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

We present a new approach for the $f(R,T)$ formalism, by exploring the extra terms of its effective energy-momentum tensor $T_{\mu\nu}^{eff}$, namely $\tilde{T}_{\mu\nu}$. Those arise from the consideration of quantum effects, which are usually neglected in general relativity and $f(R)$ theories, and are summed to the usual matter energy-momentum tensor, yielding $T_{\mu\nu}^{eff}=T_{\mu\nu}+\tilde{T}_{\mu\nu}$. Purely from the Bianchi identities, the conservation of both parts of the effective energy-momentum tensor is obtained, rather than the non-conservation of the matter one, originally approached in the $f(R,T)$ theories. In this way, the intriguing scenario of matter-creation, which still lacks observational evidences, 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 $\tilde{T}_{\mu\nu}$. We present a physical interpretation for the conservation of the extra terms of the effective energy-momentum tensor, which is related to the presence of stiff matter in the universe. The cosmological features of this approach are presented and discussed as well as the benefits of evading the matter energy-momentum tensor non-conservation.