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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]

In Questions 3–4, use primes for conformal-time derivatives and $\mathcal{H}=a^{\prime }/a$; proper-time derivatives will be stated separately. Let ${\xi }^{\mu }=(T,L^i)$. The metric is a covariant tensor, so under the passive coordinate change ${\tilde{x}}^{\mu }=x^{\mu }+{\xi }^{\mu }$,The inverse Jacobian is ${\delta }_{\mu }^{\alpha }-{\partial }_{\mu }{\xi }^{\alpha }$ to first order. At the same new coordinate argument, the homogeneous background satisfies ${\bar{g}}_{\alpha \beta }(x)={\bar{g}}_{\alpha \beta }(\tilde{x})-T{\bar{g}}_{\alpha \beta }^{\prime }$. Shifting the argument of $\delta g$ or multiplying it by a Jacobian perturbation produces only second-order terms, which we discard. Subtracting the unperturbed metric therefore givesThis is the printed Jacobian expression expanded to linear order, equivalently $\delta \tilde{g}=\delta g-{\mathcal{L}}_{\xi }\bar{g}$. It is a gauge transformation in cosmological perturbation theory, not a change of the physical spacetime.