An analytical approximation of the luminosity distance in flat cosmologies with a cosmological constant

Abstract

We present an analytical approximation formula for the luminosity distance in spatially flat cosmologies with dust and a cosmological constant. Apart from the overall factor, the effect of non-zero cosmological constant in our formula is written simply in terms of a rational function. We also show the approximate formulae for the Dyer-Roeder distance (the empty beam case) and the generalized angular diameter distance from redshift z 1 to z 2, which are particularly useful in analyzing the gravitational lens effects. Our formulae are widely applicable over the range of the density parameter and the redshift with sufficiently small uncertainties. In particular, in the range of density parameter 0.3 ≤ Ω m ≤ 1 and redshift 0.03 ≤ z ≤ 1000, the relative error for the luminosity distance by our formula is always smaller than that of the recent work by Wickramasinghe and Ukwatta (2010). Hence, we hope that our formulae will be an efficient and useful tool for exploring various problems in observational cosmology.