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

Write $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{p}}$ and use the Legendre polynomial recurrence relation. The definition in the question is inverted byFor Newtonian-gauge potentials the scanned geodesic equation is the standard identityIts monopole and dipole are $S_0={\Phi }^{\prime }$ and $S_1=k\Psi /3$. Projecting the photon Boltzmann hierarchy onto $P_0$ and $P_1$ therefore givesThe first is the photon continuity equation; the second is the photon Euler equation, with the photon quadrupole providing the anisotropic-stress term. Indeed, with photon density contrast ${\delta }_{\gamma }=4{\Theta }_0$ and velocity divergence ${\theta }_{\gamma }=3k{\Theta }_1$,