A hierarchical Bayesian framework for cosmology using Type 1 AGN variability

Independent luminosity-distance probes beyond the Type Ia supernova range are needed to test cosmic expansion at high redshift. Type 1 AGN are abundant at $z>2$, but their use for cosmology requires standardizable observables with controlled scatter, redshift dependence, and measurement uncertainty. We present a hierarchical Bayesian framework for cosmology using AGN variability, based on the empirical anti-correlation between optical/UV variability amplitude and luminosity. The method targets the moderate-baseline regime of current wide-field time-domain surveys, where individual light curves cannot typically identify the full long-timescale stochastic process, but can constrain finite-window brightness and short-lag variability. Each light curve is fitted independently to obtain posterior samples of these summaries, which are then importance-reweighted under a population model relating variability to luminosity, rest-frame wavelength, intrinsic scatter, and the assumed distance-redshift relation. This framework propagates object-level uncertainty while avoiding repeated light-curve likelihood evaluations during cosmological inference, making catalogue-scale analyses feasible. Using Gaia DR3-like G-band simulations matched to real Gaia cadences, noise properties, and quality cuts, we show that finite-baseline light curves are more robustly summarized by window-averaged brightness and short-lag variability than by the separate long-timescale parameters of stochastic models. End-to-end closure tests recover the injected variability-luminosity relation, intrinsic scatter, and distance-redshift parameters up to the expected calibration degeneracies. This Gaia G-band analysis establishes a proof of concept for AGN-variability distances in the moderate-baseline survey regime, with the main gains expected from Gaia DR4, ZTF, DESI-selected AGN samples, and Rubin/LSST-era data.