Solution
= Solution
As $\Omega_M\downarrow0$, use $\operatorname{arsinh}z=\log(z+\sqrt{1+z^2})\sim\log(2z)$ to obtain
$$
\boxed{H_0t_0\sim\frac13\log\frac4{\Omega_M}\ \longrightarrow\ \infty.}
$$
At fixed <Hubble constant>, the <age of a flat matter-Lambda universe> diverges logarithmically. The limiting purely vacuum-dominated <de Sitter spacetime> has an exponential <scale factor> and no <Big Bang> at a finite value of <cosmic time> in this flat slicing.