FLRW Transit Cosmological Model in F (R,T) Gravity

1. Introduction

Observational data from Type-Ia supernova (SNe-Ia), the cosmic microwave background (CMB), and baryon acoustic oscillations (BAO) has become a vital pillar in comprehending modern cosmology, where the evidences suggests that the universe has undergone of accelerated expansion twice, viz., early inflation [1,2,3], and late-time acceleration [4,5,6]. The late-time acceleration is supposed to be driven by dark energy (DE) that occupies approximately two-third of the total energy budget of the universe [7,8]. The most widely acknowledged $\Lambda$CDM (cold dark matter) model based Einstein’s theory of general relativity (GR) explains the late-time acceleration phenomenon via a cosmological constant (CC), $\Lambda$, which is characterized by an equation of state (EoS) parameter $(\omega =-1)$ [9,10,11].

Though the standard model explains various physical phenomena such as the formation and evolution of the large scale structure in the universe, the early universe, and the abundance of matter and energy [12,13] etc., it experiences setbacks such as the coincidence and fine-tuning problems [14,15]. Due to these setbacks of the $\Lambda$CDM model, researchers [16,17] seek other alternatives to explain late-time cosmic acceleration. One way is the modification of the Einstein-Hilbert (EH) gravitational action. Initially, the focus were only on altering the geometrical part of the EH action. In 2011, Harko et. al. [18] through general non-minimal coupling between matter and geometry introduced $f(R,T)$ gravity. Over the years, the theory has gained a lot of interest and has been extensively studied (for details see [19,20] and references therein).

It is also now well known that prior to the current accelerating expansion the universe has undergone through a decelerated phase in the past. However, constructing viable scaling models which allow the universe to transit from decelerating to an accelerating phase is still a challenging task. An alternative is to seek the solutions of the field equations under an assumption which would be consistent with the observed kinematics of the universe. This phenomenon has piqued the interest of many researchers. Some theorist working in this area have attempted to study it by constructing the cosmological models using geometrical parameters, such as a parametrization of the deceleration parameter, Hubble parameter or scale factor which can provide the transition from past deceleration to present acceleration [21,22,23,24]. Most of these models have been studied in homogeneous and isotropic background [25,26,27,28,29,30,31,32,33], but some studies also have been considered in homogeneous but anisotropic background [34,35,36,37,38,39,40].

Considering all curvatures $k=-1,0,+1$, the present study deals a transit FLRW model in $f(R,T)$ gravity. We consider a deceleration parameter proposed in Ref. [41]. The work is organized as follows. The model and field equations followed by their solutions are presented in Sect. 2. The geometrical behavior of the deceleration parameter is also presented in the same section and the constraints for a physically realistic scenario are obtained therein. The nature of matter is examined therein. As compared to GR, in $f(R,T)$ gravity due to coupling some extra terms appear on the right-hand side of the field equations. These extra terms may be treated as coupled matter or energy which may behave either as perfect fluid or DE. The nature of coupled matter is examined in Sect. 3. The results are summarised in Sect. 5.

2. The Model in $f(R,T)$ Gravity

The spatially homogeneous and isotropic FLRW space-time metric is given as

$$d{s}^2=d{t}^2-a^2\left(t\right)\left[\frac{d{r}^2}{1-k{r}^2}+r^2(d{\theta }^2+{sin}^2\theta d{\varphi }^2)\right],$$

(1)

where $0\le \theta \le \pi$, $0\le \varphi \le 2\pi$, a is the scale factor, and k represents the geometrical curvature of the universe, i.e., $k=0$ implies a flat universe, $k=+1$ a closed universe, and $k=-1$ an open universe. We consider the energy-momentum tensor of the matter as

$$T_{ij}=(\rho +p)u_i{u}_j-p{g}_{ij},$$

(2)

where $\rho$ is the energy density and p is the thermodynamic pressure of the matter. In comoving coordinates, $u^i={\delta }_0^i$, where $u_i$ is the four-velocity of the fluid that satisfies the condition $u_i{u}^j=1$.

