 Age of a flat matter-Lambda universe

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.