Accurate redshift determination of standard sirens by the luminosity distance space-redshift space large scale structure cross correlation

We point out a new possibility to determine the average redshift distribution of a large sample of gravitational wave standard sirens, without spectroscopic follow-ups. It is based on the cross correlation between the luminosity-distance space large scale structure (LSS) traced by standard sirens, and the redshift space LSS traced by galaxies in preexisting electromagnetic wave observations. We construct an unbiased and model independent estimator $E_z$ to realize this possibility. We demonstrate with BBO and Euclid that, $0.1\mathrm{\%}$ accuracy in redshift determination can be achieved. This method can significantly alleviate the need of spectroscopic follow-up of standard sirens, and enhance their cosmological applications.