The large scale structure in the 3D luminosity-distance space and its
  cosmological applications
  Future gravitational wave (GW) observations are capable of detecting millions
of compact star binary mergers in extragalactic galaxies, with $1\%$
luminosity-distance ($D_L$) measurement accuracy and better than arcminute
positioning accuracy. This will open a new window of the large scale structure
(LSS) of the universe, in the 3D {\bf luminosity-distance space (LDS)}, instead
of the 3D redshift space of galaxy spectroscopic surveys. The baryon acoustic
oscillation and the AP test encoded in the LDS LSS constrain the $D_L$-$D^{\rm
co}_A$ (comoving angular diameter distance) relation and therefore the
expansion history of the universe. Peculiar velocity induces the LDS
distortion, analogous to the redshift space distortion, and allows for a new
structure growth measure $f_L\sigma_8$. When the distance duality is enforced
($1+z=D_L/D^{\rm co}_A$), the LDS LSS by itself determines the redshift to
$\sim 1\%$ level accuracy, and alleviates the need of spectroscopic follow-up
of GW events.But a more valuable application is to test the distance duality to
$1\%$ level accuracy, in combination with conventional BAO and supernovae
measurements. This will put stringent constraints on modified gravity models in
which the gravitational wave $D^{\rm GW}_L$ deviates from the electromagnetic
wave $D^{EM}_L$. All these applications require no spectroscopic follow-ups.
