# Bouncing cosmological solutions from \(f(\mathsf{R,T})\) gravity

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## Abstract

In this work we study classical bouncing solutions in the context of \(f(\mathsf{R},\mathsf{T})=\mathsf{R}+h(\mathsf{T})\) gravity in a flat FLRW background using a perfect fluid as the only matter content. Our investigation is based on introducing an effective fluid through defining effective energy density and pressure; we call this reformulation as the “*effective picture*”. These definitions have been already introduced to study the energy conditions in \(f(\mathsf{R},\mathsf{T})\) gravity. We examine various models to which different effective equations of state, corresponding to different \(h(\mathsf{T})\) functions, can be attributed. It is also discussed that one can link between an assumed \(f(\mathsf{R},\mathsf{T})\) model in the effective picture and the theories with generalized equation of state (EoS). We obtain cosmological scenarios exhibiting a nonsingular bounce before and after which the Universe lives within a de-Sitter phase. We then proceed to find general solutions for matter bounce and investigate their properties. We show that the properties of bouncing solution in the effective picture of \(f(\mathsf{R},\mathsf{T})\) gravity are as follows: for a specific form of the \(f(\mathsf{R,T})\) function, these solutions are without any future singularities. Moreover, stability analysis of the nonsingular solutions through matter density perturbations revealed that except two of the models, the parameters of scalar-type perturbations for the other ones have a slight transient fluctuation around the bounce point and damp to zero or a finite value at late times. Hence these bouncing solutions are stable against scalar-type perturbations. It is possible that all energy conditions be respected by the real perfect fluid, however, the null and the strong energy conditions can be violated by the effective fluid near the bounce event. These solutions always correspond to a maximum in the real matter energy density and a vanishing minimum in the effective density. The effective pressure varies between negative values and may show either a minimum or a maximum.

## 1 Introduction

Today, the standard cosmological model (SCM) or the big-bang cosmology has become the most acceptable model which encompasses our knowledge of the Universe as a whole. For this reason it is called also the “concordance model” [1]. This model which allows one to track the cosmological evolution of the Universe very well, has matured over the last century, consolidating its theoretical foundations with increasingly accurate observations. We can numerate a number of the successes of the SCM at the classic level. For example, it accounts for the expansion of the Universe (Hubble law), the black body nature of cosmic microwave background (CMB) within the framework of the SCM can be understood and the predictions of light-element abundances which were produced during the nucleosynthesis. It also provides a framework to study the cosmic structure formation [2]. However, though the SCM works very well in fitting many observations, it includes a number of deficiencies and weaknesses. For instance some problems which are rooted in cosmological relics such as magnetic monopoles [3, 4], gravitons [5, 6, 7, 8, 9], moduli [10, 11, 12, 13] and baryon asymmetry [14, 15]. Despite the self-consistency and remarkable success of the SCM in describing the evolution of the Universe back to only one hundredth of a second, a number of unanswered questions remain regarding the initial state of the Universe, such as flatness and horizon problems [16, 17, 18, 19]. Moreover, there are some unresolved problems related to the origin and nature of dark matter (DM) [20, 21, 22]. Notwithstanding the excellent agreement with the observational data there still exists a number of challenging open problems associated with the late time evolution of the Universe, namely the nature of dark energy (DE) and cosmological constant problem [23, 24, 25]. Though the inflation mechanism has been introduced to treat some of the mentioned issues such as, the horizon, flatness and magnetic monopole problems at early Universe [26, 27, 28], the SCM suffers from a more fundamental issue, i.e., the *initial cosmological singularity* that the existence of which has been predicted by the pioneering works of Hawking, Penrose and Geroch in 1960s, known as the singularity theorems [29, 30, 31, 32, 33, 34, 35, 36, 37] and their later extensions by Tipler in 1978 [38, 39] and by Borde, Vilenkin and Guth in the 1990s [40, 41, 42, 43, 44, 45] (see also [46] for a comprehensive study). According to these theorems, a cosmological singularity is unavoidable if spacetime dynamics is described by General Relativity (GR) and if matter content of the Universe obeys certain energy conditions. A singular state is an extreme situation with infinite values of physical quantities, like temperature, energy density, and the spacetime curvature from which the Universe has started its evolution at a finite past. The existence of such an uncontrollable initial state is irritating, since “*a singularity can be naturally considered as a source of lawlessness*” [47]. A potential solution to the issue of cosmological singularity can be provided by “*non-singular bouncing cosmologies*” [48, 49]. Beside a huge interest in the solutions that do not display singular behavior, there can be more motivations to seek for non-singular cosmological models. The first reason for removing the initial singularity is rooted in the initial value problem since a consistent gravitational theory requires a well-posed Cauchy problem [50, 51]. However, owing to the fact that the gravitational field diverges at a spacetime singularity, we could not have a well formulated Cauchy problem as we cannot set the initial values at a singular spatial hypersurface given by \(t=const.\) Another related issue is that the existence of a singularity is inconsistent with the entropy bound \(S/E=(2\pi R)/c\hbar \), where *S*, *E*, *R*, \(\hbar \) and *c* being entropy, proper energy, the largest linear dimension, Planck’s constant and the velocity of light, respectively [47].

