Faraday Rotation Effect of Intracluster Magnetic Field on Cosmic Microwave Background Polarization

The observed magnetic field of microgauss strength in clusters of galaxies should induce the Faraday rotation effect on the linearly polarized cosmic microwave background (CMB) radiation when the CMB radiation propagates through a cluster at low redshift. Employing the Press-Schechter prescription for the cluster abundance under the cold dark matter (CDM) scenario and a plausible isothermal $\beta$-model for the gas distribution, we compute angular power spectra of the CMB polarization fields including the Faraday rotation mixing effect under the simple assumption of uniform magnetic field configuration across a cluster. As a result, we find that a parity-odd B-type polarization pattern is statistically generated on the observed sky, even when the primary polarization only contains the parity-even E-mode component, such as in the case of pure scalar perturbations. The generated B-type polarization has a peak with amplitude of $\sim0.1 \mu{\rm K}(B_0/0.1 \mu{\rm G})(\nu_0/10 {\rm GHz})^{-2}$ at angular scales of $l\approx 1000$ for the currently favored adiabatic $\Lambda$CDM model. This result also implies that, if the magnetic field has a $0.5 \mu{\rm G}$ strength and we observe at lower frequencies such as $\nu_0\simlt 5 {\rm GHz}$, the secondary signal due to the Faraday rotation effect could be comparable to the magnitude of the primary polarization. The frequency dependence of the Faraday rotation can be then used to discriminate the effect from primary and other secondary signals on the CMB polarizations. Our results therefore offer a new empirical opportunity to measure or constrain the intracluster magnetic field in the average sense.