Accuracy of estimating the multipole moments of a massive body from the gravitational waves of a binary inspiral

If the gravitational field of a massive, compact body is stationary, axially symmetric, and reflection symmetric across the equatorial plane, and if a much less massive compact object (such as a neutron star or a small black hole) were to orbit in a circle on the equatorial plane of the central, compact body, then the produced gravitational waves would carry the values of the central body's multipole moments. By detecting those waves and extracting from them the central body's lowest few moments, gravitational-wave detectors have the potential to test the black-hole no-hair theorem and search for exotic objects such as naked singularities and boson or soliton stars. This paper estimates how accurately we can expect to measure the central body's moments. The measurement errors are estimated using a combination of, first, the leading-order (of a post-Newtonian series) contribution of each moment to the gravitational-wave phase, second, an a priori probability distribution that constrains each moment's magnitude to a range appropriate for a compact body, and third, any relations that the multipole moments satisfy among themselves, which reduce the number of degrees of freedom for the waves (this is useful in cases when one is searching for a specific type of compact body). We find that the Earth-based LIGO detector cannot provide sufficiently precise measurements of enough multipole moments to search for exotic objects, but the space-based LISA detector can do so.