Equal-expansion curvature overdensity ($\delta =K/(a^2{H}^2)$)

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 $\delta \rho =3K/(8\pi G{a}^2)$, or $\delta =K/(a^2{H}^2)$. 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.