Rotation of the cosmic microwave background polarization from weak
  gravitational lensing
  When a cosmic microwave background (CMB) photon travels from the surface of
last scatter through spacetime metric perturbations, the polarization vector
may rotate about its direction of propagation. This gravitational rotation is
distinct from, and occurs in addition to, the lensing deflection of the photon
trajectory. This rotation can be sourced by linear vector or tensor metric
perturbations and is fully coherent with the curl deflection field. Therefore,
lensing corrections to the CMB polarization power spectra as well as the
temperature-polarization cross-correlations due to non-scalar perturbations are
modified. The rotation does not affect lensing by linear scalar perturbations,
but needs to be included when calculations go to higher orders. We present
complete results for weak lensing of the full-sky CMB power spectra by general
linear metric perturbations, taking into account both deflection of the photon
trajectory and rotation of the polarization. For the case of lensing by
gravitational waves, we show that the B modes induced by the rotation largely
cancel those induced by the curl component of deflection.
