The study of the Cosmic Microwave Background (CMB) radiation holds the key to understanding the seeds of the structure we see around us in the universe, and could potentially enable precision measures for most of the important cosmological parameters. For this reason, as well as because of its intrinsic interest, one would like a physically transparent framework for the study of CMB anisotropies which is as general, powerful, and flexible as possible.

Theoretically the calculation of CMB anisotropies is ``clean'', involving as it does only linear perturbation theory. However the calculations can become quite complex once one allows for the possiblilty of non-flat universes, non-scalar perturbations to the metric, and polarization as well as temperature anisotropies. Recently Hu & White [1] presented a formalism for calculating CMB anisotropies which treats all types of perturbations, temperature and polarization anisotropies, and hierarchy and integral solutions on an equal footing. The formalism, named the total angular momentum method, greatly simplifies the physical interpretation of the equations and the form of their solutions (see e.g. [2]). However it was presented in detail only for the case of flat spatial hypersurfaces. Here we generalize the treatment for the curved spaces of open and closed Friedman-Robertson-Walker (FRW) universes.

Aspects of this method in open (hyperbolic, negatively curved) geometries have been introduced in Hu & White [3] and Zaldarriaga, Seljak & Bertschinger [4] for the cases of tensor temperature and scalar polarization respectively. The latter work also addressed methods for efficient implementation through the line of sight integration technique [5]. In this paper, we complete the total angular momentum method for arbitrary perturbation type and FRW metric, paying particular attention to the case of open universes because of its strong observational motivation. As an example we use this formalism to compute the temperature and polarization angular power spectra of both scalar and tensor modes in critical density and open inflationary models. We incorporated the formalism into the CMBFAST code of Seljak & Zaldarriaga [5], which has been made publically available. With this work, the perturbation theory of CMB temperature and polarization anisotropy formation through gravitational instability in an FRW universe may be considered complete.

The outline of the paper is as follows: we begin by establishing our notation for fluctuations about a FRW background cosmology in §II. We then present the Boltzmann equation in our formalism in §III, which contains the main results. We give some examples and discuss applications in §IV. Some of the more technical parts of the derivations (the Einstein, radial and hierarchy equations) are presented in a series of three Appendices.