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The cosmic stellar mass density is the mass moment of a galaxy stellar mass function in a stated volume convention. With a comoving mass function it is mass per comoving volume, while proper density is larger by $(1+z{)}^3$. Comparing it to a critical or matter density requires keeping the epoch and volume convention explicit.

The galaxy stellar mass function counts galaxies per stellar-mass interval and volume, often using comoving volume. Its stellar-mass moment gives the cosmic stellar mass density, while its number moment counts galaxies only if the low-mass population is integrable or truncated. It differs from the initial mass function of individual stars.

Integrate galactic stellar mass, rather than galaxy count, against the galaxy stellar mass function. With $x=M_{\ast }/M_{\ast ,\mathrm{ch}}$, the comoving cosmic stellar mass density, evaluated by the stellar mass density from a square-root exponential cutoff, isPut $u=\sqrt{x}$, so $dx=2udu$ and $x^{-3/4}=u^{-3/2}$. The integral is $2\Gamma (1/2)=2\sqrt{\pi }$, givingThe supplied present mean matter density and ${\Omega }_m=0.25$ imply ${\rho }_{\mathrm{crit},0}=3.5\times 10^{10}/0.25=1.4\times 10^{11}{M}_{\odot }\mathrm{Mp}{\mathrm{c}}^{-3}$. ThereforeThis is about $0.25\%$ of today's critical density, or $1\%$ of today's mean matter density, expressed per present comoving volume. The proper stellar density at $z=3$ is $64{\rho }_{\ast }^{\mathrm{com}}$; if comparing that proper density directly with today's critical density its ratio is $0.162$. It must not be confused with the usual comoving-density comparison. The low-mass number integral diverges if extrapolated to zero mass, but the mass integral converges; real populations require a low-mass cutoff.