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

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- Solution b

Solution

The Friedmann acceleration equation and

H = \overset{a}{˙} / a

give the deceleration parameter

q = - \frac{a ¨}{a H ^{2}} = \frac{4 π G}{3 H ^{2}} \sum_{i} ρ_{i} (1 + 3 w_{i}) = \frac{1}{2} i \sum Ω_{i} (z) (1 + 3 w_{i}) .

An Einstein-de Sitter universe contains only pressureless matter, so

q_{0} = 1/2

and cannot explain the observed negative value. For the stated Lambda-CDM model,

q_{0} = \frac{1}{2} [0.3 + 0.7 (1 - 3)] = - 0.55,

so its cosmological constant produces the required accelerating universe.

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