 Solution

Normalize the scale factor to $a(t)=R(t)/R(t_0)$, with $a(t_0)=1$. Along a radial null geodesic, $d\chi =cdt/a(t)$. Two successive wave crests sent by the same comoving emitter and received by the same comoving observer traverse the same comoving distance, soTo 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 redshiftExpansion stretches wavelengths, so a spectral line of known rest wavelength gives $z={\lambda }_{\mathrm{obs}}/{\lambda }_{\mathrm{rest}}-1$.

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 $d\simeq c(t_0-t_e)$, henceCombine 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.