Faraday rotation of the cosmic microwave background polarization by a stochastic magnetic field

A primordial cosmological magnetic field induces Faraday rotation of the cosmic microwave background polarization. This rotation produces a curl-type polarization component even when the unrotated polarization possesses only gradient-type polarization, as expected from scalar density perturbations. We compute the angular power spectrum of curl-type polarization arising from small Faraday rotation due to a weak stochastic primordial magnetic field with a power-law power spectrum. The induced polarization power spectrum peaks at arc minute angular scales. Faraday rotation is one of the few cosmological sources of curl-type polarization, along with primordial tensor perturbations, gravitational lensing, and the vector and tensor perturbations induced by magnetic fields; the Faraday rotation signal peaks on significantly smaller angular scales than any of these, with a power spectrum amplitude which can be comparable to that from gravitational lensing. Prospects for detection are briefly discussed.

Figure 1

The angular power spectrum of the Faraday rotation angle, $C_l^{\alpha }$, induced by a stochastic magnetic field. The curves in order of decreasing amplitude on the right side of the plot correspond to magnetic field power spectral indices $n_B=2$, 1, 0, $-1$, and $-2$. The magnetic fields have been normalized to a nanogauss at the smoothing scale $\lambda =1\mathrm{M}\mathrm{p}\mathrm{c}$.

Figure 2

The $C$-polarization power spectrum of the microwave background induced by the Faraday rotation field in Fig. 1, again with the magnetic field normalization scale $\lambda =1\mathrm{M}\mathrm{p}\mathrm{c}$.