Cosmological bounce ($\dot{a}(t_b)=0,\ddot{a}(t_b)>0$)

A smooth cosmological bounce is a positive minimum of the scale factor: contraction changes to expansion, with $a(t_b)>0$, $\dot{a}(t_b)=0$ and $\ddot{a}(t_b)>0$ in the nondegenerate case. The closed slicing of de Sitter spacetime gives $a=H_{\Lambda }^{-1}\cosh [H_{\Lambda }(t-t_b)]$, whose minimum is $H_{\Lambda }^{-1}$. The Friedmann acceleration equation with zero separate cosmological constant requires $\varrho +3p<0$ there.