As we have seen in §2, the polarization pattern reflects the local quadrupole anisotropy at last scattering. Hence the temperature and polarization anisotropy patterns are correlated in a way that can distinguish between the scalar, vector and tensor sources.
There are two subtleties involved in establishing the correlation. First, the quadrupole moment of the temperature anisotropy at last scattering is not generally the dominant source of anisotropies on the sky, so the correlation is neither 100% nor necessarily directly visible as patterns in the map.
The second subtlety is that the correlation occurs through the E-mode
unless the polarization has been Faraday or otherwise rotated between the
last scattering surface and the present.
As we have seen an E-mode is modulated in the direction of, or perpendicular
to, its polarization axis. To be correlated with the temperature, this
modulation must also correspond to the modulation of the temperature
perturbation.
The two options are that E is parallel or perpendicular to crests in the
temperature perturbation.
As modes of different direction are superimposed, this translates
into a radial or tangential polarization pattern around hot spots (see
Fig. 12a).
On the other hand B-modes do not correlate with the temperature.
In other words, the rotation of the pattern in Fig. 12a
by 45 into those of Fig. 12b (solid and dashed
lines) cannot be generated by Thomson scattering.
The temperature field that generates the polarization has no way to
distinguish between points reflected across the symmetric hot spot and so
has no way to choose between the
rotations.
This does not however imply that B vanishes.
For example, for a single plane wave fluctuation B can change signs across
a hot spot and hence preserve reflection symmetry
(e.g. Fig. 6 around the hot spot
,
).
However superposition of oppositely directed waves as in
Fig. 12b would destroy the correlation with the hot spot.
The problem of understanding the correlation thus breaks down into
two steps: (1) determine how the quadrupole moment of the temperature
at last scattering correlates with the dominant source of
anisotropies; (2) isolate the E-component (Q-component
in coordinates) and determine whether it represents
polarization parallel or perpendicular to crests and so
radial or tangential to hot spots.