# Extended Gravity Cosmography

## TL;DR

This paper reviews the use of cosmography as a model-independent method to analyze dark energy and modified gravity, introducing advanced polynomial techniques to improve the reconstruction of cosmic expansion history from observational data.

## Contribution

It introduces a novel rational polynomial approach to cosmography that enhances model-independent analysis of the universe's expansion, especially for $f(R)$ and $f(T)$ gravity models.

## Key findings

- The new method provides accurate, model-independent approximations of the Hubble flow.
- Numerical bounds on $f(R)$ and $f(T)$ functions are obtained from cosmic data.
- Comparison of $ ext{Lambda}$CDM with modified gravity models demonstrates the approach's effectiveness.

## Abstract

Cosmography can be considered as a sort of a model-independent approach to tackle the dark energy/modified gravity problem. In this review, the success and the shortcomings of the $\Lambda$CDM model, based on General Relativity and standard model of particles, are discussed in view of the most recent observational constraints. The motivations for considering extensions and modifications of General Relativity are taken into account, with particular attention to $f(R)$ and $f(T)$ theories of gravity where dynamics is represented by curvature or torsion field respectively. The features of $f(R)$ models are explored in metric and Palatini formalisms. We discuss the connection between $f(R)$ gravity and scalar-tensor theories highlighting the role of conformal transformations in the Einstein and Jordan frames. Cosmological dynamics of $f(R)$ models is investigated through the corresponding viability criteria. Afterwards, the equivalent formulation of General Relativity (Teleparallel Equivalent General Relativity) in terms of torsion and its extension to $f(T)$ gravity is considered. Finally, the cosmographic method is adopted to break the degeneracy among dark energy models. A novel approach, built upon rational Pad\'e and Chebyshev polynomials, is proposed to overcome limits of standard cosmography based on Taylor expansion. The approach provides accurate model-independent approximations of the Hubble flow. Numerical analyses, based on Monte Carlo Markov Chain integration of cosmic data, are presented to bound coefficients of the cosmographic series. These techniques are thus applied to reconstruct $f(R)$ and $f(T)$ functions and to frame the late-time expansion history of the universe with no \emph{a priori} assumptions on its equation of state. A comparison between the $\Lambda$CDM cosmological model with $f(R)$ and $f(T)$ models is reported.