Euclid preparation: XII. Optimizing the photometric sample of the Euclid
  survey for galaxy clustering and galaxy-galaxy lensing analyses
  The accuracy of photometric redshifts (photo-zs) particularly affects the
results of the analyses of galaxy clustering with photometrically-selected
galaxies (GCph) and weak lensing. In the next decade, space missions like
Euclid will collect photometric measurements for millions of galaxies. These
data should be complemented with upcoming ground-based observations to derive
precise and accurate photo-zs. In this paper, we explore how the tomographic
redshift binning and depth of ground-based observations will affect the
cosmological constraints expected from Euclid. We focus on GCph and extend the
study to include galaxy-galaxy lensing (GGL). We add a layer of complexity to
the analysis by simulating several realistic photo-z distributions based on the
Euclid Consortium Flagship simulation and using a machine learning photo-z
algorithm. We use the Fisher matrix formalism and these galaxy samples to study
the cosmological constraining power as a function of redshift binning, survey
depth, and photo-z accuracy. We find that bins with equal width in redshift
provide a higher Figure of Merit (FoM) than equipopulated bins and that
increasing the number of redshift bins from 10 to 13 improves the FoM by 35%
and 15% for GCph and its combination with GGL, respectively. For GCph, an
increase of the survey depth provides a higher FoM. But the addition of faint
galaxies beyond the limit of the spectroscopic training data decreases the FoM
due to the spurious photo-zs. When combining both probes, the number density of
the sample, which is set by the survey depth, is the main factor driving the
variations in the FoM. We conclude that there is more information that can be
extracted beyond the nominal 10 tomographic redshift bins of Euclid and that we
should be cautious when adding faint galaxies into our sample, since they can
degrade the cosmological constraints.
