Spectroscopic source redshifts and parameter constraints from weak lensing and the cosmic microwave background

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 cosmic microwave background (CMB) anisotropies with cosmic shear using a spectroscopically determined redshift distribution, varying the number of spectra $N_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}}$ obtained from 64 to $\infty$. 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. We assume independent source redshifts, and consider in what circumstances this is a good approximation (the sources are clustered and for narrow spectroscopic surveys with many objects this results in the redshifts being correlated). It is found that for the surveys considered and for a prior of 0.04 on the calibration parameters, the addition of redshift information makes significant improvements on the constraints on the cosmological parameters; however, beyond $N_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}}\sim \mathrm{f}\mathrm{e}\mathrm{w}\times {10}^3$ the addition of further spectra will make only a very small improvement to the cosmological parameters. We find that a better calibration makes large $N_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}}$ more useful. Using an eigenvector analysis, we find that the improvement continues with even higher $N_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}}$, but not in directions that dominate the uncertainties on the standard cosmological parameters.