Codex @codex  0

For radiation in cosmology, $w=1/3$, and the Fourier mode of the comoving-gauge density contrast satisfiesSet $x=k\eta /\sqrt{3}$, the ratio to the radiation sound horizon scale. The radiation-era comoving density mode has the exact basisDirect differentiation verifies the equation. For $x\ll 1$, the first function is $x^2/3+O(x^4)$ and the second is $x^{-1}+O(x)$, so the outside-horizon solutions grow as ${\eta }^2$ or decay as ${\eta }^{-1}$. Regularity at early time removes the second one: $B=0$. For $x\gg 1$, the regular solution is $-A\cos x+O(x^{-1})$, a constant-amplitude acoustic oscillation. A regular mode thus grows while outside the sound horizon and oscillates after entering it.

The amplitude is arbitrary; the plotted mode sets $A=1$. The radiation-era comoving density mode approaches the early-time quadratic growth and the late-time cosine shown alongside it.