Prospect of Precision Cosmology and Testing General Relativity using Binary Black Holes- Galaxies Cross-correlation

Modified theories of gravity predict deviations from General Relativity (GR) in the propagation of gravitational waves (GW) across cosmological distances. A key prediction is that the GW luminosity distance will vary with redshift, differing from the electromagnetic (EM) luminosity distance due to varying effective Planck mass. We introduce a model-independent, data-driven approach to explore these deviations using multi-messenger observations of dark standard sirens (Binary Black Holes, BBH). By combining GW luminosity distance measurements from dark sirens with Baryon Acoustic Oscillation (BAO) measurements, BBH redshifts inferred from cross-correlation with spectroscopic or photometric galaxy surveys, and sound horizon measurements from the Cosmic Microwave Background (CMB), we can make a data-driven test of GR (jointly with the Hubble constant) as a function of redshift. Using the multi-messenger technique with the spectroscopic DESI galaxy survey, we achieve precise measurements of deviations in the effective Planck mass variation with redshift. For the Cosmic Explorer and Einstein Telescope (CEET), the best precision is approximately 3.6%, and for LIGO-Virgo-KAGRA (LVK), it is 7.4% at a redshift of $\rm {z = 0.425}$. Additionally, we can measure the Hubble constant with a precision of about 1.1% from CEET and 7% from LVK over five years of observation with a 75% duty cycle. We also explore the potential of cross-correlation with photometric galaxy surveys from the Rubin Observatory, extending measurements up to a redshift of $\rm {z \sim 2.5}$. This approach can reveal potential deviations from models affecting GW propagation using numerous dark standard sirens in synergy with DESI and the Rubin Observatory.