# Analytical expressions and numerical evaluation of the luminosity distance in a flat cosmology

## TL;DR

This paper introduces a new hypergeometric function-based expression for calculating luminosity distances in flat mbda CDM cosmology, offering improved computational efficiency especially at high redshifts, and compares it with existing methods.

## Contribution

It presents a novel, simple hypergeometric function expression for luminosity distance in flat mbda CDM cosmology, enhancing computational efficiency over existing elliptic integral methods.

## Key findings

- The new expression is the most efficient for redshifts z  1.
- Combining the new formula with Carlson's elliptic integral approach covers the full redshift range.
- Approximate fitting formulae provide acceptable accuracy for practical purposes.

## Abstract

Accurate and efficient methods to evaluate cosmological distances are an important tool in modern precision cosmology. In a flat $\Lambda$CDM cosmology, the luminosity distance can be expressed in terms of elliptic integrals. We derive an alternative and simple expression for the luminosity distance in a flat $\Lambda$CDM based on hypergeometric functions. Using a timing experiment we compare the computation time for the numerical evaluation of the various exact formulae, as well as for two approximate fitting formulae available in the literature. We find that our novel expression is the most efficient exact expression in the redshift range $z\gtrsim1$. Ideally, it can be combined with the expression based on Carlson's elliptic integrals in the range $z\lesssim1$ for high precision cosmology distance calculations over the entire redshift range. On the other hand, for practical work where relative errors of about 0.1% are acceptable, the analytical approximation proposed by Adachi & Kasai (2012) is a suitable alternative.