Expansion-weighted derivative of a passive density perturbation ($a^2\dot{\delta }=\mathrm{constant}$)

When the self-gravity source in the pressureless linear cosmological density perturbation equation is negligible, $\ddot{\delta }+2(\dot{a}/a)\dot{\delta }=0$. Multiplication by $a^2$ makes it $d(a^2\dot{\delta })/dt=0$, so $\delta =C_1+C_2\int dt/a^2(t)$. This explains how Hubble friction changes growth for an arbitrary positive scale factor. For $a\propto t^p$, the nonconstant mode is $t^{1-2p}$ when $p\ne 1/2$ and $\log t$ when $p=1/2$.