Probing gravity and growth of structure with gravitational waves and galaxies' peculiar velocity

The low-redshift velocity field is a unique probe of the growth of cosmic structure and gravity. We propose to use distances from gravitational wave (GW) detections, in conjunction with the redshifts of their host galaxies from wide field spectroscopic surveys (e.g. DESI, 4MOST, TAIPAN), to measure peculiar motions within the local Universe. Such measurement has the potential to constrain the growth rate $f\sigma_8$ and test gravity through determination of the gravitational growth index $\gamma$, complementing constraints from other peculiar velocity measurements. We find that binary neutron star mergers with associated counterpart at $z\lesssim 0.2$ that will be detected by the Einstein Telescope (ET) will be able to constrain $f\sigma_8$ to $\sim 3\%$ precision after $10$ years of operations when combined with galaxy overdensities from DESI and TAIPAN. If a larger network of third generation GW detectors is available (e.g. including the Cosmic Explorer), the same constraints can be reached over a shorter timescale ($\sim 5$ years for a 3 detectors network). The same events (plus information from their hosts' redshifts) can constrain $\gamma$ to $\sigma_\gamma\lesssim 0.04$. This constraint is precise enough to discern General Relativity from other popular gravity models at $3\sigma$. This constraint is improved to $\sigma_\gamma\sim 0.02-0.03$ when combined with galaxy overdensities. The potential of combining galaxies' peculiar velocities with gravitational wave detections for cosmology highlights the need for extensive optical to near--infrared follow--up of nearby gravitational wave events, or exquisite GW localization, in the next decade.