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

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. Different kinds of kludges have already been used to scope out data analysis issues for LISA. The models we study here are based on computing a particle’s inspiral trajectory in Boyer-Lindquist coordinates, and subsequent identification of these coordinates with flat-space spherical polar coordinates. A gravitational waveform may then be computed from the multipole moments of the trajectory in these coordinates, using well-known solutions of the linearised gravitational perturbation equations in flat space time. We compute waveforms using a standard slow-motion quadrupole formula, a quadrupole/octupole formula, and a fast-motion, weak-field formula originally developed by Press. We assess these approximations by comparing 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.

Figure 1

Expected sensitivity curve $\sqrt{S_h(f)}$ for LISA; black curve: numerical curve as generated in 65, 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 gray curves in the b&w version), respectively) for equatorial orbits and for an observer at a latitudinal position $\theta =45°$ or 90°. 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 gray curves in the b&w version), respectively) for circular-inclined orbits and for an observer at a latitudinal position $\theta =90°$. 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 gray 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 $p=12M$, $e=0.1$, $\iota =120°$ and spin $a=0.9M$. One can see that Press waveform performs better than quadrupole-octupole waveform at high frequencies.

Figure 6

The orbital evolution, $r(t)$, and $h_{+}$ polarization of the GW signal, for an inspiralling quasicircular-inclined orbit with initial parameters $p=10M$, $e=0$, $\iota =45°$ and $a=0.9M$, $\mu /M={10}^{-2}$, $\Theta =90°$. The black curve corresponds to the TB waveform and the red (gray in b&w version) line represents the quadrupole-octupole NK waveform. The inlaid box shows the part of the waveform where the TB and NK results start to deviate, which correlates with the deviation in the inspiral.

Figure 7

Overlap between TB and NK waveforms ($h_{+}$ and $h_{\times }$) for an inspiralling (quasi)circular-inclined orbit as a function of truncation time. We show overlaps for different masses of the central BH: $M={10}^5$, ${10}^6$, ${10}^7{M}_{\odot }$ and for sensitivity curves with and without white dwarf (WD) confusion noise. One can see that the overlap drops as we increase the truncation time and the mass $M$. On the top horizontal axis we show the orbital radius $r/M$ corresponding to the truncation time. For $M={10}^5{M}_{\odot }$ the lines with and without WD confusion lie on top of each other.

Figure 8

The integrand of the overlap between TB and NK waveforms in the frequency domain (solid black line). Breaks in the curve correspond to a negative correlation. We also overplot the accumulative overlap (red solid line) and scaled $S_h(f)$ as a dashed blue line.

Figure 9

In this plot we present SNR obtained by applying the truncated NK waveform as a template to the full inspiral represented by TB waveform: $\mathrm{SNR}(h_{\mathrm{tr}}^{\mathrm{NK}},s^{\mathrm{TB}})=\frac{(h_{\mathrm{tr}}^{\mathrm{NK}}|s^{\mathrm{TB}})}{(h_{\mathrm{tr}}^{\mathrm{NK}}|h_{\mathrm{tr}}^{\mathrm{NK}}{)}^{1/2}}$ normalized by the total SNR: $\mathrm{SNR}(s^{\mathrm{TB}},s^{\mathrm{TB}})=(s^{\mathrm{TB}}|s^{\mathrm{TB}}{)}^{1/2}$. The primary mass was taken to be ${10}^6{M}_{\odot }$ and the sensitivity curved contained WD confusion noise. The total SNR for the source at 10 Gpc is approximately 66 (high SNR is due to large (${10}^{-2}$) mass ratio).

Figure 10

Convergence of the averaged energy flux $⟨\dot{E}⟩$ with respect to the averaging time $T$ for a pair of equatorial-eccentric orbits. Note the increased convergence rate with decreasing eccentricity.

Figure 11

Angular radiation pattern—energy radiated per unit solid angle as a function of the colatitude of the observer, ${\Theta }_{\mathrm{obs}}$, for orbits with $p=5M$, $e=0.4$, $a=0.9M$ and a sequence of inclinations, as labeled.

Figure 12

Total radiated energy for Kerr parabolic orbits; as function of periastron.