Impact of the returning radiation in current tests of the Kerr black hole hypothesis using X-ray reflection spectroscopy

The past 10 years have seen remarkable progress in our capability of analyzing reflection features in the X-ray spectra of accreting black holes. Today X-ray reflection spectroscopy is a mature technique and a powerful tool for studying the accretion process around black holes, measuring black hole spins, and testing Einstein's theory of General Relativity in the strong field regime. However, current reflection models still rely on a number of simplifications and caution is necessary when we derive very precise measurements. In this paper, we study the impact of the returning radiation on our capability of measuring the properties of black holes using X-ray reflection spectroscopy, and in particular on our capability of testing the Kerr black hole hypothesis. While the returning radiation alters the reflection spectrum of the disk, from the analysis of our simulations we find that models without returning radiation can normally recover well the correct black hole spin parameters and can test the Kerr metric. Our study thus confirms that current tests of the Kerr hypothesis using X-ray reflection spectroscopy can be robust.