CMB lensing beyond the leading order: Temperature and polarization anisotropies

We investigate the weak lensing corrections to the cosmic microwave background temperature and polarization anisotropies. We consider all the effects beyond the leading order: post-Born corrections, LSS corrections, and, for the polarization anisotropies, the correction due to the rotation of the polarization direction between the emission at the source and the detection at the observer. We show that the full next-to-leading order correction to the B-mode polarization is not negligible on small scales and is dominated by the contribution from the rotation; this is a new effect not taken into account in previous works. Considering vanishing primordial gravitational waves, the B-mode correction due to rotation is comparable to cosmic variance for 3500, in contrast to all other spectra where the corrections are always below that threshold for a single multipole. Moreover, the sum of all the effects is larger than cosmic variance at high multipoles, showing that higher-order lensing corrections to B-mode polarization are in principle detectable.