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In hierarchical galaxy formation, low-mass dark-matter structures typically collapse before larger structures because the smoothed linear density variance is larger on smaller mass scales. Accretion and mergers assemble increasingly massive haloes. Cooling, feedback and angular momentum determine how baryons form galaxies within that halo hierarchy; the ordering is statistical rather than an absolute rule for every object.

Monolithic galaxy formation idealizes a galaxy as forming in one early rapid dissipative collapse, followed mainly by passive evolution. Rapid formation of the stars is distinct from early assembly of all those stars into their final object; hierarchical galaxy formation can assemble old stellar populations later.

The relevant comparison is continuing hierarchical galaxy formation in the Lambda-CDM model against an idealized monolithic galaxy formation model in which both formation and assembly were essentially complete early. Each observation below concerns a population or a fossil record, rather than a currently merging pair. None individually proves that every galaxy formed through the same route.

Fossil substructure in stellar halos. Old stars in the Milky Way's stellar halo include coherent groups in orbital-integral space, and external halos contain extended debris and shells. Phase mixing can erase spatial concentration while retaining correlations of orbital energy or angular momentum. Such stellar streams, when linked to a disrupted dwarf progenitor, record accretion of an already formed stellar population. A smooth single collapse followed by quiet passive evolution has no equally direct mechanism for creating several dynamically and chemically distinct accreted populations. This supports a composite assembly history, without requiring that every halo star was accreted or counting a current merging pair as evidence.

Growth of the quiescent population. The galaxy red sequence contains old stellar populations, yet its inferred stellar mass density grows between approximately redshift one and the present. Pure fading of an already fixed early population changes luminosities and colours, but cannot increase the mass in that population. Quenching adds newly passive systems, while subsequent assembly combines older stellar populations into larger galaxies. The observed evolution therefore requires demographic evolution after the putative completed collapse epoch. It is consistent with hierarchical galaxy formation plus quenching, although red-sequence growth alone does not distinguish assembly from all possible histories of in-situ star formation; stellar-mass estimates and survey selection must be treated consistently.

Evolution of galaxy morphology. Distant galaxy samples contain a larger prevalence of irregular and peculiar structures than a simple non-evolving extrapolation of the local population. Settled discs and smooth spheroids become common outcomes as time proceeds. Continued gas supply, dynamical interactions, and internal rearrangement in hierarchical galaxy formation naturally allow this galaxy morphological evolution. A model in which structure became fixed during one early collapse and then only stellar light faded has difficulty producing the population change. Rest-frame wavelength, resolution, and surface-brightness selection can affect the observed classifications, and clumpiness is not unique evidence for a galaxy merger: unstable gas-rich discs can also be irregular.

Evolution of sizes at fixed stellar mass. Massive high-redshift galaxies, including concentrated systems, are often more compact than comparable populations at low redshift. Galaxy size evolution implies structural growth beyond passive luminosity fading. In particular, deposition of accreted stars at large radii in minor galaxy mergers can enlarge the outer stellar body while preserving an old central stellar population; gas accretion can grow discs from the inside outward. A completed early collapse with a fixed stellar structure does not naturally supply those new outskirts. Changes in sample membership and expansion associated with stellar mass loss also contribute, so this is evidence requiring continued structural evolution rather than a unique quantitative measurement of a merger rate.

Diffuse intracluster stars. Intracluster light and extended outer envelopes show that some stars no longer belong to individual cluster galaxies. Their irregular distribution and substructure retain evidence of redistribution within an assembling cluster. Tidal stripping and disruption of member systems supply stars to the common environment, as expected when smaller bound systems are incorporated into a growing hierarchy. Passive evolution of independent galaxies formed in isolated early collapses cannot itself move their stars into the space between galaxies. Early stellar formation followed by later stripping remains possible, but admitting that later dynamical assembly already departs from the strict completed-collapse picture. This evidence concerns the cumulative stellar distribution, not merely the existence of an occasional present-day merging pair.

The dark matter halo population follows gravitational hierarchical galaxy formation: many low-mass dark-matter haloes and an exponentially rare high-mass tail. The Press-Schechter halo mass function captures this broad form. A constant conversion from dark matter halo mass to luminosity would simply rescale that distribution and predict too much light from both very small and very large systems. The luminosity function instead has a shallower faint end and a sharp bright-end decline, often described by a Schechter function,Define the halo star-formation efficiency as ${ϵ}_{\ast }(M_h)=M_{\ast }/(f_b{M}_h)$, distinguishing this integrated baryon conversion from the star-formation efficiency per free-fall time. If the stellar mass-to-light ratio is ${{\rm Y}}_{\ast }$, a simple one-central-galaxy mapping is $L={ϵ}_{\ast }{f}_b{M}_h/{{\rm Y}}_{\ast }$. For a monotone mapping without scatter,Thus a mass-dependent conversion efficiency and its Jacobian change the shape, not just the horizontal scale, of the luminosity function. Scatter, satellites and the stellar mass-to-light ratio add further differences.

