Error formulae for the energy-dependent cross-spectrum

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.

arXiv commentAccepted for publication in MNRAS. 12 pages, 6 figures

References in corpus (35)

• On the Absorption of X-rays in the Interstellar Medium
• On characterising the variability properties of X-ray light curves from active galaxies
• Active Galactic Nuclei as scaled-up Galactic black holes
• RXTE Observation of Cygnus X-1: II. Timing Analysis
• X-ray reverberation around accreting black holes
• Non-linear X-ray variability in X-ray binaries and active galaxies
• On the X-ray time lags in the black hole candidates
• The corona contracts in a black-hole transient
• Discovery of a relation between black hole mass and soft X-ray time lags in active galactic nuclei
• X-Ray Variability Coherence: How to Compute it, What it Means, and How it Constrains Models of Cyg X-1 and GX 339-4
• A global look at X-ray time lags in Seyfert Galaxies
• A quasi-periodic modulation of the iron line centroid energy in the black hole binary H 1743-322
• Relativistic iron K X-ray Reverberation in NGC 4151
• Accretion disc variability in the hard state of black hole X-ray binaries
• An exact analytic treatment of propagating mass accretion rate fluctuations in X-ray binaries
• Modelling the broad Fe Kα reverberation in the AGN NGC 4151
• Modelling variability in black hole binaries: linking simulations to observations
• Discovery of high-frequency iron K lags in Ark 564 and Mrk 335
• A public relativistic transfer function model for X-ray reverberation mapping of accreting black holes
• Frequency resolved spectroscopy of Cyg X-1: fast variability of the reflected emission in the soft state
• Evolution of the reverberation lag in GX 339-4 at the end of an outburst
• Multi-timescale X-ray reverberation mapping of accreting black holes
• Relativistic X-ray reverberation modelling of the combined time-averaged and lag-energy spectra in AGN
• A Physical Model for the Spectral-Timing Properties of Accreting Black Holes
• An X-ray reverberation mass measurement of Cygnus X-1
• Modelling hard and soft state of Cygnus X-1 with propagating mass accretion rate fluctuations
• The reverberation lag in the low mass X-ray binary H1743-322
• Modelling the cross-spectral variability of the black hole binary MAXI J1659-152 with propagating accretion rate fluctuations
• An imperfect double: probing the physical origin of the low-frequency QPO and its harmonic in black hole binaries
• Coupling between QPOs and broadband noise components in GRS 1915+105
• Fast variability as a probe of the smallest regions around accreting black holes
• Breaking the spectral degeneracies in black hole binaries with fast timing data
• Cross-spectral modelling of the black hole X-ray binary XTEJ1550-564: challenges to the propagating fluctuations paradigm
• An observational method for fast stochastic X-ray polarimetry-timing
• The stochastic X-ray variability of the accreting millisecond pulsar MAXI J0911-655

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