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

TL;DR

This paper introduces a simple analytical approximation for the luminosity distance in flat cosmologies with a cosmological constant, improving accuracy over previous formulas across a wide parameter range.

Contribution

It provides a new, accurate, and easy-to-use analytical formula for luminosity and angular diameter distances in flat cosmologies with a cosmological constant.

Findings

Relative error less than previous models for specified parameter ranges.

Applicable to redshifts from 0.03 to 1000 and density parameters 0.3 to 1.

Useful for gravitational lensing and observational cosmology analyses.

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 (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 $0.3 \leq \Omega_{\rm m} \leq 1$ and redshift $0.03 \leq z \leq 1000$, 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.