Physical Foundations of Cosmology

404 Cosmic microwave background anisotropies

Problem 9.23Using (9.121), verify that after averaging over the random polariza-
tionseik, the component〈Pθφ^2 〉does not vanish and depends only onθ. Calculate
theBmode polarization in the case of nondiagonalPab(θ)and show that it is
generically different from zero.

The polarization vectorpainduced by the gravitational wave is a linear combi-
nation ofeθandeφ.Therefore, the polarization vectors are oriented in circulating
patterns, as illustrated in Figure 9.6. In this case theBmode, which has odd parity
(handedness), does not vanish. This is due to the fact that the gravitational wave
is not symmetric with respect to the rotations aroundk.Hence, the gravitational
waves present at recombination can be detected indirectly via theBmode of the
CMB polarization.
To characterize the polarization field on today’s sky one can use the appropriate
correlation functions, for instance,

CET(θ)≡

〈

E(n 1 )

δT

T

(n 2 )

〉

, (9.158)

where the averaging is performed over all directions on the sky satisfying the
conditionn 1 ·n 2 =cosθ.The other correlation functions areCBT,CEE,CBBand
CEB.As in the case of temperature fluctuations, the polarizationsE(n)andB(n)
can be expanded in terms of the scalar spherical harmonics:

E=

∑

l,m

a ̃lmEYlm(θ,φ), B=

∑

l,m

a ̃lmBYlm(θ,φ). (9.159)

Since we directly measure the polarization tensor itself, however, it is not very prac-
tical to take second derivatives of the experimental data to calculate the coefficients
a ̃lm.Instead we note that

a ̃lmE =

∫

E(n)Ylm∗(n)d^2 n=

∫

Pab;abYlm∗d^2 n=

1

Nl

∫

PabYlmE∗(ab)d^2 n, (9.160)

where

Nl≡

√

2(l−2)!

(l+2)!

and

YlmE(ab)≡Nl

(

Ylm;ab−^12 gabYlm c;c

)

are theE-type tensor harmonics which obey orthogonality relations analogous to
the scalar spherical harmonics:
∫
YlmE∗(ab)YlE′m′(ab)d^2 n=δll′δmm′. (9.161)