Free Access
Volume 520, September-October 2010
Article Number L7
Number of page(s) 5
Section Letters
Published online 29 September 2010

Online Material

Appendix A: The determination of $\Delta $ p $_{\rm lin}$

Although partly discussed in previous papers (e.g. Hutsemékers et al. 2005), we provide some details of the simulations performed to estimate $\Delta p_{\rm lin}$, 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.

\end{figure} Figure A.1:

The effect of the addition of a small systematic polarization $\Delta p_{\rm lin}$ on the distributions of the polarization degree $p_{\rm lin}$ and of the polarization angle $\theta _{\rm lin}$. From top to bottom, $\Delta p_{\rm lin}$ = 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 $p < 0.6\%$). We found that it was reasonably well reproduced by a half-gaussian 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 $\theta$, uniformly distributed between 0 $\hbox{$^\circ$ }$ and 180 $\hbox{$^\circ$ }$. From p and $\theta$, we compute the normalized Stokes parameters q and u to which we add a systematic polarization $\Delta q > 0$ (we assume for simplicity that $\Delta u = 0$, which corresponds to add linear polarization at $\theta = 0\hbox{$^\circ$ }$). We also add random noise uniformly generated between -$0.2\%$ and +$0.2\%$, in agreement with the uncertainties in the measurements (Hutsemékers et al. 2005). Finally, from the modified q and u we recompute p, $\theta$, and $\sigma_{\theta}$ in the usual way and select the good quality measurements with the criteria previously used, i.e., $p \geq 0.6\%$ and $\sigma_{\theta} \leq 14\hbox{$^\circ$ }$. This leaves us with $\sim$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 $\Delta q =
0$%, 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 $\Delta p_{\rm lin} = \Delta q =
0.5\%$. 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, $\Delta p_{\rm lin}$ 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 photon-pseudoscalar mixing

In the weak mixing case, photons with polarizations parallel to an external magnetic field $\vec{B}$ that propagate through the distance L can decay into pseudoscalars with a probability

\begin{displaymath}P_{\gamma a} \simeq (\varg Bl)^{2} ~ \sin^{2} (\xi/2),
\end{displaymath} (B.1)

where $l = 2\omega/(\omega^2_p-m^2_a)$ and $\xi = L/l$, $\omega$ is the photon frequency, $\omega_p$ the plasma frequency, ma the pseudoscalar mass, and $\varg$ the photon-pseudoscalar coupling constant (Raffelt & Stodolky 1988; Jain et al. 2002). As long as $P_{\gamma a}$ is small, the linear polarization perpendicular to $\vec{B}$ generated by dichroism can be approximated by $\Delta p_{\rm lin} = P_{\gamma a}$. The mixing also induces a polarization-dependent phase shift (retardance)

\begin{displaymath}\phi_a \simeq \left( \frac{\varg Bl}{2} \right) ^{2} ~ (\xi - \sin \xi )
\end{displaymath} (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 $(\varg Bl)^{2}$.

Assuming $m_a \ll \omega_p$, we have $l \simeq 4 \times 10^{-14}~\nu ~
n_{\rm e}^{-1}$ Mpc, where $\nu$ is the frequency in GHz and $n_{\rm e}$ the electronic density in cm-3. At optical wavelengths ( $\nu =
5 \times 10^{5}$ GHz) and under various conditions (e.g. $n_{\rm e} \sim
10^{-6}$ cm-3 and $L \sim 10$ Mpc in superclusters, or $n_{\rm e} \sim 10^{-8}$ cm-3 and $L \sim 1$ Gpc in the intergalactic medium), $\xi = L/l \sim 500$. With a frequency bandwidth $\Delta\nu / \nu
\sim 0.2$ and $\xi \gg 1$, we find that $\phi_a \simeq \langle \Delta
p_{\rm lin} \rangle ~ \xi /2 \sim 10^{2} ~ \langle \Delta p_{\rm
lin} \rangle$, where $\langle \Delta p_{\rm lin} \rangle$ represents the average value of $\Delta p_{\rm lin}$. 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 $\vec{B}$, the dichroism and birefringence induced by photon-pseudoscalar mixing modify the polarization according to

             q = $\displaystyle q_{_0} - \Delta p_{\rm lin} ~ ,$  
u = $\displaystyle u_{_0} \cos \phi_a ~ ,$ (B.3)
$\displaystyle \varv$ = $\displaystyle u_{_0} \sin \phi_a ~ ,$  

where q0 and u0 are the normalized Stokes parameters representing the initial linear polarization state and $\varv_{_0} =
0$. Assuming that the sources are initially polarized at $p_{_0}
\simeq 2\%$ with randomly oriented polarization angles and that $\langle \Delta p_{\rm lin} \rangle < 0.01$ (Appendix A), we finally obtain the average circular polarization expected to result from the photon-pseudoscalar mixing: $\langle \vert \varv \vert \rangle ~
\sim (2/ \pi) p_{_0} ~ \phi_a$, i.e., $\langle \vert p_{\rm circ}\vert \rangle
~ \sim \langle \Delta p_{\rm lin} \rangle \sim 0.5\%$. 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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