General relativistic modelling of the negative reverberation X-ray time delays in AGN

We present the first systematic physical modelling of the time-lag spectra between the soft (0.3-1 keV) and the hard (1.5-4 keV) X-ray energy bands, as a function of Fourier frequency, in a sample of 12 active galactic nuclei which have been observed by XMM-Newton. We concentrate particularly on the negative X-ray time-lags (typically seen above $10^{-4}$ Hz) i.e. soft band variations lag the hard band variations, and we assume that they are produced by reprocessing and reflection by the accretion disc within a lamp-post X-ray source geometry. We also assume that the response of the accretion disc, in the soft X-ray bands, is adequately described by the response in the neutral iron line (Fe k$\alpha$) at 6.4 keV for which we use fully general relativistic ray-tracing simulations to determine its time evolution. These response functions, and thus the corresponding time-lag spectra, yield much more realistic results than the commonly-used, but erroneous, top-hat models. Additionally we parametrize the positive part of the time-lag spectra (typically seen below $10^{-4}$ Hz) by a power-law. We find that the best-fitting BH masses, M, agree quite well with those derived by other methods, thus providing us with a new tool for BH mass determination. We find no evidence for any correlation between M and the BH spin parameter, $\alpha$, the viewing angle, $\theta$, or the height of the X-ray source above the disc, $h$. Also on average, the X-ray source lies only around 3.7 gravitational radii above the accretion disc and the viewing angles are distributed uniformly between 20 and 60 degrees. Finally, there is a tentative indication that the distribution of spin parameters may be bimodal above and below 0.62.