The dominant component is smooth, so only the subdominant matter density clusters. The cold-matter perturbation equation is therefore
$$
\ddot\delta+2H\dot\delta-4\pi G\bar\rho_m\delta=0,
\qquad
\bar\rho_m\propto a^{-3}.
$$
At late times, $a\gg1$, the source term is negligible compared with the expansion terms. Setting $\tau=t-t_*$ and using $H=2/\tau$ gives
$$
\ddot\delta+\frac4\tau\dot\delta\simeq0.
$$
It integrates to
$$
\boxed{\delta=C_1+C_2\tau^{-3}
=C_1+C_2a^{-3/2}}.
$$
The constant mode shows the <suppression of matter growth by smooth accelerated expansion>; the second mode decays.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 4 / e / ii / Solution

If a smooth component dominates the expansion but does not cluster, pressureless matter still satisfies
$$
\ddot\delta+2H\dot\delta-4\pi G\bar\rho_m\delta=0.
$$
As $\bar\rho_m$ becomes negligible, Hubble friction leaves a constant mode and a decaying mode, so structure growth freezes.


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