Probability models of chance fluctuations in spectra of astronomical sources with applications to X-ray absorption lines

The search for faint emission or absorption lines in astronomical spectra has received considerable attention in recent years, especially in the X-ray wavelength range. These features usually appear as a deficit or excess of counts in a single-resolution element of the detector, and as such they are referred to as unresolved fluctuations. The general problem under investigation is the probability of occurrence of chance fluctuations. A quantitative answer is necessary to determine whether detected fluctuations are a real (astronomical, in this case) signal, or if they can be attributed to chance. This application note provides a new comprehensive method to answer this question as function of the instrument's resolution, the wavelength coverage of the spectrum, the number of fluctuations of interest, and the confidence level chosen. The method is based on the binomial distribution and addresses also the probability of multiple simultaneous fluctuations. A critical aspect of the model is the a priori knowledge of the location of possible fluctuations, which significantly affects the probability of detection. In fact, a wider wavelength range for the ‘blind’ search of possible fluctuations results in a larger number of ‘tries’ for the detection of a fluctuation, lowering the overall significance of a specific feature. The method is illustrated with a case study and examples using X-ray data.