How many supermassive black hole binaries are detectable through tracking relative motions by sub/millimeter VLBI

The sub/millimeter wavelengths (86-690 GHz) very long baseline interferometry (VLBI) will provide $\sim5-40\ \mu$as angular resolution, $\sim10$ mJy baseline sensitivity, and $\sim 1\ \mu$as/yr proper motion precision, which can directly detect supermassive black hole binary (SMBHB) systems by imaging the two visible sources and tracking their relative motions. Such a way exhibits an advantage compared to indirect detect methods of observing periodic signals in motion and light curves, which are difficult to confirm from competing models. Moreover, tracking relative motion at sub/millimeter wavelengths is more reliable, as there is a negligible offset between the emission region and the black hole center. In this way, it is unnecessary to correct the black hole location by a prior of jet morphology as it would be required at longer wavelengths. We extend the formalism developed in D'Orazio & Loeb (2018) to link the observations with the orbital evolution of SMBHBs from the $\lesssim$10 kpc dynamical friction stages to the $\lesssim 0.01$ pc gravitational radiation stages, and estimate the detectable numbers of SMBHBs. By assuming 5\% of AGNs holding SMBHBs, we find that the number of detectable SMBHBs with redshift $z\le 0.5$ and mass $M\leq 10^{11}M_\odot$ is about 20. Such detection relies heavily on proper motion precision and sensitivity. Furthermore, we propose that the simultaneous multi-frequency technique plays a key role in meeting the observational requirements.