A curious relation between the flat cosmological model and the elliptic integral of the first kind

TL;DR

This paper derives a closed-form expression for the luminosity distance-redshift relation in a flat universe with a non-zero cosmological constant, using elliptic integrals to avoid numerical integration.

Contribution

It introduces a novel analytical relation involving elliptic integrals for the flat cosmological model with a non-zero cosmological constant.

Findings

Provides a closed-form solution for luminosity distance in flat universe with cosmological constant.

Eliminates the need for numerical integration in relevant cosmological calculations.

Enhances analytical tools for cosmological distance measurements.

Abstract

The dependence of the luminosity distance on the redshift has a key importance in the cosmology. This dependence can well be given by standard functions for the zero cosmological constant. The purpose of this article is to present such a relation also for the non-zero cosmological constant, if the universe is spatially flat. A definite integral is used. The integration ends in the elliptic integral of the first kind. The result shows that no numerical integration is needed for the non-zero cosmological constant, if the universe is spatially flat.