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In a radiation-dominated background with comoving cold dark matter, neglect its contribution to gravity but retain the radiation perturbation. The regular adiabatic initial conditions select $F(x)={\int }_0^x(1-\cos t)dt/t+\sin x/x-(1-\cos x)/x^2-1/2$. It obeys $x^2{F}^{(4)}+5x{F}^{(3)}+x^2{F}^{\prime \prime }+x{F}^{\prime }=0$, with $F\sim x^2/8$ at zero and $F\sim \ln x+\gamma -1/2$ at infinity. The accompanying radiation contrast is $K[-2\cos x+4\sin x/x-4(1-\cos x)/x^2]/3$. This exact leading-radiation solution explains the transition from common superhorizon growth to acoustic radiation oscillations and logarithmic matter growth.

In Synchronous gauge in cosmology with coordinates comoving with cold dark matter, the linearized Einstein field equations and matter conservation give the displayed density equation. Primes denote conformal time derivatives. The radiation term contains its pressure contribution and cannot be dropped on superhorizon scales merely because radiation oscillates after cosmological horizon entry.

An instantaneous kinetic suppression estimate replaces sound speed by the velocity dispersion of cold dark matter. Nonrelativistic particles in thermal contact with photons have $\sigma \propto a^{-1/2}$; after kinetic decoupling, their momenta redshift and $\sigma \propto a^{-1}$. A background dynamical estimate using total density is then constant during radiation domination and decreases as $a^{-1/2}$ during matter domination. Using only the species' own matter density instead makes the self-gravitating collisionless Jeans length decrease as $a^{-1/2}$ throughout the decoupled nonrelativistic era. Neither instantaneous estimate is the accumulated distance traveled by collisionless free streaming.

A cosmological entropy perturbation describes relative composition changes between cosmic components, rather than only a common displacement along the background evolution. For cold dark matter and photons, define fractional density perturbations ${\delta }_c=\delta {\rho }_c/{\bar{\rho }}_c$ and ${\delta }_{\gamma }=\delta {\rho }_{\gamma }/{\bar{\rho }}_{\gamma }$. Their relative entropy variable isThe factor $3/4$ follows from the different background dilution laws ${\bar{\rho }}_c\propto a^{-3}$ and ${\bar{\rho }}_{\gamma }\propto a^{-4}$, with scale factor $a$. Under a first-order time shift $\tilde{\eta }=\eta +T$, the density perturbation gauge transformation gives ${\tilde{\delta }}_c={\delta }_c+3\mathcal{H}T$ and ${\tilde{\delta }}_{\gamma }={\delta }_{\gamma }+4\mathcal{H}T$, where $\mathcal{H}=a^{\prime }/a$. These shifts cancel in $S_{c\gamma }$.

Thus this relative entropy is gauge invariant at first order. The condition $S_{c\gamma }=0$ gives the common adiabatic relation ${\delta }_c=3{\delta }_{\gamma }/4$. A pure isocurvature perturbation initial condition is chosen with vanishing initial comoving curvature perturbation; a mixed initial state can instead have both nonzero curvature and nonzero $S_{c\gamma }$. Vanishing initial curvature is therefore a condition on a pure initial mode, not a requirement for defining the relative entropy variable.

Cosmological structure formation is the gravitational amplification of initial linear cosmological density perturbations into the cosmic web and bound objects. Overdense regions attract extra matter while surrounding regions become relatively empty. In the cold dark matter picture, small initial particle velocities preserve fluctuations that stronger free streaming would erase. The resulting spectrum of seed fluctuations influences the populations of galaxies and clusters; gas cooling and feedback add further physics to their visible contents.^[ref]

The galactic angular-momentum catastrophe is the excessive loss or redistribution of baryonic specific angular momentum in early simulated galaxy formation, producing overly compact disks or large central concentrations. Premature cooling, gravitational torques, interactions, and resolution contributed. Improved stellar feedback and gas accretion histories can mitigate it; it is a historical simulation problem rather than a theorem against cold dark matter.

George R. Blumenthal helped formulate the cold dark matter account of the formation of galaxies and larger cosmic structures. The 1984 paper with Sandra Faber, Joel Primack and Martin Rees considered matter whose early velocity dispersion was sufficiently small that free streaming would not erase fluctuations on galactic scales.

This connects a proposed microscopic property of dark matter to observable cosmic structure: surviving density fluctuations grow through gravity, and their initial spectrum constrains the later distribution of galaxies. The paper evaluated a model with calculable consequences, rather than merely adding unseen mass to individual systems.^[ref]

A matter component with relativistic motion during an important part of structure formation. Its free streaming suppresses small-scale perturbations. Making a light thermal neutrino Majorana rather than Dirac does not by itself turn this component into cold dark matter.

