Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 37E / c / i / Solution

With $\Lambda >0$ and $k=-1$, the Friedmann effective potential for pressureless matter isAlthough $V$ has one maximum, it is negative, so $C=1$ lies above it and ${\dot{a}}^2=C-V(a)$ never vanishes. An initially small expanding universe therefore expands forever. At large $a$,so the expansion approaches the exponential de Sitter scale factor in flat slicing. There is no turning point.