Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 53 / 1 / ii / b / Solution

For $t>0$, $\dot{a}=n{t}^{n-1}>0$ and $\ddot{a}=n(n-1)t^{n-2}$. Consequently inflation occurs exactly when $\boxed{n>1}$. The case $n=1$ is coasting, and $0<n<1$ is decelerating. The Hubble slow-roll parameter is ${ϵ}_H=-\dot{H}/H^2=1/n$, giving the same acceleration criterion ${ϵ}_H<1$.