Subhorizon radiation acoustic oscillations (${\Delta }_r\sim C\cos (k\tau /\sqrt{3}+\vartheta )$)

Pressure-supported radiation perturbations well inside the horizon oscillate rather than exhibit secular gravitational growth. In the ideal relativistic fluid, $c_s^2=1/3$ and the leading conformal frequency is $k/\sqrt{3}$. Their rapidly changing source has a small late-time response in the slowly varying cold-matter density mode.

In deep radiation domination, an oscillatory radiation density source in the matter equation has order ${\Delta }_r/{\tau }^2$ and frequency of order $k$. Its rapidly varying particular response is therefore of order ${\Delta }_r/(k\tau {)}^2$. For $k\tau \gg 1$ it does not control the secular matter growth, though forcing near horizon entry sets the constant and logarithmic homogeneous amplitudes.