#  The cosmological model in $f(R,T^{\phi})$ gravity with Scalar Field
conformity

J. K. Singh<EMAIL_ADDRESS>Department of Mathematics, Netaji Subhas
University of Technology, New Delhi-110 078, India  Akanksha Singh
<EMAIL_ADDRESS>Department of Mathematics, Netaji Subhas University
of Technology, New Delhi-110 078, India  Shaily<EMAIL_ADDRESS>Department of Mathematics, Netaji Subhas University of Technology, New
Delhi-110 078, India  J. Jena<EMAIL_ADDRESS>Department of
Mathematics, Netaji Subhas University of Technology, New Delhi-110 078, India

###### Abstract

The homogeneous and isotropic cosmological model in generalized
$f(R,T^{\phi})$ theories associated with scalar field is discussed, which is
motivated by the $f(R,T)$ theory of gravity studied by Harko et al. [1, 2].
The $f(R,T^{\phi})$ gravity can be explained as $f(R,T)$ gravity with a self-
interacting scalar field $\phi$, where $T^{\phi}$ is the trace of the energy-
momentum tensor. The parametrization of Hubble parameter $H(t)$ is taken as
$\alpha-\beta e^{-\gamma t}$, where $\alpha$, $\beta$ and $\gamma$ are
arbitrary constants such that $\alpha,\gamma>0$ and $\beta<0$. The model shows
no space-time singularity and the expansion of the universe will continue
forever, i.e., the future scenario of the universe attains Big Freeze. The
model predicts the moderate inflationary scenario at the time of the evolution
of the universe and it is consistent with $\Lambda$CDM in late times. The
consistency of the model has also been examined using recent observational
Hubble dataset and supernovae dataset. Finally, the physical features of the
model have been discussed in some detail.

PACS number: 04.50.-h, 98.80.-k.

Keywords: FLRW Universe, $f(R,T^{\phi})$ gravity, Parametrization, Big Freeze,
Dark energy, Statefinder diagnostic

##  I  Introduction

The cosmic triangle provided a format for depicting the universe’s past,
present, and future states. The creation of structures in the universe during
the era when matter predominated necessitates a period of decelerated
expansion in which gravity ought to be the dominant factor. Cosmic
acceleration is a late-time occurrence. The acceleration suggests the presence
of cosmic dark energy, which accelerates the expansion by outpacing the
gravitational self-attraction of matter. The current accelerated expansion
phase of the universe is indicated by the findings from cosmic microwave
background radiation, large scale structure, and supernovae surveys [3, 4, 5,
6, 7]. By using only supernova data, Riess et al. [8] presented proof for the
presence of a shift from deceleration to acceleration with a certainty of
greater than $99\%$.

The dynamics of the universe may be considerably influenced by exotic types of
energy, even though it is known that the universe contains a substantial
portion of ordinary matter, which slows the expansion. General relativity can
account for cosmic acceleration by proposing the inexplicable cosmic fluid
known as dark energy, which has a high negative pressure. The Einstein
cosmological constant $\Lambda$ [9, 10], which captures the contribution of
vacuum energy to the spatial curvature of space-time, is apparently the most
obvious choice for dark energy. Some authors have used the reconstruction
formula to describe the dark energy models [11, 12]. The apparent magnitude-
redshift relationship has been discussed using Type Ia supernovae [10, 13].
The basic underlying premise of dark energy models is that Einstein’s General
Relativity is the right gravity theory. However, it is possible that both
cosmic acceleration and dark matter are indicators of a disruption in our
knowledge of gravitational principles, prompting one to investigate the idea
of modifying the Hilbert-Einstein Lagrangian. Modified gravity generally
appears to be quite appealing since it provides qualitative solutions to a
variety of fundamental issues about the late accelerating universe and dark
energy. The study of the models related to the dark energy and dark matter
concerns are the real challenges in the modern astrophysics and cosmology. The
dark energy models discussed in the modified theory of gravity can provide
viable solutions which are consistent with the recent observational dataset.

The standard Einstein Lagrangian can be modified by substituting any arbitrary
function of $R$ for the scalar curvature $R$, and this modified gravity is
known as $f(R)$ gravity. Furthermore, replacing the Ricci scalar $R$ with the
scalar torsion $T$ yields $f(T)$ gravity [14, 15], and replacing with the
gravitational constant $G$ yields $f(G)$ gravity [16, 17, 18, 19, 20, 21].
Several additional modified theories of gravity, such as $f(R,G)$ [22] and
$f(T,B)$ [23] have recently been investigated. A wide range of phenomena can
be produced from modified theories of gravity by adopting different functions.
Although $f(R)$ gravity is the most feasible alternative theory of gravity
[24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], it does not hold up to some
observational tests, such as the solar system regime [35, 36]. A much general
extension of $f(R)$ gravity could be $f(R,S_{m})$, where the matter Lagrangian
$S_{m}$ is a function of the energy momentum tensor’s trace $T$ and is taken
as $f(R,T)$ gravity. To account for exotic imperfect fluids and quantum
effects, the term $T$ is introduced. The $f(R,T)$ gravity can also explain the
late-time cosmic acceleration. In order to address the problems raised by
$f(R)$ gravity, some observational tests [37, 38] have been applied to
$f(R,T)$ gravity. One can view some notable works [1, 2, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60] to fully
comprehend the $f(R,T)$ theory.

