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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 38D / a / Solution

For

k = - 1

, the Friedmann-Lemaitre-Robertson-Walker metric is

d s^{2} = - d t^{2} + a^{2} (t) [d χ^{2} + sinh^{2} χ (d θ^{2} + sin^{2} θ d ϕ^{2})] .

The empty first Friedmann equation, with

Λ = 0

, gives

\frac{a ˙ ^{2} - 1}{a ^{2}} = 0.

Choosing the expanding branch and shifting the origin of proper time yields

a (t) = t, t > 0 .

Solved by gpt-5.6-sol high.

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