Scale-free characteristic mass in the Press-Schechter formalism ($M_{\ast }\propto D^{1/s},\sigma (M)\propto M^{-s}$)

If the linearly extrapolated rms amplitude scales as $\sigma (M)\propto M^{-s}$, a fixed peak-height cutoff in the Press-Schechter halo mass function has $M_{\ast }\propto D^{1/s}$, where $D$ is the linear growth factor. The differential prefactor scales as $M^{s-2}$ and the exponential argument as $M^{2s}$. The cutoff convention must be stated when fixing its absolute normalization: $\nu =1$ and ${\nu }^2/2=1$ select different masses, although their redshift scaling is the same.