The Smallest Haloes That Can Cool Their Gas When a Uniform Heating Term Is Present

The galaxy population has a lower edge: below a halo mass of a few times 1e8 solar masses, gas can no longer cool within a Hubble time and haloes stay dark or nearly so. That threshold — set by the requirement that virialised gas reach the atomic cooling regime at 1e4 K and radiate faster than the universe ages — is a clean function of cooling physics, and this paper computes how a uniform dissolution heating term Q = f u_L / t_H, from the surviving vacuum density u_L = 5.25e-09 erg per cubic centimetre, moves it. Two redshift scalings of the term are carried in parallel: variant A, constant at today's rate, and variant B, tracking the ambient density as (1 + z)^3. Without the term, the minimum cooling halo mass is 1.78e09 solar masses at z = 0, 5.62e08 at z = 2, and 2.00e08 at z = 5. At the forest-permitted coupling the thresholds are identical in both variants. At f = 1e-05, variant A leaves the z = 5 threshold untouched (2.00e08) and lifts z = 2 modestly to 7.08e08, while variant B lifts z = 2 to 3.55e09 and z = 5 to 3.98e08; at z = 0, f = 1e-05 in either variant leaves no halo in the scanned range (to 1e11 solar masses) able to cool its mean-density gas at all. Since dwarf galaxies with old stellar populations demonstrably did cool their gas in haloes of a few times 1e8 solar masses, thresholds computed here convert directly into an exclusion: a uniform term at f of 1e-05 with density-tracking scaling is incompatible with the dwarf population's existence at z = 2, and even the constant variant is excluded at low redshift by the same logic applied to late-forming systems. Computed from stated assumptions; not a measurement.