Peak-height calibration from an exponential mass-function cutoff ($\sigma (M,z)=\frac{{\delta }_{\mathrm{sc}}}{\sqrt{2}}(M/M_{h,\mathrm{ch}}{)}^{-\beta }$)

Defining characteristic mass by an exponential factor $\exp [-(M/M_{h,\mathrm{ch}}{)}^{2\beta }]$ fixes its peak height to $\nu =\sqrt{2}$, rather than one. Hence $\sigma (M_{h,\mathrm{ch}},z)={\delta }_{\mathrm{sc}}/\sqrt{2}$ and the variance at other masses follows the same power-law scaling. The precise characteristic-mass convention determines this factor.

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; $M_{\ast }\propto D^{1/q}$.