For successive wavecrests from the same comoving source, $1+z=a(t_0)/a(t_s)$. Therefore
$$
\Delta z=\frac{a(t_0+\Delta t_0)}{a(t_s+\Delta t_s)}
-\frac{a(t_0)}{a(t_s)}.
$$
The same radial null path implies equality of the elapsed <conformal time>,
$$
\frac{\Delta t_s}{a(t_s)}=\frac{\Delta t_0}{a(t_0)},
\qquad
\Delta t_s=\frac{\Delta t_0}{1+z_1}.
$$
Expanding both scale factors to first order gives the <redshift drift>
$$
\boxed{\Delta z
=\left[(1+z_1)H_0-H(z_1)\right]\Delta t_0
=H_0\Delta t_0\left[1+z_1-E(z_1)\right]}.
$$

Solution

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

{title2=$\dot z$}
{wiki}

Redshift drift is the change in a comoving source's cosmological redshift with observer time. It directly probes $\dot z=(1+z)H_0-H(z)$.


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 Incoming links: Redshift drift