 Solution

For a flat matter-dominated universe, the Friedmann solution has $a\propto t^{2/3}$, $H=2/(3t)$ and $\bar{\rho }=1/(6\pi G{t}^2)$. Thus the density equation isTrying $t^s$ gives $s(s-1)+4s/3-2/3=0$, with roots $2/3$ and $-1$. ThereforeThe growing mode is proportional to the scale factor. When smooth dark energy or a cosmological constant dominates, the matter source density falls while expansion provides continued damping, suppressing growth. In the late constant-$\Lambda$ limit, $H\to H_{\Lambda }$ and $\bar{\rho }\to 0$, so $\ddot{\delta }+2H_{\Lambda }\dot{\delta }\simeq 0$ and $\delta$ tends to a constant plus a decaying $e^{-2H_{\Lambda }t}$ term. Matter perturbations therefore cease growing appreciably rather than continuing the matter-era law indefinitely.