 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-310/3/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

a \propto t^{2/3}

H = 2/ (3 t)

, and

4 π G \overset{ρ}{ˉ}_{m} = 2/ (3 t^{2})

. Substituting

Δ_{m} \propto t^{s}

s (s - 1) + \frac{4}{3} s - \frac{2}{3} = 0,

whose roots are

s = 2/3

and

s = - 1

. Therefore

Δ_{m} = C_{+} t^{2/3} + C_{-} t^{- 1} = C_{+} a + C_{-} a^{- 3/2} .

Δ_{m} \propto a .

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