Cross-spectral analysis of the X-ray variability of Markarian 421

Using the cross-spectral method, we confirm the existence of the X-ray hard lags discovered with cross-correlation function technique during a large flare of Mrk 421 observed with BeppoSAX. For the 0.1–2 versus 2–10 keV light curves, both methods suggest sub-hour hard lags. In the time domain, the degree of hard lag, i.e. the amplitude of the 3.2–10 keV photons lagging the lower energy ones, tends to increase with the decreasing energy. In the Fourier frequency domain, by investigating the cross-spectra of the 0.1–2/2–10 keV and the 2–3.2/3.2– 10 keV pairs of light curves, the flare also shows hard lags at the lowest frequencies. However, with the present data, it is impossible to constrain the dependence of the lags on frequencies even though the detailed simulations demonstrate that the hard lags at the lowest frequencies probed by the flare are not an artefact of sparse sampling, Poisson and red noise. As a possible interpretation, the implication of the hard lags is discussed in the context of the interplay between the (diffusive) acceleration and synchrotron cooling of relativistic electrons responsible for the observed X-ray emission. The energy-dependent hard lags are in agreement with the expectation of an energy-dependent acceleration time-scale. The inferred magnetic field (B ∼ 0.11 G) is consistent with the value inferred from the spectral energy distributions of the source. Future investigations with higher quality data that show whether or not the time-lags are energy-/frequency-dependent will provide a new constraint on the current models of the