On the relation between the non-flat cosmological models and the elliptic integral of first kind

TL;DR

This paper explores the mathematical relationship between non-flat cosmological models and elliptic integrals, extending previous flat model results to more general geometries using Taylor series and numerical methods.

Contribution

It introduces an analytic formula involving elliptic integrals for non-flat models, building on prior flat model formulas, and suggests combined numerical-analytic approaches for nearly flat universes.

Findings

Elliptic integrals can describe non-flat cosmological distances.

A Taylor series approach extends flat model formulas to non-flat cases.

Combined numerical and analytic methods improve calculations for nearly flat models.

Abstract

Recently we have found an analytic integration formula that describes the dependence of the luminosity distance on the redshift for the flat cosmological model with a non-zero cosmological constant. The purpose of this article is to search for a similar relation for the non-flat models. A Taylor series was used. The elliptic integral of the first kind can indeed be applied. The result shows that a combination of the numerical integration with the previous analytic formula can also be useful for the nearly flat cosmological models.