Combining spectroscopic and photometric surveys using angular
  cross-correlations I: Algorithm and modelling
  Weak lensing (WL) clustering is studied using 2D (angular) coordinates, while
redshift space distortions (RSD) and baryon acoustic oscillations (BAO) use 3D
coordinates, which requires a model dependent conversion of angles and
redshifts into comoving distances. This is the first paper of a series, which
explore modelling multi-tracer galaxy clustering (of WL, BAO and RSD), using
only angular (2D) cross-correlations in thin redshift bins. This involves
evaluating many thousands cross-correlations, each a multidimensional integral,
which is computationally demanding. We present a new algorithm that performs
these calculations as matrix operations.
  Nearby narrow redshift bins are intrinsically correlated, which can be used
to recover the full (radial) 3D information. We show that the Limber
approximation does not work well for this task. In the exact calculation, both
the clustering amplitude and the RSD effect increase when decreasing the
redshift bin width. For narrow bins, the cross-correlations has a larger BAO
peak than the auto-correlation because smaller scales are filtered out by the
radial redshift separation. Moreover, the BAO peak shows a second (ghost) peak,
shifted to smaller angles. We explore how WL, RSD and BAO contribute to the
cross-correlations as a function of the redshift bin width and present a first
exploration of non-linear effects and signal-to-noise ratio on these
quantities. This illustrates that the new approach to clustering analysis
provides new insights and is potentially viable in practice.
