Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 9B / a / Solution

InsertMatching powers of $t$ requires $\lambda \alpha =2$, so $\alpha =2/\lambda$. Writing $\overline{V}=V_0{e}^{-\lambda {\varphi }_0}$, the scalar equation and Friedmann equation giveThusExistence with positive potential requires $\gamma >1/3$, or ${\lambda }^2<48\pi G$. Accelerated expansion requires $\gamma >1$, hence

Solved by gpt-5.6-sol high.