= Solution
Set $L=8\pi G\rho_\Lambda/3>0$. The <Friedmann effective potential for matter, radiation and vacuum> now reads
$$
V(a)=-\frac{M}{a}-\frac{Q}{a^2}-La^2,\qquad a_t^2+V(a)=E,\qquad E=-k.
$$
With nonzero nonnegative dust or radiation, $V$ tends to $-\infty$ at both ends. Its stationary point satisfies
$$
2La_E^4-Ma_E-2Q=0.
$$
Equivalently $V'(a)=0$ when $2L=M/a^3+2Q/a^4$; the right-hand side decreases strictly, so there is exactly one stationary point. It is a maximum because
$$
V''(a)=-\frac{2M}{a^3}-\frac{6Q}{a^4}-2L<0.
$$
The energy line in the preceding sketch therefore gives \b[five broad types of history]:
* \b[Monotonic expansion.] For $E>V(a_E)$, the universe expands from $a=0$ to arbitrarily large $a$.
* \b[Monotonic contraction.] Reversing <cosmic time> gives the corresponding contracting history.
* \b[Recollapse.] For $E<V(a_E)$, the inner allowed interval ends at $a_-$: the universe expands from zero to $a_-$ and contracts back to zero.
* \b[Bounce.] The outer allowed interval starts at $a_+$. A universe contracting from large $a$ reaches $a_+$ and re-expands. Here $V'(a_+)<0$, so $a_{tt}>0$ at the <turning point>.
* \b[Static equilibrium.] For $E=V(a_E)$, $a=a_E$ and $a_t=0$ give the <Einstein static universe>.
At the critical energy there are also marginal branches within the monotonic classes: expansion from zero can approach $a_E$ asymptotically, or expansion can start asymptotically at $a_E$ and proceed to infinity; contraction gives their reversals. They must not be mistaken for reaching the static solution in finite time. Indeed near the maximum,
$$
E-V(a)\simeq-\tfrac12V''(a_E)(a-a_E)^2,
$$
so $dt\simeq da/[\sqrt{-V''(a_E)/2}\,|a-a_E|]$ diverges logarithmically. The static equilibrium is unstable. Since $V(a_E)<0$, the recollapsing, bouncing and static cases require positive spatial curvature $k>0$.
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