TensorBmode and stochastic Faraday mixing

Abstract

This paper investigates the Faraday effect as a different source of $B$-mode polarization. The $E$-mode polarization is Faraday rotated provided a stochastic large-scale magnetic field is present prior to photon decoupling. In the first part of the paper we discuss the case where the tensor modes of the geometry are absent and we argue that the $B$ mode recently detected by the BICEP2 Collaboration cannot be explained by a large-scale magnetic field rotating, through the Faraday effect, the well-established $E$-mode polarization. In this case, the observed temperature autocorrelations would be excessively distorted by the magnetic field. In the second part of the paper the formation of Faraday rotation is treated as a stationary, random and Markovian process with the aim of generalizing a set of scaling laws originally derived in the absence of the tensor modes of the geometry. We show that the scalar, vector and tensor modes of the brightness perturbations can all be Faraday rotated even if the vector and tensor parts of the effect have been neglected, so far, by focussing the attention on the scalar aspects of the problem. The mixing between the power spectra of the $E$-mode and $B$-mode polarizations involves a unitary transformation depending nonlinearly on the Faraday rotation rate. The present approach is suitable for a general scrutiny of the polarization observables and of their frequency dependence.