Negative-vacuum Friedmann lifetime bound ($t_{\mathrm{life}}\le \pi \sqrt{3/(8\pi G|{\rho }_{\Lambda }|)}$)

For a FRW metric with positive scale factor, nonnegative dust and radiation, and negative vacuum density, the Friedmann acceleration equation gives $a_{tt}\le -{\omega }^{2}a$, 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 $a(t)\le a_{\max }\cos (\omega t)$ until the first zero, since its forcing is nonpositive and its sine kernel is nonnegative for $0\le t\le \pi /(2\omega )$. 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.