Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 1 / a

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- Solution a

Solution

The cosmological continuity equation and

P_{i} = w_{i} ρ_{i}

give

\frac{d ρ _{i}}{d a} + \frac{3 ( 1 + w _{i} )}{a} ρ_{i} = 0, ρ_{i} (a) = ρ_{i, 0} a^{- 3 (1 + w_{i})} .

With

a_{0} = 1

,

1 + z = a^{- 1}

and

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

the Friedmann equation becomes

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

Spatial curvature may be included as an effective component with

w = - 1/3

and

Ω_{k, 0} = - k / H_{0}^{2}

.

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