The Palatini formalism of the $f(R,\mathcal{L}_{m},T)$ theory of gravity

TL;DR

This paper formulates the $f(R, ext{L}_m,T)$ gravity theory using the Palatini approach, deriving new field equations and exploring their implications for cosmology and potential observational tests.

Contribution

It is the first to apply the Palatini formalism to the $f(R, ext{L}_m,T)$ gravity theory, revealing distinct properties and phenomenological implications.

Findings

Derived new Palatini field equations for $f(R, ext{L}_m,T)$ gravity.

Presented the Newtonian limit and Friedmann-like equations in this framework.

Suggested observational signatures to distinguish between metric and Palatini formalisms.

Abstract

We present the first formulation of the recently proposed $f(R,\mathcal{L}_m,T)$ theory of gravity within the Palatini formalism, a well-known alternative variational approach where the metric and connection are treated as independent variables. By applying this formalism, we derive a new set of field equations that exhibit, as expected, distinct properties compared to their metric formalism counterparts. We particularly present the Newtonian limit of this formalism, as well as the resulting Friedmann-like equations. We highlight that potential observational signatures may distinguish between the metric and Palatini frameworks. Our results open new pathways for exploring the phenomenology of modified gravity theories and their testability with observational data.