The https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS_NOTES_GR/gr-cosmology-exam-paper-2011-solutions.pdf#page=14[official notes, PDF pp. 14–16, handwritten pp. 7–9] integrate the exponential <scale factor>, discuss the sign required for an incoming <photon>, and obtain the distance–<redshift> relation. They use $a(t_e)$ for the paper's otherwise undefined $R(t_e)$.

Solution

Past exam of the general relativity and cosmology and condensed matter physics paper of Part B of the physics course of the University of Oxford / 2011 / Section V / 2 / Unlettered 4 / Solution

 Solution (source code)