Matter bound from a Friedmann bounce (${\Omega }_m\le 2/[z_{\ast }^2(z_{\ast }+3)]$)

In a dust-plus-cosmological-constant universe with an FRW metric with $a_0=1$, a nondegenerate minimum at $a_{\ast }<1$ has $H(a_{\ast })=0$ and $\ddot{a}(a_{\ast })>0$. Combining these conditions with ${\Omega }_m+{\Omega }_{\Lambda }+{\Omega }_k=1$ gives the strict version of the displayed bound; equality is the limiting stationary case. A source at cosmological redshift $z$ requires $z_{\ast }=a_{\ast }^{-1}-1\ge z$, so a high-redshift source bounds the allowed present matter density parameter even without directly observing the bounce.