Vector perturbations represent vortical motions of the matter, where and , similar to ``eddies'' in water. There is no associated density perturbation, and the vorticity is damped by the expansion of the universe as are all motions that are not enhanced by gravity. However, the associated temperature fluctuations, once generated, do not decay as both and T scale similarly with the expansion. For a plane wave perturbation, the velocity field with direction reversing in crests and troughs (see Fig. 5). The radiation field at these extrema possesses a dipole pattern due to the Doppler shift from the bulk motion. Quadrupole variations vanish here but peak between velocity extrema. To see this, imagine sitting between crests and troughs. Looking up toward the trough, one sees the dipole pattern projected as a hot and cold spot across the zenith; looking down toward the crest, one sees the projected dipole reversed. The net effect is a quadrupole pattern in temperature with
The lobes are oriented at 45 from and since the line of sight velocity vanishes along and at 90 degrees to here. The latter follows since midway between the crests and troughs itself is zero. The full quadrupole distribution is therefore described by , where i represents the spatial phase shift of the quadrupole with respect to the velocity.
Fig. 5: The vector quadrupole moment (l=2, m=1). Since v is perpendicular to k, the Doppler effect generates a quadrupole pattern with lobes 45 degrees from v and k that is spatially out of phase (interplane peaks) with v
Thomson scattering transforms the quadrupole temperature anisotropy into a local polarization field as before. Again, the pattern may be visualized from the intersection of the tangent plane to with the lobe pattern of Fig. 5. At the equator ( ), the lobes are oriented from the line of sight and rotate into and out of the tangent plane with . The polarization pattern here is a pure U-field which varies in magnitude as . At the pole , there are no temperature variations in the tangent plane so the polarization vanishes. Other angles can equally well be visualized by viewing the quadrupole pattern at different orientations given by .
The full , m=1 pattern,
is displayed explicitly in Fig. 6 (yellow lines, real part). Note that the pattern is dominated by U-contributions especially near the equator. The similarities and differences with the scalar pattern will be discussed more fully in §3.
Fig. 6: Polarization pattern for l=2, m=1. The scattering of a vector (m=1) quadrupole perturbation generates the E pattern (yellow, thick lines) as opposed to the B pattern, (purple, thin lines).
Animation: Same as for scalars
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