principle of
general relativity (GR) and accordingly, the GR field equations were used.

With this, a spate of work was begun in which attempts were made to get an
accelerating universe by modifying Einstein’s field equations. It has been
suggested that the Ricci scalar $R$ be replaced by an arbitrary function
$f(R)$ of $R$ in the Einstein-Hilbert action, and that the new theory and its
field equations be called $f(R)$ theory [43] $-$ [56]. The idea is that a non
linear Ricci scalar may be helpful in developing negative pressure in the
universe, producing acceleration. Seeing the complexity of solutions due to
non linear Ricci scalar [57], more alternatives were proposed. In one of the
options, $f(R)$ was replaced by an arbitrary function $f(R,T)$ of $R$ and $T$
where $T$ is the trace of the energy momentum tensor. This theory is called
$f(R,T)$ gravity [58] $-$ [71]. The authors of the theory [58] have dealt
three options for the specific functional form of $f(R,T$), especially for
cosmological interpretations. They are
$R+2f(T),f_{1}(R)+f_{2}(T)~{}\text{and}~{}f_{1}(R)+f_{2}(R)f_{3}(T)$. The
purpose of this work is to model a universe in $f(R,T)$ gravity which meets
observational constraints [20]$-$[22]. For this, we consider the first simple
alternative of $f(R,T)$, i.e., $f(R,T)=R+\lambda$ T, taking $\lambda$ as an
arbitrary constant. We consider an FLRW spacetime with perfect fluid. Our
three model parameters, viz., the Hubble, deceleration and equation of state,
are estimated with the help of three data sets: (i) the 77 Hubble OHD data set
[72]$-$[87] (ii) the 580 SNIa supernova distance modulus data set [17] and
(iii) the 66 pantheon SNIa data set, which includes the high red shift data in
the range $0\leq z\leq 2.36$ [88]$-$[90]. We solve the $f(R,T)$ field
equations by making the simplest possible parametrization of the equation of
state parameter as given by Gong and Zhang [35].
At this junction, it is desirable to describe the new results/important points
presented in this paper which are different from the past studies. In the past
there had been some very interesting reconstruction and review works in the
modified theories of gravity [69]$-$ [73]. The learned authors also noticed
certain finite time singularities and rips in certain models. However as an
important and noticeable feature in our model, we have developed two energy
parameters $\Omega_{m}$ and $\Omega_{\mu}$ and found that $\Omega_{m}$ \+
$\Omega_{\mu}$=1. The parameter $\Omega_{m}$ is associated with the matter,
whereas $\Omega_{\mu}$ is associated with $f(R,T)$ gravity. We have
statistically estimated using high red shift data set in the range $0\leq
z\leq 2.36$ that at present $\Omega_{\mu}$ is dominant, and that the two
energy densities are approximately in the ratio 3:1 to 3:2. Moreover we have
also performed a state finder diagnostic of our model and found that our model
is at present in quintessence. We have presented 35 plots and tables to
describe the findings of our work more effectively.

The following is a short description of the work, broken down into sections:
In section $2$, the $f(R,T)$ field equation for a perfect fluid filled FLRW
space time are described. Section $3$ is the architect of the rest of the part
of the paper. In this section, we get expressions for the Hubble and
deceleration parameters as functions of red shift. In sections $4$, $5$ and
$6$, the model parameters have been estimated on the basis of the three data
sets. In sections 7, we have presented the error bar plots, confidence regions
and likelihood plots for the Hubble, distance modulus and apparent magnitude.
The purpose here is to show the proximity of observational and theoretical
results. The jerk and snap parameters are discussed in section 8, and it is
found that our model is at present a dark energy quintessence model. In
section $9$, we have obtained expressions for the densities and pressures. It
is found that at present $\mu$ density ($\rho_{\mu}$) is dominant over baryon
density and they are nearly in the ratio 1:3 or 2:3 as per different observed
data sets. The last section $10$ is devoted to the concluding remarks.

##  2 f(R,T) Field Equations for Perfect Fluid Filled FLRW Space Time:

We refer to [58] for the field equations of f(R,T) gravity:

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

where we take the energy momentum tensor $T_{ij}$ as that of perfect fluid:

$T_{ij}=(\rho+p)u_{i}u_{j}-pg_{ij}.$ (2)

The other symbols in the field equations (1) have their usual meanings. The
Einstein Hilbert action is:

${}S=\int(\frac{1}{16\pi G}(R+2\lambda)+L_{m})\sqrt{-g}dx^{4},$ (3)

where $L_{m}$ is the matter Lagrangian. The field equations (1) are obtained
replacing the Ricci scalar R by a arbitrary function $f(R,T)$ of $R$ and trace
$T$ of energy momentum tensor $T_{ij}$. Accordingly, the $f(R,T)$ action is:

${}S=\int{\left(\frac{1}{16\pi G}f(R,T)+L_{m}\right)\sqrt{-g}}dx^{4}.$ (4)

The FLRW metric is:

${}ds^{2}=dt^{2}-a^{2}(t)(dx^{2}+dy^{2}+dz^{2}).$ (5)

We solve Eqs. (1) and (2) for this metric and get:

${}2\dot{H}+3H^{2}=-(8\pi+3\lambda)p+\lambda\rho$ (6)

and

${}3H^{2}=(8\pi+3\lambda)\rho-\lambda p,$ (7)

where $H=\frac{\dot{a}}{a}$ is the Hubble parameter.

