Extended Gravity Cosmography

TL;DR

This review analyzes how extended theories of gravity, notably $f(R)$ and $f(T)$, can address late-time cosmic acceleration within metric, Palatini, and teleparallel frameworks, while employing cosmography as a model‑independent diagnostic. It Introduces and tests rational approximation techniques (Padé and Chebyshev) to overcome convergence limits of standard cosmography and to reconstruct gravity actions from late-time data. Through model‑independent reconstructions in both curvature and torsion sectors, the work finds mild departures from $\Lambda$CDM compatible with current data, and demonstrates the viability of reconstructing $f(R)$ and $f(T)$ from cosmographic constraints. The authors emphasize the need for higher‑redshift observations and perturbation information to decisively falsify $\Lambda$CDM with cosmography and to further constrain the space of extended gravity models.

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 $Λ$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 Padé 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 $Λ$CDM cosmological model with $f(R)$ and $f(T)$ models is reported.

Figures (35)

• Figure 1: Representation of parallelograms determined by torsion.
• Figure 2: Dimensionless luminosity distance in terms of the redshift in the case of rational Chebyshev approximations ($R_{1,1}$), ($R_{1,2}, R_{2,1}$) and ($R_{1,3}, R_{2,2}, R_{3,1}$) orders; it is possible to notice the comparison with the $\Lambda$CDM model. Choosing a set of values for the free parameters enables to get the correct expansion orders in Chebyshev analyses.
• Figure 3: Convergence radii for different orders. In particular, for second-order Taylor (dashed curve) and equivalent (1,1) Padé (dotted curve) and (1,1) rational Chebyshev (solid curve). In the picture of Chebyshev approximation we took $j_0=2$, $s_0=-1$.
• Figure 4: 1$\sigma$ and 2$\sigma$ confidence level contours and posterior distributions inferred from the MCMC analysis by combining SN+OHD+BAO data surveys. The results have been obtained for fourth-order Taylor approximation of $d_L$. The units of $H_0$ are Km/s/Mpc, whereas $r_d$ in Mpc.
• Figure 5: 68% and 95% confidence levels and corresponding contours with posterior distributions determined from the MCMC analysis. Here, we considered a combined SN+OHD+BAO survey for the (2,2) Padé approximation of $d_L$. $H_0$ is written in Km/s/Mpc, while $r_d$ in Mpc.