White dwarf-neutron star binaries: A plausible pathway for long-duration gamma-ray bursts from compact object mergers

Two long-duration gamma-ray bursts (GRBs) were recently discovered with kilonovae, the signature of r-process element production in a compact binary merger, rather than supernovae. This has forced a re-evaluation of the long-established dichotomy between short bursts (2s, a class of massive star core-collapse events). We aim to determine whether white dwarf--neutron star (WDNS) mergers and white dwarf--black hole (WDBH) mergers are plausible explanations for long-duration compact merger GRBs, in terms of their galactocentric merger offsets and cosmological rates. We modelled the host galaxies of GRBs, 211211A and 230307A, and employed binary population synthesis to predict the merger offset distributions of compact binaries. We compared them with the observed (projected) offsets of GRBs, 211211A and 230307A. We also investigated the evolutionary pathways to WDNS and WDBH mergers, predicted their cosmological rates, and compared them with inferred volumetric GRB rates. We find that WDNS mergers occur at lower host offsets than binary neutron star mergers, but that in the specific cases of GRBs, 211211A and 230307A, the observed offsets are consistent with either scenario. We predict that WDNS mergers occur at a similar rate to binary neutron star mergers and long GRBs, and that WDBH mergers are a factor of ten rarer, with the caveat that these rates currently carry uncertainties of the order of the magnitude level. We demonstrate, solely in terms of galactocentric offsets and event rates, that WDNS mergers are a plausible explanation for GRBs, 211211A and 230307A, and long-duration gamma-ray bursts from compact object mergers more generally. WDNS binaries have lower systemic velocities than binary neutron stars, but longer delay times, and ultimately merge with an offset distribution that is not measurably different without large sample sizes. Therefore, offsets and rates alone cannot currently distinguish between compact binary progenitor models for supernova-less long-duration GRBs.