The effective equation of state in Palatini $f({{\mathcal {R}}})$ cosmology

We investigate how the cosmological equation of state can be used for scrutinizing extended theories of gravity, in particular, the Palatini $$f({{\mathcal {R}}})$$ f ( R ) gravity. Specifically, the approach consists, at first, in investigating the effective equation of state produced by a given model. Then, the inverse problem can also be considered in view of determining which models are compatible with a given effective equation of state. We consider and solve some cases and show that, for example, power-law models are (the only models) capable of transforming barotropic Equations of State into effective barotropic ones. Moreover, the form of equation of state is preserved (only) for $$f({{\mathcal {R}}})={{\mathcal {R}}}$$ f ( R ) = R , as expected. In this perspective, modified Equations of State are a feature capable of distinguishing extended gravity with respect to general relativity. We investigate also quadratic and non-homogeneous effective Equations of State showing, in particular, that they contain the Starobinsky model and other ones.