Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 73 / 2 / ii / Solution

The age of an FLRW universe at an epoch is the elapsed cosmic time since the Big Bang, not the lookback time from today. For positive matter and positive cosmological constant, integrating $dt=da/\dot{a}$ givesPut $u=\sqrt{{\Omega }_{\Lambda ,0}/{\Omega }_{m,0}}{a}^{\prime 3/2}$; then $du=(3/2)\sqrt{{\Omega }_{\Lambda ,0}/{\Omega }_{m,0}}\sqrt{a^{\prime }}d{a}^{\prime }$. The integral becomes the age of a flat matter-Lambda universe:Choose $0<\theta <\pi /2$ such that $\tan \theta =\sqrt{{\Omega }_{m,0}/{\Omega }_{\Lambda ,0}}{a}^{-3/2}$. The upper-limit value of $u$ is $\cot \theta$, andConsequentlyThe integration constant is fixed by $t\to 0$ as $z\to \infty$, where $\theta \to \pi /2$. Equivalently, differentiating $u=\cot \theta$ changes the last integral to $-\int d\theta /\sin \theta$, recovering the supplied trigonometric identity. The formula assumes positive ${\Omega }_{\Lambda ,0}$ and ${\Omega }_{m,0}$; its zero-vacuum limit is regular even though this parametrization is singular.