  Theory predicts that a supermassive black hole binary (SMBHB) could be
observed as a luminous active galactic nucleus (AGN) that periodically varies
on the order of its orbital timescale. In X-rays, periodic variations could be
caused by mechanisms including relativistic Doppler boosting and shocks. Here
we present the first systematic search for periodic AGNs using $941$ hard X-ray
light curves (14-195 keV) from the first 105 months of the Swift Burst Alert
Telescope (BAT) survey (2004-2013). We do not find evidence for periodic AGNs
in Swift-BAT, including the previously reported SMBHB candidate
MCG+11$-$11$-$032. We find that the null detection is consistent with the
combination of the upper-limit binary population in AGNs in our adopted model,
their expected periodic variability amplitudes, and the BAT survey
characteristics. We have also investigated the detectability of SMBHBs against
normal AGN X-ray variability in the context of the eROSITA survey. Under our
assumptions of a binary population and the periodic signals they produce which
have long periods of hundreds of days, up to $13$% true periodic binaries can
be robustly distinguished from normal variable AGNs with the ideal uniform
sampling. However, we demonstrate that realistic eROSITA sampling is likely to
be insensitive to long-period binaries because longer observing gaps reduce
their detectability. In contrast, large observing gaps do not diminish the
prospect of detecting binaries of short, few-day periods, as 19% can be
successfully recovered, the vast majority of which can be identified by the
first half of the survey.