The field equations in $f(R,T)=R+2f\left(T\right)$ gravity with the system of units $8\pi G=1=c$, are obtained as

$$R_{ij}-\frac{1}{2}R{g}_{ij}=T_{ij}+2\left(T_{ij}+p{g}_{ij}\right)f^{\prime }\left(T\right)+f\left(T\right)g_{ij},$$

(3)

where a prime represents the derivative with respect to T. The above equation for $f\left(T\right)=\lambda T$, i.e., $f(R,T)=R+2\lambda T$, where $T=\rho -3p$, simplifies to

$$R_{ij}-\frac{1}{2}R{g}_{ij}=(1+2\lambda )T_{ij}+\lambda (\rho -p)g_{ij},$$

(4)

For the metric (1) and energy-momentum tensor (2), the above equation yields

$$\begin{array}{ccc}\hfill 3H^2+3\frac{k}{a^2}& =& {\rho }_m+\lambda (3{\rho }_m-p_m),\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill 2\dot{H}+3H^2+\frac{k}{a^2}& =& -p_m-\lambda (3p_m-{\rho }_m).\hfill \end{array}$$

(6)

It is vital to note that in $f(R,T)$ gravity, both the physical quantities ${\rho }_m$ and $p_m$ no longer epitomize the effective energy density and pressure as in GR. Indeed, the coupling between matter and $f(R,T)$ gravity adds some additional terms visible on the RHS of the field equations. These additional terms with $\lambda$ can be treated as matter, and we term them “coupled matter" or “coupled energy". Therefore, to distinguish the “main" matter from the “coupled" matter or energy, we write ${\rho }_f=\lambda (3{\rho }_m-p_m)$ and $p_f=\lambda (3p_m-{\rho }_m)$, which represents the coupled matter or energy.

Eqs. (5) and (6) consist of three unknowns, i.e., ${\rho }_m,p_m$, and H. Hence, to determine the solution, one extra assumption is required. We have considered the simple parametrization of the scale factor as recently studied in Ref. [41], viz.,

$$a\left(t\right)=a_{0}exp{\left(\alpha t+\beta \right)}^p,$$

(7)

where $a_0$, $\alpha >0$, $\beta >0$ and $0<p<1$ are arbitrary constants.

The Hubble parameter, $H=\dot{a}/a$ gives

$$H\left(t\right)=\frac{\alpha p}{{(\alpha t+\beta )}^{1-p}},$$

(8)

Using $a=a_0/(1+z)$, where $a_0$ is the present value of a, and z is the redshift, one obtains

$$H\left(z\right)=\alpha p{\left[A-log(1+z)\right]}^{1-\frac{1}{p}}.$$

(9)

The deceleration parameter, $q=-a\ddot{a}/\dot{a^2}$ in terms of the redshift is becomes

$$q\left(z\right)=-1\left(\frac{1}{p}-1\right){\left[A-log(1+z)\right]}^{-1},$$

(10)

where $A=(1/p-1)/(1+q_0)$, here $q_0$ is the present value of deceleration parameter.

Recently, using 51 points $H\left(z\right)$ data set [42] and 1048 points Pantheon SNe-Ia data set [43] with $H_0=67.77km{s}^{-1}Mp{c}^{-1}$ [44], Mishra and Dua [41] obtained the observational constraints on $q_0$ and p, and reported the best fitted values $q_0\approx -0.40$ and $p=0.47$ with $H\left(z\right)$ data set and $q_0\approx -0.54$ and $p=0.66$ with Pantheon data set.

Figure 1 plots the deceleration parameter q versus red shift z for the best fitted values mentioned above. We observe that the universe transits from deceleration ($q>0$) to acceleration ($q<0$) at a red shift $z=0.8$ for $H\left(z\right)$ data, and $z=1.2$ for Pantheon data. Also, the present values of deceleration parameter are $q_0=-0.40$ for $H\left(z\right)$ data, and $q_0=-0.54$ for Pantheon data.

Now, the the main objectives of our study are following:

• To determine the matter in the presence of which the model can yield the desired evolution of the universe.
• Examine the role of the additional terms of $f(R,T)$ in these models.
• Compare/distinguish the outcomes from those of Einstein’s gravity.

Using (7) and (8) in (5) and (6), we obtain the energy density and pressure of matter

$$\begin{array}{ccc}\hfill {\rho }_m& =& \frac{e^{-2{(\beta +\alpha t)}^p}\left[k(8\lambda +3)+{\alpha }^{2}p{e}^{2{(\beta +\alpha t)}^p}{(\beta +\alpha t)}^{p-2}\left(2\lambda -2\lambda p+3(2\lambda +1)p{(\beta +\alpha t)}^p\right)\right]}{(2\lambda +1)(4\lambda +1)},\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill p_m& =& -\frac{k{e}^{-2{(\beta +\alpha t)}^p}+{\alpha }^{2}p{(\beta +\alpha t)}^{p-2}\left(2(3\lambda +1)(p-1)+3(2\lambda +1)p{(\beta +\alpha t)}^p\right)}{(1+4\lambda )(1+2\lambda )}.\hfill \end{array}$$

(12)

The vital requirement is now to corroborate that the ad hoc assumption made to find the solutions is consistent to yield a realistic cosmological scenario, viz., the energy density of matter ought to be positive. We note that ${\rho }_m\ge 0$ requires $\lambda \ge 0$ in all the three models, viz., $k=0,\pm 1$.

