Circular polarization in a spherical basis
  Circular polarization of the cosmic microwave background (CMB) arises in the
standard cosmological model from Faraday conversion of the linear polarization
generated at the surface of last scatter by various sources of birefringence
along the line of sight. If the sources of birefringence are generated at
linear order in primordial density perturbations the principal axes of the
index-of-refraction tensor are determined by gradients of the primordial
density field. Since linear polarization at the surface of last scatter is
generated at linear order in density perturbations, the circular polarization
thus arises at second order in primordial perturbations. Here, we re-visit the
calculation of the circular polarization using the total-angular-momentum
formalism, which allows for some simplifications in the calculation of the
angular power spectrum of the circular polarization---especially for the
dominant photon-photon scattering contribution---and also provides some new
intuition.
