= Matter-era growing and decaying density modes
{title2=$\delta=A\tau^2+B\tau^{-3}$}
During <matter domination> in a flat <FLRW metric>, the <cosmological continuity equation> gives $\rho\propto a^{-3}$, and the <Friedmann equation> gives $a\propto\tau^2$ and $8\pi G\rho a^2=12/\tau^2$. The <CDM density equation in a matter-radiation universe>, with radiation neglected, is therefore $\delta''+2\delta'/\tau-6\delta/\tau^2=0$. Its <Euler-Cauchy equation> powers satisfy $(p-2)(p+3)=0$, proving the displayed general solution. The growing <density contrast> is proportional to the <scale factor>, while the decaying mode is proportional to $a^{-3/2}$. This all-scale form uses the CDM-comoving <synchronous gauge>; other density slices require their corresponding gauge transformation.
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