Role of Extended Gravity Theory in Matter Bounce Dynamics

Standard cosmological model has many success stories in providing information on the evolution of the Universe. Over the time, inflationary model has solved problems like flatness, cosmological horizon etc. The existence of dark energy, dark matter is also some way answered. However, one of the long standing issue, the initial singularity, is still preventing the researchers to move forward. It can be noted that this particular issue leads to the breakdown of the space time description. As a result, the physical laws presuppose the space time. So, an extensive research has been suggested to overcome this singularity issue. Though the cosmological bounce is not new, but Novello and Salim Novello79 and Melnikov and Orlov Melnikov79 have first suggested the exact solution of bouncing geometry. Then after a gap of almost three decades, Novello and Perez Bergliaffa Novello08 have reviewed the initial singularity issue and indicated the significance of non-singular Universe in the modern cosmology. It can also be in non local gravity models, the non-singular bouncing scenarios have also been reported Calcagni14 . Battefeld and Peter Battefeld15 have reviewed the classical bouncing cosmologies to find the strength and weakness of the models that aim to explain the observation by mechanism in the bouncing models.

It is well known that the expansion of the Universe is a consistent result in all the cosmological observations and the process of expansion is from the initial stage of evolution to the present. In a flat isotropic space time the Universe keeps on expanding as long as the energy density is positive. But, several theoretical models have indicated that before inflation, the Universe might have experienced a contraction phase extending the beginning of time into far past. Cosmological model with negative potential or positive curvature can cause the contraction phase, since the Hubble parameter changes its sign after crossing zero Linde01 ; Felder02 . Whenever, the kinetic energy of scalar field increases for the potential, the contraction phase of the Universe starts and once it dominates, it leads to the collapsing of the Universe to a singularity. This sudden collapse can be avoided if the Universe starts expanding before it collapses, such a scenario is termed as bouncing Universe. In fact, the existence of bouncing solution depends on the spatial curvature. The bouncing can happen in the closed Universe, when the total energy of the Universe balances with the curvature term whereas for a negatively curved Universe, the possibility is prevented by the null energy condition. In the recent research, several bouncing models are being proposed to give a satisfactory answer to the singularity issue necessarily not from an isotropic and homogeneous background.

Most of the cosmological models in modern cosmology are based on the fundamental cosmological principle assuming the homogeneity and isotropy of the Universe on cosmic scales. However, at cosmic scales $\gtrsim 1$ Gpc, the cosmological principle is still to be well conceived Caldwell2008 ; Deng2018 . Anomalies like the lack of correlations on large angular scales, hemispherical power asymmetry and the quadrupole($C_{2}$)-octupole($C_{3}$) alignment seem to hint for a possible violation of statistical isotropy and scale-invariance of primordial perturbations. Precise measurements of WMAP showed the $C_{2}$ and $C_{3}$ are aligned and are concentrated in a plane about $30^{0}$ to the galactic plane suggesting an asymmetric expansion Buiny2006 ; Tripathy2014 . Also there have been claims of cosmic anisotropy from observations such as the preferred direction dubbed as the axis of evil Deng2018 ; Tripathy2014 and references therein. Besides these claims, peculiar motions due to the inhomogeneity and anisotropy of surrounding structure in the Universe are also found to be non-negligible Colin2019 ; Aghanim2018 . Nevertheless, the nature and origin for the violation of the homogeneity and isotropy has yet resolved Schwarz2016 . In view of this, in the present work, we are motivated to take up this bouncing issue in a homogeneous and anisotropic background of the metric in the framework of an extended theory of gravity Mishra18a ; Mishra21a .

