A Cross-Band (X-ray $\times$ Optical) Periodicity Search for Supermassive Black Hole Binaries: A Null Result and the First Completeness-Corrected Constraint

Abstract: 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-BAT hard X-ray AGN (Stage 1) and 175 4XMM-DR14 AGN (Stage 2). No source is a co-periodic 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 disc, we require both bands individually significant with periods coincident within 5%, and gate the survivors with the model-independent null-signal-template test of Robnik et al. (2024). Over $P=100$-$900$ d the completeness-corrected 95% upper limit on the co-periodic fraction is amplitude-dependent: $\lesssim 3\%$ for hard-X-ray fractional modulation $\gtrsim 0.3$, $\approx 15\%$ (precision-limited) at 0.2, and uninformative below $\sim 0.15$ (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 co-periodic fraction is $$\sim 3\times10^{-2}\, f_{\rm bin}\, δ<em>{\rm mod}$$ (modulo a factor $g&lt;1$), with $f</em>{\rm bin}$ the sub-pc binary fraction, $δ_{\rm mod}$ 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 co-periodic fraction, ready for the denser X-ray monitoring of Einstein Probe and eROSITA.

• The paper conducts the first sample-level cross-band periodicity search for supermassive black hole binaries, analyzing 1,194 Swift-BAT AGN alongside optical light curves and validating false-positive rates with end-to-end injections.
• The search finds zero convincing candidates, while cross-band period matching suppresses optical red-noise false positives by approximately 1,000-fold beyond single-band analyses.
• The completeness-corrected upper limit is approximately 3% for binaries producing hard-X-ray fractional modulation of at least 0.3, with sensitivity limited primarily by X-ray cadence rather than optical data quality.

Motivation and design

The paper addresses the identification of electromagnetic counterparts to the sub-parsec supermassive black hole binary (SMBHB) population inferred from pulsar-timing-array detections of the nanohertz gravitational-wave background. Existing photometric periodicity searches have nominated of order $10^2$ candidates, but essentially none survive model-independent red-noise scrutiny. The central methodological idea is that hard X-rays (coronal emission) and optical continuum (disc emission) have largely independent stochastic variability, whereas a genuine binary modulating the global accretion flow should imprint a common period on both bands. Requiring period coincidence across two quasi-independent channels multiplies single-band false-alarm probabilities; in this sample it suppresses the optical red-noise false-positive excess by an additional factor of $\sim10^3$ beyond what any single light curve can achieve.

Two samples are analysed: Stage 1 comprises 1194 Swift-BAT AGN with monthly 14–195 keV light curves (1022 accretion-driven, 172 jet-dominated), cross-matched to ZTF and ASAS-SN optical photometry; Stage 2 comprises 175 4XMM-DR14 serendipitous AGN, included as a pipeline portability demonstration rather than an independent constraint.

Pipeline and validation

Each band is fitted with a damped random walk (DRW) via Gaussian-process marginal likelihood, searched with a generalised Lomb–Scargle periodogram over observed-frame periods $P\in[100,3000]$ d (truncated to $\approx100$–900 d after cycle-coverage cuts and seasonal-alias masking), and assigned a look-elsewhere-corrected Monte-Carlo significance against DRW nulls matched to each band's sampling. Candidate preconditions are both bands flagged at $p<10^{-3}$ with periods matching within 5% (a permutation-calibrated chance-coincidence rate of 3.7%); survivors would then face Fisher combination and the model-independent null-signal-template (NST) test of Robnik et al.

Validation is unusually thorough for this class of search. False-positive calibration on synthetic DRW realisations tracks nominal rates at all three survey cadences. End-to-end completeness is measured by injecting sinusoids and Doppler-boost profiles onto the real light curves and running the identical production path — 960 multi-band injections plus a dedicated 1413-injection BAT set — rather than bypassing the per-band fit. The result is strongly band-asymmetric: optical recovery reaches 50–90% above $\sim$0.2 mag semi-amplitude, while the noise-dominated monthly BAT cadence recovers only $\lesssim18\%$ even at the largest amplitudes, and the sparse 4XMM cadence essentially zero. Noise-model robustness is checked by comparing DRW against a more flexible DRW+SHO kernel: 99.6% of rescaled BAT bands favour plain DRW, so the null result does not depend on the noise-model assumption.

The null result

After a robust count-rate rescaling that cures a documented BAT DRW-fit pathology (89.9% of raw fits hit a method-of-moments fallback bound; rescaling restores interior convergence for ~96%), no source in either stage is both-flagged, period-matched, or NST-confirmed: 0 tier-1 and 0 tier-2 candidates, holding separately in the accretion-driven and jet-dominated subsets. The expected number of chance co-periodic false positives under the null is $\approx0.015$, consistent with the observed zero and confirming the search is not threshold-limited.

