University of Chicago

Consider first the large-angle scalar perturbations. Here the dominant source of correlated anisotropies is the temperature perturbation on the last scattering surface itself. The Doppler contributions can be up to half of the total contribution but as we have seen in §2.3 do not correlate with the quadrupole moment. Contributions after last scattering, while potentially strong in isocurvature models for example, also rapidly lose their correlation with the quadrupole at last scattering.

As we have seen, the temperature gradient associated with the scalar fluctuation makes the photon fluid flow from hot regions to cold initially. Around a point on a crest therefore the intensity peaks in the directions along the crest and falls off to the neighboring troughs. This corresponds to a polarization perpendicular to the crest (see Fig. 12). Around a point on a trough the polarization is parallel to the trough. As we superpose waves with different we find the pattern is tangential around hot spots and radial around cold spots ([Crittenden et al.] 1995). It is important to stress that the hot and cold spots refer only to the temperature component which is correlated with the polarization. The correlation increases at scales approaching the horizon at last scattering since the quadrupole anisotropy that generates polarization is caused by flows.

Fig. 12:
Temperature-polarization cross correlation. *E*-parity polarization
perpendicular (parallel) to crests generates a tangential
(radial) polarization field around hot spots. *B* polarization
does not correlate with temperature since the 45 degree rotated contributions from
oppositely directed modes cancel.

For the vectors, no temperature perturbations exist on the last scattering
surface and again Doppler contributions do not correlate with the quadrupole.
Thus the main correlations with the temperature will come from the quadrupole
moment itself.
The correlated signal is reduced since the strong *B*-contributions of vectors
play no role.
Hot spots occur in the direction , where the hot lobe
of the quadrupole is pointed at the observer (see Fig. 5).
Here the *Q* (*E*) component lies in the N-S direction perpendicular to the
crest (see Fig. 6).
Thus the pattern is tangential to hot spot, like scalars
([Hu & White] 1997).
The signature peaks near the horizon at last scattering for reasons similar
to the scalars.

For the tensors, both the temperature and polarization perturbations arise from the quadrupole moment, which fixes the sense of the main correlation. Hot spots and cold spots occur when the quadrupole lobe is pointed at the observer, , and . The cold lobe and hence the polarization then points in the E-W direction. Unlike the scalars and vectors, the pattern will be mainly radial to hot spots ([Crittenden et al.] 1995). Again the polarization and hence the cross-correlation peaks near the horizon at last scattering since gravitational waves are frozen before horizon crossing ([Polnarev] 1985).