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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]

The normalized cycloid hasThe initially expanding perturbation reaches maximum radius at $\eta =\pi$, $t=t_m$, then formally collapses to zero radius at $\eta =2\pi$, $t=2t_m$. A finite virialized object is assigned the conventional formation epoch $t_f=2t_m$; random motions replace the pressureless point-collapse singularity. The cycloid alone does not describe that relaxation.

For the standard top-hat virialization estimate, write $U=-C_{G}G{M}^2/R$ and retain the same gravitational structure factor $C_G$ at turnaround and final equilibrium. The coherent kinetic energy vanishes at turnaround, whereas the virial theorem gives $2K_v+U_v=0$ in the final isolated equilibrium. conservation of energy givesThe final density is eight times the turnaround density. Meanwhile $\bar{\rho }\propto a^{-3}\propto t^{-2}$, so $\bar{\rho }(t_f)=\bar{\rho }(t_m)/4$. The Einstein-de Sitter virial overdensity follows by combining these factors with the supplied turnaround contrast:The value 200 is a convenient rounded overdensity convention, not exactly the Einstein-de Sitter result.

The virial radius of a dark-matter halo follows from $M_v=4\pi R_v^3{\rho }_v/3$:The hydrostatic temperature of an inverse-square halo follows from hydrostatic equilibrium and the ideal gas law:Since ${\rho }_g\propto r^{-2}$ and $M(r)=M_{v}r/R_v$, cancellation of ${\rho }_g/r$ gives $2k_B{T}_v/\mu =G{M}_v/R_v$. ThereforeHere $\mu$ is a dimensionful mean particle mass, as specified in the paper; a dimensionless mean molecular weight would need multiplication by a particle-mass unit. The isothermal interior is a further halo idealization, not an exact continuation of the uniform top-hat profile used in the virial-overdensity estimate. Changing its gravitational structure factor in a literal energy comparison would change the turnaround-radius relation.

The formation redshift from halo mean density uses the cluster data, which imply ${\rho }_v=3M_v/(4\pi R_v^3)=5.37148\times 10^{13}{M}_{\odot }{\mathrm{Mpc}}^{-3}$. Thus, using the requested rounded convention,Using $18{\pi }^2$ instead gives $z_f\simeq 1.00$. In the Einstein-de Sitter time scaling the rounded-convention result is $t_f/t_0=(1+z_f{)}^{-3/2}\simeq 0.376$. Directly from the mass and radius,For the estimate, take mean particle mass $\mu =0.6m_p$, appropriate approximately to a hydrogen-helium mixture with complete ionization. This gives $T_v\simeq 1.41\times 10^8\mathrm{K}$, or $k_B{T}_v\simeq 12.1\mathrm{keV}$. The composition choice is additional input, not a value supplied by the examination; the boxed expression retains the result for any $\mu$.

Write the dimensionless scale factor as $a=R/R_0$ and the density contrast as $\delta =(\rho -\bar{\rho })/\bar{\rho }$. Gravity amplifies overdensities while expansion and pressure oppose growth. In a sub-Hubble Newtonian single-fluid approximation, a comoving Fourier mode obeysFor pressureless matter in an Einstein-de Sitter background, the growing solution is $\delta \propto t^{2/3}\propto a$ and the decaying solution is $t^{-1}$. Jeans support suppresses growth when the pressure term dominates. The equation is not a substitute for relativistic perturbation theory on horizon-sized scales.

cold dark matter clusters without baryonic pressure support. Sub-horizon growth is much weaker during radiation domination, and baryons are initially tied to the photon fluid, producing acoustic oscillations rather than unrestricted collapse. Horizon entry, radiation-matter equality, acoustic evolution and diffusion therefore give a scale-dependent transfer function. The primordial curvature spectrum and the later matter-density spectrum are not the same function: evolution and the gravitational constraints relate them. Late dark-energy domination reduces the growth rate.

When $|\delta |$ approaches unity, linear theory fails. Collapse, virialization, mergers and tidal interactions form a hierarchy of haloes. Gas must cool and lose angular momentum to make dense stellar systems; feedback then changes its supply and distribution. Simulations combine collisionless gravitational dynamics with increasingly complex gas physics rather than extending the linear growing mode indefinitely. Galaxy clustering measures biased tracers of matter; gravitational lensing and CMB constraints provide complementary tests. Approximately adiabatic, nearly scale-invariant initial perturbations explain the coherent acoustic pattern and supply seeds for this later growth.^[ref]

virialization is relaxation toward a gravitationally supported bulk state satisfying an appropriate virial balance. For an isolated approximately stationary Newtonian system with negligible boundary terms, $2K+W\simeq 0$. It does not require every collisionless constituent to reach collisional thermal equilibrium, and ongoing infall, tides or pressure at a boundary modify the balance.