= 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 $\rho_C/\rho_R\ll1$. To leading order the <linear cosmological density perturbation> equation is therefore
$$
\delta''+\frac1\tau\delta'\simeq0,
\qquad (\tau\delta')'\simeq0.
$$
Two integrations give
$$
\boxed{\delta(\tau)\simeq A+B\ln\left(\frac{\tau}{\tau_H}\right),\qquad \tau_H\sim k^{-1}.}
$$
The reference time makes the logarithm dimensionless. A growing superhorizon mode reaches entry with a nonzero derivative, so matching generally excites $B\ne0$. For example, an abrupt leading-order match from $\delta\propto\tau^2$ gives $B=\tau_H\delta'_H\simeq2\delta_H$, 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 $4\pi Ga^2\rho_C\delta$. 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.
Back to article page