##  3 Deceleration and Hubble Parameters:

We assume $\lambda=8\pi\mu$. Then the field equations (6) and (7) are
simplified as:

${}H^{2}(1-2q)=-8\pi(p+\mu(3p-\rho))$ (8)

and

${}3H^{2}=8\pi(\rho+\mu(3\rho-p)),$ (9)

where $q=-\frac{\ddot{a}}{aH^{2}}$ is deceleration parameter and
$\dot{H}=-(q+1)H^{2}$. It is clear from the field equations (8) and (9) that
for $\mu=0$, these equations reduce to those of general relativity. So the
additional terms containing $\mu$ are there due to $f(R,T)$ gravity. We
require $q\leq 0$ (acceleration) at present due to the presence of these
terms. We let:

$p_{\mu}\equiv\mu(3p-\rho),~{}\rho_{\mu}\equiv\mu(3\rho-p).$ (10)

and call these the dominant pressure and dominant energy density,
respectively, for reasons which will become clear later. These may be regarded
as the contribution of $f(R,T)$ gravity to the pressure and density. We get
the energy parameters $\Omega_{m}$ and $\Omega_{\mu}$ and the equations of
state parameter $\omega$ and $\omega_{\mu}$ as follows:

$\Omega_{m}=\frac{8\pi\rho}{3H^{2}},~{}\Omega_{\mu}=\frac{8\pi\rho_{\mu}}{3H^{2}},~{}\omega_{m}=\frac{p}{\rho},~{}\omega_{\mu}=\frac{p_{\mu}}{\rho_{\mu}}.$
(11)

Here suffixes $m$ and $\mu$ stands for the GR effect, and $f(R,T)$ effect,
respectively. We may also interpret the parameters with the $\mu$ suffix as
terms arising due to the curvature dominance of $f(R,T)$ gravity.

From Eqs. (8) $-$ (11), we get the following:

$H^{2}(1-2q)=-8\pi(p+p_{\mu}),~{}~{}3H^{2}=8\pi(\rho+\rho_{\mu}),~{}~{}\Omega_{m}+\Omega_{\mu}=1$
(12)

and

$\omega_{\mu}=\frac{3\omega_{m}-1}{3-\omega_{m}}~{}~{}~{}~{}\omega_{m}=\frac{3\omega_{\mu}+1}{\omega_{\mu}+3}.$
(13)

Eqs. (12) and (13) give us the following expression for $q$ :

$2q=1+\frac{3(8\mu+3)\omega_{\text{$\mu$}}+3}{8\mu+\omega_{\text{$\mu$}}+3}.$
(14)

From this, we get the present value of $\mu$ as:

$\mu_{0}=\frac{5\omega_{\text{$\mu$0}}-q_{0}\left(\omega_{\text{$\mu$0}}\right)-3q_{0}+3}{4\left(-3\omega_{\text{$\mu$0}}+2q_{0}-1\right)}$
(15)

We observe that there are two equations and four unknowns $H$, $q$, $p$ and
$\rho$. Hence, one cannot solve these equation in general. However, to get an
explicit solution of the above equations, we have to assume at least one
reasonable relation among the variables. For this, we consider the simplest
parametrization of the equation of state parameter $\omega_{\mu}$ as given by
Gong and Zhang [35]:

$\omega_{\mu}=\frac{\omega_{\mu 0}}{(1+z)},$ (16)

where $z$ is the red shift and $\omega_{\mu 0}$ is the present value of the
equation of state parameter $\omega_{\mu}$ . With this we can now solve for
$q$ in term of $z$ as follows:

$q=\frac{q_{0}\left(1.5\omega_{\text{$\mu$0}}^{2}+0.5z\omega_{\text{$\mu$0}}-1.5z-1.5\right)+2z\omega_{\text{$\mu$0}}}{1.5\omega_{\text{$\mu$0}}^{2}+\left(q_{0}-0.5\right)z\omega_{\text{$\mu$0}}-1.5z-1.5}$
(17)

Then H may be obtained from q by solving the following differential equation:

$H_{z}(1+z)=(q+1)H$ (18)

We obtain the expression for $H$ as follows:

$H=H_{0}~{}\alpha_{1}~{}\alpha_{2}~{}exp{\left[-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}+\frac{\alpha_{6}\alpha_{7}}{\alpha_{8}}\right]},$
(19)

where, $H_{0}$ is the present value of the Hubble parameter and:

