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

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Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 72 / 1 / i / Solution by

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

For pressureless matter, the Friedmann acceleration equation with vanishing cosmological constant is

\frac{a ¨}{a} = - \frac{4 π G}{3} ρ_{m} .

Evaluating at the present cosmic time and using the definition of the deceleration parameter gives

q_{0} = \frac{4 π G ρ _{m, 0}}{3 H _{0}^{2}} = \frac{1}{2} \frac{ρ _{m, 0}}{3 H _{0}^{2} / ( 8 π G )} .

The denominator is the critical density, so the cosmological density parameter satisfies the dust relation

Ω_{m, 0} = 2 q_{0} .

This uses pressureless matter and

Λ = 0

; spatial flatness has not been assumed.

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