= Open slicing of de Sitter spacetime
{title2=$a(t)=\dfrac{\sqrt{-k}}{H_\Lambda}\sinh(H_\Lambda t)$}
With no matter or radiation, $k<0$ and $H_\Lambda=\sqrt{\Lambda/3}>0$, the <Friedmann equation> in units $c=1$ is $\dot a^2=H_\Lambda^2a^2-k$. Separating variables on the expanding branch and choosing its coordinate origin at $a=0$ gives the displayed <scale factor>. Normalization $a(t_0)=1$ imposes $\sinh(H_\Lambda t_0)=H_\Lambda/\sqrt{-k}$ and $H_0^2=H_\Lambda^2-k$. The zero of the <scale factor> is the boundary of this open coordinate patch, rather than a curvature singularity of <de Sitter spacetime>.
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