When baryons are a small fraction of the gravitating matter, their perturbation is forced by the cold dark matter potential. Comparing pressure frequency $c_s^2{k}^2$ with $4\pi G{a}^2{\bar{\rho }}_C$ yields the scale ${\lambda }_J=c_s\sqrt{\pi /(G{\bar{\rho }}_C)}$ for following the growing matter mode. It is not an absolute statement that every sub-Jeans baryon perturbation is zero or cannot be forced: a slowly varying response is ${\delta }_B\simeq (k_J^2/k^2){\delta }_C$. In the pressureless limit the difference from CDM is a constant plus a decaying mode during matter domination, so baryons catch up in relative amplitude.

Joel Primack helped formulate cold dark matter as a quantitative framework for galaxy formation. The 1984 study with George R. Blumenthal, Sandra Faber and Martin Rees linked a primordial fluctuation spectrum to the formation of galaxies and galaxy clusters when collisionless particles have sufficiently small early velocity dispersion.^[ref]

Here “cold” concerns the particles' early random velocities, not whether the later galaxies are thermally cold.

The Lambda-CDM model combines a cosmological constant with cold dark matter and the standard baryonic and radiation components in an expanding Friedmann-Lemaître-Robertson-Walker metric.

Let ${\delta }_c=C_g{\tau }^2+C_d{\tau }^{-3}$ be the matter-era growing and decaying density modes, with radiation neglected in the background gravity. The radiation density contrast isThe formula follows by the Green's function of the forced harmonic oscillator. For $k\tau \gg 1$, the slowly varying particular response to the decaying mode begins at $16C_d{\tau }^{-5}/{\omega }^2$; lower-endpoint oscillations are absorbed into $C_1,C_2$. The growing mode drives a constant radiation offset, while cold dark matter continues growing as ${\tau }^2$.

cold dark matter is nonrelativistic and effectively collisionless on the scales of interest. Its small velocity dispersion limits free streaming, allowing small-scale gravitational wells to survive. baryons coupled to photons undergo acoustic motion and damping before recombination; dark matter does not share that photon pressure, and provides potential wells into which baryons can later fall.

For a subhorizon pressureless perturbation, once a Newtonian treatment is applicable,During radiation domination the matter contribution to the background gravity is small; neglecting it gives $\delta \simeq C_1+C_2\log t$. This slow subhorizon growth is the Mészáros effect. During matter domination,in an Einstein-de Sitter universe. Once $\delta$ approaches unity, linear theory fails and collapse, virialization and merging become important.

Modes entering the horizon during radiation domination therefore grow less before equality than longer modes entering later. This scale dependence is encoded in a transfer function, with a schematic linear matter power spectrum $P(k,a)=A{k}^{n_s}{T}^2(k)D^2(a)$. cold dark matter supports hierarchical assembly; a sufficiently large warm- or hot-dark-matter free-streaming length suppresses small-scale structure. Superhorizon density contrasts change under a gauge transformation in cosmological perturbation theory, so the subhorizon growth laws should not be applied indiscriminately outside the horizon. The original cold-dark-matter formation model explicitly connects weak free streaming to the spectrum that seeds galaxies and clusters.^[ref]

Growing and decaying modes. Neglect pressure and insert the Einstein-de Sitter background. A trial $\Delta \propto t^p$ givesThereforeThe quoted growth law selects the growing mode; arbitrary initial data can also contain the decaying one. In a static homogeneous Jeans model, $\ddot{\Delta }=(4\pi G\bar{\rho }-k^2{c}_s^2)\Delta$, so an unstable mode grows exponentially, $e^{\sqrt{4\pi G\bar{\rho }-k^2{c}_s^2}t}$. Expansion introduces the $2H\dot{\Delta }$ damping term and a falling background density, yielding power-law growth instead. If the wavevector in the supplied pressure term is comoving, that term would need $R^{-2}$; the pressureless result does not depend on this convention.

Amplitude problem. Using the stated recombination epoch, matter-era growth from then until today is only about $1501$. A baryonic density perturbation of order $10^{-5}$ would reach merely $1.5\times 10^{-2}$, far below the order-unity regime needed for nonlinear collapse. photon coupling also prevents baryons from freely collapsing before recombination. The observed microwave temperature contrast is not an exact direct measurement of baryon density contrast at every scale: acoustic transfer and gravitational redshift matter. The estimate expresses the insufficient growth of tiny photon-coupled baryonic seeds, not an exact conversion between the two observables.

