Scale-free halo cutoff normalization ($A=2q{\bar{\rho }}_{m,0}{M}_{\ast }^{-q}/\sqrt{\pi }$)

For $\sigma (M,0)=S{M}^{-q}$ with $q=(n+3)/6>0$, matching the Press-Schechter halo mass function to $A{M}^{q-2}\exp [-(M/M_{\ast }{)}^{2q}]$ requires $\sigma (M_{\ast },0)={\delta }_c(z)/\sqrt{2}$. The comoving normalization is $A=2q{\bar{\rho }}_{m,0}{M}_{\ast }^{-q}/\sqrt{\pi }$. Consequently both $A$ and $M_{\ast }$ evolve with the linear growth factor; $M_{\ast }\propto D^{1/q}$.