Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 15B / c

Put $x=a^2$. Apart from the positive factor $H_0^2/a^4$,For $\beta >4$, its discriminant $\beta (\beta -4)$ is positive and its two roots areBoth are positive. Moreover $F(0)>0$ and $F(2)=4-\beta <0$, soA universe expanding from $a=0$ follows the allowed region $0\le a^2\le x_{-}$ and reaches $a^2=x_{-}$ with $H=0$. There,so $H$ crosses from positive to negative: expansion reverses into contraction.

As $a\to 0$ on the contracting branch,Hence $ada\sim -H_0\sqrt{\beta }dt$, so zero scale factor is reached in finite time. The universe therefore undergoes the supercritical closed radiation--cosmological-constant recollapse to a Big Crunch.