Using LISA extreme-mass-ratio inspiral sources to test off-Kerr deviations in the geometry of massive black holes

Inspirals of stellar-mass compact objects (COs) into ~106M[sun] black holes are especially interesting sources of gravitational waves for the planned Laser Interferometer Space Antenna (LISA). The orbits of these extreme-mass-ratio inspirals (EMRIs) are highly relativistic, displaying extreme versions of both perihelion precession and Lense-Thirring precession of the orbital plane. We investigate the question of whether the emitted waveforms can be used to strongly constrain the geometry of the central massive object, and in essence check that it corresponds to a Kerr black hole (BH). For a Kerr BH, all multipole moments of the spacetime have a simple, unique relation to M and S, the BH mass, and spin; in particular, the spacetime's mass quadrupole moment Q is given by Q=-S2/M. Here we treat Q as an additional parameter, independent of S and M, and ask how well observation can constrain its difference from the Kerr value. This was already estimated by Ryan, but for the simplified case of circular, equatorial orbits, and Ryan also neglected the signal modulations arising from the motion of the LISA satellites. We consider generic orbits and include the modulations due to the satellite motions. For this analysis, we use a family of approximate (basically post-Newtonian) waveforms, which represent the full parameter space of EMRI sources, and which exhibit the main qualitative features of true, general relativistic waveforms. We extend this parameter space to include (in an approximate manner) an arbitrary value of Q, and then construct the Fisher information matrix for the extended parameter space. By inverting the Fisher matrix, we estimate how accurately Q could be extracted from LISA observations of EMRIs. For 1 yr of coherent data from the inspiral of a 10M[sun] black hole into rotating black holes of masses 105.5M[sun], 106M[sun], or 106.5M[sun], we find Delta(Q/M3)~10-4, 10-3, or 10-2, respectively (assuming total signal-to-noise ratio of 100, typical of the brightest detectable EMRIs). These results depend only weakly on the eccentricity of the inspiral orbit or the spin of the central object.