Odintsov et al. Odintsov15 have investigated two bouncing scenarios such as the loop quantum cosmological ekpyrotic and super bounce in $f(R)$, $f(T)$ and $f(G)$ gravity. Amani Amani16 calculated the Friedmann equations with the action of modified gravity and investigated the behaviour of bouncing cosmology. To provide a description to the early Universe, Brandenberger and Peter Brandenberger16 have reviewed cosmological inflation as an alternative to bouncing cosmology. Ijjas and Steinhardt Ijjas18 have introduced a mechanism with the non-singular bouncing cosmological model to resolve some fundamental issues of cosmology. Caruana et al. Caruana20 have obtained the cosmological bounce solution in an extended gravity that characterizes with two scalars. Saidov and Zhum Saidov10 have studied the bouncing inflation in non-linear model. In Palatini $f(R)$ gravity Barragan and Olmo Barragan10 have investigated the bouncing cosmology for an isotropic and Bianchi I space-time. Bamba et al. Bamba14 have presented the cosmological model with power law and exponential law from where the bouncing cosmology can be realized. Moreover, from the background solution the perturbations has been analyzed. It can be inferred that in several early Universe scenario, the bouncing cosmology has been emerged as a natural consequence Bamba16 ; Hohmann17 . Cai et al. Cai11a have shown that $f(T)$ gravity can provide a mechanism to avoid the phenomenal Big Bang singularity issue. In extended teleparalellel gravity, de la Cruz-Dombriz et al. Dombriz18 have studied the cosmological bouncing solutions. In an anisotropic and homogeneous Universe, Matsui et al. Matsui19 have obtained a class of solutions without encountering any singularity or violating null energy condition in which the Universe expands at the initial stage then contract and again expand resulting in a bounce in the GR framework. In the background of an anisotropic LRSBI Universe, Tripathy et al. have discussed a bouncing model within the framework of Brans-Dicke theory SKT2020 .

In the last decade, plethora of cosmological models were presented in $f(R,T)$ theory of gravity to address the late time cosmic acceleration issue e.g. cosmological aspects Moraes16 ; Xu16 ; Sharif17 ; Godani20 ; Tripathy20 ; Saridakis20 , anisotropic issueSharif14 ; Zubair18 ; Mishra18 ; Rahaman20 , wormhole geometry Azizi13 ; Yousaf17 ; Elizalde19 ; Mishra20 and so on. Several researchers have studied the bouncing cosmological models in $f(R,T)$ gravity. Singh et al. have generated a new parametrization for the Hubble parameter to demonstrate the bouncing scenario in the cosmological model Singh18 . Classical bouncing solutions were examined in the setting of $f(R,T)$ gravity in a flat FLRW background with a perfect fluid as the only matter content by Shabani and Ziaie Shabani18 . Tripathy and Mishra Tripathy19 and Tripathy et al. Tripathy20a have used different bouncing scale factors to frame the model in extended gravity theory. Mishra et al., Mishra19 looked at the ramifications of including the Bianchi type I metric into the $f(R,T)$ gravity theory field equations, focusing on solutions for a matter-dominated Universe. In the isotropic backdrop, Agrawal et al., investigated the matter bounce scenario in $f(R,T)$ gravity Agrawal21a and $f(Q,T)$ gravity Agrawal21b with numerous scale factors.

Tolman Tolman31a studies the theoretical conditions for nonstatic models of the universe to display periodic behaviour in time using relativistic mechanics and relativistic thermodynamics. In simplistic models, loop quantum cosmology predicts that the big bang will be replaced by a quantum bounce. A natural question is whether the Universe preserves its memory of the preceding period after the bounce. Corichi and Singh Corichi08a show that this question can be addressed unambiguously, at least within a model that can be solved accurately. Machando et al. create a nonperturbative flow equation for modified gravity theories of the type $S=\int d^{4}x\sqrt{-g}f(R)$ using the functional renormalization group equation for quantum gravity. The Machado and Saueressig Machado08a use this equation to show that certain gravitational interaction monomials can be decoupled from the renormalization group flow in a coherent manner, and they duplicate recent results on the asymptotic safety conjecture. Cailleteau et al. Cailleteau09a used the effective framework of loop quantum cosmology in the presence of anisotropies to investigate the impact of nonperturbative quantum gravity effects in the ekpyrotic/cyclic model. Although various flaws in $f(R)$ gravity have been demonstrated, Sotiriou and Faraoni Sotiriou10a assessed it as a class of toy theories. Ashtekar and Singh Ashtekar11a discussed loop quantum cosmology (LQC), which is the consequence of applying loop quantum gravity (LQG) concepts to cosmological settings, and gave an outline of the current status of the field. A spin-spin interaction is generated by the minimum coupling between the torsion tensor and Dirac spinors, which is relevant in fermionic matter at extremely high densities. Poplawski Poplawski12a show that such an interaction prevents the unphysical Big Bang singularity by causing a cusp like bounce at a finite minimum scale factor, which prevents the Universe from contracting. This theory also explains why the current Universe appears spatially flat, homogeneous and isotropic at the largest scales. Benedetti and Caravelli Benedetti12a proposed that a generic $f(R)$ ansatz for the effective action serves a role akin to the local potential approximation (LPA) in scalar field theory in the context of the functional renormalization group flow of gravity.

