= Matter-vacuum age endpoint limits
{title2=$H_0t_0\sim\tfrac13\log(4/\Omega_m)\ (\Omega_m\to0),\quad H_0t_0\to2/3\ (\Omega_m\to1)$}
The <age of a flat matter-Lambda universe> is $H_0t_0=2\operatorname{arsinh}\sqrt{(1-\Omega_m)/\Omega_m}/(3\sqrt{1-\Omega_m})$. The large-argument expansion $\operatorname{arsinh}z=\log(2z)+O(z^{-2})$ gives the first limit. Its small-argument expansion $\operatorname{arsinh}z=z+O(z^3)$ 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.
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