Adiabatic waveforms for extreme mass-ratio inspirals via multivoice decomposition in time and frequency

We compute adiabatic waveforms for extreme mass-ratio inspirals (EMRIs) by “stitching” together a long inspiral waveform from a sequence of waveform snapshots, each of which corresponds to a particular geodesic orbit. We show that the complicated total waveform can be regarded as a sum of “voices.” Each voice evolves in a simple way on long timescales, a property which can be exploited to efficiently produce waveform models that faithfully encode the properties of EMRI systems. We look at examples for a range of different orbital geometries: spherical orbits, equatorial eccentric orbits, and one example of generic (inclined and eccentric) orbits. To our knowledge, this is the first calculation of a generic EMRI waveform that uses strong-field radiation reaction. We examine waveforms in both the time and frequency domains. Although EMRIs evolve slowly enough that the stationary phase approximation (SPA) to the Fourier transform is valid, the SPA calculation must be done to higher order for some voices, since their instantaneous frequency can change from chirping forward ($\dot{f}>0$) to chirping backward ($\dot{f}<0$). The approach we develop can eventually be extended to more complete EMRI waveform models—for example, to include effects neglected by the adiabatic approximation, such as the conservative self-force and spin-curvature coupling.

Figure 1

An example of the time-domain and frequency-domain structure of one voice: $l=m=2$, $k=0$, $n=20$, for an equatorial Kerr inspiral with $a=0.9M$, $p_{\mathrm{init}}=12M$, $e_{\mathrm{init}}=0.7$, and mass ratio $ϵ={10}^{-3}$. On the left, we show this voice’s amplitude (top) and frequency (bottom) as a function of time over the inspiral. The voice’s frequency increases until it reaches $f_{\max }\simeq 0.037/M$; it then reverses and falls to $f_{\text{final}}\simeq 0.024/M$ at the end of inspiral. The spiky features occur when this amplitude passes through zero (note the log scale). On the right, we show the frequency-domain structure computed with the extended SPA. In the top right, we examine contributions to the SPA along the two branches of $F(t)=f$. The curve with long dashes is the magnitude of the SPA for the branch with $\dot{f}>0$; the curve with short dashes is for the branch with $\dot{f}<0$. The bottom right shows the final SPA Fourier transform, obtained by summing contributions from the two branches, and compares this to a discrete Fourier transform (DFT). Because of the large dynamic range between the low-frequency and high-frequency behavior of this voice, we separately examine the DFT for the early-time, low-frequency portion (corresponding to $t^{\mathrm{i}}\lesssim 5.5\times {10}^{5}M$, plotted with magenta dashes) and the DFT for the late, high-frequency portion ($t^{\mathrm{i}}\gtrsim 5.5\times {10}^{5}M$, plotted with blue dots). Modulo lobes at the boundaries of the two DFTs (associated with the Tukey window used to taper the time-domain signal near these boundaries), the DFTs coincide perfectly with the SPA. Notice the interesting behavior in the band $0.024\lesssim Mf\lesssim 0.037$. This arises from beating between the contributions along the two branches.

Figure 2

The waveform $h_{+}$ for inspiral into a Schwarzschild black hole with $(p_{\mathrm{init}},e_{\mathrm{init}},x_{I,\mathrm{init}})=(12M,0.2,1)$ (left-hand panels) and $(p_{\mathrm{init}},e_{\mathrm{init}},x_{I,\mathrm{init}})=(12M,0.7,1)$ (right-hand panels). The waves are observed in the plane of the orbit, so $h_{\times }=0$, and the mass ratio is $ϵ={10}^{-3}$ in all cases. The top panels show the complete inspiral waveform over this domain, with the initial radial anomaly angle ${\chi }_{r0}=2\pi /3$. The bottom panels compare $h_{+}$ including the phase corrections ${\xi }_{m0n}$ (solid [red] curves) with $h_{+}$ neglecting these corrections (i.e., incorrectly using the fiducial amplitudes ${\check{A}}_{lm0n}$; dashed [blue] curves). In both cases, we compare early times (the first $3000M$ of inspiral) and late times (an interval of $1000M$ near the end of inspiral). Neglecting ${\xi }_{m0n}$ leads to significant differences in the waveforms.

