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

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.