  We propose a novel method to test the binary black hole (BBH) nature of
compact binaries detectable by gravitational wave (GW) interferometers and
hence constrain the parameter space of other exotic compact objects. The spirit
of the test lies in the "no-hair" conjecture for black holes where all
properties of a black hole are characterised by the mass and the spin of the
black hole. The method relies on observationally measuring the quadrupole
moments of the compact binary constituents induced due to their spins. If the
compact object is a Kerr black hole (BH), its quadrupole moment is expressible
solely in terms of its mass and spin. Otherwise, the quadrupole moment can
depend on additional parameters (such as equation of state of the object). The
higher order spin effects in phase and amplitude of a gravitational waveform,
which explicitly contains the spin-induced quadrupole moments of compact
objects, hence uniquely encodes the nature of the compact binary. Thus we argue
that an independent measurement of the spin-induced quadrupole moment of the
compact binaries from GW observations can provide a unique way to distinguish
binary BH systems from binaries consisting of exotic compact objects.
