Dynamics of quasi de Sitter and linear combination of exponential models in extended gravity

In modern cosmology, the main theme of research is on the issue of late time cosmic acceleration phenomena. Supernovae type Ia provide strong evidence that the Universe is currently undergoing an accelerated phase of expansion. Plethora of observations during the last two decades have confirmed the late time cosmic acceleration of the Universe [1, 2, 3, 4, 5, 6, 7]. These observations have developed a curiosity among the cosmologists to explain this late time dynamics. In the purview of General Relativity (GR), it becomes difficult to explain this issue. Hence the idea of modifying GR has taken momentum in the last decade. Researchers changed both the geometrical and matter part of field equations of GR. The changes made in the matter part lead to the inclusion of some exotic form of energy known as dark energy and the changes in the geometric part lead to the extended gravity. Several geometrical extensions of GR have been made in recent past such that $f(R)$ gravity [8, 9, 10], $f(R,T)$ gravity [11], $f(R,G)$ gravity [12], $f(T)$ gravity [13], $f(Q,T)$ gravity [14]. The $f(R,T)=f_{1}(R)+f_{2}(T)$ gravity consists of the functions of Ricci scalar $R$ and trace of energy momentum tensor $T$ and have been suggested in three different functional forms [11] such as (a) $R+2f(T)$; (b) $f_{1}(R)+f_{2}(T)$ and (c) $f_{1}(R)+f_{2}(R).f_{3}(T)$. Each functional has its own merit to answer the late time cosmic acceleration issue. It is believed that, the dark sector of the Universe is responsible for the cosmic evolution and the dynamics at least at its late phase. The dark sector responsible for cosmic evolution may be the dark matter or the dark energy. It still remains as an open question, whether the dark matter is mediated through a weakly interacting massive particle or is a manifestation of GR modification. The dark energy on the other hand is an exotic energy form that violates the strong energy condition and is associated with a fluid of negative pressure. Within the purview of GR, the dark energy is required to explain the late time cosmic speed up phenomena. Till date, the quest to unfold the mystery of late time cosmic acceleration through different dark energy models and modified gravity theories still remains unquenched. Besides, there have been questions and possible efforts to find solutions concerning the initial singularity problem. In this context, there have been many interesting discussions in recent times [15, 16, 17, 18, 19].

Several cosmological models under $f(R,T)$ gravity have been proposed in last decade. We have discussed some prominent models that are successful in giving some insight to the issue of late time cosmic acceleration phenomena. Houndjo [20] has investigated the cosmological reconstruction with the power law cosmology. Santos [22] obtained the Godel solution and Azizi [21] studied the wormhole geometry in the $f(R,T)$ gravity. In the metric formalism, Alavarenga et al. [23] have studied the scalar perturbations. Pasqua et al. [24] have shown the quintessence behaviour of modified holographic dark energy model. Yadav [25] has shown the existence of string cosmological model in power law cosmology. Considering the existence of conformal killing vector, Das et al. [26] have shown the behaviour of compact stars in $f(R,T)$ gravity. Yousaf et al. [27] have examined the irregularity factors of spherical star in anisotropic and isotropic fluid. Tripathy et al. [28] have discussed some cosmological models speculating a bouncing scenario in $f(R,T)$ gravity. Zaregonbadi et al. [29] derived the metric components in the galactic halo with the minimal coupling of matter and geometry. Alhamzawi and Alhamzawi [30] have calculated the effects of $f(R,T)$ gravity in gravitational lensing and compared with the effects of general relativity. Fayaz [31] have analysed the ghost model of dark energy in $f(R,T)$ gravity and GR in an anisotropic background. Xu et al. [32] have studied the quantum cosmology and obtained the general form of the quantum potential, gravitational Hamiltonian and the canonical momenta. Shabani and Ziaie [33] have obtained the late time cosmological solution. In the non-minimal coupled scenario, Sharif and Nawazish [34] have determined the existence of Noether symmetry and performed the cosmological analysis.