During the past decades, models which describe bouncing behavior have been designed and studied as an approach to resolve the problem of initial singularity. These models suggest that the Universe existed even before the big-bang and underwent an accelerated contraction phase towards reaching a non-vanishing minimum radius. The transition from a preceding cosmic contraction regime to the current accelerating expansion phase (as already predicted in SCM) is the so called “*Big Bounce*”. From this perspective, the idea that the expansion phase is preceded by a contraction phase paves a new way towards modeling the early Universe and thus, may provide a suitable setting to obviate some of the problems of the SCM without the need to an inflationary scenario. Although an acceptable model can be considered as the one being capable of explaining the issues that have been treated by inflationary mechanism, e.g., most inflationary scenarios can give the scale-invariant spectrum of the cosmological perturbations [52], problems of the SCM may find solutions in the contracting regime before the bounce occurs. The horizon problem, for example, is immediately resolved if the far separated regions of the present Universe were in causal connection during the previous contraction phase. Similarly, the homogeneity, flatness, and isotropy of the Universe may also be addressed by having a smoothing mechanism in the contraction phase, see e.g., [53, 54, 55] for more details. Moreover, though the fine-tuning is required to keep a stable contracting regime, the nonsingular bounce succeeds in sustaining a nearly scale-invariant power spectrum [56, 57].

Another type of theories are called non-singular “*matter bounce*” scenarios which is a cosmological model with an initial state of matter-dominated contraction and a non-singular bounce [81]. Such a model provides an alternative to inflationary cosmology for generating the observed spectrum of cosmological fluctuations [53, 54, 55, 82, 83, 84, 85]. In these theories some matter fields are introduced in such a way that the WEC is violated in order to make \(\dot{H}>0\) at the bounce. From Eq. (2), it is obvious that putting aside the correction term leads to negative values for the time derivative of the Hubble parameter for all fluids which respect WEC. Therefore, in order to obtain a bouncing cosmology it is necessary to either go beyond the GR framework, or else to introduce new forms of matter which violate the key energy conditions, i.e., the null energy condition (NEC) and the WEC. For a successful bounce, it can be shown that within the context of SCM the NEC and thus the WEC, are violated for a period of time around the bouncing point. In the context of matter bounce scenarios, many studies have been performed using quintom matter [86, 87, 88], Lee–Wick matter [89], ghost condensate field [90], Galileon fields [91, 92] and phantom field [93, 94, 95, 96, 97]. Cosmological bouncing models have also been constructed via various approaches to modified gravity such as \(f(\mathsf{R})\) gravity [98, 99, 100, 101, 102], teleparallel \(f(\mathsf{T})\) gravity [103, 104], brane world models [105], Einstein–Cartan theory [106, 107, 108, 109, 110, 111, 112, 113], Horava–Lifshitz gravity [114], nonlocal gravity [115, 116] and others [117]. There are also other cosmological models such as Ekpyrotic model [118, 119] and string cosmology [120, 121, 122, 123, 124] which are alternatives to both inflation and matter bounce scenarios.

*modified equation of state*” (MEoS) models. This branch of research presumes a mysterious fluid(s) specified by an unusual EoS with the hope of dealing with some unanswered questions in the cosmological realm. For example some relevant works in the literature can be addressed as follows; in [148] the author has employed an EoS of the form \(p=-\rho +\gamma \rho ^{\lambda }\) in order to obtain power-law and exponential inflationary solutions. The case with \(\lambda =1/2\) has been analyzed in [147, 150, 151] to focus on the future expansion of the Universe. Emergent Universe models have been studied in [149] by taking into account an exotic component with \(p=A\rho -B \rho ^{1/2}\) and in [152] with \(A=-1\). Different cosmological aspects of DE with more simple form of EoS, i.e, \(p_{\mathsf{DE}}=\alpha (\rho _{\mathsf{DE}}-\rho _{0})\) have been investigated in [153] and the study of cosmological bouncing solutions can be found in [52].

