= Age of a dust universe without a cosmological constant
{title2=$H_0t_0=\int_0^1\left(1-2q_0+2q_0/x\right)^{-1/2}\,dx$}
For an expanding <pressureless matter> universe with zero <cosmological constant>, normalize $x=a/a_0$. <Dust relation between deceleration and density> and mass conservation give $\dot x^2=H_0^2(1-2q_0+2q_0/x)$. Integrating from the <Big Bang> yields the age integral above.
For $q_0>1/2$, set $x=q_0(1-\cos\theta)/(2q_0-1)$ to obtain $H_0t_0=q_0(\theta_* -\sin\theta_*)/(2q_0-1)^{3/2}$, with $\cos\theta_*=(1-q_0)/q_0$ on the expanding branch. For $0<q_0<1/2$, the corresponding hyperbolic expression is $q_0(\sinh\eta_* -\eta_*)/(1-2q_0)^{3/2}$ with $\cosh\eta_*=(1-q_0)/q_0$. The flat-dust and empty <Milne universe> limits are respectively $2/3$ and $1$.
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