Comoving volume ($V=a^3{V}_c$)

A comoving volume has fixed boundaries in comoving coordinates. Its physical volume grows as the cube of the scale factor. In a homogeneous adiabatic gas, cosmological entropy conservation makes $sV$ constant.

On a present FLRW spatial slice, a shell has volume $R_0^3{S}^2(\chi )d\chi d\Omega$. The radial light-ray relation gives $d\chi /dz=c/(R_{0}H)$, yielding the displayed expression in terms of transverse comoving distance. At emission the proper volume is smaller by $(1+z{)}^3$. For a conserved comoving population its proper density is greater by the same factor, so either epoch gives the same count.

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.