Physics of polarization fluctuations 231orientation of polarization. Note that the transformation law in equation (7.15) is
characteristic not of a vector but of the second-ranktensor
ρ=1
2
(
I+QU−iV
U+iVI−Q)
, (7.17)
which also corresponds to the quantum mechanical density matrix for an
ensemble of photons (Kosowsky 1996). In kinetic theory, the photon distribution
functionf(x,p,t)discussed in section 7.2.2 must be generalized toρij(x,p,t),
corresponding to this density matrix.
7.3.2 Thomson scattering and the quadrupolar source
Non-zero linear polarization in the microwave background is generated around
decoupling because the Thomson scattering which couples the radiation and the
electrons is not isotropic but varies with the scattering angle. The total scattering
cross-section, defined as the radiated intensity per unit solid angle divided by the
incoming intensity per unit area, is given by
dσ
d=
3 σT
8 π∣
∣εˆ′·ˆε∣
∣^2 (7.18)
whereσTis the total Thomson cross section and the vectorsˆεandεˆ′ are
unit vectors in the planes perpendicular to the propogation directions which
are aligned with the outgoing and incoming polarization, respectively. This
scattering cross section can give no net circular polarization, soV = 0for
cosmological perturbations and will not be discussed further. Measurements of
V polarization can be used as a diagnostic of systematic errors or microwave
foreground emission.
It is a straightforward but slightly involved exercise to show that these
relations imply that an incoming unpolarized radiation field with the multipole
expansion equation (7.6) will be Thomson scattered into an outgoing radiation
field with Stokes parameters
Q(nˆ)−iU(nˆ)=3 σT
8 πσB√
π
5a 20 sin^2 β (7.19)if the incoming radiation field has rotational symmetry around its direction of
propagation, as will hold for individual Fourier modes of scalar perturbations.
Explicit expressions for the general case of no symmetry can be derived in terms
of Wigner D-symbols (Kosowsky 1999).
In simple and general terms, unpolarized incoming radiation will be
Thomson scattered into linearly polarized radiation if and only if the incoming
radiation has a non-zero quadrupolar directional dependence. This single fact
is sufficient to understand the fundamental physics behind polarization of the
microwave background. During the tight-coupling epoch, the radiation field has