Extended Gravity Cosmography
  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.
