 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 isThe curvature term represents the spatial curvature of an FLRW universe in the local patch; for the convention $a_s$ dimensionless, $K=k{c}^2$ 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 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.