Issue 
A&A
Volume 520, SeptemberOctober 2010



Article Number  L7  
Number of page(s)  5  
Section  Letters  
DOI  https://doi.org/10.1051/00046361/201015359  
Published online  29 September 2010 
Online Material
Appendix A: The determination of p
Although partly discussed in previous papers (e.g. Hutsemékers et al. 2005), we provide some details of the simulations performed to estimate , the small additional linear polarization needed to reproduce the observed alignments of quasar polarization vectors. These simulations extend those discussed in Hutsemékers & Lamy (2001), accounting for the measurement errors.
Figure A.1: The effect of the addition of a small systematic polarization on the distributions of the polarization degree and of the polarization angle . From top to bottom, = 0%, 0.25%, and 0.5% (see text for details). 

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We first modeled the distribution of the debiased polarization degree in the full quasar sample (from Hutsemékers et al. 2005, including objects with ). We found that it was reasonably well reproduced by a halfgaussian distribution of zero mean and unitary variance. According to this distribution, we randomly generate 80 values of the polarization degree p, most of them lying between 0% and 3%. We also generate 80 values of the polarization angle , uniformly distributed between 0 and 180 . From p and , we compute the normalized Stokes parameters q and u to which we add a systematic polarization (we assume for simplicity that , which corresponds to add linear polarization at ). We also add random noise uniformly generated between  and +, in agreement with the uncertainties in the measurements (Hutsemékers et al. 2005). Finally, from the modified q and u we recompute p, , and in the usual way and select the good quality measurements with the criteria previously used, i.e., and . This leaves us with 60 polarization values. This is comparable to the number of objects in the alignment region (cf. Fig. 7 of Hutsemékers et al. 2005, to which these simulations should be compared).
The results are illustrated in Fig. A.1 for %, 0.25%, and 0.5% from top to bottom. We see that, in the distribution ofpolarization angles, a significant deviation to uniformity is only obtained for . At the same time, the distribution of the polarization degree does not appear significantly modified. Since this additional polarization at a single polarization angle is likely unrealistic, should be seen as a lower limit, although it cannot be much higher. Indeed, much larger values would make the distribution of the polarization degree incompatible with the observations (e.g. Hutsemékers & Lamy 2001).
Appendix B: Circular polarization due to photonpseudoscalar mixing
In the weak mixing case, photons with polarizations parallel to an
external magnetic field
that propagate through the distance
L can decay into pseudoscalars with a probability
(B.1) 
where and , is the photon frequency, the plasma frequency, m_{a} the pseudoscalar mass, and the photonpseudoscalar coupling constant (Raffelt & Stodolky 1988; Jain et al. 2002). As long as is small, the linear polarization perpendicular to generated by dichroism can be approximated by . The mixing also induces a polarizationdependent phase shift (retardance)
(B.2) 
acquired by the photons during propagation, which results in circular polarization. As noted by Raffelt & Stodolky (1988), both effects are on the order of .
Assuming , we have Mpc, where is the frequency in GHz and the electronic density in cm^{3}. At optical wavelengths ( GHz) and under various conditions (e.g. cm^{3} and Mpc in superclusters, or cm^{3} and Gpc in the intergalactic medium), . With a frequency bandwidth and , we find that , where represents the average value of . Similar estimates are derived when accounting for density fluctuations (Jain et al. 2002).
Adopting the convention of u=0 and q>0 for polarization vectors
parallel to ,
the dichroism and birefringence induced by
photonpseudoscalar mixing modify the polarization according to
q  =  
u  =  (B.3)  
= 
where q_{0} and u_{0} are the normalized Stokes parameters representing the initial linear polarization state and . Assuming that the sources are initially polarized at with randomly oriented polarization angles and that (Appendix A), we finally obtain the average circular polarization expected to result from the photonpseudoscalar mixing: , i.e., . This estimate applies to a variety of plausible situations, in agreement with the simulations shown in Das et al. (2005), Hutsemékers et al. (2008), and Payez et al. (2008).
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