Detecting non-sinusoidal periodicities in observational data: the von Mises periodogram for variable stars and exoplanetary transits

This paper introduces an extension of the linear least-squares (or Lomb-Scargle) periodogram for the case when the model of the signal to be detected is non-sinusoidal and depends on unknown parameters in a non-linear manner. The attention is paid to the problem of estimating the statistical significance of candidate periodicities found using such non-linear periodograms. This problem is related to the task of quantifying the distributions of maximum values of these periodograms. Based on recent results in the mathematical theory of extreme values of random field (the generalized Rice method), we give a general approach to find handy analytic approximation for these distributions. This approximation has the general form $e^{-z} P(\sqrt z)$, where $P$ is an algebraic polynomial and $z$ being the periodogram maximum. The general tools developed in this paper can be used in a wide variety of astronomical applications, for instance in the studies of variable stars and extrasolar planets. For this goal, we develop and consider in details the so-called von Mises periodogram: a specialized non-linear periodogram where the signal is modelled by the von Mises periodic function $\exp(\nu \cos \omega t)$. This simple function with an additional non-linear parameter $\nu$ can model lightcurves of many astronomical objects that show periodic photometric variability of different nature. We prove that our approach can be perfectly applied to this non-linear periodogram. We provide a package of auxiliary C++ programs, attached as the online-only material. They should faciliate the use of the von Mises periodogram in practice.