= Closed slicing of de Sitter spacetime
{title2=$ds^2=\ell^2[-d\tau^2+\cosh^2\tau\,d\Omega_3^2],\quad\ell^2=3/\Lambda$}
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 $dT/d\tau=\operatorname{sech}\tau$ and $\cos T=\operatorname{sech}\tau$, so the metric is $\ell^2\sec^2T[-dT^2+d\Omega_3^2]$. The conformal metric is a finite-time slab of the <Einstein static universe>, with spacelike conformal infinities $T=\pm\pi/2$.
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