Towards a Precision Measurement of Binary Black Holes Formation Channels Using Gravitational Waves and Emission Lines

The formation of compact objects--neutron stars, black holes, and supermassive black holes--and its connection to the chemical composition of the galaxies is one of the central questions in astrophysics. We propose a novel data-driven, multi-messenger technique to address this question by exploiting the inevitable correlation between gravitational waves and atomic/molecular emission line signals. We show that the minimum delay time of 0.5 Gyr and the power-law index $κ=1$ of the fiducial scenario of the probability distribution function $p(t_d) \propto t_d^{-κ}$ is possible to measure with a standard deviation 0.12 (and 0.45) and 0.06 (and 0.34) respectively from five years of LIGO-Virgo-KAGRA observation in synergy with SPHEREx line intensity mapping (and DESI emission-line galaxies).