= Solution
For $\Lambda>0$ and $k=1$,
$$
V(a)=-\frac{\mu}{a}-\frac{\Lambda a^2}{3},
\qquad C=-1.
$$
The unique maximum occurs at
$$
a_E=\left(\frac{3\mu}{2\Lambda}\right)^{1/3},
\qquad
V_{\max}=-\Lambda a_E^2.
$$
The <critical closed matter-cosmological-constant universe> is obtained from $V_{\max}=-1$:
$$
\boxed{\Lambda_{\rm crit}=\frac{4}{9\mu^2},
\qquad a_E=\frac{3\mu}{2}=\Lambda_{\rm crit}^{-1/2}.}
$$
If $\Lambda>\Lambda_{\rm crit}$, 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 $\Lambda=\Lambda_{\rm crit}$, 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 $0<\Lambda<\Lambda_{\rm crit}$, 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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-2-friedmann-effective-potentials.png]
{title=Friedmann effective potentials for pressureless matter}
{description=Energy-line diagrams for zero, positive, and negative cosmological constant, including the subcritical, critical, and supercritical closed cases.}
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