= Equal-expansion curvature overdensity
{title2=$\delta=K/(a^2H^2)$}
Compare a flat background <Friedmann equation> with a nearly homogeneous curved region on a slicing of equal local <Hubble parameter>. To first order in its curvature parameter $K$, subtraction gives $\delta\rho=3K/(8\pi Ga^2)$, or $\delta=K/(a^2H^2)$. Conserved curvature therefore gives $\delta\propto a^2$ in <radiation domination> and $\delta\propto a$ in <matter domination>. This construction specifies the slicing: two patches at equal proper time generally do not have the same <Hubble parameter>. For growing dust perturbations, the <continuity equation> gives $H_s-H=-\dot\delta/3$ to first order. These different time comparisons can share growth powers without sharing the same curvature-to-density coefficient.
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