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Testing the Bardeen metric with the black hole candidate in Cygnus X-1

In general, it is very difficult to test the Kerr-nature of an astrophysical black hole candidate, because it is not possible to have independent measurements of both the spin parameter $a_*$ and possible deviations from the Kerr solution. Non-Kerr objects may indeed look like Kerr black holes with different spin. However, it is much more difficult to mimic an extremal Kerr black hole. The black hole candidate in Cygnus X-1 has the features of a near extremal Kerr black hole, and it is therefore a good object to test the Kerr black hole paradigm. The 3$σ$-bounds $a_* > 0.95$ and $a_* > 0.983$ reported in the literature and valid in the Kerr spacetime become, respectively, $a_* > 0.78$ and $|g/M| < 0.41$, and $a_* > 0.89$ and $|g/M| < 0.28$ in the Bardeen metric, where $g$ is the Bardeen charge of the black hole.