An analytical approximation of the growth function in Friedmann-Lema\^itre universes

TL;DR

This paper introduces a simple analytical approximation for the growth function in various Friedmann-Lemaître universes, achieving high accuracy without complex elliptic functions, applicable across different cosmological models.

Contribution

The authors derive a universal rational function approximation for the growth function applicable to all spatially flat, open, and closed universes, simplifying calculations in cosmology.

Findings

Relative error less than 0.2% for flat cosmologies with dust and cosmological constant.

Error less than 0.4% for dust cosmologies without cosmological constant.

Applicable over a wide range of scale factors and density parameters.

Abstract

We present an analytical approximation formula for the growth function in a spatially flat cosmology with dust and a cosmological constant. Our approximate formula is written simply in terms of a rational function. We also show the approximate formula in a dust cosmology without a cosmological constant, directly as a function of the scale factor in terms of a rational function. The single rational function applies for all, open, closed and flat universes. Our results involve no elliptic functions, and have very small relative error of less than 0.2 per cent over the range of the scale factor $1/1000 \la a \lid 1$ and the density parameter $0.2 \la \Omega_{\rmn{m}} \lid 1$ for a flat cosmology, and less than $0.4$ per cent over the range $0.2 \la \Omega_{\rmn{m}} \la 4$ for a cosmology without a cosmological constant.