Probing the nature of central objects in extreme-mass-ratio inspirals with gravitational waves

We extend the work of Ryan [F. Ryan, Phys. Rev. D 52, 5707 (1995), , 10.1103/PhysRevD.52.5707F. Ryan, Phys. Rev. D 56, 1845 (1997), 10.1103/PhysRevD.56.1845.] on mapping the spacetime of the central object of an extreme mass-ratio inspiral (EMRI) by using gravitational waves emitted by the system, which may be observed in future missions such as LISA. Whether the central object is a black hole or not can be probed by observing the phasing of these waves, which carry information about its mass and spin multipole moments. We go beyond the phase terms found by Ryan, which were obtained in the quadrupolar approximation of the point-particle limit, and derive terms up to the fifth post-Newtonian (PN) order. Since corrections due to horizon absorption (i.e., if the central object is a black hole) and tidal heating appear by that order, at 2.5PN and 5PN, respectively, we include them here. Corrections due to the motion of the central object, which was addressed only partially by Ryan, are included as well. Additionally, we obtain the contribution of the higher order radiative multipole moments. For the tidal interaction, our results have been derived in the approximation of the Newtonian tidal field. Therefore, in the potential for tidal field, only the contribution due to the mass of the central object has been included as well. Using these results, we argue that it might be possible for LISA to probe if the central object in an EMRI has a horizon or not. We also discuss how our results can be used to test the no-hair theorem from the inspiral phase of such systems.