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 $p=2/3,-1$, hence
$$
\boxed{\delta_m=C_1a+C_2a^{-3/2}}.
$$
During <radiation domination>, neglecting the rapidly oscillating radiation perturbation and the subdominant matter source leaves
$$
\ddot\delta_m+2H\dot\delta_m=0,
\qquad a\propto t^{1/2}.
$$
Therefore
$$
\boxed{\delta_m=C_3+C_4\log a}.
$$
This logarithmic behavior is the <Mészáros effect> for subhorizon cold dark matter.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-310/3/a/solution