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.
