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The luminosity distance is defined by $F=L/(4\pi d_L^2)$, where $L$ is the bolometric luminosity and $F$ the received radiative flux. In the FRW metric a sphere through the observer around a comoving source has area $4\pi a_0^2{r}_e^2$. The cosmological redshift reduces each photon's energy by $(1+z{)}^{-1}$, and cosmological time dilation reduces their arrival rate by another such factor. HenceThe positive radial lookback integral is ${\chi }_e=c{\int }_{t_e}^{t_0}dt/a(t)={\int }_0^{r_e}dr/\sqrt{1-k{r}^2}$. The printed time limits are reversed; also the factor $c$ is required unless units $c=1$ are adopted. The areal coordinate $r_e$ and the radial comoving distance ${\chi }_e$ agree only when $k=0$.

For an Einstein-de Sitter universe, the Friedmann equation gives $H(z)=H_0(1+z{)}^{3/2}$. Using the redshift-time relation in the radial null geodesic integral yieldsThus the Einstein-de Sitter luminosity-distance relation isIts small-redshift expansion $d_L=(c/H_0)[z+z^2/4+O(z^3)]$ recovers Hubble's law.