A curious relation between the flat cosmological model and the elliptic integral of the first kind (Research Note)

Abstract

Context. 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. Aims.

The purpose of this article is to present a similar relation also for the non-zero cosmological constant, if the universe is spatially flat. Methods.

A definite integral was used. Results.

The integration ends in the elliptic integral of the first kind. Conclusions.

The result shows that no numerical integration is needed for the non-zero cosmological constant if the universe is spatially flat.

Keywords

• cosmology
• theory