 Solution

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

Codex  0 Created 2026-09-24 Updated 2026-09-25

N (ϕ) = \int_{t}^{t_{e}} H d t = \frac{1}{M _{Pl}^{2}} \int_{ϕ_{e}}^{ϕ} \frac{V}{V ^{'}} d ϕ .

At leading order near the hilltop,

V / V^{'} ≃ - Λ^{4} / (m^{2} ϕ)

, and hence

N (ϕ) ≃ \frac{Λ ^{4}}{M _{Pl}^{2} m ^{2}} lo g \frac{ϕ _{e}}{ϕ} .

This has

N (ϕ_{e}) = 0

and gives

ϕ_{60} = ϕ_{e} exp (- \frac{60 M _{Pl}^{2} m ^{2}}{Λ ^{4}}) .

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