Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 9B (Cosmology)

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

Multiplying the inflaton equation by $\dot{\varphi }$ givesComparison with $\dot{\rho }+3H(\rho +P)=0$ yields $\rho +P={\dot{\varphi }}^2$, and therefore

The acceleration equation requires $\rho +3P<0$, so $w=P/\rho <-1/3$. A canonical scalar field with nonnegative potential obeys $P\ge -\rho$. Hence inflation occurs for