Under $f(R,T)$ gravity, Islam and Basu [35] described the compact star model in the presence of magnetic field and indicated the importance of the solution in the context of interior of compact objects. Elizalde and Khurshudyan [36] have discussed the static wormhole with the possible violation of weak and dominant energy conditions. Mishra et al. [37] have introduced the reconstruction method to obtain an anisotropic cosmological model of the Universe. In metric approach of $f(R,T)$ gravity, Khan et al. [38] have studied the higher dimensional collapse of perfect fluid with spherically symmetric space-time. Wu et al. [39] introduced the non-minimal coupling between the matter and geometry in Palatini formulation by assuming the affine connection and metric as independent field variables. Using metric-affine theories, Barrientos et al. [40] found the field equations and shown its relevance with the field equations of $f(R)$ gravity. Tretyakov [41] introduced the higher derivative matter field in $f(R,T)$ gravity and discussed the stability conditions. Mishra et al. [42] obtained the bounce solution in an anisotropic space-time. Fisher and Carlson [43] have re-examined the $f(R,T)$ gravity and claimed that the term $f_{2}(T)$ should be included in the matter Lagrangian. Tripathy and Mishra [44] have presented the phantom cosmological in $f(R,T)$ gravity in different rip cosmologies. Maurya and Tello-Ortiz [45] have investigated the nature of charged anisotropic fluid spheres and also have shown the similarities and differences between modified gravity and GR. Mishra et al.[46] presented the wormhole solution of the model with the square of the trace of the energy momentum tensor. Recently, Tripathy has explored the possibility of Casimir wormholes in an extended gravity theory [47]. Rahaman et al. [48] explained the Karmarkar condition in $f(R,T)$ gravity and studied the existence of compact spherical systems whereas with the same condition, Ahmad and Abbas [49] have studied the non-adiabatic gravitational collapse in an anisotropic fluid. Mishra and Tripathy [50] compared the little rip and specific form of Hubble parameter cosmological models in an anisotropic space-time. Gamonal [51] studied with minimal coupling, the slow-roll approximation of cosmic inflation and obtained the modified slow-roll parameters.

The paper is organized as follows: we have presented a brief overview of the $f(R,T)$ gravity and derived the field equations and the basic parameters in Sec II. In Sec III, we have formulated two cosmological models based on quasi de Sitter (QDS) and linear combination of exponential scale factors (LCE). The analysis of different dynamical parameters of the models are performed. Also, we have carried out certain tests of the models through the calculations of the energy conditions and the state finder diagnostic pair. The results and the conclusion are given in Sec IV.

Harko et al. [11] have proposed the geometrically extended $f(R,T)$ theory of gravity, whose action can be described as,

$$S=\int d^{4}x\sqrt{-g}\left[\frac{1}{16\pi}(f_{1}(R)+f_{2}(T))+\mathcal{L}_{m}\right].$$

(1)

In above the function $f(R,T)$ has been split into two functions $f_{1}(R)$ and $f_{2}(T)$ i.e the function of Ricci scalar $R$ and trace of energy momentum tensor $T=g^{ij}T_{ij}$ respectively. The Lagrangian has been considered as, $\mathcal{L}_{m}=-p$. We assumed here, $f_{1}(R)=R$ and $f_{2}(T)=2\beta T+2\Lambda_{0}$, such that $f(R,T)=R+2\beta T+2\Lambda_{0}$; $\beta$ be the coupling constant and $\Lambda_{0}$ may be considered as the usual cosmological constant in GR. The matter field considered to be a perfect fluid whose energy momentum tensor can be expressed as, $T_{ij}=(\rho+p)u_{i}u_{j}-pg_{ij}$ and $u^{i}u_{i}=1$. Here $p$ and $\rho$ respectively denote the pressure and energy density of the cosmic fluid and the trace of the energy momentum tensor is given by $T=\rho-3p$. Now, the field equations of $f(R,T)$ gravity can be derived as [11, 44],

$$R_{ij}-\frac{1}{2}Rg_{ij}=(8\pi+2\beta)T_{ij}+\left[(\rho-p)\beta+\Lambda_{0}\right]g_{ij}.$$

(2)

