Secondary B-mode polarization from Faraday rotation in clusters and galaxies

We revisit the polarisation induced by Faraday rotation when Cosmic Microwave Background photons traverse magnetised plasma. We compute the secondary B-mode angular power spectrum from Faraday rotation due to magnetic fields in galaxies and galaxy clusters with masses ranging from $10^{11}$ to $10^{16.5} M_\odot$. We investigate its dependence on the electron and the magnetic field profiles. Namely, we consider both the beta-profile of electron density as well as an electron density distribution based on the Navarro-Frenk-White dark matter profile. We model the magnetic field structure in galaxies and clusters motivated by recent observations. We further account for its redshift evolution and we examine the importance of its coherence length. We find that the B-mode polarisation from Faraday rotation depends on the normalisation parameter $C_l\propto \sigma_8^{5-6}$. At 30 GHz for $\sigma_8=0.8$, the B-modes from Faraday rotation range between $0.01 {\mu \rm K}^2$ and $4 \times 10^{-3} {\mu \rm K}^2$ at $l=10^4$ in the case of a maximally coherent fields. For smaller coherence lengths, those amplitudes are smaller and they peak at higher multipoles.