Closed slicing of de Sitter spacetime ($d{s}^2={\ell }^2[-d{\tau }^2+{\cosh }^2\tau d{\Omega }_3^2],{\ell }^2=3/\Lambda$)

The vacuum Einstein field equations with positive cosmological constant admit a closed spherical slicing with scale factor $a(t)=\ell \cosh [(t-t_0)/\ell ]$. The time-symmetric slice is at $t=t_0$. The substitution $T=2\arctan (e^{\tau })-\pi /2$ gives $dT/d\tau =\mathrm{sech}\tau$ and $\cos T=\mathrm{sech}\tau$, so the metric is ${\ell }^2{\sec }^{2}T[-d{T}^2+d{\Omega }_3^2]$. The conformal metric is a finite-time slab of the Einstein static universe, with spacelike conformal infinities $T=\pm \pi /2$.

For a complete comoving observer at the north pole of the closed slicing of de Sitter spacetime, radial null rays have $d\chi =\pm dT$. The finite past conformal interval gives the displayed particle horizon; the finite remaining future interval gives the displayed observer event horizon. The angular distances range strictly between zero and $\pi$ at every finite proper time. Complete constant-$T$ three-spheres are Cauchy hypersurfaces, so their domains of dependence cover the global spacetime and they have no Cauchy horizons. Cosmological horizons are observer-dependent and are not failures of global hyperbolicity.