Palatini Approach to Modified Gravity Based on Determinantal Invariants of the Ricci and Energy-Momentum Tensors

The [Formula: see text] theory of gravity, formulated in terms of the invariants [Formula: see text] and [Formula: see text] associated with the determinants of the Ricci and energy-momentum tensors, is examined. In this framework, which couples geometry to matter in a nontrivial manner, both the Ricci and energy-momentum tensors are assumed to be invertible, and the field equations are derived by extremizing the action with respect to the space-time metric and the independent Palatini connection. The theory naturally gives rise to an intrinsic bimetric structure through a disformal transformation governed jointly by curvature and matter. This structure leads to the non-conservation of energy and momentum, as well as to non-geodesic motion driven by an emergent force expressed in terms of [Formula: see text] and [Formula: see text], thereby extending the dynamics well beyond those of conventional gravitational models. A perfect fluid, serving as a cosmological model with an invertible energy-momentum tensor, provides a suitable matter sector for the model, enabling a detailed characterization of the geometry-matter interplay in terms of the pressure and density. We then extend our investigation to the barotropic equation of state and to the Ricci tensor, the latter being a conformal transformation of the geometric metric dictated by the deformation function [Formula: see text]. Our analysis shows that the theory dynamically induces an effective cosmological constant [Formula: see text] that depends on the state parameters, thereby providing a geometric mechanism capable of generating late-time cosmic acceleration.