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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 9E / a / Solution

Successive radial wave crests travel the same fixed comoving distance, so

\int_{t_{e}}^{t_{0}} \frac{d t}{a ( t )} = \int_{t_{e} + δ t_{e}}^{t_{0} + δ t_{0}} \frac{d t}{a ( t )} .

To first order,

δ t_{0} / a (t_{0}) = δ t_{e} / a (t_{e})

. Since wavelength is proportional to the period,

1 + z = \frac{λ _{0}}{λ _{e}} = \frac{δ t _{0}}{δ t _{e}} = \frac{a ( t _{0} )}{a ( t _{e} )} .

Solved by gpt-5.6-sol high.

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