Friedmann bounce for pressureless matter and positive cosmological constant

When the energy line $-k$ lies below the maximum of the pressureless-matter effective potential, its large-$a$ allowed region is separated from the Big-Bang region. A contracting large universe reaches the outer turning point at positive scale factor and re-expands, producing a nonsingular bounce.

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.