 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-3/9b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Insert

a = (t / t_{0})^{γ}, ϕ = ϕ_{0} + α lo g t .

Matching powers of

t

requires

λ α = 2

, so

α = 2/ λ

\overline{V} = V_{0} e^{- λ ϕ_{0}}

λ \overline{V} = α (3 γ - 1), γ = \frac{16 π G}{λ ^{2}} .

Thus

\overline{V} = \frac{2 ( 3 γ - 1 )}{λ ^{2}}, ϕ_{0} = λ^{- 1} lo g (V_{0} / \overline{V}) .

Existence with positive potential requires

γ > 1/3

, or

λ^{2} < 48 π G

. Accelerated expansion requires

γ > 1

, hence

λ^{2} < 16 π G .

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