 Solution

For $\Lambda >0$ and $k=1$,The unique maximum occurs atThe critical closed matter-cosmological-constant universe is obtained from $V_{\max }=-1$:

If $\Lambda >{\Lambda }_{\mathrm{crit}}$, then $V_{\max }<-1$: the energy line lies above the barrier, and a universe expanding from $a=0$ passes over it and expands forever toward de Sitter behavior. If $\Lambda ={\Lambda }_{\mathrm{crit}}$, the exact solution at $a=a_E$ is the Einstein static universe; the Big-Bang branch approaches it only asymptotically, while a separate branch can emerge from it and expand forever. If $0<\Lambda <{\Lambda }_{\mathrm{crit}}$, the equation $V(a)=-1$ has two positive roots $a_{-}<a_E<a_{+}$. The Big-Bang branch expands to $a_{-}$ and recollapses, while the disconnected large-$a$ branch contracts to $a_{+}$ and undergoes a Friedmann bounce for pressureless matter and positive cosmological constant. These are all qualitatively different possibilities.