Prospects for determining the nature of the secondaries of extreme mass-ratio inspirals using the spin-induced quadrupole deformation

The measurement of multipole moments of astrophysical objects through gravitational wave (GW) observations provides a novel way to distinguish black holes from other astrophysical objects. This paper studies the gravitational wave radiation from an extreme mass ratio inspiral (EMRI) system consisting of a supermassive Kerr black hole (the primary object) and a spinning stellar-mass compact object (the secondary object). The quadrupolar deformation induced by the spin of the secondary is different for different astrophysical objects. We compute the effect of the quadrupolar deformation on the GW phase and provide an order of magnitude estimate of whether LISA can distinguish different astrophysical objects through GW phase measurement. We find that although LISA can not distinguish between a black hole and a neutron star, it can distinguish black holes from a large variety of highly spinning astrophysical objects like superspinars and highly deformable exotic compact objects like boson stars for EMRI systems with relatively large mass ratio ($q\sim 10^{-4}$). Furthermore, we show that the effect of spin-induced quadrupolar deformation on the GW phase for white dwarf and brown dwarf-EMRI systems can be quite significant even for small values of mass ratio ($q\lesssim 10^{-6}$).