Emission-angle and polarization-rotation effects in the lensed CMB
  Lensing of the CMB is an important effect, and is usually modelled by
remapping the unlensed CMB fields by a lensing deflection. However the lensing
deflections also change the photon path so that the emission angle is no longer
orthogonal to the background last-scattering surface. We give the first
calculation of the emission-angle corrections to the standard lensing
approximation from dipole (Doppler) sources for temperature and quadrupole
sources for temperature and polarization. We show that while the corrections
are negligible for the temperature and E-mode polarization, additional
large-scale B-modes are produced with a white spectrum that dominates those
from post-Born field rotation (curl lensing). On large scales about one percent
of the total lensing-induced B-mode amplitude is expected to be due to this
effect. However, the photon emission angle does remain orthogonal to the
perturbed last-scattering surface due to time delay, and half of the
large-scale emission-angle B modes cancel with B modes from time delay to give
a total contribution of about half a percent. While not important for planned
observations, the signal could ultimately limit the ability of delensing to
reveal low amplitudes of primordial gravitational waves.We also derive the
rotation of polarization due to multiple deflections between emission and
observation. The rotation angle is of quadratic order in the deflection angle,
and hence negligibly small: polarization typically rotates by less than an
arcsecond, orders of magnitude less than a small-scale image rotates due to
post-Born field rotation (which is quadratic in the shear). The field-rotation
B modes dominate the other effects on small scales.
