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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 15A / d / Solution

Neglecting the pressure and self-gravity bracket leaves

\ddot{δ} + 2 H \dot{δ} = 0.

During radiation domination,

a (t) \propto t^{1/2}

and

H = 1/ (2 t)

, so

\ddot{δ} + \frac{1}{t} \dot{δ} = 0.

Consequently

δ (t) = A + B lo g t = A + 2 B lo g a .

Matter perturbations therefore grow only logarithmically while radiation dominates.

Solved by gpt-5.6-sol high.

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