Cosmological eras with the neutrino non-relativistic transition in RTB gravity

Theories based on the Ricci and the trace of the energy-momentum tensor, or the short name Ricci-trace-based (RTB) theories, represent a gravitational approach based on the Ricci scalar $R$, and the trace of the energy-momentum tensor $g^{\mu\nu}T_{\mu\nu} = T$. This theory fits into gravitational models of the type $f(R, T) = R + f(T)$, where $f(T)$ is an arbitrary function of $T$. In this study, we explore a cosmological scenario within the context of RTB models, investigating in detail the cosmological consequences of the coupling between matter and geometry. In order to address this issue, we propose a toy model in which the non-relativistic cosmological neutrino transition plays a role in the cosmic evolution since the effective total energy-momentum tensor trace is affected in this process. We raise questions about the coupling of neutrinos with geometry during this transition, providing a detailed analysis of how RTB gravity deals with this phenomenon and the impact of neutrinos on cosmological dynamics. In summary, we show that the coupling of cosmological neutrinos with $T$ dependent cosmologies is severely challenged.