Power-law Press-Schechter stellar mass function ($n_{\ast }(M_{\ast })\propto M_{\ast }^{\beta -2}{e}^{-(M_{\ast }/M_{\ast ,\mathrm{ch}}{)}^{2\beta }}$)

For a power-law matter spectrum, smoothed variance scales as $\sigma \propto M^{-\beta }$ with $\beta =(n+3)/6$. Differentiating the Press-Schechter collapsed fraction gives a halo mass function proportional to $M^{\beta -2}\exp [-(M/M_{h,\mathrm{ch}}{)}^{2\beta }]$. A constant stellar-to-halo mass ratio preserves both exponents and rescales characteristic mass and normalization.