3 Statistical analysis and results

Figure 1: A map of the polarization vectors of all significantly polarized ( $p \geq 0.6 \%$ and $\sigma _{\theta } \leq 14\hbox {$^\circ $ }$) quasars with right ascensions $11^{\rm h}$ $15^{\rm m} \leq \alpha \leq 14^{\rm h}$ $29^{\rm m}$, and redshifts $1.0 \leq z \leq 2.3$. The vector length is arbitrary. The 13 new objects are indicated by additional points

We first want to test the hypothesis that the polarization position angles of quasars located in region A1 preferentially lie in the interval [146 $\hbox{$^\circ$ }$-46 $\hbox{$^\circ$ }$] instead of being uniformly distributed. This angular sector was selected prior to the new observations - on the basis of the results of Paper I-, and the polarization position angles have been measured for a sample of quasars different from that one at the origin of the detection of the effect. Out of the 13 new significantly polarized quasars in region A1 (Table 1), 10 have their polarization position angles in the expected range. To test the null hypothesis H₀ of uniform distribution of circular observations against the alternative of sectoral preference, we may use a simple binomial test (e.g. Lehmacher & Lienert 1980; Siegel 1956). If $P_{\scriptscriptstyle \rm A}$ is the probability under H₀ that a polarization position angle falls in the angular sector [146 $\hbox{$^\circ$ }$-46 $\hbox{$^\circ$ }$], then $P_{\scriptscriptstyle \rm A} = 80\hbox{$^\circ$ }/180\hbox{$^\circ$ }$. If L denotes the number of polarization position angles falling in [146 $\hbox{$^\circ$ }$-46 $\hbox{$^\circ$ }$], L has a binomial distribution under H₀, such that the probability to have $L_{\star}$ or more polarization angles in [146 $\hbox{$^\circ$ }$-46 $\hbox{$^\circ$ }$] is

$$P(L \geq L_{\star}) = \sum_{l=L_{\star}}^{N} \left(\begin{ar... ...tyle \rm A}^{l} \; (1-P_{\scriptscriptstyle \rm A})^{N-l}\, .$$ (1)

With N=13 and $L_{\star}=10$, we compute $P(L \geq 10)$ = 1.8%. This indicates that the hypothesis of uniform distribution of polarization position angles may be rejected at the 1.8% significance level in favour of coherent orientation. A map of the quasar polarization vectors is illustrated in Fig. 1, including the objects from Paper I. An alignment is clearly seen, with a net clustering of polarization vectors around $\theta \sim 165\hbox{$^\circ$ }$- $170\hbox{$^\circ$ }$. Altogether, there are 29 significantly polarized quasars in this region, and 25 of them have their polarization vectors aligned i.e. their polarization position angles in the range 146 $\hbox{$^\circ$ }$-46 $\hbox{$^\circ$ }$ (Tables 1 and 2, and Paper I). It is interesting to note that the effect is stable when we increase the polarization degree cutoff (then decreasing the probability of a possible contamination): out of 22 quasars with $p \geq 0.8\%$, 19 have their polarization vectors aligned, and out of 17 quasars with $p \geq 1.0\%$, 16 have their polarization vectors aligned.

Since in total 43 new polarized objects were found all over the sky, it is also interesting to re-run the global statistical tests used in Paper I. These tests are applied to the whole sample of 213 objects. The statistics basically measure the dispersion of polarization position angles for groups of $n_{\rm v}$ neighbours in the 3-dimensional space, the significance being evaluated through Monte-Carlo simulations, shuffling angles over positions. It is not our purpose to repeat here what was done in Paper I, but only to illustrate the trend with the larger sample. The significance levels of the statistical tests, i.e. the probabilities that the test statistics would have been exceeded by chance only, are given in Fig. 2 for the four tests considered in Paper I. Compared to Figs. 9 and 10 of Paper I, all the statistical tests indicate a net decrease of the significance level for the larger sample, strengthening the view that polarization vectors are not randomly distributed over the sky but are coherently oriented in groups of 20-30 objects. We note a shift of the minimum significance level towards slightly higher values of $n_{\rm v}$, as expected from the increase of the number density of the objects.

Figure 2: The logarithmic significance level (S.L.) of the four statistical tests defined in Paper I, S with $\Delta \theta _{\rm c} = 60\hbox {$^\circ $ }$ $(\ifmmode\hbox{\rlap{$\sqcap$ }$\sqcup$ }\else{\unskip\nobreak\hfil \penalty50\... ...{\rlap{$\sqcap$ }$\sqcup$ } \parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi)$, $S_{\rm D} (\diamondsuit )$, $Z_{\rm c}^{\rm m} (+)$, $Z_{\rm c} (\times )$, when applied to the new sample of 213 polarized quasars. $n_{\rm v}$ is the number of nearest neighbours around each quasar; it is involved in the calculation of averaged quantities. The dashed horizontal line indicates the 1% significance level

All these results confirm the existence of orientation effects in the distribution of quasar polarization vectors, and more particularly in the high-redshift region A1 where an independent test was performed.