= Vacuum-energy FLRW solution
{title2=$\dot a^2+k=H_\Lambda^2a^2$}
The <equation of state> $p=-\varrho$ makes the <cosmological continuity equation> imply constant <energy density> $\varrho_0$. For $\varrho_0>0$, put $H_\Lambda^2=8\pi G\varrho_0/3$. Solving both the <Friedmann equation> and <Friedmann acceleration equation> gives
$$
a=Ae^{H_\Lambda t}+Be^{-H_\Lambda t},\qquad 4H_\Lambda^2AB=k.
$$
Up to shifts of <cosmic time> and time reversal, the positive <scale factors> are $Ae^{H_\Lambda t}$ for $k=0$, $H_\Lambda^{-1}\cosh(H_\Lambda t)$ for $k=1$, and $H_\Lambda^{-1}\sinh(H_\Lambda t)$ on $t>0$ for $k=-1$. They are the flat, closed and open slicings of <de Sitter spacetime>. The open zero is a coordinate boundary with finite <Ricci scalar> $12H_\Lambda^2$ and finite <Kretschmann scalar> $24H_\Lambda^4$.
For the expanding flat branch, the <cosmological event horizon> has <comoving radial distance> $\int_t^\infty ds/a(s)=1/[H_\Lambda a(t)]$ and <proper distance in cosmology> $H_\Lambda^{-1}$. An event on the horizon reaches the central <comoving observer> only at infinite reception time; events beyond it never do.
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