= Negative-vacuum Friedmann lifetime bound
{title2=$t_{\rm life}\leq\pi\sqrt{3/(8\pi G|\rho_\Lambda|)}$}
For a <FRW metric> with positive <scale factor>, nonnegative dust and radiation, and negative vacuum density, the <Friedmann acceleration equation> gives $a_{tt}\leq-\omega^2a$, with $\omega^2=8\pi G|\rho_\Lambda|/3$. At maximum expansion, $a(0)=a_{\max}$ and $a_t(0)=0$. The forced-oscillator integral formula gives $a(t)\leq a_{\max}\cos(\omega t)$ until the first zero, since its forcing is nonpositive and its sine kernel is nonnegative for $0\leq t\leq\pi/(2\omega)$. Thus each half of the expansion-recollapse history lasts at most $\pi/(2\omega)$. Nonzero dust or radiation makes the endpoint a physical density singularity; the matter-free limit instead bounds the open anti-de Sitter coordinate patch.
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