Generalized Ricci and energy–momentum tensors squared gravity within the framework of Palatini formalism

In this paper, we study a modified gravity of the form [Formula: see text] in the Palatini formalism, whose Lagrangian depends on the Ricci scalar, Ricci and energy–momentum squared tensors. In this approach, we derive the field equations and show the connection is Levi-Civita with respect to an auxiliary metric related to the physical metric via a disformal transformation with factors depending on curvature and matter. For a perfect fluid, we obtain explicit, exact solutions for the auxiliary metric and the Ricci tensor, demonstrating that the vacuum solutions naturally yield an effective cosmological constant. Analysis of the energy–momentum tensor conservation reveals a generic violation, which results in the emergence of an extra force that affects the trajectory of the test particle. The derived force highlights a distinctive feature of the non-minimal coupling between matter and geometry. Specializing to [Formula: see text], the field equations directly impose an upper bound on the combination of energy density and pressure. For the quadratic model [Formula: see text], General Relativity is recovered at low energies, with a maximum in [Formula: see text]. The theory exhibits a phenomenology significantly richer than its individual components.