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Gravitational waves from inspiraling compact objects provide us withinformation of the distance scale since we can infer the absolute luminosity ofthe source from analysis of the wave form, which is known as standard sirens.The first detection of the gravitational wave signal of the binary black holemerger event by Advanced LIGO has opened up the possibility of utilizingstandard sirens as cosmological probe. In order to extract information of thedistance-redshift relation, we cross-correlate weak lensing, which is anunbiased tracer of matter distribution in the Universe, with the projectednumber density of gravitational wave sources. For weak lensing, we employtomography technique to efficiently obtain information of large-scalestructures at wide ranges of redshifts. Making use of the cross-correlationsalong with the auto-correlations, we present forecast of constraints on fourcosmological parameters, i.e., Hubble parameter, matter density, the equationof state parameter of dark energy, and the amplitude of matter fluctuation. Tofully explore the ability of cross-correlations, which require largeoverlapping sky coverage, we consider the specific case with the upcomingsurveys by \textit{Euclid} for weak lensing and Einstein Telescope for standardsirens. We show that cosmological parameters can be tightly constrained solelyby these auto- and cross-correlations of standard sirens and weak lensing. Forexample, the $1\text{-}\sigma$ error of Hubble parameter is expected to be$\sigma (H_0) = 0.33 \, \mathrm{km} \, \mathrm{s}^{-1} \, \mathrm{Mpc}^{-1}$.Thus, the proposed statistics will be a promising probe into the distancescale.

Exploring the distance-redshift relation with gravitational wave standard sirens and tomographic weak lensing