Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 67 / 1 / i / Solution

Use units $c=1$, and let $r$ be the areal radial coordinate of the FRW metric, rather than the radial comoving distance when $k\ne 0$. On a constant-cosmic time slice the spatial metric tensor isThe determinant of the spatial metric tensor supplies the factor $(1-k{r}^2{)}^{-1/2}$. Integrating over the solid angle gives $d{V}_{\mathrm{shell}}=4\pi a^3{r}^2|dr|/\sqrt{1-k{r}^2}$. An incoming radial null geodesic obeysConsequently the proper shell volume sampled by an emission-time interval of positive width $dt$ is $4\pi a^2{r}^{2}dt$. Multiplying by the proper number density proves the source counts on an FLRW past light cone:Here $r(t)$ is evaluated on the observer's past light cone, not along a fixed source worldline. The curvature factor cancels between the spatial volume and the radial light speed; it has not been set to one.