For the <hilltop inflation> potential,
$$
V'=-m^2\phi,
\qquad
V''=-m^2.
$$
The two <potential slow-roll parameters> are consequently
$$
\epsilon_V
=\frac{M_{\rm Pl}^2m^4\phi^2}
{2(\Lambda^4-\tfrac12m^2\phi^2)^2}
\simeq\frac{M_{\rm Pl}^2m^4\phi^2}{2\Lambda^8}
$$
and
$$
\eta_V
=-\frac{M_{\rm Pl}^2m^2}{\Lambda^4-\tfrac12m^2\phi^2}
\simeq-\frac{M_{\rm Pl}^2m^2}{\Lambda^4}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 2 / a / Solution

{wiki}

Hilltop inflation begins near a local maximum of the inflaton potential. For
$$
V(\phi)=\Lambda^4-\frac12m^2\phi^2,
$$
a positive field rolls away from the maximum at $\phi=0$.


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 Incoming links: Hilltop inflation