Numerical Relativity papers published by members of the SXS Collaboration in reverse chronological order.
2025 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000
#2025
Effective Resistivity in Relativistic Reconnection: A Prescription Based on Fully Kinetic Simulations
Astrophys.J.Lett. 978, L45 (2025)
Abstract
A variety of high-energy astrophysical phenomena are powered by the release—via magnetic reconnection—of the energy stored in oppositely directed fields. Single-fluid resistive magnetohydrodynamic (MHD) simulations with uniform resistivity yield dissipation rates that are much lower (by nearly 1 order of magnitude) than equivalent kinetic calculations. Reconnection-driven phenomena could be accordingly modeled in resistive MHD employing a nonuniform, “effective” resistivity informed by kinetic calculations. In this work, we analyze a suite of fully kinetic particle-in-cell (PIC) simulations of relativistic pair-plasma reconnection—where the magnetic energy is greater than the rest mass energy—for different strengths of the guide field orthogonal to the alternating component. We extract an empirical prescription for the effective resistivity, ηeff=αB0∣J∣p/∣J∣p+1+entcp+1, where B$_{0}$ is the reconnecting magnetic field strength, J is the current density, n$_{t}$ is the lab-frame total number density, e is the elementary charge, and c is the speed of light. The guide field dependence is encoded in α and p, which we fit to PIC data. This resistivity formulation—which relies only on single-fluid MHD quantities—successfully reproduces the spatial structure and strength of nonideal electric fields and thus provides a promising strategy for enhancing the reconnection rate in resistive MHD simulations.
Surrogate modeling of gravitational waves microlensed by spherically symmetric potentials
Abstract
The anticipated observation of the gravitational microlensing of gravitational waves (GWs) promises to shed light on a host of astrophysical and cosmological questions. However, extracting the parameters of the lens from the modulated GWs requires accurate modeling of the lensing amplification factor, accounting for wave- optics effects. Analytic solutions to the lens equation have not been found to date, except for a handful of simplistic lens models. While numerical solutions to this equation have been developed, the time and computational resources required to evaluate the amplification factor numerically make large-scale parameter estimation of the lens (and source) parameters prohibitive. On the other hand, surrogate modeling of GWs has proven to be a powerful tool to accurately, and rapidly, produce GW templates at arbitrary points in parameter space, interpolating from a finite set of available waveforms at discrete parameter values. In this work, we demonstrate that surrogate modeling can also effectively be applied to the evaluation of the time-domain microlensing amplification factor $\widetilde{F}(t)$, for arbitrary lens configurations. We show this by constructing $\widetilde{F}(t)$ for two lens models, viz. point-mass lens, and singular isothermal sphere, which notably includes logarithmic divergence behaviour. We find both surrogates reproduce the original lens models accurately, with mismatches $\lesssim 5 \times 10^{-4}$ across a range of plausible microlensed binary black hole sources observed by the Einstein Telescope. This surrogate is between 5 and $10^3$ times faster than the underlying lensing models, and can be evaluated in about 100 ms. The accuracy and efficiency attained by our surrogate models will enable practical parameter estimation analyses of microlensed GWs.
#2024
Accurate waveforms for eccentric, aligned-spin binary black holes: The multipolar effective-one-body model SEOBNRv5EHM
Abstract
The measurement of orbital eccentricity in gravitational-wave (GW) signals will provide unique insights into the astrophysical origin of binary systems, while ignoring eccentricity in waveform models could introduce significant biases in parameter estimation and tests of General Relativity. Upcoming LIGO-Virgo-KAGRA observing runs are expected to detect a subpopulation of eccentric signals, making it vital to develop accurate waveform models for eccentric orbits. Here, employing recent analytical results through the third post- Newtonian order, we develop SEOBNRv5EHM: a new time-domain, effective-one-body, multipolar waveform model for eccentric binary black holes with spins aligned (or antialigned) with the orbital angular momentum. Besides the dominant (2,2) mode, the model includes the (2,1), (3,3), (3,2), (4,4) and (4,3) modes. We validate the model's accuracy by computing its unfaithfulness against 99 (28 public and 71 private) eccentric numerical-relativity (NR) simulations, produced by the Simulating eXtreme Spacetimes Collaboration. Importantly, for NR waveforms with initial GW eccentricities below 0.5, the maximum (2,2)-mode unfaithfulness across the total mass range 20-200 $M_\odot$ is consistently below or close to $1 \%$, with a median value of $ \sim 0.02 \% $, reflecting an accuracy improvement of approximately an order of magnitude compared to the previous-generation SEOBNRv4EHM and the state-of-the-art TEOBResumS-Dalí eccentric model. In the quasi- circular-orbit limit, SEOBNRv5EHM is in excellent agreement with the highly accurate SEOBNRv5HM model. The accuracy, robustness, and speed of SEOBNRv5EHM make it suitable for data analysis and astrophysical studies. We demonstrate this by performing a set of recovery studies of synthetic NR-signal injections, and parameter- estimation analyses of the events GW150914 and GW190521, which we find to have no eccentricity signatures.
