Testing the Kerr black hole hypothesis: Comparison between the gravitational wave and the iron line approaches

The recent announcement of the detection of gravitational waves by the LIGO/Virgo Collaboration has opened a new window to test the nature of astrophysical black holes. Konoplya & Zhidenko have shown how the LIGO data of GW 150914 can constrain possible deviations from the Kerr metric. In this letter, we compare their constraints with those that can be obtained from accreting black holes by fitting their X-ray reflection spectrum, the so-called iron line method. We simulate observations with eXTP, a next generation X-ray mission, finding constraints much stronger than those obtained by Konoplya & Zhidenko. Our results can at least show that, contrary to what is quite commonly believed, it is not obvious that gravitational waves are the most powerful approach to test strong gravity. In the presence of high quality data and with the systematics under control, the iron line method may provide competitive constraints.