Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 312 / 4

In the absence of scalar anisotropic stress, the traceless spatial Einstein field equations giveThe photon mass-shell condition $g_{\mu \nu }{P}^{\mu }{P}^{\nu }=0$, together with $p^2=g_{ij}{P}^i{P}^j$ and $ϵ=ap$, gives to first orderUse the time component of the geodesic equation, divide it by $d\eta /d\lambda =P^0$, and substitute the stated Christoffel symbols. Keeping the perturbed ratios $P^i/P^0=(1+\Phi +\Psi ){\hat{p}}^i$ where they multiply the background $\mathcal{H}$ terms makes those terms cancel. Comparing the result with the total derivative of $P^0=ϵa^{-2}(1-\Psi )$ yieldsThe first term is the time-dependent gravitational redshift and the second is the gravitational frequency shift along the photon direction.

Write the total derivative of the phase-space distribution function asSince $a\bar{T}$ is constant for the background photon gas, expand the Bose-Einstein distribution asAt first order the four terms are respectivelyThe angular-deflection velocity is already first order and multiplies the first-order angular dependence of $f$, so its contribution is second order. Thus the collisionless left-hand side of the Free-streaming photon Boltzmann equation is

The conventional vertical variable is $D_{\ell }=\ell (\ell +1)C_{\ell }/(2\pi )$ in $\mu {\mathrm{K}}^2$, plotted against the dimensionless angular multipole $\ell$. The Sachs-Wolfe effect supplies a nearly flat large-angle plateau for $\ell \lesssim 30$. The total spectrum has its first acoustic peak near $\ell \simeq 220$ with $D_{\ell }$ of order $5\times 10^3\mu {\mathrm{K}}^2$, followed by further Cosmic microwave background acoustic peaks; the Doppler CMB anisotropy is phase shifted relative to the photon-density oscillation. At $\ell \gtrsim 1000$, Cosmic microwave background diffusion damping lets photons random-walk across perturbations during recombination and produces the rapidly falling damping tail.