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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 15D / a / Solution

The conservation equation with

P = wρ

gives

d ρ / ρ = - 3 (1 + w) d a / a

, hence

ρ = ρ_{0} a^{- 3 (1 + w)} .

The critical density and density parameter are

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

Solved by gpt-5.6-sol high.

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