Model-independent limits and constraints on extended theories of gravity from cosmic reconstruction techniques

The onset of dark energy domination depends on the particular gravitational theory driving the cosmic evolution. Model independent techniques are crucial to test both the present $\Lambda$CDM cosmological paradigm and alternative theories, making the least possible number of assumptions about the Universe. In this paper we investigate whether cosmography is able to distinguish between different gravitational theories, by determining bounds on model parameters for three different extensions of General Relativity, i.e. $k-$essence, $F(T)$ and $f(R)$ theories. We expand each class of theories in powers of redshift $z$ around the present time, making no additional assumptions. This procedure is an extension of previous work and can be seen as the most general approach for testing extended theories of gravity with cosmography. In the case of $F(T)$ and $f(R)$ theories, we show that some assumptions on model parameters often made in previous works are superfluous or unjustified. We use data from the Union2.1 SN catalogue, BAO data and $H(z)$ differential age compilations, which probe cosmology on different scales. We perform a Monte Carlo analysis using a Metropolis-Hastings algorithm with a Gelman-Rubin convergence criterion, reporting 1-$\sigma$ and 2-$\sigma$ confidence levels. We perform two distinct fits, first taking only data at $z<1$ and then for all $z$. We obtain the corresponding parameter intervals, and find that the data is compatible the $\Lambda$CDM limit of all three theories at the 1-$\sigma$ level, while still compatible with quite a large portion of parameter space. We compare our results to the truncated $\Lambda$CDM paradigm, showing that the permitted regions of coefficients are significantly modified and in general widened compared to values reported in the existing literature. Finally, we test the extended theories through the Bayesian selection criteria AIC and BIC.