= Solution
Treat the long-wavelength nearly uniform spherical perturbation as a separate locally homogeneous expanding region, neglecting pressure gradients and tidal corrections. Its local <Friedmann equation> is
$$
\boxed{H_s^2+\frac{K}{a_s^2}=\frac{8\pi G}{3}\rho_s.}
$$
The curvature term represents the <spatial curvature of an FLRW universe> in the local patch; for the convention $a_s$ dimensionless, $K=kc^2$ has units of inverse time squared. Positive $K$ corresponds to a closed, more tightly bound patch. Here and below the printed $\rho$ is interpreted as <mass density> or <energy density> divided by $c^2$; literal SI <energy density> requires $G\rho/c^2$ on the right.
The region generally has its own <scale factor> and <Hubble parameter>. The two printed equations with an identical $H(t)$ can be compared as a leading-order \i[equal-expansion] construction: on a slicing where $H_s=H$, use $a_s=a$ in the already first-order curvature term to obtain $K/a^2=(8\pi G/3)\delta\rho$. This <equal-expansion curvature overdensity> construction does not make them both exact evolution equations with a permanently identical expansion history at common proper time. For example, a growing dust <density contrast> obeys the <continuity equation> $H_s-H=-\dot\delta/3$, which is nonzero when $\delta\propto a$. This explicit example prevents an inconsistent identification of the two local histories.
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