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Let $X$ be the hydrogen mass fraction, $f_g={\rho }_g/{\rho }_{\mathrm{tot}}$ the gravitating gas fraction, and $\chi =n_{\mathrm{tot}}/n_H$ the total-particle-to-hydrogen ratio. The optically thin gas cooling time and free-fall time of a uniform sphere arewhere ${\rho }_{\mathrm{tot}}=m_p{n}_H/(X{f}_g)$. Equating them gives the cooling-to-free-fall equality curve:At fixed composition, its shape is $n_{\mathrm{crit}}\propto T^2/{\Lambda }^2$. Gas above this curve cools faster than it falls; gas below it cools more slowly. Strong line cooling produces a deep low-density trough. In the high-temperature free-free regime, $\Lambda \propto T^{1/2}$ gives $n_{\mathrm{crit}}\propto T$.

The displayed uniform self-gravitating-cloud example uses $f_g=1$, $X=0.76$, $\mu \simeq 0.59$ and $\chi \simeq 2.24$. Holding these ionized-gas factors fixed makes the diagram transparent, but near the atomic threshold their actual temperature dependence changes numerical normalizations. With these conventions the supplied cooling values imply $n_{\mathrm{crit}}(1.5\times 10^4\mathrm{K})\simeq 9.6\times 10^{-10}\mathrm{c}{\mathrm{m}}^{-3}$ and $n_{\mathrm{crit}}(10^7\mathrm{K})\simeq 0.107\mathrm{c}{\mathrm{m}}^{-3}$. For gas supported in a dark-matter-dominated dark matter halo, take $f_g<1$: at fixed $n_H$ the free-fall time of a uniform sphere is shorter and the critical curve shifts upward by $1/f_g$.

To add virial mass contours in a cooling diagram, use the uniform self-gravitating sphere consistently. Its Newtonian gravitational potential energy is $-3G{M}^2/(5R)$ and its thermal kinetic energy is $(3/2)M{k}_{B}T/(\mu m_p)$. The virial theorem gives $k_B{T}_{\mathrm{vir}}=\mu m_{p}GM/(5R)$, while $M=(4\pi /3)R^3{\rho }_{\mathrm{tot}}$. Eliminating $R$ givesThus constant total gravitating mass gives slope three in the logarithmic density-temperature plane, with larger mass contours lying at smaller density. A different density profile changes the structural coefficient; the customary halo temperature scale $k_B{T}_{\mathrm{vir}}=\mu m_{p}GM/(2R)$ shifts the mass contours but preserves their slopes. The horizontal illustrative formation-density lines are $n_{H,\mathrm{vir}}\simeq {\Delta }_v(2\times 10^{-7})(1+z{)}^3\mathrm{c}{\mathrm{m}}^{-3}$, using ${\Delta }_v=18{\pi }^2$. They show why increasing formation cosmological redshift helps gas reach the rapid-cooling region.

At the low-mass end, a dark matter halo's virial temperature may not reach the atomic threshold. The atomic and molecular cooling thresholds for galaxy formation therefore favor an atomic-cooling mass of order $10^8{M}_{\odot }$ at early formation epochs, with its precise value varying as approximately $(1+z{)}^{-3/2}$. Smaller primordial systems require molecules; photoheating and stellar feedback further suppress their luminous output.

At the high-mass end, the virial temperature at a fixed assembly density scales as $M^{2/3}$. Beyond the line-cooling interval the gas becomes increasingly hot, while thermal bremsstrahlung improves only as $T^{1/2}$. Hence at fixed assembly density $t_{\mathrm{cool}}/t_{\mathrm{ff}}\propto T^{1/2}\propto M^{1/3}$ in that regime. Large systems can remain pressure-supported hot atmospheres for many dynamical times, instead of promptly fragmenting into one stellar object. This is the upper galaxy mass from gas cooling: it associates efficient condensation with roughly galactic, rather than group or cluster, masses and motivates the broad $10^8$–$10^{12}{M}_{\odot }$ interval.

The diagram gives characteristic cooling scales, not exact universal mass bounds. Density profiles, gas fraction, enrichment and formation epoch move the boundaries; the two supplied values alone do not determine a complete numerical cutoff. Some gas can cool in dense cores of larger dark-matter haloes, early smaller progenitors can form stars before merging, and subsequent galaxy mergers can assemble more massive galaxies. Long-lived hot atmospheres also require heating to prevent excessive late-time cooling. These qualifications explain why the cooling criterion for galaxy formation is useful without equating each dark dark matter halo to one galaxy.

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 writtenHere $T(k)$ 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 ${\Delta }^2=k^{3}P/(2{\pi }^2)$, 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 temperatureUnlike 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 exampleThe 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 $j\sim R{v}_c$ 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 $10^8{M}_{\odot }$ at a redshift of order ten, with approximate dependence $M_{\mathrm{atomic}}\propto (1+z{)}^{-3/2}$ 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, $t_{\mathrm{cool}}\propto T^{1/2}\propto M_h^{1/3}$, while $t_{\mathrm{dyn}}$ 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,If $d{n}_h/d{M}_h\propto M_h^{-a}$ 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. The halo hierarchy supplies the gravitational framework; cooling, angular momentum and feedback determine which parts become luminous galaxies.