Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 67 / 2 / d

First fix a genuine normalization error in the original PDF: the Gaussian exponent uses ${\sigma }^2$, so $\sigma$ must be the standard deviation, the variance is ${\sigma }^2$, and the normalized probability density function isThe printed prefactor has only $\sigma$ under its square root; its integral is $\sqrt{\sigma }$ rather than one. Use the normalized normal distribution above.

The one-scale probability of exceeding the linear spherical-collapse threshold is $P_M=\frac{1}{2}\mathrm{erfc}[{\delta }_c/(\sqrt{2}\sigma )]$. In the Press-Schechter formalism, the mass fraction above $M$ is assigned toThe factor two is necessary to recover all the mass in the hierarchical limit $\sigma \to \infty$: the uncorrected one-point upper tail tends to only one half. A precise justification is available in the excursion-set description of halo formation with a sharp-k smoothing filter. The smoothed Gaussian random field then becomes Brownian motion in its variance $S$. Reflecting a trajectory after its first crossing of ${\delta }_c$ pairs trajectories which crossed but end below the threshold with those which end above it. The Brownian reflection principle therefore gives $\mathrm{Pr}({\max }_{s\le S}\delta (s)>{\delta }_c)=2\mathrm{Pr}(\delta (S)>{\delta }_c)$.

This multiscale mass assignment is an extra modelling prescription: a Gaussian one-point probability and a collapse threshold alone do not mathematically determine a halo mass function. The “all matter” normalization additionally assumes arbitrarily large small-scale variance; a finite variance cutoff leaves an uncollapsed fraction.

Differentiate at fixed $z$ and fixed ${\delta }_c$:For decreasing $\sigma (M,z)$, a mass interval contains the positive fraction $-F^{\prime }(M)dM$. Its comoving mass is ${\rho }_{m,0}[-F^{\prime }(M)]dM$, so division by halo mass gives the Press-Schechter halo mass function:The absolute value, or equivalently a minus sign multiplying ${\partial }_M\sigma$, is essential for a positive abundance. The original PDF correctly places absolute values around the abundance and the full derivative expression; these bars are lost in the converted TeX. The density is comoving, so the prefactor uses today's matter mass density, rather than the physical density at redshift $z$.