= Solution
<Thomson scattering> generates linear <Cosmic microwave background polarization> when the incident radiation has a <photon quadrupole>. The cross-section prefers electric fields perpendicular to the scattering direction. Contributions from an isotropic field cancel, and a dipole alone does not supply the required spin-two angular pattern. Schematically the outgoing Stokes combination is proportional to
$$
Q\pm iU\ \propto\ \int d\Omega'\,\sin^2\vartheta\,e^{\pm2i\varphi}\Theta(\widehat n'),
$$
which selects the incident quadrupolar angular dependence. Tight coupling nearly isotropizes the radiation, making the quadrupole small; it develops when the photon mean free path becomes finite near decoupling. At leading tight-coupling order its size scales as $(k/\dot\kappa)v_\gamma$.
The polarization is therefore weaker than the temperature anisotropy: a characteristic amplitude is of order \b[a few percent to roughly ten percent of the temperature signal], or several microkelvin compared with temperature fluctuations of several tens of microkelvin on acoustic scales. This is an order-of-magnitude comparison, not a scale-independent ratio of every power-spectrum multipole.
The <photon monopole> follows the acoustic displacement, approximately $\cos(kr_s)$; the velocity sourcing the quadrupole follows $\sin(kr_s)$. Hence polarization maxima occur near displacement zeros, between the main temperature peaks. In the leading coherent-oscillator approximation the shift is half the peak spacing, $\Delta\ell/2$, although projection and gravitational driving modify exact locations. Scalar perturbations generate <E-mode polarization> at linear order, not <B-mode polarization>. Tensor perturbations and lensing can give B modes, and <reionization> generates an additional large-angle polarization feature.
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