Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 14B / c / Solution

Put $a=\alpha \tilde{a}$ withThenFor $y={\tilde{a}}^2$ this becomesThe expanding big-bang branch hasDifferentiation givesThus, with $\omega =\sqrt{4\Lambda /3}$,where the initial derivative givesFor $\lambda >1$, $\tilde{a}$ expands forever and becomes asymptotically exponential. For $\lambda <1$, it reaches a maximum and recollapses to zero. At $\lambda =1$, $y$ increases monotonically to $3/(2\Lambda )$, the limiting static radius.