Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 15E / b / iv / Solution

From the explicit solution, $\ddot{a}=\gamma /2$. Since $a(t_0)=1$ and $\dot{a}(t_0{)}^2=\beta +\gamma$,Because an open universe has $\beta >0$, this is strictly greater than $-1/2$. It approaches $-1/2$ only in the zero-curvature limit, so this model cannot attain $q_0\le -0.5$ for the permitted parameters and therefore cannot reproduce $q_0\approx -0.55$.