A comprehensive study of Modulation effects on CMB Polarization

The Cosmic Microwave Background is characterized by temperature and linear polarization fields. Dipole modulation in the temperature field has been extensively studied in the context of hemispherical power asymmetry. In this article, we show that a dipole modulation, and in general, any kind of modulation isn't allowed in the $E$ and $B$ modes. This is the main result of this paper. This result explains why no evidence of modulation in $E$ mode has been found in the literature. On the contrary, the linear polarization fields $Q$ and $U$ have no such restrictions. We show that modulation under certain situations can be thought of as local $U(1)$ gauge transformations on the surface of a sphere. As far as the modulation function is concerned, we show that physical considerations enforce it to be (i) a spin $0$ field and (ii) a scalar under parity. As masking is a specific type of modulation, our study suggests that direct masking of $E$ mode isn't also possible. Masking in the $E$ map can only be applied through $Q$ and $U$ fields. This means that in principle, leaking of $E$ and $B$ mode powers into each other is unavoidable.