= Solution
Normalize the <scale factor> to $a(t)=R(t)/R(t_0)$, with $a(t_0)=1$. Along a radial <null geodesic>, $d\chi=c\,dt/a(t)$. Two successive wave crests sent by the same comoving emitter and received by the same comoving observer traverse the same <comoving distance>, so
$$
\int_{t_e}^{t_0}\frac{c\,dt}{a(t)}
=\int_{t_e+\delta t_e}^{t_0+\delta t_0}\frac{c\,dt}{a(t)}.
$$
To first order, $\delta t_0/a(t_0)=\delta t_e/a(t_e)$. Frequencies are inverse proper periods for the <comoving observers>, giving the <cosmological redshift>
$$
\boxed{1+z=\frac{\lambda_0}{\lambda_e}
=\frac{R(t_0)}{R(t_e)}=\frac1{a(t_e)}.}
$$
Expansion stretches wavelengths, so a spectral line of known rest wavelength gives $z=\lambda_{\rm obs}/\lambda_{\rm rest}-1$.
At small $z$, the light travel time is short compared with the expansion time. Expanding $R(t_e)$ gives $z\simeq H_0(t_0-t_e)$ and $d\simeq c(t_0-t_e)$, hence
$$
\boxed{cz\simeq H_0d,\qquad H_0=\left.\frac{\dot R}{R}\right|_{t_0}.}
$$
Combine independently calibrated <luminosity distances> with galaxy spectral redshifts and fit the slope. Nearby peculiar velocities must be allowed for, while high-redshift data require the full distance-redshift relation rather than the linear approximation.
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