Optimizing spectroscopic and photometric galaxy surveys: same-sky benefits for dark energy and modified gravity

The combination of multiple cosmological probes can produce measurements of cosmological parameters much more stringent than those possible with any individual probe. We examine the combination of two highly correlated probes of late-time structure growth: (i) weak gravitational lensing from a survey with photometric redshifts and (ii) galaxy clustering and redshift space distortions from a survey with spectroscopic redshifts. We choose generic survey designs so that our results are applicable to a range of current and future photometric and spectroscopic redshift surveys. Combining the surveys greatly improves their power to measure both dark energy and modified gravity. An independent, non-overlapping combination sees a dark energy figure of merit more than four times larger than that produced by either survey alone. The powerful synergies between the surveys are strongest for modified gravity, where their constraints are orthogonal, producing a non-overlapping joint figure of merit nearly two orders of magnitude larger than either alone. Our projected angular power spectrum formalism makes it easy to model the cross-correlation observable when the surveys overlap on the sky, producing a joint data vector and full covariance matrix. We calculate a same-sky improvement factor, from the inclusion of these cross-correlations, relative to non-overlapping surveys. We find nearly a factor of 4 for dark energy and more than a factor of 2 for modified gravity. We have also forecast constraints on the linear growth factor, finding a factor of 3.4 improvement for an independent combination of probes and a modest 20 per cent same-sky benefit.