Cosmology in scalar-tensor $f(R,T)$ gravity

In this work, we use reconstruction methods to obtain cosmological solutions in the recently developed scalar-tensor representation of $f(R,T)$ gravity. Assuming that matter is described by an isotropic perfect fluid and the spacetime is homogeneous and isotropic, i.e., the Friedmann-Lema\^itre-Robsertson-Walker (FLRW) universe, the energy density, the pressure, and the scalar field associated with the arbitrary dependency of the action in $T$ can be written generally as functions of the scale factor. We then select three particular forms of the scale factor: an exponential expansion with ${a(t)\propto e^t}$ (motivated by the de Sitter solution); and two types of power-law expansion with ${a(t)\propto t^{1/2}}$ and ${a(t)\propto t^{2/3}}$ (motivated by the behaviors of radiation- and matter-dominated universes in general relativity, respectively). A complete analysis for different curvature parameters ${k=\{-1,0,1\}}$ and equation of state parameters ${w=\{-1,0,1/3\}}$ is provided. Finally, the explicit forms of the functions $f\left(R,T\right)$ associated with the scalar-field potentials of the representation used are deduced.