Measuring the distance-redshift relation with the cross-correlation of gravitational wave standard sirens and galaxies

Gravitational waves from inspiraling compact binaries are known to be an excellent absolute distance indicator, yet it is unclear whether electromagnetic counterparts of these events are securely identified for measuring their redshifts, especially in the case of black hole–black hole mergers such as the one recently observed with the Advanced LIGO. We propose to use the cross-correlation between spatial distributions of gravitational wave sources and galaxies with known redshifts as an alternative means of constraining the distance-redshift relation from gravitational waves. In our analysis, we explicitly include the modulation of the distribution of gravitational wave sources due to weak gravitational lensing. We show that the cross-correlation analysis in next-generation observations will be able to tightly constrain the relation between the absolute distance and the redshift and therefore constrain the Hubble constant as well as dark energy parameters.

Figure 1

The cross-correlation power spectrum $C^{\mathrm{wg}}$ between GW sources and galaxies [Eq. (18)]. The luminosity distance range of GW sources is fixed to that corresponds to $0.9<z<1.1$ (gray shaded region) in our fiducial cosmology. The spectroscopic galaxy sample has the redshift range $z_{\mathrm{g}}-\mathrm{\Delta }z/2<z<z_{\mathrm{g}}+\mathrm{\Delta }z/2$ with $\mathrm{\Delta }z=0.1$. Solid line shows the cross-correlation power spectrum at multipole $\ell =100$ as a function of the central redshift of the galaxy sample $z_{\mathrm{g}}$. Dotted and dashed lines show contributions of $C^{\mathrm{sg}}$ [Eq. (19)] and $C^{\mathrm{tg}}$ [Eq. (20)] to $C^{\mathrm{wg}}$, respectively.

Figure 2

Projected 68% C.L. constraints in the ${\mathrm{\Omega }}_{\mathrm{m}}\text{−}h$ (top) and ${\mathrm{\Omega }}_{\mathrm{m}}\text{−}{w}_{\mathrm{de}}$ (bottom) planes. In each panel, the other model parameters are marginalized over. Solid lines show constraints for ${\ell }_{\max }=300$, whereas dotted lines show constraints for ${\ell }_{\max }=100$.