A direct method to compute the galaxy count angular correlation function
  including redshift-space distortions
  In the near future, cosmology will enter the wide and deep galaxy survey area
allowing high-precision studies of the large scale structure of the universe in
three dimensions. To test cosmological models and determine their parameters
accurately, it is natural to confront data with exact theoretical expectations
expressed in the observational parameter space (angles and redshift). The
data-driven galaxy number count fluctuations on redshift shells, can be used to
build correlation functions $C(\theta; z_1, z_2)$ on and between shells which
can probe the baryonic acoustic oscillations, the distance-redshift distortions
as well as gravitational lensing and other relativistic effects. Transforming
the model to the data space usually requires the computation of the angular
power spectrum $C_\ell(z_1, z_2)$ but this appears as an artificial and
inefficient step plagued by apodization issues. In this article we show that it
is not necessary and present a compact expression for $C(\theta; z_1, z_2)$
that includes directly the leading density and redshift space distortions terms
from the full linear theory. It can be evaluated using a fast integration
method based on Clenshaw-Curtis quadrature and Chebyshev polynomial series.
This new method to compute the correlation functions without any Limber
approximation, allows us to produce and discuss maps of the correlation
function directly in the observable space and is a significant step towards
disentangling the data from the tested models.
