Stochastic Modeling Handbook for Optical AGN Variability

This work develops application techniques for stochastic modeling of Active Galactic Nuclei (AGNs) variability as a probe of accretion disk physics. Stochastic models, specifically Continuous Auto-Regressive Moving Average (CARMA) models, characterize light curves with a perturbation spectrum and an Impulse-Response function, which crucially provides an interpretation for variability timescales. CARMA timescales are not physical but rather, they describe correlation structure and ordered information in stochastic processes. We begin this tutorial by reviewing discrete auto-regressive and moving-average processes, we bridge these components to their continuous analogs, and lastly we investigate the significance of CARMA timescales, obtained by modeling a light curve in the time domain, in relation to the shape of the power spectrum (PSD) and structure function. We determine that higher order CARMA models, for example the Damped Harmonic Oscillator (DHO or CARMA(2, 1)) are more sensitive to deviations from a single-slope power-law description of AGN variability; unlike Damped Random Walks (DRW or CAR(1)) where the PSD slope is fixed, the DHO slope is not. Higher complexity stochastic models than the DRW capture additional covariance in data and output additional characteristic timescales that probe the driving mechanisms of variability. We provide code using Kali software to generate simulations of diverse complexity stochastic light curves. We also provide a heuristic discussion of aliasing effects in ground-based cadences and the importance of light curve length in regards to uncertainty and limitations in timescale estimation.