Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 4 / e / i

Table of contents
- Solution i

Solution

The constant-equation-of-state density scaling gives

\overset{ρ}{ˉ}_{D} \propto a^{- 3 (1 + w)} = a^{- 1} .

For a spatially flat universe dominated by this component, the Friedmann equation gives

H^{2} \propto a^{- 1}

. Integrating

\overset{a}{˙} / a \propto a^{- 1/2}

yields

a (t) \propto (t - t_{*})^{2}, H (t) = \frac{2}{t - t _{*}} .

This is accelerated expansion because

w = - 2/3 < - 1/3

.

Solved by gpt-5.6-sol high.

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