Recent observations suggest the existence of a scalar field which is presently
evolving on cosmological time scales. Numerous fundamental theories also
report the existence of the scalar field $\phi$, which encourages us to
explore the scalar field’s dynamical characteristics in cosmology. There have
been many different scalar-field cosmological models proposed thus far [61,
62, 63, 64, 65, 66, 67, 68, 69]. A dark energy component with negative
pressure is causing the universe to expand at an accelerated rate right now.
The dark energy component of the universe is often interpreted either in terms
of a cosmological constant or as a scalar field. Even though some research
suggests that dark energy is a cosmological constant, it is more generally
modeled as a quintessence: a scalar field rolling down a flat potential [64,
70, 71, 72, 73, 74]. Distinctively, quintessence models exhibiting ‘tracker’
properties enable the scalar field to dominate the current universe regardless
of the initial conditions, avoiding the coincidence and fine-tuning issues
[75, 76]. Johari [77] has introduced the concept of integrated tracking, which
essentially shows that the tracker potentials follow a definite path of
evolution of the universe, in compatibility with the observational
constraints. In the literary works, various models of the original
quintessence theory have been discussed. A few of these models include the
possibility of a scalar field evolution driven by a non-canonical kinetic term
[78] and a non-minimal coupling between quintessence and dark matter [74, 79,
80, 65]. Having an unclumped form of energy density permeating the entire
cosmos is the only way to achieve a flat universe and, consequently, account
for the CMB measurements, if we accept the estimates of matter density derived
from observations of clusters [81]. The idea that the unknown, unclumped
energy is due to a scalar field $\phi$ that has not yet reached its ground
state, which is called dynamical lambda or quintessence, has received much
attention [70, 64, 82, 83]. In contrast to the $\Lambda$CDM, which is static
and never changes, the quintessence model is dynamic and changes over time
[67, 64]. The spintessence [84], the k-essence [85, 86], the quintom [87], and
the tachyon [88, 89] are some additional scalar fields dark energy models that
have been presented. The $f(R,T^{\phi})$ technique simply presents the
$f(R,T)$ gravity as a function of $R$ and the trace of the energy-momentum
tensor of a self-interacting scalar field $\phi$, while the matter field
energy-momentum tensor gets entered into the field equations in a general
manner. By incorporating a scalar field, the $f(R,T^{\phi})$ theory was
accounted for in [38]. The $T^{\phi}$ term makes an appropriate contribution
to the paper, and as a result, even for regimes where $T=0$, the theory can be
separated from the $f(R)$ gravity.

The paper is structured as follows: In section II, we discuss $f(R,T^{\phi})$
theory of gravity and the solutions of the Einstein field equations with some
physical features of the model. Section III covers some specific geometrical
interpretations of the model using error bar plots, statefinder diagnostic
tools, the massless scalar potential $V(\phi)$, the quintessence-like and
phantom-like scalar field kinetic term $\dot{\phi}^{2}$ during the cosmic
evolution. Finally, we conclude our results in Section IV.

##  II  Field equations and cosmic evolution in $f(R,T^{\phi})$ gravity

The following action results in the gravitational field equations in
$f\left(R,T^{\phi}\right)$ gravity.

$\displaystyle{I=\frac{1}{2}\int\left(f\left(R,T^{\phi}\right)+2L_{\phi}\right)\sqrt{-g}~{}dx^{4},}$
(1)

where $f\left(R,T^{\phi}\right)=-\left(R+mR^{2}+nT^{\phi}\right)$ and
$L_{\phi}=-\left(\frac{\epsilon}{2}~{}\phi_{,\mu}\phi^{,\mu}-V\left(\phi\right)\right)$.

Here, we have used $8\pi G=1$.

We vary action in Eq. (1) w.r.t. $g_{\mu\nu}$ and get the following equation

$f_{R}\left(R,T^{\phi}\right)R_{\mu\nu}-\frac{1}{2}f\left(R,T^{\phi}\right)g_{\mu\nu}+\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)f_{R}\left(R,T^{\phi}\right)=T_{\mu\nu}-f_{T^{\phi}}\left(R,T^{\phi}\right)\left(T_{\mu\nu}+\Theta_{\mu\nu}\right),$
(2)

where $\Box$ is the d’ Alembert operator defined by
$\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ and $\nabla_{\mu}$ indicates the
covariant derivative w.r.t. $g_{\mu\nu}$ associated with the symmetric Levi-
Civita connection.

$\displaystyle
T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}L_{\phi}\right)}{\delta
g^{\mu\nu}}=g_{\mu\nu}L_{\phi}-2\frac{\delta L_{\phi}}{\delta g^{\mu\nu}}$ (3)

and

$\displaystyle\Theta_{\mu\nu}=g^{\alpha\beta}\frac{\delta
T_{\alpha\beta}}{\delta
g^{\mu\nu}}=-2T_{\mu\nu}+g_{\mu\nu}L_{\phi}-2g^{\alpha\beta}\frac{\delta^{2}L_{\phi}}{\delta
g_{\mu\nu}\delta g^{\alpha\beta}}.$ (4)

We have

$\displaystyle
L_{\phi}=-\left(\frac{\epsilon}{2}~{}\phi_{,\mu}\phi^{,\mu}-V(\phi)\right)=-\frac{1}{2}\epsilon\dot{\phi}^{2}+V(\phi),$
(5)

where dot represents the differentiation of the function w.r.t. cosmic time
$t$.

Using Eq. (5) in Eq. (4) we get,

$\Theta_{\mu\nu}=-2T_{\mu\nu}+g_{\mu\nu}\left(-\frac{1}{2}\epsilon\dot{\phi}^{2}+V\left(\phi\right)\right).$
(6)

Using Eq. (6) in Eq. (2) we get,

$f_{R}\left(R,T^{\phi}\right)R_{\mu\nu}-\frac{1}{2}f\left(R,T^{\phi}\right)g_{\mu\nu}+\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)f_{R}\left(R,T^{\phi}\right)=T_{\mu\nu}+f_{T^{\phi}}\left(R,T^{\phi}\right)\left(T_{\mu\nu}+g_{\mu\nu}\left(\frac{1}{2}\epsilon\dot{\phi}^{2}-V\left(\phi\right)\right)\right).$
(7)

Contracting Eq. (7) w.r.t. $g^{\mu\nu}$ we get,

$Rf_{R}\left(R,T^{\phi}\right)-2f\left(R,T^{\phi}\right)+3\Box
f_{R}\left(R,T^{\phi}\right)=T^{\phi}+f_{T^{\phi}}\left(R,T^{\phi}\right)\left(T^{\phi}+4\left(\frac{1}{2}\epsilon\dot{\phi}^{2}-V\left(\phi\right)\right)\right).$
(8)

From Eqs. (7) and (8) we get,

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{1}{f_{R}\left(R,T^{\phi}\right)}~{}\left(T_{\mu\nu}+T_{\mu\nu}^{{}^{\prime}}\right),$
(9)

where

$T_{\mu\nu}^{{}^{\prime}}=f_{T^{\phi}}\left(R,T^{\phi}\right)\left(T_{\mu\nu}+g_{\mu\nu}\left(\frac{1}{2}\epsilon\dot{\phi}^{2}-V\left(\phi\right)\right)\right)+\frac{1}{2}\left(f\left(R,T^{\phi}\right)-Rf_{R}\left(R,T^{\phi}\right)\right)g_{\mu\nu}+\left(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box\right)f_{R}\left(R,T^{\phi}\right).$
(10)