Tarai et al., Tarai18 studied extended gravity, namely the $f(R,T)$ gravity, in a Bianchi VI_h=-1 space time filled with magnetised anisotropic matter. In General Relativity (GR) and $f(R,T)$ gravity, locally rotationally symmetric (LRS) Bianchi type-I viscous and non viscous cosmological models are investigated Jokweni21 . Energy conditions Alvarenga13a ; Sharif13a ; Sharif13b , perturbation analysis and stability analysis Alvarenga13b ; Sharif20 , cosmological solutions Shabani13 ; Moraes15 , dynamical instability Noureen14 and astrophysical scenarios Deb19 ; Maurya19 have all been widely explored using the $f(R,T)$ theory.

The paper is organized as follow: in section II, the basic formalism of the modified gravity model has been presented. In section III, the cosmological model has been set up with the bouncing scale factor by deriving the physical parameters. Within the bouncing scenario, different dynamical parameters are obtained. The energy conditions of the model has been analysed. The models are tested through the calculation of different cosmographic coefficients. Also, we have presented a scalar field reconstruction of the bouncing model. In section IV, a linear homogeneous and isotropic perturbation calculation have been carried out to test the stability of the model. At the end, the summary and conclusion of the work are presented in section V.

The $f(R,T)$ theory of gravity is another geometrical extension of GR. The action of this extended gravity can be written as,

$$S=\int d^{4}x\sqrt{-}g\mathcal{L}_{m}+\frac{1}{16\pi}\int d^{4}x\sqrt{-}gf(R,T),$$

(1)

where $\mathcal{L}_{m}$ be the matter Lagrangian and $f(R,T)$ be the function of the Ricci scalar $R$ and trace of energy momentum tensor $T$. The trace can be expressed as, $T=T_{ij}g^{ij}$. The $f(R,T)$ gravity field equations can be written as Harko11 ; Mishra18 ,

$$f_{R}(R)R_{ij}-\frac{1}{2}f(R)g_{ij}+(g_{ij}\Box-\nabla_{i}\nabla_{j})f_{R}(R)=[8\pi+f_{T}(T)]T_{ij}+\left[pf_{T}(T)+\frac{1}{2}f(T)\right]g_{ij},$$

(2)

where $f_{R}$ and $f_{T}$ respectively represent the partial derivatives with respect to $R$ and $T$. In the present work, we consider linear functions for $f(R)$ and $f(T)$ such as $f(R)=\lambda R$ and $f(T)=\lambda T$. For this choice, we have $f(R,T)=\lambda\left(R+T\right)$, where $\lambda$ is the non-zero finite rescaling factor. Now, the field equations (2) can be reduced to,

$$R_{ij}-\frac{1}{2}Rg_{ij}=\left(\frac{8\pi}{\lambda}+1\right)T_{ij}+\left(\frac{T}{2}+p\right)g_{ij}.$$

(3)

Here, $\lambda$ rescales the usual field equations in GR. If the above $f(R,T)$ gravity field equation be compared with the usual GR field equations $R_{ij}-\frac{1}{2}Rg_{ij}=\kappa T_{ij}+\Lambda g_{ij}$, then the term $\left(\frac{T}{2}+p\right)$ may be identified as a time varying cosmological constant. Also, we have a redefined Einstein constant $\kappa=\frac{8\pi}{\lambda}+1$. One should note that, this type of gravity model can not be reduced to GR. In this minimally coupled field equations (3), we shall frame and investigate the cosmological model of the Universe in an anisotropic LRS Bianchi I (LRSBI) space-time, which can be considered as,

