Cosmographic analysis with Chebyshev polynomials

TL;DR

This paper introduces Chebyshev polynomial parameterization for cosmography, significantly reducing error propagation and improving stability and accuracy over traditional Taylor and Padé series, especially at high redshift.

Contribution

The study demonstrates that rational Chebyshev polynomials outperform standard series in cosmography, providing unbiased, stable, and highly accurate estimations of cosmographic parameters.

Findings

Chebyshev polynomials reduce error propagation compared to Taylor series.

Chebyshev approach remains stable at high redshift data.

Standard Taylor series fail to predict accurately with high-redshift data.

Abstract

The limits of standard cosmography are here revised addressing the problem of error propagation during statistical analyses. To do so, we propose the use of Chebyshev polynomials to parameterize cosmic distances. In particular, we demonstrate that building up rational Chebyshev polynomials significantly reduces error propagations with respect to standard Taylor series. This technique provides unbiased estimations of the cosmographic parameters and performs significatively better than previous numerical approximations. To figure this out, we compare rational Chebyshev polynomials with Pad\'e series. In addition, we theoretically evaluate the convergence radius of (1,1) Chebyshev rational polynomial and we compare it with the convergence radii of Taylor and Pad\'e approximations. We thus focus on regions in which convergence of Chebyshev rational functions is better than standard approaches. With this recipe, as high-redshift data are employed, rational Chebyshev polynomials remain highly stable and enable one to derive highly accurate analytical approximations of Hubble's rate in terms of the cosmographic series. Finally, we check our theoretical predictions by setting bounds on cosmographic parameters through Monte Carlo integration techniques, based on the Metropolis-Hastings algorithm. We apply our technique to high-redshift cosmic data, using the JLA supernovae sample and the most recent versions of Hubble parameter and baryon acoustic oscillation measurements. We find that cosmography with Taylor series fails to be predictive with the aforementioned data sets, while turns out to be much more stable using the Chebyshev approach.