188 Cosmic Microwave Background
where휃is the scattering angle. By symmetry, Thomson scattering can generate no
circular polarization, so푉=0always.
The net polarization produced in the directionzfrom an incoming field of intensity
퐼′(휃,휙)is determined by integrating Equation (8.31) over all incoming directions. Note
that the coordinates for each incoming direction must be rotated some angle휙about
the푧-axis as in Equation (8.31), so that the outgoing Stokes parameters all refer to a
common coordinate system. The result is then [9]
퐼(z)=^1
2퐾
∫
d훺( 1 +cos^2 휃)퐼′(휃,휙), (8.32)푄(z)−i푈(z)=1
2
퐾∫ d훺sin^2 휃e2i휙퐼′(휃,휙). (8.33)Expanding the incident radiation intensity in spherical coordinates,
퐼′(휃,휙)=∑
퓁푚푎퓁푚푌퓁푚(휃,휙), (8.34)
leads to the following expressions for the outgoing Stokes parameters:
퐼(z)=1
2
퐾
(
8
3
√
휋푎 00 +
4
3
√
1
5
휋푎 20
)
, (8.35)
푄(z)−i푈(z)= 2 퐾√
2 휋
15
푎 22. (8.36)
Thus, if there is a nonzero quadrupole moment푎 22 in the incoming, unpolarized
radiation field, it will generate linear polarization in the scattering plane. To determine
the outgoing polarization in some other scattering direction,n, making an angle훽
withz, one expands the incoming field in a coordinate system rotated through훽.This
derivation requires too much technical detail to be carried out here, so we only state
the result [9]:
푄(n)−i푈(zn)=퐾√
1
5
휋푎 20 sin^2 훽. (8.37)Multipole Analysis. The tensor harmonic expansion in Equation (8.20) for the radi-
ation temperature푇 and the temperature multipole components푎T(퓁푚)in in Equa-
tion (8.21) can now be completed with the corresponding expressions for the polar-
ization tensor푃. From the expression in Equation (8.29) its components are
푃ab(n)=1
2
(
푄(n)−푈(n)sin휃
−푈(n)sin휃 −푄(n)sin^2 휃)
=푇 0
∑∞
퓁= 2∑퓁
푚=−퓁[
푎E(퓁푚)푌(E퓁푚)ab(n)+푎B(퓁푚)푌(B퓁푚)ab(n)]
. (8.38)
The existence of the two modes (superscripted) E and B is due to the fact that the
symmetric traceless tensor in Equation (8.38) describing linear polarization is spec-
ified by two independent Stokes parameters,푄and푈. This situation bears analogy
with the electromagnetic vector field, which can be decomposed into the gradient