Open slicing of de Sitter spacetime ($a(t)=\frac{\sqrt{-k}}{H_{\Lambda }}\sinh (H_{\Lambda }t)$)

With no matter or radiation, $k<0$ and $H_{\Lambda }=\sqrt{\Lambda /3}>0$, the Friedmann equation in units $c=1$ is ${\dot{a}}^2=H_{\Lambda }^2{a}^2-k$. 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.