Space-borne Atom Interferometric Gravitational Wave Detections II: Dark Sirens and Finding the One

In this paper, we investigate the potential of dark sirens by the space-borne atom interferometric gravitational-wave detectors to probe the Hubble constant. In the mid-frequency band, the sources live a long time. The motion of a detector around the Sun as well as in Earth orbit would induce large Doppler and reorientation effects, providing a precise angular resolution. Such precise localization for the GW sources makes it possible to observe the dark sirens with only one potential host galaxy, which are dubbed "golden dark sirens". We construct the catalogs of golden dark sirens and estimate that there are around 79 and 35 golden dark sirens of binary neutron stars (BNS) and binary black holes (BBH) that would be pass the detection threshold of AEDGE in 5 years. Our results show that with 5, 10, and all 79 golden dark BNS tracked by AEDGE one can constrain $H_0$ at 5.5\%, 4.1\%, and 1.8\% precision levels. With 5, 10, and all 35 golden dark BBH one can constrain $H_0$ at 2.2\%, 1.8\%, and 1.5\% precision levels, respectively. It suggests that only 5-10 golden dark BBH by AEDGE are sufficient to arbitrate the current tension between local and high-$z$ measurements of $H_0$.