The Hubble series: convergence properties and redshift variables
Abstract
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 wellknown exemplar of this phenomenon is the Hubble relation between distance and redshift. However, we now have considerable highz 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 zredshift) 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 zbased 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 yredshift, we again argue that the radius of convergence of any series expansion in y is less than or equal to 1, so that ybased expansions are likely to be good all the way back to the big bang (y = 1), but that ybased expansions must break down for y < 1, now corresponding to a universe more than twice its current size.
 Publication:

Classical and Quantum Gravity
 Pub Date:
 December 2007
 DOI:
 10.1088/02649381/24/23/018
 arXiv:
 arXiv:0710.1887
 Bibcode:
 2007CQGra..24.5985C
 Keywords:

 General Relativity and Quantum Cosmology
 EPrint:
 15 pages, 2 figures, accepted for publication in Classical and Quantum Gravity