 Solution

After cosmological horizon crossing, the rapidly oscillating radiation source can be averaged away as specified. Well inside radiation domination, the remaining cold dark matter self-gravity is also subleading because ${\rho }_C/{\rho }_R\ll 1$. To leading order the linear cosmological density perturbation equation is thereforeTwo integrations giveThe reference time makes the logarithm dimensionless. A growing superhorizon mode reaches entry with a nonzero derivative, so matching generally excites $B\ne 0$. For example, an abrupt leading-order match from $\delta \propto {\tau }^2$ gives $B={\tau }_H{\delta }_H^{\prime }\simeq 2{\delta }_H$, whereas its exact numerical coefficient depends on the transition and the radiation forcing.

This logarithmic growth is the Mészáros effect. It is an asymptotic radiation-era result: after dropping the oscillating radiation perturbation, the original equation still contains $4\pi G{a}^2{\rho }_C\delta$. Retaining that term and the changing background leads to the Mészáros equation, which provides the interpolation near matter-radiation equality. It would be incorrect to claim that a pure logarithm solves the complete mixed-background equation exactly.