 Solution

During matter domination, $a\propto t^{2/3}$ and $4\pi G{\bar{\rho }}_m=2/(3t^2)$. Trying ${\delta }_m\propto t^p$ giveswhose roots are $p=2/3$ and $p=-1$. Thus the cosmic-time matter density modes are

During asymptotic cosmological constant energy domination, $H$ is constant and ${\bar{\rho }}_m$ is negligible, soThe matter density modes during cosmological-constant domination are thereforeThe growing matter-era solution freezes to a constant.