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The temperature and polarization fluctuations
are expanded into the normal modes defined
in §IIB,^{}

| |
(47) |

A comparison with Eqn. (9) and
(43) shows
that and represent
polarization with electric and magnetic type parities
respectively [3,4]. Because the temperature
has electric type parity, only
couples
directly to the temperature in the scattering sources.
Note that and represent polarizations
with *Q* and *U* interchanged and thus represent polarization
patterns rotated by . A simple example is given by
the *m*=0 modes. In the -frame,
represents a pure *Q*,
or north/south-east/west, polarization field
whose amplitude depends on
, e.g. for . represents a pure *U*, or northwest/southeast-northeast/southwest,
polarization with the
same dependence.
The power spectra of temperature and polarization anisotropies
today are defined as, e.g. for with the average being over
the () *m*-values. Recalling the normalization of the
mode functions from Eqn. (10) and
(11), we obtain

| |
(48) |

where *X* takes on the values , *E* and *B*.
There is no cross correlation or
due to parity [see Eqns. (6) and
(24)].
We also employ the notation for the *m* contributions
individually.
Note that here due to
azimuthal symmetry in the transport problem so that .
As we shall now show, the modes are stimulated by
scalar, vector and tensor perturbations in the metric. The
orthogonality of the spherical harmonics assures us that these modes
are independent, and we now discuss the contributions separately.

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*Wayne Hu*

*9/9/1997*