Figure 3

The phase correction ${\xi }_{100}$ (bottom panel) and ${\xi }_{001}$ (top panel) for inspiral with $p_{\mathrm{init}}=12M$, $e_{\mathrm{init}}=0.2$ at mass ratio $ϵ={10}^{-3}$. In both panels, the solid (black) line corresponds to ${\chi }_{r0}=0$, dotted (red) curves show ${\chi }_{r0}=\pi /6$, short-dashed (blue) curves show ${\chi }_{r0}=\pi /2$, long-dashed (magenta) curves show ${\chi }_{r0}=11\pi /6$, and dot-dashed (green) curves show ${\chi }_{r0}=3\pi /2$. Notice that these curves show only gentle variation until nearly the end, with many changing rapidly as the inspiral approaches the last stable orbit.

Figure 4

The same as Fig. 3, but for an inspiral with $p_{\mathrm{init}}=12M$, $e_{\mathrm{init}}=0.7$. These phases show more variation in this case than when $e_{\mathrm{init}}=0.2$, although the curves still show only gentle variations until the system approaches the end of inspiral.

Figure 5

The waveform for inspiral into a Schwarzschild black hole with $p_{\mathrm{init}}=12M$, $e_{\mathrm{init}}=0.7$, ${\chi }_{r0}=0$ at mass ratio $ϵ={10}^{-3}$ computed voice by voice using the methods described here (blue crosses), and computed using a time-domain Teukolsky equation solver for the worldline of an inspiral with these initial conditions. Though only voices with $l=2$, $m=2$ are included, the multivoice data are otherwise identical to the ${\chi }_{r0}=0$ data shown in Fig. 2. The top panel shows a stretch $\mathrm{\Delta }{t}^{\mathrm{i}}=3000M$ near the beginning of inspiral; the bottom shows a stretch $\mathrm{\Delta }{t}^{\mathrm{i}}=1000M$ near $t^{\mathrm{i}}=1.5\times {10}^{5}M$, roughly the middle of this inspiral. The two calculations agree perfectly in these two snapshots; we in fact find that they hold this agreement all the way to the end at $t^{\mathrm{i}}\simeq 3\times {10}^{5}M$.

Figure 6

Same as Fig. 5, but for ${\chi }_{r0}=2\pi /3$; the multivoice data shown here are identical to the $l=2$, $m=2$ ${\chi }_{r0}=2\pi /3$ data shown in Fig. 2. The two waveforms agree very well at early times, but a secular offset accumulates in this case; as shown in the lower panel, the two waveforms are roughly 2 radians out of phase by $t^{\mathrm{i}}\simeq 1.5\times {10}^{5}M$. This grows to about 4 or 5 radians by the end of inspiral. Our hypothesis is that the time-domain integrator includes slow-time contributions (i.e., contributions from source terms that evolve on the longer inspiral timescale $T_{\mathrm{i}}$) which are left out of the adiabatic waveform.

Figure 7

Same as Fig. 6, but using the ad hoc phase modification introduced in Eq. (7.1). This correction adjusts the orbital phase with a term that depends on the inspiral rate. Though this correction has not been rigorously computed, it nonetheless substantially improves the agreement between the two waveforms, at least over this domain of $t^{\mathrm{i}}$. (As described in the text, the waveforms drift from one another as inspiral continues.) This supports the hypothesis that this drift arises due to slow-time variations neglected in the adiabatic approximation.

Figure 8

Some of the voices which contribute to $h_{+}$ for the case $(p_{\mathrm{init}},e_{\mathrm{init}})=(12M,0.2)$, as shown in the left panels of Fig. 2. The magnitude of the amplitude $|H_{lm0n}(t^{\mathrm{i}})|$ is on the left; the phase ${\mathrm{\Phi }}_{m0n}(t^{\mathrm{i}})$ is on the right. The top panels show voices with $l=4$ and $m=4$; the bottom panels show $l=2$ and $m=2$. Solid (red) curves are data with $n=0$, short-dashed (blue) curves are $n=1$, long-dashed (green) curves are $n=2$, dot-dashed (magenta) curves are $n=3$, and dotted (black) curves are $n=4$. In all cases, both the amplitude and the phase evolve smoothly and simply. As functions of time, these voices can be sampled much less densely than $h_{+}$ must be sampled in order to accurately track its behavior.

