Background
The CMB (cosmic microwave background)
consists of light emitted a mere 380 thousand years after the
big bang when the universe was less than 0.002% of its current age. This light comes from what is known as the surface
of last scattering  the time when the universe cooled enough to allow
protons to capture electrons to form atoms. This greatly
reduced the frequency with which electrons interacted with the photons, so
the photons could travel unhindered and the universe became transparent.
Recent experiments (WMAP, DASI)
have shown that this light is ever so slightly
polarized, due primarily to the motion of the primordial plasma
just before last scattering. This polarization
holds the key to seeing past the surface of last scattering to the earliest
moments in our universe.
Visualizing the Polarization
To get an idea of what we mean
by polarization, let’s say you look at a single point in the
sky (or on the map generated by WMAP). At
that point, you see an electromagnetic wave that is traveling directly
toward you. The wavevector would look like a point
since you’re viewing it headon.
The wave’s electric and magnetic
field vectors (E and B) are perpendicular, so
we can just talk about the polarization in terms of the direction
that the Efield points.
Summing over all incident waves,
the Efields are roughly equal in all directions, but
not quite. There will be one direction that has a slightly greater
magnitude of E than the other directions (see
figure to the left).
We
can represent polarization as a line with length proportional
to the excess magnitude in that direction and at an angle such
that it is aligned with the direction of largest E. This line is not a vector, however, because we have
no meaningful way of saying which way this line “points.” It only takes a rotation of 180 degrees
to come around to the same orientation, as opposed to 360 degrees
for a vector. Thus, the polarization field is some sort of tensor field
rather than a vector field. Here (below) is
what a possible polarization field looks like on a small patch of CMB. (Image by Seljak and Zaldarriaga.



This graph shows how Q and U depend on
the angle between the polarization vector and the xaxis. Since the polarization vector is invariant
under rotation by 180 degrees, it is really only necessary to look
at half of the plot. One sees that Q and U are
out of phase by 45 degre
This image shows the definition of the
angle Q used in the graph
of Q and U. Note that the value of Q depends on both the polarization direction
(shown by the bluish bar at an angle Q from the xaxis)
and the orientation of the axes. The
little red arrows on the axes and on the polarization vector
point in the direction of increasing Q if the axes
or the polarization vector were rotated.
Stokes Parameters
The Stokes parameters. I is a measure of the intensity of the radiation,
while Q and U measure the radiation’s linear polarization. Q and U can be viewed as the same measurement made with
respect to two different sets of axes with one set rotated by 45 degrees
with respect to the other. The circular polarization
of the radiation is described by V. In
these equations, E_{x} and E_{y} are the amplitudes of the x
and y components of the Efield. The
phase difference is f.