We have

$f\left(R,T^{\phi}\right)=-\left(R+mR^{2}+nT^{\phi}\right).$ (11)

When using the homogeneous, isotropic, $f(R,T^{\phi})$ gravity, the spatially
flat FLRW model’s metric is,

$ds^{2}=dt^{2}-a^{2}\left(t\right)\left(dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\right).$ (12)

For this given metric,

$\displaystyle R_{11}=2\dot{a}^{2}+a\ddot{a},$ $\displaystyle
R_{22}=r^{2}\left(2\dot{a}^{2}+a\ddot{a}\right)=r^{2}R_{11},$ $\displaystyle
R_{33}=r^{2}\sin^{2}\theta\left(2\dot{a}^{2}+a\ddot{a}\right)=r^{2}\sin^{2}\theta
R_{11},$ $\displaystyle R_{44}=-3\frac{\ddot{a}}{a},$ $\displaystyle
R=-6\left(\frac{\dot{a}^{2}}{a^{2}}+\frac{\ddot{a}}{a}\right).$ (13)

The energy$-$momentum tensor of a scalar field $\phi$ with self-interacting
scalar potential $V\left(\phi\right)$ for a perfect fluid is

$T_{\mu\nu}=\epsilon\phi_{,\mu}\phi_{,\nu}-g_{\mu\nu}\left[\frac{\epsilon}{2}\phi_{,\sigma}\phi^{,\sigma}-V\left(\phi\right)\right].$
(14)

Using Eq. (14) we get the following values,

$\displaystyle
T_{11}=a^{2}\left(\frac{\epsilon}{2}\dot{\phi}^{2}-V\left(\phi\right)\right),$
$\displaystyle
T_{22}=a^{2}r^{2}\left(\frac{\epsilon}{2}\dot{\phi}^{2}-V\left(\phi\right)\right)=r^{2}T_{11},$
$\displaystyle
T_{33}=a^{2}r^{2}\sin^{2}\theta\left(\frac{\epsilon}{2}\dot{\phi}^{2}-V\left(\phi\right)\right)=\sin^{2}\theta~{}T_{22},$
$\displaystyle T_{44}=\frac{\epsilon}{2}\dot{\phi}^{2}+V\left(\phi\right),$
$\displaystyle T^{\phi}=-\epsilon\dot{\phi}^{2}+4V\left(\phi\right).$ (15)

Using Eqs. (9), (11), (12), (II), (II), we obtain the following field
equations

$\displaystyle
2\dot{H}+3H^{2}-6m\left(26\dot{H}H^{2}+2H\dddot{H}+12H\ddot{H}+9\dot{H}^{2}\right)=\left(1-n\right)\frac{\epsilon}{2}\dot{\phi}^{2}+\left(2n-1\right)V\left(\phi\right),$
$\displaystyle
3H^{2}-18m\left(6\dot{H}H^{2}+2H\ddot{H}-\dot{H}^{2}\right)=\left(n-1\right)\frac{\epsilon}{2}\dot{\phi}^{2}+\left(2n-1\right)V\left(\phi\right).$
(16)

Resolving equations in (II) provides the following value for $V$ and
$\dot{\phi}^{2}$

$\displaystyle
V\left(\phi\right)=\frac{6H^{2}+2\dot{H}-12m\left(22\dot{H}H^{2}+9H\ddot{H}+3\dot{H}^{2}+\dddot{H}\right)}{2(2n-1)},$
$\displaystyle\dot{\phi}^{2}=\frac{-2\dot{H}+12m\left(4\dot{H}H^{2}+3H\ddot{H}+6\dot{H}^{2}+\dddot{H}\right)}{(n-1)\epsilon}.$
(17)

In the context of FLRW cosmology, the energy density $\rho_{\phi}$ and
pressure $p_{\phi}$ are given as functions of the scalar field $\phi$ by [90,
91]

$\displaystyle\rho_{\phi}=\frac{1}{2}\epsilon\dot{\phi}^{2}+V\left(\phi\right),$
$\displaystyle p_{\phi}=\frac{1}{2}\epsilon\dot{\phi}^{2}-V\left(\phi\right).$
(18)

Each equation’s right-hand side is written for a field with an arbitrary
potential $V(\phi)$.

Using Eqs. (II) in (II) we get

$\displaystyle\rho_{\phi}=\frac{-n\dot{H}+3\left(n-1\right)H^{2}+6m\left(\left(-14n+18\right)\dot{H}H^{2}+\left(-3n+6\right)H\ddot{H}+\left(9n-3\right)\dot{H}^{2}+n\dddot{H}\right)}{\left(n-1\right)\left(2n-1\right)},$
$\displaystyle
p_{\phi}=\frac{\left(-3n+2\right)\dot{H}-3\left(n-1\right)H^{2}+6m\left(\left(30n-26\right)\dot{H}H^{2}+\left(15n-12\right)H\ddot{H}+\left(15n-9\right)\dot{H}^{2}+\left(3n-2\right)\dddot{H}\right)}{\left(n-1\right)\left(2n-1\right)}.$
(19)

The parametrized form of Hubble parameter $H$ used to get the solution of the
field equations is as follows:

$H(t)=\alpha-\beta e^{-\gamma t},$ (20)

where we have arbitrary constants $\alpha$, $\beta$ and $\gamma$ such that
$\alpha>0$, $\beta<0$, $\gamma>0$. The motivation for using this
parametrization is to discuss the inflationary scenario at the early evolution
of the universe and the ultimate fate of the universe for which the current
observations show that the expansion of the universe is accelerating and
attains the Big Freeze condition in late times. The constraints imposed on the
parametrization for the Hubble parameter $H$ are to ensure that the overall
function always returns a positive value and is a decreasing function of time
$t$.

