= Open dust distance-redshift relation
{title2=$r(z)=\frac{2[\Omega z+(2-\Omega)(1-\sqrt{1+\Omega z})]}{a_0H_0\Omega^2(1+z)}$}
For an open dust-only <Friedmann-Lemaître-Robertson-Walker metric> with zero <cosmological constant> and $0<\Omega<1$, $H(z)=H_0(1+z)\sqrt{1+\Omega z}$. The present radial <cosmological proper distance> is $d_S=\int_0^z dz'/H(z')$, and the areal radial coordinate satisfies $a_0r=\sinh[H_0\sqrt{1-\Omega}\,d_S]/[H_0\sqrt{1-\Omega}]$. Integrating with $v=\sqrt{1+\Omega z}$ gives the displayed relation. It has $r\sim z/(a_0H_0)$ near the observer and $r\to2/(a_0H_0\Omega)$ as $z\to\infty$. The <luminosity distance> is $(1+z)a_0r$, not the radial <cosmological proper distance>.
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