When $\beta=0$, eqn. (2) recovers the field equations of Einstein’s GR. It is to note here that $(8\pi+2\beta)$ is the redefined Einstein constant. Due to the geometrical modification incorporated through a minimal matter-geometry coupling, the effective energy momentum tensor $T_{ij}^{EF}$ can be expressed as,

$$T_{ij}^{EF}=\left[\frac{(\rho-p)\beta+\Lambda_{0}}{8\pi+2\beta}\right]g_{ij}.$$

(3)

We consider here the Friedmann-Robertson-Walker(FRW) flat space-time,

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

(4)

where $a=a(t)$ is the average scale factor. Now, with eqn. (4), the field equations (2) for $f(R,T$) gravity can be obtained as,

	$\displaystyle 2\dot{H}+3H^{2}$	$\displaystyle=$	$\displaystyle-\alpha p+\beta\rho+\Lambda_{0},$		(5)
	$\displaystyle 3H^{2}$	$\displaystyle=$	$\displaystyle\alpha\rho-\beta p+\Lambda_{0}.$		(6)

For brevity, we write $\alpha=8\pi+2\beta$. $H=\frac{\dot{a}}{a}$ is the Hubble parameter. Now, on solving eqns. (5)-(6), we obtain the pressure and energy density of the model as,

	$\displaystyle p$	$\displaystyle=$	$\displaystyle-\frac{1}{(\alpha^{2}-\beta^{2})}\left[2\dot{H}\alpha+3H^{2}(\alpha-\beta)-\Lambda_{0}(\alpha-\beta)\right],$		(7)
	$\displaystyle\rho$	$\displaystyle=$	$\displaystyle\frac{1}{\alpha^{2}-\beta^{2}}\left[-2\dot{H}\beta+3H^{2}(\alpha-\beta)-\Lambda_{0}(\alpha-\beta)\right].$		(8)

The Equation of State (EoS) parameter $\omega=\frac{p}{\rho}$ can be calculated as,

$\displaystyle\omega=-1+\frac{2(\alpha+\beta)\dot{H}}{2\beta\dot{H}-3H^{2}(\alpha-\beta)+\Lambda_{0}(\alpha-\beta)}.$

(9)

Another significant study to constrain any cosmological model is the analysis of the energy conditions. Hawking and Ellis [52] have first formulated the energy conditions. Basically, the energy conditions are governed by the congruence of space-like, time-like and light-like curves (Hawking and Ellis[52], Tahim et al. [53]), but here we shall constrain to the space-like and time-like curves only. The energy conditions are: (a) $\rho+p\geq 0$, Null energy condition (NEC); (b) $\rho+p\geq 0$, $\rho\geq 0$, Weak energy condition (WEC); (c) $\rho+3p\geq 0$, Strong energy condition (SEC); and (d) $\rho-p\geq 0$, Dominant energy condition (DEC). The violation of SEC on cosmological scales has been a point of attention based on the recent observational data in the context of accelerating Universe [54]. In Phantom field models, there is possibility of violation of the NEC. The NEC, WEC, DEC are extensively used in modified theories of gravity since the energy conditions play a significant role in defining the cosmological evolution, particularly the acceleration or deceleration of cosmic fluid[55, 56, 57, 58, 59, 60]. The energy conditions for $f(R,T)$ gravity in terms of Hubble parameter can be obtained from eqns. (7)-(8) as,

	$\displaystyle\rho+p$	$\displaystyle=$	$\displaystyle-\frac{1}{\alpha-\beta}\left[2\dot{H}\right],~{}~{}~{}~{}(NEC,WEC)$
	$\displaystyle\rho+3p$	$\displaystyle=$	$\displaystyle\frac{1}{\alpha^{2}-\beta^{2}}\left[-(6\alpha+2\beta)\dot{H}-6(\alpha-\beta)H^{2}+2(\alpha-\beta)\Lambda_{0}\right],~{}~{}~{}~{}(SEC)$
	$\displaystyle\rho-p$	$\displaystyle=$	$\displaystyle\frac{1}{\alpha^{2}-\beta^{2}}\left[2(\alpha-\beta)\dot{H}+6(\alpha-\beta)H^{2}-2(\alpha-\beta)\Lambda_{0}\right].~{}~{}~{}~{}(DEC)$		(10)

It is certain that, a given cosmological model with a specified dynamics decides the energy conditions and all other dynamical parameters such as the pressure, energy density and the EoS parameter. It is also possible to investigate the cosmic dynamics for assumed relationship between the pressure and the energy density. In the present work, we will consider some assumed cosmic dynamics through certain scale factor and then investigate the dynamical parameters of the models.

