Angular Moments and Power

The temperature and polarization fluctuations are expanded into the normal modes defined in §IIB,  

$$\begin{array} {rcl} \Theta(\eta,\vec{x},\vec{n}) &=& \displa... ...l^{(m)} \pm i B_\ell^{(m)}) \, {}_{\pm 2} G_\ell^m .\end{array}$$ (47)

A comparison with Eqn. (9) and (43) shows that $E_\ell^{(m)}$ and $B_\ell^{(m)}$ represent polarization with electric $(-1)^\ell$ and magnetic $(-1)^{\ell+1}$type parities respectively [3,4]. Because the temperature $\Theta^{(m)}_\ell$ has electric type parity, only $E_\ell^{(m)}$ couples directly to the temperature in the scattering sources. Note that $B_\ell^{(m)}$ and $E_\ell^{(m)}$ represent polarizations with Q and U interchanged and thus represent polarization patterns rotated by $45^\circ$. A simple example is given by the m=0 modes. In the $\hat{k}$-frame, $E_\ell^{(0)}$ represents a pure Q, or north/south-east/west, polarization field whose amplitude depends on $\theta$, e.g. $\sin^2\theta$ for $\ell=2$. $B_\ell^{(0)}$represents a pure U, or northwest/southeast-northeast/southwest, polarization with the same dependence.

The power spectra of temperature and polarization anisotropies today are defined as, e.g. $C_\ell^{\Theta\Theta} \equiv \left< \vert a_{\ell m} \vert^2 \right\gt$ for $\Theta = \sum a_{\ell m} Y_\ell^m$ with the average being over the ($2\ell+1$) m-values. Recalling the normalization of the mode functions from Eqn. (10) and (11), we obtain  

$$(2\ell+1)^2 C_\ell^{X\widetilde X} = {2 \over \pi} \int {... ...3 X_\ell^{(m)*}(\eta_0,k) \widetilde X_\ell^{(m)}(\eta_0,k)\, ,$$ (48)

where X takes on the values $\Theta$, E and B. There is no cross correlation $C_\ell^{\Theta B}$ or $C_\ell^{E B}$ due to parity [see Eqns. (6) and (24)]. We also employ the notation $C_\ell^{X\widetilde X(m)}$ for the m contributions individually. Note that $B_\ell^{(0)}=0$ here due to azimuthal symmetry in the transport problem so that $C_\ell^{BB(0)} =0$.

As we shall now show, the $m=0,\pm 1,\pm 2$ modes are stimulated by scalar, vector and tensor perturbations in the metric. The orthogonality of the spherical harmonics assures us that these modes are independent, and we now discuss the contributions separately.