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The spacetime geometry around astrophysical black holes is supposed to bewell approximated by the Kerr metric, but deviations from the Kerr solution arepredicted in a number of scenarios involving new physics. Broad iron K$\alpha$lines are commonly observed in the X-ray spectrum of black holes and originateby X-ray fluorescence of the inner accretion disk. The profile of the iron lineis sensitively affected by the spacetime geometry in the strong gravity regionand can be used to test the Kerr black hole hypothesis. In this paper, weextend previous work in the literature. In particular: i) as test-metric, weemploy the parametrization recently proposed by Konoplya, Rezzolla, andZhidenko, which has a number of subtle advantages with respect to the existingapproaches; ii) we perform simulations with specific X-ray missions, and weconsider NuSTAR as a prototype of current observational facilities and eXTP asan example of the next generation of X-ray observatories. We find a significantdifference between the constraining power of NuSTAR and eXTP. With NuSTAR, itis difficult or impossible to constrain deviations from the Kerr metric. WitheXTP, in most cases we can obtain quite stringent constraints (modulo we havethe correct astrophysical model).

Testing the Kerr metric with the iron line and the KRZ parametrization