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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-70/4/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 70 / 4 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

Each separately conserved component obeys the constant-equation-of-state density scaling

ρ \propto a^{- 3 (1 + w)}

. For the flat Friedmann-Lemaître-Robertson-Walker metric, the Friedmann equations therefore give

\frac{H ( z ) ^{2}}{H _{0}^{2}} = Ω_{m} (1 + z)^{3} + Ω_{r} (1 + z)^{4} + Ω_{X} (1 + z)^{3 (1 + w)} .

Write

H (z) / H_{0} = 1 + h_{1} z + O (z^{2})

, where

h_{1} = [3 Ω_{m} + 4 Ω_{r} + 3 (1 + w) Ω_{X}] /2

. Expanding the reciprocal inside the luminosity distance integral gives

H_{0} d_{L} = (1 + z) [z - \frac{h _{1}}{2} z^{2} + O (z^{3})] = z + (1 - \frac{h _{1}}{2}) z^{2} + O (z^{3}) .

Matching its quadratic coefficient gives the deceleration parameter

q_{0} = h_{1} - 1

. Using flatness in this part,

q_{0} = \frac{Ω _{m}}{2} + Ω_{r} + \frac{1 + 3 w}{2} Ω_{X} = \frac{1}{2} (1 + Ω_{r} + 3 w Ω_{X}) .

A negative deceleration parameter signifies accelerated expansion; negative pressure must overcome the positive matter and radiation terms. Restoring units replaces

H_{0} d_{L}

by

H_{0} d_{L} / c

in the expansion, leaving

q_{0}

unchanged.

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