Fast extreme-mass-ratio-inspiral waveforms: New tools for millihertz gravitational-wave data analysis

We present the fastemriwaveforms (FEW) package, a collection of tools to build and analyze extreme mass ratio inspiral (EMRI) waveforms. Here, we expand on [A. J. K. Chua et al., Phys. Rev. Lett. 126, 051102 (2021).] that introduced the first fast and accurate fully-relativistic EMRI waveform template model. We discuss the construction of the overall framework; constituent modules; and the general methods used to accelerate EMRI waveforms. Because the fully relativistic FEW model waveforms are for now limited to eccentric orbits in the Schwarzschild spacetime, we also introduce an improved augmented analytic kludge (AAK) model that describes generic Kerr inspirals. Both waveform models can be accelerated using graphics processing unit (GPU) hardware. With the GPU-accelerated waveforms in hand, a variety of studies are performed including an analysis of EMRI mode content, template mismatch, and fully Bayesian Markov Chain Monte Carlo-based EMRI parameter estimation. We find relativistic EMRI waveform templates can be generated with fewer harmonic modes ($\sim 10–100$) without biasing signal extraction. However, we show for the first time that extraction of a relativistic injection with semirelativistic amplitudes can lead to strong bias and anomalous structure in the posterior distribution for certain regions of parameter space.

Figure 1

Conventions for the source frame, solar system baricenter frame, and polarization angle are shown in the schematic diagram above. For the source-frame diagram (a), the coordinate basis is defined in Eq. (7). The direction toward the solar system baricenter $(-\widehat{\mathbf{R}})$ is shown as the red arrow which defines ${\widehat{\mathbf{z}}}_w$. The source-frame polarization vectors are shown in blue and the basis vectors in the wave frame [see Eq. (6)] are shown in orange. Note the rotation by $\pi /2$ to align the source-frame polarization vectors with the wave-frame basis. Also, due to conventions chosen, ${\varphi }_{•}=-\pi /2$ always. In the solar system baricenter frame (b), the solar system baricenter frame polarization basis is shown in blue. The red arrow shows the angle to the source ($+\widehat{\mathbf{R}}$). Finally, in plot (c), the polarization determination is illustrated. The polarization angle represents the rotation down the line-of-sight from the wave-frame (red) to the solar system baricenter polarization basis (black).

Figure 2

The overall modular framework for FEW is shown in the top schematic diagram. This diagram describes the high-level progression of module usage during waveform production. Each segment of waveform production is labelled at the top of each column and assigned a color. The bottom diagram shows the waveforms and modules that have been implemented to-date in the FEW package. Stock waveforms are shown in yellow in the first column. Their individual module progression toward the final waveform is shown with arrows. The modules are assigned the color of the segment of waveform production they belong to. The utility functions that are used both for waveform production and overall analysis are also shown toward the bottom in gray.

Figure 3

The effect of adjusting the $\epsilon$ parameter on the mismatch, mode count, and waveform generation time. This parameter determines the threshold for the fractional added power of the harmonic modes. For each plot shown, three sets of parameters are run, labeled in the legend of the central plot as $(p_0,e_0)$. The left graph shows the mismatch versus $\epsilon$ compared against slow FEW waveforms (Sec. 4c). The mismatch asymptotically approaches a minimum value specific to each set of parameters. This minimum value above zero is due to the noisy floor of the RomanNet. The center and right plots show the corresponding number of modes and speed, respectively, for each set of parameters tested. A dashed vertical line at $\epsilon ={10}^{-5}$ is added to each plot to indicate the $\epsilon$ chosen for the base injection waveform in the following sections.

Figure 4

The mode power content as a fraction of the total power emitted for a given geodesic orbit with various eccentricities and $p_0=10$. The mode data was computed with bicubic spline-generated amplitudes (see Sec. 4c). Plots in each row have the same eccentricity (labeled along the left edge). Columns represent a singular $l$ mode value (labeled along the top edge). Within each plot, $m$ and $n$ mode values are given along the horizontal and vertical axes, respectively. We show $n\in [-30,30]$ and $0\le m\le l$. For $m<0$, the plot would be the mirror image of above reflected around $n=0$. Modes with $P_{lmn}/P_{\mathrm{tot}}<{10}^{-10}$ are not shown.

