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The onset of dark energy domination depends on the particular gravitationaltheory driving the cosmic evolution. Model independent techniques are crucialto test both the present $\Lambda$CDM cosmological paradigm and alternativetheories, making the least possible number of assumptions about the Universe.In this paper we investigate whether cosmography is able to distinguish betweendifferent gravitational theories, by determining bounds on model parameters forthree 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 anextension of previous work and can be seen as the most general approach fortesting extended theories of gravity with cosmography. In the case of $F(T)$and $f(R)$ theories, we show that some assumptions on model parameters oftenmade in previous works are superfluous or unjustified. We use data from theUnion2.1 SN catalogue, BAO data and $H(z)$ differential age compilations, whichprobe cosmology on different scales. We perform a Monte Carlo analysis using aMetropolis-Hastings algorithm with a Gelman-Rubin convergence criterion,reporting 1-$\sigma$ and 2-$\sigma$ confidence levels. We perform two distinctfits, first taking only data at $z<1$ and then for all $z$. We obtain thecorresponding parameter intervals, and find that the data is compatible the$\Lambda$CDM limit of all three theories at the 1-$\sigma$ level, while stillcompatible with quite a large portion of parameter space. We compare ourresults to the truncated $\Lambda$CDM paradigm, showing that the permittedregions of coefficients are significantly modified and in general widenedcompared to values reported in the existing literature. Finally, we test theextended theories through the Bayesian selection criteria AIC and BIC.

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