 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-3/14d/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Under the stated approximation the perturbation equation is

\ddot{δ} + 2 H \dot{δ} = 0, \dot{δ} \propto a^{- 2} .

a \propto t^{1/2}

, so

\dot{δ} \propto t^{- 1}

and

δ = C_{1} + C_{2} lo g t = C_{1} + 2 C_{2} lo g a .

a \propto t

, so

\dot{δ} \propto t^{- 2}

and

δ = C_{1} + C_{2} t^{- 1} = C_{1} + C_{2} a^{- 1} .

The nondecaying mode is constant: curvature domination gives freezing of matter perturbations during curvature domination, rather than continued growth.

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