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

Write $P^i=(ϵ/a^2){\hat{p}}^i$ and setwith $A$ first order. The photon on-shell condition isUsing ${\delta }_{ij}{\hat{p}}^i{\hat{p}}^j=1$ and retaining linear terms gives $A=-\delta N+{\hat{p}}^i{\partial }_i\psi$. Hence

Divide the time component of the geodesic equation by $(P^0{)}^2$ and use $d/d\lambda =P^{0}d/d\eta$. To first order, $P^i/P^0={\hat{p}}^i(1+\delta N-\hat{\mathbf{p}}\cdot \nabla \psi )$. Substitution of the supplied Christoffel symbols makes the background $2\mathcal{H}$ terms cancel the derivative of $a^{-2}$. The derivatives of the lapse and shift also combine, leavingThus the comoving photon energy is conserved in the unperturbed spacetime and changes through the gradient of the scalar gravitational source.

The collisionless equation states that $f$ is constant along a photon geodesic:At linear order $d{x}^i/d\eta ={\hat{p}}^i$. The last term is second order because the background distribution is isotropic. Inserting the stated temperature parametrization into this Free-streaming photon Boltzmann equation and using the energy equation givesFor a Fourier mode, with $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{p}}$, this is

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.