Integral Solution for the Microwave Background Anisotropies in Nonflat Universes
Abstract
We present an efficient method of computing cosmic microwave background (CMB) anisotropies in nonflat universes. First, we derive the Boltzmann equation for CMB temperature and polarization fluctuations produced by scalar perturbations in a general RobertsonWalker universe. We then apply the integral method in order to solve this equation, writing temperature and polarization anisotropies as a time integral over a geometrical term and a source term. The geometrical terms can be written using ultraspherical Bessel functions, which depend on curvature. These cannot be precomputed, as for flat space. Instead, we solve their differential equation directly for selected values of the multipoles. The resulting computational time is comparable to the flatspace case, improving over previous methods by 23 orders of magnitude. This allows one to compute highly accurate CMB temperature and polarization spectra, matter transfer functions, and their CMB normalizations for any cosmological model, thereby avoiding the need to use the various inexact fitting formulae that exist in the literature.
 Publication:

The Astrophysical Journal
 Pub Date:
 February 1998
 DOI:
 10.1086/305223
 arXiv:
 arXiv:astroph/9704265
 Bibcode:
 1998ApJ...494..491Z
 Keywords:

 COSMOLOGY: COSMIC MICROWAVE BACKGROUND;
 COSMOLOGY: LARGESCALE STRUCTURE OF UNIVERSE;
 GRAVITATION;
 COSMOLOGY: DARK MATTER;
 Cosmology: Cosmic Microwave Background;
 Cosmology: Dark Matter;
 Gravitation;
 Cosmology: LargeScale Structure of Universe;
 Astrophysics
 EPrint:
 29 pages, 2 figure, AAS LateX, minor revisions to match the accepted version, code available at http://arcturus.mit.edu:80/~matiasz/CMBFAST/cmbfast.html