#
waynehu

## Professor,
Department of Astronomy and Astrophysics

University of Chicago

Tensor fluctuations are transverse-traceless perturbations to the metric,
which can be viewed as gravitational waves. A plane gravitational wave
perturbation represents a quadrupolar ``stretching'' of space in the plane
of the perturbation (see Fig. 7).
As the wave passes or its amplitude changes, a circle of test particles in
the plane is distorted into an ellipse whose semi-major axis
semi-minor axis as the spatial phase changes from crest
trough (see Fig. 7, yellow ellipses).
Heuristically, the accompanying stretching of the wavelength of photons
produces a quadrupolar temperature variation with an pattern

in the coordinates defined by .

Fig. 7: The tensor quadrupole moment (*l*=2, *m*=2).
Since gravity waves distort space in the plane of the perturbation, changing a
circle of test particles into an ellipse, the radiation acquires an
*m*=2 quadrupole moment.

Thomson scattering again produces a polarization pattern from the quadrupole
anisotropy. At the equator, the quadrupole pattern intersects the tangent
( ) plane with hot and cold lobes rotating
in and out of the direction with the azimuthal angle .
The polarization pattern is therefore purely *Q* with a
dependence.
At the pole, the quadrupole lobes lie completely in the polarization plane
and produces the maximal polarization unlike the scalar and vector cases.
The full pattern,

is shown in Fig. 8 (real part).
Note that *Q* and *U* are present in nearly equal amounts for the tensors.

Fig. 8: Polarization pattern for *l*=2, *m*=2.
Scattering of a tensor perturbation generates the *E* pattern
(yellow, thick lines) as opposed to the *B* (purple, thin lines) pattern.

Animation: Same as for scalars.

** Next:** Polarization Patterns