$f(R)$ gravity modifications: from the action to the data
It is a very well established matter nowadays that many modified gravity models can offer a sound alternative to General Relativity for the description of the accelerated expansion of the universe. But it is also equally well known that no clear and sharp discrimination between any alternative theory and the classical one has been found so far. In this work, we attempt at formulating a different approach starting from the general class of $f(R)$ theories as test probes: we try to reformulate $f(R)$ Lagrangian terms as explicit functions of the redshift, i.e., as $f(z)$. In this context, the $f(R)$ setting to the consensus cosmological model, the $Λ$CDM model, can be written as a polynomial including just a constant and a third-order term. Starting from this result, we propose various different polynomial parameterizations $f(z)$, including new terms which would allow for deviations from $Λ$CDM, and we thoroughly compare them with observational data. While on the one hand we have found no statistically preference for our proposals (even if some of them are as good as $Λ$CDM by using Bayesian Evidence comparison), we think that our novel approach could provide a different perspective for the development of new and observationally reliable alternative models of gravity.