Comoving-gauge density contrast ($\Delta =\delta -3\mathcal{H}(1+w)v$)

For a perfect fluid with $P=w\rho$, in Newtonian gauge in cosmology the density perturbation on fluid-comoving slices is the displayed combination of the fractional density perturbation and velocity potential. The Einstein energy and momentum constraints combine into ${\nabla }^2\varphi =4\pi G{a}^2\bar{\rho }\Delta$ when scalar anisotropic stress vanishes. This gives a relativistic Poisson constraint valid beyond the subhorizon limit.

During matter domination in a flat universe, the growing comoving-gauge density contrast is proportional to $a\propto {\eta }^2$. Its Newtonian potential is constant by the Poisson constraint. The momentum constraint gives $v=-2\varphi /(3\mathcal{H})=-\eta \varphi /3$. Thus the peculiar velocity potential grows as $a^{1/2}$ in the convention where the peculiar velocity is $\nabla v$.

For a flat single-fluid background with constant barotropic equation of state, $w\ne -1/3$, adiabatic perturbations and zero anisotropic stress, $a\propto {\eta }^{2/(1+3w)}$ and $a^2\bar{\rho }\propto {\eta }^{-2}$. The scalar potential obeys ${\varphi }^{\prime \prime }+[6(1+w)/(1+3w)]{\varphi }^{\prime }/\eta -w{\nabla }^2\varphi =0$. Substituting the Poisson constraint $\Delta \propto {\eta }^2{\nabla }^2\varphi$ gives the displayed equation. The homogeneous mode and the degenerate vacuum fluid $w=-1$ require separate interpretation.

For radiation in cosmology, the comoving-gauge density contrast satisfies ${\Delta }_k^{\prime \prime }+(k^2/3-2/{\eta }^2){\Delta }_k=0$. The displayed basis solves it. The first mode is regular and grows as $A{x}^2/3$ outside the sound horizon; the second diverges as $B/x$ toward the initial singularity. At large $x$, the regular mode approaches $-A\cos x$ and has constant-amplitude acoustic oscillations.