 Solution

For the shifted density perturbation in Newtonian gauge, set ${\Delta }_R=\delta -4\Psi$. The radiation continuity equation gives ${\Delta }_R^{\prime }=-4\theta /3$, and its Euler equation givesWriting $D_R=k^2+6{\mathcal{H}}^2$, the relativistic constraint becomesEliminating the potential therefore givesThis has the requested $\mathcal{F}=12{\mathcal{H}}^2/(k^2+6{\mathcal{H}}^2)$. Well inside the sound horizon, $k\gg \mathcal{H}$, the potential contribution is suppressed relative to the density term and $\delta \simeq {\Delta }_R$. ThusThese are sound waves: radiation-pressure gradients restore a compression, while inertia carries the fluid through equilibrium. The perfect-fluid approximation neglects diffusion damping and freely streaming species' anisotropic stress. It gives the high-frequency limiting oscillations, not a claim that every radiation-era perturbation is an undriven sinusoid at horizon entry.