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

(4)

where the metric potentials $A$ and $B$ are the functions of the cosmic time $t$. We consider the matter field of the Universe as,

$$T_{ij}=(p+\rho)u_{i}u_{j}-pg_{ij}-\rho_{B}x_{i}x_{j},$$

(5)

where $p$ and $\rho$ respectively be the matter pressure and energy density. $\rho_{B}$ represents the anistropic fluid so that the the energy density $\rho$ is due to the perfect fluid and the anisotropic fluid. In a co-moving coordinate system, $u^{i}u_{i}=1$, $u^{i}x_{i}=0$ and $x^{i}x_{i}=-1$. Now, we can derive the field equations (3) as,

	$\displaystyle 2\frac{\ddot{B}}{B}+\frac{\dot{B}^{2}}{B^{2}}$	$\displaystyle=$	$\displaystyle-\alpha\left(p-\rho_{B}\right)+\dfrac{\rho}{2},$		(6)
	$\displaystyle\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}}{A}.\frac{\dot{B}}{B}$	$\displaystyle=$	$\displaystyle-\alpha p+\frac{\rho_{B}}{2}+\frac{\rho}{2},$		(7)
	$\displaystyle 2\frac{\dot{A}}{A}.\frac{\dot{B}}{B}+\frac{\dot{B}^{2}}{B^{2}}$	$\displaystyle=$	$\displaystyle-\frac{p}{2}+\frac{\rho_{B}}{2}+\alpha\rho.$		(8)

Where $\alpha=\frac{8\pi}{\lambda}+\frac{3}{2}$. An overhead dot on the field variables represents the ordinary derivative with respect to the cosmic time. The above set of field equations (6)-(8) can also be expressed in terms of the directional Hubble parameters that are defined as, $H_{x}=\frac{\dot{A}}{A}$ and $H_{y}=H_{z}=\frac{\dot{B}}{B}$. Since the field equations (6)-(8) contain more number of unknowns than the equations, we have the flexibility to make some assumptions among the field variables or the state of matter. We prefer to use an anisotropic relation between the field variables, as $H_{x}=kH_{y}$, for $k\neq 1$. Further the Hubble parameter and scale factor can be related as, $H=\frac{\dot{\mathcal{R}}}{\mathcal{R}}$ and the average Hubble parameter is, $H=\left(\frac{k+2}{3}\right)H_{y}$. With these transformations, we can write the field equations (6)-(8) as,

$$\left[\frac{6}{k+2}\right]\dot{H}+\left[\frac{27}{(k+2)^{2}}\right]H^{2}=-\alpha p+\alpha\rho_{B}+\frac{\rho}{2},$$

(9)

$$\left[\frac{3(k+1)}{k+2}\right]\dot{H}+\left[\frac{9(k^{2}+k+1)}{(k+2)^{2}}\right]H^{2}=-\alpha p+\frac{\rho_{B}}{2}+\frac{\rho}{2},$$

(10)

$$\left[\frac{9(2k+1)}{(k+2)^{2}}\right]H^{2}=-\frac{p}{2}+\frac{\rho_{B}}{2}+\alpha\rho.$$

(11)

Solving the above set of equations (9)-(11), we obtain the pressure and energy density as,

$$p=\dfrac{6}{(1-4\alpha^{2})\left(k+2\right)^{2}}\left[\left\{\left(k-1\right)+2\alpha\left(k+1\right)\right\}\left(k+2\right)\dot{H}+\left\{3\left(k^{2}-k-3\right)+6\alpha\left(k^{2}+k+1\right)\right\}H^{2}\right],$$

(12)

$$\rho=\dfrac{6}{(1-4\alpha^{2})\left(k+2\right)^{2}}\left[2(k+2)\dot{H}+3\left\{3-2\alpha\left(2k+1\right)\right\}H^{2}\right].$$

(13)

The equation of state parameter is defined as $\omega=\frac{p}{\rho}$ which may be obtained from the expressions of the pressure and energy density as

$\displaystyle\omega$

$\displaystyle=$

$\displaystyle-1+(1+2\alpha)\left[\dfrac{(k^{2}+3k+2)\dot{H}+3k(k-1)H^{2}}{(2k+4)\dot{H}+({9-6\alpha(2k+1))}H^{2}}\right].$

The effective cosmological constant, $\Lambda=\frac{T}{2}+p$ may be obtained in a straightforward manner as

$\displaystyle\Lambda$

$\displaystyle=$

$\displaystyle\dfrac{6}{\left(1+2\alpha\right)\left(k+2\right)}\left(\dot{H}+3H^{2}\right).$

(15)

In the present work, we wish to investigate a matter bounce scenario within the framework of the extended gravity and the formalism developed in this section. In view of this, we wish to consider a scale factor that mimics a matter bounce scenario in the foregoing expressions of the dynamical parameters and study the influence of the model parameters on their dynamical behaviour.

