GRB 211211A-like Events and How Gravitational Waves May Tell Their Origins

GRB 211211A is a rare burst with a genuinely long duration, yet its prominent kilonova association provides compelling evidence that this peculiar burst was the result of a compact binary merger. However, the exact nature of the merging objects, whether they were neutron star pairs, neutron star--black hole systems, or neutron star--white dwarf systems, remains unsettled. This {\it Letter} delves into the rarity of this event and the possibility of using current and next-generation gravitational wave detectors to distinguish between the various types of binary systems. Our research reveals an event rate density of $\gtrsim 5.67^{+13.04}_{-4.69} \times 10^{-3}\ \rm Gpc^{-3}\ yr^{-1}$ for GRB 211211A-like gamma-ray bursts (GRBs), which, assuming GRB 211211A is the only example of such a burst, is significantly smaller than that of typical long- and short-GRB populations. We further calculated that if the origin of GRB 211211A is a result of a neutron star--black hole merger, it would be detectable with a significant signal-to-noise ratio (S/N), given the LIGO-Virgo-KAGRA designed sensitivity. On the other hand, a neutron star--white dwarf binary would also produce a considerable S/N during the inspiral phase at decihertz and is detectable by next-generation spaceborne detectors DECIGO and the Big Bang Observer. However, to detect this type of system with millihertz spaceborne detectors like LISA, Taiji, and TianQin, the event must be very close, approximately 3 Mpc in distance or smaller.