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waynehu

Professor, Department of Astronomy and Astrophysics
University of Chicago

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Electric and Magnetic Modes

 

Any polarization pattern on the sky can be separated into ``electric'' (E) and ``magnetic'' (B) components. This decomposition is useful both observationally and theoretically, as we will discuss below. There are two equivalent ways of viewing the modes that reflect their global and local properties respectively. The nomenclature reflects the global property. Like multipole radiation, the harmonics of an E-mode have tex2html_wrap_inline1284 parity on the sphere, whereas those of a B-mode have tex2html_wrap_inline1288 parity. Under tex2html_wrap_inline1290 , the E-mode thus remains unchanged for even tex2html_wrap_inline1272 , whereas the B-mode changes sign as illustrated for the simplest case tex2html_wrap_inline1298 in Fig. 9 (recall that a rotation by 90 tex2html_wrap_inline1160 represents a change in sign). Note that the E and B multipole patterns are tex2html_wrap_inline1128 rotations of each other, i.e. tex2html_wrap_inline1308 and tex2html_wrap_inline1310 . Since this parity property is obviously rotationally invariant, it will survive integration over tex2html_wrap_inline1086 .

  Fig. 9: The electric (E) and magnetic (B) modes are distinguished by their behavior under a parity transformation n → -n. E modes have (-1)l parity and B modes have (-1)l+1, here (l=2, m=0), even and odd respectively. The local distinction between the two is that the polarization direction is aligned with the principal axes of the polarization amplitude for E and crossed (45 degrees) for B. Dotted lines represent a sign reversal in the polarization.

The local view of E and B-modes involves the second derivatives of the polarization amplitude (second derivatives because polarization is a tensor or spin-2 object). In much the same way that the distinction between electric and magnetic fields in electromagnetism involves vanishing of gradients or curls (i.e. first derivatives) for the polarization there are conditions on the second (covariant) derivatives of Q and U. For an E-mode, the difference in second (covariant) derivatives of U along tex2html_wrap_inline1062 and tex2html_wrap_inline1064 vanishes as does that for Q along tex2html_wrap_inline1352 and tex2html_wrap_inline1354 . For a B-mode, Q and U are interchanged. Recalling that a Q-field points in the tex2html_wrap_inline1062 or tex2html_wrap_inline1064 direction and a U-field in the crossed direction, we see that the Hessian or curvature matrix of the polarization amplitude has principle axes in the same sense as the polarization for E and 45 tex2html_wrap_inline1160 crossed with it for B (see Fig. 9). Stated another way, near a maximum of the polarization (where the first derivative vanishes) the direction of greatest change in the polarization is parallel/perpendicular and at tex2html_wrap_inline1128 degrees to the polarization in the two cases.

The distinction is best illustrated with examples. Take the simplest case of tex2html_wrap_inline1042 , m=0 where the E-mode is a tex2html_wrap_inline1384 field and the B-mode is a tex2html_wrap_inline1388 field (see Fig. 4). In both cases, the major axis of the curvature lies in the tex2html_wrap_inline1062 direction. For the E-mode, this is in the same sense; for the B-mode it is crossed with the polarization direction. The same holds true for the m=1,2 modes as can be seen by inspection of Fig. 6 and 8.

Next: Electric and Magnetic Spectra