Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 9E (Cosmology)

The potential slow-roll parameter isFrom the slow-roll approximation,ThereforeThus $ϵ\ll 1$ implies ${\dot{\varphi }}^2\ll V$, consistently with potential-energy domination.

The same equations giveOn the branch $V^{\prime }/V>0$, the definition of $ϵ$ turns this into

By definition,Using $dt=d\varphi /\dot{\varphi }$ and the result of part (a),Reversing the limits yields the slow-roll e-fold count

For the natural inflation potential, write $x=\varphi /f$. Thenand the half-angle identity givesTherefore, on the stated rolling branch,

Substitution into part (b) givesSincethe e-fold count for the natural-inflation cosine potential is