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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-310/2/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 310 / 2 / b / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

Normalize the scale factor so that

a_{0} = 1

. With separately conserved pressureless matter and radiation in cosmology, their energy densities scale as

a^{- 3}

and

a^{- 4}

. Hence the flat Friedmann equation is

H^{2} = H_{0}^{2} (Ω_{m, 0} a^{- 3} + Ω_{r, 0} a^{- 4}), H^{2} = H_{0}^{2} (Ω_{m, 0} a^{- 1} + Ω_{r, 0} a^{- 2}) .

Equality of the two densities defines matter-radiation equality, with

a_{eq} = Ω_{r, 0} / Ω_{m, 0}

. Define the dimensionless variable

y = \frac{a}{a _{eq}} = a \frac{Ω _{m, 0}}{Ω _{r, 0}} = \frac{ρ ˉ _{m}}{ρ ˉ _{r}} .

Substituting

a = a_{eq} y

gives

H^{2} = C (\frac{1}{y} + \frac{1}{y ^{2}}), C = \frac{H _{0}^{2} Ω _{m, 0}^{2}}{Ω _{r, 0}} .

Both reference density fractions are positive. In the two-component flat model their sum at the reference epoch is one; no dark-energy component is being included in this background calculation.

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