402 Cosmic microwave background anisotropies
9.10.4 EandBpolarization modes and correlation functions
To analyze the field of temperature fluctuations we computed the temperature
autocorrelation function. The polarization induced at the last scattering surface
is characterized by the tensorfieldPab(n)on the celestial sphere. The induced
polarization is correlated at different points on the sphere and, as with the
temperature fluctuation field, can be characterized by the correlation functions
〈Pab(n 1 )Pcd(n 2 )〉.
The symmetric traceless tensorPab(n)has two independent components and
therefore, instead ofPab(n)itself, it is more convenient to consider two independent
scalar functions built out of the polarization tensor:
E(n)≡Pab;ab, B(n)≡Pab;accb, (9.153)where ; denotes the covariant derivative on the two-dimensional sphere with metric
(9.140) and
cb≡√
g(
01
− 10
)
(9.154)
is the two-dimensional skew-symmetric Levi–Civita “tensor.” It behaves as a tensor
only under coordinate transformations with positive Jacobian. Under reflections,
cbchanges sign. Therefore, only theEmode of polarization is a scalar, while the
Bmode is a pseudo-scalar, reminiscent of an electric (E) and magnetic (B) field
respectively.
The most important thing is that theBmodeis not generated by scalar perturba-
tions. To prove this let us consider the polarization induced by inhomogeneity with
wavenumberk.We use the particular spherical coordinate system where thez-axis
determining the Euler angleθcoincides with the directionk.In this coordinate sys-
tem, the direction of observationnis characterized by the polar anglesθandφ,and
k·n=kcosθ.For everynwe can use the orthogonal coordinate vectorseθ(n)and
eφ(n), tangential to the coordinate lines on the celestial sphere, as the polarization
basis. As is clear from (9.152), thel-dependence of the temperature of the incident
radiation appears only in the combinationk·l.Therefore, from (9.151) we may
easily infer that the nondiagonal component of the polarization tensor should be
proportional to
Pθφ(θ,φ)∝(k·eθ)(
k·eφ)
. (9.155)
This component vanishes because in our coordinate system the vectoreφ(n)is
transverse tokat every point on the sphere. The diagonal components ofPcan
depend only onk·eθ,k·nand the metric (all of which areφ-independent) and