Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 61 / 3 / iv / Solution

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.