Cross-correlating 2D and 3D galaxy surveys
  Galaxy surveys probe both structure formation and the expansion rate, making
them promising avenues for understanding the dark universe. Photometric surveys
accurately map the 2D distribution of galaxy positions and shapes in a given
redshift range, while spectroscopic surveys provide sparser 3D maps of the
galaxy distribution. We present a way to analyse overlapping 2D and 3D maps
jointly and without loss of information. We represent 3D maps using spherical
Fourier-Bessel (sFB) modes, which preserve radial coverage while accounting for
the spherical sky geometry, and we decompose 2D maps in a spherical harmonic
basis. In these bases, a simple expression exists for the cross-correlation of
the two fields. One very powerful application is the ability to simultaneously
constrain the redshift distribution of the photometric sample, the sample
biases, and cosmological parameters. We use our framework to show that combined
analysis of DESI and LSST can improve cosmological constraints by factors of
${\sim}1.2$ to ${\sim}1.8$ on the region where they overlap relative to
identically sized disjoint regions. We also show that in the overlap of DES and
SDSS-III in Stripe 82, cross-correlating improves photo-$z$ parameter
constraints by factors of ${\sim}2$ to ${\sim}12$ over internal photo-$z$
reconstructions.
