Parametrization of the equation of state and the expanding universe

TL;DR

This paper explores the parametrization of the equation of state in cosmology using Taylor series expansions, analyzing convergence and deriving related physical quantities to improve modeling of the universe's expansion.

Contribution

It introduces a systematic approach to approximate the equation of state and related cosmological functions via Taylor series, including convergence analysis and explicit formulas.

Findings

Convergent criteria for the Taylor series are established.

Explicit power series for energy density and Hubble parameter are derived.

The approach enables flexible truncation for accurate cosmological modeling.

Abstract

The structure of the equation of state $\omega$ could be very complicate in nature while a few linear models have been successful in cosmological predictions. Linear models are treated as leading approximation of a complete Taylor series in this paper. If the power series converges quickly, one can freely truncate the series order by order. Detailed convergent analysis on the choices of the expansion parameters is presented in this paper. The related power series for the energy density function, the Hubble parameter and related physical quantities of interest are also computed in this paper.