  Supermassive black hole (SMBH) binaries are expected in a Lambda CDM
cosmology given that most (if not all) massive galaxies contain a massive black
hole at their center. So far, however, direct evidence for such binaries has
been elusive. We use cross-correlation to search for temporal velocity shifts
in the MgII broad emission lines of 0.36 < z < 2 quasars with multiple
observations in the Sloan Digital Sky Survey. For ~ 10^9 Msun BHs in SMBH
binaries, we are sensitive to velocity drifts for binary separations of ~ 0.1
pc with orbital periods of ~100 years. We find seven candidate sub-pc--scale
binaries with velocity shifts > 3.4 sigma ~ 280 km/s, where sigma is our
systematic error. Comparing the detectability of SMBH binaries with the number
of candidates (N < 7), we can rule out that most 10^9 Msun BHs exist in ~
0.03-0.2 pc scale binaries, in a scenario where binaries stall at sub-pc scales
for a Hubble time. We further constrain that < one-third of quasars host SMBH
binaries after considering gas-assisted sub-pc evolution of SMBH binaries,
although this result is very sensitive to the assumed size of the broad line
region. We estimate the detectability of SMBH binaries with ongoing or
next-generation surveys (e.g., BOSS, Subaru Prime Focus Spectrograph), taking
into account the evolution of the sub-parsec binary in circumbinary gas disks.
These future observations will provide longer time baselines for searches
similar to ours and may in turn constrain the evolutionary scenarios of SMBH
binaries.
