High-accuracy waveforms for binary black hole inspiral, merger, and ringdown (0810.1767v2)

Abstract: The first spectral numerical simulations of 16 orbits, merger, and ringdown of an equal-mass non-spinning binary black hole system are presented. Gravitational waveforms from these simulations have accumulated numerical phase errors through ringdown of ~0.1 radian when measured from the beginning of the simulation, and ~0.02 radian when waveforms are time and phase shifted to agree at the peak amplitude. The waveform seen by an observer at infinity is determined from waveforms computed at finite radii by an extrapolation process accurate to ~0.01 radian in phase. The phase difference between this waveform at infinity and the waveform measured at a finite radius of r=100M is about half a radian. The ratio of final mass to initial mass is M_f/M = 0.95162 +- 0.00002, and the final black hole spin is S_f/M_f^2=0.68646 +- 0.00004.

• The paper presents a spectral method that simulates binary black hole inspiral, merger, and ringdown with phase errors reduced to ≤0.02 radian when waveforms are aligned.
• It reports precise final state measurements, with a mass ratio M_f/M = 0.95162 and a spin parameter S_f/M_f^2 = 0.68646, which corroborate other numerical approaches.
• The high-fidelity waveforms enhance gravitational wave detector calibration, improving detection accuracy for instruments like LIGO and LISA.

Insights into High-Accuracy Waveforms for Binary Black Hole Evolution

This paper offers an in-depth analysis of numerical simulations pertaining to the inspiral, merger, and ringdown phases of binary black hole systems. The authors present their method for generating gravitational waveforms with high numerical accuracy and discuss the implications of these results for our understanding of general relativity and the capability of gravitational wave detectors.

The paper employs a spectral numerical method to simulate an equal-mass, nonspinning binary black hole system, extending previous simulations in both duration and accuracy. By conducting spectral numerical simulations of 16 orbits through the final black hole ringdown, the research team achieves accumulated numerical phase errors of ≤0.1 radian when no adjustment is applied, and ≤0.02 radian when waveforms are aligned at peak amplitude.

Key numerical results from the simulations include a final black hole mass ratio of $M_f/M = 0.95162 \pm 0.00002$ and a spin parameter of $S_f/M_f^2 = 0.68646 \pm 0.00004$. These findings align well with results from other computational approaches, reaffirming the viability of the developed model.

Methodology and Numerical Precision

The authors solve Einstein’s equations using spectral methods, which can achieve exponential convergence rates for smooth solutions. This approach contrasts with the more traditional finite difference methods, which converge polynomially. Spectral methods generate highly accurate initial data and simulate long-term binary orbits efficiently.

However, simulating the merger phase posed significant challenges. The authors address this by modifying the gauge conditions, handling the behavior near the black hole horizons more robustly, and evolving the binary black hole system until the formation of a common apparent horizon.

Implications for Gravitational Wave Detection

The waveforms derived from the spectral simulations are crucial for the calibration of gravitational wave detectors like LIGO and LISA. High-fidelity waveforms improve the ability of matched filtering techniques to detect gravitational waves accurately, facilitating precise astrophysical measurements of black hole properties.

Extrapolation to Infinity

A noteworthy aspect of this paper is the meticulous extrapolation of waveforms to an observer at infinity, thereby removing near-field effects and gauge ambiguities present in finite extraction-radius calculations. Such extrapolation is essential for interpreting physical phenomena accurately and is critical for waveforms’ use in gravitational wave astronomy.

Recommendations for Future Developments

The research team points out that while the achieved accuracy is sufficient for LIGO detections, it is potentially inadequate for precise parameter estimation and for use in LISA. Future improvements may involve refining gauge choices and extrapolation techniques, alongside widespread adoption and comparison with other numerical relativity methods. The development of techniques robust to varied mass ratios and spins in black hole systems remains a key area for future research advancements in this field.

This research lays a comprehensive groundwork for future studies, indicating promising directions for simulating more complex systems and further enhancing the precision of gravitational wave modeling.