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 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 × 106 M⊙, these lags correspond to the true light travel distance of ∼20–50 rg. Hence, by assuming a lamp-post geometry and a face-on disc, this places the corona at ∼10–25 rg 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 four observations. This means that the corona height can significantly evolve from ∼10–25 rg to ∼55rg, 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.