E-mode polarization ($P_E$)

The parity-even part of linear sky polarization can be represented by a scalar potential $P_E$ through $Q\pm iU={ð}_{\pm }^2(P_E\pm i{P}_B)$. It is naturally separated from B-mode polarization by the parity of its spherical multipoles. Linear scalar cosmological perturbations generate E modes; the potential has an unobservable low-multipole kernel.

For a scalar photon quadrupole emitted at distance ${\chi }_{\ast }$, the meridian-basis source is $Q\pm iU\propto {\Theta }_2{e}^{-ik{\chi }_{\ast }\mu }(1-{\mu }^2)$. Integrating its scalar-potential equation and removing the low-multipole kernel of polarization potentials yields $P_E\propto {\sum }_{\ell \ge 2}(-i{)}^{\ell }(2\ell +1){\Theta }_2{j}_{\ell }(k{\chi }_{\ast })P_{\ell }(\mu )/(k{\chi }_{\ast }{)}^2$, 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.

Twice-applied spin-raising or lowering annihilates the ordinary spherical-harmonic multipoles $\ell =0,1$. Polarization potentials therefore have a four-dimensional monopole/dipole ambiguity on the sphere. For an axisymmetric mode it is $A+B\cos \theta$. Setting these components to zero defines the observable $\ell \ge 2$ potentials without changing $Q$ or $U$.