Coupled gravitational response. Both pressureless components respond to the same perturbed potential. The equations areTheir difference obeys $\ddot{D}+2H\dot{D}=0$, where $D={\Delta }_D-{\Delta }_B$. Thus $\dot{D}\propto R^{-2}$ and, in Einstein-de Sitter, $D=C_0+C_1{t}^{-1/3}=C_0+{\tilde{C}}_1{R}^{-1/2}$.

Catch-up branch and its condition. With dark matter dominating the source and in its growing mode, write ${\Delta }_D=AR$. The full baryon solution isThe requested baryon catch-up approximation after recombination selects the branch with negligible relative decaying mode, ${\tilde{C}}_1=0$. Defining $C_0=AR(z_0)$ then givesThis branch applies at and after $z_0$ and has ${\Delta }_B(z_0)=0$ and matched baryon/dark-matter growth velocities at that epoch. It does not follow uniquely from vanishing initial baryon density contrast alone. If both ${\Delta }_B$ and its initial time derivative vanish at $R=R(z_0)$, the zero-initial-baryon-velocity catch-up transient instead givesBoth approach unity for $x\gg 1$. The displayed target therefore needs its stated no-relative-decaying-mode assumption; a finite baryon fraction modifies the transient further.

Physical consequence. cold dark matter is not held to the radiation by Thomson scattering, and can build gravitational wells before baryon decoupling, with appreciable matter-era growth after equality. Its growing perturbations then attract baryons once pressure and radiation drag become unimportant. The relative offset stays constant or decays while the common mode grows, so baryons catch up. Nonlinear structure need not grow solely from the tiny baryonic perturbation inferred at decoupling. cold dark matter growth during radiation domination is slower than the matter-era law, and the stated formula is neither a radiation-era nor a nonlinear-collapse solution.

Small primordial fluctuations seed later structure. For adiabatic cosmological perturbations on scales outside the horizon, the uniform-density curvature perturbation is conserved when the non-adiabatic pressure perturbation and the decaying mode can be neglected. A density contrast outside the horizon is gauge-dependent, so its apparent power-law growth should not be confused with directly measured local collapse.

For subhorizon, pressureless matter, the linearized continuity equation, Euler equation and Poisson equation combine intoHubble expansion damps peculiar motion; self-gravity amplifies overdensities. During radiation domination, radiation pressure drives acoustic oscillations and cold dark matter growth after horizon entry is only logarithmic. During matter domination the growing and decaying solutions are $\delta \propto a$ and $\delta \propto a^{-3/2}$. Dark-energy domination later suppresses growth relative to the matter-era law.

For a pressure-supported fluid the Fourier-mode equation also contains $+c_s^2{k}^2\delta /a^2$, giving a Jeans length separating acoustic support from gravitational growth. Before recombination, baryons are tightly coupled to photons and oscillate rather than freely following dark matter. Silk damping from photon diffusion damps small-scale photon-baryon fluctuations; after decoupling, baryons can fall into dark-matter potential wells. free streaming smooths species-dependent scales and distinguishes cold from warm or hot dark matter.

Schematically, the linear matter power spectrum can be written $P_{\delta }(k,a)=A{k}^{n_s}{T}^2(k)D^2(a)$ under the usual primordial-curvature normalization. The cosmological transfer function records horizon entry, matter-radiation equality, acoustic physics and damping; the linear growth factor records later amplification. density contrasts approaching unity require nonlinear collapse and halo formation beyond this approximation. cosmic microwave background anisotropies and galaxy clustering test the same initial perturbations through different transfer histories; galaxy bias, redshift-space distortions and observational selection enter the galaxy comparison.

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 scaleUnlike collisionless dark matter, it can radiate away thermal energy. A rough cooling time iswhere 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.^[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: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.

For a separately conserved homogeneous component, energy conservation gives $d(\rho c^2{a}^3)=-pd(a^3)$, equivalently $\dot{\rho }+3H(\rho +p/c^2)=0$. Dust has negligible pressure, while isotropic radiation has $p_{\gamma }={\rho }_{\gamma }{c}^2/3$. HenceThe matter factor is volume dilution; photons additionally lose energy by cosmological redshift. Nonrelativistic baryons and cold dark matter both contribute to the matter term, so ${\Omega }_M={\Omega }_B+{\Omega }_C$. At equality ${\rho }_{M0}{a}_{\mathrm{eq}}^{-3}={\rho }_{\gamma 0}{a}_{\mathrm{eq}}^{-4}$, givingThis is equality for the source's matter-plus-photon model. Additional relativistic species would contribute to total radiation and shift the equality value. Although the source calls the $\rho$'s energy densities, its Friedmann equation and later ${\rho }_{\gamma }{c}^2$ show that their consistent SI meaning is mass-equivalent density.

