Compact Binary Formation in Open Star Clusters III: Probability of Binary Black Holes Hidden Inside of Gaia Black Hole Binary

The Gaia mission and its follow-up observations have discovered a few candidates of non-interacting single black holes (BHs) and visible stars, Gaia BH1, BH2, and BH3, collectively called "astrometric BH binaries". This paper investigates whether any of these candidates harbor binary BHs (BBHs), namely, whether any such candidates are previously undiscovered "astrometric BBH triples''. Focusing on open star clusters, which are promising formation sites of astrometric BH binaries, we estimate the formation rate of astrometric BBH triples through gravitational $N$-body simulations. We find a competitively high formation efficiency of astrometric BBH triples ($\sim 10^{-6} M_\odot^{-1}$ or $\sim 10$\% of astrometric BH binaries) in low-metallicity environments but no astrometric BBH triples in solar-metallicity environments. Most of the astrometric BBH triples in our simulations were dynamically stable for $10$ Gyrs, indicating that $\sim10$\% of astrometric BH binary candidates may indeed harbor inner BBHs if they originate from open star clusters in low-metallicity environments. Astrometric BBH triples can be distinguished from astrometric BH binaries through radial velocity follow-up of the tertiary star. According to the statistics of our simulated samples, a small percent of astrometric BH binary candidates should exhibit detectable radial-velocity modulations generated by inner BBHs. Such candidates preferentially exhibit "outer'' orbital periods of $\gtrsim 10^3$ days and moderately high "outer'' orbital eccentricities ($\gtrsim 0.7$). Our current result will strongly motivate the search for astrometric BBH triples in the upcoming Gaia Data Release 4 and Gaia Final Data Release.