Error formulae for the energy-dependent cross-spectrum

TL;DR

This paper derives new analytic error formulae for the energy-dependent cross-spectrum and rms spectrum in X-ray variability studies, improving accuracy over previous methods and providing tools for better data analysis.

Contribution

It introduces novel, more accurate error formulae for the energy-dependent cross-spectrum, applicable for any intrinsic coherence, and offers an optimal reference band definition and coherence calculation.

Findings

Existing error formulae can lead to over-fitting.

New formulae are more accurate and easy to implement.

A code for applying the new formulae is publicly available.

Abstract

I present analytic error formulae for the energy-dependent cross-spectrum and rms spectrum, which are Fourier statistics widely used to probe the rapid X-ray variability observed from accreting compact objects. The new formulae cover the modulus, phase, real and imaginary parts of the cross-spectrum, and are valid for any value of intrinsic coherence between variability in different energy bands. I show that existing error formulae (including that for the phase lag), which are valid for a single cross-spectrum or power spectrum, lead to over-fitting when applied to the energy-dependent cross-spectrum - which consists of cross-spectra between individual energy channels and a common reference band. I also introduce an optimal, unbiased way to define the reference band and an accurate way to calculate the intrinsic coherence between energy bands. I find that the traditional use of the old formulae has likely had a rather benign impact on the literature, but recommend the use of the new formulae in future wherever appropriate, since they are more accurate and are no harder to implement than existing error estimates. A code to implement the new error formulae on observational data is available online.