 Age of a flat matter-coasting-fluid universe

Integrating $dt=da/(aH)$ gives $H_0{t}_0={\int }_0^1\sqrt{a}da/\sqrt{{\Omega }_m+(1-{\Omega }_m)a}$. For $b=1-{\Omega }_m$ in the interior of the physical density range, this is $b^{-1}-{\Omega }_m{b}^{-3/2}\mathrm{arsinh}\sqrt{b/{\Omega }_m}$. The pure matter and pure coasting endpoints have ages $2/(3H_0)$ and $1/H_0$.