"Kludge" gravitational waveforms for a test-body orbiting a Kerr black hole

TL;DR

This work introduces numerical kludge (NK) waveforms that fuse exact Kerr geodesic motion with flat-space GW emission to produce fast, approximate EMRI templates. The authors demonstrate that NK quadrupole-octupole waveforms achieve overlaps $\gtrsim 0.95$ with Teukolsky-based waveforms for many orbits, particularly when the periapsis lies beyond $5M$, and show reasonable performance for inspirals up to several orbits before plunge. They also quantify the accuracy of NK-derived GW fluxes and discuss the limitations due to neglecting tail effects and the conservative self-force, outlining a practical path to include conservative corrections via PN-inspired frequency shifts. The results suggest NK templates can play a valuable role in initial detection, hierarchical searches, and non-Kerr spacetime studies, while highlighting areas for refinement (tails, horizon flux, self-force) to improve precision for parameter estimation and strong-field physics.

Abstract

One of the most exciting potential sources of gravitational waves for low-frequency, space-based gravitational wave (GW) detectors such as the proposed Laser Interferometer Space Antenna (LISA) is the inspiral of compact objects into massive black holes in the centers of galaxies. The detection of waves from such "extreme mass ratio inspiral" systems (EMRIs) and extraction of information from those waves require template waveforms. The systems' extreme mass ratio means that their waveforms can be determined accurately using black hole perturbation theory. Such calculations are computationally very expensive. There is a pressing need for families of approximate waveforms that may be generated cheaply and quickly but which still capture the main features of true waveforms. In this paper, we introduce a family of such "kludge" waveforms and describe ways to generate them. We assess performance of the introduced approximations by comparing "kludge" waveforms to accurate waveforms obtained by solving the Teukolsky equation in the adiabatic limit (neglecting GW backreaction). We find that the kludge waveforms do extremely well at approximating the true gravitational waveform, having overlaps with the Teukolsky waveforms of 95% or higher over most of the parameter space for which comparisons can currently be made. Indeed, we find these kludges to be of such high quality (despite their ease of calculation) that it is possible they may play some role in the final search of LISA data for EMRIs.

Figures (12)

• Figure 1: Expected sensitivity curve $\sqrt{S_h(f)}$ for LISA; black curve: numerical curve as generated in larson, red curve: analytic approximation used in this paper, see text for details).
• Figure 2: Comparing TB and quadrupole-octupole kludge waveforms (black and red curves (bold black and grey curves in the b&w version), respectively) for equatorial orbits and for an observer at a latitudinal position $\theta = 45^\circ$ or $90^\circ$. Orbital parameters are listed above each graph. The waveforms are scaled in units of $D/\mu$ where $D$ is the radial distance of the observation point from the source and $\mu$ is the test-body's mass. The x-axis measures retarded time (in units of $M$) and we are showing the '+' polarization of the GW in each case. The overlaps between the NK and TB waveforms are $0.970, 0.987, 0.524$ going from the top figure down.
• Figure 3: Comparing TB and quadrupole-octupole kludge waveforms (black and red curves (bold black and grey curves in the b&w version), respectively) for circular-inclined orbits and for an observer at a latitudinal position $\theta = 90^\circ$. Orbital parameters are listed above each graph. The waveforms are scaled in units of $D/\mu$ where $D$ is the radial distance of the observation point from the source and $\mu$ is the test-body's mass. The x-axis measures retarded time (in units of $M$) and we again show the plus polarization of the GW. The overlaps between the NK and TB waveforms are $0.882$ for the top figure and $0.955$ for the bottom figure.
• Figure 4: Comparing TB and quadrupole-octupole kludge waveforms (black and red curves (bold black and grey curves in the b&w version), respectively) for generic orbits. Orbital parameters are listed above each graph. The waveforms are scaled in units of $D/\mu$ where $D$ is the radial distance of the observation point from the source and $\mu$ is the test-body's mass. The x-axis measures retarded time (in units of $M$). The overlaps between the NK and TB waveforms are $0.991$ and $0.966$ for the top and bottom figures respectively.
• Figure 5: Comparison of the integrand of the SNR in the frequency domain $[\tilde{h}(f)\tilde{s}^*(f)]/S_h(f)$ where $\tilde{h}(f)$ is determined from the quadrupole, quadrupole-octupole and Press waveforms, and $s(f)$ is determined from the TB waveform. We have chosen orbital parameters $p= 12M,~e=0.1,~\iota = 120^\circ$ and spin $a=0.9M$. One can see that Press waveform performs better than quadrupole-octupole waveform at high frequencies.