$\begin{array}[]{rrrrr}\alpha_{1}&=&\left(1.5\,-1.5\omega_{\text{$\mu$0}}^{2}\right){}^{\frac{q_{0}\left(0.75\,-0.75\omega_{\text{$\mu$0}}\right)-0.75\omega_{\text{$\mu$0}}+0.75}{\left(q_{0}-0.5\right)\omega_{\text{$\mu$0}}-1.5}}\\\
\\\
\alpha_{2}&=&\left(z\omega_{\text{$\mu$0}}\left(q_{0}(-1.z-1.)+0.5z+0.5\right)+(-1.5z-1.5)\omega_{\text{$\mu$0}}^{2}+z(1.5z+3)+1.5\right){}^{\frac{-0.75\omega_{\text{$\mu$0}}-0.75q_{0}\omega_{\text{$\mu$0}}+0.75q_{0}+0.75}{0.5\omega_{\text{$\mu$0}}-1.q_{0}\omega_{\text{$\mu$0}}+1.5}}\\\
\\\
\alpha_{3}&=&\left(q_{0}\left(3.75\,-1.5q_{0}\right)+5.25\right)\omega_{\text{$\mu$0}}+\left(-0.750q_{0}^{2}-3.75\right)\omega_{\text{$\mu$0}}^{3}-1.5\left(q_{0}+1\right){}^{2}\omega_{\text{$\mu$0}}^{2}+8.88^{-16}(q_{0}+1)\\\
\\\
\alpha_{4}&=&\tan^{-1}\left(\frac{\left(0.5\,-1.q_{0}\right)\omega_{\text{$\mu$0}}-1.5\omega_{\text{$\mu$0}}^{2}+3}{\sqrt{\omega_{\text{$\mu$0}}\left(\omega_{\text{$\mu$0}}\left(\left(-2.25\omega_{\text{$\mu$0}}-1.5\right)\omega_{\text{$\mu$0}}-0.25\right)+q_{0}^{2}\left(-\omega_{\text{$\mu$0}}\right)+q_{0}\left(\omega_{\text{$\mu$0}}\left(3.\omega_{\text{$\mu$0}}+1.\right)-8.88^{-16}\right)+4.44^{-16}\right)+1.78^{-15}}}\right)\\\
\\\
\alpha_{5}&=&c\sqrt{\omega_{\text{$\mu$0}}\left(\omega_{\text{$\mu$0}}\left(\left(-2.25\omega_{\text{$\mu$0}}-1.5\right)\omega_{\text{$\mu$0}}-0.25\right)-q_{0}^{2}\omega_{\text{$\mu$0}}+q_{0}\left(\omega_{\text{$\mu$0}}\left(3.\omega_{\text{$\mu$0}}+1.\right)-8.88^{-16}\right)+4.44^{-16}\right)+1.77^{-15}}\\\
c&=&\left(0.5\omega_{\text{$\mu$0}}-q_{0}\omega_{\text{$\mu$0}}+1.5\right)\\\
\\\
\alpha_{6}&=&3.\left(-1.25\omega_{\text{$\mu$0}}^{3}-0.5\omega_{\text{$\mu$0}}^{2}+1.75\omega_{\text{$\mu$0}}+q_{0}^{2}\left(\omega_{\text{$\mu$0}}^{2}-0.5\omega_{\text{$\mu$0}}-0.5\right)\omega_{\text{$\mu$0}}\right)\\\
&&+3\left(q_{0}\left(-0.25\omega_{\text{$\mu$0}}^{3}-\omega_{\text{$\mu$0}}^{2}+1.25\omega_{\text{$\mu$0}}+2.96^{-16}\right)+2.96^{-16}\right)\\\
\\\
\alpha_{7}&=&\tan^{-1}\left(\frac{-1.5\omega_{\text{$\mu$0}}^{2}+\omega_{\text{$\mu$0}}\left(q_{0}(-2.z-1.)+1.z+0.5\right)+3.z+3.}{\sqrt{-2.25\omega_{\text{$\mu$0}}^{4}+\left(3.q_{0}-1.5\right)\omega_{\text{$\mu$0}}^{3}+\left(-1.q_{0}^{2}+1.q_{0}-0.25\right)\omega_{\text{$\mu$0}}^{2}+\left(4.44^{-16}-8.88^{-16}q_{0}\right)\omega_{\text{$\mu$0}}+1.77^{-15}}}\right)\\\
\\\
\alpha_{8}&=&\left(0.5\omega_{\text{$\mu$0}}-q_{0}\omega_{\text{$\mu$0}}+1.5\right)\\\
&&\sqrt{-2.25\omega_{\text{$\mu$0}}^{4}+\left(3q_{0}-1.5\right)\omega_{\text{$\mu$0}}^{3}+\left(-q_{0}^{2}+q_{0}-0.25\right)\omega_{\text{$\mu$0}}^{2}+\left(4.44^{-16}-8.88^{-16}q_{0}\right)\omega_{\text{$\mu$0}}+1.77^{-15}}\end{array}$

##  4 Estimation of $H_{0}$, $q_{0}$ and $\omega_{\mu 0}$ from A Observed
Hubble Data Set:

In this section we will estimate the present values of $H_{0}$, $q_{0}$ and
$\omega_{\mu 0}$ statistically from a data set of $77$ observed values of H at
different red shifts using (a) The cosmic chronometric method, (b) the BAO
signal in galaxy distribution and (c) the BAO signal in the Ly$\alpha$ forest
distribution alone or cross-correlated with QSOs [72]$-$[87]. We compare the
data set values from those obtained theoretically from Eq. (19) by forming the
following chi square function of $H_{0}$, $q_{0}$ and $\omega_{\mu 0}$.

