= Solution
Set $a=1$ in the parametric solution. With $b=1-\Omega$,
$$
\beta\tau_0=2\operatorname{arsinh}\sqrt{\frac b\Omega},\qquad\sinh(\beta\tau_0)=\frac{2\sqrt b}{\Omega}.
$$
Consequently
$$
\boxed{H_0t_0=\frac1b-\frac\Omega{b^{3/2}}\operatorname{arsinh}\sqrt{\frac b\Omega}}.
$$
The endpoint values are obtained by limits. A particularly direct proof of the age bound avoids subtracting nearly equal terms: from the Friedmann equation,
$$
H_0t_0=\int_0^1\frac{\sqrt a\,da}{\sqrt{\Omega+(1-\Omega)a}}.
$$
For nonnegative component densities, $0\le\Omega\le1$ and $a\le\Omega+(1-\Omega)a\le1$. The integrand lies between $\sqrt a$ and $1$. Integrating therefore gives
$$
\boxed{\frac23H_0^{-1}\le t_0\le H_0^{-1}}.
$$
Equality holds for pure matter and pure <coasting fluid>, respectively. These are the <age of a flat matter-coasting-fluid universe> and the <coasting-fluid age bound>; the positivity range is necessary for the stated universal inequality.
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