Compact Binary Formation in Open Star Clusters III: Probability of Binary Black Holes Hidden in 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 ``astrimetric 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 ($\sim 10^{-6} {\rm M}_\odot^{-1}$ 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.