Beyond second-order convergence in simulations of binary neutron stars in full general-relativity (1306.6052v2)

Abstract: Despite the recent rapid progress in numerical relativity, a convergence order less than the second has so far plagued codes solving the Einstein-Euler system of equations. We report simulations of the inspiral of binary neutron stars in quasi-circular orbits computed with a new code employing high-order, high-resolution shock-capturing, finite-differencing schemes that, for the first time, go beyond the second-order barrier. In particular, without any tuning or alignment, we measure a convergence order above three both in the phase and in the amplitude of the gravitational waves. Because the new code is able to calculate waveforms with very small phase errors already at modest resolutions, we are able to obtain accurate estimates of tidal effects in the inspiral that are essentially free from the large numerical viscosity typical of lower-order methods, and even for the challenging large compactness and small-deformability binary considered here. We find a remarkable agreement between our Richardson-extrapolated waveform and the one from the tidally corrected post-Newtonian (PN) Taylor-T4 model, with a de-phasing smaller than 0.2 radians during the seven orbits of the inspiral and up to the contact point. Because our results can be used reliably to assess the validity of the PN or other approximations at frequencies significantly larger than those considered so far in the literature, they seem to exclude at these compactnesses significant tidal amplifications from next-to-next-to-leading--order terms in the PN expansion.

Insights into Beyond Second-Order Convergence in Simulations of Binary Neutron Stars

This paper, authored by D. Radice, L. Rezzolla, and F. Galeazzi, presents significant advancements in the numerical simulation of binary neutron star (BNS) inspirals in full General Relativity, marking a notable improvement in high-order convergence techniques. The focus is on overcoming the traditional barriers posed by second-order convergence in numerical relativity codes, which has hindered precise computational modeling within the context of the Einstein–Euler system of equations.

The authors introduce a sophisticated numerical code that achieves a convergence order greater than three in both the phase and amplitude of gravitational waves emitted by BNS systems. This leap in convergence accuracy allows for enhanced prediction and modeling of tidal effects during inspiral phases, which is critical for deriving insights into neutron star properties such as equation of state (EOS) at supra-nuclear densities. Notably, the paper highlights the capability of the new code to provide gravitational waveforms with minimal phase errors and avoid the pitfalls of large numerical viscosity typical of lower-order methods, even when simulating systems with large compactness and small deformability.

Main Contributions

1. Numerical Technique and Implementation:
   • The paper introduces the WhiskyTHC code, which represents an advancement over previous methodologies by employing high-resolution shock-capturing, finite-differencing schemes.
   • This code uses fifth-order flux-vector splitting finite-difference methods combined with BSSNOK spacetime evolution, ensuring stability through artificial dissipation applied selectively to spacetime variables.
2. Simulation Results and Convergence:
   • Demonstrations indicate superior convergence order exceeding three, marking a transition from traditionally accepted low convergence orders which impair the reliability and accuracy of computational results.
   • The code provides a means to address phase uncertainties and arbitrary waveform alignment methods, crucial for detecting and analyzing gravitational waves from BNS mergers.
3. Implications for Post-Newtonian (PN) Modeling:
   • The accurate convergence and extrapolation achieved with this code produce a close match when compared with tidally corrected PN Taylor-T4 models. The de-phasing is significantly reduced, showcasing the ability to model gravitational waves up to contact points with increased reliability.
   • This development potentially excludes significant tidal contributions from higher-order PN terms at certain frequencies, offering a pathway for more refined comparisons between NR predictions and semi-analytic models.

Implications of the Research

The implications of this research are profound in both theoretical and practical domains. The enhancement in convergence order is pivotal for the development of precise gravitational wave templates, crucial for the next generation of gravitational wave detectors like Advanced LIGO, Virgo, or KAGRA. The ability to simulate BNS mergers with reduced computational cost and greater fidelity will facilitate systematic exploration of parameter spaces relevant to tidal interactions and EOS characterization.

Moreover, the advancements enable improved testing and enhancement of semi-analytical models such as the PN and effective one body (EOB) approximations. This could lead to an increased understanding of the complex dynamical interactions during BNS inspirals and mergers, contributing directly to the refinement of gravitational wave data analysis techniques.

Speculation on Future Developments

Future work will likely build on these findings to explore more diverse and complex neutron star systems, possibly incorporating realistic EOSs and considering additional relativistic effects. This could include expanding simulations to different mass ratios, spins, and magnetic fields, which are relevant for the realistic conditions expected in observed astrophysical systems. Further development could also focus on integrating these simulation techniques more closely with observational data analysis, enhancing the ability of laser-interferometer detectors to discern tidal effects within gravitational wave signals.

In conclusion, the paper demonstrates significant progress in numerical relativity relevant to BNS simulation, pushing the boundaries of existing methodologies, and laying the groundwork for future explorations in computational astrophysics.