Dynamical aspects of generalized Palatini theories of gravity

We study the field equations of modified theories of gravity in which the Lagrangian is a general function of the Ricci scalar and Ricci-squared terms in Palatini formalism. We show that the independent connection can be expressed as the Levi-Cività connection of an auxiliary metric which, in particular cases of interest, is related with the physical metric by means of a disformal transformation. This relation between physical and auxiliary metric boils down to a conformal transformation in the case of $f(R)$ theories. We also show with explicit models that the inclusion of Ricci-squared terms in the action can impose upper bounds on the accessible values of pressure and density, which might have important consequences for the early time cosmology and black hole formation scenarios. Our results indicate that the phenomenology of $f(R,R_{\mu \nu }{R}^{\mu \nu })$ theories is much richer than that of $f(R)$ and $f(R_{\mu \nu }{R}^{\mu \nu })$ theories and that they also share some similarities with Bekenstein’s relativistic theory of MOND.