Incorporating Photometric Redshift Probability Density Information into
  Real-Space Clustering Measurements
  The use of photometric redshifts in cosmology is increasing. Often, however
these photo-zs are treated like spectroscopic observations, in that the peak of
the photometric redshift, rather than the full probability density function
(PDF), is used. This overlooks useful information inherent in the full PDF. We
introduce a new real-space estimator for one of the most used cosmological
statistics, the 2-point correlation function, that weights by the PDF of
individual photometric objects in a manner that is optimal when Poisson
statistics dominate. As our estimator does not bin based on the PDF peak it
substantially enhances the clustering signal by usefully incorporating
information from all photometric objects that overlap the redshift bin of
interest. As a real-world application, we measure QSO clustering in the Sloan
Digital Sky Survey (SDSS). We find that our simplest binned estimator improves
the clustering signal by a factor equivalent to increasing the survey size by a
factor of 2-3. We also introduce a new implementation that fully weights
between pairs of objects in constructing the cross-correlation and find that
this pair-weighted estimator improves clustering signal in a manner equivalent
to increasing the survey size by a factor of 4-5. Our technique uses
spectroscopic data to anchor the distance scale and it will be particularly
useful where spectroscopic data (e.g, from BOSS) overlaps deeper photometry
(e.g.,from Pan-STARRS, DES or the LSST). We additionally provide simple,
informative expressions to determine when our estimator will be competitive
with the autocorrelation of spectroscopic objects. Although we use QSOs as an
example population, our estimator can and should be applied to any clustering
estimate that uses photometric objects.
