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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. DefineThe Friedmann equation is a mechanical energy equation,For nonnegative dust and radiation, at least one nonzero, $V$ tends to $-\infty$ as $a\to 0$ and to $+\infty$ as $a\to \infty$, whileThus every energy line meets the potential once. An expanding solution reaches this turning point and then contracts: $a_{tt}=-V^{\prime }(a)/2<0$. The Friedmann effective potential for matter, radiation and vacuum is sketched below, alongside the positive-vacuum case.

To bound the full cosmic time from the initial zero of $a$ to its final zero, use the Friedmann acceleration equation:Put maximum expansion at $t=0$, with $a(0)=a_{\max }$ and $a_t(0)=0$. The forced-oscillator identity isUntil the first zero and for $0\le t\le \pi /(2\omega )$, the kernel is nonnegative and the bracket is nonpositive. Hence $a(t)\le a_{\max }\cos (\omega t)$, 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 isAt 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.