$B$ polarization of the CMB from Faraday rotation

We study the effect of Faraday rotation due to a uniform magnetic field on the polarization of the cosmic microwave background. Scalar fluctuations give rise only to parity-even $E$-type polarization of the cosmic microwave background. However in the presence of a magnetic field, a nonvanishing parity-odd $B$-type polarization component is produced through Faraday rotation. We derive the exact solution for the $E$ and $B$ modes generated by scalar perturbations including the Faraday rotation effect of a uniform magnetic field, and evaluate their cross correlations with temperature anisotropies. We compute the angular autocorrelation function of the $B$-modes in the limit that the Faraday rotation is small. We find that uniform primordial magnetic fields of present strength around $B_0={10}^{-9}\mathrm{G}$ rotate $E$-modes into $B$-modes with amplitude comparable to those due to the weak gravitational lensing effect at frequencies around $\nu =30\mathrm{G}\mathrm{H}z$. The strength of $B$-modes produced by Faraday rotation scales as $B_0/{\nu }^2$. We evaluate also the depolarizing effect of Faraday rotation upon the cross correlation between temperature anisotropy and $E$-type polarization.

Figure 1

Temperature-polarization ($E$-mode) cross correlation in a $\Lambda$CDM background cosmology in the case of no magnetic field (thin solid line) and in the case of Faraday rotation in a uniform magnetic field with $F=1$ (thick solid line) and $F=3$ (dashed line). $F$ is defined in Eq. (3). For large $F$ the depolarizing effect of differential Faraday rotation is quite noticeable.

Figure 2

Visibility function $g(\eta )$ (solid line) as a function of conformal time in the background $\Lambda$CDM cosmology, calculated with CMBFAST, and multiplied by $j_0(F\tau )$ with $F=3$ (dotted line), and multiplied by $\tau$ (dashed line). The left panel displays times around hydrogen recombination and photon decoupling. The right panel displays times after reionization, which was assumed to take place at an optical depth ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{i}\mathrm{o}\mathrm{n}}=0.17$.

Figure 3

Temperature-polarization ($B$-mode) cross correlation (thick solid line) in a $\Lambda$CDM background cosmology due to Faraday rotation of $E$-modes, scaled by the factor $F$ defined in Eq. (3), assumed to be smaller than unity. See the text for the definition of $C_l^{\Theta \mathrm{B}}$, which is actually a nondiagonal correlation. The correlation $\Theta E$ in the absence of magnetic field is also displayed for reference (thin solid line).

Figure 4

Angular autocorrelation multipoles of $B$-modes of CMB polarization generated by Faraday rotation of E-modes generated by scalar perturbations in a $\Lambda$CDM cosmology, scaling away the factor $F^2$ (thick solid line). The autocorrelation of $E$-modes from which they originate is also displayed (thin solid line) for comparison purposes. Also shown are the autocorrelation of $B$-modes due to weak gravitational lensing (dashed line) and to gravitational waves in a model with tensor-to scalar ratio $r={10}^{-2}$ (dotted line).