$\chi^{2}(H_{0},q_{0},\omega{\mu
0})=\frac{1}{76}\sum\limits_{i=1}^{77}\frac{[Hth(z_{i},H_{0},q_{0},\omega{\mu
0})-H_{ob}(z_{i})]^{2}}{\sigma{(z_{i})}^{2}},$ (20)

These parameters are estimated by getting the minimum value of chi squared for
the values of the parameters taken in the range ( $H_{0}$,65$-$75), ($q_{0}$,
-0.60 $-$ -0.40) and ($\omega_{\mu 0}$, -0.90 $-$ -0.70) . It is found that

$\left\\{H_{0}=69.2163,q_{0}=-0.551698,\omega_{\text{$\mu$0}}=-0.759826\right\\}$

for minimum chi squire $\chi 2$ =0.685263 which is a a good fit. The fit will
be more clear from the following error bar and livelihood plots in figure $1$.

Figures $1(a)$ and $1(b)$ describe the growth of $H$ and
$H/(1+z)=\dot{a}/a_{0}$ over red shift $z$, respectively. H is an increasing
function of z. These figures also show the closeness of the theoretical graph
to the observed values. Figures $1(c)$, $1(d)$ and $1(e)$ are likelihood
probability curves for $H_{0}$, $q_{0}$ and $\omega_{\mu 0}$. Estimated values
on the basis of the 77 OHD data set are
$\left\\{H_{0}=69.2163,q_{0}=-0.551698,\omega_{\text{$\mu$0}}=-0.759826\right\\}$.

(a)  (b)  (c)  (d)
(e)

Figure 1: Figures (a) and (b) are the error bar plots for Hubble parameter $H$
and expansion rate $H/(1+z)=\dot{a}/a_{0}$ over red shift $z$, respectively.
Figures (c), (d) and (e) are likelihood probability curves for $H_{0}$,
$q_{0}$ and $\omega_{\mu 0}$.

Figures $2(a)$, $2(b)$, and $2(c)$ are the 1$\sigma$, 2$\sigma$ and 3$\sigma$
confidence region plots for the pair of parameters ($H_{0}$, $q_{0}$ ),
($H_{0}$, $\omega_{\mu 0}$) and ($q_{0}$, $\omega_{\mu 0}$). The estimated
value points
$(H_{0}=69.2163,q_{0}=-0.551698)$,$(H_{0}=69.2163,\omega_{\text{$\mu$0}}=-0.759826)$
and $(q_{0}=-0.551698,\omega_{\text{$\mu$0}}=-0.759826)$ are red spotted.
These are just there to show that our estimated values are within the
statistically specified regions.

(a)  (b)  (c)

Figure 2: Figures (a), (b) and (c) are the 1$\sigma$, 2$\sigma$ and 3$\sigma$
confidence region plots for the pair of parameters ($H_{0}$, $q_{0}$ ),
($H_{0}$, $\omega_{\mu 0}$) and ($q_{0}$, $\omega_{\mu 0}$). The estimated
value points are red spotted.

##  5 Distance Modulus and Apparent Magnitude for the Model:

The luminosity distance ($d_{L}$) plays a very important parameter for
observing distances of luminous objects like standard kindles, population I
and II stars. The luminosity distance of the SN Ia supernovae was found to be
more than expected, which has given rise to the concept of the accelerating
universe [11, 12]. $d_{L}$ is given by:

$\mu(z)=m_{b}-M=5Logd_{L}(z)+\mu_{0},$ (21)

where, $M$ and $m_{b}$ are the absolute and apparent magnitudes, respectively.
$\mu$ is called distance modulus. $\mu_{0}$ and $d_{L}$ are defined by

$\mu_{0}=25+5Log\big{(}\frac{c}{H_{0}}\big{)},$ (22)

and

$d_{L}(z)=(1+z)H_{0}\int_{0}^{z}{\frac{1}{H(z*)}dz*}.$ (23)

As we have already obtained expressions for H, we can determine $d_{L}$ and
$\mu(z).$ We can also obtain expressions for apparent magnitude $m_{b}$ from
these expressions. For this we use two important observations (i.) The
absolute magnitude of all standard candles are assumed to be the same, and
(ii) The luminosity distance of very low red shift SNIa are approximated as
$d_{L}=\frac{cz}{H_{0}}.$ Using these facts and taking a low red shift
supernova with red shift $z=0.014$ and $m_{b}=14.57$, we find $M=-19.30.$ With
this, we obtain the expression for the apparent magnitude as follows:

$m_{b}=5Log(1+z)H_{0}\int_{0}^{z}{\frac{1}{H(z*)}dz*}+5Log\left(\frac{c}{H_{0}}\right)\mu_{0}+5.70$
(24)

##  6 Estimation of Model parameters from distance modulus and apparent data
sets:

We use two standard data sets (i) 580 SN Ia data set of distance modulus (DM)
in the red shift range $0\leq z\leq 1.5$ (ii) 66 pantheon data set of SN Ia
apparent magnitude (AP) comprising 40 bin plus 16 high red shift data in the
range $0.014\leq z\leq 2.26$. We consider theoretical results of distance
modulus and apparent magnitude as obtained from Eqs. (21) and (24), as
function of model parameters $H_{0}$, $q_{0}$ and $\omega_{\mu 0}$. In these
expressions we take red shifts from the data sets. Thus we form a parallel
data set of theoretical results. With these, we form two chi squire functions
separately for the DM and AP as follows:

