The cosmic microwave background (CMB) is fast becoming the premier laboratory for early universe and classical cosmology. With the flood of high quality data expected in the coming years, most notably from the new MAP [1] and Planck Surveyor [2] satellite missions, it is imperative that theoretical tools for their interpretation be developed. The corresponding techniques involved should be as physically transparent as possible so that the implications for cosmology will be readily apparent from the data.
Toward this end, we reconsider the general problem of temperature and polarization anisotropy formation in the CMB. These anisotropies arise from gravitational perturbations which separate into scalar (compressional), vector (vortical), and tensor (gravity wave) modes. In previous treatments, the simple underlying geometrical distinctions and physical processes involved in their appearance as CMB anisotropies has been obscured by the choice of representation for the angular distribution of the CMB. In this paper, we systematically develop a new representation, the total angular momentum representation, which puts vector and tensor modes for the temperature and all polarization modes on an equal footing with the familiar scalar temperature modes. For polarization, this completes and substantially simplifies the ground-breaking work of [3,4]. Although we consider only flat geometries here for simplicity, the framework we establish allows for straightforward generalization to open geometries [5,6,7,8] unlike previous treatments.
The central idea of this method is to employ only observable quantities, i.e. those which involve the total angular dependence of the temperature and polarization distributions. By applying this principle from beginning to end, we obtain a substantial simplification of the radiation transport problem underlying anisotropy formation. Scattering terms couple only the quadrupole moments of the temperature and polarization distributions. Each moment of the distribution corresponds to angular moments on the sky which allows a direct relation between the fundamental scattering and gravitational sources and the observable anisotropy through their integral solutions.
We study the means by which gravitational perturbations of the scalar, vector, or tensor type, originating in either the cosmological fluids or seed sources such as defects, form temperature and polarization anisotropies in the CMB. As is well established [3,4], scalar perturbations generate only the so-called electric parity mode of the polarization. Here we show that conversely the ratio of magnetic to electric parity power is a factor of 6 for vectors, compared with 8/13 for tensors, independent of their source. Furthermore, the large angle limits of polarization must obey simple geometrical constraints for its amplitude that differ between scalars, vectors and tensors. The sense of the temperature-polarization cross correlation at large angle is also determined by geometric considerations which separate the scalars and vectors from the tensors [9]. These constraints are important since large-angle polarization unlike large-angle temperature anisotropies allow one to see directly scales above the horizon at last scattering. Combined with causal constraints, they provide robust signatures of causal isocurvature models for structure formation such as cosmological defects.
In §II we develop the formalism of the total angular momentum representation and lay the groundwork for the geometric interpretation of the radiation transport problem and its integral solutions. We further establish the relationship between scalars, vectors, and tensors and the orthogonal angular modes on the sphere. In §III, we treat the radiation transport problem from first principles. The total angular momentum representation simplifies both the derivation and the form of the evolution equations for the radiation. We present the differential form of these equations, their integral solutions, and their geometric interpretation. In §IV we specialize the treatment to the tight-coupling limit for the photon-baryon fluid before recombination and show how acoustic waves and vorticity are generated from metric perturbations and dissipated through the action of viscosity, polarization and heat conduction. In §V, we provide specific examples inspired by seeded models such as cosmological defects. We trace the full process that transfers seed fluctuations in the matter through metric perturbations to observable anisotropies in the temperature and polarization distributions.