Therefore, recasting the \(f(\mathsf{R},\mathsf{T})\) field equations into the “*effective picture*” may provide a bridge to the cosmological models supported by MEoS. Via this connection the problem of an exotic fluid turns into the problem of a usual fluid with exotic gravitational interactions. However, contrary to the former, in the latter case we start with a predetermined Lagrangian, i.e., \(f(\mathsf{R},\mathsf{T})\) gravitational Lagrangian. The importance of the effective picture becomes more clear when one considers the energy conditions in \(f(\mathsf{R},\mathsf{T})\) gravity. As discussed in [154], in \(f(\mathsf{R},\mathsf{T})\) gravity the energy conditions would be obtained for effective pressure and effective energy density. Therefore, it is reasonable to define a fluid as a source with effective pressure and energy density. As we shall see, the bouncing solutions in \(f(\mathsf{R},\mathsf{T})\) gravity (using only one perfect fluid in a flat FLRW background) in the framework of our effective fluid approach, exhibit nonsingular properties such that in a finite value of the bounce time \(t_{\mathsf{b}}\), non of the cosmological quantities would diverge. More exactly, as \(t\rightarrow t_{\mathsf{b}}\) we observe that the scale factor decreases to a minimum non-vanishing value, i.e., \(a\rightarrow a_{\mathsf{b}}\), \(H\big |_{t\rightarrow t_\mathsf{b}}\!\!\!\rightarrow 0\), \(\rho \big |_{t\rightarrow t_\mathsf{b}}\!\!\!\rightarrow \rho _{\mathsf{b}}\), \(\rho _{\mathsf{(eff)}}\big |_{t\rightarrow t_\mathsf{b}}\!\!\!\rightarrow 0\) and \(p_{\mathsf{(eff)}}\big |_{t\rightarrow t_\mathsf{b}}\!\!\!\rightarrow {p_{\mathsf{(eff)}}}_{\mathsf{b}}\). Therefore, non of the future singularities would appear. Also, in all cases we have \(\mathcal {W}\big |_{t\rightarrow t_\mathsf{b}}\!\!\!\rightarrow -\infty \), where \(\mathcal {W}\) being the effective EoS parameter and subscript “b” stands for the value of quantities at the time at which the bounce occurs. We then observe that if we want to describe nonsingular bouncing solutions in \(f(\mathsf{R},\mathsf{T})=\mathsf{R}+h(\mathsf{T})\) gravity using a minimally coupled scalar field, a phantom field should be employed. These solutions show a violation of the NEC in addition to the strong energy condition SEC. Such a behavior is predicted in GR for a perfect fluid in FLRW metric with \(k=-1,0\) [155].

The current research is planned as follows. In Sect. 2 we briefly present the effective fluid picture. Sect. 3 is devoted to the bouncing solutions with asymptotic de Sitter behavior before and after the bounce. We first analyze models with constant effective pressure in Sect. 3.1, they are called models of type A. We then proceed to investigate the corresponding bouncing solutions, the energy conditions, the scalar field representation and finally the stability of these type of solutions. In Sect. 3.2 models which correspond to two different EoSs assuming \(p_{(\mathsf{eff})}=\mathcal {Y}(\rho _{(\mathsf{eff})})\) will be discussed. These models are named as B, C, D and E models. An example of the matter bounce solution is considered in Sect. 4 which is labeled as model E. The connection of A-E models with MEoS theories will be presented through the effective picture. Section 5, is devoted to study of scalar-type cosmological perturbations. In Sect. 6 we give a brief review of singular models in the context of \(f(\mathsf{R,T})\) gravity and obtain a class of solutions exhibiting singular behavior. Finally, in Sect. 7 we summarize our conclusions.

## 2 Reformulation of \(f(\mathsf{R},\mathsf{T})\) field equations in terms of a conserved effective fluid

*a*(

*t*) is the scale factor of the universe and \(d\Omega ^2\) is the standard line element on a unit two sphere. Applying metric (11) to field equation (8) together with using the definition for \(\mathbf {\Theta _{\mu \nu }}\) for a perfect fluid leads to

*H*indicates the Hubble parameter. Note that, we have used \(\mathsf{L}^\mathsf{{\mathsf{(m)}}}=p\) for a perfect fluid in the expression (9). Applying the Bianchi identity to the field equation (8) gives following covariant equation

## 3 Asymptotic de-Sitter bouncing solutions in \(f(\mathsf{R},\mathsf{T})\) gravity

In this section we study different bouncing cosmological solutions of \(f(\mathsf{R},\mathsf{T})\) gravity. We extract those solutions which correspond to some properties of the effective fluid. Such an approach may help us to understand how these solutions can emerge in \(f(\mathsf{R},\mathsf{T})\) gravity. Furthermore, there can be obtained more bouncing solutions, however, to show that \(f(\mathsf{R},\mathsf{T})\) gravity theories are capable of describing a non-singular pre-Big Bang era, we are restrict ourselves to study only few examples. We set \(\kappa ^{2}=1\) in the rest of the work.

