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 the 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, namely quintessence, F (T) and f (R) gravitational 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 through the use of 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 superflous or even unjustified. We use data from the Union 2.1 supernovae catalogue, baryonic acoustic oscillation data and H (z) differential age compilations, which probe cosmology on different scales of the cosmological evolution. 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. To do so, we perform two distinct fits, assuming only data within z < 1 first and then without limitations in redshift. We obtain the corresponding numerical intervals in which coefficients span, 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, demonstrating that our bounds divert from the expectations of previous works, showing that the permitted regions of coefficients are significantly modified and in general widened with respect to values usually reported in the existing literature. Finally, we test the extended theories through the Bayesian selection criteria AIC and BIC.