9.10 Polarization of the cosmic microwave background 403therefore, the general form of the polarization tensor is
Pab(θ,φ)=(
Q(θ) 0
0 −Q(θ)sin^2 θ)
, (9.156)
where we have taken into account thatPaa= 0.
Problem 9.22CalculateEandBfor the polarization tensor given by (9.156) and
verify thatB=0 in this case.
BecauseB(θ,φ)is a pseudo-scalar function, it does not depend on the coordinate
system used to calculate it and vanishes for every mode of the density perturbations.
Thus density perturbations generate onlyEmode polarization, which describes the
component of the polarization with even parity (the scalar perturbation with given
kis symmetric with respect to rotations aroundkand therefore has no handedness).
It is easy to see that forPabgiven by (9.156) the appropriate polarization vectors
are proportional toeθat every point whereQ(θ)>0 and proportional toeφfor
Q(θ)< 0 .Therefore, in terms of polarization patterns, the Emode produces
arrangements of polarization vectors which are oriented radially or tangentially to
the circles with respect to the density perturbations, respecting their axial symmetry,
as illustrated in Figure 9.6.
In contrast with scalar perturbations, gravitational waves also generateBmode
polarization. To see this, let us consider a gravitational wave with wavenumberk
in a coordinate system where thez-axis is aligned alongk. Taking into account the
general structure of the temperature fluctuations induced by gravitational waves
(see (9.119), (9.120)), we can infer from (9.151) that the nondiagonal component
of the polarization tensor is proportional to
Pθφ∝eik(eθ)i(
eφ)k
, (9.157)whereeikis the polarization tensor of the gravitational waves.
Epolarization Bpolarization
Fig. 9.6.