Modelling spatial variations of the speed of light

In this paper we extend a new method to measure possible variation of the speed of light by using Baryon Acoustic Oscillations and the Hubble function presented in our earlier paper [V. Salzano, M. P. D\c{a}browski, and R. Lazkoz, Phys. Rev. D93, 063521 (2016)] onto an inhomogeneous model of the universe. The method relies on the fact that there is a simple relation between the angular diameter distance $(D_{A})$ maximum and the Hubble function $(H)$ evaluated at the same maximum-condition redshift, which includes speed of light $c$. One limit of such method was the assumption of null spatial curvature (even if we showed that even a non-zero curvature would have negligible effects). Here, we move one step further: we explicitly assume a model with intrinsic non-null curvature, and calculate the exact relation between $D_{A}$ and $H$ in this case. Then, we evaluate if current or future missions such as SKA can be sensitive enough to detect any such kind of spatial variation of $c$ which can perhaps be related to the recently observed spatial variation of the fine structure constant (an effect known as $\alpha$-dipole).