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Once a mode is well inside the horizon, radiation pressure opposes gravitational collapse. For an ideal radiation fluid its cosmological sound speed is $c_s=1/\sqrt{3}$, and its density perturbation undergoes subhorizon radiation acoustic oscillations with frequency approximately $k/\sqrt{3}$. Unlike the superhorizon growing mode, its amplitude does not acquire sustained gravitational growth. Free-streaming relativistic components can additionally phase-mix the perturbation; the acoustic-fluid description is the appropriate idealization for the stated system.

During deep radiation domination, the radiation source in the dark-matter equation oscillates rapidly compared with a Hubble time. Its accumulated late-time effect averages away. More quantitatively, an oscillatory source of magnitude ${\Delta }_r/{\tau }^2$ gives a rapidly varying particular solution of order ${\Delta }_r/(k\tau {)}^2$ when $k\tau \gg 1$. It can therefore be neglected in computing the slowly varying homogeneous dark-matter growth, although its forcing near horizon entry fixes the integration constants.

The dark-matter self-gravity term is also subleading while $a/a_{\mathrm{eq}}\ll 1$. Keeping the leading expansion drag and averaging the radiation source yieldsConsequentlyThis logarithmic growth of matter perturbations during radiation domination is the Mészáros effect. It is the leading behavior well after horizon entry but before equality; close to equality the neglected self-gravity matters. The radiation perturbation is not small simply because it is radiation: its rapid oscillations, rather than a small radiation background density, justify dropping its source for this slow mode.

Under the stated approximation the perturbation equation isDuring radiation domination, $a\propto t^{1/2}$, so $\dot{\delta }\propto t^{-1}$ andMatter perturbations therefore have only logarithmic growth of matter perturbations during radiation domination.

In a curvature-dominated universe, $a\propto t$, so $\dot{\delta }\propto t^{-2}$ andThe nondecaying mode is constant: curvature domination gives freezing of matter perturbations during curvature domination, rather than continued growth.