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.

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.