General approach to the Lagrangian ambiguity in $f(R, T)$ gravity

TL;DR

This paper proposes a novel approach to resolve ambiguities in the Lagrangian formulation of $f(R,T)$ gravity, enabling a consistent description of cosmological evolution without dark energy.

Contribution

It generalizes the treatment of the matter Lagrangian in $f(R,T)$ gravity by removing its dependence and considering the energy-momentum trace as an unknown variable.

Findings

Successfully describes transition from decelerated to accelerated universe

Eliminates dependence on matter Lagrangian in field equations

Provides a consistent framework for $f(R,T)$ cosmology

Abstract

The $f(R,T)$ gravity is a theory whose gravitational action depends arbitrarily on the Ricci scalar, $R$, and the trace of the stress-energy tensor, $T$; its field equations also depend on matter Lagrangian, $\mathcal{L}_{m}$. In the modified theories of gravity where field equations depend on Lagrangian, there is no uniqueness on the Lagrangian definition and the dynamics of the gravitational and matter fields can be different depending on the choice performed. In this work, we have eliminated the $\mathcal{L}_{m}$ dependence from $f(R,T)$ gravity field equations by generalizing the approach of Moraes [Eur. Phys. J. C 79(8), 674 (2019)]. We also propose a general approach where we argue that the trace of the energy-momentum tensor must be considered an "unknown" variable of the field equations. The trace can only depend on fundamental constants and few inputs from the standard model. Our proposal resolves two limitations: first the energy-momentum tensor of the $f(R,T)$ gravity is not the perfect fluid one; second, the Lagrangian is not well-defined. As a test of our approach we applied it to the study of the matter era in cosmology, and the theory can successfully describe a transition between a decelerated Universe to an accelerated one without the need for dark energy.