### 3.1 Type A models: solutions which correspond to a constant effective pressure, \(p_{(\mathsf{eff})}(\mathsf{T})=\mathcal {P}\)

As can be seen from Fig. 1, the effective density reduces from a constant value and tends to zero near the bounce. From Eq. (19) we see that the vanishing of Hubble parameter at the bounce demands that the effective density becomes zero. Also for the same reason, the effective EoS diverges at the bounce. Such behaviors are common for all bouncing models that we shall present in the framework of minimal \(f(\mathsf{R},\mathsf{T})\) gravity. the matter energy density itself increases from small values to a maximum value near the bounce. Based on the exchange of energy between gravitational field and matter constituents (that the mechanism of which is explained in [128]) one may explain the process of bouncing behavior; the interaction of the real fluid with curvature leads to some transformations of energy from gravitational field to matter before the bounce where the spacetime curvature is dominant in comparison to matter energy density. Such a transmutation, that the start of which is triggered at the time far past the bounce, gives rise to an increase in the energy density as the bounce event is approached. At the bounce time the energy density of matter grows to a maximum value after which the process of transmutation is reversed until the density falls back to zero (the post-bounce regime). Note that the effective energy density remains constant in the de-Sitter era. However, some physics is needed in order to explain the process of matter production from curvature component which disturbs the stability of de-Sitter era to enter the bounce event.

*w*. We plot the diagrams for \(w=0.6\) in Fig. 1. This figure shows that the NEC and SEC are violated in this case. Our studies show that the bouncing behavior is achieved from solution (28) for \(w>1/3\). Note that, as we have mentioned before, solution (28) is only one of the three possible solutions of Eq. (27). Investigating other solutions may validate the cases with \(w<1/3\) from energy conditions point of view.

*t*, which gives

*H*and \(\rho \) as dynamical variables, we arrive at the following dynamical system

*w*. Due to the appearance of the zero eigenvalues, one may not decide about the stability properties of these fixed points, however, by inspecting Eqs. (46) and (47) it is possible to figure out the nature of the fixed points. For fixed point \(\mathsf{P}^{(-)}\) Eq. (47) becomes

*H*on vertical and horizontal axises, respectively, we have \( \dot{H}>0~ \& ~\dot{\rho }>0\) for all points with \(\rho >0\) and \( \dot{H}<0~ \& ~ \dot{\rho }<0\) for points with \(\rho <0\). These show that when \(\rho \rightarrow \rho +\delta \rho \) for \(t\rightarrow t+\delta t\), the solution at \(\mathsf{P}^{(-)}\) will not stay stationary and hence it is a repulsive fixed point. On the other hand, for \(\mathsf{P}^{(+)}\) we have

### 3.2 Solutions which correspond to a general effective EoS, \(p_{(\mathsf {eff})}=\mathcal {Y}(\rho _{(\mathsf {eff})})\)

These class of models can be constructed by imposing a particular condition on the effective profiles. This approach can be viewed as a sort of classification of \(f(\mathsf{R},\mathsf{T})\) gravity models based on the properties of the effective quantities. Generally, one can obtain a class of \(h(\mathsf{T})\) functions for a determined property which is specified by an effective EoS. In the following sections we consider two subclasses based on conditions on the effective densities. We find that each class of \(h(\mathsf{T})\) solutions that exhibit bouncing behavior correspond to an effective EoS which is already introduced or obtained for an exotic fluid in the literature [52, 152, 153, 157, 158].

#### 3.2.1 Type B models: solutions which follow the relation \(d\rho _\mathsf{{(eff)}}/d\mathsf{T}=[n/(1+w)\mathsf{T}](\rho _\mathsf{{(eff)}}+p_\mathsf{{(eff)}})\)

*n*is an arbitrary constant. Substituting the relation \(d\rho _\mathsf{{eff}}/d\mathsf{T}=[n/(1+w)\mathsf{T}](\rho _\mathsf{{eff}}+p_\mathsf{{eff}})\) into Eq. (18) gives

*w*and

*n*. We can check that, there are only two cases which correspond to Eq. (53) and thus to the scale factor \(a_\mathsf{{B}}\) as the solution; when \(w=-1/5,n=12/5\) and \(w=-1/5,n=6/5\). However, the latter leads to similar physics to the former. The physical quantities constructed out of the bouncing solution for \(w=-1/5,~n=12/5\) are given as follows

*ansatz*\(a_{\mathsf{B}}(t)\) satisfies Eq. (53) we must have

Seeking for a general solution demands that one substitutes the solution (52) into (70) (using the fact that \(\mathsf{T}=(3w-1)\rho \)) and solves for the resulting differential equation to find the scale factor. However, the resulting equation cannot be solved analytically for arbitrary values of *w* and *n*. Nevertheless, for particular values of these parameters a non-singular solution can be obtained as given in expressions (54)–(58). But, we are still able to find more general solutions.

