 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-4/10d/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

H = \overset{a}{˙} / a = 2/ (3 t)

H^{2} = 8 π G ρ_{m} /3

gives

ρ_{m} = 1/ (6 π G t^{2})

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

Substituting

t^{s}

yields

s (s - 1) + 4 s /3 - 2/3 = 0

, or

(3 s - 2) (s + 1) = 0

. Hence

δ_{k} = A_{k} t^{2/3} + B_{k} t^{- 1},

with growing mode proportional to the scale factor and decaying mode proportional to

t^{- 1}

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