Age of a flat matter-coasting-fluid universe ($H_0{t}_0=\frac{1}{b}-\frac{{\Omega }_m}{b^{3/2}}\mathrm{arsinh}\sqrt{\frac{b}{{\Omega }_m}}$)

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 $1/H_0$.

For nonnegative matter and coasting-fluid densities, $0\le {\Omega }_m\le 1$. Since $a\le {\Omega }_m+(1-{\Omega }_m)a\le 1$ on the normalized integration interval, the age integrand lies between $\sqrt{a}$ and one. The age of a flat matter-coasting-fluid universe therefore decreases monotonically with the matter fraction and stays between the pure-fluid endpoint ages.