Einstein-de Sitter luminosity-distance relation ($d_L=2(c/H_0)[1+z-\sqrt{1+z}]$)

In an Einstein-de Sitter universe normalized by $a_0=1$, the Friedmann equation gives $H(z)=H_0(1+z{)}^{3/2}$. Integrating the radial null geodesic gives $r(z)=c{\int }_0^{z}d{z}^{\prime }/H(z^{\prime })=2(c/H_0)[1-(1+z{)}^{-1/2}]$. The luminosity distance is $(1+z)r(z)$, yielding the displayed expression. Near the observer it is $(c/H_0)[z+z^2/4+O(z^3)]$; the limiting radial comoving distance is finite even though the luminosity distance diverges.