Inferring the role of binary neutron star mergers in r-process nucleosynthesis with multi-messenger observations using Cosmic Explorer and Einstein Telescope

Identifying the cosmic origin of rapid neutron-capture (r-process) elements remains an open problem. Binary neutron-star (BNS) mergers and rare classes of core-collapse supernovae (CCSNe) represent the main contenders as major r-process production sites. Although BNS mergers could exclusively account for r-process nucleosynthesis, results from chemical evolution studies taking into account their delays with respect to star formation, observed BNS rates by gravitational-wave (GW) detectors, as well as issues with retention in low-mass halos suggest otherwise. Here, we propose a method to measure the contribution of BNS mergers to cosmic r-process nucleosynthesis with the third-generation GW detectors Cosmic Explorer and Einstein Telescope. It exploits the redshift-dependent correlation between the total number of BNS GW events and the average r-process abundances at redshifts $z \lesssim 1$. We apply this correlation technique to mock GW and abundance data, accounting for expected observational uncertainties in two limiting scenarios: GW events with electromagnetic counterpart (multi-messenger 'bright-sirens') and without ('dark-sirens'). Using Fisher forecasts, we demonstrate that the fractional cumulative contribution of BNS mergers to the total cosmic r-process $F_{\rm{BNS,z0}}$ can be estimated to the $\lesssim 5-6\%$ precision level for both scenarios at $1\sigma$ for fiducial astrophysical scenarios with $F_{\rm{BNS,z0}} \gtrsim 0.1-1$. Furthermore, the method also yields estimates of the BNS delay-time distribution parameters comparable to other approaches. Although cosmic r-process abundances may be reconstructed from local observations at low metallicity, this method also provides a science case to identify signatures of neutron-capture elements beyond the local Universe.