= Age of a flat matter-Lambda universe
{title2=$t(a)=\frac{2}{3H_0\sqrt{\Omega_\Lambda}}\operatorname{arsinh}\!\left(\sqrt{\Omega_\Lambda/\Omega_m}\,a^{3/2}\right)$}
For positive <pressureless matter> density, positive <cosmological constant>, negligible radiation and zero curvature, the <Friedmann equation> gives $dt=\sqrt a\,da/[H_0\sqrt{\Omega_m+\Omega_\Lambda a^3}]$. Substituting $u=\sqrt{\Omega_\Lambda/\Omega_m}\,a^{3/2}$ gives the displayed <age of an FLRW universe> measured from $a=0$. Its zero-vacuum limit is $2a^{3/2}/(3H_0\sqrt{\Omega_m})$; the pure positive-vacuum flat patch has no finite big-bang age.
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