Ian Hinder

The Einstein Toolkit: a community computational infrastructure for relativistic astrophysics

We describe the Einstein Toolkit, a community-driven, freely accessible computational infrastructure intended for use in numerical relativity, relativistic astrophysics, and other applications. The toolkit, developed by a collaboration involving researchers from multiple institutions around the world, combines a core set of components needed to simulate astrophysical objects such as black holes, compact objects, and collapsing stars, as well as a full suite of analysis tools. The Einstein Toolkit is currently based on the Cactus framework for high-performance computing and the Carpet adaptive mesh refinement driver. It implements spacetime evolution via the BSSN evolution system and general relativistic hydrodynamics in a finite-volume discretization. The toolkit is under continuous development and contains many new code components that have been publicly released for the first time and are described in this paper. We discuss the motivation behind the release of the toolkit, the philosophy underlying its development, and the goals of the project. A summary of the implemented numerical techniques is included, as are results of numerical test covering a variety of sample astrophysical problems.

Gravitational Recoil from Spinning Binary Black Hole Mergers

The inspiraling and merger of binary black holes will likely involve black holes with not only unequal masses but also arbitrary spins. The gravitational radiation emitted by these binaries will carry angular as well as linear momentum. A net flux of emitted linear momentum implies that the black hole produced by the merger will experience a recoil or kick. Previous studies have focused on the recoil velocity from unequal-mass, nonspinning binaries. We present results from simulations of equal-mass but spinning black hole binaries and show how a significant gravitational recoil can also be obtained in these situations. We consider the case of black holes with opposite spins of magnitude a aligned and antialigned with the orbital angular momentum, with a the dimensionless spin parameter of the individual holes. For the initial setups under consideration, we find a recoil velocity of V = 475a km s-1. Supermassive black hole mergers producing kicks of this magnitude could result in the ejection of the final hole produced by the collision from the core of a dwarf galaxy.

Testing gravitational-wave searches with numerical relativity waveforms: results from the first Numerical INJection Analysis (NINJA) project

The Numerical INJection Analysis (NINJA) project is a collaborative effort between members of the numerical relativity and gravitational-wave data analysis communities. The purpose of NINJA is to study the sensitivity of existing gravitational-wave search algorithms using numerically generated waveforms and to foster closer collaboration between the numerical relativity and data analysis communities. We describe the results of the first NINJA analysis which focused on gravitational waveforms from binary black hole coalescence. Ten numerical relativity groups contributed numerical data which were used to generate a set of gravitational-wave signals. These signals were injected into a simulated data set, designed to mimic the response of the initial LIGO and Virgo gravitational-wave detectors. Nine groups analysed this data using search and parameter-estimation pipelines. Matched filter algorithms, un-modelled-burst searches and Bayesian parameter estimation and model-selection algorithms were applied to the data. We report the efficiency of these search methods in detecting the numerical waveforms and measuring their parameters. We describe preliminary comparisons between the different search methods and suggest improvements for future NINJA analyses.

Constraint damping in the Z4 formulation and harmonic gauge

We show that by adding suitable lower-order terms to the Z4 formulation of the Einstein equations, all constraint violations except constant modes are damped. This makes the Z4 formulation a particularly simple example of a ?-system as suggested by Brodbeck et al (1999 J. Math. Phys. 40 909). We also show that the Einstein equations in harmonic coordinates can be obtained from the Z4 formulation by a change of variables that leaves the implied constraint evolution system unchanged. Therefore, the same method can be used to damp all constraints in the Einstein equations in harmonic gauge.

Unequal mass binary black hole plunges and gravitational recoil

We present results from fully nonlinear simulations of unequal mass binary black holes plunging from close separations well inside the innermost stable circular orbit with mass ratios q ≡ M1/M2 = {1, 0.85, 0.78, 0.55, 0.32}, or equivalently, with reduced mass parameters η ≡ M1M2/(M1 + M2)2 = {0.25, 0.248, 0.246, 0.229, 0.183}. For each case, the initial binary orbital parameters are chosen from the Cook–Baumgarte equal-mass ISCO configuration. We show waveforms of the dominant l = 2, 3 modes and compute estimates of energy and angular momentum radiated. For the plunges from the close separations considered, we measure kick velocities from gravitational radiation recoil in the range 25–82 km s−1. Due to the initial close separations our kick velocity estimates should be understood as a lower bound. The close configurations considered are also likely to contain significant eccentricities influencing the recoil velocity.

