 Solution

Let $D(0)=1$. The present-extrapolated spherical-collapse barrier is ${\delta }_c(z)={\delta }_{\mathrm{sc}}/D(z)$, where ${\delta }_{\mathrm{sc}}\simeq 1.686$ is the linear contrast extrapolated to spherical collapse in a matter-dominated universe. It is not the nonlinear interior overdensity $18{\pi }^2$. The cosmological mass variance ${\sigma }^2(M,0)$ is the variance of the present-extrapolated linear density contrast smoothed over a mass $M$. It is not the variance of already virialized halo densities.

For a fixed-shape smoothing filter and $P(k)=P_0{k}^n$, rescale the variance integral with $x=kR$:Thus $\sigma (M,0)=S{M}^{-q}$, with $q=(n+3)/6$ and $S$ a normalization carrying the appropriate mass units. For a real-space top-hat filter, convergence requires $-3<n<1$; a spectrum outside this interval needs physical cutoffs and does not have this unrestricted scale-free result.

If $n(M,z)dM$ counts halos per comoving volume, the Press-Schechter halo mass function follows by differentiating the cumulative mass fraction and converting mass fraction to number:The minus sign is needed because the cumulative fraction decreases with $M$. To match the printed exponential coefficient exactly, define $M_{\ast }(z)$ byThis peak-height calibration from an exponential mass-function cutoff differs from the also-common convention $\nu (M_{\ast })=1$. Here $\nu =\sqrt{2}(M/M_{\ast }{)}^q$, and consequentlyEquivalently $A=\sqrt{2/\pi }q{\bar{\rho }}_{m,0}{\delta }_c/S$. The scale-free halo cutoff normalization therefore evolves with $z$; the displayed $A$ is not a redshift-independent universal constant. For physical number density, replace ${\bar{\rho }}_{m,0}$ by ${\bar{\rho }}_m(z)$ instead. Since $M_{\ast }=(\sqrt{2}S/{\delta }_c{)}^{1/q}$, it grows as $D^{6/(n+3)}$; in matter domination it is proportional to $(1+z{)}^{-6/(n+3)}$. At the collapse epoch itself, $\sigma (M_{\ast },z)={\delta }_{\mathrm{sc}}/\sqrt{2}$.