Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 310 / 3 / b / Solution

During matter domination, $a\propto t^{2/3}$, $H=2/(3t)$, and $4\pi G{\bar{\rho }}_m=2/(3t^2)$. Substituting ${\Delta }_m\propto t^s$ into the linear growth equation giveswhose roots are $s=2/3$ and $s=-1$. ThereforeThe growing one of the matter-dominated density-perturbation modes is