Source counts on an FLRW past light cone ($dN=4\pi a^2{r}^{2}ndt$)

For proper number density $n(t)$ in an FLRW metric $d{s}^2=d{t}^2-a^2[d{r}^2/(1-k{r}^2)+r^{2}d{\Omega }^2]$, a radial shell has proper volume $4\pi a^3{r}^2|dr|/\sqrt{1-k{r}^2}$. A radial null geodesic obeys $|dr|=\sqrt{1-k{r}^2}dt/a$. Multiplication gives the displayed shell count on a local radial branch of the past light cone. Spatially uniform number density need not be time independent or have conserved comoving number density.