Hot-spot model for accretion disc variability as random process

Aims. Theory of random processes provides an attractive mathematical tool to describe the fluctuating signal from accreting sources, such as active galactic nuclei and Galactic black holes observed in X-rays. These objects exhibit featureless variability on different timescales, probably originating from an accretion disc. Methods. We study the basic features of the power spectra in terms of a general framework, which permits semi-analytical determination of the power spectral density (PSD) of the resulting light curve. We consider the expected signal generated by an ensemble of spots randomly created on the accretion disc surface. Spot generation is governed by Poisson or by Hawkes processes. The latter one represents an avalanche mechanism and seems to be suggested by the observed form of the power spectrum. We include general relativity effects shaping the signal on its propagation to a distant observer. Results. We analyse the PSD of a spotted disc light curve and show the accuracy of our semi-analytical approach by comparing the obtained PSD with the results of Monte Carlo simulations. The asymptotic slopes of PSD are 0 at low frequencies and they drop to -2 at high frequencies, usually with a single frequency break. More complex two-peak solutions also occur. The amplitude of the peaks and their frequency difference depend on the inherent timescales of the model, i.e., the intrinsic lifetime of the spots and the typical duration of avalanches. Conclusions. At intermediate frequencies, the intrinsic PSD is influenced by the individual light curve profile as well as by the type of the underlying process. However, even in cases when two Lorentzians seem to dominate the PSD, it does not necessarily imply that two single oscillation mechanisms operate simultaneously. Instead, it may well be the manifestation of the avalanche mechanism. The main advantage of our approach is an insight in the model functioning and the fast evaluation of the PSD.