Using Cross-Correlations to Calibrate Lensing Source Redshift
  Distributions: Improving Cosmological Constraints from Upcoming Weak Lensing
  Surveys
  Cross-correlations between the galaxy number density in a lensing source
sample and that in an overlapping spectroscopic sample can in principle be used
to calibrate the lensing source redshift distribution. In this paper, we study
in detail to what extent this cross-correlation method can mitigate loss of
cosmological information in upcoming weak lensing surveys (combined with a CMB
prior) due to lack of knowledge of the source distribution. We consider a
scenario where photometric redshifts are available, and find that, unless the
photometric redshift distribution p(z_{ph}|z) is calibrated very accurately a
priori (bias and scatter known to ~0.002 for, e.g., EUCLID), the additional
constraint on p(z_{ph}|z) from the cross correlation technique to a large
extent restores the cosmological information originally lost due to the
uncertainty in dn/dz(z). Considering only the gain in photo-z accuracy and not
the additional cosmological information, enhancements of the dark energy figure
of merit of up to a factor of 4 (40) can be achieved for a SuMIRe (Subaru
Measurement of Images and Redshifts, the combination of the Hyper Suprime Cam
lensing survey and the Prime Focus Spectrograph redshift survey)-like
(EUCLID-like) combination of lensing and redshift surveys. However, the success
of the method is strongly sensitive to our knowledge of the galaxy bias
evolution in the source sample. If this bias is modeled by a free parameter in
each of a large number of redshift bins, we find that a prior of order 0.01 is
needed on b_i \sqrt{\Delta z} in each redshift slice (where \Delta z is the bin
width and b_i the value of the galaxy bias in the i-th source bin) to optimize
the gains from the cross-correlation method (i.e. to approach the cosmology
constraints attainable if the bias were known exactly).[abridged]
