Unveiling the architecture of a pulsar - binary black-hole triple system with pulsar arrival time analysis

A large number of binary black holes (BBHs) with longer orbital periods are supposed to exist as progenitors of BBH mergers recently discovered with gravitational wave (GW) detectors. In our previous papers, we proposed to search for such BBHs in triple systems through the radial-velocity modulation of the tertiary orbiting star. If the tertiary is a pulsar, high precision and cadence observations of its arrival time enable an unambiguous characterization of the pulsar -- BBH triples located at several kpc, which are inaccessible with the radial velocity of stars. The present paper shows that such inner BBHs can be identified through the short-term R{\o}mer delay modulation, on the order of $10$ msec for our fiducial case, a triple consisting of $20~M_\odot$ BBH and $1.4~M_\odot$ pulsar with $P_\mathrm{in}=10$ days and $P_\mathrm{out}=100$ days. If the relativistic time delays are measured as well, one can determine basically all the orbital parameters of the triple. For instance, this method is applicable to inner BBHs of down to $\sim 1$ hr orbital periods if the orbital period of the tertiary pulsar is around several days. Inner BBHs with $\lesssim 1$ hr orbital period emit the GW detectable by future space-based GW missions including LISA, DECIGO, and BBO, and very short inner BBHs with sub-second orbital period can be even probed by the existing ground-based GW detectors. Therefore, our proposed methodology provides a complementary technique to search for inner BBHs in triples, if exist at all, in the near future.