Figure 9

Some voices with $l=2$ and $m=-2$ which contribute to $h_{+}$ for the small-eccentricity case shown in Fig. 2. Colors and definitions are exactly as in Fig. 8. The behavior of modes with $n=3$ and $n=4$ are interesting: During the inspiral, there are moments at which the phase evolution reverses direction, corresponding to the voice’s instantaneous frequency changing sign. The amplitudes corresponding to these voices pass through zero at times very close to these moments. The zero passage of these amplitudes leads to the sharp appearance that we see here. The real and imaginary parts of $H_{2-23}$ and $H_{2-24}$ are perfectly smooth.

Figure 10

Some voices with $l=2$, $m=2$ which contribute to the large-eccentricity waveform shown in Fig. 2. The top panel shows the voices’ phase; the lower panel shows their amplitudes. Solid (red) curves are data with $n=0$, short-dashed (blue) curves are $n=3$, long-dashed (green) curves are $n=6$, dot-dashed (magenta) curves are $n=9$, and dotted (black) curves are $n=12$. In contrast to the small-eccentricity case, the voice with $n=0$ does not dominate over most of the inspiral. In fact, of the voices shown, the one which starts weakest ($n=3$) evolves to become the loudest by the end of inspiral. Nonetheless, all amplitudes and phases evolve in a smooth and simple way, just as in the low-eccentricity case.

Figure 11

Frequency-domain representation of the voices shown in Figs. 8 and 9. The top-left panel shows voices with $l=m=2$; the bottom-left panel shows voices with $l=m=4$; both panels on the right show voices with $l=2$, $m=-2$. In all cases, the solid (red) curves are the frequency-domain amplitudes with $n=0$; short-dashed (blue) curves are for $n=1$; long-dashed (green) curves are $n=2$; dot-dashed (magenta) curves are $n=3$; and dotted (black) curves are $n=4$. (We use two panels for the voices with $m=-2$ to showcase the range in amplitude of this case.) Because we generate the waveform in the time domain and use the stationary-phase Fourier transform, each voice spans a slightly different range of frequency.

Figure 12

Frequency-domain representation of the voices shown in Fig. 10. All curves show voices with $l=m=2$. The solid (red) curve is the frequency-domain amplitude with $n=0$; the short-dashed (blue) curve is for $n=3$; the long-dashed (green) curve is for $n=6$; the dot-dashed (magenta) curve is for $n=9$; and the dotted (black) curve is for $n=12$.

Figure 13

Left panels: the waveform $h_{+}$ for an example of spherical inspiral into a black hole with $a=0.9M$. This example has $(r_{\mathrm{init}},x_{I,\mathrm{init}})=(10M,0.5)$, is at mass ratio $ϵ={10}^{-3}$, and is observed in the black hole’s equatorial plane. During inspiral, the orbit’s radius steadily shrinks, while $x_I$ decreases very slightly (see text for details). The polar anomaly angle ${\chi }_{\theta 0}=2\pi /3$. The top-left panel shows the waveform over the entire inspiral; the bottom-left panels zoom in to early and late times. The lower panels also illustrate the influence of the anomaly angle, comparing ${\chi }_{\theta 0}=2\pi /3$ (solid [red] curves) with inspiral on the fiducial geodesic (dashed [blue] curves). We again see that this angle has a large effect on the waveform, in this case changing the initial orientation so that the frame-dragging induced modulation begins at a different phase. Right panels: some of the $l=m=2$ voices which contribute to this waveform, both time domain (top) and frequency domain (bottom). Each line is a different $k$ index: solid (red) is $k=0$, short-dashed (blue) is $k=1$, long-dashed (green) is $k=-1$, dot-dashed (magenta) is $k=2$, and dotted (black) is $k=-2$. Across the examples of zero-eccentricity inspiral we have examined, the tendency is that voices with $|k+m|=l$ are the strongest, and they fall off as $|k+m|$ moves away from this peak.

