= Press-Schechter halo mass function
{c}
The <Press-Schechter halo mass function> estimates the comoving <number density> of <dark matter halos> per <mass> interval. Let $\bar\rho$ be the mean comoving matter <mass density>, $\sigma(M,z)$ the root-mean-square linearly evolved <density contrast> smoothed on mass scale $M$, and $\delta_c$ the <linear spherical-collapse threshold>. Define $\nu=\delta_c/\sigma(M,z)$. For a <Gaussian distribution> of the initial smoothed <density contrast>, the doubled upper-tail prescription gives
$$
F(>M)=2\int_\nu^\infty\frac{e^{-x^2/2}}{\sqrt{2\pi}}\,dx.
$$
Assume $\sigma$ decreases with $M$. Identifying the <mass> fraction in a narrow interval with $[M/\bar\rho](dn/dM)dM$ gives
$$
\frac{dn}{dM}=-\frac{\bar\rho}{M}\frac{dF(>M)}{dM}
=-\sqrt{\frac2\pi}\frac{\bar\rho}{M^2}
\nu e^{-\nu^2/2}\frac{d\log\sigma}{d\log M}.
$$
The minus sign makes the abundance positive for the stated decreasing variance. The exponential suppresses rare high-$\nu$ objects, while the derivative converts the smoothing scale into a <mass> interval. This is a statistical collapse model: spherical dynamics and its treatment of nested collapsed regions limit the accuracy of the predicted abundance.<Halo Models of Large Scale Structure by Asantha Cooray and Ravi Sheth>[Halo abundance and Gaussian-threshold mass-function construction]{ref}
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