  We propose a novel method to test the consistency of the multipole moments of
compact binary systems with the predictions of General Relativity (GR). The
multipole moments of a compact binary system, known in terms of symmetric and
trace-free tensors, are used to calculate the gravitational waveforms from
compact binaries within the post-Newtonian (PN) formalism. For nonspinning
compact binaries, we derive the gravitational wave phasing formula, in the
frequency domain, parametrizing each PN order term in terms of the multipole
moments which contribute to that order. Using GW observations, this
{\it{parametrized multipolar phasing}} would allow us to derive the bounds on
possible departures from the multipole structure of GR and hence constrain the
parameter space of alternative theories of gravity. We compute the projected
accuracies with which the second generation ground-based detectors, such as
Advanced Laser Interferometer Gravitational-wave Observatory (LIGO), the third
generation detectors such as Einstein Telescope and Cosmic Explorer, as well as
space-based detector Laser Interferometer Space Antenna (LISA) will be able to
measure these multipole parameters. We find that while Advanced LIGO can
measure the first two or three multipole coefficients with good accuracy,
Cosmic Explorer and Einstein Telescope may be able to measure the first four
multipole coefficients which enter the phasing formula. Intermediate mass ratio
inspirals, with mass ratio of several tens, in the frequency band of planned
space-based LISA mission should be able to measure all the seven multipole
coefficients which appear in the 3.5PN phasing formula. Our finding highlights
the importance of this class of sources for probing the strong-field gravity
regime. The proposed test will facilitate the first probe of the multipolar
structure of Einstein's general relativity.
