Comoving number density ($n_p=(1+z{)}^3{n}_c$)

Comoving number density is object count per comoving volume. With scale factor normalized to unity today, proper density is $n_p=a^{-3}{n}_c$. A conserved population moving with homogeneous expansion has constant $n_c$ despite declining proper density. Formation, mergers or changing sample selection can make $n_c$ evolve.

For constant comoving number density and complete selection in a spatially flat Einstein-de Sitter universe, integrate the present-epoch shell volume up to the population's formation redshift. The radial distance is $D_M=2c[1-(1+z{)}^{-1/2}]/H_0$, so the count per unit solid angle is $n_0{D}_M^3(z_f)/3$. It tends to a finite value as $z_f$ tends to infinity because this model's present comoving particle-horizon radius is $2c/H_0$. Formation, mergers or an incomplete observed sample change this ideal count.

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.