Integral Solutions

  The Boltzmann equations have formal integral solutions that are simple to write down by considering the properties of source projection from §IIB. The hierarchy equations for the temperature distribution Eqn. (60) merely express the projection of the various plane wave temperature sources $S_\ell^{(m)} G_\ell^m$ on the sky today (see Eqn. (61)). From the angular decomposition of $G_\ell^m$ in Eqn. (14), the integral solution immediately follows  

$${\Theta_\ell^{(m)}(\eta_0,k) \over 2\ell + 1}\, = \int_0^... ...S_{\ell'}^{(m)}(\eta) \, j_\ell^{(\ell'm)}(k(\eta_0-\eta)) \, .$$ (61)

Here

$$\tau(\eta) \equiv \int_\eta^{\eta_0} \dot\tau(\eta') d\eta'$$ (62)

is the optical depth between $\eta$ and the present. The combination $\dot\tau e^{-\tau}$ is the visibility function and expresses the probability that a photon last scattered between $d\eta$ of $\eta$ and hence is sharply peaked at the last scattering epoch.

Similarly, the polarization solutions follow from the radial decomposition of the

$$-\sqrt{6} \dot\tau P^{(m)} \left[ {}_{2}^{\vphantom{2}} G_2^m {\bf M}_+ +{}_{-2}^{\vphantom{2}} G_2^m {\bf M}_- \right]$$ (63)

source. From Eqn. (24), the solutions, 

$$\begin{array} {rcl}\displaystyle{}{E^{(m)}_\ell(\eta_0,k) \o... ...m{\ell}} (\eta)\beta_\ell^{(m)}(k(\eta_0-\eta)) \, .\end{array}$$

immediately follow as well.

Thus the structures of $j_\ell^{(\ell'm)}$, $\epsilon_\ell^{(m)}$,and $\beta_\ell^{(m)}$ shown in Figs. 3 and 4 directly reflects the angular power of the sources $S_{\ell'}^{(m)}$ and P^(m). There are several general results that can be read off the radial functions. Regardless of the source behavior in k, the B-parity polarization for scalars vanishes, dominates by a factor of 6 over the electric parity at $\ell \gg 2$for the vectors, and is reduced by a factor of 8/13 for the tensors at $\ell \gg 2$ [see Eqn. (23)].

Furthermore, the power spectra in $\ell$ can rise no faster than  

$$\begin{array} {rcl}\displaystyle{}\ell^2 C_\ell^{\Theta\Thet... ...&& \quad \ell^2 C_\ell^{\Theta E(m)} \propto \ell^4,\end{array}$$

due to the aliasing of plane-wave power to $\ell \ll k(\eta_0-\eta)$(see Eqn. 25) which leads to interesting constraints on scalar temperature fluctuations [22] and polarization fluctuations (see §VC).

Features in k-space in the $\ell = \vert m\vert$ moment at fixed time are increasingly well preserved in $\ell$-space as |m| increases, but may be washed out if the source is not well localized in time. Only sources involving the visibility function $\dot\tau e^{-\tau}$ are required to be well localized at last scattering. However even features in such sources will be washed out if they occur in the $\ell = \vert m\vert+1$ moment, such as the scalar dipole and the vector quadrupole (see Fig. 3). Similarly features in the vector E and tensor B modes are washed out.

The geometric properties of the temperature-polarization cross power spectrum $C_\ell^{\Theta E}$ can also be read off the integral solutions. It is first instructive however to rewrite the integral solutions as $(\ell \ge 2)$ 

$$\begin{array} {rcl}\displaystyle{}{\Theta_\ell^{(0)}(\eta_0,... ...eft[ \dot\tau P^{(2)}- \dot H \right] j_\ell^{(22)},\end{array}$$

where we have integrated the scalar and vector equations by parts noting that $d e^{-\tau} /d\eta = \dot\tau e^{-\tau}$.Notice that $\Theta_0^{(0)}+\Psi$ acts as an effective temperature by accounting for the gravitational redshift from the potential wells at last scattering. We shall see in §IV that $v_B^{(1)}\approx V$ at last scattering which suppresses the first term in the vector equation. Moreover, as discussed in §IIB and shown in Fig. 5, the vector dipole terms ($j^{(11)}_\ell$)do not correlate well with the polarization ($\epsilon^{(1)}_\ell$)whereas the quadrupole terms ($j^{(21)}_\ell$) do.

The cross power spectrum contains two pieces: the relation between the temperature and polarization sources $S_{\ell'}^{(m)}$ and P^(m) respectively and the differences in their projection as anisotropies on the sky. The latter is independent of the model and provides interesting consequences in conjunction with tight coupling and causal constraints on the sources. In particular, the sign of the correlation is determined by [21] 

$$\begin{array} {rcl}\displaystyle{}{\rm sgn}[C_\ell^{\Theta E... ...} {\rm sgn}[P^{(2)}(\dot\tau P^{(2)} - \dot H)] \, ,\end{array}$$

where the sources are evaluated at last scattering and we have assumed that $\vert\Theta_0^{(0)}+ \Psi\vert \gg \vert P^{(0)}\vert$ as is the case for standard recombination (see § IV). The scalar Doppler effect couples only weakly to the polarization due to differences in the projection (see §IIB). The important aspect is that relative to the sources, the tensor cross spectrum has an opposite sign due to the projection (see Fig. 5).

These integral solutions are also useful in calculations. For example, they may be employed with approximate solutions to the sources in the tight coupling regime to gain physical insight on anisotropy formation (see §IV and [22,23]). Seljak & Zaldarriaga [24] have obtained exact solutions through numerically tracking the evolution of the source by solving the truncated Boltzmann hierarchy equations. Our expression agree with [3,4,24] where they overlap.