From Eq. (20), the cosmic scale factor $a(t)$ and the deceleration parameter
$q(t)$ can be calculated as

$a(t)=e^{\alpha t+\frac{\beta}{\gamma}e^{-\gamma t}},\mbox{\hskip
20.00003pt}q(t)=-1-\frac{\beta\gamma e^{\gamma t}}{\left(\beta-\alpha
e^{\gamma t}\right)^{2}}.$ (21)

Fig. 1 depicts a visual representation of the scale factor $a$, Hubble
parameter $H$, and deceleration parameter $q$ with respect to cosmic time $t$.
We have chosen $\alpha$, $\beta$ as $0.05$ and $-1$ respectively.

(a)  (b)

(c)

Figure 1: Depiction of graphs of $a$, $H$ and $q$ vs. $t$ with $\alpha=0.05$
and $\beta=-1$. The model shows inflation at the early epoch and accelerating
in late times in all cases of $\gamma=0.2,\,0.3,\,1.1$.

According to Fig. 1(a), for $\gamma=0.2$ and $0.3$, the values of parameter
$a$ increase continuously as we approach the late universe, whereas for
$\gamma=1.1$, the value of scale factor $a$ increases, then decreases for some
time in the early universe, and finally increases as we approach the late
universe. This model is free of initial singularity because $a$ has a finite
non-zero value for $t=0$ for all of the $\gamma$ values mentioned, and this is
easily observable for $\gamma=0.3$ and $1.1$ from the plot embedded in the
Fig. 1(a), while for $\gamma=0.2$, it can be easily verified with the help of
value

$\lim_{t\to\ 0}a(t)=0.00673795.$

The Hubble parameter, $H$, aids in determining the rate of cosmic expansion.
The value of $H$ is positive for all values of $t$ for $\gamma=0.2,0.3,1.1$ as
shown in Fig. 1(b), and this suggests that the universe is expanding. As, for
all values of $\gamma$ mentioned in the model, the Hubble parameter $H(t)$
satisfies the following property

$\lim_{t\to\infty}H(t)=0.05.$

Therefore, we can conclude that our model will experience the Big Freeze
condition.

For all values of $\gamma$, the universe is expanding at an accelerated rate
with mild inflation and attains the de-Sitter regime $(q=-1)$ [92] in late
time, as shown in Fig. 1(c). The deceleration parameter $q$ shows transition
to an accelerating state for $\gamma=0.3$ and $1.1$ while exhibiting eternal
acceleration for $\gamma=0.2$.

The scalar field $\phi$ is independent of the space coordinates because the
metric space is homogeneous. So, we consider $\phi$ to be a function of $t$
only, and this can be used to get the time derivative $d\phi/dt=\dot{\phi}$,
whereas all the spatial components of the gradients, $\partial\phi/\partial
x,\partial\phi/\partial y$ and $\partial\phi/\partial z$, are identically
zero, which is consistent with the isotropic and expanding properties of the
model. From symmetry analysis, it is evident that the energy density
$\rho_{\phi}$ and the isotropic pressure $p_{\phi}$ together make up the
mechanical properties of the field.

(a)  (b)

(c)

Figure 2: Depiction of $\rho_{\phi}-t$, $p_{\phi}-t$ and $\omega-t$ plots with
$\alpha=0.05$, $\beta=-1$, $m=0.6$ and $n=0.7$. The scalar field energy
density $\rho_{\phi}$ and pressure $p_{\phi}$ approach to finite quantity as
$t\rightarrow\infty$. The EoS parameter $\omega\rightarrow-1$ i.e. the model
is consistent with $\Lambda$CDM in late times.

Fig. 2(a) provides the variation of energy density $\rho_{\phi}$ for
$\gamma=0.2,0.3$ and $1.1$ with respect to $t$. As time passes since the
universe creation, for $\gamma=0.2$ and $0.3$, the value of $\rho_{\phi}$
falls and gets closer to a finite positive value as time $t$ approaches
$\infty$ while, for $\gamma=1.1$, the value of $\rho_{\phi}$ decreases,
reaches its minimum, and then increases for a while in the early universe,
followed by a decrease in its value at present and late times of the universe,
it eventually approaches the same finite positive value as for the
$\gamma=0.2,0.3$. Although the idea that a non-minimally coupled scalar field
energy density can be negative has been understood for a long time [93, 94,
95, 96], the problematic ramifications have been brought to light by the works
of [97, 98]. Here, $\rho_{\phi}$ has negative values only for $\gamma=1.1$ for
a short interval of time at the early stage of evolution. The function
$\rho_{\phi}$ has values of $52.6391$, $68.4837$, and $59.7011$ for
$\gamma=0.2$, $0.3$, and $1.1$ at $t=0$, respectively, indicating that our
model is free of initial singularity.

The scalar field pressure $p_{\phi}$ for this model is shown in Fig. 2(b) for
all the values of $\gamma$. The scalar field pressure $p_{\phi}$ increases
monotonically and approaches a finite negative value as $t\rightarrow\infty$
in the case of $\gamma=0.2,0.3$, whereas in the case of $\gamma=1.1$,
$p_{\phi}$ increases from negative to positive, reaches its maximum value, and
finally decreases to the same finite negative value as for the
$\gamma=0.2,0.3$ in late times. Therefore, our model appears to be
accelerating both now and in the near future.

The equation of state (EoS) parameter explains the various cosmic regimes of
the evolution of the universe. Using Eqs. (II), we see the evolution of EoS
parameter $\omega=p_{\phi}/\rho_{\phi}$, for $\gamma=0.2,0.3$ and $1.1$ in
Fig. 2(c). In the case of $\gamma=0.2,0.3$, $\omega$ begins in the
quintessence region $\left(-1<\omega<0\right)$, crosses the quintom line
$\left(\omega=-1\right)$, enters into the phantom region, and then approaches
$\Lambda$CDM $\left(\omega=-1\right)$ in the late times. The model fluctuates
in short intervals at the early epoch for $\gamma=1.1$, but later exhibits
properties of a quintessence model and eventually tends towards the
$\Lambda$CDM. The model is consistent with $\Lambda$CDM in late times for all
cases.

##  III  Dynamics of the model

We depict the error bar plots using datasets $H(z)$ and $SNeIa$ to examine the
consistency of our model with $\Lambda$CDM for the cases of $\gamma=0.2,0.3$
and $1.1$. Also, we use other parameters like statefinder diagnostic, scalar
potential and $\dot{\phi}^{2}$, to distinguish our model from the different
models and to know more about the characteristics of our model.