Let us examine the behavior of actual matter. The EoS parameter of actual matter ${\omega }_m=p_m/{\rho }_m$ gives

$${\omega }_m=-\frac{k{(\beta +\alpha t)}^2+{\alpha }^{2}p{e}^{2{(\beta +\alpha t)}^p}{(\beta +\alpha t)}^p\left(2(3\lambda +1)(p-1)+3(2\lambda +1)p{(\beta +\alpha t)}^p\right)}{k(8\lambda +3){(\beta +\alpha t)}^2+{\alpha }^{2}p{e}^{2{(\beta +\alpha t)}^p}{(\beta +\alpha t)}^p\left(2\lambda -2\lambda p+3(2\lambda +1)p{(\beta +\alpha t)}^p\right)},$$

(13)

Since $\alpha$ is a scaling parameter and $\beta$ is just a shifting parameter. The only crucial parameter in this study is p. Therefore, we set $\alpha =1$ and $\beta =0$ to study the behavior of EoS parameter for the best fitted values $p=0.47$ ($H\left(z\right)$ data) and $p=0.66$ (Pantheon data). Figure 2 and Figure 3 depict the behavior of matter with $H\left(z\right)$ and Pantheon data, respectively.

Analytically, we find that for all the three spatial curvature models, ${\omega }_m=\frac{1}{\lambda }+3$ at $t=0$, and ${\omega }_m\to -1$ as $t\to \infty$. Therefore, ${\omega }_m$ starts with a finite value ${\omega }_m\ge 3$ with the evolution and approaches towards the cosmological constant at late time. Hence, the matter exhibits the unified description of all kind matter, including hard matter (${\omega }_m\ge 1$), radiation (${\omega }_m=1/3$), dust (${\omega }_m=0$), quintessence ($-1<{\omega }_m\le -1/3$), and a cosmological constant (${\omega }_m=-1$), in the same order as it is required for the cosmological evolution.

3. The Behavior of Coupled Matter

The energy density and pressure of coupled matter are obtained as

$$\begin{array}{ccc}\hfill {\rho }_f& =& \frac{2\lambda \left(k(12\lambda +5)e^{-2{(\beta +\alpha t)}^p}+{\alpha }^{2}p{(\beta +\alpha t)}^{p-2}\left(6(2\lambda +1)p{(\beta +\alpha t)}^p+p-1\right)\right)}{8{\lambda }^2+6\lambda +1},\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill p_f& =& \frac{2\lambda \left({\alpha }^{2}p{(\beta +\alpha t)}^{p-2}\left(8\lambda +p\left(-8\lambda -6(2\lambda +1){(\beta +\alpha t)}^p-3\right)+3\right)-k(4\lambda +3)e^{-2{(\beta +\alpha t)}^p}\right)}{8{\lambda }^2+6\lambda +1}.\hfill \end{array}$$

(15)

In all the three spatial curvature models we note that the energy density of coupled matter is negative at very early stage of evolution and turns out to be positive after an instance. Since ${\rho }_f$ becomes zero at the transition time, it is not worthy to use EoS parameter to study the behavior of coupled matter as ${\omega }_f=p_f/{\rho }_f$ would diverge at that instance of time. Alternatively, we study the behavior of coupled matter via the energy conditions which are stated as

• Null Energy Conditions (NEC) : $\rho +p\ge 0$
• Weak Energy Conditions (WEC) : $\rho \ge 0$, $\rho +p\ge 0$
• Strong Energy Conditions (SEC) : $\rho +3p\ge 0$
• Dominant Energy Conditions (DEC) : $\rho \ge \left|p\right|$

Figure 4 shows the behavior of energy density for a flat model which shows that the WEC violates at the very early time. We observe the same behavior in closed and open models.