$\chi^{2}(H_{0},q_{0},\omega_{\mu 0})=\sum\limits_{i=1}^{580}\frac{[\mu
th(z_{i},H_{0},q_{0},\omega_{\mu 0})-\mu
ob(z_{i})]^{2}}{\sigma{(z_{i})}^{2}},$ (25)

and

$\chi^{2}(H_{0},q_{0},\omega_{\mu
0})=\sum\limits_{i=1}^{66}\frac{[m_{b}th(z_{i},H_{0},q_{0},\omega_{\mu
0})-m_{b}ob(z_{i})]^{2}}{\sigma{(z_{i})}^{2}},$ (26)

If the parameters $H_{0},q_{0}$ and $\omega_{\mu 0}$ be given in the range (
$H_{0}$, 65$-$75), ($q_{0}$, -0.60 $-$ -0.40) and ($\omega_{\mu 0}$, -0.90 $-$
-0.70), we find for the $\mu$ data set,
$~{}H_{0}=70.0749,~{}q_{0}=-0.628623~{}~{}\text{and}~{}~{}\omega_{\text{$\mu$0}}=-0.804005.$
$\text{for minimum}~{}~{}\chi^{2}=562.245$, whereas for AP data set it is
$H_{0}=71.7266,,~{}q_{0}=-0.602399~{}\text{and}~{}~{}\omega_{\text{$\mu$0}}=-0.798498$
$\text{for minimum}~{}~{}\chi^{2}=71.861214.$ More over if we combine Hubble
77 data set and 580 distant modulus data set make a pool of 657 data set we
get the following result:
$~{}H_{0}=69.949,~{}q_{0}=-0.61708~{}~{}\text{and}~{}~{}\omega_{\text{$\mu$0}}=-0.806833.$
$\text{for minimum}~{}~{}\chi^{2}=615.311.$ We present the following table to
display all of our statistical findings:

Table 1:  Statistical estimations for model parameters $H_{0}$, $q_{0}$ and $\omega_{\mu 0}$ Datasets |  $H_{0}$ |  $q_{0}$ |  $\omega_{\text{$\mu$0}}$ |  $\chi^{2}$
---|---|---|---|---
77 $OHD$ H |  $69.2163$ |  $-0.551698~{}$ |  -0.759826 |  52.0805
580 D.M $\mu$ |  $70.0749$ |  $-0.628623~{}$ |  -0.804005 |  562.245
66 A.M $m_{b}$ |  $71.7266$ |  $-0.602399~{}$ |  -0.798498 |  71.861214
657 $OHD$ H + D.M $\mu$ |  $69.949$ |  $-0.61708~{}$ |  -0.806833 |  615.311

Figure $3(a)$ describes the growth of distant modulus $\mu$ over red shift
$`z^{\prime}$. The distant modulus is increasing function of red shift which
means that in the past distant modulus was more. It is gradually decreasing
over time. The figure also shows that theoretical graph passes near by through
the 580 vertical error lines whose middle points are observed values at
different red shifts. Figure $3(b)$, $3(c)$ and $3(d)$ are the 1$\sigma$,
2$\sigma$ and 3$\sigma$ confidence region plots for the pair of parameters
($H_{0}$, $q_{0}$ ), ($H_{0}$, $\omega_{\mu 0}$) and ($q_{0}$, $\omega_{\mu
0}$). The estimated value points $(H_{0}=70.0749,q_{0}=-0.628623)$,
$(H_{0}=70.0749,\omega_{\text{$\mu$0}}=-0.804005)$ and
$(q_{0}=-0.628623,\omega_{\text{$\mu$0}}=-0.804005)$ are red spotted. These
are just there to show that our estimated values are within the statistically
specified regions.

(a)  (b)  (c)  (d)

Figure 3: Figure (a) is the error bar plot for $\mu$ over $z$. Figures (b),
(c) and (d) are the 1$\sigma$, 2$\sigma$ and 3$\sigma$ confidence region plots
for ($H_{0}$, $q_{0}$ ), ($H_{0}$, $\omega_{\mu 0}$) and ($q_{0}$,
$\omega_{\mu 0}$). The estimated value points
$(H_{0}=70.0749,q_{0}=-0.628623)$,
$(H_{0}=70.0749,\omega_{\text{$\mu$0}}=-0.804005)$ and
$(q_{0}=-0.628623,\omega_{\text{$\mu$0}}=-0.804005)$ are red spotted.

Figures $4(a)$, $4(b)$ and $4(c)$ are likelihood probability curves for Hubble
$H_{0}$, deceleration $q_{0}$ and equation of state $\omega_{\mu 0}$
parameters. Estimated values on the basis of 580 SNIa D.M.($\mu$) data set
$H_{0}=70.0749,q_{0}=-0.628623,\omega_{\text{$\mu$0}}=-0.804005$ are at the
peak.

