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In deep matter domination, the scale factor is $a\propto {\tau }^2$, so $a^{\prime }/a=2/\tau$. Neglect radiation in the gravity source, including its backreaction, to obtainFor a power ${\tau }^p$ the Euler-Cauchy equation becomes $p(p-1)+2p-6=(p-2)(p+3)=0$. Its two independent matter-era growing and decaying density modes give the full approximate solutionThere is no $k$-dependent pressure term for cold dark matter. Thus this leading equation is valid both outside and inside the horizon in the stated comoving synchronous gauge, as long as the perturbations remain linear and the omitted radiation gravity is negligible. A different time slicing would give a different density variable.

For $k>0$, put $\omega =k/\sqrt{3}$, $C_g=A/{\tau }_i^2$ and $C_d=B{\tau }_i^3$. The remaining radiation equation is a forced harmonic oscillator:The matter-era forced radiation response can be written exactly within the matter-background approximation asThe last term is a Green's function particular solution; changing its lower limit only changes the two oscillatory constants. For the decaying source, a slowly varying particular solution obeys ${\delta }_{r,p}\simeq (4/3){\delta }_c^{\prime \prime }/{\omega }^2$, since a second time derivative is smaller than ${\omega }^2{\delta }_{r,p}$ by order $(k\tau {)}^{-2}$. Consequently the subhorizon asymptotic form isFor a pure growing mode, the constant offset and acoustic terms are an exact solution of this reduced radiation equation. In particular, the late radiation contrast does not follow the ever-growing ${\delta }_c$.

Before crossing, a regular growing adiabatic cosmological perturbation has both contrasts proportional to ${\tau }^2$, with ${\delta }_r=(4/3){\delta }_c$ to leading order. Afterwards cold dark matter keeps this quadratic growth, whereas radiation undergoes subhorizon radiation acoustic oscillations about a constant forced offset. To make the requested sketch concrete, one regular leading matter-era solution on both sides isIts small-$k\tau$ expansion gives ${\delta }_r=(4/3){\delta }_c+O((k\tau {)}^4)$, and its oscillation frequency is the radiation sound frequency. The sketch plots amplitudes in units of $\epsilon$; $\epsilon$ can be chosen small enough that the whole plotted interval remains linear. The line at ${\tau }_h$ uses the wavelength-to-particle-horizon convention in the question, rather than asserting that the smooth transition occurs at a sharp surface.

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.