Equivalence of matter-type modified gravity theories to general relativity with nonminimal matter interaction

In this study, we first establish that gravity models incorporating matter-related terms, such as $f(\mathcal{L}_{\rm m})$, $f(g_{\mu\nu} T^{\mu\nu})$, and $f(T_{\mu\nu} T^{\mu\nu})$, into the usual matter Lagrangian density $\mathcal{L}_{\rm m}$, are equivalent to general relativity with nonminimal matter interactions. Through the redefinition $\mathcal{L}_{\rm m}+f \rightarrow \mathcal{L}_{\rm m}^{\rm tot}$, these models are exactly GR, yet the usual material field $T_{\mu\nu}$ and its accompanying partner, the modification field $T_{\mu\nu}^{\rm mod}$, engage in nonminimal interactions. Specifically, $\nabla^{\mu}T_{\mu\nu}=-Q_{\nu}=-\nabla^{\mu}T_{\mu\nu}^{\rm mod}$, where $Q_{\nu}$ is the interaction kernel that governs the rate of energy transfer. Our focus narrows on the specific model of $f(T_{\mu\nu} T^{\mu\nu})$, known as Energy-Momentum Squared Gravity, where the usual material field $T_{\mu\nu}$ is accompanied by an \textit{energy-momentum squared field} (EMSF), $T_{\mu\nu}^{\rm emsf}$, along with a sui generis nonminimal interaction between them. We demonstrate that a particular $T_{\mu\nu}^{\rm emsf}$ can be introduced by \textit{removing} $\frac{\partial^2 \mathcal{L}_{\rm m}}{\partial g^{\mu\nu} \partial g^{\sigma\epsilon}}$ (the new term emerging in models that incorporate scalars formed from $T_{\mu\nu}$), thanks to the freedom in determining the interaction kernel, but this approach compromises the Lagrangian formulation of EMSG. Additionally, we address the ambiguities regarding the perfect fluid stemming from this new term. We show the proper way of calculating this term for a perfect fluid, revealing that it is indeed non-zero, contrary to common assumption in the literature. Finally, we re-examine cosmological models within the realm of EMSG, offering new insights into the applicability and interpretation of our findings in EMSG and similar theoretical frameworks.