 Solution

For a $1\mathrm{Mpc}$ mode, evolve the regular superhorizon density contrast as $\delta \propto a^2$ until cosmological horizon entry. In the radiation domination approximation its entry scale is $a_{\mathrm{ent}}\simeq (H_{0}R/c)\sqrt{{\Omega }_{\gamma }}=2.2\times 10^{-6}$, well before matter-radiation equality. At that point use the Mészáros equation basis of d, fixing both constants by continuity of $\delta$ and $\dot{\delta }$, equivalently $\delta$ and $d\delta /d\ln a$.

Explicitly, for entry data ${\delta }_e$ and ${\delta }_e^{\prime }=2{\delta }_e/y_e$, solveThe nonzero Wronskian guarantees a unique match. This produces slow logarithmic growth before equality, then $\delta \propto a$ with a decaying transient after equality. For example, at $y_e\ll 1$, $A_2\simeq -2{\delta }_e$ and $A_1\simeq {\delta }_e+2{\delta }_e[\ln (4/y_e)-3]$. An abrupt match estimates the growth history; a precise amplitude requires a common gauge and the coupled relativistic perturbation equations through entry. Linear evolution also ceases to apply once $|\delta |$ approaches unity, and late dark energy further suppresses growth.