(a)  (b) (c)

Figure 4: Figures (a), (b) and (c) are likelihood probability curves for
$H_{0}$, $q_{0}$ and $\omega_{\mu 0}$ parameters. Estimated values on the
basis of 580 SNIa DM ($\mu$) data set
$H_{0}=70.0749,q_{0}=-0.628623,\omega_{\text{$\mu$0}}=-0.804005$

Figure $5(a)$ describes the growth of AM $m_{b}$ over red shift $`z^{\prime}$.
The AM is increasing function of red shift which means that in the past DM was
more. It is gradually decreasing over time. The figure also shows that
theoretical graph passes near by through the 66 vertical error lines whose
middle points are observed values of AP at different red shifts. Figure
$5(b),5(c)~{}\text{and}~{}5(d)$ are the 1$\sigma$, 2$\sigma$ and 3$\sigma$
confidence region plots for the pair of parameters ($H_{0}$, $q_{0}$ ),
($H_{0}$, $\omega_{\mu 0}$) and ($q_{0}$, $\omega_{\mu 0}$). The estimated
value points $(H_{0}=71.7266,q_{0}=-0.602399)$,
$(H_{0}=71.7266,\omega_{\text{$\mu$0}}=-0.798498)$ and
$(q_{0}=-0.602399,\omega_{\text{$\mu$0}}=-0.798498)$ are red spotted.

(a)  (b) (c) (d)

Figure 5: Figure (a) is the error bar plot for AM over $z$. Figures (b), (c)
and (d) are the 1$\sigma$, 2$\sigma$ and 3$\sigma$ confidence region plots for
($H_{0}$, $q_{0}$ ), ($H_{0}$, $\omega_{\mu 0}$) and ($q_{0}$, $\omega_{\mu
0}$). The estimated value points$(H_{0}=71.7266,q_{0}=-0.602399)$,
$(H_{0}=71.7266,\omega_{\text{$\mu$0}}=-0.798498)$ and
$(q_{0}=-0.602399,\omega_{\text{$\mu$0}}=-0.798498)$ are red spotted.

Figures $6(a)$, $6(b)$ and $6(c)$ are likelihood probability curves for Hubble
$H_{0}$, deceleration $q_{0}$ and equation of state $\omega_{\mu 0}$
parameters. Estimated values on the basis of 66 Pantheon apparent modulus data
set $H_{0}=71.7266,q_{0}=-0.602399,\omega_{\text{$\mu$0}}=-0.798498$ are at
the peak.

(a) (b) (c)

Figure 6: Figures (a), (b) and (c) are likelihood probability curves for
$H_{0}$, $q_{0}$ and $\omega_{\mu 0}$ parameters. Estimated values
$H_{0}=71.7266,q_{0}=-0.602399,\omega_{\text{$\mu$0}}=-0.798498$, which are
obtained on the basis of 66 Pantheon apparent modulus data set, are at the
peaks.

###  6.1 Determination of the present values of Energy parameters
$\Omega_{m0}$ and $\Omega_{\text{$\mu$0}}$:

In sections $4$ and $6$, we have estimated the present values of $H_{0}$,
$q_{0}$, and $\omega_{\mu 0}$ on the basis of four data sets which are
described in Table-1. From Eqs. (13) and (14), we obtain values of
$\mu=8\pi\lambda$ and $\omega_{m}$ for matter. From Eq. (9), we get energy
parameters $\Omega_{\mu 0}$ and $\Omega_{m0}$ as follows:

$\Omega_{m0}=\frac{8\pi\rho}{3H_{0}^{2}}=\frac{1}{1+\mu(3-\omega_{m0})};~{}~{}\Omega_{\text{$\mu$0}}=1-\Omega_{m0}$
(27)

Table 2: Present Value of Energy Parameters $\Omega_{m}$ and $\Omega_{\mu_{0}}$ Datasets |  $\mu$ |  $\omega_{m}$ |  $\Omega_{m0}$ |  $\Omega_{\text{$\mu$0}}$
---|---|---|---|---
77 $OHD$ H |  $0.620112$ |  $-0.571151~{}$ |  0.311089 |  0.688911
580 D.M $\mu$ |  $0.582203$ |  $-0.642996~{}$ |  0.320414 |  0.679586
66 A.M $m_{b}$ |  $0.437467$ |  $-0.633883~{}$ |  0.386145 |  0.613855
657 $OHD$ H + D.M $\mu$ |  $0.428288$ |  $-0.647688~{}$ |  0.39028 |  0.60972

In this table, we find that at present $\Omega_{\mu}$ is dominant and it is
nearly in the ratio 1:3 or 2:3 as per different observed data sets. We note
that $\Omega_{\mu}$ is due to the curvature and energy momentum dominance in
$f(R,T)$ gravity.

