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The formation of compact objects-neutron stars, black holes, and supermassiveblack holes-and its connection to the chemical composition of the galaxies isone of the central questions in astrophysics. We propose a novel data-driven,multi-messenger technique to address this question by exploiting the inevitablecorrelation between gravitational waves and atomic/molecular emission linesignals. For a fiducial probability distribution function $p(t_d)\proptot_d^{-\kappa}$ of time delays, this method can provide a measurement of theminimum delay time of $0.5$ Gyr and the power-law index $\kappa=1$ with astandard deviation $0.12$ (and $0.45$) and $0.06$ (and $0.34$), respectivelyfrom five years of LIGO-Virgo-KAGRA observation in synergy with SPHEREx lineintensity mapping (and DESI emission-line galaxies). Such measurements willprovide data-driven multi-messenger constraints on the delay time distributionwhich is currently not well known.

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