Cosmographic analysis with Chebyshev polynomials

Abstract

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