The expression for ${\rho }_f+p_f$ is obtained as

$${\rho }_f+p_f=\frac{4\lambda \left(k{e}^{-2{(\beta +\alpha t)}^p}+{\alpha }^{2}p(1-p){(\beta +\alpha t)}^{p-2}\right)}{2\lambda +1}.$$

(16)

Since $\lambda >0$ and $0<p<1$, the above expression implies ${\rho }_f+p_f\ge 0$ for flat and closed models. For $k=-1$, Figure 8 plots ${\rho }_f+p_f\ge 0$ which shows that the NEC also holds good for open model.

Further, following is the expression of ${\rho }_f-p_f$

$${\rho }_f-p_f=\frac{8\lambda \left(2k{e}^{-2{(\beta +\alpha t)}^p}+{\alpha }^{2}p\left(3p{(\beta +\alpha t)}^p+p-1\right){(\beta +\alpha t)}^{p-2}\right)}{4\lambda +1}.$$

(17)

Figure 6 and Figure 7, plot ${\rho }_f-p_f$ with $H\left(z\right)$ and Pantheon data, respectively. Due to the domination of $p-1$ term at very early evolution ${\rho }_f-p_f$ starts from a negative value, later on it becomes positive and after attaining a maximum value it starts decreasing and finally approaches towards zero at late time. Hence, the DEC are satisfied except at very early stage of evolution.

Lastly, we have

$${\rho }_f+3p_f=\frac{8\lambda \left({\alpha }^{2}p{(\beta +\alpha t)}^{p-2}\left(-2(3\lambda +1)(p-1)-3(2\lambda +1)p{(\beta +\alpha t)}^p\right)-k{e}^{-2{(\beta +\alpha t)}^p}\right)}{8{\lambda }^2+6\lambda +1}.$$

(18)

Figure 8 and Figure 9, depict the behavior of ${\rho }_f+3p_f$ with $H\left(z\right)$ data and Pantheon data, respectively. In both models ${\rho }_f+3p_f$ become negative at late time. Hence, the SEC violate at the late time evolution in all the three both models.

4. The Behavior of Effective Matter

The energy density and pressure of effective matter can be obtained from ${\rho }_{eff}={\rho }_m+{\rho }_f$ and $p_{eff}=p_m+p_f$ which also can be read by Eqs. (11) and (12) with $\lambda =0$. Similarly, the EoS of effective matter can be read by Eq. (13) with $\lambda =0$ which implies that the behavior of effective matter remains the same as in GR. We note that the effective matter obeys WEC for $k=0,\pm 1$. The behavior of the effective matter with $H\left(z\right)$ and Pantheon data is illustrated in Figure 10 and Figure 11, respectively.

The behavior of effective matter is similar to the matter as discussed in Sec. 2. This can be observed by comparing Figure 2 and Figure 3 with Figure 10 and Figure 11. Therefore, the interpretation made for matter is also hold true for effective matter as well.

5. Conclusion

In this paper, we have studied an FLRW spacetime model filled with a perfect fluid in the framework of $f(R,T)$ gravity, where $f(R,T)=R+2f\left(T\right)$, for all spatially curved models. In order to find solutions, we have chosen a parametrization of the scale factor, $a\left(t\right)=a_0{e}^{{(\alpha t+\beta )}^p}$, where $\alpha ,\beta >0$, and $0<p<1$ that yields a deceleration parameter transiting from an early deceleration to a late time acceleration phase. The kinematical dynamics of the model is independent of the theory and is studied earlier in Ref. [41]. To eschew repetition, the kinematical study is not presented in this paper.

The main attributes of the models are as follows:

• A physically realistic cosmological model is possible only for $\lambda \ge 0$.
• As compared to GR, in $f(R,T)$ gravity due to coupling there are extra terms on the right-hand side of the field equations. These extra terms may be termed as coupled matter which may behave either as perfect fluid or DE.
• The main matter exhibits the characteristics of all kind matter, viz., hard matter (${\omega }_m\ge 1$), radiation (${\omega }_m=1/3$), dust (${\omega }_m=0$), quintessence ($-1<{\omega }_m\le -1/3$), and a cosmological constant (${\omega }_m=-1$), in the same order as it is required to depict the cosmic history including the transition from a decelerated to an accelerating universe.
• The coupled matter satisfies the NEC, however, it violates the WEC and the NEC at very early stage of evolution. It also violates the SEC at late times which shows that the coupled matter contributes as quintessence DE. The model explains late time acceleration without including any form of hypothetical exotic matter, indicating that the $f(R,T)=R+2\lambda T$ gravity can be a good alternative to DE.