 Radiation-era comoving density mode

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.