Scattering Matrix

  The calculation of Thomson scattering including polarization was first performed by Chandrasekhar [16]; here we show a much simpler derivation employing the spin harmonics. The Thomson differential scattering cross section depends on angle as $\vert\hat{\epsilon}' \cdot \hat{\epsilon}\vert^2$ where $\hat{\epsilon}'$ and $\hat{\epsilon}$ are the incoming and outgoing polarization vectors respectively in the electron rest frame. Radiation polarized perpendicular to the scattering plane scatters isotropically, while that in the scattering plane picks up a factor of $\cos^2\beta$ where $\beta$ is the scattering angle. If the radiation has different intensities or temperatures at right angles, the radiation scattered into a given angle will be linearly polarized.

Now let us evaluate the scattering term explicitly. The angular dependence of the scattering gives

$$\left( \begin{array} {c} \Theta_{\parallel} \\ \Theta_{\perp... ...{\parallel} \\ \Theta_{\perp} \\ U \\ \end{array} \right) \, ,$$ (38)

where the U transformation follows from its definition in terms of the difference in intensities polarized $\pm 45^\circ$ from the scattering plane. With the relations $\Theta=\Theta_\parallel + \Theta_\perp$ and $Q \pm iU = \Theta_\parallel - \Theta_\perp \pm iU$, the angular dependence in the $\vec{T}$ representation of Eqn. (44) becomes,  

$$\vec{T}'= {\bf S}\vec{T} = {3 \over 4}\left( \begin{array} {... ...displaystyle{\dot a \over a}}\\ \end{array}\right) \vec{T} \, ,$$ (39)

where the overall normalization is fixed by photon conservation in the scattering. To relate these scattering frame quantities to those in the frame defined by $\hat{k} = \hat{e}_3$, we must first perform a rotation from the $\hat{k}$ frame to the scattering frame. The geometry is displayed in Fig. 1, where the angle $\alpha$ separates the scattering plane from the meridian plane at $(\theta',\phi')$ spanned by $\hat{e}_r$ and $\hat{e}_\theta$. After scattering, we rotate by the angle between the scattering plane and the meridian plane at $(\theta,\phi)$ to return to the $\hat{k}$ frame. Eqn. (43) tells us these rotations transform $\vec{T}$ as ${\bf R}(\psi)\vec{T} = {\rm diag} (1, e^{2i\psi}, e^{-2i\psi}) \vec{T}$. The net result is thus expressed as  

$${\bf R}(\gamma){\bf S}(\beta){\bf R}(-\alpha) = {1 \over 2}{... ...\vphantom{\displaystyle{\dot a \over a}}\end{array}\right) \, ,$$ (40)

where we have employed the explict spin-2, $\ell=2$ forms in Tab. 1. Integrating over incoming angles, we obtain the collision term in the electron rest frame  

$$\vec{C}[{\vec {T}}]_{\rm rest} = -\dot\tau \vec{T}(\Omega) ... ...f R}(\gamma){\bf S}(\beta){\bf R}(-\alpha) \vec{T}(\Omega')\, ,$$ (41)

where the two terms on the rhs account for scattering out of and into a given angle respectively. Here the differential optical depth $\dot \tau = n_e \sigma_T a$ sets the collision rate in conformal time with n_e as the free electron density and $\sigma_T$ as the Thomson cross section.

The transformation from the electron rest frame into the background frame yields a Doppler shift $\hat{n} \cdot \vec{v}_B$ in the temperature of the scattered radiation. With the help of the generalized addition relation for the harmonics Eqn. (7), the full collision term can be written as  

$$\vec{C}[{\vec {T}}] = -\dot\tau {\vec I}(\Omega) + {1 \ove... ...sum_{m=-2}^2 {\bf P}^{(m)}(\Omega,\Omega') \vec{T}(\Omega')\, .$$ (42)

The vector $\vec{I}$ describes the isotropization of distribution in the electron rest frame and is given by

$$\vec{I}(\Omega) = \vec{T}(\Omega) - \left( \int {d\Omega'\o... ...pi} \Theta' + \hat{n} \cdot \vec{v}_B \, , 0 , \, 0 \right) .$$ (43)

The matrix ${\bf P}^{(m)}$ encapsulates the anisotropic nature of Thomson scattering and shows that as expected polarization is generated through quadrupole anisotropies in the temperature and vice versa 

$$\begin{array} {rcl}\displaystyle{}{\bf P}^{(m)} = \left( \be... ...\displaystyle{\dot a \over a}}\\ \end{array}\right),\end{array}$$ (44)

where $Y_\ell^m {}'\equiv Y_\ell^{m*} (\Omega')$ and $\, {}_{s}^{\vphantom{m}} Y_{\ell}^{m} {}' \equiv \, {}_{s}^{\vphantom{m*}} Y_{\ell}^{m*}(\Omega')$and the unprimed harmonics are with respect to $\Omega$.These $m=0,\pm 1,\pm 2$ components correspond to the scalar, vector and tensor scattering terms as discussed in §IIC and IIIC.