Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 310 / 1 / d

The identityshows that accelerating expansion has $H>0$ and $ϵ<1$, whereas accelerated contraction has $H<0$, $\ddot{a}<0$, and $ϵ>1$.

For the monomial inflation potential $V=\lambda {\varphi }^p$, the two potential slow-roll parameters areThus ${ϵ}_V<1$ requires $|\varphi |>p{M}_{\mathrm{Pl}}/\sqrt{2}$, while $|{\eta }_V|<1$ requires $|\varphi |>\sqrt{p|p-1|}{M}_{\mathrm{Pl}}$.

At ${\varphi }_{\ast }$, the specified velocity gives kinetic energyHence ${\rho }_{\varphi }=3V$, $P_{\varphi }=V$, and the first Hubble slow-roll parameter isThe universe is therefore decelerating, despite the flatness of the potential: the chosen initial kinetic energy violates the slow-roll approximation.