Palatini formulation of $f(R,T)$ gravity theory, and its cosmological implications

TL;DR

This paper explores the Palatini formulation of $f(R,T)$ gravity, deriving field equations, analyzing test particle motion, and investigating cosmological models that predict late-time acceleration of the universe.

Contribution

It introduces the Palatini approach to $f(R,T)$ gravity, deriving new field equations and cosmological implications, including generalized Friedmann equations and specific accelerating universe models.

Findings

Independent connection expressed as Levi-Civita of an auxiliary metric

Energy-momentum tensor non-conservation in the theory

Cosmological models exhibit late-time acceleration

Abstract

We consider the Palatini formulation of $f(R,T)$ gravity theory, in which a nonminimal coupling between the Ricci scalar and the trace of the energy-momentum tensor is introduced, by considering the metric and the affine connection as independent field variables. The field equations and the equations of motion for massive test particles are derived, and we show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy-momentum trace dependent metric, related to the physical metric by a conformal transformation. Similarly to the metric case, the field equations impose the non-conservation of the energy-momentum tensor. We obtain the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra-force, which is identical to the one obtained in the metric case. The thermodynamic interpretation of the theory is also briefly discussed. We investigate in detail the cosmological implications of the theory, and we obtain the generalized Friedmann equations of the $f(R,T)$ gravity in the Palatini formulation. Cosmological models with Lagrangians of the type $f=R-\alpha ^2/R+g(T)$ and $f=R+\alpha ^2R^2+g(T)$ are investigated. These models lead to evolution equations whose solutions describe accelerating Universes at late times.