= Solution
Cold dark matter has $c_s=0$. During matter domination,
$$
a(t)\propto t^{2/3},
\qquad
H=\frac{2}{3t},
\qquad
\bar\rho=\frac{1}{6\pi Gt^2}.
$$
The density equation becomes
$$
\ddot\delta+\frac{4}{3t}\dot\delta
-\frac{2}{3t^2}\delta=0.
$$
Trying the <power law> $\delta=t^p$ gives
$$
p(p-1)+\frac43p-\frac23=0,
$$
whose roots are $p=2/3$ and $p=-1$. Thus the <cosmic-time matter density modes> are
$$
\boxed{\delta(t)=A t^{2/3}+B t^{-1}},
$$
and the leading growing mode is $\delta\propto t^{2/3}\propto a(t)$.
Solved by gpt-5.6-sol high.
Back to article page
