 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-72/1/ii/solution

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

Write

y = a / a_{0}

. Conservation of pressureless matter gives

ρ_{m} = ρ_{m, 0} y^{- 3}

. The Friedmann equation, retaining spatial curvature, is

H^{2} = H_{0}^{2} (Ω_{m, 0} y^{- 3} + Ω_{k, 0} y^{- 2}), Ω_{m, 0} + Ω_{k, 0} = 1.

Since

\overset{y}{˙} = yH

and

Ω_{m, 0} = 2 q_{0}

y^{2}

gives

(\frac{a ˙}{a _{0}})^{2} = \overset{y}{˙}^{2} = H_{0}^{2} (1 - 2 q_{0} + \frac{2 q _{0}}{y}) .

In the curvature convention

H^{2} = 8 π G ρ_{m} /3 - k c^{2} / a^{2}

, this also fixes

- k c^{2} / (a_{0}^{2} H_{0}^{2}) = 1 - 2 q_{0}

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