Galaxy stellar mass function ($d{n}_{\mathrm{gal}}/d{M}_{\ast }$)

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.

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.

A generalized mass function with low-mass slope $-7/4$ and cutoff $e^{-\sqrt{M_{\ast }/M_{\ast ,\mathrm{ch}}}}$ differs from the ordinary exponential Schechter cutoff. Its total stellar-mass density is finite, even though its untruncated total galaxy count diverges. This form arises from a scale-free Press-Schechter mass function with a constant stellar-to-halo mass mapping.

Integrating galactic stellar mass against a $x^{-7/4}{e}^{-\sqrt{x}}$ mass function requires ${\int }_0^{\infty }{x}^{-3/4}{e}^{-\sqrt{x}}dx$. Substitution $u=\sqrt{x}$ turns it into $2\Gamma (1/2)=2\sqrt{\pi }$. Thus the stellar density is $2\sqrt{\pi }$ times characteristic number density times characteristic galactic stellar mass, in the same volume convention as the mass function.