= Solution
During <matter domination>, $\rho_C\propto a^{-3}$. The flat <Friedmann equation> in <conformal time> implies $a'^2\propto a$, and hence
$$
\boxed{a\propto\tau^2,\qquad8\pi G\rho_Ca^2=\frac{12}{\tau^2}.}
$$
With radiation neglected, the <CDM density equation in a matter-radiation universe> reduces to
$$
\delta_C''+\frac2\tau\delta_C'-\frac6{\tau^2}\delta_C=0.
$$
For this <Euler-Cauchy equation>, $\delta_C\propto\tau^p$ requires $p(p-1)+2p-6=(p-2)(p+3)=0$. Thus the <matter-era growing and decaying density modes> are $\tau^2$ and $\tau^{-3}$, respectively, and
$$
\boxed{\delta_C(\tau,k)\propto\tau^2\propto a.}
$$
There is no explicit $k$ in this pressureless equation, so the result holds on all scales within the stated CDM-comoving <synchronous gauge> description. The amplitude can still depend on $k$ through its earlier evolution; the result does not equate <density contrasts> on different time slicings.
Back to article page
