Professor, Department of Astronomy and Astrophysics
University of Chicago

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Tensor Perturbations


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 tex2html_wrap_inline1226 semi-minor axis as the spatial phase changes from crest tex2html_wrap_inline1226 trough (see Fig. 7, yellow ellipses). Heuristically, the accompanying stretching of the wavelength of photons produces a quadrupolar temperature variation with an tex2html_wrap_inline1230 pattern


in the coordinates defined by tex2html_wrap_inline1086 .

  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 ( tex2html_wrap_inline1098 ) plane with hot and cold lobes rotating in and out of the tex2html_wrap_inline1064 direction with the azimuthal angle tex2html_wrap_inline1198 . The polarization pattern is therefore purely Q with a tex2html_wrap_inline1246 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