Cooling criterion for galaxy formation ($t_{\mathrm{cool}}\lesssim t_{\mathrm{dyn}}$)

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.