Combining Spectroscopic and Photometric Surveys: Same or different sky?

Abstract: This article looks at the combined constraints from a photometric and spectroscopic survey. These surveys will measure cosmology using weak lensing (WL), galaxy clustering, baryon acoustic oscillations (BAO) and redshift space distortions (RSD). We find, contrary to some findings in the recent literature, that overlapping surveys can give important benefits when measuring dark energy. We therefore try to clarify the status of this issue with a full forecast of two stage-IV surveys using a new approach to properly account for covariance between the different probes in the overlapping samples. The benefit of the overlapping survey can be traced back to two factors: additional observables and sample variance cancellation. Both needs to be taken into account and contribute equally when combining 3D power spectrum and 2D correlations for lensing. With an analytic example we also illustrate that for optimal constraints, one should minimize the (Pearson) correlation coefficient between cosmological and nuisance parameters and maximize the one among nuisance parameters (e.g. galaxy bias) in the two samples. This can be achieved by increasing the overlap between the spectroscopic and photometric surveys. We show how BAO, WL and RSD contribute to this benefit and also look at some other survey designs, such as photometric redshift errors and spectroscopic density.