How to Observe a Non-Kerr Spacetime Using Gravitational Waves

We present a generic criterion which can be used in gravitational-wave data analysis to distinguish an extreme-mass-ratio inspiral into a Kerr background spacetime from one into a non-Kerr spacetime. We exploit the fact that when an integrable system, such as the system that describes geodesic orbits in a Kerr spacetime, is perturbed, the tori in phase space which initially corresponded to resonances disintegrate so as to form Birkhoff chains on a surface of section. The KAM curves of the islands in such a chain share the same ratio of frequencies, even though the frequencies themselves vary from one KAM curve to another inside an island. However the KAM curves, which do not lie in a Birkhoff chain, do not share this characteristic property. Such a temporal constancy of the ratio of frequencies during the evolution of the gravitational-wave signal will signal a non-Kerr spacetime.

Figure 1

The surface of section of the outer region on the ($\rho$,$\dot{\rho }$) plane for the parameter set $E=0.95$, $L_z=3M$, $\chi =0.9$, $Q=0.95$ of MN. $E$, $L_z$ are the energy and the angular momentum of the inspiral, respectively, while $\chi$ is the spin of the central body (for details about these quantities see 11).

Figure 2

The evolution of the ratio of fundamental frequencies $f_{\rho }/f_z$ for a MN EMRI with $Q=0.95$ and ratio of masses $\mu /M=5\times {10}^{-5}$. The initial conditions were chosen to get quite a long plateau. The fundamental frequencies were computed numerically from the Fourier analysis of the orbital motion in time intervals of length $5000M$. The oscillations of the ratio—mainly at the plateau interval—are artifacts due to Fourier analyzing a finite part of the orbit.