Spectroscopic source redshifts and parameter constraints from weak lensing and CMB

Weak lensing is a potentially robust and model-independent cosmological probe, but its accuracy is dependent on knowledge of the redshift distribution of the source galaxies used. The most robust way to determine the redshift distribution is via spectroscopy of a subsample of the source galaxies. We forecast constraints from combining CMB anisotropies with cosmic shear using a spectroscopically determined redshift distribution, varying the number of spectra $N_{spec}$ obtained from 64 to infinity. The source redshift distribution is expanded in a Fourier series, and the amplitudes of each mode are considered as parameters to be constrained via both the spectroscopic and weak lensing data. It is found that for the surveys considered and for a prior of 0.04 on the calibration parameters, the degradation of the standard deviations of the model parameters is at worst 65% even when only $N_{spec}=64$ spectra are obtained. We find that the usefulness of large $N_{spec}$ is slightly improved when the calibration is better constrained. We examine the degradation of the combinations of parameters using an eigenvector analysis, finding that small $N_{spec}$ is significantly worsening the constraints in some directions, but that these directions are already so well determined from weak lensing that they do not dominate the uncertainties on the standard cosmological parameters.