For a Gaussian smoothed density contrast, the Press-Schechter factor of two gives the collapsed mass fraction
$$
F(>M)=2\int_{\delta_c}^{\infty}
\frac{d\delta}{\sqrt{2\pi}\sigma}
e^{-\delta^2/(2\sigma^2)}
=\operatorname{erfc}\left(\frac\nu{\sqrt2}\right),
\qquad
\nu=\frac{\delta_c}{\sigma(M)}.
$$
The fraction in the interval $[M,M+dM]$ is $-(dF/dM)dM=(M/\bar\rho)(dn/dM)dM$. Since $d\nu/dM=-\nu\,d\log\sigma/dM$,
$$
\boxed{
\frac{dn}{dM}
=-\sqrt{\frac2\pi}\frac{\bar\rho}{M^2}
\nu e^{-\nu^2/2}\frac{d\log\sigma}{d\log M}}
$$
or, equivalently, the same expression with $|d\log\sigma/d\log M|$. The minus sign is needed because $\sigma(M)$ decreases with $M$; this is the positive <Press-Schechter halo mass function>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-312/3/a/solution