Statistical isotropy of CMB polarization maps

We formulate statistical isotropy of CMB anisotropy maps in its most general form. We also present a fast and orientation independent statistical method to determine deviations from statistical isotropy in CMB polarization maps. The importance of having statistical tests of departures from SI for CMB polarization maps lies not only in interesting theoretical motivations but also in testing cleaned CMB polarization maps for observational artifacts such as residuals from polarized foreground emission. We propose a generalization of the bipolar power spectrum (BiPS) to polarization maps. Application to the observed CMB polarization maps will be soon possible after the release of WMAP 3 yr data. As a demonstration we show that for $E$-polarization this test can detect the breakdown of statistical isotropy due to polarized synchrotron foreground.

Figure 1

CMB polarization maps with no foregrounds are statistically isotropic and have null bipolar power spectrum (top). Adding polarized synchrotron emission violates statistical isotropy and results in a detectable nonzero bipolar power spectrum (bottom). Dots show the BiPS after bias subtraction and lines show the $1-\sigma$ of the cosmic variance.

Figure 2

Adding 5% of the polarized synchrotron emission just violates statistical isotropy (top). Adding 7.5% of the polarized synchrotron emission clearly violates statistical isotropy and results in a detectable nonzero bipolar power spectrum (bottom). Dots show the BiPS after bias subtraction and lines show the $1-\sigma$ of the cosmic variance.