We present a method to implement the idea of ( ) to constrain cosmological parameters with weak gravitational lensing .
Photometric redshift information on foreground galaxies is used to produce templates of the mass structure at foreground slices z _ { \ell } , and the predicted distortion field is cross-correlated with the measured shapes of sources at redshift z _ { s } .
The variation of the cross-correlation with z _ { s } depends purely on ratios of angular diameter distances .
We propose a formalism for such an analysis that makes use of all foreground-background redshift pairs , and derive the Fisher uncertainties on the dark energy parameters that would result from such a survey .
Surveys from the proposed SNAP satellite or the LSST observatory could constrain the dark energy equation of state to \sigma _ { w _ { 0 } } \approx 0.01 f _ { sky } ^ { -1 / 2 } and \sigma _ { w _ { a } } \approx 0.035 f _ { sky } ^ { -1 / 2 } after application of a practical prior on \Omega _ { m } .
Advantages of this method over power-spectrum measurements are that it is unaffected by residual PSF distortions , is not limited by sample-variance , and can use non-linear mass structures to constrain cosmology .
The signal is , however , very small , amounting to a change of a few parts in 10 ^ { 3 } of the lensing distortion .
In order to realize the full sensitivity to cosmological parameters , the calibration of lensing distortion must be independent of redshift to comparable levels , and photometric redshifts must be similarly free of bias .
Both of these tasks require substantial advance over the present state of the art , but we discuss how such accurate calibrations might be achieved using internal consistency tests .
Elimination of redshift bias would require spectroscopic redshifts of \sim 10 ^ { 4 } -10 ^ { 5 } high redshift galaxies—fewer for lensing surveys less ambitious than SNAP or LSST .