 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-71/1/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

k < 0

in the metric convention. Evaluating the Friedmann equations today gives

H_{0}^{2} = H_{0}^{2} Ω_{0} - k / a_{0}^{2}

, so

- k = a_{0}^{2} H_{0}^{2} (1 - Ω_{0}), 0 < Ω_{0} < 1.

On a constant-time slice the radial line element is

d ℓ = a_{0} d r / 1 - k r^{2}

. Therefore

d_{S} = a_{0} \int_{0}^{r} \frac{d r ^{'}}{1 + ( - k ) r ^{'2}} = \frac{arsinh [ a _{0} H _{0} 1 - Ω _{0} r ]}{H _{0} 1 - Ω _{0}} .

This is a cosmological proper distance on today's slice, not the distance traveled during the photon flight. The printed condition

k < 1

is insufficient to specify an open model; the derivation uses its intended

k < 0

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