= Solution
The continuity equation and the cosine solution give
$$
\Theta_1=-\frac{\Theta_0'}k
\propto c_s\sin(kr_s)
$$
when the potentials are constant. Part (d) then implies
$$
\Theta_2\propto\frac{k}{\Gamma}\sin(kr_s).
$$
Since <Cosmic microwave background polarization> is sourced by the local quadrupole at last scattering,
$$
\boxed{\mathcal P_*(k)\propto\sin(kr_{s*})}.
$$
The monopole contribution to the CMB temperature oscillates as $\cos(kr_{s*})$, so its acoustic extrema occur near $kr_{s*}=m\pi$, whereas polarization extrema occur near $kr_{s*}=(m+\tfrac12)\pi$. 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.
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