$F(R,w)$ Gravity: A new gravity framework

In this work we present a new framework of the gravity sector by considering the extension $F(R,w)$, in which $R$ is the Ricci scalar and $w$ is the equation of state. Three different choices of function $F(R,w)$ are investigated under the Palatini formalism. The models appear equivalent to $F(R)$ models of gravity with effective momentum-energy tensors. For linear dependence of Ricci scalar in which $F(R,w)=k(w)R$, the model appears equivalent to Einstein-Hilbert action with effective momentum-energy tensor. Recovering the minimal coupling case of the last choice does not face Jordan-Einstein frame ambiguities and exhibits natural alignments with general relativity results in the matter\text{/} radiation dominated eras. We discuss some astrophysical implications of the model by considering scalar fields as dominant matter forms. We show that the Higgs inflation could be saved within the $F(R,w)$ model. We suggest some future investigations exemplified by constant-roll inflation and universe evolution for $F(R)=f(R)k(w)$ where $f(R)$ represents the Starobinsky gravitational form. Using the model and comparing it with pure $F(R)$ gravity, we provide preliminary indications of $F(R,w)$'s impact. As a final note, we suggest using the Polytropic equation of state in future works to investigate $F(R,w)$.