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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, $\sigma (M_{\ast },z)={\delta }_{\mathrm{sc}}/\sqrt{2}$.

The characteristic halo mass inferred from the stellar mapping isThe peak-height calibration from an exponential mass-function cutoff and $\beta =1/4$ giveUsing ${\delta }_{\mathrm{sc}}\simeq 1.686$, appropriate to the nearly matter-dominated collapse at this redshift,The barrier in the original complementary-error-function expression is extrapolated to the present, but this answer is the variance at $z=3$. Rewriting the ratio with ${\delta }_{\mathrm{sc}}$ and $\sigma (M,3)$ consistently cancels the growth normalization; using the present-extrapolated barrier with the redshift-three variance would double-count growth.

The baryon conversion efficiency of a halo is not observationally constant. The stellar-to-halo mass ratio rises from low-mass haloes to a broad maximum near galactic halo masses, then decreases toward groups and clusters; it also varies with redshift and has intrinsic scatter. Photoheating and stellar-feedback-driven outflows suppress baryon retention and star formation in small haloes. Long cooling times, hot atmospheres and feedback from active nuclei limit efficient conversion in massive haloes. The peak commonly occurs around halo masses of order $10^{12}{M}_{\odot }$, with its position evolving with epoch.

Therefore a mass-independent $f_{\ast }\simeq 0.05$ is a simplified average, not a realistic stellar-to-halo relation. Observed ratios must be compared with ${ϵ}_{\ast }=f_b{f}_{\ast }$, not with $f_{\ast }$ alone. Scatter, satellite populations and the mass-dependent efficiency change the stellar mass-function shape and invalidate an exact constant-rescaling of the halo mass function.