  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 in black-hole X-ray
binaries in two different scenarios. In the case that a pair of quasi-periodic
oscillations 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, on the other hand, the 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.
