 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-1/9d/a/solution

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

The slow-roll equations imply

\frac{1}{a} \frac{d a}{d ϕ} = \frac{H}{ϕ ˙} = - \frac{3 H ^{2}}{c ^{2} V ^{'} ( ϕ )} = - \frac{8 π G}{c ^{4}} \frac{V}{V ^{'}} .

For

V = λ ϕ^{n} / n

V / V^{'} = ϕ / n

. Integrating from

ϕ_{i}

and using

a (ϕ_{i}) = 1

gives

a (ϕ) = exp [\frac{4 π G}{c ^{4} n} (ϕ_{i}^{2} - ϕ^{2})] .

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