Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 9D / a

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- Solution a

Solution

The Friedmann equation immediately gives

Ω - 1 = \frac{k c ^{2}}{H ^{2}} .

Using

P = wρ

in the acceleration equation and eliminating

ρ a^{2}

with Friedmann gives

2 H^{'} + (1 + 3 w) (H^{2} + k c^{2}) = 0.

Differentiate

Ω - 1 = k c^{2} H^{- 2}

and substitute this equation:

Ω^{'} = - 2 (Ω - 1) \frac{H ^{'}}{H} = (1 + 3 w) H Ω (Ω - 1),

where

H^{2} + k c^{2} = Ω H^{2}

was used in the last step.

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