  According to the "no-hair" conjecture, a Kerr black hole (BH) is completely
described by its mass and spin. In particular, the spin-induced quadrupole
moment of a Kerr BH with mass $m$ and dimensionless spin $\chi$ can be written
as $Q=-\kappa\,m^3\chi^2$, where $\kappa_{\rm BH}=1$. Thus by measuring the
spin-induced quadrupole parameter $\kappa$, we can test the binary black hole
nature of compact binaries and distinguish them from binaries comprised of
other exotic compact objects, as proposed in [N. V. Krishnendu et al., PRL 119,
091101 (2017)]. Here, we present a Bayesian framework to carry out this test
where we measure the symmetric combination of individual spin-induced
quadrupole moment parameters fixing the anti-symmetric combination to be zero.
The analysis is restricted to the inspiral part of the signal as the
spin-induced deformations are not modeled in the post-inspiral regime. We
perform detailed simulations to investigate the applicability of this method
for compact binaries of different masses and spins and also explore various
degeneracies in the parameter space which can affect this test. We then apply
this method to the gravitational wave events, GW151226 and GW170608 detected
during the first and second observing runs of Advanced LIGO and Advanced Virgo
detectors. We find the two events to be consistent with binary black hole
mergers in general relativity. By combining information from several more of
such events in future, this method can be used to set constraints on the black
hole nature of the population of compact binaries that are detected by the
Advanced LIGO and Advanced Virgo detectors.
