Timelike geodesic in the expanding flat slicing of de Sitter spacetime

For a timelike geodesic moving in the $x$ direction with $t(0)=x(0)=0$ and $\dot{x}(0)=v_0>0$, spatial translation symmetry gives $e^{2Ht}\dot{x}=v_0$. Proper-time normalization then gives

The geodesic with nonzero conserved spatial momentum reaches $t=-\infty$ after the finite past proper timeThis is incompleteness of the expanding coordinate patch, whose geodesics extend smoothly in global de Sitter spacetime; it is not a curvature singularity.