Geometry of weak lensing of CMB polarization

Abstract

Hu [Phys. Rev. D62 (2000) 043007] has presented a harmonic-space method for calculating the effects of weak gravitational lensing on the cosmic microwave background (CMB) over the full sky. Computing the lensed power spectra to first order in the deflection power requires one to formulate the lensing displacement beyond the tangent-space approximation. We point out that for CMB polarization this displacement must undergo geometric corrections on the spherical sky to maintain statistical isotropy of the lensed fields. Although not discussed by Hu, these geometric effects are implicit in his analysis. However, there they are hidden by an overly-compact notation that is both unconventional and rather confusing. Here we aim to ameliorate this deficiency by providing a rigorous derivation of the lensed spherical power spectra.