Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 3 / a / Solution

During matter domination, $4\pi G{\bar{\rho }}_m=3H^2/2$ and $a\propto t^{2/3}$. Substitution of ${\delta }_m\propto t^p$ gives $p=2/3,-1$, henceDuring radiation domination, neglecting the rapidly oscillating radiation perturbation and the subdominant matter source leavesThereforeThis logarithmic behavior is the Mészáros effect for subhorizon cold dark matter.