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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 14B / a / Solution

During matter domination,

a (t) \propto t^{2/3}, \frac{a ˙}{a} = \frac{2}{3 t}, \frac{4 π G ρ ˉ _{c}}{c ^{2}} = \frac{2}{3 t ^{2}} .

The perturbation equation becomes

\ddot{δ}_{k} + \frac{4}{3 t} \dot{δ}_{k} - \frac{2}{3 t ^{2}} δ_{k} = 0.

Substitution of the power law

δ_{k} \propto t^{α}

gives

3 α^{2} + α - 2 = 0,

whose roots are

α = 2/3

and

α = - 1

. Hence the cosmic-time matter density modes give

δ_{k} (t) = A (k) (\frac{t}{t _{eq}})^{2/3} + B (k) (\frac{t}{t _{eq}})^{- 1} .

Solved by gpt-5.6-sol high.

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