Rapid Generation of Fully Relativistic Extreme-Mass-Ratio-Inspiral Waveform Templates for LISA Data Analysis

The future space mission LISA will observe a wealth of gravitational-wave sources at millihertz frequencies. Of these, the extreme-mass-ratio inspirals of compact objects into massive black holes are the only sources that combine the challenges of strong-field complexity with that of long-lived signals. Such signals are found and characterized by comparing them against a large number of accurate waveform templates during data analysis, but the rapid generation of templates is hindered by computing the $\sim {10}^3–{10}^5$ harmonic modes in a fully relativistic waveform. We use order-reduction and deep-learning techniques to derive a global fit for the $\approx 4000$ modes in the special case of an eccentric Schwarzschild orbit, and implement the fit in a complete waveform framework with hardware acceleration. Our high-fidelity waveforms can be generated in under 1 s, and achieve a mismatch of $\lesssim 5\times {10}^{-4}$ against reference waveforms that take $\gtrsim {10}^4$ times longer. This marks the first time that analysis-length waveforms with full harmonic content can be produced on timescales useful for direct implementation in LISA analysis algorithms.

Figure 1

Evolution of mismatch between fast and fiducial waveforms from $(p_0,e_0)$ to $(p,e)$, for 12 EMRIs with $M={10}^6{M}_{\odot }$, $\mu \in [15,304]M_{\odot }$, and $(p_0,e_0)$ along the model domain boundary. Each small mass is chosen such that the EMRI plunges after a year. These results are for $(\theta ,\phi )=(\pi /2,0)$, but do not depend strongly on the viewing angle. In the worst case (top-left curve), the final 0.01% of the waveform causes the mismatch to increase from under $4\times {10}^{-4}$ to $5\times {10}^{-4}$.

Figure 2

Six hour snapshots of fast (orange) and fiducial (blue) waveforms, 1 yr before plunge (top) and just before plunge (bottom). Waveforms are for the worst-case EMRI $(M,\mu ,p_0,e_0)=({10}^6{M}_{\odot },15M_{\odot },10,0.7)$, with a 1 yr mismatch of $5\times {10}^{-4}$. Small amplitude deviations are visible just before plunge at $(p,e)\approx (7,0.5)$, where the mode-distribution error approaches its maximum across the domain of validity.

Figure 3

Computational wall time for fast and fiducial waveforms, broken down into individual modules. All times are averaged over $\ge 5$ evaluations of the worst-case waveform on a single CPU core (and GPU), where the CPU is an Intel Xeon Gold 6132 and the GPU is an NVIDIA Tesla V100.