*Q*is an arbitrary constant and for the Hubble parameter we have

^{1}

^{2}is assumed for a perfect fluid. We therefore observe that if we apply \(\rho _{(\mathsf {eff})}\rightarrow 3\rho _\mathsf{{DE}}\) within Eq. (19), we obtain the same Friedman equation as the one given in [153]. Also, by redefining the parameters as \(n\rightarrow 1/\gamma \) and \(\alpha (1+n)\Lambda _\mathsf{{C}}/2\rightarrow \theta \) in (73) we will obtain the corresponding solution for the scale factor. These considerations show that the problem of dark fluid with an unusual EoS (which may not clearly correspond to a definite Lagrangian) can be explained in the framework of \(f(\mathsf{R},\mathsf{T})\) gravity.

#### 3.2.2 Type D models: solutions which are consistent with the relation \(d\rho _\mathsf{{(eff)}}/d\mathsf{T}=m\)

*m*is an arbitrary constant and \(\Gamma _\mathsf{{D}}\), \(\Lambda _\mathsf{{D}}\) are integration constants. Substituting (79) into the conservation equation (18), we obtain a first order differential equation for the matter energy density in terms of the scale factor. However, since the mentioned equation cannot be solved for an exact general solution for arbitrary values of

*w*and

*m*, we proceed with particular cases. Note that, further investigations may give other exact solutions or even numerical simulations can be utilized to study other solutions. At the present, we work on a particular case \(w=1\). The conservation equation (18) then yields

*sinh*in (81) disappears, we arrive at a different model. In this case, the behavior of cosmological quantities is the same as the case for which \(\Upsilon \ne 0\). However, the evolution of matter energy density and the effective pressure are different. We typically plot these quantities for both situations in Fig. 6. The thick curves belong to the solution (81) and the thin ones show the solution with \(\Upsilon =0\). The energy condition considerations show that models of type D lead to the violation of NEC near the bounce event. Also the phase space and the scalar field representation of this model are the same as model A.

## 4 Matter bounce solutions in \(f(\mathsf{R},\mathsf{T})\) gravity

*w*(except for \(w=1/3,1\)). The value \(w=-1\) can be accessed for large values of \(\mathbf {M}\) (see relations (95)). From (52) and (95) we see that for models of type E we have \(\rho _\mathsf{{E}}=\rho _{0}a_\mathsf{{E}}^{-1/\mathbf {M}}\). The effective quantities are then obtained as

## 5 Stability of the bouncing models

*t*,

*x*,

*y*,

*z*), generally. In the current work we shall obtain necessary equations for models including a barotropic perfect fluid with equation of state \(p=w\rho \) and a general \(h(\mathsf{T})\) function. In this respect, the authors of [126] have already considered the matter perturbations in a narrow class of \(f(\mathsf{R},\mathsf{T})\) models

^{3}for a pressure-less perfect fluid. The perturbations of EMT in the longitudinal gauge are given by [160]

*v*is a covariant velocity perturbation [161]. Using the background Eqs. (16) and (17) we obtain the following equations for the scalar perturbations in Fourier space

^{4}Note that Eqs. (116)–(122) are the most general equations describing scalar perturbations in minimal \(f(\mathsf{R},\mathsf{T})\) gravity for condition \(F=1\) when a barotropic perfect fluid is included. These equations are not independent, so that it is possible to obtain one equation from another one; for example (120) follows from multiplying (116) by 2 then adding it to (119) and using (127). In the above equations and relations we have used the following definitions for the source terms, which appear in the right hand sides of field equation (8), its trace and in Eq. (14) when is written as \(\nabla _{\beta }\mathsf {T}_{\,\alpha }^{\beta }=\Sigma _{\alpha }\), respectively

*i*shows the initial values required for integrations. As we see, the solutions are stable in the period of bounce. Note that other coefficients except those which are shown in Eqs. (138) and (139) vanish in the limit \(t\rightarrow 0\). In the limit of large times, we have numerically plotted the evolution of the matter contrast \(\delta \) and the potential \(\Phi \) in Fig. 11. As can be seen, from (140) and (141), it is obvious that there happens no instability at the period of bounce in models D and E, however, far away from the bounce point, both \(\delta \) and \(\Phi \) increase dramatically in model D, see Fig. 11.

