Power density spectrum of nonstationary short-lived light curves

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 nonstationary 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 is known to be a chi2(2) in the Leahy normalisation. However, this is no more the case when a nonstationary signal is also present. As a consequence, the uncertainties on the power spectrum cannot be calculated based on the chi2(2) properties, as assumed by tools such as XRONOS powspec. We generalise the result in the case of a nonstationary signal affected by uncorrelated white noise and show that the new distribution is a non-central chi2(2,lambda), whose non-central value lambda is the power spectrum of the deterministic function describing the nonstationary 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 nonstationary short-lived processes affected by uncorrelated statistical noise, for which ensemble averaging does not make any physical sense.

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