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.
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