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