Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 4 / e

The continuity equation and the cosine solution givewhen the potentials are constant. Part (d) then impliesSince Cosmic microwave background polarization is sourced by the local quadrupole at last scattering,The monopole contribution to the CMB temperature oscillates as $\cos (k{r}_{s\ast })$, so its acoustic extrema occur near $k{r}_{s\ast }=m\pi$, whereas polarization extrema occur near $k{r}_{s\ast }=(m+\frac{1}{2})\pi$. The principal polarization peaks are therefore interleaved with the principal temperature peaks. Projection, baryon loading, time-varying potentials, and the temperature Doppler contribution shift the observed angular peaks from this idealized phase relation.

Solved by gpt-5.6-sol high.