The homogeneous <inflaton> obeys
$$
\ddot\phi+3H\dot\phi+V'(\phi)=0.
$$
Under the <slow-roll approximation>, the acceleration and kinetic-energy corrections may be neglected, so the field equation and <Friedmann equation> become
$$
3H\dot\phi\simeq-V'=m^2\phi,
\qquad
H^2\simeq\frac{V}{3M_{\rm Pl}^2}
\simeq\frac{\Lambda^4}{3M_{\rm Pl}^2}.
$$
Thus a positive field rolls away from the hilltop while $H$ is approximately constant.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-310/2/c/solution