Comparing Gravitational Waveform Extrapolation to Cauchy-Characteristic Extraction in Binary Black Hole Simulations (1309.3605v1)

Abstract: We extract gravitational waveforms from numerical simulations of black hole binaries computed using the Spectral Einstein Code. We compare two extraction methods: direct construction of the Newman-Penrose (NP) scalar $\Psi_4$ at a finite distance from the source and Cauchy-characteristic extraction (CCE). The direct NP approach is simpler than CCE, but NP waveforms can be contaminated by near-zone effects---unless the waves are extracted at several distances from the source and extrapolated to infinity. Even then, the resulting waveforms can in principle be contaminated by gauge effects. In contrast, CCE directly provides, by construction, gauge-invariant waveforms at future null infinity. We verify the gauge invariance of CCE by running the same physical simulation using two different gauge conditions. We find that these two gauge conditions produce the same CCE waveforms but show differences in extrapolated-$\Psi_4$ waveforms. We examine data from several different binary configurations and measure the dominant sources of error in the extrapolated-$\Psi_4$ and CCE waveforms. In some cases, we find that NP waveforms extrapolated to infinity agree with the corresponding CCE waveforms to within the estimated error bars. However, we find that in other cases extrapolated and CCE waveforms disagree, most notably for $m=0$ "memory" modes.

• The paper compares waveform extrapolation and Cauchy-characteristic extraction (CCE) for extracting gravitational waves from binary black hole simulations.
• The study finds that CCE produces more accurate, gauge-invariant waveforms than extrapolation, particularly for m=0 "memory" modes and in high-mass ratio systems.
• Choosing the appropriate method depends on the required precision; CCE is preferred for high fidelity, despite its higher computational cost, driving research into optimizing its efficiency.

Comparison of Gravitational Waveform Extrapolation and Cauchy-Characteristic Extraction in Binary Black Hole Simulations

The paper examines two different methods for extracting gravitational waveforms from numerical simulations of binary black holes: waveform extrapolation and Cauchy-characteristic extraction (CCE). Both methods use the Spectral Einstein Code to simulate black hole mergers, but they differ in their approach to identifying wave patterns at significant distances from the source where gravitational waves reach Earth.

Methods Overview

• Waveform Extrapolation: This technique involves calculating the Newman-Penrose scalar $\Psi_4$ at multiple finite radii and extrapolating it to future null infinity to closely approximate waveforms as they would be detected by observatories. While this method leverages an efficient construction procedure, it relies on the assumption that $rM\Psi_4$ asymptotically converges, which can sometimes lead to inaccuracies because of gauge effects and near-zone effects.
• Cauchy-Characteristic Extraction (CCE): CCE directly computes gravitational waves at future null infinity using a compatible combination of Cauchy and characteristic evolutions. This approach provides gauge-invariant waveforms by construction, thus ensuring accuracy at future null infinity. However, it requires more complex processing and longer computation times.

Results

The authors report that the extrapolated waveforms have deficiencies that CCE does not share, particularly for modes with $m=0$, often referred to as "memory" modes. Their results demonstrate that CCE results are consistent across different gauge conditions while extrapolated waveforms can diverge due to gauge selection. Significantly compromised situations, such as in high-mass ratio binary systems, suggest that CCE should be prioritized when high fidelity in waveform extraction is crucial.

While both methods are subject to numerical errors—waveform extrapolation due to numerical fit uncertainty and CCE due to initial data selection and worldtube placement—the authors highlight that the limitations and higher complexity of CCE are often justified to achieve robust gravitational waveform prediction.

Implications and Future Directions

These findings emphasize the need for choosing the appropriate waveform extraction method with consideration toward the specific use case. For computational radiation models that cover extended parameter spaces, the use of CCE is advisable to ensure the precision of numerical relativity data. However, given its computational expense, CCE usage should be carefully balanced by cost-benefit analyses depending on the intended precision, particularly in high-stakes scenarios such as multimessenger astronomy.

The paper contributes significantly to understanding the necessary conditions for effective waveform extrapolation from numerical relativity data and raises considerations for potential reductions in computational costs while maintaining accuracy, such as improving the methods for waveform extrapolation by using varying polynomial extrapolation orders depending on the phase of the simulation.

Conclusion

The comprehensive comparative analysis of extraction methods presented in this paper offers valuable insights into optimizing the accuracy of gravitational waveform models, fundamentally supporting the detection and interpretation of gravitational waves by observatories. This is critical as multimodal astrophysical observations become an increasing focus in celestial dynamics and gravitational physics research. Further advancements in reducing computational requirements of CCE, without sacrificing accuracy, remain a promising area of research to enhance the utility and accessibility of this method.