The fundamental notion of *anisotropy* is unconventional, but not
new. In 1946, G. Gamow pointed out the ubiquitous presence of rotation
of successively larger accumulations of matter, such as planets, stars,
and galaxies [Gamow, 1946]. The rotation of stars presumably originated
from the rotating spiral arms of the protogalaxy that the stars formed
from, but what is the origin of the galactic rotation? Gamow proposed
that the origin lies in some type of "universal rotation" about some
axis along a certain direction in space. This implies that space is
*anisotropic,* or inequivalent with respect to directions. And in
1949, K. Gödel at the Institute for Advanced Study in Princeton,
New Jersey, showed that anisotropic solutions of A. Einstein's field
equations of general relativity exist [Gödel, 1949]. His solution
was equivalent to a rotation of matter relative to what he named "the
compass of inertia" of the universe.

In 1982, P. Birch studied the polarization of electromagnetic radiation from various galaxies. Although his data sample was limited, and the statistical method he used was fairly simple, he found indications of a certain type of electromagnetic anisotropy in the data [Birch, 1982]. Birch took this as evidence for a universal rotation. Further work related to anisotropy and rotational asymmetry of the cosmos has followed [Ahluwalia and Goldman, 1993; Kendall and Young, 1984; Sachs, 1989].

Electromagnetic anisotropy could also be the signature of so-called axions or other pseudoparticles interacting with the electromagnetic field of charged particles. Several papers discuss this [Peccei and Quinn, 1977; Sikivie, 1984; Weinberg, 1978; Wilczek, 1978].

In this article, I will show how a new and well documented statistical
analysis of a relatively large data set indicates a *new type of
electromagnetic anisotropy* over the largest distance scales in the
cosmos. The emphasis will be on describing the data analysis, since at
this point, the correct explanation for our anisotropic effect
is not known.

In the cosmos, there are many galaxies that emit highly plane-polarized electromagnetic radiation. In 1950, H. Alfven and K. Herlofson predicted that the strong plane-polarization of radio waves emitted from certain galaxies was produced by synchrotron motion of charged particles within such galaxies [Alven and Herlofson, 1950]. Such motion is a high-speed (close to the velocity of light), circular motion, which is the same type of motion imparted to elementary particles in so-called "synchrotrons," which are machines used to study such particles. Alven and Herlofson hypothesized that the charged particles revolved rapidly around a strong galactic magnetic field, emitting higly plane-polarized electromagnetic waves, with their polarization plane being perpendicular to the magnetic field.

Today, astronomers have accumulated a fair amount of data on the plane-polarization of radio waves that have traveled over cosmological distances. Such polarization data is generally quite valuable, since the extraordinarily vast distances involved in such data allow the detection of possible electromagnetic effects that presently would be unmeasurable over Earthly distance scales.

This linear relationship between and ^{2} is a characteristic feature of
so-called "Faraday rotation," and is shown in Figure 1 below, where
data from several different galaxies are plotted [Gardner and Whiteoak,
1963]. In its journey through the cosmic expanse, a plane-polarized
wave passes through localized regions of space that are filled
with magnetized plasmas of charged particles, like ions and electrons.
The interaction between a magnetized plasma and the plane-polarized
wave produces the rotation of the polarization plane of the wave. This
so-called "Faraday effect" is a well-understood physical process.

The handedness and strength of the Faraday polarization rotation depend on the orientation and strength of the magnetic field in the plasma, the plasma density, and the wavelength of the wave. There will always be a component of the magnetic field in the plasma that is parallel to the wave's line of travel. If this component points in the same direction as the propagation direction of the wave, the rotation of the wave's polarization plane will be counterclockwise, as observed from a point on the wave's line of travel where the wave is approaching you. If the magnetic field component points oppositely to the wave's propagation direction, the Faraday polarization rotation is clockwise. The magnitude of the polarization rotation depends on the magnitude of the magnetic field component along the wave's line of propagation, the density of charged particles making up the plasma, and on the wavelength of the wave. Experimentally, it is found that the amount of Faraday rotation is proportional to the square of the wavelength of the wave.

The constant in Equation 1 is generally different for different galaxies. It is called the "Faraday rotation measure" of a galaxy, and represents the strength of the polarization rotation of waves emitted by the galaxy. Its magnitude depends upon the magnetic field strength and the electron density along the line of sight from the galaxy to Earth. Note that conventional Faraday rotation does not account for the angle , the orientation of the polarization plane of the wave at = 0, as found by extrapolation of the Faraday rotation lines in Figure 1. represents the orientation of the polarization plane before the Faraday mechanism rotates it. , as opposed to , is the angle we work with, since it represents the part of the polarizational data that does not involve the known Faraday effect.

