Open dust distance-redshift relation ($r(z)=\frac{2[\Omega z+(2-\Omega )(1-\sqrt{1+\Omega z})]}{a_0{H}_0{\Omega }^2(1+z)}$)

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 $d_S={\int }_0^{z}d{z}^{\prime }/H(z^{\prime })$, and the areal radial coordinate satisfies $a_{0}r=\sinh [H_0\sqrt{1-\Omega }{d}_S]/[H_0\sqrt{1-\Omega }]$. Integrating with $v=\sqrt{1+\Omega z}$ gives the displayed relation. It has $r\sim z/(a_0{H}_0)$ near the observer and $r\to 2/(a_0{H}_0\Omega )$ as $z\to \infty$. The luminosity distance is $(1+z)a_{0}r$, not the radial cosmological proper distance.