The Fast and the Frame-Dragging: Efficient waveforms for asymmetric-mass eccentric equatorial inspirals into rapidly-spinning black holes

TL;DR

This work extends the FEW framework to efficiently generate fully relativistic adiabatic waveforms for eccentric equatorial inspirals into rapidly spinning Kerr black holes, enabling $|a|\le 0.999$, $e<0.9$, and $p<200$ with ~100 ms wall-time. The model precomputes flux and mode-amplitude data on multi-dimensional grids and uses fast interpolation (tricubic spline for fluxes and bicubic+linear for amplitudes) coupled with a continuous ODE solution to deliver accurate waveforms suitable for LISA data analysis. Validation against independent datasets shows mismatches around $\sim 10^{-5}$ across most of parameter space, with larger interpolation- and separatrix-related errors in extreme regions; the study also investigates SNR impacts, parameter-recovery biases from approximate amplitude models, horizon redshifts, and eccentricity measurability. Collectively, the FEW v2 framework offers a robust, scalable tool for studying asymmetric-mass EMRI/IMRI signals, supporting Bayesian inference and enabling exploration of formation channels, cosmological reach, and data challenges for LISA.

Abstract

Observations of gravitational-wave signals emitted by compact binary inspirals provide unique insights into their properties, but their analysis requires accurate and efficient waveform models. Intermediate- and extreme-mass-ratio inspirals (I/EMRIs), with mass ratios $q \gtrsim 10^2$, are promising sources for future detectors such as the Laser Interferometer Space Antenna (LISA). Modelling waveforms for these asymmetric-mass binaries is challenging, entailing the tracking of many harmonic modes over thousands to millions of cycles. The FastEMRIWaveforms (FEW) modelling framework addresses this need, leveraging precomputation of mode data and interpolation to rapidly compute adiabatic waveforms for eccentric inspirals into zero-spin black holes. In this work, we extend FEW to model eccentric equatorial inspirals into black holes with spin magnitudes $|a| \leq 0.999$. Our model supports eccentricities $e < 0.9$ and semi-latus recta $p < 200$, enabling the generation of long-duration IMRI waveforms, and produces waveforms in $\sim 100$ ms with hardware acceleration. Characterising systematic errors, we estimate that our model attains mismatches of $\sim 10^{-5}$ (for LISA sensitivity) with respect to error-free adiabatic waveforms over most of parameter space. We find that kludge models introduce errors in signal-to-noise ratios (SNRs) as great as $^{+60\%}_{-40\%}$ and induce marginal biases of up to $\sim 1σ$ in parameter estimation. We show LISA's horizon redshift for I/EMRI signals varies significantly with $a$, reaching a redshift of $3$ ($15$) for EMRIs (IMRIs) with only minor $(\sim10\%)$ dependence on $e$ for an SNR threshold of 20. For signals with SNR $\sim 50$, spin and eccentricity-at-plunge are measured with uncertainties of $δa \sim 10^{-7}$ and $δe_f \sim 10^{-5}$. This work advances the state-of-the-art in waveform generation for asymmetric-mass binaries.

Figures (32)

• Figure 1: Waveform for a retrograde eccentric into a spinning in the frequency (upper-left), time-frequency (upper-right) and time domain (early and late times in lower-left and lower-right respectively). This system has parameters $\{m_1, m_2, a, p_0,e_0, d_\mathrm{L}\} = \{10^5\,M_\odot, 30\,M_\odot, -0.998, 28.3, 0.85, 1\,\mathrm{Gpc}\}$, and plunges after one year. When observed with over this duration, this signal has an of 41. The rich harmonic mode structure of the waveform evolves as the binary circularizes and the trajectory enters the strong-field region of the (see also \ref{['sec:amplitudes']}). Despite the waveform's complexity and size, it is generated by in less than $100\,\mathrm{ms}$ of wall-time for an A100 , sufficiently fast enough for full parameter estimation studies to be performed on a timescale of hours.
• Figure 2: Continuous solutions for the orbital phase, frequency and frequency derivative (left) of the trajectory obtained with an adaptive stepping integrator, and their absolute difference (right) with respect to a densely-stepped (fixed step size) integration. The considered binary system has component masses $(10^6, 10 M_\odot)$ and $(a, e_0)=(0.998, 0.5)$. The polar phase and its derivatives (denoted here by $\theta$) do not enter waveform generation for equatorial inspirals, but are included here for completeness.
• Figure 3: Trajectory characteristics as a function of integrator absolute tolerance $\sigma_\mathrm{tol}$ for $100$ draws of four-year duration inspirals with $\epsilon$ of $10^{-4}$ (blue circles), $10^{-5}$ (orange diamonds) and $10^{-6}$ (green squares) respectively. Top panel: Deviation of the phase $\Phi_\phi$ at end of inspiral with respect to a trajectory with $\sigma_ \mathrm{tol} = 10^{-13}$. Middle panel: Wall-time per trajectory evaluation. Bottom panel: Number of adaptive points in the trajectory solution.
• Figure 4: Top panel: The blue curve represents a highly eccentric and equatorial inspiral trajectory for $\epsilon = 10^{-5}$ and $a = 0.998$. The black dotted line describes separatrix $p_{\text{sep}}$, which evolves as a function of eccentricity. The orange diamonds represent individual snapshots of the trajectory. Bottom panel: A plot of the mode spectrum normalised by the total mode power $P_{\ell mn,\rm tot}^{(p = 2, e = 0.09)}$ at the points in the trajectory indicated by the orange diamonds. Modes with a normalised power below $10^{-10}$ are not shown. As the orbital parameters evolve, the distribution of the mode power with respect to mode index changes significantly, presenting the need for wide mode index coverage in a typical inspiral. The number of modes per row that account for $99\%$ of the total power in that row are $(129, 40, 24, 26)$; the union of these mode sets consists of $160$ elements (due to the evolving shape of the mode spectrum). For $99.999\%$ of the total power (our default for waveform generation) these counts become $(718, 345, 199, 174)$ and $988$ respectively.
• Figure 5: Relative error in $\hat{f}_{p}^{(0)}$ between an interpolation of our data grid and the corresponding data of Ref. Fujita:2020zxe, available from the BHPC data lake BHPClub. The two datasets agree to better than $8$ digits over the majority of the parameter space. Here we examine prograde orbits with $a = 0.5$; we observe similar agreement for other spin values $a \in [-0.9, 0.9]$.