General Statistical Properties of the Cosmic Microwave Background Polarization Field

The distribution of the polarization of the cosmic microwave background (CMB) in the sky is determined by the hypothesis of random Gaussian distribution of the primordial density perturbations. This hypotheses is well motivated by the inflationary cosmology. Therefore, a test of the consistency of the statistical properties of the CMB polarization field with the Gaussianity of primordial density fluctuations is a realistic way to study the nature of primordial inhomogeneities in the universe. This paper contains theoretical predictions of the general statistical properties of the CMB polarization field. All results were obtained under the assumption of the Gaussian nature of the signal. We pay special attention to the following two problems: first, the classification and statistics of the singular points of the polarization field where polarization is equal to zero, and second, the topology of the contours of the value of the degree of polarization. We have investigated the percolation properties for the zones of strong and weak polarization. We have also calculated Minkowski functionals for the CMB polarization field. All results are analytical.