###  III.1  Error bar plots using $H(z)$ and $SNeIa$ datasets

The Hubble parameter $H$ in terms of redshift $z$ for the model is as follows:

$H(z)=\alpha\left(1+ProductLog\left[-\frac{\beta\left(\frac{a_{0}}{1+z}\right)^{-\frac{\gamma}{\alpha}}}{\alpha}\right]\right)$
(22)

The distance modulus $\mu(z)$ plotted in Fig. 3(b) is defined as

$\displaystyle\mu(z)=m-M=5~{}log_{10}D_{L}(z)+\mu_{0},$ (23)

where $m$ and $M$ represent the apparent and absolute magnitudes of the object
respectively and

$D_{L}(z)=(1+z)c\int_{0}^{z}\frac{dz^{*}}{H(z^{*})},\mbox{\hskip
20.00003pt}\mu_{0}=25+5~{}log_{10}\left(\frac{c}{H_{0}}\right),$ (24)

are the luminosity distance and nuisance parameter respectively. Using the
Planck 2018 results [99], we have $H_{0}=67.4~{}km/s/Mpc$.

(a)  (b)

Figure 3: Error bar plots comparing our model with standard $\Lambda$CDM using
$H(z)$ and $SNeIa$ datasets respectively, with $\alpha=0.05$, $\beta=-1$. Our
model is best fitted with $\Lambda$CDM for $\mu(z)$ data at present for all
cases of $\gamma$.

In Fig. 3(a), the present model has been fitted using 77 points of $H(z)$
dataset [100] and union 2.1 compilation observational dataset [101] of 580
points has been used to fit the present model in Fig. 3(b). The Fig. 3(a) is
not well fitted so much but the Fig. 3(b) is better fitted when we compare our
model with $\Lambda$CDM using $H(z)$ and $SNeIa$ data respectively.

###  III.2  Statefinder diagnostic

The statefinder diagnostic is a geometrical analysis to describe the phenomena
of distinct dark energy (DE) models by the parameters $r$ and $s$, which are
defined as

$\displaystyle r=\frac{\dddot{a}}{aH^{3}},\mbox{\hskip
20.00003pt}s=\frac{r-1}{3(q-\frac{1}{2})},$ (25)

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

The sequence of parameters $\\{s,r\\}$ was proposed by Sahni et al. [102, 103]
and Alam et al. [104], where the variable represented by $r$ is identical to
the jerk parameter $j$. A specific linear combination of the jerk parameter
and the deceleration parameter $q$ is used as another variable known as
statefinder and represented by $s$. For the value of scale factor $a$ in Eq.
(21), the parameters $r$ and $s$ can be expressed in terms of $t$ as

$\displaystyle r=\frac{\alpha^{3}e^{3\gamma
t}-\beta^{3}-3\beta^{2}\left(\gamma-\alpha\right)e^{\gamma
t}-\beta\left(3\alpha^{2}-3\alpha\gamma+\gamma^{2}\right)e^{2\gamma
t}}{\left(\alpha e^{\gamma t}-\beta\right)^{3}},$ $\displaystyle
s=\frac{2\beta\gamma e^{\gamma
t}\left(3\beta+\left(\gamma-3\alpha\right)e^{\gamma t}\right)}{3\left(\alpha
e^{\gamma t}-\beta\right)\left(3\alpha^{2}e^{2\gamma
t}+3\beta^{2}+2\beta\left(\gamma-3\alpha\right)e^{\gamma t}\right)}.$ (26)

(a)  (b)

Figure 4: Depiction of $s-r$ and $q-r$ trajectories with $\alpha=0.05$ and
$\beta=-1$.

The $r$ and $s$ parameters are plotted in the three trajectories in the $s-r$
plane for all values of $\gamma$ as shown in Fig. 4(a). The arrows represent
the directions of the $s-r$ trajectories in the figure. The trajectory for
$\gamma=0.2$ starts from the Chaplygin gas region ($r>1,s<0$), proceeds
through the quintessence region ($r<1,s>0$) and then approaches the
$\Lambda$CDM ($r=1,s=0$). The trajectory for $\gamma=0.3$ starts from the
Chaplygin gas region, overlaps the line $r=1$ and progresses through the
quintessence region, and converges to $\Lambda$CDM finally. The curve for
$\gamma=1.1$ begins from the region $r>1,s>0$, moves through the quintessence
region, crosses the $\Lambda$CDM point and passes through the Chaplygin gas
region before tracing the same path again.

In the $q-r$ plane, Fig. 4(b) depicts the three trajectories with time for all
cases of $\gamma$. The trajectory for $\gamma=0.2$ completes its journey in
the region of $q<0$. The trajectories for $\gamma=0.3$ and $1.1$ begin in
$q<0$, move across the region $q>0$, and eventually complete their evolution
in the $q<0$ region. The $\gamma=0.3$ trajectory also passes through the SCDM
($r=1,q=1/2$). These trajectories cross the $\Lambda$CDM line, which is the
line $r=1$ parallel to the $q$-axis in the $q-r$ plane. All the trajectories
converge to steady state model (SS) ($r=1,q=-1$) at the end. Thus, our model
behaves like the steady state model based on the evolution of the trajectories
to SS.

###  III.3  Scalar Field Correspondence

The scalar potential $V\left(\phi\right)$ does not depend on $\epsilon$ as
given in Eq. (II). Now, it is very difficult to express the scalar field
$\phi$ in closed form, so the potential function $V\left(\phi\right)$ can not
be expressed in terms of $\phi$ explicitly. In Fig. 5 the graph of scalar
potential with time for some particular values of parameters shows that
$V\left(\phi\right)$ remains positive and decreasing for $\gamma=0.2$, $0.3$,
and settles to a positive constant in late times of the evolution of the
universe. For $\gamma=1.1$, the scalar potential $V\left(\phi\right)$
approaches a positive constant as $t\rightarrow\infty$, but it deviates to the
region of negative potential for a short span of time at the initial stage of
the evolution.

(a)  Figure 5: Depiction of graph of scalar potential vs. $t$ with
$\alpha=0.05$, $\beta=-1$, $m=0.6$ and $n=0.7$.

