After analyzing the data in a variety of ways, we are able to conclude with confidence that there is no evidence for a chiral effect on the propagation of photons from distant radio sources. Despite this negative result, there are still good reasons to further pursue observations such as those examined in this paper.
Figure 6: Positions of radio sources on the sky, including all redshifts.
The symbols indicate deviations from 
; squares
are sources with 
, and 
's are sources
with 
. The size of the symbol indicates the
amount of deviation from 
.
Figure 7: Same as Figure Six, including only galaxies with 
.
In Figures Six and Seven we have plotted the position of the
sources in the sky, indicated by symbols related to the deviation
of 
from . Figure Six includes all of the
galaxies, while Figure Seven is limited to the distant sources
with . The squares represent sources with
, while the 's are sources
with . The size of the symbol is related
linearly to the deviation from , although for clarity
there is an offset so that points with very close
to still have a nonzero size. One conclusion to be
drawn immediately from these graphs is that there is a need for
additional data to be collected in the southern celestial
hemisphere, especially at high redshifts. In the future, observations
of polarization of the cosmic microwave background may be the
best source of data for constraining phenomena such as these [13, 26].
In characterizing the limits one can place on chiral effects, for
convenience we hypothesized a fixed four-vector 
which
would represent a violation of Lorentz invariance. If an effect
were to be found, however, it is by no means necessary that such
a profound conclusion would have to be drawn. A more plausible
hypothesis would be that of a very slowly-varying scalar field 
with a coupling as in (1); the application of the data
discussed in this paper to this possibility was examined in [12].
Such a field could arise as an ultralight axion, with mass of order
the Hubble constant today (or less). Interestingly, such axions may appear
naturally in the strongly coupled limit of heterotic string theory
[27, 28]. Another possibility is the detection of
axion-like cosmic strings; in the vicinity of such a string,
the polarization angle of two light rays passing on either side will
undergo rotations in opposite directions [9]. Although
there is no obvious sign of such a signal in Figures Six and Seven,
the importance of such a finding encourages us to continue the
search.