= Solution
Normalize $a(t_0)=1$, taking $a$ dimensionless and $r$ as a length, so the source's curvature constant $k$ has units of inverse length squared. With densities in the mass-equivalent convention, the first <Friedmann equation> is
$$
\boxed{H^2=\frac{8\pi G}{3}\sum_i\rho_i-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3}.}
$$
The sum can include dust, radiation and other material components; vacuum energy must be represented either in that sum or by $\Lambda$, without double counting. For physical energy densities $u_i$, replace $\rho_i$ by $u_i/c^2$. The companion acceleration equation is $\ddot a/a=-(4\pi G/3)\sum_i(\rho_i+3p_i/c^2)+\Lambda c^2/3$, with energy conservation $\dot\rho_i+3H(\rho_i+p_i/c^2)=0$ for separately conserved components.
For an empty universe with $\Lambda=0$, $H^2=-kc^2/a^2$. On the expanding branch with $H_0>0$,
$$
\boxed{k=-\frac{H_0^2}{c^2}<0.}
$$
Equivalently, in the convention where the normalized curvature label is $k=-1$ and the scale factor has length units, $a_0=c/H_0$. A rescaling of comoving coordinates changes the numerical magnitude of $k$ but not its negative sign. The degenerate empty case $H_0=0$ has $k=0$ and a constant scale factor; it does not have a finite age $1/H_0$.
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