= Solution
Within the equal-expansion construction of <6/a>, subtract the flat-background <Friedmann equation> to obtain
$$
\delta\rho=\frac{3K}{8\pi Ga^2},\qquad
\delta=\frac{3K}{8\pi Ga^2\rho_b}.
$$
The background <density> scales as $\rho_b\propto a^{-4}$ during <radiation domination> and $\rho_b\propto a^{-3}$ during <matter domination>. Therefore
$$
\boxed{\delta\propto a^2\quad(a\ll a_{\rm eq}),\qquad
\delta\propto a\quad(a\gg a_{\rm eq}).}
$$
These are the regular adiabatic growing-mode scalings on scales outside the <Hubble radius>, with a specified time slicing. In a consistent comoving description the numerical relation between curvature and <density contrast> differs from the naive subtraction, but the powers agree. The result does not imply that every gauge's <density contrast>, every decaying mode, or subhorizon dark matter during <radiation domination> has these powers.
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