10.2. Simulated radio halo polarization
Different values of the power spectrum spectral index will generate very different total intensity and polarization brightness distributions for the radio emission of a halo. So far, polarization emission from radio halos has not been detected. The current upper limits to the polarization at 1.4 GHz are a few percent (~ 5%).
Murgia et al.  simulated the expected halo total intensity and polarization brightness distributions at 1.4 GHz and 327 MHz, as they would appear when observed with a beam of 45", by introducing in the 3-dimensional magnetic field an isotropic population of relativistic electrons. Different values of the magnetic field strength and power spectrum index were assumed.
Fig. 11 (top) shows simulated radio halo brightness and polarization percentage distributions at 1.4 GHz (see caption for more details). Fig. 11 (bottom) shows the expected fractional polarization profiles at 1.4 GHz and 327 MHz for the different values of the average magnetic field strength and power spectrum spectral index. Simulations indicate that a power spectrum slope steeper than n = 3 and a magnetic field strength lower than ~ 1 µG result in a radio halo polarization percentage at 1.4 GHz far in excess of the current observational upper limits. This means that, in agreement with the RM simulations, either the power spectrum spectral index is flatter than n = 3 or the magnetic field strength is significantly higher than ~ 1 µG. The halo depolarization at 327 MHz is particularly severe and the expected polarization percentage at this frequency is always below 1%. Moreover it is also evident that the magnetic field power spectrum slope has a significant effect in shaping the radio halo. In particular, flat power spectrum indexes (n < 3) give raise to very smooth radio brightness images (under the assumption that the radiating electrons are uniformly distributed).
Figure 11. Simulated halo brightness and polarization degree  for cluster at z = 0.075 as it would appear when observed with a beam of 45". Top: simulated halo images at 1.4 GHz for different values of the magnetic field power spectrum slope n and B0 = 1 µG; the vectors lengths are proportional to the degree of polarization, with 100 percent corresponding to 100 kpc on the sky. Field directions are those of the E-vector. Bottom: radially averaged profiles of the polarization percentage at 327 MHz and 1.4 GHz for three values of the magnetic field strength, namely <B>0 = 0.5, 1 and 5 µ G.