The polarization data we analyzed consisted of the angle that labels the observed, "Faraday-compensated"
orientation of the plane of polarization of radio waves emitted by 160
galaxies. These polarization orientations are quite meaningless by
themselves however, unless they are compared with some other similar,
physical characteristic of their respective galaxies. Since all of the
observed galaxies were elliptical in shape, astrophysicists measured
the orientation of the axis of elongation - also called the "major
axis" - of the galaxies for this purpose. This orientation is
specified as an angle . The two angles are
illustrated in Figure 2 below. They are by definition restricted to the
be between 0^{} and 180^{}, since an angle greater than 180^{} is superfluous.
= 191^{} for example, represents the
same orientation of the polarization plane as
= 11^{}.

The measurement uncertainties were less than 5^{} for and
typically 5^{} for . In our analysis, is a
central parameter - it represents a possible rotation of a wave's
polarization plane that is not explicable in terms of the Faraday
effect. The frequency of the radio waves emitted from the 160 galaxies
in our data set varies, but typically spans a range of 1 to 3 GHz.
Their visual magnitudes are between 8 and 23.

Positional information on galaxies abounds [Burbidge and Crowne, 1979;
Spinrad, Djorgovski, Marr and Aguilar, 1985]. As part of our total data
set, we recorded the positional coordinates of the 160 galaxies above
that we had found polarization data for. The positional coordinates of
galaxies are given in the astronomical literature as the redshift,
declination and right ascension of a galaxy. We used the redshift
*z* to compute the distance *r* to a galaxy from the
expression appropriate for a universe of "critical average mass
density," namely

where *h*_{0} = (2/3) (10^{-10}
years^{-1}), and *h* is the Hubble constant.

Right ascension and declination are so-called "equatorial
coordinates" for specifying spatial directions in the cosmos. The
equatorial coordinate system in astronomy is analogous to Earth's
cordinate system of latitudes and longitudes. Declination corresponds
to latitude, and right ascencion corresponds to longitude. A direction
of 90^{} declination points along
Earth's polar axis toward the North pole (extending beyond Earth to
outer space), 0^{} declination
points somewhere along Earth's extended equatorial plane, and
-90^{} declination points along
Earth's polar axis toward the South pole. Positive declination values
refers to directions in the northern celestial hemisphere, while
negative declination values refers to directions in the southern
celestial hemisphere. Right ascension values are celestial latitudes,
running from 0 to 24 hours. The direction of 0^{} declination and 0 hours right ascension
points from Earth to the point in space where the Sun's ecliptic
intersects Earth's equatorial plane at Spring equinox.

It is worth noting that the positional distribution of the 160 galaxies we studied is not uniform over the sky. Rather, the majority of the galaxies come from the northern sky, since most of the world's radio observatories are located in Earth's northern hemisphere.

(1) *galaxies* - since the angles are part of the polarization
properties of the waves emitted by the galaxies,

(2) *electromagnetism* - since the waves are electromagnetic waves,
and

(3) *space* - since the waves travel through immense distances,
and from all directions in the universe.

Some studies of the and angles have promted astronomers to propose a
so-called "two-population hypothesis" to explain the observed values
of and . This
hypothesis asserts that there are two populations of galaxies out
there: one population in which a galaxy's observed radiation has its
polarization plane oriented approximately parallel to the major axis of
the galaxy ( | - | =
0^{} ), and one in which a galaxy's
observed radiation has its polarization plane oriented approximately
perpendicular to the major axis of the galaxy ( | - | = 90^{} ) [Clarke, Kronberg, and Simard-Normandin,
1980]. But these conclusions were based on a very small subset of the
galaxies that have polarization measurements taken on them. The full
data set available indicates that the observed polarization plane
orientation relative to the major axis take all possible values, so one
really need to assume that several "populations" exist, in order to
explain the data. This is admittedly not a very satisfying
explanation.

As an alternative to the invokation of arbitrary of ad-hoc galaxy populations, we asked ourselves whether there is a pattern in the and angle data that can be explained by some general, unifying relationship. We were not able to find any relationship that could arise from any obvious, conventional physical theory, as discussed toward the end of this article. We therefore decided to explore unconventional relationships - relationships that, if present in the data, would force Physics to move to new frontiers.

This mode of scientific investigation is extraordinarily useful, since it contains in it the seeds for scientific improvement. If one always studied phenomena within the framework of conventional science, the chance of hitting upon something new that could not be explained within our present understanding would be minimal. And if one never observed contradictions to a theory - because one was too complacent to care to look for them - the theory could never be improved upon.

