Statistical hierarchy of varying speed of light cosmologies

Variation of the speed of light is quite a debated issue in cosmology with some benefits, but also with some controversial concerns. Many approaches to develop a consistent varying speed of light (VSL) theory have been developed recently. Although a lot of theoretical debate has sprout out about their feasibility and reliability, the most obvious and straightforward way to discriminate and check if such theories are really workable has been missed out or not fully employed. What is meant here is the comparison of these theories with observational data in a fully comprehensive way. In this paper we try to address this point i.e., by using the most updated cosmological probes, we test three different candidates for a VSL theory (Barrow \& Magueijo, Avelino \& Martins, and Moffat) signal. We consider many different ans\"{a}tze for both the functional form of $c(z)$ (which cannot be fixed by theoretical motivations) and for the dark energy dynamics, in order to have a clear global picture from which we extract the results. We compare these results using a reliable statistical tool such as the Bayesian Evidence. We find that the present cosmological data is perfectly compatible with any of these VSL scenarios, but in one case (Moffat model) we have a higher Bayesian Evidence ratio in favour of VSL than in the standard $c=$ constant $\Lambda$CDM scenario. Moreover, in such a scenario, the VSL signal can help to strengthen constraints on the spatial curvature (with indication toward an open universe), to clarify some properties of dark energy (exclusion of a cosmological constant at $2\sigma$ level) and is also falsifiable in the nearest future due to some peculiar issues which differentiate this model from the standard model. Finally, we have applied some priors which come from cosmology and, in particular, from information theory and gravitational thermodynamics.