Age of a flat matter-Lambda universe ($t(a)=\frac{2}{3H_0\sqrt{{\Omega }_{\Lambda }}}\mathrm{arsinh}\left(\sqrt{{\Omega }_{\Lambda }/{\Omega }_m}{a}^{3/2}\right)$)

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.

The age of a flat matter-Lambda universe is $H_0{t}_0=2\mathrm{arsinh}\sqrt{(1-{\Omega }_m)/{\Omega }_m}/(3\sqrt{1-{\Omega }_m})$. The large-argument expansion $\mathrm{arsinh}z=\log (2z)+O(z^{-2})$ gives the first limit. Its small-argument expansion $\mathrm{arsinh}z=z+O(z^3)$ gives the second. The divergence at the pure-vacuum endpoint reflects the absence of a finite Big Bang time in the expanding flat de Sitter spacetime patch.