= Solution
For <radiation in cosmology>, $w=1/3$, and the Fourier mode of the <comoving-gauge density contrast> satisfies
$$
\Delta_k''+\left(\frac{k^2}{3}-\frac2{\eta^2}\right)\Delta_k=0.
$$
Set $x=k\eta/\sqrt3$, the ratio to the radiation <sound horizon> scale. The <radiation-era comoving density mode> has the exact basis
$$
\boxed{\Delta_k=A\left(\frac{\sin x}{x}-\cos x\right)
+B\left(\frac{\cos x}{x}+\sin x\right).}
$$
Direct differentiation verifies the equation. For $x\ll1$, 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 $x\gg1$, 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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-55-radiation-density-mode.png]
{title=The regular radiation-fluid density mode grows outside the sound horizon and oscillates inside 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.
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