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

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.

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 ${\theta }_{\gamma }=3k{\Theta }_1$,

The line-of-sight solution for free-streaming photons follows the unperturbed ray $\mathbf{x}(\eta )={\mathbf{x}}_0-\hat{\mathbf{p}}({\eta }_f-\eta )$. If $\Phi =\Psi$ is spatially homogeneous, its gradient vanishes and $d\log ϵ/d\eta ={\Phi }^{\prime }$. Integration along the ray givesThe needed initial condition is the temperature pattern on the initial hypersurface; for a superhorizon adiabatic mode during matter domination one may use ${\Theta }_0({\eta }_i)=-2\Psi ({\eta }_i)/3$.

This calculation contains free streaming of the initial pattern and the gravitational temperature shift caused by the evolving potential, the homogeneous limit of the Integrated Sachs-Wolfe effect. A perfectly homogeneous potential changes only the unobservable sky monopole and therefore contributes no CMB angular power spectrum for $\ell \ge 1$. Direction-dependent Sachs-Wolfe effect, Doppler, acoustic, polarization, lensing, and rescattering contributions require spatially varying perturbations or collision terms and are outside this special calculation.