####  III.3.1  Quintessence Scalar Field $\left(\epsilon=+1\right)$

In the quintessence scalar field, Eq. (II) yields

$\dot{\phi}^{2}=\frac{-2\dot{H}+12m\left(4\dot{H}H^{2}+3H\ddot{H}+6\dot{H}^{2}+\dddot{H}\right)}{n-1}.$
(27)

We observe that $\dot{\phi}^{2}$ is negative during the evolution in the late
universe for $\gamma=0.2,0.3$ as shown in Fig. 6(a) and hence $\phi$ becomes
imaginary. Thus, the kinetic term $\left(\frac{1}{2}\dot{\phi}^{2}\right)$ is
negative for the quintessence scalar field in the late times, but it must be
positive. Therefore, the massless scalar field $\phi$ can not be evaluated in
cases of $\gamma=0.2,0.3$. However, $\dot{\phi}^{2}$ is negative only for a
short interval of time at the initial stage of the evolution for $\gamma=1.1$,
and then its value increases suddenly and eventually decreases to zero in late
times.

(a)  (b)

Figure 6: Depiction of graphs of $\dot{\phi}^{2}$ for $\epsilon=1$ and $-1$
respectively vs. $t$ with $\alpha=0.05$, $\beta=-1$, $m=0.6$ and $n=0.7$.

####  III.3.2  Phantom Scalar Field $\left(\epsilon=-1\right)$

In the phantom scalar field, Eq. (II) yields

$\dot{\phi}^{2}=\frac{-2\dot{H}+12m\left(4\dot{H}H^{2}+3H\ddot{H}+6\dot{H}^{2}+\dddot{H}\right)}{-(n-1)}.$
(28)

We observe that the kinetic energy $\dot{\phi}^{2}$ is valid for $\gamma=0.2$,
$0.3$ as shown in Fig. 6(b) except for a very short time interval at the
initial stage of the evolution. The value of $\dot{\phi}^{2}$ increases in the
beginning but decreases to zero in late times for $\gamma=0.2,0.3$. In
contrast, except for a brief period, $\dot{\phi}^{2}$ is negative during the
evolution initially and eventually converges to zero for $\gamma=1.1$.

##  IV  Conclusions

In this paper, we consider the isotropic homogeneous cosmological model, which
contains a massless scalar field $\phi$ obeying a potential $V(\phi)$. We have
taken specific values for the arbitrary constants $\alpha$, $\beta$, $\gamma$,
$m$ and $n$. The graphical representations of various parameters are drawn
after taking $\alpha=0.05$, $\beta=-1$, $\gamma=0.2,0.3$ and $1.1$, $m=0.6$,
$n=0.7$. We summarise our work for this model as follows:

  * •

The model has no space-time singularity as the expansion scale factor $a$
evolves from a finite size in the infinite past and increases exponentially
with cosmic time $t$.

  * •

The parametrized Hubble parameter $H$ shows expansion forever and the future
scenario of the universe attains Big Freeze.

  * •

The deceleration parameter $q$ exhibits inflation at the initial stage of
evolution for all values of $\gamma$ and transition from the early decelerated
expanding phase to the late time accelerated expanding phase for
$\gamma=0.3,\,1.1$. Also, the model shows eternal acceleration in the case of
$\gamma=0.2$.

  * •

The model starts with a finite positive value of the energy density of
massless scalar field $\rho_{\phi}$ and eventually decreases monotonically to
a finite positive quantity in late times, except for some short intervals of
time at the initial stage. Also, the model starts with a finite negative value
of the pressure of massless scalar field $p_{\phi}$ and eventually
monotonically increases to a finite negative quantity in late times, except
for some short intervals of time at the initial stage.

  * •

From Fig. 2(c), we observe that the model shows highly unstable behaviour at
the initial epoch for some values of $\gamma$, but the model shows consistency
with $\Lambda$CDM in the late times for all values of $\gamma$.

  * •

Our model is well fitted at the redshift $z\simeq 1$ only, but it deviates for
the values other than $z\simeq 1$ when we compare it with the $\Lambda$CDM
using the $H(z)$ dataset in the error bar plots. The model is well fitted with
the $\Lambda$CDM using the $SNeIa$ dataset for $\gamma=0.2$, but it deviates
from the $\Lambda$CDM for the values of redshift $z$ that are not close to $0$
in the cases of $\gamma=0.3$ and $1.1$.

  * •

The plotted statefinder diagnostic shows that the $s-r$ trajectories for
$\gamma<1$ start from the Chaplygin gas region, and all three trajectories
pass through the $\Lambda$CDM model. All three trajectories in the $q-r$ plane
in the form of loops pass through the point, indicating the steady state (SS)
model. In both the graphs, the trajectory for $\gamma=0.3$ passes through the
SCDM, while the other trajectories deviate from it.

  * •

The massless scalar potential $V(\phi)$ begins with a high potential at the
initial stage of evolution and gradually decreases until it approaches a very
small finite value for $\gamma<1$, but for $\gamma>1$, the potential $V(\phi)$
begins with a high potential and gradually decreases until it approaches its
minimum value, then increases up to a certain positive value, and finally
approaches a finite value after the gradual decrement. For the case
$\gamma=1.1$, the massless scalar potential $V(\phi)$ becomes negative for a
short period of time.

  * •

The quintessence-like scalar field kinetic term $\dot{\phi}^{2}$ violates all
over the time except for some intervals at the beginning of the cosmic
evolution for $\gamma<1$, but in the case of $\gamma>1$, it is valid
throughout the evolution except for some intervals at the beginning. On the
other hand, for the same values of $\gamma$, the phantom-like scalar field
kinetic term $\dot{\phi}^{2}$ behaves in the exact opposite manner of the
quintessence-like scalar field kinetic term $\dot{\phi}^{2}$.

Therefore, in this work, we conclude that our model in $f(R,T^{\phi})$ gravity
is a well explained model with the massless self-interacting scalar field
$\phi$. The model has no space-time singularity. The model is accelerated
expanding and attains the Big Freeze scenario in the future evolution of the
universe. The model predicts the moderate inflationary scenario at the time of
the evolution of the universe, and it is consistent with $\Lambda$CDM in late
times. Our model turns out viable at present-day and in late times
observations of $H(z)$ and $SNeIa$ dataset. Hence, we can say that our model
plays an important role to describe the early as well as late times of the
evolution of the universe.

