Revealing the intrinsic X-ray reverberation lags in IRAS 13224-3809 through the Granger causality test

The Granger causality is an econometric test for determining whether one time series is useful for forecasting another one with a certain Granger lag. Here, the light curves in the 0.3-1 keV (reflection dominated, soft) and 1.2-5 keV (continuum dominated, hard) bands of Active Galactic Nuclei (AGNs) are produced, and the Granger lags are estimated and compared to the traditional lag-frequency spectra. We find that the light curves in the hard band Granger-cause (lead) those in the soft band, whereas the obtained Granger lags could be interpreted as the intrinsic reverberation lags. Then, we extract the Granger-lag profiles from 14 XMM-Newton observations of IRAS 13224-3809, and find that the lags are significant in 12 observations. The majority of the obtained Granger (intrinsic) lags are ~ 200-500 s. With the IRAS 13224-3809 mass of 2 $\times 10^6$ $M_{\odot}$, these lags correspond to the true light-travel distance of ~ 20-50 $r_g$. Hence, by assuming a lamp-post geometry and a face-on disc, this places the corona at ~ 10-25 $r_{g}$ above the central black hole. Moreover, multiple Granger lags consisting of the small and large lags of < 500 s and > 1000 s are detected in 4 observations. This means that the corona height can significantly evolve from ~ 10-25 $r_{g}$ to ~ 55 $r_{g}$, or vice versa, towards the end of the observation. Granger lags are a promising way to measure the intrinsic lags, and provide evidence of coronal height variability within each individual observation.