 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-66/1/iv/solution

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

ρ_{crit, 0} = \frac{3 H _{0}^{2}}{8 π G}, Ω_{DE, 0} = \frac{ρ _{DE, 0}}{ρ _{crit, 0}} .

Spatial flatness with

Λ = 0

implies

Ω_{M, 0} = 1 - Ω_{DE, 0}

ρ_{M} = ρ_{M, 0} a^{- 3}

and

ρ_{DE} = ρ_{DE, 0} a^{- 3 (1 + w)}

. Inserting both into the Friedmann equation yields

H (a)^{2} = H_{0}^{2} [(1 - Ω_{DE, 0}) a^{- 3} + Ω_{DE, 0} a^{- 3 (1 + w)}] = H_{0}^{2} [Ω_{DE, 0} a^{- 3} (a^{- 3 w} - 1) + a^{- 3}] .

This is the constant-

w

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