= Total conformal lifetime of a flat matter-Lambda universe
{title2=$\chi_{\rm max}=\frac{\Omega_m^{-1/3}\Omega_\Lambda^{-1/6}}{3H_0}B(1/3,1/6)$}
Use $c=1$, $a_0=1$, positive <pressureless matter> and positive <cosmological constant>, with $\Omega_m+\Omega_\Lambda=1$ and no radiation. The <Friedmann equation> gives $\chi_{\rm max}=H_0^{-1}\int_0^\infty dy/(\Omega_\Lambda+\Omega_my^3)^{1/2}$. Set $y=(\Omega_\Lambda/\Omega_m)^{1/3}u$, then $v=u^3$. The remaining integral is $\tfrac13\int_0^\infty v^{-2/3}(1+v)^{-1/2}\,dv=\tfrac13B(1/3,1/6)$, proving the formula by the <beta function> integral. Its two endpoints converge for positive densities, but the lifetime diverges in either pure-fluid limit.
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