Stokes Parameters

The Boltzmann equation for the CMB describes the transport of the photons under Thomson scattering by the electrons. The radiation is described by the intensity matrix: the time average of the electric field tensor E_i^* E_j over a time long compared to the frequency of the light or equivalently as the components of the photon density matrix (see [19] for reviews). For radiation propagating radially $\vec{E} \perp \hat{e}_r$, so that the intensity matrix exists on the $\hat{e}_\theta \otimes \hat{e}_\phi$ subspace. The matrix can further be decomposed in terms of the $2 \times 2$ Pauli matrices $\sigma_i$ and the unit matrix ${\bf 1}$ on this subspace.

For our purposes, it is convenient to describe the polarization in temperature fluctuation units rather than intensity, where the analogous matrix becomes,  

$${\bf T} = \Theta {\bf 1} + Q \sigma_3 + U \sigma_1 + V \sigma_2.$$ (34)

$\Theta = {\rm Tr}({\bf T} {\bf 1})/2 = \Delta T/T$ is the temperature perturbation summed over polarization states. Since $Q = {\rm Tr}({\bf T}\sigma_3)/2$,it is the difference in temperature fluctuations polarized in the $\hat{e}_\theta$ and $\hat{e}_\phi$ directions. Similarly $U={\rm Tr}({\bf T} \sigma_1)/2$ is the difference along axes rotated by 45$^\circ$, $(\hat{e}_\theta \pm \hat{e}_\phi)/\sqrt{2}$,and $V={\rm Tr}({\bf T} \sigma_2)/2$ that between $(\hat{e}_\theta \pm i \hat{e}_\phi)/\sqrt{2}$.Q and U thus represent linearly polarized light in the north/south-east/west and northeast/southwest-northwest/southeast directions on the sphere respectively. V represents circularly polarized light (in this section only, not to be confused with vector metric perturbations).

Under a counterclockwise rotation of the axes through an angle $\psi$ the intensity ${\bf T}$ transforms as ${\bf T}'={\bf R}{\bf T}{\bf R}^{-1}$. $\Theta$ and V remain distinct while Q and U transform into one another. Since the Pauli matrices transform as $\sigma_3' \pm i \sigma_1' = e^{\mp 2i \psi} (\sigma_3 \pm i \sigma_1)$ a more convenient description is  

$${\bf T} = \Theta {\bf 1} + V \sigma_2 + (Q + iU) {\bf M}_+ + (Q - iU) {\bf M}_- \, ,$$ (35)

where recall that ${\bf M}_{\pm} = (\sigma_3 \mp i \sigma_1)/2$(see Eqn. 1), so that $Q \pm i U$ transforms into itself under rotation. Thus Eqn. (2) implies that $Q \pm i U$ should be decomposed into $s=\pm 2$ spin harmonics [3,4].

Since circular polarization cannot be generated by Thomson scattering alone, we shall hereafter ignore V. It is then convenient to reexpress the matrix as a vector  

$$\vec{T} = (\Theta, Q+iU, Q-iU) \, .$$ (36)

The Boltzmann equation describes the evolution of the vector $\vec{T}$ under the Thomson collisional term $C[\vec{T}]$and gravitational redshifts in a perturbed metric $G[h_{\mu\nu}]$ 

$${d \over d\eta} \vec{T} (\eta,\vec{x},\hat{n}) \equiv {\part... ...bla_i \vec{T} = \vec{C} [{\vec T}] + \vec{G} [h_{\mu\nu}] \, ,$$ (37)

where we have used the fact that $\dot x_i = n_i$ and that in a flat universe photons propagate in straight lines $\dot n=0$.We shall now evaluate the Thomson scattering and gravitational redshift terms.