Merging black holes with Cauchy-characteristic matching: Computation of late-time tails
Abstract
We present successful evolutions of binary black hole mergers using a novel numerical-relativity technique known as Cauchy- characteristic matching (CCM). This approach eliminates systematic errors associated with boundary conditions, effectively extending the computational domain to infinity. As an important application, we use CCM to resolve a late-time power-law tail in the gravitational wave from a head-on collision, and show that the tail is highly suppressed in a quasi-circular binary. Our results for the two extreme cases (orbital eccentricity $=0,1$) support the fact that tails increase with orbital eccentricity. Therefore, CCM paves the way for a detailed understanding of tails in eccentric systems. For the head-on case, we find that the tail behavior is consistent with predictions in the intermediate regime from black hole linear perturbation theory. However, we also raise the possibility that the power-law tail could be generated nonlinearly by quasinormal modes. The nonlinear contribution is expected to decay slower than predicted by Price's law, potentially dominating the signal at late times. If confirmed as nonlinear, this would be an example where nonlinearity prevails over linearity in the late-time regime of black hole dynamics.
Late-time tails in nonlinear evolutions of merging black holes
Abstract
We uncover late-time gravitational-wave tails in fully nonlinear 3+1 dimensional numerical relativity simulations of merging black holes, using the highly accurate SpEC code. We achieve this result by exploiting the strong magnification of late-time tails due to binary eccentricity, recently observed in perturbative evolutions, and showcase here the tail presence in head-on configurations for several mass ratios close to unity. We validate the result through a large battery of numerical tests and detailed comparison with perturbative evolutions, which display striking agreement with full nonlinear ones. Our results offer yet another confirmation of the highly predictive power of black hole perturbation theory in the presence of a source, even when applied to nonlinear solutions. The late-time tail signal is much more prominent than anticipated until recently, and possibly within reach of gravitational-wave detectors measurements, unlocking observational investigations of an additional set of general relativistic predictions on the long-range gravitational dynamics.
Black hole pulsars and monster shocks as outcomes of black hole-neutron star mergers
Abstract
The merger of a black hole (BH) and a neutron star (NS) in most cases is expected to leave no material around the remnant BH; therefore, such events are often considered as sources of gravitational waves without electromagnetic counterparts. However, a bright counterpart can emerge if the NS is strongly magnetized, as its external magnetosphere can experience radiative shocks and magnetic reconnection during/after the merger. We use magnetohydrodynamic simulations in the dynamical spacetime of a merging BH-NS binary to investigate its magnetospheric dynamics. We find that the magnetosphere develops compressive waves that steepen into shocks. After swallowing the NS, the BH acquires a magnetosphere that quickly evolves into a split monopole configuration and then undergoes an exponential decay (balding), enabled by magnetic reconnection and also assisted by the ring-down of the remnant BH. This spinning BH drags the split monopole into rotation, forming a transient pulsar-like state. It emits a striped wind if the swallowed magnetic dipole moment is inclined to the spin axis. We predict two types of transients from this scenario: (1) a fast radio burst emitted by the shocks as they expand to large radii and (2) an X/gamma-ray burst emitted by the $e^\pm$ outflow heated by magnetic dissipation.
Overtones and Nonlinearities in Binary Black Hole Ringdowns
Abstract
Using high-accuracy numerical relativity waveforms, we confirm the presence of numerous overtones of the \(\ell=2\), \(m=2\) quasinormal mode early in the ringdown of binary black hole mergers. We do this by demonstrating the stability of the mode amplitudes at different fit times, ruling out the possibility that a linear superposition of modes unphysically fits a highly nonlinear part of the waveform. We also find a number of previously unidentified subdominant second- order quasinormal modes in the $(2,2)$ mode. Even though these modes are mathematically nonlinear, they nevertheless confirm the validity of perturbation theory as a good approximation for describing much of the ringdown.
