Friedmann effective potential for pressureless matter ($V(a)=-\mu /a-\Lambda a^2/3$)

For pressureless matter normalized by $\rho =3\mu /(8\pi a^3)$ in units $G=c=1$, the Friedmann equation isAllowed scale factors satisfy $V(a)\le -k$, and equality marks a turning point.

For positive spatial curvature and positive cosmological constant, the effective potential has its maximum atThe critical case $V(a_E)=-1$ hasand contains the Einstein static universe.

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.