A Cross-band (X-Ray × Optical) Periodicity Search for Supermassive Black Hole Binaries: A Null Result and the First Completeness-corrected Constraint

We present the first sample-level search for supermassive black hole binaries (SMBHBs) requiring coherent quasi-periodicity at a common period in the X-ray and optical bands, over 1194 Swift–Burst Alert Telescope (BAT) hard X-ray active galactic nuclei (AGN) (Stage 1) and 175 4XMM-DR14 AGN (Stage 2). No source is a coperiodic candidate. Each light curve is modelled as a damped random walk (DRW) and searched with a Lomb–Scargle periodogram and a look-elsewhere-corrected Monte Carlo significance. Because DRW red noise is largely independent between corona and disk, we require both bands individually significant with periods coincident within 5%, and gate the survivors with the model-independent null-signal-template test of J. Robnik et al. Over P = 100–900 days the completeness-corrected 95% upper limit on the coperiodic fraction is amplitude-dependent: ≲3% for hard-X-ray fractional modulation ≳0.3, ≈15% (precision-limited) at 0.2, and uninformative below ∼0.15 (the ϵ = 1 floor is 0.25%). The sensitivity is set by the hard X-ray monitoring, not the optical photometry or the statistics, the opposite of the usual assumption. Daily MAXI and RXTE/ASM monitoring of the brightest AGN raises the X-ray completeness 5–8-fold, and the search remains null. Integrated over the BAT black hole mass function, the expected all-amplitude coperiodic fraction is ∼ 3 × 1 0 − 2 f bin δ mod (modulo a factor g < 1), with f bin the subparsec binary fraction, δ mod the modulating duty cycle, and g the fraction reaching recoverable hard-X-ray amplitude, so a null is expected. We deliver a validated cross-band framework and the first completeness-corrected constraint on the coperiodic fraction, ready for the denser X-ray monitoring of Einstein Probe and eROSITA.