## References

  * [1] T. Harko, F. S. N. Lobo, S. Nojiri and S. D. Odintsov, Phys. Rev. D 84, 024020 (2011).
  * [2] T. Harko, Phys. Rev. D 90, no.4, 044067 (2014).
  * [3] A. G. Riess et al. [Supernova Search Team], Astron. J. 116, 1009-1038 (1998).
  * [4] S. Perlmutter et al. [Supernova Cosmology Project], Astrophys. J. 517, 565-586 (1999).
  * [5] D. N. Spergel et al. [WMAP], Astrophys. J. Suppl. 148 (2003), 175-194.
  * [6] E. Komatsu et al. [WMAP], Astrophys. J. Suppl. 180, 330-376 (2009).
  * [7] P. Astier et al. [SNLS], Astron. Astrophys. 447, 31-48 (2006).
  * [8] A. G. Riess et al. [Supernova Search Team], Astrophys. J. 607, 665-687 (2004).
  * [9] S. M. Carroll, W. H. Press and E. L. Turner, Ann. Rev. Astron. Astrophys. 30, 499-542 (1992).
  * [10] B. P. Schmidt et al. [Supernova Search Team], Astrophys. J. 507, 46-63 (1998).
  * [11] S. Nojiri and S. D. Odintsov, J. Phys. Conf. Ser. 66, 012005 (2007).
  * [12] J. Matsumoto and S. Nojiri, Phys. Lett. B 687, 236-242 (2010).
  * [13] S. Perlmutter et al. [Supernova Cosmology Project], Nature 391, 51-54 (1998).
  * [14] G. R. Bengochea and R. Ferraro, Phys. Rev. D 79, 124019 (2009).
  * [15] R. Myrzakulov, Gen. Rel. Grav. 44, 3059-3080 (2012).
  * [16] S. Nojiri and S. D. Odintsov, Phys. Lett. B 631, 1-6 (2005).
  * [17] S. Nojiri, S. D. Odintsov and O. G. Gorbunova, J. Phys. A 39, 6627-6634 (2006).
  * [18] G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov and S. Zerbini, Phys. Rev. D 73, 084007 (2006).
  * [19] I. H. Brevik and J. Quiroga Hurtado, Int. J. Mod. Phys. D 16, 817-826 (2007).
  * [20] E. Elizalde, A. N. Makarenko, V. V. Obukhov, K. E. Osetrin and A. E. Filippov, Phys. Lett. B 644, 1-6 (2007).
  * [21] A. De Felice and S. Tsujikawa, Phys. Lett. B 675, 1-8 (2009).
  * [22] K. Bamba, S. D. Odintsov, L. Sebastiani and S. Zerbini, Eur. Phys. J. C 67, 295-310 (2010).
  * [23] S. Bahamonde, M. Zubair and G. Abbas, Phys. Dark Univ. 19, 78-90 (2018).
  * [24] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451-497 (2010).
  * [25] E. E. Flanagan, Class. Quant. Grav. 21, 417-426 (2003).
  * [26] G. Allemandi, A. Borowiec and M. Francaviglia, Phys. Rev. D 70, 103503 (2004).
  * [27] G. Allemandi, A. Borowiec and M. Francaviglia, Phys. Rev. D 70, 043524 (2004).
  * [28] S. Nojiri and S. D. Odintsov, eConf C0602061, 06 (2006).
  * [29] S. Capozziello, Int. J. Mod. Phys. D 11, 483-492 (2002).
  * [30] S. M. Carroll, V. Duvvuri, M. Trodden and M. S. Turner, Phys. Rev. D 70, 043528 (2004).
  * [31] S. Capozziello, V. F. Cardone and A. Troisi, JCAP 08, 001 (2006).
  * [32] C. G. Boehmer, T. Harko and F. S. N. Lobo, Astropart. Phys. 29, 386-392 (2008).
  * [33] C. G. Boehmer, T. Harko and F. S. N. Lobo, JCAP 03, 024 (2008).
  * [34] H. A. Buchdahl, Mon. Not. Roy. Astron. Soc. 150, 1 (1970).
  * [35] T. Chiba, Phys. Lett. B 575, 1-3 (2003).
  * [36] A. L. Erickcek, T. L. Smith and M. Kamionkowski, Phys. Rev. D 74, 121501 (2006).
  * [37] R. Myrzakulov, Eur. Phys. J. C 72, 2203 (2012).
  * [38] P. H. R. S. Moraes and J. R. L. Santos, Eur. Phys. J. C 76, 60 (2016).
  * [39] V. K. Bhardwaj and A. K. Yadav, Int. J. Geom. Meth. Mod. Phys. 17, no.10, 2050159 (2020).
  * [40] A. K. Yadav, M. Mondal and F. Rahaman, Pramana 94, no.1, 90 (2020).
  * [41] L. K. Sharma, A. K. Yadav and B. K. Singh, New Astron. 79, 101396 (2020).
  * [42] N. Singla, M. Kumar Gupta and A. Kumar Yadav, Grav. Cosmol. 26, no.2, 144-152 (2020).
  * [43] L. K. Sharma, A. K. Yadav, P. K. Sahoo and B. K. Singh, Results Phys. 10, 738-742 (2018).
  * [44] A. K. Yadav and A. T. Ali, Int. J. Geom. Meth. Mod. Phys. 15, no.02, 1850026 (2017).
  * [45] J. K. Singh, A. Singh, G. K. Goswami and J. Jena, Annals Phys. 443, 168958 (2022).
  * [46] F. G. Alvarenga, A. de la Cruz-Dombriz, M. J. S. Houndjo, M. E. Rodrigues and D. Sáez-Gómez, Phys. Rev. D 87, no.10, 103526 (2013) [erratum: Phys. Rev. D 87, no.12, 129905 (2013)].
  * [47] M. J. S. Houndjo, Int. J. Mod. Phys. D 21, 1250003 (2012).
  * [48] M. Jamil, D. Momeni, M. Raza and R. Myrzakulov, Eur. Phys. J. C 72, 1999 (2012).
  * [49] Z. Yousaf, K. Bamba and M. Z. u. H. Bhatti, Phys. Rev. D 93, no.12, 124048 (2016).
  * [50] P. H. R. S. Moraes, G. Ribeiro and R. A. C. Correa, Astrophys. Space Sci. 361, no.7, 227 (2016).
