Using LISA EMRI sources to test off-Kerr deviations in the geometry of massive black holes

TL;DR

This paper explores how gravitational wave observations from LISA can test whether massive black holes are described by the Kerr metric by constraining deviations in their quadrupole moments using EMRI signals.

Contribution

It extends previous analyses by considering generic orbits and signal modulations, providing a more realistic estimate of LISA's ability to measure deviations from Kerr black hole geometry.

Findings

LISA can measure the quadrupole moment deviation to within 10^{-4} for certain EMRIs.

The measurement accuracy depends weakly on orbit eccentricity and black hole spin.

Results suggest strong potential for testing the Kerr nature of massive black holes with future gravitational wave data.

Abstract

Inspirals of stellar-mass compact objects into $\sim 10^6 M_{\odot}$ black holes are especially interesting sources of gravitational waves for LISA. We investigate 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's mass and spin; 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 Kerr value. This was already estimated by Ryan, but for simplified (circular, equatorial) orbits, and neglecting signal modulations due to the motion of the LISA satellites. Here we consider generic orbits and include these modulations. We use a family of approximate (post-Newtonian) waveforms, which represent the full parameter space of Inspiral sources, and 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 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. For 1 year of coherent data from the inspiral of a $10 M_{\odot}$ BH into rotating BHs of masses $10^{5.5} M_{\odot}$, $10^6 M_{\odot}$, 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 of 100, typical of the brightest detectable EMRIs). These results depend only weakly on the eccentricity of the orbit or the BH's spin.