  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. For a fiducial probability distribution function $p(t_d)\propto
t_d^{-\kappa}$ of time delays, this method can provide a measurement of the
minimum delay time of $0.5$ Gyr and the power-law index $\kappa=1$ 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). Such measurements will
provide data-driven multi-messenger constraints on the delay time distribution
which is currently not well known.
