Evolution Equations

It is now straightforward to rewrite the Boltzmann equation (19) as the evolution equations for the amplitudes of the normal modes of the temperature and polarization $\vec{T}_\ell^{(m)} = (\Theta^{(m)}_\ell, E^{(m)}_\ell, B_\ell^{(m)})$.The gravitational sources and scattering sources of these equations follow from Eq. (20) and (21) by noting that the spin harmonics are orthogonal,

$$\int d\Omega\ (\, {}_{s}^{\vphantom{m}} {Y}_{\ell}^{m})(\, {... ...'}} {Y}_{\ell'}^{m'}{}^*) = \delta_{\ell,\ell'} \delta_{m m'}.$$ (26)

The term $n^i \vec{T}_{\vert i}$ is evaluated by use of the coupling relation Eq. (25) for $n^i (\, {}_{s}^{\vphantom{m}} {G}_{\ell}^{m})_{\vert i}$.It represents the fact that spatial gradients in the distribution become orbital angular momentum as the radiation streams along its trajectory $\vec{x}(\hat{n})$.For example, a temperature variation on a distant surface surrounding the observer appears as an anisotropy on the sky. This process then simply reflects a projection relation that relates distant sources to present day local anisotropies.

With these considerations, the temperature fluctuation evolves as  

$$\dot\Theta_\ell^{(m)} = q \Bigg[ {\, {}_{0}^{\vphantom{m}} {... ...ot\tau \Theta_\ell^{(m)} + S_\ell^{(m)}, \qquad (\ell \ge m),$$ (27)

and the polarization as 

$$\begin{array} {rcl}\dot E_\ell^{(m)} &=& q \Bigg[ {\, {}_{... ...} B_{\ell + 1}^{(m)} \Bigg] -\dot\tau B_\ell^{(m)}.\end{array}$$

The temperature fluctuation sources in Newtonian gauge are

$$\begin{array} {lll} S_0^{(0)} = \dot\tau \Theta_0^{(0)} - \d... ... \dot H \vphantom{\displaystyle{\dot a \over a}}\, ,\end{array}$$ (28)

and in synchronous gauge,  

$$\begin{array} {lll} S_0^{(0)} = \dot\tau \Theta_0^{(0)} - \d... ... \dot H \vphantom{\displaystyle{\dot a \over a}}\, ,\end{array}$$ (29)

The $\ell=m=2$ source doesn't contain a curvature factor because we have recursively defined the basis functions in terms of the lowest member, which is $\ell=2$ in this case. In the above  

$$P^{(m)} = {1 \over 10} \left[ \Theta_2^{(m)} - \sqrt{6} E_2^{(m)} \right].$$ (30)

and note that the photon density and velocities are related to the $\ell=0,1$moments as

$$\delta_\gamma = 4\Theta_0^{(0)}$\,, \qquad $v_\gamma^{(m)}= \Theta_1^{(m)} \, ;$$ (31)

whereas the anisotropic stresses are given by

$$\pi_\gamma^{(m)} Q_{ij}^{(m)} = 12 \int {d \Omega \over 4\pi}\ (n_i n_j-{1\over 3}\gamma_{ij})\Theta^{(m)},$$ (32)

which relates them to the quadrupole moments ($\ell=2$) as

$$(1-3K/k^2)^{1/2} \pi_\gamma^{(0)}= {12 \over 5}\Theta_2^{(0)... ...a_2^{(1)}, \qquad \pi_\gamma^{(2)}= {8\over 5} \Theta_2^{(2)}.$$ (33)

The evolution of the metric and matter sources are given in Appendices A3--A5.