Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 310 / 1 / a

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Solution

For a component with constant barotropic equation of state

P = wρ

, the cosmological perfect-fluid continuity equation gives

\frac{d ρ}{ρ} = - 3 (1 + w) \frac{d a}{a} .

Hence

ρ_{i} = ρ_{i, 0} a^{- 3 (1 + w_{i})}

. Choose the present scale factor to be

a_{0} = 1

, so the cosmological redshift obeys

1 + z = a^{- 1}

, and define the present cosmological density parameter by

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

Substitution in the spatially flat Friedmann equation then yields

H (z) = H_{0} [i \sum Ω_{i, 0} (1 + z)^{3 (1 + w_{i})}]^{1/2} .

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