Boltzmann Equation

  The Boltzmann equation describes the evolution in time $(\eta)$ of the spatial ($\vec{x}$) and angular ($\hat{n}$) distribution of the radiation under gravity and scattering processes. In the notation of [1], it can be written implicitly as  

$${d \over d\eta} \vec{T}(\eta,\vec{x},\hat{n}) \equiv {\part... ...vec{T}_{\vert i} = \vec{C}[\vec{T}] + \vec{G}[h_{\mu\nu}] \, ,$$ (14)

where $\vec{T} = (\Theta, Q+iU, Q-iU)$ encapsulates the perturbation to the temperature $\Theta=\Delta T/T$ and the polarization (Stokes Q and U parameters) in units of the temperature fluctuation. The term $\vec{C}$ accounts for collisions, here Compton scattering of the photons with the electrons, while the term $\vec{G}$ accounts for gravitational redshifts.