 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-3/14a/b/solution

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

\overset{ρ}{˙} = \dot{ϕ} \ddot{ϕ} + V^{'} (ϕ) \dot{ϕ}, ρ + P = \dot{ϕ}^{2} .

The continuity equation therefore becomes

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

Away from an isolated turning point division by

\dot{ϕ}

gives, and continuity extends through such points,

\ddot{ϕ} + 3 H \dot{ϕ} + \frac{d V}{d ϕ} = 0 .

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