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

The official notes, PDF p. 11, handwritten p. 4 give the Lagrangian for the geodesic equations.

The official notes, PDF pp. 11–13, handwritten pp. 4–6 derive all four geodesic equations and list the twelve nonzero Christoffel symbols up to symmetry of the lower indices, using time coordinate $t$.

The official notes, PDF pp. 13–14, handwritten pp. 6–7 integrate the radial geodesic equations and establish the vanishing integration constant for a null geodesic.

The 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 $R(t_e)$.