= Matter-vacuum conformal age integral
{title2=$\tau_0\simeq H_0^{-1}\Omega_m^{-1/3}\Omega_\Lambda^{-1/6}J$}
In a flat universe containing <pressureless matter> and a positive <cosmological constant>, normalize the <scale factor> to one today. The <Friedmann equation> and $d\tau=dt/a$ give
$$
\tau_0=H_0^{-1}\int_0^1\frac{da}{\sqrt{\Omega_m a+\Omega_\Lambda a^4}}
=H_0^{-1}\Omega_m^{-1/3}\Omega_\Lambda^{-1/6}
\int_0^{(\Omega_\Lambda/\Omega_m)^{1/3}}\frac{dy}{\sqrt{y(1+y^3)}}.
$$
The change of variable is $a=(\Omega_m/\Omega_\Lambda)^{1/3}y$. Including <radiation in cosmology> makes this an approximation, with its omitted early contribution of order $H_0^{-1}\sqrt{\Omega_r}/\Omega_m$. When the upper limit is large, the integral approaches $I=\Gamma(1/6)\Gamma(1/3)/(3\sqrt\pi)$; this follows by setting $z=y^3$ and using the <beta function> integral.
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