 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/15e/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

With

P = - 2 ρ /3

\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.

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