A strategy to search for an inner binary black hole from the motion of the tertiary star

There are several on-going projects to search for stars orbiting around an invisible companion. A fraction of such candidates may be a triple, instead of a binary, consisting of an inner binary black hole (BBH) and an outer orbiting star. In this paper, we propose a methodology to search for a signature of such an inner BBH, possibly a progenitor of gravitational-wave sources discovered by {\it LIGO}, from the precise radial velocity (RV) follow-up of the outer star. We first describe a methodology using an existing approximate RV formula for coplanar circular triples. We apply this method and constrain the parameters of a possible inner binary objects in 2M05215658+4359220, which consists of a red giant and an unseen companion. Next we consider co-planar but non-circular triples. We compute numerically the RV variation of a tertiary star orbiting around an inner BBH, generate mock RV curves, and examine the feasibility of the BBH detection for our fiducial models. We conclude that the short-cadence RV monitoring of a star-BH binary provides an interesting and realistic method to constrain and/or search for possible inner BBHs. Indeed a recent discovery of a star--BH binary system LB-1 may imply that there are a large number of such unknown objects in our Galaxy, which are ideal targets for the methodology proposed here.