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Cross-correlations between the galaxy number density in a lensing sourcesample and that in an overlapping spectroscopic sample can in principle be usedto calibrate the lensing source redshift distribution. In this paper, we studyin detail to what extent this cross-correlation method can mitigate loss ofcosmological information in upcoming weak lensing surveys (combined with a CMBprior) due to lack of knowledge of the source distribution. We consider ascenario where photometric redshifts are available, and find that, unless thephotometric redshift distribution p(z_{ph}|z) is calibrated very accurately apriori (bias and scatter known to ~0.002 for, e.g., EUCLID), the additionalconstraint on p(z_{ph}|z) from the cross correlation technique to a largeextent restores the cosmological information originally lost due to theuncertainty in dn/dz(z). Considering only the gain in photo-z accuracy and notthe additional cosmological information, enhancements of the dark energy figureof merit of up to a factor of 4 (40) can be achieved for a SuMIRe (SubaruMeasurement of Images and Redshifts, the combination of the Hyper Suprime Camlensing survey and the Prime Focus Spectrograph redshift survey)-like(EUCLID-like) combination of lensing and redshift surveys. However, the successof the method is strongly sensitive to our knowledge of the galaxy biasevolution in the source sample. If this bias is modeled by a free parameter ineach of a large number of redshift bins, we find that a prior of order 0.01 isneeded on b_i \sqrt{\Delta z} in each redshift slice (where \Delta z is the binwidth and b_i the value of the galaxy bias in the i-th source bin) to optimizethe gains from the cross-correlation method (i.e. to approach the cosmologyconstraints attainable if the bias were known exactly).[abridged]

Using Cross-Correlations to Calibrate Lensing Source Redshift Distributions: Improving Cosmological Constraints from Upcoming Weak Lensing Surveys