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

Abstract Accurate and efficient methods to evaluate cosmological distances are an important tool in modern precision cosmology. In a flat Λ cold dark matter (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 ΛCDM 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 of z ≳ 1. Ideally, it can be combined with the expression based on Carlson's elliptic integrals in the range of z ≲ 1 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 per cent are acceptable, the analytical approximation proposed by Adachi & Kasai (2012) is a suitable alternative.