We investigate the cosmological solutions of $f(R,T)$ modified theories of gravity for a perfect fluid in a spatially Friedmann-Lemaître-Robertson-Walker 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 nonminimal $g(R)h(T)$, and nonminimal $g(R)(1+h(T))$ couplings through the dynamical systems approach. We introduce a few variables and dimensionless parameters to simplify the equations to 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, and hence relates the dynamical variables. In this case, 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 numerical solutions for a few functions of $g(R)$, and the results of the corresponding models confirm the predictions. We separate the solutions into six classes which demonstrate more acceptable solutions and there is more freedom to have the matter-dominated era than in $f(R)$ 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 nonminimal theory suffers from the absence of a standard matter era, though we illustrate that the nonminimal theory can have acceptable cosmological solutions.

• Received 25 June 2013

DOI:https://doi.org/10.1103/PhysRevD.88.044048