A search for periodic AGN variability in $\textit{Gaia}$ Data Release 3
Supermassive black hole binaries (SMBHB) are expected to produce periodic modulations in active galactic nuclei (AGN) light curves, but distinguishing such signals from stochastic red-noise variability remains a major challenge. We present the first systematic search for statistically significant AGN periodicities using the optical photometry from the Gaia space mission Data Release 3 (DR3), with the goal of identifying SMBHB candidates and establishing a methodological data analysis framework that can be scaled to the forthcoming Data Release 4 (DR4). We analyse Gaia G band light curves of 377,128 sources from the Gaia celestial reference frame (CRF3). Stochastic variability is modelled as a damped random walk Gaussian process, and empirical false alarm probabilities are derived by comparing observed Lomb-Scargle periodogram peaks against 100,000 synthetic red-noise realisations. Candidates from this first stage are then re-evaluated using full Markov chain Monte Carlo inference under both exponential and powered-exponential kernels. We find 13 sources surviving our statistical criterion ($p < α= 10^{-5}$) after both stages of filtering, which is consistent with the expected false-positive rate. All candidates cover fewer than 2.5 cycles of the candidate period and are systematically concentrated in a region of the parameter space indicative of model misspecification. No reliable periodic SMBHB candidates are retained. The ${\sim}950$-day baseline of Gaia DR3 confines all detections to the few-cycle regime where red noise most convincingly mimics periodicity, a limitation that photometric precision alone cannot overcome. The longer baseline of Gaia DR4 will be essential to push beyond this regime. We offer our data analysis software pipeline in open access to the community.