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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.

For a radial null geodesic in a spatially flat universe, the comoving radial distance increment is $d\chi =cdz/H(z)$. If the cloud and quasar redshifts differ only slightly, take $H$ at their common approximate epoch $z_{\ast }$, givingMultiplying by the scale factor at that epoch gives the local proper distance in cosmology relevant to the quasar's radiation flux:If distance is instead expressed in comoving units, it is $(1+z_{\ast })d_{cq}$. Equivalently the small rest-frame velocity difference is $\delta v=c\delta z/(1+z_{\ast })$, and $d_{cq}\simeq \delta v/H(z_{\ast })$. This quasar-cloud redshift separation assumes the redshift difference is due to homogeneous expansion, with negligible peculiar velocities. Since the two observations lie on a light cone, the proper separation is understood as the short separation on a nearly common cosmic-time slice; specifying the epoch matters at higher order.