Power-density spectrum of non-stationary short-lived light curves

Abstract

The power-density spectrum of a light curve is often calculated as the average of a number of spectra derived on individual time intervals the light curve is divided into. This procedure implicitly assumes that each time interval is a different sample function of the same stochastic ergodic process. While this assumption can be applied to many astrophysical sources, there remains a class of transient, highly non-stationary and short-lived events, such as gamma-ray bursts, for which this approach is often inadequate. The power spectrum statistics of a constant signal affected by statistical (Poisson) noise are known to be a χ22 in the Leahy normalization. However, this is no more the case when a non-stationary signal is also present. As a consequence, the uncertainties on the power spectrum cannot be calculated on the basis of the χ22 properties, as assumed by tools such as xronospowspec. We generalize the result in the case of a non-stationary signal affected by uncorrelated white noise and show that the new distribution is a non-central χ22(λ), whose non-central value λ is the power spectrum of the deterministic function describing the non-stationary signal. Finally, we test these results in the case of synthetic curves of gamma-ray bursts. We end up with a new formula for calculating the power spectrum uncertainties. This is crucial in the case of non-stationary short-lived processes affected by uncorrelated statistical noise, for which ensemble averaging does not make any physical sense.