 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/9e/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

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} )} .

 New to topics? Read the docs here!