Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 73 / 1 / iv / Solution

In the intended comparison of emission and reception epochs, with negligible peculiar velocities, the Earth-observed cosmological redshifts correspond to $a_A=a_0/2$ and $a_B=a_0/10$. A photon emitted at the B epoch and received at the A epoch consequently hasThese factors divide, whereas the redshifts themselves do not subtract. There is a geometrical qualification to this usual answer: the two Earth-observed redshifts alone do not specify an arbitrary actual B-to-A observation. The assumed events must be connected by a null geodesic, as when A is on the same radial photon path between B and Earth. If the galaxies have an unspecified angular separation, or A observes at a different cosmic time, the B emission event reaching A is different and its redshift is not determined by these two numbers alone.