Angular Modes

In this section, we derive the basic properties of the angular modes of the temperature and polarization distributions that will be useful in §III to describe their evolution. In particular, the Clebsch-Gordan relation for the addition of angular momentum plays a central role in exposing the simplicity of the total angular momentum representation.

A scalar, or spin- field on the sky such as the temperature can be decomposed into spherical harmonics $Y_\ell^m$. Likewise a spin-s field on the sky can be decomposed into the spin-weighted spherical harmonics $\, {}_{s}^{\vphantom{m}} Y_{\ell}^{m}$ and a tensor constructed out of the basis vectors $\hat{e}_\theta \pm i\hat{e}_\phi$,$\hat{e}_r$[12]. The basis for a spin-2 field such as the polarization is $\, {}_{\pm 2}^{\vphantom{m}} Y_{\ell}^{m}{\bf M}_{\pm}$ [3,4] where  

$${\bf M}_{\pm} \equiv {1 \over 2}(\hat{e}_\theta \mp i \hat{e}_\phi) \otimes (\hat{e}_\theta \mp i \hat{e}_\phi) \, ,$$ (1)

since it transforms under rotations as a $2 \times 2$ symmetric traceless tensor. This property is more easily seen through the relation to the Pauli matrices, ${\bf M}_\pm =\sigma_3 \mp i \sigma_1$, in spherical coordinates $(\theta,\phi)$.The spin-s harmonics are expressed in terms of rotation matrices as [12] 

$$\begin{array} {rcl}\displaystyle{}\, {}_{s}^{\vphantom{m}} Y... ...(-1)^{\ell-r-s} e^{im\phi} (\cot\theta/2)^{2r+s-m} .\end{array}$$

The rotation matrix ${\cal D}_{-s,m}^{\ell}(\phi,\theta,\psi) = \sqrt{4\pi/(2\ell+1)} \, {}_{s}^{\vphantom{m}} Y_{\ell}^{m}(\theta,\phi)e^{-is\psi}$represents rotations by the Euler angles $(\phi,\theta,\psi)$.Since the spin-2 harmonics will be useful in the following sections, we give their explicit form in Table 1 for $\ell=2$; the higher $\ell$ harmonics are related to the ordinary spherical harmonics as

$$\, {}_{\pm 2}^{\vphantom{m}} Y_{\ell}^{m} = \left[ {(\ell-2)... ...\phi - {1 \over \sin^2\theta} \partial^2_\phi \right] Y_\ell^m.$$ (2)

By virtue of their relation to the rotation matrices, the spin harmonics satisfy: the compatibility relation with spherical harmonics, $\, {}_{0}^{\vphantom{m}} Y_{\ell}^{m} = Y_{\ell}^m$;the conjugation relation $\, {}_{s}^{\vphantom{m*}} Y_{\ell}^{m*} = (-1)^{m+s} \, {}_{-s}^{\vphantom{-m}} Y_{\ell}^{-m}$; the orthonormality relation,  

$$\int d\Omega \, \left(\, {}_{s}^{\vphantom{m*}} Y_{\ell}^{m*... ...{m}} Y_{\ell}^{m} \right)= \delta_{\ell,\ell'} \delta_{m,m'};$$ (3)

the completeness relation,  

$$\sum_{\ell,m} \left[ \, {}_{s}^{\vphantom{m*}} Y_{\ell}^{m*}... ...right] = \delta(\phi-\phi')\delta(\cos\theta-\cos\theta') \, ;$$ (4)

the parity relation,  

$$\, {}_{s}^{\vphantom{m}} Y_{\ell}^{m} \rightarrow (-1)^\ell \, {}_{-s}^{\vphantom{m}} Y_{\ell}^{m} ;$$ (5)

the generalized addition relation,  

$$\sum_m \left[ \, {}_{s_1}^{\vphantom{m*}} Y_{\ell}^{m*} (\th... ... Y_{\ell}^{-s_1} (\beta,\alpha) \right] e^{-i s_2 \gamma} \, ,$$ (6)

which follows from the group multiplication property of rotation matrices which relates a rotation from $(\theta',\phi')$through the origin to $(\theta,\phi)$ with a direct rotation in terms of the Euler angles $(\alpha,\beta,\gamma)$ defined in Fig. 1; and the Clebsch-Gordan relation, 

$$\begin{array} {rcl}\displaystyle{}\left(\, {}_{s_1}^{\vphant... ...t( \, {}_{s}^{\vphantom{m}} Y_{\ell}^{m}\right) \, .\end{array}$$

  

Figure 1: Addition theorem and scattering geometry. The addition theorem for spin-s harmonics Eqn. (7) is established by their relation to rotations Eqn. (2) and by noting that a rotation from $(\theta',\phi')$ through the origin (pole) to $(\theta,\phi)$ is equivalent to a direct rotation by the Euler angles $(\alpha,\beta,\gamma)$. For the scattering problem of Eqn. (48), these angles represent the rotation by $\alpha$ from the $\hat{k} = \hat{e}_3$ frame to the scattering frame, by the scattering angle $\beta$, and by $\gamma$ back into the $\hat{k}$ frame.

It is worthwhile to examine the implications of these properties. Note that the orthogonality and completeness relations Eqns. (4) and (5) do not extend to different spin states. Orthogonality between $s=\pm 2$states is established by the Pauli basis of Eqn. (1) ${\bf M}_{\pm}^* {\bf M}_{\pm}^{\vphantom{*}} = {\bf 1}$ and ${\bf M}_\pm^* {\bf M}_\mp^{\vphantom{*}} = {\bf 0}$. The parity equation (6) tells us that the spin flips under a parity transformation so that unlike the s=0 spherical harmonics, the higher spin harmonics are not parity eigenstates. Orthonormal parity states can be constructed as [3,4]  

$${1 \over 2}[\, {}_{2}^{\vphantom{m}} Y_{\ell}^{m}{\bf M}_+ \pm \, {}_{-2}^{\vphantom{m}} Y_{\ell}^{m}{\bf M}_-] \, ,$$ (7)

which have ``electric'' $(-1)^\ell$ and ``magnetic'' $(-1)^{\ell+1}$ type parity for the $(\pm)$ states respectively. We shall see in §IIIC that the polarization evolution naturally separates into parity eigenstates. The addition property will be useful in relating the scattering angle to coordinates on the sphere in §IIIB. Finally the Clebsch-Gordan relation Eqn. (8) is central to the following discussion and will be used to derive the total angular momentum representation in §IIB and evolution equations for angular moments of the radiation §IIIC.