The Hubble series: convergence properties and redshift variables

In cosmography, cosmokinetics and cosmology, it is quite common to encounter physical quantities expanded as a Taylor series in the cosmological redshift z . Perhaps the most well-known exemplar of this phenomenon is the Hubble relation between distance and redshift. However, we now have considerable high- z data available; for instance, we have supernova data at least back to redshift z ≈ 1.75. This opens up the theoretical question as to whether or not the Hubble series (or more generally any series expansion based on the z -redshift) actually converges for large redshift. Based on a combination of mathematical and physical reasonings, we argue that the radius of convergence of any series expansion in z is less than or equal to 1, and that z -based expansions must break down for z > 1, corresponding to a universe less than half of its current size. Furthermore, we shall argue on theoretical grounds for the utility of an improved parametrization y = z /(1 + z ). In terms of the y -redshift, we again argue that the radius of convergence of any series expansion in y is less than or equal to 1, so that y -based expansions are likely to be good all the way back to the big bang ( y = 1), but that y -based expansions must break down for y < −1, now corresponding to a universe more than twice its current size.