Pad\'e Approximant and Minimax Rational Approximation in Standard Cosmology

Abstract

The luminosity distance in the standard cosmology as given by $\Lambda$CDM and consequently the distance modulus for supernovae can be defined by the Pad\'e approximant. A comparison with a known analytical solution shows that the Pad\'e approximant for the luminosity distance has an error of $4\%$ at redshift $= 10$. A similar procedure for the Taylor expansion of the luminosity distance gives an error of $4\%$ at redshift $=0.7 $; this means that for the luminosity distance, the Pad\'e approximation is superior to the Taylor series. The availability of an analytical expression for the distance modulus allows applying the Levenberg--Marquardt method to derive the fundamental parameters from the available compilations for supernovae. A new luminosity function for galaxies derived from the truncated gamma probability density function models the observed luminosity function for galaxies when the observed range in absolute magnitude is modeled by the Pad\'e approximant. A comparison of $\Lambda$CDM with other cosmologies is done adopting a statistical point of view.