Session X10: Numerical Relativity Methodology

General relativistic magneto-hydrodynamics with the Einstein Toolkit

The Einstein Toolkit Consortium is developing and supporting open software for relativistic astrophysics. Its aim is to provide the core computational tools that can enable new science, broaden our community, facilitate interdisciplinary research and take advantage of petascale computers and advanced cyberinfrastructure. The Einstein Toolkit currently consists of an open set of over 100 modules for the Cactus framework, primarily for computational relativity along with associated tools for simulation management and visualization. The toolkit includes solvers for vacuum spacetimes as well as relativistic magneto-hydrodynamics. This talk will present the current capabilities of the Einstein Toolkit with a particular focus on recent improvements made to the general relativistic magneto-hydrodynamics modeling and will point to information how to leverage it for future research.   

Exploring the outer limits of Numerical Relativity

We perform a first exploration of black-hole binary evolutions using full nonlinear numerical relativity techniques at separations large enough that low-order post-Newtonian expansions are expected to be very accurate. As a case study, we evolve an equal-mass nonspinning black-hole binary in a quasicircular orbit at an initial coordinate separation of $r=100M$. We measured the orbital period of the binary and find $T=6422M$. We perform convergent simulations at three different grid resolutions and complete two, one and a half, and one and a quarter orbits for the low, medium and high resolutions, respectively. The orbital motion agrees with post-Newtonian predictions to within $1\%$. We discuss on how to improve this accuracy in future simulations. The results are relevant for the generation of long-term waveforms for detection and analysis of gravitational waves by the next generation of detectors.   

Accuracy Issues for Numerical Waveforms

We analyze the gravitational waveform error from the late inspiral, merger, and ringdown, and find that using several lower-order techniques for increasing the speed of numerical relativity simulations actually lead to apparently nonconvergent errors. Even when using standard high-accuracy techniques, rather than seeing clean convergence, where the waveform phase is a monotonic function of grid resolution, we find that the phase tends to oscillate with resolution, possibly due to stochastic errors induced by grid refinement boundaries.   

Estimating gauge errors in Newman-Penrose extrapolated waveforms via comparison with Cauchy Characteristic Extraction

Several methods exist for extracting gravitational waveforms (GW) from numerical simulations of compact object binaries. Understanding the uncertainties in these methods is essential for obtaining trustworthy waveforms. A popular method of obtaining waveforms is to extract the Newman-Penrose scalar Psi4 at several finite radii in a simulation, and then to extrapolate these data to future null infinity in order to remove near-field effects. However, the waveforms thus obtained may still be contaminated by unknown gauge (coordinate) effects. In order to estimate these gauge errors, we consider Cauchy Characteristic Extraction (CCE). Although computationally more expensive, this method yields, by construction, gauge-invariant waveforms at future null infinity. Using data from several binary black hole simulations performed with the Spectral Einstein Code (SpEC), we compare extrapolation of Psi4 to CCE. We examine the various sources of uncertainty in these two extraction methods and confirm the gauge invariance of CCE. We then use the CCE waveforms as a basis for estimating the unknown gauge errors in the extrapolated Psi4 waveforms.   

General relativistic null-cone evolutions with a high-order scheme

The accurate modeling of gravitational radiation is a key issue for gravitational wave astronomy. As simulation codes reach higher accuracy, systematic errors inherent in current numerical relativity wave-extraction methods become evident, and may lead to a wrong astrophysical interpretation of the data. Gravitational radiation is properly defined only at future null infinity, scri+. The method of Cauchy-characteristic extraction (CCE) has been successful at evolving metric data from a finite radius along null hypersurfaces to future null infinity, scri+. Current characteristic Einstein evolution codes, however, are only second-order accurate, thus requiring unnecessary high resolution to reach a given accuracy goal. Unfortunately, due to the nature of the Einstein equations in characteristic form, extending the algorithm to higher than second-order is non-trivial and requires a different approach than the so called ``null-parallelogram'' scheme. In this talk, I present a new fully non-linear 3D characteristic evolution algorithm based on spectral angular derivatives and fourth-order radial and time integration. Using linearized solutions (+noise), I show that the scheme is stable and efficient for solving the characteristic Einstein equations.   

Steps towards the well-posedness of the characteristic evolution for the Einstein equations

The correct modeling of gravitational radiation is a key requirement for a meaningful detection and interpretation of data. The Cauchy-characteristic technique connects the strong-field Cauchy evolution of the space-time near the black-hole merger to the characteristic evolution at future null infinity, where the waveform is properly defined. The PITT Null code, publicly available, is the most precise and refined computational method for the extraction of gravitational waves, but is not well-posed. The numerical relativity community recognizes that a well-posed problem is the only way to ensure that a code is stable and dependable. The well-posedness of the null-timelike problem for the Einstein equations is not yet established. We present our progress towards developing and testing a new computational evolution algorithm based on the well-posedness of characteristic initial value and boundary problems for a scalar wave. We strive to demonstrate analytically and to verify numerically the well-posedness of our algorithm for quasilinear scalar waves propagating on an asymptotically flat curved space background with source, in Bondi null coordinates. We design and test a new boundary algorithm. Tests confirm the stability properties, and reveal interesting qualitative features.   

Physical properties of a quasi-Kinnersley tetrad

Without fixing tetrad freedom, the physical interpretation of components of curvature tensors such as the Newman-Penrose $\Psi_4$ is ambiguous. Expanding on earlier literature on quasi-Kinnersley tetrads, we suggest a particular tetrad fixing procedure based on the characteristic structure of the Weyl tensor. This talk focuses on the physical properties of the resulting tetrad.   

Numerical Relativity Ringdown Waveforms: From Spherical to Spheroidal Mode Decomposition

Numerical Relativity waveforms are traditionally decomposed into spin -2 spherical multipoles. On the other hand, the quasi-normal mode ringdown of black holes is more naturally described by spin -2 spheroidal multipoles. As a consequence, numerical relativity ringdown waveforms consist of a superposition of spheroidal multipoles. We present a robust method that identifies the spheroidal multipole content in numerical relativity waveforms. We demonstrate the efficacy of the method in identifying interesting quasi-normal mode structures in simulations of unequal-mass binary black hole mergers.