Physical Foundations of Cosmology

9.10 Polarization of the cosmic microwave background 405

In deriving (9.160) we integrated twice by parts and took into account that the

polarization tensor is traceless.

Likewise, we have

a ̃lmB=

1

Nl

∫

PabYlmB∗(ab)d^2 n, (9.162)

in terms of the normalizedB-type tensor harmonics

YlmB(ab)≡

Nl

2

(

Ylm;accb+Ylm;cbca

)

. (9.163)

Note that theE- andB-type tensor harmonics only exist forl>1 and, taken

together, form a complete orthonormal basis for second rank tensors on the sphere.

Therefore, the polarization tensor can be expanded as

Pab=

∑

lm

[

almEYlmE(ab)+almBYlmB(ab)

]

, (9.164)

where, as follows from (9.160) and (9.162),almE,B=Nla ̃lmE,B. Thus, instead of first

calculating the second derivative of the polarization tensor and then expanding it

in terms of the scalar spherical harmonics, we can simply expand the polarization

tensor itself in terms of the tensor harmonics. Then, in addition to the usualCl≡

〈alm∗alm〉characterizing the temperature fluctuations, the polarization of the CMB

fluctuations can be described in terms of the sequence of multipoles:

ClBT=〈almB∗alm〉,ClET=〈almE∗alm〉,ClEE=〈almE∗almE〉,

ClBB=〈almB∗almB〉,ClEB=〈almE∗almB〉.

(9.165)

Problem 9.24Find the explicit expressions for the tensor spherical harmonics in
terms of the usual scalar spherical harmonics.
Although the tensor harmonics are technically more complicated than the scalar
harmonics, the point is that, given the orthogonality relations, the analysis of the
polarization correlation functions is exactly parallel to that of the correlation func-
tion for the temperature fluctuations. In Figure 9.7 we present the numerical result
for the concordance model. In the models without reionization thel-dependence of
l(l+ 1 )ClEEandl(l+ 1 )ClBBcan easily be understood if we take into account that
the polarization is proportional to the quadrupole component of the temperature
fluctuations at recombination. In turn, this quadrupole component is mainly due to
the perturbations which, at this time, have scales of the order of the horizon and
smaller. Therefore, the correlation functionsl(l+ 1 )ClEEandl(l+ 1 )ClBBdrop
off forl< 100 ,corresponding to the superhorizon scales at recombination. We
have seen above that the amplitude of the gravitational waves, and their contribu-
tion to the quadrupole component of the temperature fluctuations at recombination,