= Scalar E-mode radial projection
{title2=$P_{E,\ell}\propto\Theta_2j_\ell(k\chi_*)/(k\chi_*)^2$}
For a scalar <photon quadrupole> emitted at distance $\chi_*$, the meridian-basis source is $Q\pm iU\propto\Theta_2e^{-ik\chi_*\mu}(1-\mu^2)$. Integrating its scalar-potential equation and removing the <low-multipole kernel of polarization potentials> yields $P_E\propto\sum_{\ell\ge2}(-i)^\ell(2\ell+1)\Theta_2j_\ell(k\chi_*)P_\ell(\mu)/(k\chi_*)^2$, with no scalar $P_B$. The apparent $k^{-2}$ divergence lies in discarded low multipoles; $j_\ell(x)=O(x^\ell)$ makes the physical small-$x$ limit regular.
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