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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.

After the decay products dominate the background, radiation domination gives ${\rho }_r\propto a^{-4}$. Neglecting curvature and vacuum terms, the Friedmann equation implies $\dot{a}/a=C/a^2$, so $a\propto (t-t_B{)}^{1/2}$ andThe displayed $1/(2t)$ form uses a cosmic-time origin with $t_B=0$. After a late transition the extrapolated radiation-era origin need not be the original Big Bang; this time shift does not change the growth law.

For the remaining pressureless matter, neglect its self-gravity to leading order in ${\rho }_m/{\rho }_r$. Its density contrast obeys $\ddot{\delta }+(t-t_B{)}^{-1}\dot{\delta }=0$. Using the conserved expansion-weighted derivative of a passive density perturbation givesThus logarithmic growth of matter perturbations during radiation domination replaces the static exponential growth and the matter-era $t^{2/3}$ growth. The expansion persists while its dominant radiation component supplies little sustained clustering in this pressureless test-component approximation. The residual matter can eventually become important again because ${\rho }_m/{\rho }_r\propto a$; the approximation is for the radiation-dominated interval. The equation is not a pressureless growth equation for the relativistic decay products themselves, whose pressure and, when a fluid description applies, radiation acoustic oscillations must be retained.