  We present a new method for calculating linear cosmic microwave background
(CMB) anisotropy spectra based on integration over sources along the photon
past light cone. In this approach the temperature anisotropy is written as a
time integral over the product of a geometrical term and a source term. The
geometrical term is given by radial eigenfunctions which do not depend on the
particular cosmological model. The source term can be expressed in terms of
photon, baryon and metric perturbations, all of which can be calculated using a
small number of differential equations. This split clearly separates between
the dynamical and geometrical effects on the CMB anisotropies. More
importantly, it allows to significantly reduce the computational time compared
to standard methods. This is achieved because the source term, which depends on
the model and is generally the most time consuming part of calculation, is a
slowly varying function of wavelength and needs to be evaluated only in a small
number of points. The geometrical term, which oscillates much more rapidly than
the source term, does not depend on the particular model and can be precomputed
in advance. Standard methods that do not separate the two terms and require a
much higher number of evaluations. The new method leads to about two orders of
magnitude reduction in CPU time when compared to standard methods and typically
requires a few minutes on a workstation for a single model. The method should
be especially useful for accurate determinations of cosmological parameters
from CMB anisotropy and polarization measurements that will become possible
with the next generation of experiments. A programm implementing this method
can be obtained from the authors.