Figure 14

Left panels: the waveform $h_{+}$ for an example of eccentric equatorial inspiral into a black hole with $a=0.9M$. This example has $(p_{\mathrm{init}},e_{\mathrm{init}})=(12M,0.7)$, is at mass ratio $ϵ={10}^{-3}$, and is observed in the black hole’s equatorial plane. The initial conditions are similar to those used to make the large-eccentricity case shown in Fig. 2, and the two waveforms are similar at early times. However, at this rapid spin, inspiral goes very deep into the strong field, becoming nearly circular before the plunge; we find $(p_{\text{final}},e_{\text{final}})=(2.40M,0.057)$. The late waveform is consistent with low eccentricity late in the inspiral. The top-left panel shows the waveform for ${\chi }_{r0}=2/3$ over the inspiral; in addition to zooming in to early and late times, the bottom-left panels illustrate the influence of the anomaly angle, comparing ${\chi }_{r0}=2\pi /3$ (solid [red] curves) with inspiral on the fiducial geodesic (dashed [blue] curves). Right panels: some of the $l=m=2$ voices which contribute to this waveform, both time domain (top) and frequency domain (bottom). Solid (red) curves show the voices with $n=0$; short-dashed (blue) curves are for $n=3$; long-dashed (green) curves are $n=6$; dot-dashed (magenta) curves are $n=9$; and dotted (black) curves show $n=12$.

Figure 15

The inclined and eccentric trajectory in $(p,e,x_I)$ for the generic inspiral we examine. This trajectory, illustrated by the blue curve, begins at $(p_{\mathrm{init}},e_{\mathrm{init}},x_{I,\mathrm{init}})=(12M,0.25,0.5)$ and proceeds until the smaller body encounters the LSO (a section of which is illustrated by the orange plane) at $(p_{\text{final}},e_{\text{final}},x_{I,\text{final}})=(4.64M,0.084,0.488)$. A projection of this trajectory into the $(p,e)$ plane is illustrated by the red curve, along with the projection of the LSO at the final value of $x_I$ (black line in lower plane); a projection of this trajectory into the $(p,x_I)$ plane is illustrated by the green curve, along with the projection of the LSO at the final value of $e$ (black line on back “wall” of the box).

Figure 16

Left panel: the waveform $h_{+}$ for an example of generic inspiral into a black hole with $a=0.7M$. This example corresponds to $(p_{\mathrm{init}},e_{\mathrm{init}},x_{I,\mathrm{init}})=(12M,0.25,0.5)$, and has a mass ratio $ϵ={10}^{-3}$; the small body inspirals until it encounters the LSO at $(p_{\text{final}},e_{\text{final}},x_{I,\text{final}})=(4.64M,0.084,0.488)$. We show the waveform as observed in the black hole’s equatorial plane. The upper-left panel shows the waveform over the inspiral; the lower-left panels zoom in to early and late times. The lower-left panels also illustrate the influence of the anomaly angle, comparing ${\chi }_{r0}={\chi }_{\theta 0}=2\pi /3$ (solid [red] curves) with the inspiral on the fiducial geodesic (dashed [blue] curves). Right panel: some of the voices that contribute to this waveform. The voices shown here are for $l=m=2$, $k=0$; the solid (red) curve is $n=0$, the short-dashed (blue) curve is $n=1$, the long-dashed (green) curve is $n=2$, the dot-dashed (magenta) curve is $n=3$, and the dotted (black) curve is $n=4$. The behavior of these voices is quite simple, with $n=0$ being the strongest, and the voices falling away as $n$ increases.

Figure 17

Additional voices that contribute to the waveform shown in Fig. 16. The left panel shows voices with $k=2$; the right panel shows voices with $k=-2$. As in the right-hand panel of Fig. 16, the solid (red) curves are $n=0$, short-dashed (blue) curves are $n=1$, long-dashed (green) curves are $n=2$, dot-dashed (magenta) curves are $n=3$, and dotted (black) curves are $n=4$. The behavior is somewhat more complicated for these voices; although larger values of $n$ tend to be weaker, the evolution for small values of $n$ evolves rather differently than in the $k=0$ case. For both the cases shown here, the $n=0$ voice in particular evolves in importance.