Beyond $\Lambda$CDM with $f(z)$CDM: criticalities and solutions of Pad\'e Cosmography

Recently, cosmography emerged as a valuable tool to effectively describe the vast amount of astrophysical observations without relying on a specific cosmological model. Its model-independent nature ensures a faithful representation of data, free from theoretical biases. Indeed, the commonly assumed fiducial model, the $\Lambda$CDM, shows some shortcomings and tensions between data at late and early times that need to be further investigated. In this paper, we explore an extension of the standard cosmological model by adopting the $f(z)$CDM approach, where $f(z)$ represents the cosmographic series characterizing the evolution of recent universe driven by dark energy. To construct $f(z)$, we take into account the Pad\'e series, since this rational polynomial approximation offers a better convergence at high redshifts than the standard Taylor series expansion. Several orders of such an approximant have been proposed in previous works, here we want to answer the questions: What is the impact of the cosmographic series choice on the parameter constraints? Which series is the best for the analysis? So, we analyse the most promising ones by identifying which order is preferred in terms of stability and goodness of fit. Theoretical predictions of the $f(z)$CDM model are obtained by the Boltzmann solver code and the posterior distributions of the cosmological and cosmographic parameters are constrained by a Monte Carlo Markov Chains analysis. We consider a joint data set of cosmic microwave background temperature measurements from the Planck collaboration, type Ia supernovae data from the latest Pantheon+ sample, baryonic acoustic oscillations and cosmic chronometers data. In conclusions, we state which series can be used when only late time data are used, while which orders has to be considered in order to achieve the necessary stability when large redshifts are considered.