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

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- Solution c

Solution

The age of an FLRW universe follows from

d t = d a / (a H)

. Neglecting radiation and spatial curvature gives

t_{0} = \frac{1}{H _{0}} \int_{0}^{1} \frac{d a}{a Ω _{m, 0} a ^{- 3} + Ω _{Λ, 0}} = \frac{1}{H _{0}} \int_{0}^{1} \frac{a ^{1/2} d a}{Ω _{m, 0} + Ω _{Λ, 0} a ^{3}} .

For an Einstein-de Sitter universe,

Ω_{m, 0} = 1

and

Ω_{Λ, 0} = 0

, so

t_{0} = H_{0}^{- 1} \int_{0}^{1} a^{1/2} d a = \frac{2}{3} H_{0}^{- 1} .

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