One unconventional relationship we investigated was whether the
measured values for the and angles of a galaxy depended on the direction of
the line of sight to the galaxy on the sky. More specifically, we
investigated whether the observed angles can
be reproduced by assuming that the polarization plane of a wave emitted
by a galaxy is initially oriented at a fixed angle relative to the
galaxy's major axis, and then undergoes a rotation (as specified by an
angle ) that depends on the wave's direction
of travel. My Ph. D. dissertation [Nodland, 1995], and also to some
degree our article in Physical Review Letters [Nodland and Ralston,
1997], present theoretical calculations which predict a specific
mathematical form for such a rotation. These calculations are quite
involved, so I will mention them only briefly toward the end of this
article. The final result of the calculations is that, to first order
in ^{-1}, the rotation angle for a particular galaxy is given by

where is the angle between a fixed
direction "* s*" in space and the line of sight to the
galaxy.

We handled the ambiguity by making the fairly reasonable assumption that any anisotropic polarization rotation must be small. This assumption, and the requirement that the rotation be signed, led us to restrict the angles to have values between - and + only. In order to calculate a value for a polarization rotation , one must assume some initial orientation of the polarization plane at the galaxy. Our choice was the simplest possible - that the polarization plane was initially oriented parallel to the galaxy's major axis.

These relatively simple conditions allow the data analysis to be
manageable. One may assume other initial orientations, or a larger
range for , but such conditions are not
fundamentally different from the simple conditions we employed, and
would not fundamentally change results. With the simple conditions
above, two possible rotations of the initial polarization plane of a
wave emitted from a galaxy will reproduce the observed polarization
plane orientations. One rotation is positive (^{+}) and one is negative (^{-}), as seen in Figure 4.

From Figure 4, we see that the mathematical expressions for
^{+} and ^{-} in terms of
and are

A rotation given by Equation 3 is either positive or negative depending on the angle . Furthermore, depends on the direction of the fixed direction s, and the direction to the galaxy for which is computed. To allow to be either clockwise or counterclockwise, we therefore computed it from the galaxy's , , and values according to the natural assignment

=

where ^{+} and ^{-} are computed from the galaxy's and values according to
Equation 4.

{ [

The assignment in Equation 5 of positive
angles when *r* cos is positive, and
negative angles when *r* cos is negative, necessarily introduces artificial
linear correlations into the data set, because two quadrants of the
data plane of and *r* cos are excluded. This causes the values of the
computed correlation coefficients *R*_{data} to be too
high. In addition, the spatial non-uniformity of the galaxies'
distribution over the sky may have an artificial effect on the value of
a correlation coefficient.

Because of this, the actual value of *R*_{data} is not
very informative. What one needs to do is to compare the correlation
coefficient *R*_{data} of the true data set
(*r*_{i} cos _{i}, _{i}) with the correlation coefficient
*R*_{rand} of a data set that has random values computed in the same
"correlation-producing" way as that of the real data, and that has
the same "non-uniform" *r*_{i} cos _{i} values as those of the true data
set.

We achieved this by computing the correlation coefficient
*R*_{rand} of the set (*r*_{i} cos _{i}, [_{i, rand}, _{i, rand}]), where [_{i, rand}, _{i, rand}] are obtained by substituting
random major axis angles (_{i, rand})
and polarization angles (_{i, rand})
into Equations 4 and 5. We drew _{i,
rand} and _{i, rand} from
uniform, random distributions, consistent with the fact that the _{i} and _{i} in the observed data set are also
uniformly distributed. To account for the non-uniformity of the galaxy
distribution, we did not randomize the *r*_{i} cos _{i} part of the data.

In order to make a reliable comparison with *R*_{data},
one needs to calculate a large number of *R*_{rand} values
by repeatedly drawing _{i, rand} and
_{i, rand} angles to produce several
sets (*r*_{i} cos _{i},
[_{i, rand},
_{i, rand}]), and then calculating
*R*_{rand} for each set. In this way, we computed 1000
"random" correlation coefficients *R*_{rand} to be
compared with the true correlation coefficient *R*_{data}.
The comparison consisted of determining the fraction *P* of the
1000 *R*_{rand} values that equaled or exceeded the
corresponding *R*_{data} value. In statistics, this
fraction is called a "P-value," and the method of computing it by
randomizing the data and computing a large number of
*R*_{rand} values, is called a "Monte Carlo method."
*P* estimates the probability that the correlation
*R*_{data} arises from random fluctuations in the data. An
other way to state this is to say that *P* estimates the
probability that a correlation given by Equation 3 does not exist in
the data.