## 6 Singular solutions

Different bouncing solutions and their main properties*

Models |
| \(p_\mathsf{{(eff)}}\) | \(h(\mathsf{T})\) | \(\rho \) | |
---|---|---|---|---|---|

A | \(\begin{array}{l} \frac{1}{\mathfrak {A}}\left[ \cosh \left( \sqrt{-\frac{\mathcal {P}}{3}t}\right) -\sinh \left( \sqrt{-\frac{\mathcal {P}}{3}t}\right) \right] \\ \times \left\{ \mathfrak {B}+\frac{(w^2-1)\mathcal {P}\rho _{0}}{2(3w-1)}\left[ \sinh \left( \sqrt{-3\mathcal {P}} t\right) +\cosh \left( \sqrt{-3\mathcal {P}} t\right) -1\right] \right\} ^{2/3}\\ \end{array}\) | \(\mathcal {P}=cons.\) | \(\frac{2}{\alpha }\left( \mathcal {P}+\frac{w}{1-3 w}\mathsf{T}\right) \) | \(\rho _{0}a^{-3}\) | \(>1/3\) |

B | \(\mathcal {R}\left[ \cosh \left( \frac{t}{\mathcal {R}}\right) -\mathcal {S}\right] \) | \(\begin{array}{l} -\frac{\rho }{3}+\frac{2 \mathcal {S} \sqrt{3\mathcal {R}^2 \left( \mathcal {S}^2-1\right) \rho +9}}{3\mathcal {R}^2 \left( \mathcal {S}^2-1\right) }\\ +\frac{2}{\mathcal {R}^2\left( \mathcal {S}^2-1\right) }\\ \end{array}\) | \(\frac{\Gamma \mathsf{T}^{2}}{2}-\frac{7}{4\alpha }\mathsf{T}+\Lambda \) | \(\rho _{0}a^{-1}\) | \(-1/5\) |

C | \(a_{0} \left[ (Q+1)\cosh \left( \sqrt{\frac{3\alpha \Lambda }{2}}\frac{(n+1)}{2n}t\pm \frac{\cosh ^{-1}(Q)}{2}\right) \right] ^{2n/3(n+1)}\) | \(\frac{1}{n}\left[ \rho -\frac{\alpha \Lambda _\mathsf{{C}}}{2}(1+n) \right] \) | \(\frac{2\Gamma \mathsf{T}^{\frac{n+1}{2}}}{n+1}-\frac{\mathsf{T}}{\alpha }+\Lambda \) | \(\rho _{0}a^{-6/n}\) | 1 |

D | \(\left[ \frac{\Gamma \zeta }{\omega }-\frac{\Delta }{\omega } \cosh \left( \sqrt{\frac{3 \alpha \omega }{4 m}}t\right) - \Upsilon \sinh \left( \sqrt{\frac{3 \alpha \omega }{4 m}}t\right) \right] ^{1/3}\) | \(\rho -\frac{\alpha \Gamma }{\sqrt{m}}\sqrt{\rho +\frac{\alpha \Lambda }{2}}+\alpha \Lambda \) | \(2\Gamma \sqrt{\mathsf{T}}-\frac{(2m-1)\mathsf{T}}{\alpha }+\Lambda \) | \(\frac{\left( \zeta -\alpha \Gamma a^{3}\right) ^2}{8 a^6 m^2}\) | 1 |

E | \(\left( \mathbf {Q} t^2+\mathbf {Z}\right) ^\mathbf {M}\) | \(\begin{array}{l} \left( \frac{2}{3 \mathbf {M}}-1\right) \rho \\ \pm \frac{2 \mathbf {Q}}{\mathbf {Z}}\sqrt{\mathbf {M}^2-\frac{\mathbf {Z}}{3 \mathbf {Q}}\rho }-\frac{2 \mathbf {M} \mathbf {Q}}{\mathbf {Z}}\\ \end{array}\) | \(\begin{array}{l} \frac{2 \Gamma (w+1)}{2 n+3 w-1} \mathsf{T}^{\frac{n-2}{w+1}+\frac{3}{2}}\\ -\frac{2 (n-1) }{\alpha (2 n+w-3)}\mathsf{T}+\Lambda \\ \end{array}\) | \(\rho _{0}a^{-3 (w+1)/n}\) | |

