University of Chicago

Thomson scattering can only produce an *E*-mode locally since the spherical
harmonics that describe the temperature anisotropy have
electric parity.
In Figs. 4, 6, and 8,
the , *m*=0,1,2 *E*-mode patterns are shown in yellow. The *B*-mode
represents these patterns rotated by and are shown in purple
and cannot be generated by scattering.
In this way, the scalars, vectors, and tensors are similar in that scattering
produces a *local* *E*-mode only.

However, the pattern of polarization on the sky is not simply this local
signature from scattering but is modulated over the last scattering surface
by the plane wave spatial dependence of the perturbation
(compare Figs. 3 and 10).
The modulation changes the amplitude, sign, and angular structure of the
polarization but not its nature, e.g. a *Q*-polarization remains *Q*.
Nonetheless, this modulation generates a *B*-mode from the local *E*-mode
pattern.

Fig. 10:
Modulation of the local scalar pattern in Fig. 3 by plane wave fluctuations on the
last scattering surface. Yellow points represent polarization out of the plane of the
page with magnitude proportional to sign. The plane wave odulation changes the
amplitude and the sign of the polarization but does not mix *Q* and *U*.
Modulation can mix *E* and *B* however if *U* is also present.

The reason why this occurs is best seen from the local distinction between
*E* and *B*-modes. Recall that *E*-modes have polarization amplitudes that
change parallel or perpendicular to, and *B*-modes in directions
away from, the polarization direction. On the other hand, plane wave
modulation always changes the polarization amplitude in the direction
or N-S on the sphere.
Whether the resultant pattern possesses *E* or *B*-contributions depends on
whether the local polarization has *Q* or *U*-contributions.

For scalars, the modulation is of a pure *Q*-field and thus its *E*-mode
nature is preserved ([Kamionkowski et al.] 1997; [Zaldarriaga & Seljak] 1997).
For the vectors, the *U*-mode dominates the pattern and the modulation is
crossed with the polarization direction.
Thus vectors generate mainly *B*-modes for short wavelength fluctuations
([Hu & White] 1997).
For the tensors, the comparable *Q* and *U* components of the local pattern
imply a more comparable distribution of *E* and *B* modes at short wavelengths
(see Fig. 11a).

These qualitative considerations can be quantified by noting that plane wave
modulation simply represent the addition of angular momentum from the plane
wave ( ) with the local spin angular dependence.
The result is that plane wave modulation takes the local angular
dependence to higher (smaller angles) and splits the signal into *E*
and *B* components with ratios which are related to Clebsch-Gordan
coefficients. At short wavelengths, these ratios are *B*/*E*=0,6,8/13 in power
for scalars, vectors, and tensors
(see Fig. 11b and [Hu & White] 1997).

The distribution of power in multipole -space is also important.
Due to projection, a single plane wave contributes to a range of angular
scales where *r* is the comoving distance to the last
scattering surface. From Fig. 10, we see that the smallest
angular, largest variations occur on lines of sight
or though a small amount of power
projects to as .
The distribution of power in multipole space of Fig. 11b can be
read directly off the local polarization pattern.
In particular, the region near shown in Fig. 11a
determines the behavior of the main contribution to the polarization power
spectrum.

Fig. 11: The *E* and *B* components of a planewave perturbation
for scalars, vectors and tensors. (a) Modulation of the local *E*-quadrupole
patter (yellow) from scattering by a planewave. Modulation in the direction of (or orthogonal to)
the polarization generates an *E*-mode with higher *l*; modulation in the
crossed (45 degree) direction generates a *B*-mode with higher *l*.
Scalars generate only *E*-modes, vectors mainly *B*-modes, and tensors
comparable amounts of both. (b) Distribution of power in a single plane wave with *kr*=100
in multipole *l* from the addition of spin and orbital angular momentum. Features
in the power spectrum can be read directly off of the pattern in (a).

The full power spectrum is of course obtained by summing these plane wave
contributions with weights dependent on the source of the perturbations and
the dynamics of their evolution up to last scattering.
Sharp features in the *k*-power spectrum will be preserved in the multipole
power spectrum to the extent that the projectors
in Fig. 11b
approximate delta functions.
For scalar *E*-modes, the sharpness of the projection is enhanced due to strong
*Q*-contributions near ( ) that then diminish
as .
The same enhancement occurs to a lesser extent for vector
*B*-modes due to *U*
near and tensor *E*-modes due to *Q* there.
On the other hand, a supression occurs for vector *E* and tensor *B*-modes due
to the
absence of *Q* and *U* at respectively. These considerations have
consequences for the sharpness of features in the polarization power spectrum,
and the generation of asymptotic
``tails'' to the polarization spectrum at low- (see §4.4 and [Hu & White] 1997).