The new aspect of this approach is the isolation of the total angular dependence of the modes by combining the intrinsic angular structure with that of the plane-wave spatial dependence. This property implies that the normal modes correspond directly to angular structures on the sky as opposed to the commonly employed technique that isolates portions of the intrinsic angular dependence and hence a linear combination of observable modes [10]. Elements of this approach can be found in earlier works (e.g. [6,7,11] for the temperature and [3] for the scalar and tensor polarization). We provide here a systematic study of this technique which also provides for a substantial simplification of the evolution equations and their integral solution in §IIIC, including the terms involving the radiation transport of the CMB. We discuss in detail how the monopole, dipole and quadrupole sources that enter into the radiation transport problem project as anisotropies on the sky today.
Readers not interested in the formal details may skip this section
on first reading
and simply note
that the temperature and polarization distribution is
decomposed into the modes and
with
for scalar, vector and tensor metric
perturbations respectively. In this representation, the geometric
distinction between scalar, vector and tensor contributions to
the anisotropies is clear as is the reason why they do not mix.
Here the
are the spin-2 spherical harmonics [12] and were introduced
to the study of CMB polarization by [3]. The
radial decompositions
of the modes
and
(for
)
isolate the total angular dependence
by combining the intrinsic and plane wave angular momenta.