Echoes from Beyond: Detecting Gravitational Wave Quantum Imprints with LISA
Abstract
We assess the prospects for detecting gravitational wave echoes arising due to the quantum nature of black hole horizons with LISA. In a recent proposal, Bekenstein's black hole area quantization is connected to a discrete absorption spectrum for black holes in the context of gravitational radiation. Consequently, for incoming radiation at the black hole horizon, not all frequencies are absorbed, raising the possibility that the unabsorbed radiation is reflected, producing an echo-like signal closely following the binary coalescence waveform. In this work, we further develop this proposal by introducing a robust, phenomenologically motivated model for black hole reflectivity. Using this model, we calculate the resulting echoes for an ensemble of Numerical Relativity waveforms and examine their detectability with the LISA space-based interferometer. Our analysis demonstrates promising detection prospects and shows that, upon detection, LISA provides a direct probe of the Bekenstein-Hawking entropy. In addition, we find that the information extractable from LISA data offers valuable constraints on a wide range of quantum gravity theories.
Relieving scale disparity in binary black hole simulations
Abstract
Worldtube excision is a method of reducing computational burden in Numerical Relativity simulations of binary black holes in situations where there is a good analytical model of the geometry around (one or both of) the objects. Two such scenarios of relevance in gravitational-wave astronomy are (1) the case of mass-disparate systems, and (2) the early inspiral when the separation is still large. Here we illustrate the utility and flexibility of this technique with simulations of the fully self-consistent radiative evolution in the model problem of a scalar charge orbiting a Schwarzschild black hole under the effect of scalar-field radiation reaction. We explore a range of orbital configurations, including inspirals with large eccentricity (which we follow through to the final plunge and ringdown) and hyperbolic scattering.
Approximate helical symmetry in compact binaries
Abstract
The inspiral of a circular, non-precessing binary exhibits an approximate helical symmetry. The effects of eccentricity, precession, and radiation reaction break the exact symmetry. We estimate the failure of this symmetry using the flux of the BMS charge corresponding to helical symmetry carried away by gravitational waves. We analytically compute the helical flux for binaries moving on eccentric orbits and quasi-circular orbits without precession using post-Newtonian theory. The helical flux is non-vanishing at the 0PN order for eccentric orbits as expected. We analytically predict the helical flux to be at a relative 5PN order for quasi-circular non-precessing binaries. This prediction is compared with 113 quasi-circular non-precessing numerical relativity waveforms from the SXS catalog. We find good agreement between analytical and numerical results for quasi-circular non-precessing binaries establishing that helical symmetry starts to break at 5PN for these binaries.
Mass Transfer in Eccentric Black Hole–Neutron Star Mergers
Abstract
Black hole–neutron star binaries are of interest in many ways: they are intrinsically transient, radiate gravitational waves detectable by LIGO, and may produce γ-ray bursts. Although it has long been assumed that their late-stage orbital evolution is driven entirely by gravitational wave emission, we show here that in certain circumstances, mass transfer from the neutron star onto the black hole can both alter the binary's orbital evolution and significantly reduce the neutron star's mass: when the fraction of its mass transferred per orbit is ≳\(10^{−2}\), the neutron star's mass diminishes by order unity, leading to mergers in which the neutron star mass is exceptionally small. The mass transfer creates a gas disk around the black hole before merger that can be comparable in mass to the debris remaining after merger, i.e., ~0.1 \(M_{⊙}\). These processes are most important when the initial neutron star–black hole mass ratio q is in the range ≈0.2–0.8, the orbital semimajor axis is 40 ≲ \(a_{0}/r_{g}\) ≲ 300 (\(r_{g}\) ≡ \(GM_{BH}/c^{2}\)), and the eccentricity is large at \(e_{0}\) ≳ 0.8. Systems of this sort may be generated through the dynamical evolution of a triple system, as well as by other means.
