A Stochastic Model for the Luminosity Fluctuations of Accreting Black Holes

In this work we have developed a new stochastic model for the fluctuations in lightcurves of accreting black holes. The model is based on a linear combination of stochastic processes and is also the solution to the linear diffusion equation perturbed by a spatially correlated noise field. This allows flexible modeling of the power spectral density (PSD), and we derive the likelihood function for the process, enabling one to estimate the parameters of the process, including break frequencies in the PSD. Our statistical technique is computationally efficient, unbiased by aliasing and red noise leak, and fully accounts for irregular sampling and measurement errors. We show that our stochastic model provides a good approximation to the X-ray lightcurves of galactic black holes, and the optical and X-ray lightcurves of AGN. We use the estimated time scales of our stochastic model to recover the correlation between characteristic time scale of the high frequency X-ray fluctuations and black hole mass for AGN, including two new `detections’ of the time scale for Fairall 9 and NGC 5548. We find a tight anti-correlation between the black hole mass and the amplitude of the driving noise field, which is proportional to the amplitude of the high frequency X-ray PSD, and we estimate that this parameter gives black hole mass estimates to within ~ 0.2 dex precision, potentially the most accurate method for AGN yet. We also find evidence that ~ 13% of AGN optical PSDs fall off flatter than 1 / f^2, and, similar to previous work, find that the optical fluctuations are more suppressed on short time scales compared to the X-rays, but are larger on long time scales, suggesting the optical fluctuations are not solely due to reprocessing of X-rays.

💡 Research Summary

The authors present a novel stochastic framework for modeling the variability observed in the light curves of accreting black holes, both stellar‑mass (Galactic Black Holes, GBHs) and supermassive (Active Galactic Nuclei, AGN). The core of the model is a linear diffusion equation perturbed by a spatially correlated noise field. In mathematical terms, the governing equation is ∂ψ/∂t = D∇²ψ + ξ(x,t), where D is a diffusion coefficient and ξ is a zero‑mean Gaussian field with an exponential spatial correlation function C(r)=σ² exp(−r/ℓ). By Fourier transforming the equation, each spatial mode k acquires a transfer function proportional to 1/(Dk²+γ), which translates into a power spectral density (PSD) that is flat at low frequencies, breaks at characteristic frequencies set by D and ℓ, and falls off as a power law (≈k⁻² or steeper) at higher frequencies.

To make the model applicable to real astronomical time series, the authors embed it within a Gaussian Process (GP) formalism. The GP covariance kernel is derived directly from the diffusion‑noise solution, and measurement uncertainties are added to the diagonal of the covariance matrix. This approach naturally handles irregular sampling, avoids aliasing and red‑noise leakage, and yields an exact likelihood function:

L(θ) = (2π)^{-N/2} |C(θ)|^{-1/2} exp