Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 1 / d

Table of contents
- Solution d

Solution

The small-

θ

expansions are

a = \frac{A}{2} θ^{2} (1 - \frac{θ ^{2}}{12} + \dots), t = \frac{B}{6} θ^{3} (1 - \frac{θ ^{2}}{20} + \dots) .

x = (6 t / B)^{1/3}

and reverting the second series gives

θ = x (1 + x^{2} /60 + \dots)

. Therefore

\frac{a}{t ^{2/3}} = \frac{A}{2} (\frac{6}{B})^{2/3} [1 - \frac{1}{20} (\frac{6 t}{B})^{2/3} + \dots],

so

C = (A /2) (6/ B)^{2/3}

. The leading

a \propto t^{2/3}

is the flat matter-dominated limit.

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