High-redshift cosmography: auxiliary variables versus Pad\'e polynomials
Abstract
Cosmography becomes non-predictive when cosmic data span beyond the red shift limit z1 . This leads to a strong convergence issue that jeopardizes its viability. In this work, we critically compare the two main solutions of the convergence problem, i.e. the y-parametrizations of the redshift and the alternatives to Taylor expansions based on Pad\'e series. In particular, among several possibilities, we consider two widely adopted parametrizations, namely y1=1-a and y2=(a-1-1), being a the scale factor of the Universe. We find that the y2-parametrization performs relatively better than the y1-parametrization over the whole redshift domain. Even though y2 overcomes the issues of y1, we get that the most viable approximations of the luminosity distance dL(z) are given in terms of Pad\'e approximations. In order to check this result by means of cosmic data, we analyze the Pad\'e approximations up to the fifth order, and compare these series with the corresponding y-variables of the same orders. We investigate two distinct domains involving Monte Carlo analysis on the Pantheon Superovae Ia data, H(z) and shift parameter measurements. We conclude that the (2,1) Pad\'e approximation is statistically the optimal approach to explain low and high-redshift data, together with the fifth-order y2-parametrization. At high redshifts, the (3,2) Pad\'e approximation cannot be fully excluded, while the (2,2) Pad\'e one is essentially ruled out.