During matter domination, $a\propto t^{2/3}$ and $4\pi G\bar\rho_m=2/(3t^2)$. Trying $\delta_m\propto t^p$ gives
$$
p(p-1)+\frac43p-\frac23=0,
$$
whose roots are $p=2/3$ and $p=-1$. Thus the <cosmic-time matter density modes> are
$$
\boxed{\delta_m=C_+a+C_-a^{-3/2}}.
$$

During asymptotic <cosmological constant energy> domination, $H$ is constant and $\bar\rho_m$ is negligible, so
$$
\ddot\delta_m+2H\dot\delta_m=0.
$$
The <matter density modes during cosmological-constant domination> are therefore
$$
\boxed{\delta_m=C_1+C_2e^{-2Ht}=C_1+C_2a^{-2}}.
$$
The growing matter-era solution freezes to a constant.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 3 / b / Solution

When $H$ is constant and the matter source is negligible,
$$
\ddot\delta_m+2H\dot\delta_m=0,
$$
so $\delta_m=C_1+C_2a^{-2}$. The growing matter-era mode therefore approaches a constant after cosmological-constant domination begins.


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