Testing the space-time geometry around black hole candidates with the analysis of the broad K$\alpha$ iron line

Astrophysical black hole candidates are thought to be the Kerr black holes predicted by General Relativity, but there is not yet clear evidence that the geometry of the space-time around these objects is really described by the Kerr metric. In order to confirm the Kerr black hole hypothesis, we have to observe strong gravity features and check that they are in agreement with the ones predicted by General Relativity. In this paper, I study the broad K$\alpha$ iron line, which is often seen in the X-ray spectrum of both stellar-mass and super-massive black hole candidates and whose shape is supposed to be strongly affected by the space-time geometry. As found in previous studies in the literature, there is a strong correlation between the spin parameter and the deformation parameter; that is, the line emitted around a Kerr black hole with a certain spin can be very similar to the one coming from the space-time around a non-Kerr object with a quite different spin. Despite that, the analysis of the broad K$\alpha$ iron line is potentially more powerful than the continuum-fitting method, as it can put an interesting bound on possible deviations from the Kerr geometry independently of the value of the spin parameter and without additional measurements.