Neutrinos in colliding neutron stars and black holes
Abstract
In this chapter, we provide an overview of the physics of colliding black holes and neutron stars and of the impact of neutrinos on these systems. Observations of colliding neutron stars play an important role in nuclear astrophysics today. They allow us to study the properties of cold nuclear matter and the origin of many heavy elements (gold, platinum, uranium). We show that neutrinos significantly impact the observable signals powered by these events as well as the outcome of nucleosynthesis in the matter that they eject into the surrounding intergalactic medium.
Parameter control for eccentric, precessing binary black hole simulations with SpEC
Abstract
Numerical relativity simulations of merging black holes provide the most accurate description of the binary dynamics and the emitted gravitational wave signal. However, practical considerations such as imperfect initial data and initial parameters mean that achieving target parameters, such as the orbital eccentricity or the black hole spin directions, at the beginning of the usable part of the simulation is challenging. In this paper, we devise a method to produce simulations with specific target parameters, namely the Keplerian orbital parameters—eccentricity, semimajor axis, mean anomaly—and the black hole spin vectors using SpEC. The method is an extension of the current process for achieving vanishing eccentricity and it is based on a parameter control loop that iteratively numerically evolves the system, fits the orbit with analytical post-Newtonian equations, and calculates updated input parameters. Through SpEC numerical simulations, we demonstrate and convergence for the orbital eccentricity and the spin directions respectively in iterations. These tests extend to binaries with mass ratios , eccentricities , and spin magnitudes . Our method for controlling the orbital and spin parameters of numerical simulations can be used to produce targeted simulations in sparsely covered regions of the parameter space or study the dynamics of relativistic binaries.
Simulating binary black hole mergers using discontinuous Galerkin methods
Class.Quant.Grav. 42, 035001 (2025)
Abstract
Binary black holes are the most abundant source of gravitational- wave observations. Gravitational-wave observatories in the next decade will require tremendous increases in the accuracy of numerical waveforms modeling binary black holes, compared to today’s state of the art. One approach to achieving the required accuracy is using spectral-type methods that scale to many processors. Using the SpECTRE numerical-relativity (NR) code, we present the first simulations of a binary black hole inspiral, merger, and ringdown using discontinuous Galerkin (DG) methods. The efficiency of DG methods allows us to evolve the binary through ∼ 18 orbits at reasonable computational cost. We then use SpECTRE’s Cauchy Characteristic Evolution (CCE) code to extract the gravitational waves at future null infinity. The open-source nature of SpECTRE means this is the first time a spectral-type method for simulating binary black hole evolutions is available to the entire NR community.
Einstein-Klein-Gordon system via Cauchy-characteristic evolution: Computation of memory and ringdown tail
Abstract
Cauchy-characteristic evolution (CCE) is a powerful method for accurately extracting gravitational waves at future null infinity. In this work, we extend the previously implemented CCE system within the numerical relativity code SpECTRE by incorporating a scalar field. This allows the system to capture features of beyond-general- relativity theories. We derive scalar contributions to the equations of motion, Weyl scalar computations, Bianchi identities, and balance laws at future null infinity. Our algorithm, tested across various scenarios, accurately reveals memory effects induced by both scalar and tensor fields and captures Price's power-law tail (\(u^{-l-2}\)) in scalar fields at future null infinity, in contrast to the \(t^{-2l-3}\) tail at future timelike infinity.
High-Precision Ringdown Surrogate Model for Non-Precessing Binary Black Holes
Abstract
Highly precise and robust waveform models are required as improvements in detector sensitivity enable us to test general relativity with more precision than ever before. In this work, we introduce a spin-aligned surrogate ringdown model. This ringdown surrogate, NRSur3dq8_RD, is built with numerical waveforms produced using Cauchy-characteristic evolution. In addition, these waveforms are in the superrest frame of the remnant black hole allowing us to do a correct analysis of the ringdown spectrum. The novel prediction of our surrogate model is complex-valued quasinormal mode (QNM) amplitudes, with median relative errors of \(10^{-2}-10^{-3}\) over the parameter space. Like previous remnant surrogates, we also predict the remnant black hole's mass and spin. The QNM mode amplitude errors translate into median errors on ringdown waveforms of \(10^{-4}\). The high accuracy and QNM mode content provided by our surrogate will enable high-precision ringdown analyses such as tests of general relativity. Our ringdown model is publicly available through the python package surfinBH.
