Mapping spacetimes with LISA: inspiral of a test body in a ‘quasi-Kerr’ field

The future LISA detector will constitute the prime instrument for high-precision gravitational wave observations. Among other goals, LISA is expected to materialize a 'spacetime-mapping' program that is to provide information for the properties of spacetime in the vicinity of supermassive black holes which reside in the majority of galactic nuclei. Such black holes can capture stellarmass compact objects, which afterwards slowly inspiral under the emission of gravitational radiation. The small body's orbital motion and the associated waveform observed at infinity carry information about the spacetime metric of the massive black hole, and in principle it is possible to extract this information and experimentally identify (or not!) a Kerr black hole. In this paper we lay the foundations for a practical spacetime-mapping framework. Our work is based on the assumption that the massive body is not necessarily a Kerr black hole, and that the vacuum exterior spacetime is stationary axisymmetric, described by a metric which deviates slightly from the known Kerr metric. We first provide a simple recipe for building such a 'quasi-Kerr' metric by adding to the Kerr metric the leading order deviation which appears in the value of the spacetime's quadrupole moment. We then study geodesic motion of a test body in this metric, mainly focusing on equatorial orbits, but also providing equations describing generic orbits formulated by means of canonical perturbation theory techniques. We proceed by computing approximate 'kludge' gravitational waveforms which we compare with their Kerr counterparts. We find that a modest deviation from the Kerr metric is sufficient for producing a significant mismatch between the waveforms, provided we fix the orbital parameters. This result suggests that an attempt to use Kerr waveform templates for studying extreme mass ratio inspirals around a non-Kerr object might result in serious loss of signal-to-noise ratio and total number of detected events. The waveform comparisons also unveil a 'confusion' problem, that is the possibility of matching a true non-Kerr waveform with a Kerr template of different orbital parameters.