  * [51] M. E. S. Alves, P. H. R. S. Moraes, J. C. N. de Araujo and M. Malheiro, Phys. Rev. D 94, no.2, 024032 (2016).
  * [52] N. K. Sharma and J. K. Singh, Int. J. Theor. Phys. 53, no.9, 2912-2922 (2014).
  * [53] Z. Yousaf, M. Ilyas and M. Z. Bhatti, Mod. Phys. Lett. A 32, no.30, 1750163 (2017).
  * [54] A. Das, F. Rahaman, B. K. Guha and S. Ray, Eur. Phys. J. C 76, no.12, 654 (2016).
  * [55] J. K. Singh and N. K. Sharma, Int. J. Theor. Phys. 53, 1424-1433 (2014).
  * [56] R. Nagpal, J. K. Singh and S. Aygün, Astrophys. Space Sci. 363, 114 (2018).
  * [57] R. Myrzakulov, [arXiv:1205.5266 [physics.gen-ph]].
  * [58] R. Myrzakulov, Entropy 14, 1627-1651 (2012).
  * [59] M. Sharif, S. Rani and R. Myrzakulov, Eur. Phys. J. Plus 128, 123 (2013).
  * [60] S. Chattopadhyay, Proc. Indian Natl. Sci. Acad. 84, no.1, 87-93 (2014).
  * [61] A. Vilenkin, Phys. Lett. B 117, 25-28 (1982).
  * [62] D. S. Salopek, J. R. Bond and J. M. Bardeen, Phys. Rev. D 40, 1753 (1989).
  * [63] I. M. Khalatnikov and A. Mezhlumian, Phys. Lett. A 169, 308-312 (1992).
  * [64] R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582-1585 (1998).
  * [65] J. Khoury and A. Weltman, Phys. Rev. Lett. 93, 171104 (2004).
  * [66] J. K. Singh, N. K. Sharma and A. Beesham, Theor. Math. Phys. 193, no.3, 1865-1879 (2017).
  * [67] M. Sami and T. Padmanabhan, Phys. Rev. D 67, 083509 (2003) [erratum: Phys. Rev. D 67, 109901 (2003)].
  * [68] V. Singh and A. Beesham, Eur. Phys. J. C 78, no.7, 564 (2018).
  * [69] C. P. Singh and V. Singh, Astrophys. Space Sci. 339, 101-109 (2012).
  * [70] B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988).
  * [71] E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753-1936 (2006).
  * [72] J. Martin, Mod. Phys. Lett. A 23, 1252-1265 (2008).
  * [73] W. Zimdahl and D. Pavon, Phys. Lett. B 521, 133-138 (2001).
  * [74] L. Amendola, Phys. Rev. D 62, 043511 (2000).
  * [75] I. Zlatev, L. M. Wang and P. J. Steinhardt, Phys. Rev. Lett. 82, 896-899 (1999).
  * [76] P. J. Steinhardt, L. M. Wang and I. Zlatev, Phys. Rev. D 59, 123504 (1999).
  * [77] V. B. Johri, Phys. Rev. D 63, 103504 (2001).
  * [78] C. Armendariz-Picon, V. F. Mukhanov and P. J. Steinhardt, Phys. Rev. Lett. 85, 4438-4441 (2000).
  * [79] L. Amendola and D. Tocchini-Valentini, Phys. Rev. D 64, 043509 (2001).
  * [80] M. Gasperini, F. Piazza and G. Veneziano, Phys. Rev. D 65, 023508 (2002).
  * [81] S. D. M. White, J. F. Navarro, A. E. Evrard and C. S. Frenk, Nature 366, 429-433 (1993).
  * [82] J. A. Frieman, C. T. Hill, A. Stebbins and I. Waga, Phys. Rev. Lett. 75, 2077-2080 (1995).
  * [83] K. Coble, S. Dodelson and J. A. Frieman, Phys. Rev. D 55, 1851-1859 (1997).
  * [84] L. A. Boyle, R. R. Caldwell and M. Kamionkowski, Phys. Lett. B 545, 17-22 (2002).
  * [85] C. Armendariz-Picon, T. Damour and V. F. Mukhanov, Phys. Lett. B 458, 209-218 (1999).
  * [86] T. Chiba, T. Okabe and M. Yamaguchi, Phys. Rev. D 62, 023511 (2000).
  * [87] B. Feng, X. L. Wang and X. M. Zhang, Phys. Lett. B 607, 35-41 (2005).
  * [88] A. Sen, JHEP 07, 065 (2002).
  * [89] T. Padmanabhan, Phys. Rev. D 66, 021301 (2002).
  * [90] J. D. Barrow, Nucl. Phys. B 310, 743-763 (1988).
  * [91] J. D. Barrow, Phys. Lett. B 235, 40-43 (1990).
  * [92] Y. L. Bolotin, V. A. Cherkaskiy, O. A. Lemets, D. A. Yerokhin and L. G. Zazunov, [arXiv:1502.00811 [gr-qc]] (2015).
  * [93] W. A. Hiscock, Class. Quant. Grav. 7, L35 (1990).
  * [94] J. D. Bekenstein, Phys. Rev. D 11, 2072-2075 (1975).
  * [95] L. H. Ford, Phys. Rev. D 35, 2339 (1987).
  * [96] M. S. Madsen, Class. Quant. Grav. 5, 627-639 (1988).
  * [97] C. Barcelo and M. Visser, Class. Quant. Grav. 17, 3843-3864 (2000).
  * [98] L. H. Ford and T. A. Roman, Phys. Rev. D 64, 024023 (2001).
  * [99] N. Aghanim et al. [Planck], Astron. Astrophys. 641, A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)].
  * [100] Shaily, M. Zeyauddin and J. K. Singh, [arXiv:2207.05076 [gr-qc]].
  * [101] N. Suzuki et al. [Supernova Cosmology Project], Astrophys. J. 746, 85 (2012).
  * [102] V. Sahni, T. D. Saini, A. A. Starobinsky and U. Alam, JETP Lett. 77, 201-206 (2003).
  * [103] V. Sahni, [arXiv:astro-ph/0211084 [astro-ph]] (2002).
  * [104] U. Alam, V. Sahni, T. D. Saini and A. A. Starobinsky, Mon. Not. Roy. Astron. Soc. 344, 1057 (2003).