 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-1/9b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Let

H = a^{'} / a = \overset{a}{˙}

, where a prime denotes differentiation with respect to conformal time

d τ = d t / a

. Multiplying the Friedmann equation by

a^{2}

gives

H^{2} + k c^{2} = \frac{8 π G}{3} ρ a^{2} .

The critical density is the value obtained when

k = 0

, so

Ω = \frac{ρ}{ρ _{crit}} = \frac{8 π Gρ a ^{2}}{3 H ^{2}}, \frac{k c ^{2}}{H ^{2}} = Ω - 1 .

Since

H^{'} = a \overset{a}{¨}

2 H^{'} = - \frac{8 π G}{3} ρ a^{2} = - (H^{2} + k c^{2}),

2 H^{'} + H^{2} + k c^{2} = 0.

Differentiating

Ω = 1 + k c^{2} / H^{2}

and substituting this equation yields

Ω^{'} = H Ω (Ω - 1) .

For an expanding universe

H > 0

Ω = 1

is an unstable constant solution: curves starting with

0 < Ω < 1

decrease towards

0

, while curves starting with

Ω > 1

rise away from

1

and diverge as the closed model reaches maximum expansion. This is the Flatness problem.

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