Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 312 / 3 / i / Solution

Let the photon phase-space distribution be a Planck distribution whose local temperature is $T(\eta )[1+\Theta (\eta ,\mathbf{x},\hat{\mathbf{p}})]$. In the absence of collisions, Liouville theorem says that the distribution is constant along a photon null geodesic. Linearizing $df/d\eta =0$ about the homogeneous distribution and using $d\mathbf{x}/d\eta =\hat{\mathbf{p}}$ givesA spatial Fourier transform sends $\hat{\mathbf{p}}\cdot \nabla$ to $i\hat{\mathbf{p}}\cdot \mathbf{k}$, soThis is the Free-streaming photon Boltzmann equation: the second term transports angular structure, while the last term is the gravitational redshift source.