Cosmological bounce
= Cosmological bounce
{title2=$\dot a(t_b)=0,\quad\ddot a(t_b)>0$}
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.