Secondary CMB anisotropies from magnetized halos --I: Power spectra of
  the Faraday rotation angle and conversion rate
  Magnetized plasmas within halos of galaxies leave their footprint on the
polarized anisotropies of the cosmic microwave background. The two dominant
effects for astrophysical halos are Faraday rotation generating rotation of the
plane of linear polarization, and Faraday conversion inducing a leakage from
linear polarization to circular polarization. We revisit these sources of
secondary anisotropies by computing the angular power spectra of the Faraday
rotation angle and of the Faraday conversion rate by the large scale
structures. To this end, we use the halo model and we pay special attention to
the impact of magnetic field projections. Assuming magnetic fields of halos to
be uncorrelated, we found a vanishing 2-halo term, and angular power spectra
peaking at multipoles $\ell\sim10^4$. The Faraday rotation angle is dominated
by the contribution of thermal electrons. For the Faraday conversion rate, we
found that both thermal electrons and relativistic, non-thermal electrons
contribute equally in the most optimistic case for the density and Lorentz
factor of relativistic electrons, while in more pessimistic cases the thermal
electrons give the dominant contribution. Assuming the magnetic field to be
independent of the halo mass, the angular power spectra for both effects
roughly scale with the amplitude of matter perturbations as $\sim\sigma_8^3$,
and with a very mild dependence with the density of cold dark matter.
Introducing a dependence of the magnetic field strength with the halo mass
leads to an increase of the scaling with the amplitude of matter fluctuations,
up to $\sim\sigma_8^{9.5}$ for Faraday rotation and $\sim\sigma_8^{15}$ for
Faraday conversion for a magnetic field strength scaling linearly with the halo
mass.
