Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 1 / 9C / Solution

For comoving galaxies their relative comoving vector $\mathbf{\chi }$ is constant, and the physical displacement is $\mathbf{r}(t)=a(t)\mathbf{\chi }$, where $a$ is the scale factor. Differentiating gives the Hubble lawPeculiar velocity relative to this comoving frame is excluded in this relation.

Transport a wave crest between neighbouring comoving observers separated by a small physical distance $d\ell$. Their recession speed is $Hd\ell$, while the light travel time is $dt=d\ell /c$ to first order. The local Doppler shift therefore changes its wavelength by $d\lambda /\lambda =Hdt$. Multiplying these infinitesimal shifts along the ray, or integrating their logarithms, yieldsThe use of the local first-order Doppler law is exact in the limit of infinitesimal steps: the accumulated second-order errors tend to zero. A single special-relativistic recession velocity assigned to widely separated galaxies would not justify this cosmological redshift relation.