In order to understand the background cosmology, we need to incorporate the scale factor to obtain the solution to the field equations. We have studied the cosmological model by incorporating two types of scale factor such as (i))quasi de Sitter (QDS) and (ii) linear combinations of exponential (LCE) functions.

We have presented two cosmological models of the Universe based on different forms of de-Sitter dynamics. The dynamical parameters for both the models are obtained and have been tested with the energy conditions criteria and the geometrical diagnostics. The wormhole throat radius of both the models are also calculated. Some of the key results are:

• •

  The QDS and LCE scale factors are employed to analyse the dynamical behaviour of the EoS parameter. The effect of the parameters of both the QDS and LCE models are observed on the evolutionary aspects of the EoS parameter. While the LCE model reduces to the $\Lambda$CDM behaviour at the present epoch, the QDS model shows a departure from the $\Lambda$CDM behaviour. The LCE model behaves like a Phantom field dominated model where the EoS parameter increases from a higher negative values to converge to $\omega=-1$ at late times. However, for the QDS model, for a substantial time zone, the EoS parameter appears to be a constant quantity close to $-1$ and then increases at late epoch. The effect of the coupling constant is quite visible in the dynamical behaviour of the EoS parameter. For the QDS model, with an increase in $\beta$, the EoS parameter increases but for the LCE model, $\omega$ at a given epoch decreases with $\beta$. The interesting aspect of both the models are that, both the model predict the EoS parameter at the present epoch which are quite compatible to recent results.

• •

  In order to obtain the good dynamical and geometrical description of the cosmological model, the cosmographic coefficients are more useful. It involves the higher order derivative of the scale factor. The first two derivative at present epoch provides the Hubble parameter and deceleration parameter respectively. The third and fourth derivatives lead to the geometrical parameters $j$ and $s$. The $(j,s)$ diagnostic pair for the LCE model approach to $(1,0)$ at late times indicating its compatiblity with the concordant $\Lambda$CDM behaviour. Also, the values predicted at the present epoch from LCE model are closer to $\Lambda$CDM values $\{1,0\}$. However, for the QDS model, from the analysis of the statefinder diagnostics, we obtain a slight departure from the $\Lambda$CDM behaviour.

• •

  Energy conditions has a major role in identifying the physical frame [71] of any theoretical model. So the physical meaning of the extended gravity depends on the energy conditions and initial conditions. In addition to the observations, this might be a formidable tool to test the dark components of the Universe. In this work, we have obtained and analysed the graphical behaviour of respective energy conditions of both the models that show a clear violation of SEC and satisfaction of the DEC. However, while the QDS model satisfies NEC, the LCE scenario violates the null energy conditions. This violation of NEC in LCE model leads to a phantom field like behaviour of the model. The result is as expected in the geometrically extended gravity [72, 73].

As a final remark, we say that, the de Sitter like models as discussed in the present work may be useful in understanding the hidden mystery of the Universe concerning the late time cosmic acceleration issue as well as a possible bouncing feature at the initial epoch. Although, the QDS model shows a little departure from the $\Lambda$CDM behaviour, the LCE model seems to be compatible to the $\Lambda$CDM model at least at the present epoch and is quite incompatible with $\Lambda$CDM behaviour in the recent past upto $z\simeq 3$.

BM and SKT acknowledge IUCAA, Pune (India) for providing the necessary support under the visiting associateship program. BM and EG acknowledge DST, New Delhi, India for providing facilities through DST-FIST lab, Department of Mathematics, where a part of this work was done. The authors are thankful to the anonymous referee for the constructive suggestions and comments for the improvement of the paper.