X-ray time lags from a pivoting power law in black holes

Abstract

Most black hole candidate X-ray binaries show Fourier time lags between softer and harder X-rays. The hard photons seem to arrive up to a few ms after the soft for a given Fourier frequency of the perturbation. The energy dependence of the time lags has a roughly logarithmic behavior. Up to now most theories fail to explain the observed magnitude and Fourier frequency dependence of the lags or fail other statistical tests. We show that the time lags can arise from a simple pivoting power law model, which creates the logarithmic dependence on the photon energy at once. A pivoting power law arises naturally from jet/synchrotron models for the X-ray emission, but may also be applicable to corona models. A hint to the coherence features of the light-curves can be obtained from the power spectral density, which can be decomposed into a few broad Lorentzians that could arise from a couple of strongly damped oscillators with low quality factors below one. Using small variations of the power law index for each Lorentzian separately the lags can be derived analytically. They show the correct Fourier frequency dependence of the time lags. If one assumes variations of the power law index by +- 0.2 the model can account for the observed magnitude of the time lags in Cyg X-1. As a further test we calculated the cross- and auto-correlation functions for our model, which also show qualitatively the observed behavior. The coherence function for the model is in agreement with the observed data in the Fourier regime, where the model is valid.