In order to find the behaviour of the Universe, we need to know the physical and dynamical parameters. All the physical parameters are expressed in form of the Hubble parameter. It is to mention here that the inflationary scenario fails to get the complete past history of the Universe. So, as a possible solution to this issue, the matter bounce has been proposed. In the present work, we are interested to the matter bounce scenario with the corresponding scale factor $\mathcal{R}=\left(1+\frac{3}{4}\rho_{c}t^{2}\right)^{\frac{n}{3}}$, such that the Hubble parameter can be presented in the form Odintsov16 ,

$$H(t)=\frac{\dot{\mathcal{R}}}{\mathcal{R}}=\dfrac{\frac{n}{2}\rho_{c}t}{(\frac{3}{4}\rho_{c}t^{2}+1)}.$$

(16)

Here we have considered a bouncing cosmological model described by a given scale factor motivated from loop quantum gravity (LQG) and have studied the consequent dynamical behaviour of the model. Also, $\rho_{c}$ is a constant parameter that captures the quantum nature of the space-time. This nature observed to be promising in the study of matter bounce scenario Cai11b ; Ewing13 ; Odintsov14 . $n$ is an integer and for the sake of brevity, we consider $n=1$ and 2 in our work. The bouncing scale factor ensures a bouncing scenario at $t_{b}=0$. For this bouncing scale factor we have $\dot{H}=\dfrac{\frac{n}{2}\rho_{c}}{(\frac{3}{4}\rho_{c}t^{2}+1)}-\dfrac{\frac{3n}{4}\rho_{c}^{2}t^{2}}{(\frac{3}{4}\rho_{c}t^{2}+1)^{2}}$. The above bouncing scale factor satisfies the bouncing conditions $H<0$ for $t<t_{b}$, $H>0$ for $t>t_{b}$, at $t=t_{b}$, $H=0$. Also, at $t=t_{b}$, we have $\dot{H}=\frac{n}{2}\rho_{c}>0$.

The deceleration parameter for the above mentioned bouncing scenario is given by

$$q=-1-\frac{\dot{H}}{H^{2}}=-1-\frac{4-3\rho_{c}t^{2}}{2n\rho_{c}t^{2}}.$$

(17)

We show the evolution of the bouncing scale factor, the Hubble parameter and the deceleration parameter in FIG.1 (left panel),FIG.1 (right panel) and FIG.2 respectively. One should note that, the bouncing scale factor, the Hubble parameter and the deceleration parameter depend on the parameters $n$ and $\rho_{c}$. $n$ decides the stiffness in the scale factor. In order to plot the figures, we chose a representative value of the parameter as $\rho_{c}=0.12$. The three figures show a symmetric bounce around $t_{b}=0$. For a given value of $n$, the scale factor symmetrically increases as we move away from the bouncing epoch. The Hubble parameter assumes negative values in the pre-bounce epoch and positive values after the bounce and vanishes at the bounce. Consequently, the deceleration parameter becomes a negative quantity in the pre and post bounce scenario. With the advancement in time away from the bounce, it goes on increasing to become $q=-0.5$ at late times. Near the bounce, due to the characteristic behaviour of the bouncing model, the deceleration parameter shows a singularity.