One instructive case is SWIFT J0507.7+6732, which appeared as a tier-1 candidate at broad-pass Monte-Carlo resolution ($p_{\rm joint}\sim4\times10^{-6}$) but dissolved when its BAT $p$-value was refined from the $\sim10^3$0 floor to 0.020 — a Monte-Carlo-resolution artefact atop an unreliable fallback fit. This demonstrates concretely that broad-pass flags are upper bounds requiring resolution before significance is claimed.

The completeness-corrected limit

A direct joint injection experiment (1800 co-injections into 300 BAT+optical pairs at common physical fractional amplitude $\sim10^3$1) removes both the factorisation assumption and the rescaled-unit ambiguity. Joint completeness is steeply amplitude-dependent — 0.017 at $\sim10^3$2, 0.09 at 0.3, 0.14 at 0.4 — so the limit must be quoted as a function of amplitude:

$\sim10^3$3

with the raw $\sim10^3$4 value of 0.25% explicitly identified as an optimistic floor rather than a conservative bound. The amplitude dependence is dominated entirely by the X-ray leg: optical completeness alone would give $\sim10^3$50.3%. Under the Doppler-boost mechanism the hard-X-ray-to-optical amplitude ratio is $\sim10^3$6, reinforcing that cadence and noise, not amplitude deficit, bind the sensitivity — though the authors note this conservatism is mechanism-specific, since for accretion-rate modulation the cross-band amplitude ratio is unconstrained.

Deep-cadence test

The completeness analysis makes a falsifiable prediction: where X-ray cadence is dense, the search should cease to be X-ray-limited. For the 60 X-ray-brightest AGN, ingesting daily MAXI/GSC and RXTE/ASM monitoring (e.g., NGC 4151 gains 8805 X-ray epochs versus 157 from BAT, a $\sim10^3$7 increase) lifts the pooled X-ray efficiency to $\sim10^3$8 at $\sim10^3$9 — a 5–8-fold improvement at $P\in[100,3000]$0. The search remains null even in this high-completeness regime, though with $P\in[100,3000]$1 usable sources the resulting fraction limit (~13–27%) is weaker than the full-sample constraint. The authors candidly flag three limitations here: the monitored sources are the most-studied AGN in the sky, Compton-thick Seyfert 2s are absorbed out of the monitor bands, and the ASM+MAXI stitch could inject low-frequency power at the seam.

Population interpretation

Integrating the gravitational-wave-driven residence time $P\in[100,3000]$2 over the BASS active black-hole mass function gives a sample-averaged residence fraction $P\in[100,3000]$3 (GW-validity-restricted, stable at $P\in[100,3000]$4). The expected all-amplitude co-periodic fraction is therefore $P\in[100,3000]$5, compared like-for-like with the large-amplitude limit via the unknown fraction $P\in[100,3000]$6 reaching recoverable amplitude. Only in the most optimistic corner ($P\in[100,3000]$7, $P\in[100,3000]$8) does the expectation approach the $P\in[100,3000]$9 limit; for plausible $\approx100$0–$\approx100$1 the expectation falls 1–3 orders below sensitivity, so a null is the firmly expected outcome. The constraint does not yet test $\approx100$2 strongly.

Limitations

Several qualifications bear directly on the results. The cross-band independence assumption is weakest for the BAT×ASAS-SN pair (67% of sources), whose epochs overlap in time, permitting residual reprocessing correlation; the authors show quantitatively that even maximally correlated nulls would yield $\approx100$3 both-flagged source, which must still period-match, so the verdict is robust. The search window excludes classic long-period candidates such as PG 1302−102 and OJ 287. Below $\approx100$4 the surveys cannot detect modulation even if every AGN hosted such a binary, so the limit is uninformative there. The BAT DRW-fit pathology biases the $\approx100$5 floor optimistically until end-to-end completeness is folded in, and the deep-cadence efficiency extrapolation rests on a 7-source injection subsample.

Conclusion

This work delivers the first sample-level cross-band (X-ray × optical) periodicity search for SMBHBs, a validated and publicly archived pipeline, and the first completeness-corrected constraint on the co-periodic fraction: $\approx100$6 for hard-X-ray fractional modulation $\approx100$7, weakening steeply below. Its principal empirical finding is counterintuitive and consequential: the sensitivity of such searches is set by hard-X-ray monitoring cadence, not optical photometry or statistics — demonstrated directly by the several-fold completeness lift under daily MAXI/ASM monitoring. The framework is positioned for the denser X-ray monitoring of Einstein Probe and eROSITA, where the multiplied-false-alarm advantage of the cross-band requirement can convert clean nulls into tight population constraints.