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

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- Solution c

Solution

The homogeneous inflaton obeys

\ddot{ϕ} + 3 H \dot{ϕ} + V^{'} (ϕ) = 0.

Under the slow-roll approximation, the acceleration and kinetic-energy corrections may be neglected, so the field equation and Friedmann equation become

3 H \dot{ϕ} ≃ - V^{'} = m^{2} ϕ, H^{2} ≃ \frac{V}{3 M _{Pl}^{2}} ≃ \frac{Λ ^{4}}{3 M _{Pl}^{2}} .

Thus a positive field rolls away from the hilltop while

H

is approximately constant.

Solved by gpt-5.6-sol high.

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