Free Streaming

As the radiation free streams, gradients in the distribution produce anisotropies. For example, as photons from different temperature regions intersect on their trajectories, the temperature difference is reflected in the angular distribution. This effect is represented in the Boltzmann equation (45) gradient term,

$$\hat{n} \cdot \vec{\nabla} \rightarrow i\hat{n} \cdot \vec{k} = i\sqrt{4\pi \over 3}k Y_1^0 \, .$$ (49)

which multiplies the intrinsic angular dependence of the temperature and polarization distributions, $Y_\ell^m$ and $\, {}_{\pm 2}^{\vphantom{m}} Y_{\ell}^{m}$ respectively, from the expansion Eqn. (55) and the angular basis of Eqns. (10) and (11). Free streaming thus involves the Clebsch-Gordan relation of Eqn. (8) 

$$\begin{array} {rcl}\displaystyle{}\sqrt{4 \pi \over 3} Y_1^0... ...\left(\, {}_{s}^{\vphantom{m}} Y_{\ell+1}^{m}\right)\end{array}$$ (50)

which couples the $\ell$ to $\ell\pm 1$ moments of the distribution. Here the coupling coefficient is  

$$\, {}_{s}^{\vphantom{m}} \kappa_{\ell}^{m} = \sqrt{(\ell^2-m^2)(\ell^2-s^2)/\ell^2}\, .$$ (51)

As we shall now see, the result of this streaming effect is an infinite hierarchy of coupled $\ell$-moments that passes power from sources at low multipoles up the $\ell$-chain as time progresses.