We computed *R*_{data}, the corresponding 1000
*R*_{rand} values, and the corresponding P-value for the
set of pairs (*r*_{i} cos _{i}, _{i}) for over 400 trial orientations of
the direction * s,* which systematically covered all
directions in space. As one varies

To explore this indication of anisotropy, we selected the galaxies in
the data set that had redshifts greater than 0.3, roughly the most
distant half of the sample (71 galaxies). The electromagnetic radiation
from these galaxies has traversed a large fraction of the universe, and
would serve as an ideal sample for a check of whether the apparent
anisotropy seen in Figure 6 is more or less prevalent when the effects
of distant regions of space are given more emphasis. Surprisingly, we
found an even stronger signal of anisotropy for this galaxy set. We see
in Figure 7 a well-connected cluster of peaks in 1/*P* when
* s* is in the region

As seen in Figures 6 and 7, analysis of the synchrotron radiation data
pinpoints only approximately the orientation * s* which
yields a signal of anisotropy in the data, as quantified by a very
small associated P-value. We may call this s-direction the "direction
of anisotropy." We may visualize the direction by an infinitely long
line, or "anisotropy axis" that runs through the universe through the
two points Earth and Sextans. We see that the data strongly indicate
that the anisotropy direction lies within an "anisotropy cone" that
has its vertex at Earth, and its central axis pointing from Earth to
the constellation Aquila, which is in the direction (declination, right
ascension) = (0

In Figure 8, the double anisotropy cone is shown in red, positioned with its vertex at Earth, at the center of the figure, and opening up toward the constellation Aquila in one direction, and toward the constellation Sextans in the opposite direction. Our data, consisting of 160 radio galaxies, are shown as yellow dots. The most distant galaxies in the data set are about 7 billion light years away.

For the s-direction with highest 1/*P* value of the full data set,
the distribution of *R*_{rand} is a Gaussian, having a
mean = 0.60 and standard deviation = 0.032, with *R*_{data} = 0.66 =
+ 1.88 . In
contrast, in a typical direction away from the anisotropy direction,
like (declination, right ascension) = (60^{}, 12 hours) for example,
the distribution is given by = 0.47 and = 0.04, with *R*_{data} = 0.48 =
+ 0.25 .

For the data set of distant galaxies with *z* 0.3, and for an s-direction yielding a high
1/*P* value, the distribution of *R*_{rand} is also
Gaussian, with a typical mean value of = 0.76
and standard deviation = 0.027, with
*R*_{data} = 0.86 = + 3.7 . Distributions of *R*_{rand} with
long tails were not seen. As mentioned above, the spatial distribution
of galaxies in the sample is non-uniform, so that the number of
galaxies assigned to ^{} in
Equation 4, and the and values for *R*_{rand}, depend on
the trial s-direction. The P-values, displayed in Figures 4 and 5, are
therefore much more meaningful than the values of the correlation
coefficients *R*_{data} themselves.

The average best fit value for the proportionality constant in Equation 3 is =
(1.1 0.08) 10^{25} (*h*_{0} / *h*)
meters for an s-direction of * s* = (declination, right
ascension)

We also employed a second statistical test on the data, a test that is
conceptually disparate from the one described above. For each
"random" data set (*r*_{i} cos _{i}, [_{i, rand}, _{i, rand}]), we varied * s*
over the celestial sphere (410 directions) to maximize

The important test is for the far-half galaxies with redshift *z*
0.3, since that galaxy set exhibited a
P-value of order 0.001 in the first statistical procedure. For the
far-half sample with *z* 0.3, we found
that the fraction of the largest-*R*_{rand}'s that
exceeded *R*_{data} when * s* = (declination,
right ascension)

In particular, the rate of rotation of the polarization plane caused by
the new effect depends on the angle (denoted above) between the direction of travel of the
polarized wave and a fixed direction in space (denoted * s*
above), pointing approximately toward the constellation Aquila from
Earth. The more parallel the direction of straight-line travel of the
wave is with this fixed direction, the greater the rotation of the
polarization plane of the wave (as given by Equation 3 above). The
amount of polarization rotation is also proportional to the distance of
travel of the wave. These are the only two dependencies of the
rotation.

The anisotropic effect is illustrated in Figure 9. In this diagram, Earth is at the center, and the direction toward Sextans is represented by an infinitely long "anisotropy axis" (red). The axis extends from Earth toward Sextans in one direction, and toward the constellation Aquila in the opposite direction. A plane-polarized radio wave emitted by Galaxy A (green) travels in a straight line toward Earth in a direction almost parallel to the anisotropy axis (red). On the other hand, a plane-polarized radio wave emitted by Galaxy B (blue) approaches Earth in a direction almost perpendicular to the anisotropy axis.

