= Solution
\b[<Galaxy> formation combines gravitational assembly of dark-matter haloes with dissipative baryonic physics.] Primordial density fluctuations supply a statistical seed distribution. In linear theory, each <Fourier mode> evolves through a growth factor and a scale-dependent <transfer function>; the <power spectrum> describes the variance by scale. Radiation pressure, <free streaming> and the time of horizon entry influence the small-scale transfer, so one should not replace the spectrum by a single universal growing power of time.
When fluctuations become nonlinear, overdense regions collapse, virialize and build larger structures through <accretion> and mergers. <Cold dark matter> permits a broad hierarchy of bound haloes. A spherical-collapse threshold and a mass-variance prescription, such as the <Press-Schechter formalism>, organize an approximate halo abundance; departures from <spherical collapse> and correlated environments limit its literal interpretation.
Gas falling into a halo is compressed and may be shock-heated toward a virial scale
$$
k_BT_{\rm vir}\sim\frac{\mu m_pV_c^2}{2}.
$$
Unlike collisionless dark matter, it can radiate away <thermal energy>. A rough cooling time is
$$
t_{\rm cool}\sim
\frac{\tfrac32n_{\rm tot}k_BT}{n_en_i\Lambda(T,Z)},
$$
where the denominator specifies a chosen optically thin <astrophysical cooling function> convention. Cooling is effective for condensation when this time is short compared with the relevant dynamical or supply time. Atomic transitions, molecular processes, <ionization> state, density and metal abundance matter. A stable virial shock is not obligatory for all <accretion>: cold dense inflows can supply gas in regimes where shock-heated gas would cool rapidly.
The distinction between a gravitationally assembled massive halo and a radiatively condensing luminous component is fundamental. In the classic two-stage picture, non-dissipative clustering builds potential wells while cooling allows gas to concentrate much further than the halo component.<Core condensation in heavy halos by Simon D. M. White and Martin J. Rees>[1978 abstract, gravitational clustering plus dissipative gas condensation]{ref}
<Angular momentum> inherited from tidal torques and later <accretion> helps gas settle into a rotationally supported disc after cooling; disc size depends on angular-momentum retention and transport. Locally self-gravitating gas fragments and forms <stars>. Stellar radiation, winds and supernovae then heat gas, drive <turbulence>, enrich it with heavy elements and can expel it from shallow potential wells. Metal enrichment can also accelerate later cooling, so feedback is more than a single positive or negative energy term.
<Galaxy> growth is usefully expressed as a gas-budget competition:
$$
\dot M_g=\dot M_{\rm in}-(1-\mathcal R)\psi-\dot M_{\rm wind}
+\dot M_{\rm reaccrete}.
$$
Here $\mathcal R$ is the promptly returned stellar mass fraction; unlike the simplification in Question1, a general evolutionary model need not set it to zero. The equation separates newly supplied, converted, expelled and recycled gas rather than assuming that halo mass uniquely fixes stellar mass.
Accreting central <black holes> can provide radiative and mechanical feedback. Heating a halo, ejecting gas or maintaining a hot atmosphere can reduce fresh fuel; removal of a satellite's gas supply and <ram pressure> in a surrounding medium provide additional environmental routes. <Quiescent galaxy assembly> can still add stellar mass through dry mergers, so cessation of <star formation> is not identical to cessation of all mass assembly.
Mergers and disc instabilities can rearrange morphology and trigger bursts, while smooth <accretion> can rebuild discs. Cosmological expansion changes the supply and merger environment; accelerated late-time expansion also slows linear growth. A predictive model must connect these mechanisms to the stellar mass function, gas fractions, star-formation histories, sizes, chemical abundances and clustering. Those observables test different pieces of the assembly-and-feedback picture; none alone implies a one-to-one instantaneous relation between <galaxy> <luminosity> and total halo mass.
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