In shallow potential wells, photoheating can prevent gas accretion, while stellar feedback from radiation, stellar winds and supernovae heats or ejects gas. These effects lower halo star-formation efficiency toward low mass, helping flatten the faint end. The need for atomic or molecular line cooling also affects the smallest systems. Intermediate-mass dark-matter haloes retain gas and cool efficiently, so the integrated conversion reaches a maximum around galactic dark matter halo masses.

In massive dark-matter haloes, long optically thin gas cooling times and stable hot atmospheres reduce fresh cold-gas supply. Active-galactic-nucleus feedback can replenish the thermal energy lost in cooling and suppress a cooling flow, reducing halo star-formation efficiency and sharpening the bright cutoff. Massive group or cluster dark-matter haloes contain many distinct galaxies rather than one object with luminosity proportional to their entire mass. Galaxy mergers redistribute stars and can grow already formed bright galaxies, but do not turn all hot dark matter halo gas into new stars. The luminosity function is the gravitational dark matter halo population filtered through strongly mass-dependent baryonic physics. The plotted abundance and efficiency curves are qualitative examples, not observational fits or calibrated mass-to-light relations.

Let $\delta =(\rho -\bar{\rho })/\bar{\rho }$ and define the cosmological density power spectrum by $⟨{\delta }_{\mathbf{k}}{\delta }_{{\mathbf{k}}^{\prime }}^{\ast }⟩=(2\pi {)}^3{\delta }_{\mathrm{D}}(\mathbf{k}-{\mathbf{k}}^{\prime })P(k)$. The contribution per logarithmic wave-number interval is the dimensionless cosmological power spectrum ${\Delta }^2(k)=k^{3}P(k)/(2{\pi }^2)$. The linear cosmological mass variance on mass scale $M$ iswith $R$ a comoving smoothing radius and $W$ the top-hat filter in Fourier space. On halo scales the observed spectrum is consistent with greater variance at smaller masses. Growth therefore brings smaller objects to the collapse threshold earlier, statistically; larger structures assemble by accretion and dark-matter halo mergers. This statistical growth from smaller bound systems to larger ones is hierarchical galaxy formation. It is an ordering of dark-matter assembly, not a rule that every small visible galaxy must precede every large visible galaxy.

For linear modes the linear growth factor gives $P(k,z)=D^2(z)P(k,0)$. During matter domination $D\propto a$, so $P\propto a^2$ and the characteristic nonlinear mass increases. Once modes become nonlinear, coupling between scales and halo formation change the shape, so the same multiplication cannot be used for the entire late-time spectrum. At low redshift accelerated expansion suppresses linear growth. Galaxy clustering traces the matter spectrum with galaxy bias; it should not be equated directly with an unbiased matter measurement.

The broad linear shape was set by the cosmological transfer function before and around matter-radiation equality, with baryonic acoustic structure also imprinted before recombination. A nearly scale-invariant primordial curvature spectrum has ${\mathcal{P}}_{\mathcal{R}}(k)\propto k^{n_s-1}$, with $n_s$ close to one. After converting curvature perturbations to matter-density perturbations,Thus nearly scale-invariant primordial curvature does not mean a constant density $P(k)$. Modes with $k\ll k_{\mathrm{eq}}$ enter the horizon after equality and have $T\simeq 1$. Modes with $k\gg k_{\mathrm{eq}}$ enter during radiation domination, when radiation controls the expansion and cold-matter perturbations grow only slowly. The cold-dark-matter transfer function behaves approximately as $T\propto \ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ at large $k$. Hence the density spectrum turns over near $k_{\mathrm{eq}}=a_{\mathrm{eq}}{H}_{\mathrm{eq}}/c$:The turnover records the equality horizon, while the late nonlinear excess records gravitational clustering. On galactic scales the effective slope is greater than $-3$, giving the growing small-scale variance needed for hierarchical galaxy formation.

Cold dark matter has negligible primordial thermal velocities and a very short collisionless free streaming length on galactic scales. Warm dark matter has appreciable residual velocities while structure is being seeded; particles stream across small fluctuations and reduce their contrast. Its cosmological transfer function is consequently cut off below a characteristic length, suppressing low-mass halos and delaying their formation. Above that cutoff its assembly can still be hierarchical. The important distinction is the free-streaming scale, rather than the present temperature or an arbitrary particle-mass label.

