Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 4 / iv / Solution

The integrating factor for the Fourier-space transport equation is $e^{ik\mu \eta }$. Therefore the line-of-sight solution for free-streaming photons between an initial time ${\eta }_i$ and observation at ${\eta }_0$ isThe first term freely streams the initial angular distribution; the integral accumulates the lapse and shift source along the unperturbed photon path.