Redshift in varying speed of light cosmology

In standard cosmology, redshift is related to scale factor by z = a-1 - 1. Varying speed of light cosmologies have also applied this relationship, in which c does not explicitly appear, with the assumption that ć. Measured redshift is not a comparison of an observed spectrum with the spectrum as it was emitted at a distant location, but a comparison with a reference spectrum generated more locally. This distinction suggests decomposition into two parts: (a) change during the flight of a photon and (b) difference in physics at the time of emission and at the time of observation of a photon associated with an electron transition between specific bound states of an atom. Based on atomic units consistent with data and a relativistic atomic model, redshift is given by z = β(θ)θa-1 - 1, where θ = c/c0, with c0 the present value of c, and β is a function of the atomic parameters describing the transition. The modified form appears to have a modest effect (a difference in scale factor <2 per cent) for redshifts that are not much greater than 10. However, the modification can have a major effect for an early universe with c significantly larger than the present. The simplified form z = θa-1 - 1, which results from a non-relativistic model, provides an approximation for redshift that is not transition-specific.