##  7 Plots of Deceleration, Jerk and Snap Parameters:

The jerk (j) and snap (s) parameters are related to the third and fourth order
derivatives of the scale factor. They play a very important role in examining
the instability of a cosmological model. The jerk also plays a role in
statefinder diagnostics. They are defined as: $j=\frac{\dddot{a}}{aH^{3}}$ and
$s=-\frac{\ddddot{a}}{aH^{4}}.$ j in the terms of q can be written as:

${}j(z)=q(z)+2{q(z)}^{2}+(1+z)\frac{dq(z)}{dz}.$ (28)

whereas s in terms of q and j is computed as:

${}s(z)=(3q(z)+2)j(z)+\frac{dj(z)}{dz}(1+z)$ (29)

Having estimated the present values of $H_{0}$, $q_{0}$, $\omega_{\mu 0}$ and
$\mu$ in sections $4$ and $6$, we can plot and analyse q, j and s which are
presented in the following figures.

(a) (b) (c)

Figure 7: Figures (a), (b) and (c) are plots for $q$, $j$ and $s$. Each plot
contains three curves which corresponds to the three data sets described in
sections (4) and (6). Transition red shifts for $q$ and $s$ are displayed. $j$
is always positive.

We make following observations from the three plots of Figure $7$

  * •

q and s show a transition from negative to positive, which indicates that the
universe is accelerating at present, and it was decelerating in the past.

  * •

The Transition red shifts are well within the observational results

  * •

j is always positive. Its present values are 1.42783, 1.6416 and 1.64327 as
per the OHD, SNIa and pantheon data sets. These values are greater than one,
which shows that this model behaves differently from the $\Lambda$CDM
concordance model where j=1.

##  8 Statefinder Diagnostic:

In this important section, we use a very useful diagnostic technique given by
Sahni et al. [41]. They have used a pair of statefinder parameters $(r,s)$
depending on the scale factor $a$. The purpose is to differentiate
cosmological models from the standard $\Lambda$CDM concordance model. The
techniques also describe the evolution of the models in the sense that it
indicates through which phases the model has passed through. $s(z)$ is defined
as follows:

$s(\text{z})=\frac{r(z)-1}{3\left(q(z)-\frac{1}{2}\right)},$ (30)

where $r$ is the jerk parameter.

We plot two parametric curves, one with $s$ against $r$, and the other with
$q$ against $r$.

(a) (b)

Figure 8: Figures (a) and (b) are plots of $s$ versus $r$ and $q$ versus $r$.
These show that at present our model is in quintessence. In the past too, it
was not Einstein De sitter as reflected in the figures.

We make following observations as reflected in the two plots of Figure $8$:

  * •

Fig (a) shows that the three parametric (s,r) curves based on the the
estimated values of the OHD, SNIa and phantom data sets meet at $\Lambda$CDM
point(1,0) from both directions, i.e., from the Chaplygin gas model to
quintessence.

  * •

At present, our model is in quintessence.

  * •

The Fig. (b) also tells us that our model does not approximate the
$\Lambda$CDM model, nor was it the Einstein-Desitter model in the past.

##  9 Density and Pressure in the Model:

From Eqs. (8) and (9), we get, $\rho$, $p$ and $\omega_{m}=\frac{p}{\rho}$,
for the matter as:

$8\pi\rho=\frac{H^{2}(2\mu(q+4)+3)}{(2\mu+1)(4\mu+1)}$ (31)

$8\pi p=\frac{H^{2}((6\mu+2)q-1)}{(2\mu+1)(4\mu+1)}$ (32)

and

$\omega_{m}=\frac{p}{\rho}=\frac{(6\mu+2)q-1}{2\mu(q+4)+3}$ (33)

We can express the density and pressure in a more convenient way as follows:

$\rho=\Omega_{m0}\rho_{c0}\frac{H^{2}(2\mu(q+4)+3)}{H_{0}^{2}(2\mu(q_{0}+4)+3)}$
(34)
$p=\omega_{m0}\Omega_{m0}\rho_{c0}\frac{H^{2}((6\mu+2)q-1)}{H_{0}^{2}((6\mu+2)q_{0}-1)},$
(35)

where $\rho_{c0}=\frac{3H_{0}^{2}}{8\pi G}$ is the critical density and it has
the value $1.88\times 10^{-29}h^{2}gm/cm^{3}$, h=$H_{0}/100$. The other
parameters like $\omega_{m0}$, $\Omega_{m0}$, $H_{0}$ and $q_{0}$ have been
estimated by us. So we can plot the matter density and pressure. From Eqs.
(34), (32) and (10), we can plot $\rho_{\mu}$ and $p_{\mu}$. We note that
these develop due to the curvature and trace of the energy momentum tensor
dominance in $f(R,T)$ gravity. The followings are the plots:

(a)  (b) (c) (d)

Figure 9: Fig(a) to (d) are plots for the matter and $\mu$ densities and
pressures. The $\mu$ pressure is negative, which is responsible for the
present day acceleration in the universe.

We make following observations from these the four plots of the Figure
${\color[rgb]{1,0,0}9}$.

  * •

We can compare the matter and $\mu$ densities in the following way:

$\rho_{\mu}=\frac{\Omega_{\mu}}{\Omega_{m}}\rho_{m}.$

At present the ratio of the two densities is obtained from Table-1 as
$2.22581,~{}2.125~{}\text{and}~{}1.60526$ as per the OHD, SNIa and Pantheon
data sets, respectively. This shows that quantitatively, the $\mu$ density is
more than the matter density.

  * •

The matter pressure is very small, i.e., dust.

