1. Introduction
Observational data from refs. [
1,
2,
3] suggest that the universe is currently in an accelerating epoch. A plethora of attempts have been made to explain this phenomenon, but neither of them is compelling. The first attempt is dark energy (DE), which is the hypothesis of exotic matter with the unique feature of anti-gravity due to highly negative pressure which hence accelerates the expansion of the universe [
4]. On the other hand, there is insufficient information about DE from the ΛCDM model in general relativity (GR). The cosmological constant (CC) is the primary candidate for DE and the second candidate is modified gravity. The shortcomings [
5] of the ΛCDM model enable authors to consider other alternatives to fundamental theories of astrophysics and cosmology. These include the dynamical candidates of DE and modified theories of gravity, e.g., higher derivative theories, Gauss-Bonnet f(G) gravity, f(R) theory, f(T) and f(R,T) gravity theory. Harko et al. [
6] introduced f(R,T) gravity, where f(R,T) is an arbitrary function of the Ricci scalar R, and the trace T of the energy-momentum tensor.
Over the years, cosmologists have solved the field equations by means of assuming some cosmological parameters, i.e., Hubble parameter, scale factor, and even some form of deceleration parameter, based on the current understanding in cosmology that the universe has undergone stages of evolution, i.e., inflation, radiation, matter, and late time acceleration. Based on that, the notion of varying deceleration parameter, which changes the signature from deceleration to acceleration, has been applied to many cosmological models. In ref. [
7], the authors developed a hyperbolic scale factor. This form of scale factor has attracted a lot of attention over the years, where it was applied for both homogeneous and isotropic or anisotropic space-times through various contexts in cosmology, i.e., see ref. [
8]. Recently, the Bianchi-I model with a perfect fluid with various cases of cosmological constant was considered [
7].
2. The Formalism of the Model in f(R,T) Gravity
The general action of f (R,T) gravity with units in which
is given by ref. [
6].
We consider
, and get the following field equations
where a prime represents an ordinary derivative of
f(T) with respect to
T. In this work we have chosen
, i.e.,
. The energy-momentum tensor (EMT) of a perfect fluid is
with
and
p the energy density and thermodynamic pressure, respectively. The trace of the energy-momentum tensor is
. Equation (8) yields
In this work, we have considered the LRS Bianchi-I spacetime:
where
A and
B are the scale factors and functions of cosmic time
. The average scale factor for the metric (16) is defined as
The average Hubble parameter is given by
where a dot denotes the derivative with respect to
.
It is crucial to mention that the coupling between geometry and matter in f (R,T) gravity adds some additional terms visible on the RHS of the field equations. These terms must be treated as matter that can be called “coupled matter”. Therefore, to distinguish between the main matter and coupled matter, we replace with and with which represents the primary or main matter.
Using the line element (4) and the energy-momentum tensor (EMT) of a perfect fluid, the field Equation (3), yield
These equations consist of four unknowns, namely,
. Therefore, to find exact solutions, one supplementary constraint is required. We assume the following relationship between the Hubble parameter and cosmic time [
7]:
where
are positive constants.
On solving, Equation (10) gives
Then, the deceleration parameter
is given by
and is illustrated in
Figure 1.
To eschew repetition,
have been articulated in ref. [
7]. The energy density and pressure are given by
where
is a constant of integration.
For a physically realistic cosmological model, the density
must be positive. Unfortunately, at early times, we find that the density
is negative. We shall comment on this later. The pressure is negative throughout the evolution. The equation of state parameter (EoS),
, is given by:
2.1. The Behavior of Coupled Matter
As elucidated above,
and
do not represent the effective matter in this model. The terms containing
λ in the Equations (7)–(9) can be assumed to be associated with the coupled matter. By separating these terms, the equations can be expressed as
where
and
, respectively, represents the energy density and pressure of the coupled matter, and are obtained as
The EOS for the coupled matter is:
2.2. State-Finder Parameter, Energy Conditions and Stability
2.2.1. State-Finder Analysis
The state finder pair {
r,
s} allows the examining the features of DE for the model, and to compare with the ΛCDM model. They rely on the third derivative of the scale factor, as they were introduced in ref. [
9]. In this model, they are given by
We observe from the above that as , {r, s} → {0, 0.6}, and as , {r, s} → {1, 0}. In this model, initially, we have and , which imply quintessence and phantom, respectively. At late times, the model mimics the ΛCDM model.
2.2.2. Energy Conditions
The behaviour of the energy conditions such as the Weak Energy Condition (WEC:
), Dominant Energy Condition (DEC:
) and Strong Energy Condition (SEC:
), have been studied. They are illustrated in
Figure 2.
2.2.3. Stability of the Models through the Speed of Sound
It is crucial to study the stability of the theory, and here, we make use of the technique of speed of sound to study stability [
10]. Numerous other techniques can be used to understand the stability of the solutions/methods [
11]. The speed of sound is given by
. If
or
, the system is unstable, whereas if
, the system is stable. In this model
We find that for the values (m = 1, n = 1, Q =1,
), (m = 2, n = 2, Q =2,
), (m = 3, n = 3, Q =3,
), our model is stable (during the early phase of the model), whereas, at present, it is unstable. This is illustrated in
Figure 3.
3. Discussion
We have studied an LRS Bianchi I model with a special Hubble parameter
, yielding
, where
are positive constants. We find that the model transits from early deceleration (
for
) to late time acceleration
which is in line with observations [
1,
2,
3]. In any physically realistic cosmological model, the energy density (
) ought to be positive throughout the evolution. Here, we observe that for (
) initially, both densities are negative but later positive. Due to the complicated nature of the expressions, we have not checked the density for all values of the parameters. Therefore, it is quite possible that the density could be positive for some parameters, and we are investigating this further, and hope to report elsewhere. However, our results do indicate that models have to be checked very carefully to ensure that all reasonable conditions are met. The pressure is negative, which is associated with late-time acceleration.
The DEC is satisfied, but not the SEC. Again, this is in keeping with the late-time acceleration of the model (
Figure 2). The other important aspect of the model is for
, the solutions of GR are recovered. The effective matter behaves like in
f (
R,
T) gravity due to similar metric potentials in both theories. We can conclude that the model accelerates at late times in
f (
R,
T).
The material presented here is brief, and an extended version will be presented elsewhere.