Depending on the generation era of perturbation modes, bounce models can be divided into two categories Nojiri19 ; Odintsov20 . First the Hubble radius ($r_{h}=\frac{1}{aH}$) asymptotically diverges to infinity and the perturbations in the sub-Hubble regime generate far away before the bounce. Second, when the Hubble radius approaches zero asymptotically. This is because all of the perturbation modes are within the horizon at that time and the perturbations occur near the bounce. A diverging Hubble radius Fig.3 (left panel), asymptotically correlates to a decelerating Universe in late time for $n=1$. In such scenarios, the perturbation modes are created at very large negative cosmic times, corresponding to the low curvature regime of the contracting era. The Hubble radius drops monotonically on both sides of the bounce before asymptotically shrinking to zero Fig.3 (right panel), on the other hand, indicating an accelerating late time Universe. As a result, in such instances, the Hubble horizon shrinks to zero for large values of cosmic time and only the Hubble horizon has an infinite dimension near the bouncing point. So, the primordial perturbation modes relevant to the present time era are formed for cosmic times near the bouncing point, because all the primordial modes are contained in the horizon only at that time. The modes escape the horizon when the horizon shrinks and become relevant for present-day observations. The $f(R)$ gravity theory is not compatible with the Planck constraints Nojiri19 ; Odintsov20 . Actually the tensor to scalar ratio becomes of the order unity, which is not consistent with the Planck constraints. One should note from FIG.3 that, the Hubble horizon form the bouncing scenario with $n=2$ drops monotonously on both the sides of the bounce signifying a late time accelerated expansion. Also, such a behaviour is compatible with recent Planck’s constraints Cai11b ; Ewing13 ; Odintsov14 ; Nojiri19 ; Odintsov20 ; Balbi07 ; Xu12 ; Mishra21 .

In the present section, we study the stability of the presented model against homogeneous and isotropic linear perturbation for which we consider an FRW background with a general solution $H(t)=H_{b}(t)$. Following Ref.Dombriz2012 ; Agrawal21a , we consider perturbations of the Hubble parameter and the energy density around the arbitrary solutions $H_{b}(t)$ respectively as $H(t)=H_{b}\left(1+\delta(t)\right)$ and $\rho(t)=\rho_{b}\left(1+\delta_{m}(t)\right)$. Here $\delta_{m}(t)$ and $\delta(t)$ are the respective deviations from the background energy density and the Hubble parameter. For the extended gravity theory considered in the present work, we have $f(R,T)=\lambda(R+T)$ which may be expanded in powers of $R_{b}$ and $T_{b}$ as

$$f(R,T)=f_{b}+\lambda(R-R_{b})+\lambda(T-T_{b})+\mathcal{O}^{2},$$

(36)

where the term $\mathcal{O}^{2}$ includes all the higher powers of $R$ and $T$. Considering the linear homogeneous and isotropic perturbation in the equivalent FRW equation, we obtain

$$6H_{b}^{2}\delta(t)=\alpha\rho_{b}\delta_{m}(t),$$

(37)

where $\rho_{b}=\frac{6H_{b}^{2}}{1+2\alpha}$. The above equation clearly correlates the perturbation in geometry with the matter perturbation.

It is straightforward to express the matter perturbation in terms of the geometry perturbation as

$$\delta_{m}(t)=\left(\frac{1+2\alpha}{\alpha}\right)\delta(t).$$

(38)

From the conservation equation, we may obtain

$$\dot{\delta}_{m}(t)+3H_{b}(t)\delta(t)=0,$$

(39)

which may be expressed as

$$\dot{\delta}(t)+\frac{3\alpha H_{b}}{1+2\alpha}\delta(t)=0.$$

(40)

On integrating the above equation (40), we obtain the geometrical perturbation as

$$\delta(t)=\delta_{0}\mathcal{R}_{b}^{-\frac{3\alpha}{1+2\alpha}},$$

(41)

where $\delta_{0}$ is an integration constant. For the matter bounce scenario considered in the present work, we get the geometry perturbation as

$$\delta(t)=\delta_{0}\left(1+\frac{3}{4}\rho_{c}t^{2}\right)^{-\frac{n\alpha}{1+2\alpha}},$$

(42)

and the matter perturbation as

$$\delta_{m}(t)=\frac{(1+2\alpha)\delta_{0}}{\alpha}\left(1+\frac{3}{4}\rho_{c}t^{2}\right)^{-\frac{n\alpha}{1+2\alpha}}.$$

(43)

It may be noted here that, these linear perturbations depend on the parameters $n$ and $\alpha$. The stability behaviour depends on the factor $\alpha$ in equation (43). When $\alpha\geq-0.50$ the matter and geometry perturbations tend to increase leading to an instability of the model. However, for $\alpha<-0.50$, the perturbations decay out both sides of the bounce ensuring the stability of the model.

