 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-67/1/iv/solution

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

Ω_{m, 0} a_{0}^{3} / a_{m i n} = 2 Ω_{Λ, 0} a_{m i n}^{2}

. Consequently

\overset{a}{˙}_{m i n}^{2} = H_{0}^{2} (3 Ω_{Λ, 0} a_{m i n}^{2} + Ω_{k, 0} a_{0}^{2}) .

Taking the positive root on the expanding branch and eliminating

a_{0}

with the result of part (iii),

\overset{a}{˙}_{m i n} = H_{0} a_{m i n} [3 Ω_{Λ, 0} + Ω_{k, 0} (\frac{2 Ω _{Λ, 0}}{Ω _{m, 0}})^{2/3}]^{1/2} .

The bracket must be positive for a genuine passage through this minimum expansion speed in a matter-Lambda universe. Negative values mean that this formal value of the scale factor is not accessible on the corresponding physical expanding solution.

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