Figure 5

Similar to Fig. 4 but for $(p_0,e_0)=(7.5,0.5)$ and different values of $\epsilon$ which controls the mode content in the waveform. Geodesic output from all modes, modes selected with $\epsilon ={10}^{-5}$, and modes selected with $\epsilon ={10}^{-2}$ are shown in each row from top to bottom, respectively. In Sec. 6d, we show waveforms built with $\epsilon ={10}^{-2}$ show high fidelity against more accurate waveforms indicating building waveforms with less modes will increase speed without sacrificing much accuracy.

Figure 6

For EMRI signals, the harmonic phases and amplitudes create mismatch between comparison waveforms, with the phase strongly dominating. Here, we show the mismatch stemming mainly from amplitude differences between template waveforms ($\mathrm{FF}\epsilon 5$, $\mathrm{FF}\epsilon 2$, FF22, SchAAK, which are listed in the title of each plot) and the slow FEW model from Sec. 4c. All template waveforms for a given $(p_0,e_0)$ use exactly the same trajectory, but differ in their amplitudes. The trajectories for the slow injection waveforms are slightly more accurate because they use dense-stepping integration, but this small difference does not contribute to the mismatch values shown. The initial values are chosen to outline our parameter space: $(p_0,e_0)\in [(10,0.7),(11.48,0.7),(12.96,0.7),(14.44,0.7),(15.92,0.7),(17.4,0.7),(16.2,0.1),(16.4,0.2),(16.6,0.3),(16.8,0.4),(17.0,0.5),(17.2,0.6)]$. These initial trajectory points are shown with gray dots. The central black hole mass, $M$, is set to ${10}^6{M}_{\odot }$ and the secondary’s mass, $\mu$, is determined using FEW utility tools so that the inspiral takes two years to evolve from the initial orbital parameters to the separatrix (shown as the grey line). All other parameters are identical to the injection waveform parameters given at the beginning of Sec. 6. The partial mismatch is shown from $t=0$ to $t=t(p,e)$ according to the color bar. As the lines move from the initial gray points to the separatrix, the mismatch is determined over an increasing amount of time. The $\mathrm{FF}\epsilon 5$ and $\mathrm{FF}\epsilon 2$ waveforms show strong overlap with the slow and accurate waveform. The FF22 waveform performs marginally and the SchAAK waveform compares poorly, especially at high eccentricity. These differences are visualized in Fig. 7.

Figure 7

Waveforms corresponding to the four waveform models tested in Fig. 6 are shown here. Each waveform follows the top-left trajectories in the sub-panels in Fig. 6 which have $(p_0,e_0)=(10,0.7)$ and a final eccentricity $\simeq 0.5$. Each row contains a different fast waveform model labeled along the vertical axis shown with a dashed orange line. These fast models are compared against the slow waveform model appearing as a solid blue line. The left and right plots represent the beginning and end of a two-year waveform. As is expected from the mismatch results, the $\mathrm{FF}\epsilon 5$ and $\mathrm{FF}\epsilon 2$ waveforms provide the strongest match to the slow waveform model. The visual difference between these two models is almost indistinguishable. The FF22 quadrupolar waveform begins to have difficulty resolving the higher peaks in the waveform and visually shows a clear separation from the top two rows. The bottom row, with the SchAAK waveform model, clearly reveals the issues related to generating EMRI waveforms with semirelativistic amplitudes: the overlap is very poor both quantitatively and visually.