It is to be noted here that, besides the stability analysis, a bouncing model should be tested through scalar and tensor perturbations with the calculation of certain indices such as the spectral index of scalar perturbations ($n_{s}$), the tensor to scalar ratio ($r$) and the running spectral index $\frac{dn_{s}}{dln~{}k}$ evaluated at the time of horizon exit and to be confronted with observational data. The Planck 2018 results constrains these parameters as $n_{s}=0.9646\pm 0.0042$, $r<0.064$ and $\frac{dn_{s}}{dln~{}k}=-0.0085\pm 0.0073$ Planck2018 . In a recent work, Elizalde et al. have carried out a detailed analysis of the scalar and tensor perturbations in a modified gravity theory dubbed as the $f(R,G)$ gravity and have shown that, for a pure bouncing scenario, the spectral index becomes one as that in a scalar-tensor gravity theory Elizalde2020 . However, in the present work, we have not calculated all those spectral indices but we have shown that a bouncing scenario is viable in the $f(R,T)$ modified gravity theory.

Within the framework of an extended gravity theory, we have constructed a cosmological model that mimics a matter bounce scenario having a pre-bounce contraction phase followed by a matter dominated expansion at the post bounce phase. As we can see, the EoS parameter obtains a value that is smaller than $-1/3$ at late epoch. Furthermore, based on a graphical representation of the Hubble radius, it has been discovered that the Universe is experiencing a late deceleration for $n=1$ and a late time acceleration for $n=2$. It is to note that, a decelerating behaviour is incompatible with observation datasets such as Planck and WMAP, which indicate the late-time acceleration phenomenon of the Universe. However, this is not the case with $\omega_{eff}=-1-2\dot{H}/3H^{2}$; the $\omega_{eff}$ exhibits singular behaviour at the bounce epoch as well as late time deceleration behaviour. As a result, the matter bounce hypothesis with $n=1$ misses the late acceleration era, particularly the dark energy epoch. We have derived the dynamical parameters of the model such as the energy density, pressure and equation of state parameter in terms of the extended gravity parameter and the bouncing scenario factor. The dynamical evolution of these quantities are studied. Also, we have investigated the effect of the model parameters on the dynamics of these parameters. It is found that, the extended gravity parameter $\alpha$ has a substantial effect on the evolutionary aspect of the energy density, pressure and the equation of state parameter. For a given value $\alpha$, the energy density decreases symmetrically as we move away from the bouncing epoch. On the other hand, the pressure increases symmetrically in both the sides of the bounce. At the bounce, while the energy density shows a peak, the pressure curve shows a well. The peak in the energy density or the well in the pressure depends on the choice of the extended gravity parameter $\alpha$. For higher values of $\alpha$, we get, more or less flat curves for the energy density and pressure. Similarly, $\alpha$ affects the evolution of the equation of state parameter. For a given value of $\alpha$, the equation of state parameter decreases from a quintessence field like region to a phantom-field like region by crossing through the phantom divide. A higher value of $\alpha$ leads to a sharp decrease in $\omega$. From the calculation of the energy condition for the given model, it is found that, while the dominant energy condition is satisfied, the strong energy condition is violated. The model satisfies the null energy condition most of the time but marginally violates at a late epoch far away from the bounce. From the analysis of the diagnostic parameters, it is concluded that, mostly the model resembles a phantom like model. We have carried out a scalar field reconstruction of the bouncing scenario and obtain the scalar field and the self interaction potential. The scalar field is observed to roll down the potential over a period both in the pre and post bounce phase. The stability of the models are tested under linear homogeneous and isotropic perturbations.

Since, a matter bounce scenario is an interesting aspect to avoid the possible singularity occurring in the usual GR models, the present model may provide some useful insight into the issue. Also, in some earlier works Tripathy19 ; Tripathy20a , it is shown that, in GR, there occurs a omega-singularity at the bouncing epoch and the role of an extended gravity removes that singularity. In the present work also, we obtain similar feature, where at the bounce, we get finite values for the equation of state parameter.

ASA acknowledges the financial support provided by University Grants Commission (UGC) through Senior Research Fellowship (File No. 16-9 (June 2017)/2018 (NET/CSIR)), to carry out the research work. BM and SKT acknowledge the support of IUCAA, Pune (India) through the visiting associateship program. The authors are thankful to the honourable reviewers for their valuable suggestions and comments to improve the quality of the paper.