A comparative study between f(R,T) and f(Σ,T) through f(R,Σ,T) for the Big Rip model

In this study, we investigate the cosmological implications of a novel functional form for the gravitational Lagrangian within the framework of [Formula: see text] gravity. This theory extends standard modifications by incorporating not only the Ricci scalar [Formula: see text] and the trace of the energy–momentum tensor [Formula: see text], but also [Formula: see text], the Ricci scalar derived from a torsion-based connection. Motivated by the pursuit of a unified description of geometric and material contributions to gravitation, we propose a new parametrization for [Formula: see text] and derive the complete set of gravitational field equations using the metric formalism. We then apply this model to a homogeneous and isotropic universe described by the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. By considering a time-dependent deceleration parameter, we obtain exact solutions and analyze the resulting evolution of key cosmological parameters energy density, pressure, and the equation of state. The physical viability of the model is rigorously assessed through an analysis of the energy conditions and the behavior of these parameters as functions of redshift. Our results demonstrate that the proposed [Formula: see text] Lagrangian yields a cosmologically dynamic and physically consistent description of the universe. Furthermore, the model successfully reduces to well-established theories, such as [Formula: see text] and [Formula: see text] gravity, in their respective limits, thereby establishing it as a robust and comprehensive generalization.