Error-analysis and comparison to analytical models of numerical waveforms produced by the NRAR Collaboration

The Numerical–Relativity–Analytical–Relativity (NRAR) collaboration is a joint effort between members of the numerical relativity, analytical relativity and gravitational-wave data analysis communities. The goal of the NRAR collaboration is to produce numerical-relativity simulations of compact binaries and use them to develop accurate analytical templates for the LIGO/Virgo Collaboration to use in detecting gravitational-wave signals and extracting astrophysical information from them. We describe the results of the first stage of the NRAR project, which focused on producing an initial set of numerical waveforms from binary black holes with moderate mass ratios and spins, as well as one non-spinning binary configuration which has a mass ratio of 10. All of the numerical waveforms are analysed in a uniform and consistent manner, with numerical errors evaluated using an analysis code created by members of the NRAR collaboration. We compare previously-calibrated, non-precessing analytical waveforms, notably the effective-one-body (EOB) and phenomenological template families, to the newly-produced numerical waveforms. We find that when the binarys total mass is ~100–200M⊙, current EOB and phenomenological models of spinning, non-precessing binary waveforms have overlaps above 99% (for advanced LIGO) with all of the non-precessing-binary numerical waveforms with mass ratios ≤4, when maximizing over binary parameters. This implies that the loss of event rate due to modelling error is below 3%. Moreover, the non-spinning EOB waveforms previously calibrated to five non-spinning waveforms with mass ratio smaller than 6 have overlaps above 99.7% with the numerical waveform with a mass ratio of 10, without even maximizing on the binary parameters.

Binary black holes : Spin dynamics and gravitational recoil

We present a study of spinning black hole binaries focusing on the spin dynamics of the individual black holes as well as on the gravitational recoil acquired by the black hole produced by the merger. We consider two series of initial spin orientations away from the binary orbital plane. In one of the series, the spins are antialigned; for the second series, one of the spins points away from the binary along the line separating the black holes. We find a remarkable agreement between the spin dynamics predicted at 2nd post-Newtonian order and those from numerical relativity. For each configuration, we compute the kick of the final black hole. We use the kick estimates from the series with antialigned spins to fit the parameters in the Kidder kick formula, and verify that the recoil in the direction of the orbital angular momentum is {proportional_to}sin{theta} and on the orbital plane {proportional_to}cos{theta}, with {theta} the angle between the spin directions and the orbital angular momentum. We also find that the black hole spins can be well estimated by evaluating the isolated horizon spin on spheres of constant coordinate radius.

Kranc: a Mathematica package to generate numerical codes for tensorial evolution equations ✩

We present a suite of Mathematica-based computer-algebra packages, termed “Kranc”, which comprise a toolbox to convert certain (tensorial) systems of partial differential evolution equations to parallelized C or Fortran code for solving initial boundary value problems. Kranc can be used as a “rapid prototyping” system for physicists or mathematicians handling very complicated systems of partial differential equations, but through integration into the Cactus computational toolkit we can also produce efficient parallelized production codes. Our work is motivated by the field of numerical relativity, where Kranc is used as a research tool by the authors. In this paper we describe the design and implementation of both the Mathematica packages and the resulting code, we discuss some example applications, and provide results on the performance of an example numerical code for the Einstein equations. Program summary

Improved effective-one-body model of spinning, nonprecessing binary black holes for the era of gravitational-wave astrophysics with advanced detectors

We improve the accuracy of the effective-one-body (EOB) waveforms that were employed during the first observing run of Advanced LIGO for binaries of spinning, nonprecessing black holes by calibrating them to a set of 141 numerical-relativity (NR) waveforms. The NR simulations expand the domain of calibration toward larger mass ratios and spins, as compared to the previous EOBNR model. Merger-ringdown waveforms computed in black-hole perturbation theory for Kerr spins close to extremal provide additional inputs to the calibration. For the inspiral-plunge phase, we use a Markov-chain Monte Carlo algorithm to efficiently explore the calibration space. For the merger-ringdown phase, we fit the NR signals with phenomenological formulae. After extrapolation of the calibrated model to arbitrary mass ratios and spins, the (dominant-mode) EOBNR waveforms have faithfulness—at design Advanced-LIGO sensitivity—above 99% against all the NR waveforms, including 16 additional waveforms used for validation, when maximizing only on initial phase and time. This implies a negligible loss in event rate due to modeling for these binary configurations. We find that future NR simulations at mass ratios ≳4 and double spin ≳0.8 will be crucial to resolving discrepancies between different ways of extrapolating waveform models. We also find that some of the NR simulations that already exist in such region of parameter space are too short to constrain the low-frequency portion of the models. Finally, we build a reduced-order version of the EOBNR model to speed up waveform generation by orders of magnitude, thus enabling intensive data-analysis applications during the upcoming observation runs of Advanced LIGO.

Matched filtering of numerical relativity templates of spinning binary black holes

Tremendous progress has been made towards the solution of the binary-black-hole problem in numerical relativity. The waveforms produced by numerical relativity will play a role in gravitational wave detection as either test beds for analytic template banks or as template banks themselves. As the parameter space explored by numerical relativity expands, the importance of quantifying the effect that each parameter has on first the detection of gravitational waves and then the parameter estimation of their sources increases. In light of this, we present a study of equal-mass, spinning binary-black-hole evolutions through matched filtering techniques commonly used in data analysis. We study how the match between two numerical waveforms varies with numerical resolution, initial angular momentum of the black holes, and the inclination angle between the source and the detector. This study is limited by the fact that the spinning black-hole binaries are oriented axially and the waveforms only contain approximately two and a half orbits before merger. We find that for detection purposes, spinning black holes require the inclusion of the higher harmonics in addition to the dominant mode, a condition that becomes more important as the black-hole spins increase. In addition, we conduct a preliminary investigation of how well a template of fixed spin and inclination angle can detect target templates of arbitrary but nonprecessing spin and inclination for the axial case considered here.