Testing the no-hair theorem with observations in the electromagnetic spectrum. III. Quasi-periodic variability

According to the no-hair theorem, astrophysical black holes are uniquely described by their masses and spins. An observational test of the no-hair theorem can be performed by measuring at least three different multipole moments of the spacetime of a black hole and verifying whether their values are consistent with the unique combinations of the Kerr solution. In this paper, we study quasi-periodic variability observed in the emission from black holes across the electromagnetic spectrum as a test of the no-hair theorem. We derive expressions for the Keplerian and epicyclic frequencies in a quasi-Kerr spacetime, in which the quadrupole moment is a free parameter in addition to mass and spin. We show that, for moderate spins, the Keplerian frequency is practically independent of small deviations of the quadrupole moment from the Kerr value, while the epicyclic frequencies exhibit significant variations. We apply this framework to quasi-periodic oscillations (QPOs) in black hole X-ray binaries in two different scenarios. In the case that a pair of QPOs can be identified as the fundamental g-and c-modes in the accretion disk, we show that the no-hair theorem can be tested in conjunction with an independent mass measurement. If pairs of oscillations are identified with non-parametric resonance of dynamical frequencies in the accretion disk, then testing the no-hair theorem also requires an independent measurement of the black hole spin. In addition, we argue that VLBI observations of Sgr A^* may test the no-hair theorem through a combination of imaging observations and the detection of quasi-periodic variability.