Linear evolution assumes $|\delta |\ll 1$. When $\delta$ becomes of order unity, overdense regions depart strongly from the Hubble flow and can turn around and collapse. Collisionless dark matter develops multistream motion after trajectories cross; gravitational mixing redistributes energy and produces a bound dark matter halo. The formal infinite-density collapse of an ideal spherical pressureless solution is not the physical endpoint. A roughly virialized halo has $2K+W\simeq 0$ and a characteristic virial velocity $V_{\mathrm{vir}}^2=G{M}_h/r_{\mathrm{vir}}$. Its gas virial temperature is conventionallyIt measures the thermal energy associated with the gravitational potential; the numerical factor depends on the velocity-dispersion convention. It does not imply that the collisionless dark matter has a thermodynamic gas temperature.

The unheaded request about baryon conversion efficiency of a halo is also answered here. Define $f_{\ast }=M_{\mathrm{stars}}/(f_b{M}_h)$ using the cosmic baryon fraction $f_b$. In small halos, shallow potentials let stellar feedback drive outflows or repeatedly heat star-forming gas; supernova energy per stellar mass is roughly fixed while binding energy per gas mass scales as $V_{\mathrm{vir}}^2$. Photoheating during reionization also prevents very small halos from retaining or accreting cool gas. Molecular/atomic cooling thresholds further reduce star formation in the smallest systems. These effects make $f_{\ast }$ fall toward low mass.

Near $M_h\sim 10^{12}{M}_{\odot }$, gas can cool efficiently and the potential is deep enough to retain more of it, while a long-lived hot atmosphere and maintenance heating are less effective than in larger systems. At high mass, higher virial temperature and lower cooling efficiency let a substantial hot atmosphere persist. Active-galactic-nucleus feedback can prevent that atmosphere from supplying cold gas and can expel some gas; the cooling-time bottleneck alone is not an adequate explanation for the low stellar fractions of massive groups and clusters. The peak reflects a competition between gas supply/cooling and feedback, rather than complete conversion of all baryons at a sharply universal mass. Its exact location and height depend on epoch, metallicity, gas history and the stellar population included.

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.

Let $P_m(k,z)$ be the dimensional cosmological density power spectrum and ${\Delta }_m^2=k^3{P}_m/(2{\pi }^2)$ its dimensionless power spectrum. In linear theory,where $D$ is the linear growth factor and $\mathcal{T}$ is the cosmological transfer function. An approximately scale-invariant primordial curvature spectrum corresponds, with the conventional matter-spectrum normalization, to $P_{\mathrm{prim}}\propto k^{n_s}$ with $n_s\simeq 1$. This does not say that the dimensional density spectrum is white noise or that the early curvature and density perturbations are the same variable.

The crucial scale dependence develops around horizon entry and matter-radiation equality. Modes entering during radiation domination undergo only logarithmic cold-matter growth, whereas modes entering later avoid that suppression. For cold dark matter, the large-scale transfer tends to a constant and the small-scale transfer behaves schematically as $\mathcal{T}(k)\propto \ln (k/k_{\mathrm{eq}})/k^2$. ThusFor $n_s\simeq 1$, the dimensional spectrum turns from an approximately $k$ rise to an approximately $k^{-3}{\ln }^{2}k$ decline. Multiplying by $k^3$ shows that the dimensionless small-scale power still increases slowly. The resulting variance of density fluctuations smoothed on mass $M$ generally increases as $M$ decreases, over the scales relevant to galaxy assembly.

After equality, and over scales where linear growth is scale-independent, the transfer shape is approximately preserved while the amplitude grows. During matter domination, $D\propto a$; late accelerated expansion slows that growth. A region collapses when its linearly evolved density contrast reaches the linear spherical-collapse threshold. Smaller mass scales, having larger variance, typically reach this threshold earlier. Small haloes form first, then accrete matter and merge into larger haloes; galaxies form from cooling baryons in those wells and themselves merge. This small-to-large assembly is hierarchical galaxy formation. It is a statistical trend: rare high peaks can produce unusually massive early objects, and gas cooling and feedback prevent a one-to-one identification of halo collapse with star formation.

For cold dark matter, random particle velocities are small enough that collisionless free streaming erases little power on galactic scales. Warm dark matter retains a larger early velocity dispersion and a larger free-streaming length, suppressing power below a finite scale. There are consequently fewer low-mass haloes, delayed formation of the smallest galaxies, and a lower limit to the hierarchy that can develop. Above the cutoff, warm-matter structure can still assemble hierarchically. The distinction concerns the particles' velocity history and transfer function, not the thermal temperature of gas in a galaxy today. The broad late-time shape was largely established by early horizon-entry physics, with subsequent nonlinear collapse and merging altering the spectrum at high wavenumber.