Hierarchical galaxy formation ($\sigma (M)\text{ increases as }M\text{ decreases}$)

In hierarchical galaxy formation, low-mass dark-matter structures typically collapse before larger structures because the smoothed linear density variance is larger on smaller mass scales. Accretion and mergers assemble increasingly massive haloes. Cooling, feedback and angular momentum determine how baryons form galaxies within that halo hierarchy; the ordering is statistical rather than an absolute rule for every object.

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.

Gas in a collapsing halo is heated toward a virial temperature. If its cooling time is short compared with the dynamical time, thermal support is removed and rapid condensation can form a galaxy. If cooling is slower, a hot atmosphere can persist or condense only gradually. This criterion links baryonic galaxy formation to the dark-matter assembly hierarchy without making every halo into one luminous object.

Primordial atomic line cooling becomes effective near the temperature where electronic excitations are accessible, of order ten thousand kelvin. Molecular hydrogen has lower-energy transitions and can cool colder gas if it forms and survives. The corresponding virial temperature thresholds translate into epoch-dependent halo mass thresholds. Radiation, molecular chemistry, metal-line cooling and gas density qualify a single universal minimum galaxy mass.

In simple hierarchical models, increasing system mass raises the virial temperature and can carry primordial gas beyond its efficient atomic line-cooling interval. At assembly densities, the cooling-time constraint then distinguishes galaxy-sized condensations from larger hot groups or clusters. The resulting characteristic mass is approximate, depends on composition and structure, and does not prohibit subsequent mergers of already formed stellar systems.