Automated determination of the end time of junk radiation in binary black hole simulations
Abstract
When numerically solving Einstein's equations for the evolution of binary black holes, physical imperfections in the initial data manifest as a transient, high-frequency pulse of ''junk radiation.'' This unphysical signal must be removed before the waveform can be used. Improvements in the efficiency of numerical simulations now allow waveform catalogs containing thousands of waveforms to be produced. Thus, an automated procedure for identifying junk radiation is required. To this end, we present a new algorithm based on the empirical mode decomposition (EMD) from the Hilbert-Huang transform. This approach allows us to isolate and measure the high- frequency oscillations present in the measured irreducible masses of the black holes. The decay of these oscillations allows us to estimate the time from which the junk radiation can be ignored. To make this procedure more precise, we propose three distinct threshold criteria that specify how small the contribution of junk radiation has to be before it can be considered negligible. We apply this algorithm to 3403 BBH simulations from the SXS catalog to find appropriate values for the thresholds in the three criteria. We find that this approach yields reliable decay time estimates, i.e., when to consider the simulation physical, for over 98.6% of the simulations studied. This demonstrates the efficacy of the EMD as a suitable tool to automatically isolate and characterize junk radiation in the simulation of binary black hole systems.
Gravitational wave surrogate model for spinning, intermediate mass ratio binaries based on perturbation theory and numerical relativity
Abstract
We present BHPTNRSur2dq1e3, a reduced order surrogate model of gravitational waves emitted from binary black hole (BBH) systems in the comparable to large mass ratio regime with aligned spin () on the heavier mass (). We trained this model on waveform data generated from point particle black hole perturbation theory (ppBHPT) with mass ratios varying from and spins from . The waveforms are 13,500 long and include all spin-weighted spherical harmonic modes except the (4,1) and modes. We find that, for binaries with , retrograde quasinormal modes are significantly excited, thereby complicating the modeling process. To overcome this issue, we introduce a domain decomposition approach to model the inspiral and merger-ringdown portion of the signal separately. The resulting model can faithfully reproduce ppBHPT waveforms with a median time- domain mismatch error of . We then calibrate our model with numerical relativity (NR) data in the comparable mass regime (). By comparing with spin-aligned BBH NR simulations at , we find that the dominant quadrupolar (subdominant) modes agree to better than () when using a time-domain mismatch error, where the largest source of calibration error comes from the transition-to-plunge and ringdown approximations of perturbation theory. Mismatch errors are below for systems with mass ratios between and typically get smaller at larger mass ratio. Our two models—both the ppBHPT waveform model and the NR-calibrated ppBHPT model—will be publicly available through gwsurrogate and the black hole perturbation toolkit packages.
Robustness of neutron star merger simulations to changes in neutrino transport and neutrino-matter interactions
Abstract
Binary neutron star mergers play an important role in nuclear astrophysics: Their gravitational wave and electromagnetic signals carry information about the equation of state of cold matter above nuclear saturation density, and they may be one of the main sources of r-process elements in the Universe. Neutrino-matter interactions during and after merger impact the properties of these electromagnetic signals and the relative abundances of the produced r-process elements. Existing merger simulations are, however, limited in their ability to realistically model neutrino transport and neutrino-matter interactions. Here, we perform a comparison of the impact of the use of state-of-the art two-moment or Monte Carlo transport schemes on the outcome of merger simulations, for a single binary neutron star system with a short-lived neutron star remnant [(5–10) ms]. We also investigate the use of different reaction rates in the simulations. While the best transport schemes generally agree well on the qualitative impact of neutrinos on the system, differences in the behavior of the high-density regions can significantly impact the collapse time and the properties of the hot tidal arms in this metastable merger remnant. The chosen interaction rates and transport algorithm as well as recent improvements by Radice et al. to the two-moment algorithms can all contribute to changes at the (10–30)% level in the global properties of the merger remnant and outflows. The limitations of previous moment schemes fixed by Radice et al. also appear sufficient to explain the large difference that we observed in the production of heavy-lepton neutrinos in a previous comparison of Monte Carlo and moment schemes in the context of a low-mass binary neutron star system.
