Professor, Department of Astronomy and Astrophysics
University of Chicago

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CMB Polarization Field

While no polarization has yet been detected, general considerations of Thomson scattering suggest that up to $10\%$ of the anisotropies at a given scale are polarized. Experimenters are currently hot on the trail, with upper limits approaching the expected level [Hedman et al, 2001,Keating et al, 2001]. Thus, we expect polarization to be an extremely exciting field of study in the coming decade.

The polarization field can be analyzed in a way very similar to the temperature field, save for one complication. In addition to its strength, polarization also has an orientation, depending on relative strength of two linear polarization states. While classical literature has tended to describe polarization locally in terms of the Stokes parameters $Q$ and $U$[*], recently cosmologists [Seljak, 1997,Kamionkowski et al, 1997,Zaldarriaga & Seljak, 1997] have found that the scalar $E$ and pseudo-scalar $B$, linear but non-local combinations of $Q$ and $U$, provide a more useful description. Postponing the precise definition of $E$ and $B$ until §3.7, we can, in complete analogy with Equation (1), decompose each of them in terms of multipole moments, and then, following Equation (2), consider the power spectra,

$\displaystyle \langle E_{\ell m}^* E_{\ell' m'} \rangle$ $\textstyle =$ $\displaystyle \delta_{\ell \ell'}\delta_{m m'} C_{\ell}^{EE}\,,$  
$\displaystyle \langle B_{\ell m}^* B_{\ell' m'} \rangle$ $\textstyle =$ $\displaystyle \delta_{\ell \ell'}\delta_{m m'} C_{\ell}^{BB}\,,$  
$\displaystyle \langle \Theta_{\ell m}^* E_{\ell' m'} \rangle$ $\textstyle =$ $\displaystyle \delta_{\ell \ell'}\delta_{m m'} C_{\ell}^{\Theta E}\,.$ (5)

Parity invariance demands that the cross correlation between the pseudoscalar $B$ and the scalars $\Theta$ or $E$ vanishes.

The polarization spectra shown in Plate 1 [bottom, plotted in $\mu$K following Equation (3)] have several notable features. First, the amplitude of the $EE$ spectrum is indeed down from the temperature spectrum by a factor of ten. Second, the oscillatory structure of the $EE$ spectrum is very similar to the temperature oscillations, only they are apparently out of phase but correlated with each other. Both of these features are a direct result of the simple physics of acoustic oscillations as will be shown in §3. The final feature of the polarization spectra is the comparative smallness of the $BB$ signal. Indeed, density perturbations do not produce $B$ modes to first order. A detection of substantial $B$ polarization, therefore, would be momentous. While $E$ polarization effectively doubles our cosmological information, supplementing that contained in $C_\ell$, $B$ detection would push us qualitatively forward into new areas of physics.

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Next: ACOUSTIC PEAKS Up: OBSERVABLES Previous: CMB Temperature Field
Wayne Hu 2001-10-15