Complementarity of Peculiar Velocity Surveys and Redshift Space Distortions for Testing Gravity

Peculiar-velocity surveys of the low-redshift universe have significant leverage to constrain the growth rate of cosmic structure and test gravity. Wide-field imaging surveys combined with multi-object spectrographs (e.g. ZTF2, LSST, DESI, 4MOST) can use Type Ia supernovae as informative tracers of the velocity field, reaching few percent constraints on the growth rate $f\sigma_8$ at $z\lesssim0.2$ where density tracers cannot do better than $\sim10\%$. Combining the high-redshift DESI survey mapping redshift space distortions with a low-redshift supernova peculiar velocity survey using LSST and DESI can determine the gravitational growth index to $\sigma(\gamma)\approx0.02$, testing general relativity. We study the characteristics needed for the peculiar velocity survey, and how its complementarity with clustering surveys improves when going from a $\Lambda$CDM model assumption to a $w_0$-$w_a$ cosmology.