 Solution

The term $2H\dot{\delta }$ is Hubble friction in the equation for the density contrast. During expansion, $H>0$, it damps the peculiar motion producing density growth. It therefore slows gravitational instability relative to the otherwise identical static model; it does not necessarily stop growth.

Multiplying by $a^2$ shows the mechanism without confusing it with microscopic dissipative friction:When the self-gravity term is negligible, the expansion-weighted derivative of a passive density perturbation is conserved, so $\dot{\delta }\propto a^{-2}$. Expansion also reduces ${\rho }_m\propto a^{-3}$ and hence weakens the source on the right. A contracting background has $H<0$ and reverses the sign of this effect.