Evading the non-continuity equation in the f (R, T) cosmology

We present a new approach for the f (R, T) gravity formalism, by thoroughly exploring the extra terms of its effective energy-momentum tensor T-mu nu(eff), which we name (T) over tilde (mu nu), so that T-mu nu(eff) = T-mu nu + (T) over tilde (mu nu), with T-mu nu. being the usual energy-momentum tensor of matter. Purely from the Bianchi identities, we obtain the conservation of both parts of the effective energy-momentum tensor, rather than the non-conservation of T-mu nu, originally occurring in the f (R, T) theories. In this way, the intriguing scenario of matter creation, which still lacks observational evidence, is evaded. One is left, then, with two sets of cosmological equations to be solved: the Friedmann-like equations along with the conservation of T-mu nu and along with the conservation of (T) over tilde (mu nu). We present a physical interpretation for the conservation of (T) over tilde (mu nu), which can be related to the presence of stiff matter in the universe. The cosmological consequences of this approach are presented and discussed as well as the benefits of evading the matter energy-momentum tensor non-conservation.