Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 1 / c / Solution

For successive wavecrests from the same comoving source,

1 + z = a (t_{0}) / a (t_{s})

. Therefore

Δ z = \frac{a ( t _{0} + Δ t _{0} )}{a ( t _{s} + Δ t _{s} )} - \frac{a ( t _{0} )}{a ( t _{s} )} .

The same radial null path implies equality of the elapsed conformal time,

\frac{Δ t _{s}}{a ( t _{s} )} = \frac{Δ t _{0}}{a ( t _{0} )}, Δ t_{s} = \frac{Δ t _{0}}{1 + z _{1}} .

Expanding both scale factors to first order gives the redshift drift

Δ z = [(1 + z_{1}) H_{0} - H (z_{1})] Δ t_{0} = H_{0} Δ t_{0} [1 + z_{1} - E (z_{1})] .

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