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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 15E / b / i / Solution

With

P = - 2 ρ /3

, the continuity equation becomes

\overset{ρ}{˙} = - \frac{a ˙}{a} ρ,

hence

ρ a

is constant. Since

a (t_{0}) = 1

, one has

ρ = ρ_{0} / a

. The Friedmann equation is therefore

(\frac{a ˙}{a})^{2} = \frac{γ}{a} + \frac{β}{a ^{2}},

where

γ = \frac{8 π G ρ _{0}}{3 c ^{2}}, β = - k c^{2} > 0.

Solved by gpt-5.6-sol high.

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