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Inspirals of stellar-mass compact objects into $\sim 10^6 M_{\odot}$ blackholes are especially interesting sources of gravitational waves for LISA. Weinvestigate whether the emitted waveforms can be used to strongly constrain thegeometry of the central massive object, and in essence check that itcorresponds to a Kerr black hole (BH). For a Kerr BH, all multipole moments ofthe spacetime have a simple, unique relation to $M$ and $S$, the BH's mass andspin; in particular, the spacetime's mass quadrupole moment is given by $Q=-S^2/M$. Here we treat $Q$ as an additional parameter, independent of $M$ and$S$, and ask how well observation can constrain its difference from the Kerrvalue. This was already estimated by Ryan, but for simplified (circular,equatorial) orbits, and neglecting signal modulations due to the motion of theLISA satellites. Here we consider generic orbits and include these modulations.We use a family of approximate (post-Newtonian) waveforms, which represent thefull parameter space of Inspiral sources, and exhibit the main qualitativefeatures of true, general relativistic waveforms. We extend this parameterspace to include (in an approximate manner) an arbitrary value of $Q$, andconstruct the Fisher information matrix for the extended parameter space. Byinverting the Fisher matrix we estimate how accurately $Q$ could be extractedfrom LISA observations. For 1 year of coherent data from the inspiral of a $10M_{\odot}$ BH into rotating BHs of masses $10^{5.5} M_{\odot}$, $10^6M_{\odot}$, or $10^{6.5} M_{\odot}$, we find $\Delta (Q/M^3) \sim 10^{-4}$,$10^{-3}$, or $10^{-2}$, respectively (assuming total signal-to-noise ratio of100, typical of the brightest detectable EMRIs). These results depend onlyweakly on the eccentricity of the orbit or the BH's spin.Comment: 12 pages, 5 figure

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