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Kinematic reconstructions of extended theories of gravity at small and intermediate redshifts

In the last few decades, extensions of General Relativity have reached always more attention especially in view of possible breakdowns of the standard $Λ$CDM 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 acceleration. We here focus our attention on two classes of extended theories, i.e. the $f(R)$ and $f(R,G)$-gravity. We parameterize the so-obtained Hubble function by means of effective barotropic fluids, by calibrating the shapes of our curves through some of the most suitable dark energy parameterizations, XCDM, CPL, WP. Afterwards, by virtue of the correspondence between the Ricci scalar and the Gauss-Bonnet topological invariant with the redshift $z$, 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, through a coarse-grained inverse scattering procedure. Although our procedure agrees with the simplest extensions of general relativity, it leaves open the possibility that the most suitable forms of $f(R)$ and $f(R,G)$ are rational Padé polynomials of first orders. These approximations seem to be compatible with numerical reconstructions within intermediate redshift domains and match fairly well small redshift tests.