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Optimizing post-Newtonian parameters and fixing the BMS frame for numerical-relativity waveform hybridizations (2403.10278v2)

Published 15 Mar 2024 in gr-qc

Abstract: Numerical relativity (NR) simulations of binary black holes provide precise waveforms, but are typically too computationally expensive to produce waveforms with enough orbits to cover the whole frequency band of gravitational-wave observatories. Accordingly, it is important to be able to hybridize NR waveforms with analytic, post-Newtonian (PN) waveforms, which are accurate during the early inspiral phase. We show that to build such hybrids, it is crucial to both fix the Bondi-Metzner-Sachs (BMS) frame of the NR waveforms to match that of PN theory, and optimize over the PN parameters. We test such a hybridization procedure including all spin-weighted spherical harmonic modes with $|m|\leq \ell$ for $\ell\leq 8$, using 29 NR waveforms with mass ratios $q\leq 10$ and spin magnitudes $|\chi_1|, |\chi_2|\leq 0.8$. We find that for spin-aligned systems, the PN and NR waveforms agree very well. The difference is limited by the small nonzero orbital eccentricity of the NR waveforms, or equivalently by the lack of eccentric terms in the PN waveforms. To maintain full accuracy of the simulations, the matching window for spin-aligned systems should be at least 5 orbits long and end at least 15 orbits before merger. For precessing systems, the errors are larger than for spin-aligned cases. The errors are likely limited by the absence of precession-related spin-spin PN terms. Using $105\,M$ long NR waveforms, we find that there is no optimal choice of the matching window within this time span, because the hybridization result for precessing cases is always better if using earlier or longer matching windows. We provide the mean orbital frequency of the smallest acceptable matching window as a function of the target error between the PN and NR waveforms and the black hole spins.

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Summary

  • The paper introduces an iterative method to align NR and PN waveforms by fixing the BMS frame and optimizing PN parameters.
  • The hybridization technique minimizes discrepancies in mass and spin by applying nonlinear least squares optimization across all relevant harmonic modes.
  • Achieving low residual errors for spin-aligned systems, the study establishes a framework for constructing precise gravitational wave models.

Optimizing Post-Newtonian Parameters and Fixing the BMS Frame for Numerical-Relativity Waveform Hybridizations

The paper addresses the challenge of constructing waveform hybrids by merging numerical relativity (NR) waveforms with post-Newtonian (PN) approximations for the use in gravitational wave astronomy. NR simulations, while precise, are computationally intensive and often insufficient in duration to cover the entire frequency band detectable by gravitational-wave observatories. PN approaches, on the other hand, are effective for modeling the early inspiral phase but lack accuracy during the merger phase. Thus, a reliable hybridization method that seamlessly combines both approaches is critical.

Methodology

The paper emphasizes the importance of aligning the Bondi-Metzner-Sachs (BMS) frame of the NR waveforms with that of PN theory. This is pivotal for reducing discrepancies between the two waveform models. Additionally, the paper introduces a strategy to optimize PN parameters to minimize errors arising from differences in the NR and PN representations of mass and spin values. The hybridization process incorporates all spin-weighted spherical harmonic modes for ∣m∣≤ℓ|m| \le \ell for ℓ≤8\ell \le 8, addressing spin-aligned systems with mass ratios up to 10 and spin magnitudes up to 0.8.

The iterative method introduced includes:

  1. Mapping NR waveforms to their superrest frame.
  2. Setting m=0m=0 modes of NR and PN waveforms to zero to negate primary memory effects.
  3. Performing nonlinear least squares optimization to determine optimal PN parameters, rotation, and time shift.
  4. Matching NR and PN BMS frames using supertranslations and boosts.
  5. Iteratively refining these parameters to converge on an accurate hybridization.

Results

For spin-aligned systems, the optimized PN and NR waveforms showed remarkable agreement, with residual errors limited by NR wave eccentricities or lack of PN eccentric terms. The optimal result was achieved with a matching window of at least 5 orbits, ensuring the termination at least 15 orbits before merger. For precessing systems, larger deviations were observed, primarily due to missing precession-related spin-spin PN terms. An absence of a definitive matching window for precessing binaries was found, suggesting better results with earlier or extended windows.

Implications and Conclusions

This method significantly reduces hybridization errors and provides more reliable waveform models for future observations. Aligning NR and PN waveforms in a consistent BMS frame is affirmed as essential, and optimizing PN parameters can yield considerable improvements. However, the paper recognizes limitations introduced by current PN models, particularly for precessing systems due to missing higher-order PN terms.

Future research can extend this framework by developing more comprehensive PN models that address these current limitations. Furthermore, improvements in NR simulations, particularly in terms of resolving residual eccentricities, alongside utilizing recent advancements in effective-one-body formalisms, can further refine hybrid waveforms. This research thus lays the groundwork for constructing more accurate gravitational waveform models essential for the next-generation detectors like Cosmic Explorer and LISA.

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