Statistical properties of Galactic CMB foregrounds: dust and synchrotron

Recent Planck observations have revealed some of the important statistical properties of synchrotron and dust polarizations, namely, the $B$ to $E$ mode power and temperature-$E$ (TE) mode cross-correlation. In this paper, we extend our analysis in Kandel et al. (2017) to analytically study $B$ to $E$ mode power as well as TE cross-correlation for dust and synchrotron polarizations, using a realistic model of magnetohydrodynamical (MHD) turbulence. Our results suggest that the Planck results for both synchrotron and dust polarizations can be understood if the turbulence in the Galaxy is sufficiently sub-Alfv\'enic. We also show how $B$ to $E$ ratio as well as the TE cross-correlation can be used to study media magnetization, compressibility, and level of density-magnetic field correlation.