 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/9e/b/solution

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

The physical horizon size at conformal time

τ

a (τ) c τ

. Horizon crossing occurs when it equals the physical wavelength:

\frac{2 πa ( τ _{H} )}{k} = a (τ_{H}) c τ_{H},

and hence

τ_{H} = \frac{2 π}{k c} .

At late times the decaying mode is neglected, so the growing solution in part (a) obeys

δ \propto τ^{2} \propto a

. For the equality-crossing mode this gives

D_{eq} = \frac{δ ( k _{eq} , τ _{0} )}{δ ( k _{eq} , τ _{eq} )} = (\frac{τ _{0}}{τ _{eq}})^{2} = \frac{a ( τ _{0} )}{a ( τ _{eq} )} .

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