  Nearly a hundred of binary black holes (BBHs) have been discovered with
gravitational-wave signals emitted at their merging events. Thus, it is quite
natural to expect that significantly more abundant BBHs with wider separations
remain undetected in the universe, or even in our Galaxy. We consider a
possibility that star-BH binary candidates may indeed host an inner BBH,
instead of a single BH. We present a detailed feasibility study of constraining
the binarity of the currently available two targets, Gaia BH1 and Gaia BH2.
Specifically, we examine three types of radial velocity (RV) modulations of a
tertiary star in star-BBH triple systems; short-term RV modulations induced by
the inner BBH, long-term RV modulations induced by the nodal precession, and
long-term RV modulations induced by the von Zeipel-Kozai-Lidov oscillations.
Direct three-body simulations combined with approximate analytic models reveal
that Gaia BH1 system may exhibit observable signatures of the hidden inner BBH
if it exists at all. The methodology that we examine here is quite generic, and
is expected to be readily applicable to future star-BH binary candidates in a
straightforward manner.
