Halo mass function

Remarks

Often one uses $dn/d\ln M = M \, n(M, z)$ instead. Many models write it in the form $$n(M, z) = f(\sigma) \, \frac{\bar\rho}{M} \, \abs{\frac{d \ln \sigma}{dM}} .$$ Here $\bar\rho$ is the mean comoving matter density and $f$ is a function to be modelled or fitted. The quantity $\sigma = \sigma(M, z)$ is the standard deviation of the linear density field smoothed on the mass scale $M$. It is computed from the matter power spectrum.

Example: the Press–Schechter formalism gives $f(\sigma) = \sqrt{2/\pi} \, (\delta_c/\sigma) \, e^{-\delta_c^2/(2\sigma^2)}$ with $\delta_c \approx 1.686$; this $f$ is positive and integrable, and $\int n(M, z) \, M \, dM = \bar\rho$ for it.

Non-example: the galaxy luminosity function counts galaxies per unit volume and per unit luminosity. It counts different objects by a different quantity, so it is not a halo mass function, even if the two are related through models of galaxy formation.

The value of $n$ depends on the halo mass convention. For example, the mass within the radius where the mean density is $200$ times the critical density differs from the virial mass.