Recently, the authors of [166], have considered some cosmological features of \(f(\mathcal {T})\) gravity (where here \(\mathcal {T}\) denotes torsion scalar) using the dynamical system approach both generally and for some specific forms of \(f(\mathcal {T})\) functions. The core of their studies is taking the advantage of this fact that the torsion scalar can be used interchangeably with the Hubble parameter (i.e., \(\mathcal {T}=-6H^{2}\)). Thus, the field equations reduce to a single equation (in the case of pressure-less matter) in the form of \(\dot{H}=\mathcal {F}(H)\), since the matter density can also be rewritten as a function of the Hubble parameter. Briefly, they have shown that in \(f(\mathcal {T})\) gravity a single equation (which can be interpreted as a simple one dimensional dynamical system) can govern the dynamics of field equations. Benefiting this useful result they investigated phase space portraits of various cosmological evolutions such as, singular and non-singular solutions. Likewise, one may be motivated to utilize such an approach in order to investigate the cosmological solutions of \(f(\mathsf{R}, \mathsf{T})\) gravity (especially, in the case of present work, i.e., the function given in (15)) through phase portrait diagrams. However, looking at Eqs. (17) and (18) one finds that it is impossible to obtain an equation like \(\dot{H}=\mathcal {F}(H)\) so that it reflects full information of the field equations. In this case for an assumed function \(h(\mathsf {T}), \) we have a two dimensional dynamical system without any further reduction. Thus, the procedure proposed in [166] would be generally failed in \(f(\mathsf{R}, \mathsf{T})\) gravity.

## 7 Concluding remarks

In the present work we studied classical bouncing behavior of the Universe in the framework of \(f(\mathsf{R},\mathsf{T})=\mathsf{R}+h(\mathsf{T})\) gravity theories. We assumed a single perfect fluid in a spatially flat, homogeneous and isotropic FLRW background. Having obtained the resulted field equations, we employed the concept of effective fluid (which is firstly introduced in [145]) via defining an effective energy density and pressure and also reformulating the field equations in terms of these fluid components. In this picture, one could recast the field equations of \(f(\mathsf{R},\mathsf{T})\) gravity for a real perfect fluid into GR field equations for an effective fluid. It is also shown that in a modified gravity model the energy conditions are usually obtained by using the effective EMT, not the one for real fluids. In \(f(\mathsf{R},\mathsf{T})\) gravity, the definitions for effective energy density and pressure have already been used to obtain the energy conditions [154]. The effective fluid has an EoS of the form, \(p_{(\mathsf {eff})}=\mathcal {Y}(\rho _{(\mathsf {eff})})\), which corresponds to an \(h(\mathsf{T})\) function. In this method one firstly specifies an effective EoS or a condition on the effective components and then obtains the corresponding \(h(\mathsf{T})\) function and other cosmological quantities.

It is also possible to make a link between \(f(\mathsf{R},\mathsf{T})\) gravity in effective picture and models which use some exotic or dark component with unusual EoS. These models which have been widely discussed in the literature (to deal with some cosmological issues) are also called theories with “generalized EoS”. The mathematical representation of effective components provides a setting within which unusual interactions of a real perfect fluid with gravitational field can be translated as the presence of an exotic fluid which admits the EoS of the form \(p_{(\mathsf {eff})}=\mathcal {Y}(\rho _{(\mathsf {eff})})\). In this paper we have shown that it is possible to recover generalized EoS models which have been previously studied in the literature ( see e.g., [52, 152, 153, 157, 158]), in the framework of \(f(\mathsf{R},\mathsf{T})\) gravity. Therefore, the problem of exotic fluid in the context of generalized EoS models which are mostly without a determined Lagrangian may be discussed in a Lagrangian based theory of gravity like \(f(\mathsf{R},\mathsf{T})\) gravity.

