6. CMB POLARIZATION

Since Thomson scattering of an anisotropic radiation field also generates linear polarization, the CMB is predicted to be polarized at the roughly 5% level [33]. Polarization is a spin 2 field on the sky, and the algebra of the modes in -space is strongly analogous to spin-orbit coupling in quantum mechanics [34]. The linear polarization pattern can be decomposed in a number of ways, with two quantities required for each pixel in a map, often given as the Q and U Stokes parameters. However, the most intuitive and physical decomposition is a geometrical one, splitting the polarization pattern into a part that comes from a divergence (often referred to as the `E-mode') and a part with a curl (called the `B-mode') [35]. More explicitly, the modes are defined in terms of second derivatives of the polarization amplitude, with the Hessian for the E-modes having principle axes in the same sense as the polarization, while the B-mode pattern can be thought of simply as a 45° rotation of the E-mode pattern. Globally one sees that the E-modes have (- 1) parity (like the spherical harmonics), while the B-modes have (- 1)⁺¹ parity.

The existence of this linear polarization allows for 6 different cross power spectra to be determined from data that measure the full temperature and polarization anisotropy information. Parity considerations make 2 of these zero, and we are left with 4 potential observables: C^TT, C^TE, C^EE, and C^BB. Since scalar perturbations have no handedness, the B-mode power spectrum can only be generated by vectors or tensors. Hence, in the context of inflationary models, the determination of a non-zero B-mode signal is a way to measure the gravity wave contribution (and thus potentially derived the energy scale of inflation), even if it is rather weak. However, one must first eliminate the foreground contributions and other systematic effects down to very low levels.

The oscillating photon-baryon fluid also results in a series of acoustic peaks in the polarization power spectra. The main `EE' power spectrum has peaks that are out of phase with those in the `TT' spectrum, because the polarization anisotropies are sourced by the fluid velocity. The correlated component of the polarization and temperature patterns comes from correlations between density and velocity perturbations on the last scattering surface, which can be both positive and negative. There is no polarization `Sachs-Wolfe' effect, and hence no large-angle plateau. However, scattering during a recent period of reionization can create a polarization `bump' at large angular scales.

The strongest upper limits on polarization are at the roughly 10 µK level from the POLAR [36] experiment at large angular scales and the PIQUE [37] and COMPASS [38] experiments at smaller scales. The first measurement of a polarization signal came in 2002 from the DASI experiment [39], which provided a convincing detection, confirming the general paradigm, but of low enough significance that it lends little constraint to models. As well as the E-mode signal, DASI also made a statistical detection of the TE correlation.

More recently the WMAP experiment was able to measure the TE cross-correlation power spectrum with high precision [40]. The results are shown in Fig. 3, along with some estimates from the DASI experiment. The detected shape of the cross-correlation power spectrum provides supporting evidence of the adiabatic nature of the perturbations, as well as directly constraining the thickness of the last scattering surface. Since the polarization anisotropies are generated in this scattering surface, the existence of correlations at angles above about a degree demonstrate that there were super-Hubble fluctuations at the recombination epoch.

Figure 3. Cross power spectrum of the temperature anisotropies and E-mode polarization signal from WMAP (points), together with some estimates from DASI (triangles) which extend to higher . Note that the DASI bands are much wider in than those of WMAP. Also note that the y-axis is not multiplied by the additional , which helps to show both the large and small angular scale features.