The actual large-scale expansion is well described by the Lambda-CDM model: radiation in cosmology, ordinary baryonic matter, cold dark matter and late-time dark energy coexist. A component becomes dominant through continuous changes in their density ratios, rather than every previous component disappearing at an instantaneous phase boundary. Matter pressure is small in its nonrelativistic regime, but radiation persists, and massive neutrinos interpolate between relativistic and nonrelativistic behavior.

There is also a late accelerated era, absent from a model ending permanently in S3. It is consistent with nearly constant vacuum energy, whose negative pressure eventually dominates because pressureless matter dilutes as $R^{-3}$. The scale factor then ceases to follow the pure $t^{2/3}$ Einstein-de Sitter universe law.

An early phase of cosmic inflation is a well-motivated model rather than a directly observed exact $w=-1$ phase. Successful cosmic inflation must end and transfer its energy to matter and radiation through reheating; realistic models generally have slowly varying $H$ and $w$ close to, rather than exactly equal to, $-1$. Extrapolating a strictly constant $H$ to arbitrarily small $R$ is not a measurement of the universe's past duration. Finally, spatial homogeneity and cosmological isotropy apply statistically on large scales: galaxies, gravitational waves and other perturbations are absent from the exact FLRW metric, and small spatial curvature remains an observational possibility.

In deep matter domination, the scale factor is $a\propto {\tau }^2$, so $a^{\prime }/a=2/\tau$. Neglect radiation in the gravity source, including its backreaction, to obtainFor a power ${\tau }^p$ the Euler-Cauchy equation becomes $p(p-1)+2p-6=(p-2)(p+3)=0$. Its two independent matter-era growing and decaying density modes give the full approximate solutionThere is no $k$-dependent pressure term for cold dark matter. Thus this leading equation is valid both outside and inside the horizon in the stated comoving synchronous gauge, as long as the perturbations remain linear and the omitted radiation gravity is negligible. A different time slicing would give a different density variable.

For $k>0$, put $\omega =k/\sqrt{3}$, $C_g=A/{\tau }_i^2$ and $C_d=B{\tau }_i^3$. The remaining radiation equation is a forced harmonic oscillator:The matter-era forced radiation response can be written exactly within the matter-background approximation asThe last term is a Green's function particular solution; changing its lower limit only changes the two oscillatory constants. For the decaying source, a slowly varying particular solution obeys ${\delta }_{r,p}\simeq (4/3){\delta }_c^{\prime \prime }/{\omega }^2$, since a second time derivative is smaller than ${\omega }^2{\delta }_{r,p}$ by order $(k\tau {)}^{-2}$. Consequently the subhorizon asymptotic form isFor a pure growing mode, the constant offset and acoustic terms are an exact solution of this reduced radiation equation. In particular, the late radiation contrast does not follow the ever-growing ${\delta }_c$.

Before crossing, a regular growing adiabatic cosmological perturbation has both contrasts proportional to ${\tau }^2$, with ${\delta }_r=(4/3){\delta }_c$ to leading order. Afterwards cold dark matter keeps this quadratic growth, whereas radiation undergoes subhorizon radiation acoustic oscillations about a constant forced offset. To make the requested sketch concrete, one regular leading matter-era solution on both sides isIts small-$k\tau$ expansion gives ${\delta }_r=(4/3){\delta }_c+O((k\tau {)}^4)$, and its oscillation frequency is the radiation sound frequency. The sketch plots amplitudes in units of $\epsilon$; $\epsilon$ can be chosen small enough that the whole plotted interval remains linear. The line at ${\tau }_h$ uses the wavelength-to-particle-horizon convention in the question, rather than asserting that the smooth transition occurs at a sharp surface.

The perturbations are density contrasts, ${\delta }_c=\delta {\rho }_c/{\bar{\rho }}_c$ and ${\delta }_r=\delta {\rho }_r/{\bar{\rho }}_r$, evaluated in the common comoving synchronous gauge. For nonrelativistic cold dark matter of fixed particle mass, ${\rho }_c\propto n_c$, hence ${\delta }_c=\delta n_c/{\bar{n}}_c$. Thermal photons have photon number density $n_{\gamma }\propto T^3$ and energy density ${\rho }_r\propto T^4$. A small local temperature perturbation therefore givesFor adiabatic initial conditions, the ratio $n_{\gamma }/n_c$ is unperturbed, so $\delta n_{\gamma }/{\bar{n}}_{\gamma }=\delta n_c/{\bar{n}}_c$. Combining these equations provesEquivalently ${\delta }_r/(1+w_r)={\delta }_c/(1+w_c)$, with $w_r=1/3$ and $w_c=0$. The relation applies before causal pressure gradients can separate the two species on superhorizon scales; it need not remain true after cosmological horizon crossing.