Normal Modes

  In this section, we introduce the total angular momentum representation for the normal modes of fluctuations in a flat universe that are used to describe the CMB temperature and polarization as well as the metric and matter fluctuations. This representation greatly simplifies the derivation and form of the evolution equations for fluctuations in §III. In particular, the angular structure of modes corresponds directly to the angular distribution of the temperature and polarization, whereas the radial structure determines how distant sources contribute to this angular distribution.

The new aspect of this approach is the isolation of the total angular dependence of the modes by combining the intrinsic angular structure with that of the plane-wave spatial dependence. This property implies that the normal modes correspond directly to angular structures on the sky as opposed to the commonly employed technique that isolates portions of the intrinsic angular dependence and hence a linear combination of observable modes [10]. Elements of this approach can be found in earlier works (e.g. [6,7,11] for the temperature and [3] for the scalar and tensor polarization). We provide here a systematic study of this technique which also provides for a substantial simplification of the evolution equations and their integral solution in §IIIC, including the terms involving the radiation transport of the CMB. We discuss in detail how the monopole, dipole and quadrupole sources that enter into the radiation transport problem project as anisotropies on the sky today.

Readers not interested in the formal details may skip this section on first reading and simply note that the temperature and polarization distribution is decomposed into the modes $Y_\ell^m \exp(i\vec{k}\cdot\vec{x})$and $\, {}_{\pm 2}^{\vphantom{m}} Y_{\ell}^{m} \exp(i\vec{k}\cdot\vec{x})$ with $m=0,\pm 1,\pm 2$ for scalar, vector and tensor metric perturbations respectively. In this representation, the geometric distinction between scalar, vector and tensor contributions to the anisotropies is clear as is the reason why they do not mix. Here the $\, {}_{\pm 2}^{\vphantom{m}} Y_{\ell}^{m}$are the spin-2 spherical harmonics [12] and were introduced to the study of CMB polarization by [3]. The radial decompositions of the modes $Y_{\ell'}^m j_{\ell'}^{(\ell m)}(kr)$and $\, {}_{\pm 2}^{\vphantom{m}} Y_{\ell'}^{m}[\epsilon_{\ell'}^{(m)}(kr) \pm i \beta_{\ell'}^{(m)}(kr)]$ (for $\ell=2$) isolate the total angular dependence by combining the intrinsic and plane wave angular momenta.