Figure 8

Timing benchmarks for various waveforms discussed in this paper. Timing for the trajectory, angular harmonic, amplitude, mode selection, summation, and frame transformation modules is shown from top to bottom, respectively. The sum of those timings is also given. The CPU and GPU versions of the fast FEW waveforms (with $\epsilon ={10}^{-5}$) (Sec. 4) are shown in light and dark blue, respectively. The slow FEW waveform is shown in green (Sec. 4c). The 5PN AAK waveform (Sec. 5) is shown in light and dark red, representing the CPU and GPU versions, respectively. $(p_0,e_0)=(10,0.7)$ for all waveforms, depicting the worst-case in our domain of validity in terms of the number of modes necessary to represent the waveform. The central black hole mass $M$ is set to ${10}^6{M}_{\odot }$ and $\mu$ is chosen to represent a two-year inspiral from the initial $(p_0,e_0)$ to the separatrix. Note the slow waveform does not perform any mode selection. Also, the 5PN AAK waveform does not specifically employ angular harmonic, amplitude, or mode selection modules. The AAK waveform piece (not including the trajectory) is treated in its entirety as a summation module. Timings were computed using a single CPU core on a Xeon Gold 6132 2.60 GHz processor and an NVIDIA V100 GPU. The fast FEW waveform GPU version is $\sim 2500\times \mathrm{faster}$ than its CPU counterpart.

Figure 9

Characteristic strain (top row) and two-dimensional $3\sigma$ posteriors in the $e_0-\ln M$ plane (bottom row) for three different binary configurations. Above each top row plot the parameters are listed as $(M,\mu ,p_0,e_0)$ and each plot is made with $\mu =10$ and $e_0=0.5$. From left to right, injections have $M$ values of $3\times {10}^5$, $1\times {10}^6$, and $3\times {10}^6$. Within each bottom row plot, the $\mathrm{FF}\epsilon 5$, $\mathrm{FF}\epsilon 2$, FF22, and SchAAK models are shown in blue, green, red, and orange, respectively. Black horizontal and vertical lines indicate the location of the injection point. The top plots illustrate the spectral difference between $\mathrm{FF}\epsilon 5$ (blue) and SchAAK (orange) waveforms at the injection point compared against the LISA sensitivity curve. For completeness, we also tested all parameter sets in this plot again with the noise contribution from the Galactic foreground. It did not change the results by any significant amount.

Figure 10

Corner plot showing one- and two-dimensional marginalized posteriors for an injection with $(M,\mu ,p_0,e_0,{\mathrm{\Phi }}_{\phi ,0},{\mathrm{\Phi }}_{r,0})=({10}^6{M}_{\odot },3M_{\odot },8.99,0.5,3.23,4.72)$. The histograms and contours for the $\epsilon ={10}^{-5}$ fast FEW ($\mathrm{FF}\epsilon 5$) and Schwarzschild AAK (SchAAK) template waveforms are shown in blue and black, respectively. The true injection parameters are denoted with the red vertical and horizontal lines. Note the sampling methods used concentrated near the true value. The samplers are not be able to access secondary modes with a strong separation from the injection point. Due to the inaccuracy of the semi-relativistic amplitudes in the SchAAK waveform, a strong bias and multimodal behavior are observed when fitting a SchAAK waveform template against an $\mathrm{FF}\epsilon 5$ relativistic injection. The consequences of this behavior are further discussed in Sec. 7.

Figure 11

Two-dimensional $3\sigma$ posteriors for $M$ and $\mu$ parameters are shown for all injections tested using the SchAAK waveform as the fitting template to the $\mathrm{FF}\epsilon 5$ relativistic injection. Each two-dimensional posterior is arranged horizontally and vertically according to the $M$ and $\mu$ injection values, respectively. Within each subplot, posteriors are shown for every injection value of eccentricity tested with eccentricities of 0.1, 0.3, 0.5, and 0.7 shown in blue, green, red, and orange, respectively. Vertical and horizontal black lines indicate the true values of the injection. Please note injections with $(\mu ,e)=(3,0.7)$ and $M\in [1\times {10}^6,3\times {10}^6]$ do not fall within our domain of validity and, therefore, are not shown.

Figure 12

Multimodality in the $p_0\text{−}{e}_0$ plane is shown above for injections with $(M,e_0)=(3\times {10}^6,0.5)$ and $\mu$ values of 3 and 10 in the top and bottom plots, respectively. In each plot, the $1\sigma$, $2\sigma$, and $3\sigma$ contours are shown for the SchAAK template waveform. Black horizontal and vertical lines give the true injection values. The two cases shown here exhibit only a slight bias with the true injection point contained within the $2\sigma$ contour.