Likelihood for Detection of Sub-parsec Supermassive Black Hole Binaries
  in Spectroscopic Surveys
  Motivated by observational searches for sub-parsec 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 parametrized 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, $10^8M_\odot$ SBHBs must, on
average, experience angular momentum transport faster than that provided by a
disk with accretion rate $0.06\,\dot{M}_E$. Spectroscopic searches with yearly
cadence of observations are in principle sensitive to binaries with orbital
separations $< {\rm few}\times 10^4\, r_g$ ($r_g = GM/c^2$ 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 AGN-like emission lines utilized by spectroscopic searches, the
projection factors imply five undetected binaries for each observed
$10^8M_\odot$ SBHB with mass ratio $0.3$ and orbital separation $10^4\,r_g$
(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 to
provide constraints on the rate of orbital evolution of SBHBs, if observed
candidates are shown to be genuine binaries.
