Testing the Kerr Nature of Black Hole Candidates using Iron Line Spectra in the CPR Framework

The iron K$\alpha$ line commonly observed in the X-ray spectrum of both stellar-mass and supermassive black hole candidates originates from X-ray fluorescence of the inner accretion disk. Accordingly, it can be used to map the spacetime geometry around these objects. In this paper, we extend previous work using the iron K$\alpha$ line to test the Kerr black hole hypothesis. We adopt the Cardoso-Pani-Rico parametrization and we test the possibility of constraining possible deviations from the Kerr solution that can be obtained from observations across the range of black hole spins and inclination angles. We confirm previous claims that the iron K$\alpha$ line is potentially a quite powerful probe for testing the Kerr metric given sufficiently high quality data and with systematics under control, especially in the case of fast-rotating black holes and high inclination angles since both conditions serve to maximize relativistic effects. We find that some geometric perturbations from Kerr geometry manifest more strongly in the iron line profile than others. While the perturbation parameter $\epsilon^t_3$ can be well constrained by the iron line profile, an orthogonal data set is necessary to constrain departures from Kerr geometry in $\epsilon^r_3$.