Baryon conversion efficiency of a halo ($f_{\ast }=M_{\ast }/(f_b{M}_h)$)

The baryon conversion efficiency of a halo is the fraction of its baryonic mass incorporated into stars. If the halo retains the cosmic baryon fraction $f_b$, its stellar-to-total-mass ratio is $f_b{f}_{\ast }$. This efficiency depends on halo mass, epoch, gas supply and feedback and must not be confused with a star-formation efficiency per free-fall time.

A monotonic luminosity assignment to a counted halo population converts its differential abundance per mass into a luminosity function by a change of variable. If $n_h\propto M_h^{-a}$ and $L\propto M_h^p$, then $\Phi \propto L^{-1-(a-1)/p}$. This simple mapping assumes one counted galaxy per halo, no scatter and a fixed selection. A mass-dependent baryon conversion efficiency of a halo, satellites and stellar-population differences modify it.

The stellar-to-halo mass ratio compares a galaxy's galactic stellar mass with its host's total halo mass. In a simple baryon-conversion model it is ${ϵ}_{\ast }=f_b{f}_{\ast }$. Observational comparisons must distinguish this total-mass ratio from the fraction of baryons converted, and must specify whether central, satellite or all stellar populations are included.

If every counted halo hosts one counted galaxy and the baryon fraction and conversion efficiency are mass-independent without scatter, galactic stellar mass is a constant multiple of halo mass. The galaxy mass function is obtained by changing variable with $dM/d{M}_{\ast }=1/{ϵ}_{\ast }$. Satellites, efficiency variation and scatter prevent this mapping from being generally exact.