Likelihood for Detection of Subparsec Supermassive Black Hole Binaries in Spectroscopic Surveys

Motivated by observational searches for subparsec supermassive black hole binaries (SBHBs), we develop a modular analytic model to determine the likelihood for detection of SBHBs by ongoing spectroscopic surveys. The model combines the parameterized rate of orbital evolution of SBHBs in circumbinary disks with the selection effects of spectroscopic surveys and returns a multivariate likelihood for SBHB detection. Based on this model, we find that in order to evolve into the detection window of the spectroscopic searches from larger separations in less than a Hubble time, 10⁸ M _o SBHBs must, on average, experience angular momentum transport faster than that provided by a disk with accretion rate . Spectroscopic searches with yearly cadences of observations are in principle sensitive to binaries with orbital separations less than a few 10⁴ r _g (r _g = GM/c ² and M is the binary mass), and for every one SBHB in this range, there should be over 200 more gravitationally bound systems with similar properties, at larger separations. Furthermore, if spectra of all SBHBs in this separation range exhibit the active galactic nucleus-like emission lines utilized by spectroscopic searches, the projection factors imply five undetected binaries for each observed 10⁸ M _o SBHB with mass ratio 0.3 and orbital separation 10⁴ r _g (and more if some fraction of SBHBs is inactive). This model can be used to infer the most likely orbital parameters for observed SBHB candidates and provide constraints on the rate of orbital evolution of SBHBs, if observed candidates are shown to be genuine binaries.