An analytical approximation of the luminosity distance in flat cosmologies with dark energy

We present an analytical approximation formula for the luminosity distance in spatially flat cosmologies with dust and a cosmological constant. We also show the approximate formulae for the so-called Dyer-Roeder distance (empty beam case) and the generalised angular diameter distance from redshift $z=z_1$ to $z=z_2$, which are particularly useful in analysing 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 $0.3 \lid Ω_{\rmn m} \lid 1$ and redshift $0.03 \lid z \lid 1000$, the relative error for the luminosity distance by our formula is always smaller than that of the recent work by \cite{wu}.

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