In the current research, we discussed four different bouncing models in \(f(\mathsf{R},\mathsf{T})\) gravity. We labeled them as the models A, B, C, D and E and briefly mentioned their main properties in Table 1. Each model can be specified either by an \(h(\mathsf{T})\) function or by an effective EoS. Models A-D mimic an asymptotic de Sitter expansion in the far past and future of the bounce. The model A corresponds to a constant effective pressure, \(p_{(\mathsf {eff})}=\mathcal {P}\); for the model B we have \(p_\mathsf{{(eff)B}}=-\rho _\mathsf{{(eff)B}}/3+\sqrt{b_\mathsf{{B}}\rho _\mathsf{{(eff)B}}+d_\mathsf{{B}}}+e_\mathsf{{B}}\), the model C is specified by \(p_\mathsf{{(eff)}C}=j_\mathsf{{C}} \rho _{(\mathsf {eff})C}+e_\mathsf{{C}}\), the model D corresponds to \(p_\mathsf{{(eff)D}}=\rho _\mathsf{{(eff)D}}+\sqrt{b_\mathsf{{D}}\rho _\mathsf{{(eff)D}}+d_\mathsf{{D}}}+e_\mathsf{{D}}\) and finally the model E obeys the EoS, \(p_\mathsf{{(eff)E}}=a_\mathsf{{E}}\rho _\mathsf{{(eff)E}}+\sqrt{b_\mathsf{{E}}\rho _\mathsf{{(eff)E}}+d_\mathsf{{E}}}+e_\mathsf{{E}}\), where the constants *b*, *d*, *e* and *j* are written in terms of model parameters. In all models the matter density grows to a maximum value at the bounce which corresponds to a minimum for the scale factor. The effective density varies from zero at the bounce to a positive value in the far past and future of the bounce. The effective pressure varies between negative values; in model A, it is a constant, in the model B it increases at the bounce, in the models C and E it decreases and the model D admits both behaviors. The effective EoS has the property \(-\infty<\mathcal {W}<-1\) when the bounce point is approached. The Hubble parameter satisfies \(H(t)=0\) and \(dH/dt>0\) at the event of bounce and also all its time derivatives have regular behavior for all models. Therefore, these bouncing solutions do not exhibit future singularities which are classified in the literature of cosmological solutions. We can consider the inherent exoticism hidden behind \(f(\mathsf{R},\mathsf{T})\) gravity in another way. As already we mentioned, this issue can be described as an unusual interaction between gravitational field and normal matter or introducing an effective fluid. From the point of view of the energy conditions, in all discussed models the SEC and NEC are violated (note that for a normal fluid NEC is not violated [47]) near the bounce and the effective density gets minimized to zero. Such a result has been previously predicted in GR [155]. As discussed in [47], the exoticness can be understood as a minimization in the effective pressure. In the other words, a minimum in the effective energy density corresponds to a minimum in the scale factor. Such a behavior is permitted provided that \(\mathcal {W}<-1\). Note that, for a normal matter, a minimum (maximum) compression leads to a minimum (maximum) energy density. Thus, in \(f(\mathsf{R},\mathsf{T})\) gravity an abnormal or effective fluid which leads to an uncommon balance in the density and pressure can be responsible for the bouncing behavior. An interesting feature of the bouncing solution in \(f(\mathsf{R},\mathsf{T})\) gravity is that one can construct solutions in which the SEC is respected by the real perfect fluid. Such solutions cannot be found in GR [155]. Also note that the real perfect fluid with \(w>-1\) never violates the NEC. Therefore, we have solutions without the future singularities and all energy conditions can be respected by a real perfect fluid. By this discussion, one may use the definition of an (effective) phantom scalar field if one asks for the matter source to be reinterpreted as that of a scalar matter field. We obtained the equivalent scalar field \(\phi _\mathsf{{(eff)}}(t)\) and its corresponding potential \(V_\mathsf{{(eff)}}(t)\) in each case. Moreover, we have studied the dynamical system representation of these models. We found that the evolution of the Universe can be displayed by trajectories which initially start from an unstable state, passing through an unstable fixed point (the bounce event) and finally are absorbed by a stable point. The initial and final states are de-Sitter era in models A, B, C and D and the decelerated expanding Universe in model E. Another important issue discussed in this work is related to the study of stability of bouncing solutions through scalar-type cosmological matter perturbations in the bouncing universe. Our numerical analysis of density perturbations for models A, B and C revealed that, though a slight jump (depending on the initial conditions) at the bounce point, the amplitude of matter density perturbation (\(\delta \)) and perturbed potential (\(\Phi \)) behave regularly throughout the bounce phase. Therefore, since the time interval during which the fluctuations that occur within density contrast and perturbed field is short, the instabilities do not have enough time to grow to a significant magnitude. However, this case does not happen for the two remaining models.

As the final remarks we should emphasize that our models were obtained by indicating different conditions on the effective density and pressure which led to different \(h(\mathsf{T})\) functions. This means that the models A, B, C, D and E are not the only possible models for the bouncing behavior. It is obvious that one can still choose other \(h(\mathsf{T})\) functions or consider other assumptions on the effective density and pressure to obtain new bouncing solutions (with even new features). Our aim was to show the existence of varieties of bouncing solutions in \(f(\mathsf{R},\mathsf{T})\) gravity and study their properties. Especially, our study was confined to the Lagrangians of type \(f(\mathsf{R},\mathsf{T})=\mathsf{R}+h(\mathsf{T})\) though other forms of Lagrangians can be investigated. The other issue is that our study was performed in the effective picture. In case such an approach is not taken seriously, one can think of it as only an alternative mathematical method. One can still investigate a nonsingular cosmological scenario without employing the equations which are written in terms of the effective quantities. In this case it is enough to assume a Lagrangian and solve the field equations to inspect for a bouncing solution. However, cosmological solutions for the \(f(\mathsf{R,T})\) gravity model presented here are not singularity free and as we observed under certain conditions, a class singular solutions could be obtained.

## Footnotes

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