= Solution
Use <cosmic time> $t$ here; derivatives with respect to <conformal time> will be used from Question 2 onward. Conservation of <pressureless matter> and <radiation in cosmology> makes $\rho_m a^3$ and $\rho_r a^4$ constant. Define
$$
M=\frac{8\pi G}{3}\rho_m a^3,\qquad Q=\frac{8\pi G}{3}\rho_r a^4,\qquad \omega^2=\frac{8\pi G}{3}|\rho_\Lambda|.
$$
The <Friedmann equation> is a mechanical energy equation,
$$
a_t^2+V(a)=-k,\qquad V(a)=-\frac{M}{a}-\frac{Q}{a^2}+\omega^2a^2.
$$
For nonnegative dust and radiation, at least one nonzero, $V$ tends to $-\infty$ as $a\to0$ and to $+\infty$ as $a\to\infty$, while
$$
V'(a)=\frac{M}{a^2}+\frac{2Q}{a^3}+2\omega^2a>0.
$$
Thus every energy line meets the potential once. An expanding solution reaches this <turning point> and then contracts: $a_{tt}=-V'(a)/2<0$. The <Friedmann effective potential for matter, radiation and vacuum> is sketched below, alongside the positive-vacuum case.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-58-effective-potentials.png]
{title=Friedmann effective potentials for negative and positive vacuum density}
To bound the full <cosmic time> from the initial zero of $a$ to its final zero, use the <Friedmann acceleration equation>:
$$
a_{tt}=-\frac{4\pi G}{3}(\rho_m+2\rho_r+2|\rho_\Lambda|)a\leq-\omega^2a.
$$
Put maximum expansion at $t=0$, with $a(0)=a_{\max}$ and $a_t(0)=0$. The forced-oscillator identity is
$$
a(t)=a_{\max}\cos(\omega t)+\frac1\omega\int_0^t\sin[\omega(t-s)]\,[a_{ss}(s)+\omega^2a(s)]\,ds.
$$
Until the first zero and for $0\leq t\leq\pi/(2\omega)$, the kernel is nonnegative and the bracket is nonpositive. Hence $a(t)\leq a_{\max}\cos(\omega t)$, so contraction cannot remain at positive $a$ beyond $\pi/(2\omega)$. Applying the same argument backward from maximum expansion bounds the expanding half. Therefore the <negative-vacuum Friedmann lifetime bound> is
$$
\boxed{t_{\rm life}\leq\frac{\pi}{\omega}=\sqrt{\frac{3\pi}{8G|\rho_\Lambda|}}.}
$$
At either endpoint, nonzero dust or radiation has divergent density. If both are absent, the same estimate instead bounds the open <Anti-de Sitter spacetime> coordinate patch; its coordinate zeros are not a finite lifetime of the maximally extended vacuum spacetime.
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