We consider f (R, T ) modified theory of gravity in which, in general, the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar and the trace of the energy–momentum tensor. We indicate that in this type of the theory, the coupling energy–momentum tensor is not conserved. However, we mainly focus on a particular model that matter is minimally coupled to the geometry in the metric formalism and wherein, its coupling energy–momentum tensor is also conserved.We obtain the corresponding Raychaudhuri dynamical equation that presents the evolution of the kinematic quantities. Then for the chosen model, we derive the behavior of the deceleration parameter, and show that the coupling term can lead to an acceleration phase after the matter dominated phase. On the other hand, the curvature of the universe corresponds with the deviation from parallelism in the geodesic motion. Thus, we also scrutinize the motion of the free test particles on their geodesics, and derive the geodesic deviation equation in this modified theory to study the accelerating universe within the spatially flat FLRW background. Actually, this equation gives the relative accelerations of adjacent particles as a measurable physical quantity, and provides an elegant tool to investigate the timelike and the null structures of spacetime geometries. Then, through the null deviation vector,wefind the observer area–distance as a function of the redshift for the chosen model, and compare the results with the corresponding results obtained in the literature.
