Normal Modes

  The temperature and polarization distributions are functions of the position $\vec{x}$ and the direction of propagation of the photons $\vec{n}$. They can be expanded in modes which account for both the local angular and spatial variations: $\, {}_{s}^{\vphantom{m}} {G}_{\ell}^{m}(\vec{x},\hat{n})$, i.e.  

$$\begin{array} {rcl} \Theta(\eta,\vec{x},\hat{n}) &=& \displa... ...}) \, \, {}_{\pm 2}^{\vphantom{m}} G_{\ell}^{m} \,, \end{array}$$ (18)

with spin s=0 describing the temperature fluctuation and $s=\pm 2$describing the polarization tensor. $E_\ell$ and $B_\ell$ are the angular moments of the electric and magnetic polarization components. It is apparent that the effects of the local scattering process $\vec{C}$is most simply evaluated in a representation where the separation of the local angular and spatial distribution is explicit [1], with the former being an expansion in $\, {}_{s}^{\vphantom{m}} {Y}_{\ell}^{m}$.The subtlety lies in relating the local basis at two different coordinate points, say the last scattering event and the observer.

In flat space, the representation is straightforward since the parallel transport of the angular basis in space is trivial. The result is a product of spin-weighted harmonics for the local angular dependence and plane waves for the spatial dependence:  

$$\, {}_{s}^{\vphantom{m}} G_{\ell}^{m}(\vec{x},\hat{n}) = (-... ...^{m}(\hat{n})] \exp(i\vec{k} \cdot \vec{x})\,, \qquad (K=0)\,.$$ (19)

Here we seek a similar construction in an curved geometry. We will see that this construction greatly simplifies the scalar harmonic treatment of [15,16,4] and extends it to vector and tensor temperature [3] modes as well as all polarization modes.

To generalize these modes to the curved geometry, we wish to replace the plane wave with some spatially dependent phase factor $\exp[i \delta(\vec{x},\vec{k})]$ related to the eigenfunctions ${\bf Q}^{(m)}$ of §IIA while keeping the same local angular dependence (see Eq. C2). By virtue of this requirement, the Compton scattering terms, which involve only the local angular dependence, retain the same form as in flat space. In Appendix C, we derive $\, {}_{s}^{\vphantom{m}} {G}_{\ell}^{m}$ by recursion from covariant contractions of the fundamental basis ${\bf Q}^{(m)}$.The result is a recursive definition of the basis  

$$n^i (\, {}_{s}^{\vphantom{m}} {G}_{\ell}^{m})_{\vert i} = ... ...ver \ell(\ell+1)}\ \, {}_{s}^{\vphantom{m}} {G}_{\ell}^{m} \, ,$$ (20)

constructed from the lowest $\ell$-mode of Eq. (B2) with the coupling coefficient  

$$\, {}_{s}^{\vphantom{m}} {\kappa}_{\ell}^{m} = \sqrt{ \left[... ...s^2)\over\ell^2}\right] \left[1 - {\ell^2\over q^2} K \right]}.$$ (21)

The structure of this relation is readily apparent. The recursion relation expresses the addition of angular momentum and is the defining equation in the total angular momentum method. It says the ``total'' local angular dependence at (say) the origin is the sum of the local angular dependence at distant points (``spin'' angular momentum) plus the angular variations induced by the spatial dependence of the mode (``orbital'' angular momentum).

The recursion relation represents the addition of angular momentum for the case of an infinitesimal spatial separation. Here the leading order spatial variation is the gradient [$n^i (\, {}_{s}^{\vphantom{m}} {G}_{\ell}^{m})_{\vert i}$] term which has an angular structure of a dipole Y₁⁰. The first term on the rhs of equation (26) arises from the Clebsch-Gordan relation that couples the orbital Y₁⁰ with the intrinsic $\, {}_{s}^{\vphantom{m}} {Y}_{\ell}^{m}$ to form $\ell\pm 1$ states, 

$$\begin{array} {rcl}\sqrt{4 \pi \over 3} Y_1^0 (\, {}_{s}^{\v... ...s \over \ell (\ell +1)} \left({}_s Y_{\ell}^m\right)\end{array}$$ (22)

where the coupling coefficient is $\, {}_{s}^{\vphantom{m}} {c}_{\ell}^{m} = \sqrt{(\ell^2-m^2)(\ell^2-s^2)/\ell^2}$.

The second term on the rhs of the coupling equation (26) accounts for geodesic deviation factors in the conversion of spatial structure into orbital angular momentum. Consider first a closed universe with radius of curvature ${\cal R}=K^{-1/2}$.Suppressing one spatial coordinate, we can analyze the problem as geometry on the 2-sphere with the observer situated at the pole. Light travels on radial geodesics or great circles of fixed longitude. A physical scale $\lambda$ at fixed latitude (given by the polar angle $\chi$)subtends an angle $\alpha = \lambda/{\cal R}\sin\chi$.In the small angle approximation, a Euclidean analysis would infer a distance related by

$${\cal D}(d) = {\cal R} \sin \chi = K^{-1/2} \sin \chi \, , \qquad (K\gt),$$ (23)

called here the angular diameter distance. For negatively curved or open universes, a similar analysis implies  

$${\cal D}(d) = \vert K\vert^{-1/2} \sinh \chi\, , \qquad (K<0).$$ (24)

Thus the angular scale corresponding to an eigenmode of wavelength $\lambda$ is

$$\theta = {\lambda \over {\cal R} \sinh{\chi}} \, \approx {1 \over \nu\sinh{\chi}} \, .$$ (25)

For an infinitesimal change $\chi$, orbital angular momentum of order $\ell$ is stimulated when

$$\begin{array} {rcl}\chi &\approx& {1\over\nu\theta} [1+{\cal... ...pprox& {\ell\over q}[1+{\cal O}(\ell^2 K / q^2) ]\,,\end{array}$$

which explains the factors of $\ell^2 K / q^2$ in the coupling term in a curved geometry. We shall see in §IIID that these infinitesimal additions of angular momentum and geodesic deviation may be encorporated into a single step by finding the integral solutions to the coupling equation (25).