A numerical approach to model independently reconstruct $f(R)$ functions through cosmographic data

The challenging issue of determining the correct $f(R)$ among several possibilities is here revised by means of numerical reconstructions of the modified Friedmann equations around the redshift interval $z\in[0,1]$. Frequently, a severe degeneracy between $f(R)$ approaches occurs, since different paradigms correctly explain present time dynamics. To set the initial conditions on the $f(R)$ functions, we involve the use of the so called cosmography of the Universe, i.e. the technique of fixing constraints on the observable Universe by comparing expanded observables with current data. This powerful approach is essentially model independent and correspondingly we got a model independent reconstruction of $f(z)$ classes within the interval $z\in[0,1]$. To allow the Hubble rate to evolve around $z\leq1$, we considered three relevant frameworks of effective cosmological dynamics, i.e. the $\Lambda$CDM model, the CPL parametrization and a polynomial approach to dark energy. Finally cumbersome algebra permits to pass from $f(z)$ to $f(R)$ and the general outcome of our work is the determination of a viable $f(R)$ function, that effectively describes the observed Universe dynamics.