Press–Schechter formalism

Definition

Let $\bar\rho$ be the mean comoving matter density. Let the linear density contrast, extrapolated to the time $t$, be a Gaussian random field with matter power spectrum $P(k)$. For a mass $M$, let $R$ be given by $M = \frac{4\pi}{3} \bar\rho R^3$. Let $\sigma(M)$ be the standard deviation of the field smoothed with a spherical top hat of radius $R$. Let $\delta_c \approx 1.686$ be the collapse threshold of the spherical collapse model. Let $F(> M)$ be the fraction of mass in collapsed halos of mass greater than $M$. The Press–Schechter formalism sets $$F(> M) = 2 \, \Pr\left(\delta_R > \delta_c\right) = \operatorname{erfc}\left( \frac{\delta_c}{\sqrt{2} \, \sigma(M)} \right).$$ Here $\delta_R$ is the smoothed field, with a normal distribution of mean $0$ and variance $\sigma(M)^2$. The function erfc is the complementary error function. The resulting halo mass function is $n(M) = (\bar\rho / M) \abs{dF/dM}$, that is, $$n(M) = \sqrt{\frac{2}{\pi}} \, \frac{\bar\rho}{M^2} \, \nu \, \abs{\frac{d \ln \sigma}{d \ln M}} \, e^{-\nu^2/2}, \qquad \nu = \frac{\delta_c}{\sigma(M)} .$$