 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-71/3/b/solution

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

Write

Ω = Ω_{M 0}

β^{2} = H_{0}^{2} (1 - Ω)

and

H = a^{'} / a = \overset{a}{˙}

. The Friedmann equation is

H^{2} = H_{0}^{2} (\frac{Ω}{a} + 1 - Ω) .

The Raychaudhuri equation has only the matter component on its active-gravity side, so

H^{'} = a \overset{a}{¨} = - \frac{4 π G}{3} a^{2} ρ_{M} = - \frac{H _{0}^{2} Ω}{2 a} .

Combining these expressions proves

2 H^{'} + H^{2} - β^{2} = 0 .

H_{0}

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