Searching for a signal in the polarization data is complicated by
the fact that 
is only defined modulo 
.
In testing any specific hypothesis, it is necessary to choose some
reasonable procedure for resolving this ambiguity. The method
chosen by the authors of [24] was the following:
for any choice of direction for the vector
, define an angle 
which is between 
and if 
and between 
and if 
, where
(which they called 
) is the angle between
and the direction toward the source.
It was noted in [24] that this procedure necessarily introduces
correlations between 
and 
. It would be
illegitimate, therefore, to take a statistical correlation between
these two quantities as itself evidence of a signal in the data.
However, if the degree of correlation were much higher than that which
would be expected if there were no signal in the data, we might
conclude that there was a measurable effect.
It is at this point in the analysis that we find two important flaws in the
procedure followed in [24]. First, one must reliably determine
the zero point for , which would be observed in the absence
of any chiral effects. In [24], the authors searched for a best
fit to the data of the form 
, where in the notation of Eq. 3, 
.
They found that the favored value for the zero point was 
. This seems to be inconsistent with the evidence of
Figure Two, which exhibits a peak at 
. The resolution is
simply the fact that the definition of , as described above,
separates the data into two groups, one with 
and one with 
. With this procedure the
favored value for 
will always be near ; it arises
essentially from taking the average of a group of points clustered
around and another clustered around 
. This
method of resolving the ambiguity is therefore inappropriate
for data which lie naturally in the vicinity of .
Nevertheless, [24] argues that the correlation found is
statistically significant, as it was only very rarely reproduced in
artificially generated sets of data. The procedure for generating
these sets is the second important flaw that we find. Figure Two
provides evidence that, regardless of the position of the source on
the sky, is distributed approximately in a Gaussian
distribution centered on ; a best fit to the Gaussian
yields a dispersion of 
. Therefore, in searching
for position-dependent effects, it is appropriate to compare the
actual data to data which is generated by drawing from a similar
distribution. In [24], on the other hand, artificial realizations
were generated completely randomly, i.e. from a flat probability
distribution for . This has a dramatic effect on the
claimed significance of the result. We performed an independent
analysis,
using two
different methods of generating the artificial data sets: first by
drawing from a flat distribution, and then from a Gaussian with the
appropriate width. The numbers generated were values of
for the positions and redshifts of the 71 sources in the sample with
. In 1000 realizations of the data drawn from a
flat distribution, in only 7 trials was the significance of the
correlation greater than that in the actual data; this is comparable
to the 6 out of 1000 reported in [24], and if reliable would
be evidence of the existence of a signal. On the other hand, in
1000 realizations of the data drawn from the appropriate Gaussian
distribution, the artificial data was more strongly correlated with
the hypothesized test function in 911 out of 1000 trials. Even if
there were no signal at all in the data, we would expect the artificial
realizations to have a stronger correlation approximately 
of the
time; the fact that our trials had better correlations over 
of
the time is due to the fact that the Gaussian slightly underestimates
the number of data points near . This result, however,
vividly demonstrates our main point: the existence of a real enhancement
of near leads to a spuriously large correlation
coefficient if one uses the procedure described in [24].
When this enhancement, which is consistent with
conventional models of the sources, is taken into account, there is no
sign of an additional effect such as that in Eq. 3.
There is another way of quantifying our claim that a random distribution
centered around is a better fit to the data than the
correlation proposed in [24]. Figure Five is a plot of
as a function of , where is defined
using the best-fit direction quoted in [24] and is
defined to be between and .
Figure 5: The difference between polarization and position angles
as a function of for the best-fit direction of
anisotropy proposed in [24]. Angles of are
to be identified with ; the data thus live on a cylinder.
The solid line represents the predicted relationship in the absence
of any signal, while the diagonal dashed line wrapping around the
cylinder represents the model suggested in [24].
We may think of
this graph as being defined on a cylinder, where is to
be identified with . With this in mind, we have plotted
two possible relationships, a solid horizontal line at
and a dashed line at 
, where
we have measured the parameters 
and from
Figure 1(d) of [24]. If the relationship claimed in [24]
is correct, the dashed line wrapping around the cylinder should be a
better fit to the data than the solid horizontal line. This can be
measured by calculating
where we take the average error to be , although
the precise value is irrelevant for purposes of comparison. The
quantity 
, which represents the difference between the
predicted and measured value of , is of course subject
to the ambiguity; however, we can resolve this ambiguity
optimistically for each point, by defining 
. Using this procedure, we calculate that the best fit
proposed in [24] yields 
, while the hypothesis
of no effect yields 
69. Thus, the horizontal solid line
in Figure Five is a much better fit than the diagonal dashed lines.
Given that there is ample evidence that the intrinsic zero point
is centered on 
, we may ask
how good a limit we can place on an effect such as that in
Eq. (3). One approach to this problem is to define
to be between and , and to assume
that the deviation from the intrinsic value is given by
. It is then possible to do a
straightforward least-squares fit to Eq. (3), with the
four components of 
as free parameters. Using the data at
redshifts , the best-fit parameters obtained in this way are
(This procedure yields separate values for each of the three
spacelike components 
; since each value is consistent with
no preferred direction, it is more appropriate to quote the limit
on the magnitude 
.)
These values are consistent with 
, and similar to the
limit on 
from [10] quoted in Eq. (5).