Which Way is Up?
IAU conventions define the
axes used for polarization measurements. The xaxis points towards the
North Celestial Pole, the point on the sky that is directly overhead at
the Earth’s North Pole. The North Star, or Polaris, is very near to this
point on the sky. The yaxis is orthogonal to the xaxis; it is chosen
so that in a right handed coordinate system, the zaxis points from the point
on the sky to the observer as shown in the picture on the right. The polarization
directions in which Q and U are maximized or minimized
are also shown; for example, if there is a +Q next to a line, then Q is
maximized when the polarization direction is along that line. (Night sky
background from
http://www.emit.org/nightsky.html).
What are E and Bmodes?
If you think of the vector field
for a static electric field, it has the interesting characteristic
that its curl is always zero. If you look
at a vector field for a static magnetic field, the divergence is
zero. In fact, one can break a vector field down
into components  one with zero curl and one with zero divergence.
In a similar way, we can break
down the polarization tensor field into two components, which
we call E and B modes. We liken Emodes to
a field with zero curl (like the static electric field), and Bmodes
to a field with zero divergence (like the static magnetic field),
although technically divergence and curl are not defined on this
tensor field. (This has to do with the fact that
the polarization tensors don’t point in a unique direction). Appropriate
combinations of derivatives are used to define operations analagous
to divergence and curl.
Most CMB polarization is in
the form of Emodes. It is possible that
Bmodes also exist, and could be generated by two sources. One source would be gravity waves resulting
from the the violent ripping of spacetime that is predicted by inflation,
a theory that has not been confirmed. The
other source is gravitational lensing  or the bending of light
due to matter (including dark matter). Gravity
waves in the early universe show up as Bmodes on large angular
scales, while the gravitational lensing effects show up on smaller
scales (because the lensing is done by clusters, which are smaller).
Finding experimental evidence for gravity
waves in the early universe would be a monumental discovery, and
better understanding of gravitational lensing can help solve questions
about dark matter and dark energy in the postlastscattering universe. These effects have as of yet not been detected, which
is actually good, because Bmodes from both these sources are
expected to be small and none of the current experiments has come
close to the sensitivity needed to see them. The
figures below show predicted CMB power spectra. The
ordinate is temperature while the abscissa is the ell of spherical
coordinates, a quantity roughly proportional to p/q, where q is the characteristic angular size
of fluctuations.
Patterns in the Plasma
Before the surface of last
scattering, the universe was a hot plasma, with electrons and photons frantically
colliding. The CMB that we observe today is a snapshot
of the last light rays that scattered off this plasma just as it was cooling
enough for electrons to become bound in atoms. By
observing the properties of this light, we can draw conclusions about the
motion and densities of electrons (and other matter) at that time.
A fascinating thing about polarization
in the CMB is that it gives us information about the velocity
of the plasma at the surface of last scattering.
Looking at the patterns of this motion can tell
us important information about the kind of oscillations that the
plasma was undergoing  largescale cosmic sound waves. The wavelength
of a sound wave could not exceed the size of the universe at the
time of the oscillation. Thus the
frequency spectrum of these sound waves provides information about the
expansion of the universe. Also, certain patterns
in the polarization, if present, could indicate the presence of gravity
waves in the early universe. The waves that categorize
these patterns are called “Bmodes”, described in further detail above.
How Polarization is Made
To look at how velocity gives rise to polarization,
imagine light that is emitted from a specific electron. An electron in the plasma is bombarded by radiation
from all directions, and the oscillating electric fields from
different waves cause it to vibrate back and forth, emitting electromagnetic
radiation (the basic principle behind scattering). If the incident
radiation from all directions is uniform in intensity and unpolarized,
then the electron vibrates equally in all directions, and the
net radiation it emits is isomorphic and unpolarized. However, if
the incident radiation is not uniform, the emitted radiation can be
polarized, even if the incident radiation is not.
To better visualize this
effect, we can look at an electron with two waves with different
magnitudes coming at it from perpendicular directions (see figure
to the right). The electron emits light in all directions, but
for simplicity, we only look at light emitted in the Y direction.
The electric field of the emitted wave comes from the oscillations
of the electron in the XZ plane (perpendicular to the wave vector).
Only the incident Efields in this plane (blue and red) influence the
electron’s motion in a way that is translated into the scattered waves. So if the incident light is stronger in one direction,
the Efield of the emitted light in the corresponding direction will
also be stronger (like the red wave in the picture) and hence polarized.
In the early universe,
the motion of the plasma causes incident
radiation in one direction to be stronger than
in other directions. In the figure,
the wave emitted in the +Y direction has Q>0 because E_{x}^{2} > E_{y}^{2}.
If we look at the velocity field of this plasma around a gravity
well (high density), it will look like the picture below. The
yellow circle is an electron, moving along with the plasma. If we
look at this electron’s rest frame,
it will appear that the plasma (including the incident radiation) is
moving away in the vertical direction, but not in the horizontal. This means that the light in this direction is red shifted
so it has lower intensity than light in the horizontal direction (see
figure to the left). The end result is that the
electron emits light that is polarized in the horizontal direction (perpendicular
to its velocity).

Last
updated 8/4/03
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