= Solution
In the <Lambda-CDM model>, nearly Gaussian primordial <density contrasts> grow under gravity within an expanding universe containing <cold dark matter>, ordinary <baryons> and a <cosmological constant>. The <cold dark matter> is effectively collisionless and has negligible pressure on galactic scales. Before recombination, <baryons> are coupled to the photon fluid: <radiation pressure> and acoustic oscillations prevent their perturbations from behaving like pressureless matter. After recombination they can fall into the gravitational potentials already established by <dark matter>, subject to gas pressure and the <Jeans mass>.
For small <density contrasts>, evolution is linear. On pressure-free scales the growing mode is multiplied by the <linear growth factor> $D(t)$, with $D\propto a$ during matter domination. The <cosmological density power spectrum> can be written
$$
P(k,t)=D^2(t)P_{\rm prim}(k)T^2(k).
$$
Here $T(k)$ is the <cosmological transfer function>. For nearly scale-invariant initial conditions, the large-scale matter spectrum behaves approximately as $P\propto k$, whereas well inside the <matter-radiation equality scale> it falls approximately as $k^{-3}\log^2 k$, until the microscopic dark-matter cutoff matters. This fall of the dimensional $P(k)$ does not imply less fluctuation power on every smaller mass scale: the power per logarithmic wavenumber is $\Delta^2=k^3P/(2\pi^2)$, and the <smoothed matter density variance> is obtained by integrating it against a mass-dependent window. Over the relevant cold-dark-matter hierarchy, smaller mass windows generally have larger variance.
The <hierarchical galaxy formation> picture follows: fluctuations on small mass scales typically reach the nonlinear collapse threshold first, while larger objects assemble later through accretion and <dark-matter halo mergers>. It is a statistical ordering, not a claim that every small object precedes every large rare peak. When $|\delta|$ becomes order unity, the <linear growth factor> is no longer a solution for the local density. Collisionless <dark matter> develops multistream regions and bound <dark-matter halos>; <phase mixing> and <violent relaxation> redistribute orbital energies, and virialized structures approximately obey the <virial theorem>.
<Baryons> have an additional nonlinear route. Infall and shocks convert bulk kinetic energy into thermal energy, with characteristic <virial temperature>
$$
k_BT_{\rm vir}\sim\frac{\mu m_pGM_h}{2R_h}.
$$
Unlike collisionless <dark matter>, the gas can lose this energy through <radiative cooling>. The <optically thin gas cooling time> is the thermal-energy density divided by the radiative loss rate, for example
$$
t_{\rm cool}=\frac{3n_{\rm tot}k_BT/2}{n_en_i\Lambda(T)},\qquad
t_{\rm dyn}\sim(G\rho_h)^{-1/2}.
$$
The density convention in $\Lambda$ must agree with the denominator; the <astrophysical cooling function> can also be defined using $n_H^2$ instead. The <cooling criterion for galaxy formation> compares this time with the collapse or supply time. Rapidly cooling gas loses pressure support, contracts and can form stars. Slowly cooling gas remains in a hot atmosphere. Stable virial shocks are not obligatory in every low-mass system: gas can also arrive in cold streams and cool while being accreted.
<Angular momentum> prevents indefinite radial contraction. <Tidal torque theory> supplies an initial halo spin, and later mergers change it. If gas radiates energy while retaining much of its <specific angular momentum>, it settles into a rotationally supported <galactic disk> rather than reaching the centre. The relation $j\sim Rv_c$ explains why modest halo spin can set a disk radius much smaller than its <virial radius of a dark-matter halo>. Torques, bars and gravitational encounters can transport <angular momentum> outwards and feed central concentrations; <radiative cooling> alone does not remove it.
<Galaxy mergers> alter stellar structure as well as assembling mass. A <major galaxy merger> can strongly disturb or destroy an existing <galactic disk>, randomizing stellar orbits and creating a spheroid through <violent relaxation>. A <gas-rich galaxy merger> also permits dissipation, inflow and a burst of <star formation>; gas left over or accreted afterwards can rebuild a <galactic disk>. <Minor galaxy mergers> add stars to outer components, thicken disks and grow bulges. <Dry galaxy mergers> add stellar mass and can increase size without much new <star formation>. Halo merging therefore does not imply instantaneous merging of its galaxies: satellite orbital decay requires <Chandrasekhar dynamical friction> and can take a substantial time.
The <atomic and molecular cooling thresholds for galaxy formation> supply a lower characteristic scale. Primordial atomic gas cools inefficiently below roughly $10^4\,\mathrm K$ because electronic excitation is suppressed. The corresponding halo mass is of order $10^8M_\odot$ at a redshift of order ten, with approximate dependence $M_{\rm atomic}\propto(1+z)^{-3/2}$ at fixed threshold temperature. <Molecular hydrogen> can cool gas at hundreds of kelvin and permit smaller early objects, provided it forms and survives dissociating radiation. <Metal-line cooling> changes these thresholds after enrichment. Thus the atomic threshold is not an absolute minimum mass for all stellar systems.
At the other end, sufficiently massive <dark-matter halos> have high <virial temperatures> and low-density hot gas. Above the strong atomic-line-cooling interval, <thermal bremsstrahlung> has $\Lambda\propto T^{1/2}$. At comparable halo gas density, $t_{\rm cool}\propto T^{1/2}\propto M_h^{1/3}$, while $t_{\rm dyn}$ depends mainly on formation density. Cooling therefore becomes less able to condense all the gas within the available time. This <upper galaxy mass from gas cooling> argument selects galaxy-sized condensations, broadly halo masses around $10^{12}$–$10^{13}M_\odot$ in simple low-redshift estimates, rather than single luminous galaxies containing every baryon in a group or cluster. Its numerical scale depends on epoch, metallicity and gas profile. Subsequent <galaxy mergers> can build larger stellar systems; cooling is not an absolute upper bound on their final mass.
Finally, a <Press-Schechter halo mass function> has many low-mass objects and a steep high-mass cutoff, while the <luminosity function> of galaxies also has a faint component and a bright cutoff, often described by a <Schechter function>. The two shapes are related through the <halo-to-galaxy luminosity mapping>, not by identifying luminosity with total halo mass. In the idealized one-central-galaxy, no-scatter limit,
$$
\Phi(L)=\frac{dn_h}{dM_h}\left|\frac{dM_h}{dL}\right|.
$$
If $dn_h/dM_h\propto M_h^{-a}$ and $L\propto M_h^p$, then $\Phi(L)\propto L^{-1-(a-1)/p}$. A constant conversion gives similar shapes. Actual <stellar feedback>, inefficient low-temperature cooling and <reionization> suppress faint galaxies relative to small haloes, while long cooling times and <active-galactic-nucleus feedback> suppress luminosity at large halo masses. Satellites, scatter, stellar populations and dust further affect the correspondence. \b[The halo hierarchy supplies the gravitational framework; cooling, <angular momentum> and feedback determine which parts become luminous galaxies.]
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