= Solution
Let a comoving test particle have physical radius $R=a(t)\chi$, where $\chi$ is fixed. Assume <Newtonian gravity>, spherical homogeneity, negligible pressure, and no mass flux through this sphere, so its enclosed mass $M=(4\pi/3)\rho R^3$ is constant. The <shell theorem> gives a conserved specific <energy>
$$
\frac12\dot R^2-\frac{GM}{R}=\mathcal E.
$$
Substitute $R=a\chi$, divide by $R^2/2$, and write $2\mathcal E/\chi^2=-kc^2$ to obtain
$$
\boxed{H^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}.}
$$
Homogeneity makes the <energy> scale as $\chi^2$, leaving the same $k$ at every comoving radius. The sign of $k$ encodes the binding-energy sign. This Newtonian derivation applies to pressure-free matter; interpreting $k$ as spatial curvature and extending the equation to relativistic pressure components requires <general relativity>, rather than conservation of this Newtonian shell <energy> alone.
Back to article page
