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.
