Constant-equation-of-state comoving density equation (${\Delta }^{\prime \prime }+\frac{2(1-3w)}{(1+3w)\eta }{\Delta }^{\prime }-\frac{6(1-w)}{(1+3w){\eta }^2}\Delta -w{\nabla }^2\Delta =0$)

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.