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 $\kappa=1$ of the fiducial scenario of the probability
distribution function $p(t_d)\propto t_d^{-\kappa}$ 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).