  * •

The $\mu$ pressure is negative. This is responsible for the present day
acceleration in the universe.

  * •

Both matter density and pressure are increasing functions of red shift.

  * •

The $\mu$ density is also showing the same trend as that of the matter.

###  9.1 Effective density and Effective pressure:

Equation (12) may be written as

$H^{2}(1-2q)=-8\pi p_{eff},~{}~{}3H^{2}=8\pi\rho_{eff},$ (36)

where $p_{eff}=(p+p_{\mu})$ and $\rho_{eff}=(\rho+\rho_{\mu}).$ We can express
these in a more convenient way as follows:

$\rho_{eff}=\rho_{c0}\frac{H^{2}}{H^{2}_{0}}~{};~{}p_{eff}=\rho_{c0}\frac{H^{2}(2q-1)}{H^{2}_{0}}$
(37)

(a)  (b) (c)

Figure 10: Plots (a) to (c) provide an overall picture of our model. Both
pressure and equation of state parameter are showing a change from negative to
positive, and the density is increasing with red shift.

We make following observations from these the three plots of the Figure $10$:

  * •

The density is increasing decreasing with time. The present value of
$\rho_{eff}$ is equal to the critical density.

  * •

Both pressure and equation of state parameter are showing a change from
negative to positive, showing that our universe is accelerating, and in the
past it was decelerating.

###  9.2 Time versus Red Shift and Transitional times:

We can calculate the time of any event from the red shift through the
following transformation

$(t_{0}-t_{1})=\intop_{t_{1}}^{t_{0}}dt=\intop_{a_{1}}^{a_{0}}\frac{da}{aH}=\intop_{0}^{z_{1}}\frac{dz}{(1+z)H(z)},$
(38)

where $t_{0},~{}t_{1}$ is the present time and some past time, respectively.
We note that at present, t=$t_{0}$ and z=0. With the help of expression for
the Hubble parameter Eq. (18), we can plot the graph of time versus red shift.
This is given in the following figure $11$.

Figure 11: The figure describes time versus red shift relation.

We make the following observations from the plot of the Figure $11$:

  * •

The three plots are based on the estimations of the model parameters for the
three data sets (i) OHD (ii) SNIa and (iii) pantheon, and they are almost
identical.

  * •

As $H^{-1}_{0}=9.8h^{-1}_{0}\times 10^{9}$ yrs, where $h_{0}=H_{0}/100$, we
can calculate the transition time for the deceleration parameter, and they are
$4.92656\times 10^{9}$, $4.98347\times 10^{9}$. These correspond to the
transition red shifts $z_{t}=0.725,~{}0.741986~{}\text{and}~{}0.8029$.

##  10 Conclusion:

We can summarise our finds as follows:

  * •

We attempted to model an FLRW model filled with a perfect fluid in $f(R,T)$
gravity, where $f(R,T)$ is taken as $R$ \+ $\frac{\mu}{8\pi}$ T.

  * •

We developed two energy parameters $\Omega_{m}$ and $\Omega_{\mu}$ and used
the relation that $\Omega_{m}$ \+ $\Omega_{\mu}$=1. The parameter $\Omega_{m}$
is associated with the matter, whereas $\Omega_{\mu}$ is associated with
$f(R,T)$ gravity. We have statistically estimated that at present
$\Omega_{\mu}$ is dominant, and that the two energy densities are
approximately in the ratio 3:1 to 3:2.

  * •

We have also introduced $\mu$ pressure ($p_{\mu}$) and $\mu$ density
($\rho_{\mu}$) along with the matter pressure and density. We find that at
present the $\mu$ density ($\rho_{\mu}$) is dominant over the matter density
and they are nearly in the ratio 1:3 to 2:3. The negative $\mu$ pressure
($p_{\mu}$) is associated with the acceleration in the universe.

  * •

Our model parameters are the present values of the Hubble, deceleration and
equation of state parameters. These parameters were estimated with the help of
the three data sets: (i) 77 Hubble OHD data set (ii) 580 SNIa supernova
distance modulus data set and (iii) 66 pantheon SNIa which include high red
shift data in the range $0\leq z\leq 2.36$.

  * •

Our deceleration and snap parameters show transition from negative to positive
and the jerk is always positive. . We have performed a state finder diagnostic
of our model and found that our model is at present in quintessence.

  * •

All of our findings are displayed in Table-1 and Table-2. The results are well
within the observational findings.

  * •

All of our outcomes are expressed in terms of red shifts and we have presented
transformation to convert red shift into time. Accordingly we have obtained
transition times for our deceleration parameter. From the cosmological
analyses, we infer that modifying Einstein’s field equations by replacing
Ricci scalar $R$ by an arbitrary function $f(R,T)$ of $R$ ans trace $T$ of
energy momentum tensor in the Einstein-Hilbert action may lead to arrive at a
model which describe transition from deceleration to acceleration at late
time. This can be seen as a one step more ahead of the fundamental theory of
general relativity physics on gravitation.

## Acknowledgments

The authors (A. Pradhan & G.K. Goswami ) are grateful for the assistance and
facilities provided by the University of Zululand, South Africa during a visit
where a part of this article was completed.

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