For empty universe it seems to me that we can have two solutions. $$H^2=\frac {8\pi G\epsilon} {3c^2}-\frac {\kappa c^2} {R^2a^2(t)}$$
For an empty universe when we set $\epsilon=0$ we get $$H^2=\frac {-\kappa c^2} {a^2(t)}$$
since the square cannot be negative $\kappa$ must be $-1$, $\kappa=-1$
However I think, $\kappa=0$ is also a solution. But in one case $a(t)$ will be a constant, $a(t)=C$ since $\dot{a}(t)=0$,
in $\dot{a}(t)=c/R$ comes when we set $\kappa=-1$, in this case it seems to me that $a(t)=ct/R_0t_0$
Are these statements true? Since in general, empty universe which is also called the Milne universe, described by negative curvature. But cant be also described by a flat space ?
