High-redshift cosmography: auxiliary variables versus Padé polynomials

Cosmography becomes non-predictive when cosmic data span beyond the redshift limit z ≃ 1. 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 Padé series. In particular, among several possibilities, we consider two widely adopted parametrizations, namely y_1 = 1−a and | y 2 = arctan ⁡ ( a − 1 − 1 ) y 2 ​ =arctan(a −1 −1)|⁠, being a the scale factor of the Universe. We find that the y_2-parametrization performs relatively better than the y_1-parametrization over the whole redshift domain. Even though y_2 overcomes the issues of y_1, we get that the most viable approximations of the luminosity distance (z) are given in terms of Padé approximations. In order to check this result by means of cosmic data, we analyse the Padé 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) Padé approximation is statistically the optimal approach to explain low- and high-redshift data, together with the fifth-order y_2-parametrization. At high redshifts, the (3,2) Padé approximation cannot be fully excluded, while the (2,2) Padé one is essentially ruled out.