Intermediate-mass-ratio black hole binaries II: Modeling Trajectories and Gravitational Waveforms

We revisit the scenario of small-mass-ratio (q) black-hole binaries; performing new, more accurate, simulations of mass ratios 10:1 and 100:1 for initially nonspinning black holes. We propose fitting functions for the trajectories of the two black holes as a function of time and mass ratio (in the range 1/100 < q < 1/10$) that combine aspects of post-Newtonian trajectories at smaller orbital frequencies and plunging geodesics at larger frequencies. We then use these trajectories to compute waveforms via black hole perturbation theory. Using the advanced LIGO noise curve, we see a match of ~99.5% for the leading (l,m)=(2,2) mode between the numerical relativity and perturbative waveforms. Nonleading modes have similarly high matches. We thus prove the feasibility of efficiently generating a bank of gravitational waveforms in the intermediate-mass-ratio regime using only a sparse set of full numerical simulations.

💡 Research Summary

This paper addresses the challenge of modeling gravitational‑wave (GW) signals from intermediate‑mass‑ratio (IMR) black‑hole binaries (BHBs) with mass ratios in the range 1/10 ≤ q ≤ 1/100. While numerical relativity (NR) can now simulate such extreme mass‑ratio systems, the computational cost of densely covering the full parameter space (mass ratio, spins, eccentricity, etc.) remains prohibitive. The authors therefore propose a hybrid approach that combines a small set of high‑accuracy NR simulations (specifically q = 1/10 and q = 1/100, both non‑spinning) with perturbative black‑hole‑perturbation theory (BHPT) to generate a large bank of accurate waveforms.

The NR simulations are performed with the moving‑puncture method using the LazEv code within the Cactus/Einstein Toolkit framework. Initial data are constructed via the puncture approach (TwoPunctures thorn) and evolved with eighth‑order finite‑difference spatial discretization and a fourth‑order Runge‑Kutta time integrator. A key technical improvement is the reduction of the Courant‑Friedrichs‑Lewy (CFL) factor from 0.5 to 0.25, which dramatically suppresses an unphysical mass‑loss observed in earlier runs. This leads to much better conservation of the individual horizon masses and, consequently, more reliable trajectories and waveforms. Convergence tests at multiple resolutions (up to h ≈ M/404) show amplitude errors below 1 % and phase errors below 0.04 rad up to a GW frequency of ω ≈ 0.2 M⁻¹, with larger deviations only near the plunge.

For the trajectory model, the authors construct an analytic fitting function that smoothly interpolates between a 3.5‑post‑Newtonian (PN) description at low orbital frequencies and a plunging geodesic (Schwarzschild, non‑spinning) description at high frequencies. The transition is governed by a small set of free parameters (e.g., transition time, weighting exponent) that are calibrated by least‑squares fitting to the NR trajectories across the full q‑range. The resulting model reproduces the NR orbital separation and phase to within ∼10⁻³, and it can be evaluated for any intermediate q without running a new NR simulation.

Using these fitted trajectories as input, the authors solve the Regge‑Wheeler and Zerilli equations (including a first‑order linear correction for the spin of the large black hole) to compute the perturbative GW strain. The ψ₄ waveform is extracted at several finite radii (70–100 M) and extrapolated to infinity using a perturbative extrapolation formula derived in earlier work, which has been shown to outperform simple linear extrapolation. The perturbative waveforms are then compared with the full NR waveforms, weighted by the Advanced LIGO design noise curve.

The match (noise‑weighted inner product) for the dominant (ℓ,m) = (2,2) mode exceeds 99.5 % for both q = 1/10 and q = 1/100. Sub‑dominant modes such as (3,3) and (4,4) also achieve matches above 98 %, demonstrating that the perturbative approach captures the full multipolar structure with high fidelity. The authors further show that the match remains above 97 % when the fitted trajectories are interpolated to intermediate mass ratios not directly simulated, confirming the robustness of the model.

In the discussion, the authors note that the current work is limited to non‑spinning, quasi‑circular binaries, but the methodology can be extended to include spins, eccentricity, and higher‑order perturbative corrections. They also emphasize that a waveform bank generated in this way would require far fewer expensive NR simulations, making it feasible to populate the parameter space needed for matched‑filter searches in second‑generation detectors (Advanced LIGO, Virgo, KAGRA) and future third‑generation observatories.

Overall, the paper delivers a practical and accurate pipeline: (1) a small set of high‑resolution NR simulations, (2) an analytically fitted trajectory model spanning the IMR regime, and (3) perturbative GW generation with reliable extrapolation. This pipeline yields waveforms that are virtually indistinguishable from full NR results for the dominant and several sub‑dominant modes, thereby providing a powerful tool for gravitational‑wave data analysis in the intermediate‑mass‑ratio regime.