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.