Recovering a redshift-extended VSL signal from galaxy surveys

We investigate a new method to recover (if any) a possible varying speed of light (VSL) signal from cosmological data. It comes as an upgrade of [1,2], where it was argued that such signal could be detected at a single redshift location only. Here, we show how it is possible to extract information on a VSL signal on an extended redshift range. We use mock cosmological data from future galaxy surveys (BOSS, DESI, \emph{WFirst-2.4} and SKA): the sound horizon at decoupling imprinted in the clustering of galaxies (BAO) as an angular diameter distance, and the expansion rate derived from those galaxies recognized as cosmic chronometers. We find that, given the forecast sensitivities of such surveys, a $\sim1\%$ VSL signal can be detected at $3\sigma$ confidence level in the redshift interval $z \in [0.,1.55]$. Smaller signals $(\sim0.1\%)$ will be hardly detected (even if some lower possibility for a $1\sigma$ detection is still possible). Finally, we discuss the degeneration between a VSL signal and a non-null spatial curvature; we show that, given present bounds on curvature, any signal, if detected, can be attributed to a VSL signal with a very high confidence. On the other hand, our method turns out to be useful even in the classical scenario of a constant speed of light: in this case, the signal we reconstruct can be totally ascribed to spatial curvature and, thus, we might have a method to detect a $0.01$-order curvature in the same redhift range with a very high confidence.