 Closed slicing of de Sitter spacetime

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 $T=\pm \pi /2$.