In the last few decades, extensions of General Relativity have reached always more attention especially in view of possible breakdowns of the standard ΛCDM paradigm at intermediate and high redshift regimes. If General Relativity would not be the ultimate theory of gravity, modifying Einstein’s gravity in the homogeneous and isotropic universe may likely represent a viable path toward the description of current universe speed up.
We here focus our attention on two classes of extended theories, i.e. the f (R) and f (R, G) paradigms, with the only requirement that the cosmological principle holds. We thus limit our treatment by only assuming the concordance paradigm is pre-served at background cosmology. In so doing, we presume that each extended models reduce to the ΛCDM scenario at infrared energy regime. To do so, we involve a few classes of F (R, G) = R + f (R, G) modified theories of gravity, i.e in which the Ricci scalar is explicitly reported. Hence, we parameterize the so-obtained Hubble rate by means of effective barotropic fluids, by calibrating the shapes of our curves through the most suitable dark energy parameterizations, e.g. XCDM, CPL, WP, and so forth. Afterwards, by virtue of the correspondence between Ricci and Gauss-Bonnet invariants and the redshift z, i.e. R = R(z) and G = G(z), we rewrite f(R,G) in terms of corresponding f(z) auxiliary functions. This scheme enables one to get numerical shapes for f(R,G) and f(R) models. Further, we fixed our priors over the free coefficients of our frameworks by means of the most recent outcomes provided by Planck’s mission.