Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 67 / 3 / iii

At the same accuracy as the small-separation conversion, its coefficient is fixed at the cloud's well-determined epoch. Since $d_{cq}$ is proportional to $\delta z=z_q-z_c$ and $\Delta z_c$ is negligible,The given emission-redshift error corresponds near $z=3$ to $\Delta z_q\simeq (1+z)1000/c\simeq 0.0133$. HenceThe uncertainty is about $1.2h^{-1}\mathrm{Mpc}$ in proper distance. Using $1+z_q=4.1$ instead changes the percentage only to $13.7\%$. More precise propagation would also specify the reference epoch and its change in the Hubble parameter; these are higher-order details of the approximate quasar-cloud redshift separation.