Constraining the lensing dispersion from the angular clustering of binary black hole mergers

Gravitational waves from inspiraling compact binaries provide direct measurements of luminosity distances and serve as a powerful probe of the high-redshift Universe. In addition to their role as standard sirens, they offer an opportunity to constrain small-scale density fluctuations through the dispersion in the distance-redshift relation induced by gravitational lensing. We propose a method to constrain this lensing dispersion without requiring the redshift information by analyzing the angular clustering of gravitational wave sources. Our formalism incorporating second-order lensing effects in the luminosity distance shows that the amplitude of the auto-correlation angular clustering decreases with increasing lensing dispersion. While we show that the auto-correlation signal is detected with sufficient signal-to-noise ratios in future gravitational wave experiments, there exists a strong degeneracy between the lensing dispersion and the linear bias of gravitational wave sources. We demonstrate that this degeneracy is partially broken by a joint analysis of the auto-correlation of gravitational wave sources and the cross-correlation with galaxies whose redshifts are known. This approach enhances the use of gravitational waves as a cosmological probe at high redshifts.