As the two waves propagate along straight lines through space, their
planes of polarization rotate around those lines, as represented by the
green and blue helices. The distances of travel are the same for both
waves, but the wave traveling nearly parallel to the anisotropy
direction (green wave) has its polarization plane rotated more than the
wave traveling in a direction nearly perpendicular to the anisotropy
direction (blue wave). In general, we find that the polarization
rotation increases systematically as a wave's direction of travel
approaches that of the fixed anisotropy direction (red line). For
illustrative purposes, the rotation effect in this diagram is
exaggerated. The actual effect is extremely tiny: we find that, on the
average, one full revolution of the polarization plane is completed
after the wave has voyaged for about ten *billion* years (as found
from Equation 3 and the constant above).

It is important to note that the infinite anisotropy axis running
through Aquila, Earth and Sextans, as shown in Figures 7 and 8, only
represents a *direction,* or, in the vernacular of Mathematics, a
*vector,* in space. Any other axis - possibly vastly remote from
Earth, Sextans and Aquila - parallel to the anisotropy axis shown here,
will suffice in defining the anisotropy vector. No particular
*location* in space, like the location of Earth for example, is
relevant - only *directions* are relevant.

We have of course considered the possibility that a local effect of the
galaxy, via some unanticipated conventional physics, might account for
our correlation. However, the fact that the correlation is seen for
*z* 0.3, but not for *z* < 0.3,
rules out a local effect. Strong magnetic fields at a galaxy might
generate unexpected initial polarization orientations, or upset the
Faraday-based fits, and this could plausibly depend on redshift. But
since the correlation is observed over the sky angle , any such explanation requires an unnatural, if
not impossible, conspiracy between distant galaxies at widely separated
sky angles.

One is left, then, with the option of contemplating new physics. In the
language of the Quantum Field Theory branch of Physics, we show that
the anisotropic polarization rotation ,
illustrated in Figure 9, can be generated by a coupling of the
electromagnetic field of the wave, represented by its so-called
"electromagnetic field tensor" *F*^{} and "electromagnetic
four-potential" *A* ^{}, to a
new, four-dimensional vacuum field *s*^{}, whose "spatial part" * s* is
the anisotropy vector we discovered [Nodland 1995; Nodland and Ralston,
1997]. The so-called "Lagrangian density"

The second term in this equation represents an anisotropic extension to
electrodynamics. is here a scale of
dimension length, and ^{} is the
four-dimensional "Levi-Civita tensor." From the so-called
"Euler-Lagrange equations," this Lagrangian density yields a modified
set of Maxwell equations for the electromagnetic field. From these
equations, one obtains the so-called "dispersion relation" between
the wave's wavenumber *k* and frequency given by

We see from Equation 8 that the wave has two propagation modes, one
given by *k*_{+}, and one by *k*_{-}. A
rotation of the plane of polarization of the
total wave arises from the difference in phase speeds between the two
modes, and is given by

Substitution of Equation 8 into Equation 9 finally yields the
anisotropic expression in Equation 3 for the polarization rotation . When subjected to coordinate transformations
such as "time reversal" and "space inversion," the new field
*s*^{} behaves in the same manner
that the intrinsic spin of an atom or elementary particle does, when
the atom or particle is subjected to such transformations. One may
therefore affix some sort of "spin" to *s*^{}.

Other proposed theories and explanations for the anisotropic polarization rotation have recently appeared after our article in Physical Review Letters [Nodland and Ralston, 1997] was published [Bracewell and Eshleman, 1997; Dobado and Maroto, 1997; Kühne, 1997; Obukhov, Korotky and Hehl, 1997; Moffat, 1997; Mansouri and Nozari, 1997; Sachs, 1997].

Over the centuries, we have gradually learned more about the world we
live in. We once thought the Earth was flat, then realized it is a
sphere. We thought the sun revolved around the Earth, then realized the
Earth revolves around the sun. And now most people believe the universe
is *isotropic,* or *directionless* - maybe this is not so either.

In one sense, the anisotropic polarization twist that seems to take place does not really matter in our daily lives. However, part of being human is having an innate curiosity about the world. Who are we, and why are we here? Millions of people around the world ask these questions. A similar yearning drives physicists, who ask the same questions on a more cosmic scale: What is this universe, and how - and why - did it come into existence? I hope that our findings can contribute in some small way to answering these questions, and satisfying the curiosity we all share.

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