= Solution
Relativistic pressure gives the radiation fluid a <sound speed> $c_s^2=1/3$. Well inside the sound horizon, $k\gg\mathcal H$, radiation <linear cosmological density perturbations> undergo rapid pressure-supported oscillations instead of sustained gravitational growth. Their potential decays. For an ideal radiation fluid with negligible <scalar anisotropic stress>, the regular <Newtonian gauge in cosmology> solution during <radiation domination> illustrates this explicitly:
$$
\Phi_r(k,\tau)=3\Phi_p\frac{\sin u-u\cos u}{u^3},\qquad u=\frac{k\tau}{\sqrt3}.
$$
For $u\gg1$ its amplitude is $O(u^{-2})$. Thus <radiation in cosmology> remains essential to the background <Friedmann equation>, while its rapidly oscillating gravitational forcing can be smoothed out when following the slowly varying <cold dark matter> density mode. This is the approximation behind the <Mészáros equation>.
The important limitation is that “subhorizon” alone is not a pointwise theorem that $|\Phi_r|\ll|\Phi_m|$. Radiation can dominate the instantaneous potential during early <radiation domination>, and the forcing near horizon entry seeds the later logarithmic matter mode. We neglect that transient and rapidly oscillating forcing in the subsequent coarse-grained growth equation, not radiation's contribution to the expansion rate. Close to horizon entry, or for accurate oscillatory evolution, the coupled radiation and matter perturbation equations must be retained. For the slow matter potential, the metric time derivatives in part (a) are also suppressed relative to its spatial <Laplacian> by $O((\mathcal H/k)^2)$. \b[The approximation smooths radiation perturbations while retaining <radiation in cosmology> in the background <cosmic expansion>.]
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