Rational approximations of $f(R)$ cosmography through Pad\'e polynomials

We consider high-redshift $f(R)$ cosmography adopting the technique of polynomial reconstruction. In lieu of considering Taylor treatments, which turn out to be non-predictive as soon as $z>1$, we take into account the Pad\'e rational approximations which consist in performing expansions converging at high redshift domains. Particularly, our strategy is to reconstruct $f(z)$ functions first, assuming the Ricci scalar to be invertible with respect to the redshift $z$. Having the thus-obtained $f(z)$ functions, we invert them and we easily obtain the corresponding $f(R)$ terms. We minimize error propagation, assuming no errors upon redshift data. The treatment we follow naturally leads to evaluating curvature pressure, density and equation of state, characterizing the universe evolution at redshift much higher than standard cosmographic approaches. We therefore match these outcomes with small redshift constraints got by framing the $f(R)$ cosmology through Taylor series around $z\simeq 0$. This gives rise to a calibration procedure with small redshift that enables the definitions of polynomial approximations up to $z\simeq 10$. Last but not least, we show discrepancies with the standard cosmological model which go towards an extension of the $\Lambda$CDM paradigm, indicating an effective dark energy term evolving in time. We finally describe the evolution of our effective dark energy term by means of basic techniques of data mining.