During matter domination, $a\propto t^{2/3}$, $H=2/(3t)$ and $4\pi G\bar\rho_m=2/(3t^2)$. Trying $\delta_m=t^p$ gives
$$
p(p-1)+\frac43p-\frac23=0,
$$
with roots $p=2/3,-1$. The growing <linear cosmological density perturbation> is therefore
$$
\boxed{\delta_m\propto t^{2/3}\propto a}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-310/3/a/solution