We investigate the cosmological solutions of f (R, T ) modified theories of gravity for a perfect fluid in a spatially FLRW metric through the phase space analysis, where R is the Ricci scalar and T denotes the trace of the energy-momentum tensor of the matter content. We explore and analyze the three general theories with the Lagrangians of minimal g(R) + h(T ), pure non-minimal g(R)h(T ) and non-minimal g(R) (1 + h(T )) couplings through the dynamical systems approach. We introduce a few variables and dimensionless parameters to simplify the equations in more concise forms. The conservation of the energy-momentum tensor leads to a constraint equation that, in the minimal gravity, confines the functionality of h(T ) to a particular form, hence, relates the dynamical variables. In this case, the acceptable cosmological solutions that contain a long enough matter dominated era followed by a late-time accelerated expansion are found. To support the theoretical results, we also obtain the numerical solutions for a few functions of g(R), and the results of the corresponding models confirm the predictions. We classify the solutions into six classes which demonstrate more acceptable solutions and there is more freedom to have the matter dominated era than in the f (R) gravity. In particular, there is a new fixed point which can represent the late-time acceleration. We draw different diagrams of the matter densities (consistent with the present values), the related scale factors and the effective equation of state. The corresponding diagrams of the parameters illustrate that there is a saddle acceleration era which is a middle era before the final stable acceleration de Sitter era for some models. All presented diagrams determine radiation, matter and late-time acceleration eras very well. The pure non-minimal theory suffers from the absence of a standard matter era, though we illustrate that the non-minimal theory can have acceptable cosmological solutions.2 development of a general phenomenological framework, i.e. the PPN-formalism, determine that it is the best known metric theory of gravity [46].Among the extended theories of gravity, there are at least two main motivations 3 for employing the higher-order gravities, i.e., those in which the Einstein-Hilbert action is modified by higher-order curvature invariants with respect to the Ricci scalar. The first motivation has a theoretical background and is related to the non-renormalizability of GR [49,50] and to the fact that GR cannot be quantized conventionally. Regarding this issue, some authors have shown that the inclusion of higher-order terms can solve this problem [51,52]. The other motivation is related to the recently achieved data in astrophysics and cosmology. Two contemporary evidences, which are referred to as dark matter and dark energy, have challenged our knowledge about the universe and have accounted for the first signals of GR breakdown. It is also worth mentioning that the concordance or ΛCDM model [53], the simplest model which adequately fits the...

To find more deliberate f (R, T ) cosmological solutions, we proceed our previous paper further by studying some new aspects of the considered models via investigation of some new cosmological parameters/quantities to attain the most acceptable cosmological results. Our investigations are performed by applying the dynamical system approach. We obtain the cosmological parameters/quantities in terms of some defined dimensionless parameters that are used in constructing the dynamical equations of motion. The investigated parameters/quantities are the evolution of the Hubble parameter and its inverse, the "weight function", the ratio of the matter density to the dark energy density and its time variation, the deceleration, the jerk and the snap parameters, and the equation-of-state parameter of the dark energy. We numerically examine these quantities for two general models R + αR −n + √ −T and R log [αR] q + √ −T . All considered models have some inconsistent quantities (with respect to the available observational data), except the model with n = −0.9 which has more consistent quantities than the other ones. By considering the ratio of the matter density to the dark energy density, we find that the coincidence problem does not refer to a unique cosmological event, rather, this coincidence also occurred in the early universe. We also present the cosmological solutions for an interesting model R+c1 √ −T in the non-flat FLRW metric. We show that this model has an attractor solution for the late times, though with w (DE) = −1/2. This model indicates that the spatial curvature density parameter gets negligible values until the present era, in which it acquires the values of the order 10 −4 or 10 −3 . As the second part of this work, we consider the weak-field limit of f (R, T ) gravity models outside a spherical mass immersed in the cosmological fluid. We have found that the corresponding field equations depend on the both background values of the Ricci scalar and the background cosmological fluid density. As a result, we attain the parametrized post-Newtonian (PPN) parameter for f (R, T ) gravity and show that this theory can admit the experimentally acceptable values of this parameter. As a sample, we present the PPN gamma parameter for general minimal power law models, in particular, the model R + c1 √ −T .

The Einstein static (ES) universe has played a major role in various emergent scenarios recently proposed in order to cure the problem of the initial singularity of the standard model of cosmology. In the model we address, we study the existence and stability of an ES universe in the context of f (R, T ) modified theories of gravity. Considering specific forms of the f (R, T ) function, we seek for the existence of solutions representing ES state. Using dynamical system techniques along with numerical analysis, we find two classes of solutions: the first one is always unstable of the saddle type, while the second is always stable so that its dynamical behavior corresponds to a center equilibrium point. The importance of the second class of solutions is due to the significant role they play in constructing non-singular emergent models in which the universe could have experienced past-eternally a series of infinite oscillations about such an initial static state after which it enters, through a suitable physical mechanism, to an inflationary era. Considering specific forms for the functionality of f (R, T ), we show that this theory is capable of providing cosmological solutions which admit emergent universe (EU) scenarios. We also investigate homogeneous scalar perturbations for the mentioned models. The stability regions of the solutions are parametrized by a linear equation of state (EoS) parameter and other free parameters that will be introduced for the models. Our results suggest that modifications in f (R, T ) gravity would lead to stable solutions which are unstable in f (R) gravity model.

In this work we study classical bouncing solutions in the context of f (R, T) = R + h(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 (R, T) gravity. We examine various models to which different effective equations of state, corresponding to different h(T) functions, can be attributed. It is also discussed that one can link between an assumed f (R, 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 (R, T) gravity are as follows: for a specific form of the f (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.

Very recently, Josset and Perez (Phys. Rev. Lett. 118:021102, 2017) have shown that a violation of the energymomentum tensor (EMT) could result in an accelerated expansion state via the appearance of an effective cosmological constant, in the context of unimodular gravity. Inspired by this outcome, in this paper we investigate cosmological consequences of a violation of the EMT conservation in a particular class of f (R, T) gravity when only the pressureless fluid is present. In this respect, we focus on the late time solutions of models of the type f (R, T) = R + β (−T). As the first task, we study the solutions when the conservation of EMT is respected, and then we proceed with those in which violation occurs. We have found, provided that the EMT conservation is violated, that there generally exist two accelerated expansion solutions of which the stability properties depend on the underlying model. More exactly, we obtain a dark energy solution for which the effective equation of state depends on the model parameters and a de Sitter solution. We present a method to parametrize the (−T) function, which is useful in a dynamical system approach and has been employed in the model. Also, we discuss the cosmological solutions for models with (−T) = 8π G(−T) α in the presence of ultra-relativistic matter.