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According to the no-hair theorem, astrophysical black holes are uniquelydescribed by their masses and spins. An observational test of the no-hairtheorem can be performed by measuring at least three different multipolemoments of the spacetime of a black hole and verifying whether their values areconsistent with the unique combinations of the Kerr solution. In this paper, westudy quasi-periodic variability observed in the emission from black holesacross the electromagnetic spectrum as a test of the no-hair theorem. We deriveexpressions for the Keplerian and epicyclic frequencies in a quasi-Kerrspacetime, in which the quadrupole moment is a free parameter in addition tomass and spin. We show that, for moderate spins, the Keplerian frequency ispractically independent of small deviations of the quadrupole moment from theKerr value, while the epicyclic frequencies exhibit significant variations. Weapply this framework to quasi-periodic oscillations in black-hole X-raybinaries in two different scenarios. In the case that a pair of quasi-periodicoscillations can be identified as the fundamental g- and c-modes in theaccretion disk, we show that the no-hair theorem can be tested in conjunctionwith an independent mass measurement. If, on the other hand, the pairs ofoscillations are identified with non-parametric resonance of dynamicalfrequencies in the accretion disk, then testing the no-hair theorem alsorequires an independent measurement of the black-hole spin. In addition, weargue that VLBI observations of Sgr A* may test the no-hair theorem through acombination of imaging observations and the detection of quasi-periodicvariability.

Testing the No-Hair Theorem with Observations in the Electromagnetic Spectrum. III. Quasi-Periodic Variability