Towards adiabatic waveforms for inspiral into Kerr black holes. II. Dynamical sources and generic orbits

This is the second in a series of papers whose aim is to generate adiabatic gravitational waveforms from the inspiral of stellar-mass compact objects into massive black holes. In earlier work, we presented an accurate $(2+1)\mathrm{D}$ finite-difference time-domain code to solve the Teukolsky equation, which evolves curvature perturbations near rotating (Kerr) black holes. The key new ingredient there was a simple but accurate model of the singular source term based on a discrete representation of the Dirac-delta function and its derivatives. Our earlier work was intended as a proof of concept, using simple circular, equatorial geodesic orbits as a test bed. Such a source is effectively static, in that the smaller body remains at the same coordinate radius and orbital inclination over an orbit. (It of course moves through axial angle, but we separate that degree of freedom from the problem. Our numerical grid has only radial, polar, and time coordinates.) We now extend the time-domain code so that it can accommodate dynamic sources that move on a variety of physically interesting world lines. We validate the code with extensive comparison to frequency-domain waveforms for cases in which the source moves along generic (inclined and eccentric) bound geodesic orbits. We also demonstrate the ability of the time-domain code to accommodate sources moving on interesting nongeodesic worldlines. We do this by computing the waveform produced by a test mass following a kludged inspiral trajectory, made of bound geodesic segments driven toward merger by an approximate radiation loss formula.

Figure 1

Comparison of time- and frequency-domain waveforms. We show waves for the $m=2$ mode from a point particle with orbital parameters $p=6.472M$, $e=0.3$ and ${\theta }_{\mathrm{inc}}=0$ orbiting a black hole with spin $a/M=0.3$. The angle between the spin axis of the black hole and the line of sight is ${\theta }_d=\pi /2$. Time-domain results are in black, frequency-domain results in red (grey in the print edition). Top panel: “plus” polarizations in dimensionless units. Middle: “cross” polarizations. Bottom: Comparison of $|h_{+}-i{h}_{\times }|$. This last quantity gives a good visual measure of the level of agreement between the two waveforms. The correlations between the two waveforms are 0.9974 (plus) and 0.9975 (cross).

Figure 2

Comparison of time- and frequency-domain waveforms. Here, we show waves for the $m=2$ mode for a geodesic with $p=6M$, $e=0.3$ and ${\theta }_{\mathrm{inc}}=\pi /3$ about a black hole with spin $a/M=0.9$; black is time-domain results, red (grey in the print edition) is frequency-domain. The correlations in this case are 0.9961 (plus) and 0.9962 (cross).

Figure 3

Comparison of time- and frequency-domain waveforms. These waves are for the $m=3$ mode from a circular geodesic with orbital parameters $p=6M$, and ${\theta }_{\mathrm{inc}}=\pi /4$ around a hole with spin $a/M=0.9$. All symbols have the same meaning as in Fig. 1. The correlations are 0.9769 (plus) and 0.9770 (cross).

Figure 4

Waveform ($m=2$ mode) of a small body spiraling into a massive black hole. We use kludge backreaction to evolve through a sequence of orbits, but compute the waves with our time-domain solver. The large black hole has spin $a=0.5M$; the small body’s orbit initially has parameters $p=10M$, $e=0.5$, and ${\theta }_{\mathrm{inc}}=0.5$ radians. The mass ratio of the system is $\mu /M=0.016$. The top panel shows the full span that we simulated; the bottom two panels are zooms on early (bottom left) and late (bottom right) segments. Note the clear evolution of the wave’s frequency as the orbit’s mean radius shrinks.

Figure 5

Extrapolation applied to $h_{+}$ for the $m=3$ mode from a point particle in a nearly circular geodesic with orbital parameters $e={10}^{-4}$, $p=6M$, and ${\theta }_{\mathrm{inc}}=\pi /4$ around a rotating black hole with spin $a/M=0.9$. The dashed and solid black lines denote $h_{+}$ obtained with resolutions $(\delta r,\delta \theta )=(0.04,\pi /60)$ and (0.026667, $\pi /90$), respectively. The solid red (solid dark grey in the print edition) line is the extrapolated waveform; the solid green (solid light grey in the print edition) line is the equivalent frequency-domain waveform. Notice how well the extrapolated time-domain wave agrees with the frequency-domain result (which is nearly hidden by the red curve).