Binary neutron star mergers using a discontinuous Galerkin-finite difference hybrid method
Class.Quant.Grav. 41, 245002 (2024)
Abstract
We present a discontinuous Galerkin-finite difference hybrid scheme that allows high-order shock capturing with the discontinuous Galerkin method for general relativistic magnetohydrodynamics in dynamical spacetimes. We present several optimizations and stability improvements to our algorithm that allow the hybrid method to successfully simulate single, rotating, and binary neutron stars. The hybrid method achieves the efficiency of discontinuous Galerkin methods throughout almost the entire spacetime during the inspiral phase, while being able to robustly capture shocks and resolve the stellar surfaces. We also use Cauchy-characteristic evolution to compute the first gravitational waveforms at future null infinity from binary neutron star mergers. The simulations presented here are the first successful binary neutron star inspiral and merger simulations using discontinuous Galerkin methods.
General-relativistic gauge-invariant magnetic helicity transport: Basic formulation and application to neutron star mergers
Abstract
Dynamo processes are ubiquitous in astrophysical systems. In relativistic astrophysical systems, such as accretion disks around black holes or neutron stars, they may critically affect the launching of winds and jets that can power electromagnetic emission. Dynamo processes are governed by several microscopic parameters, one of them being magnetic helicity. As a conserved quantity in nonresistive plasmas, magnetic helicity is transported across the system. One important implication of helicity conservation is, that in the absence of helicity fluxes some mean-field dynamos can be quenched, potentially affecting the large-scale evolution of the magnetic field. One of the major challenges in computing magnetic helicity is the need to fix a meaningful electromagnetic gauge. We here present a fully covariant formulation of magnetic helicity transport in general-relativistic plasmas based on the concept of relative helicity by Berger and Field and Finn and Antonsen. This formulation is separately invariant under gauge-transformation of the Maxwell and Einstein equations. As an application of this new formalism we present the first analysis of magnetic helicity transport in the merger of two neutron stars. We demonstrate the presence of global helicity fluxes into the outer layers of the stellar merger remnant, which may impact subsequent large-scale dynamo amplification in these regions.
A review of gravitational memory and BMS frame fixing in numerical relativity
Class.Quant.Grav. 41, 223001 (2024)
Abstract
Gravitational memory effects and the BMS freedoms exhibited at future null infinity have recently been resolved and utilized in numerical relativity simulations. With this, gravitational wave models and our understanding of the fundamental nature of general relativity have been vastly improved. In this paper, we review the history and intuition behind memory effects and BMS symmetries, how they manifest in gravitational waves, and how controlling the infinite number of BMS freedoms of numerical relativity simulations can crucially improve the waveform models that are used by gravitational wave detectors. We reiterate the fact that, with memory effects and BMS symmetries, not only can these next- generation numerical waveforms be used to observe never-before-seen physics, but they can also be used to test GR and learn new astrophysical information about our Universe.
Improved frequency spectra of gravitational waves with memory in a binary-black-hole simulation
Abstract
Numerical relativists can now produce gravitational waveforms with memory effects routinely and accurately. The gravitational-wave memory effect contains very low-frequency components, including a persistent offset. The presence of these components violates basic assumptions about time-shift behavior underpinning standard data- analysis techniques in gravitational-wave astronomy. This poses a challenge to the analysis of waveform spectra: how to preserve the low-frequency characteristics when transforming a time-domain waveform to the frequency domain. To tackle this challenge, we revisit the preprocessing procedures applied to the waveforms that contain memory effects. We find inconsistency between the zero- frequency limit of displacement memory and the low-frequency spectrum of the same memory preprocessed using the common scheme in literature. To resolve the inconsistency, we propose a new robust preprocessing scheme that produces the spectra of memory waveforms more faithfully. Using this new scheme, we inspect several characteristics of the spectrum of a memory waveform. In particular, we find a discernible beating pattern formed by the dominant oscillatory mode and the displacement memory. This pattern is absent in the spectrum of a waveform without memory. The difference between the memory and no-memory waveforms is too small to be observed by current-generation detectors in a single binary-black-hole event. Detecting the memory in a single event is likely to occur in the era of next-generation detectors.
Imprints of changing mass and spin on black hole ringdown
Abstract
We numerically investigate the imprints of gravitational radiation- reaction driven changes to a black hole’s mass and spin on the corresponding ringdown waveform. We do so by comparing the dynamics of a perturbed black hole evolved with the full (nonlinear) versus linearized Einstein equations. As expected, we find that the quasinormal mode amplitudes extracted from nonlinear evolution deviate from their linear counterparts at third order in initial perturbation amplitude. For perturbations leading to a change in the black hole mass and spin of , which is reasonable for a remnant formed in an astrophysical merger, we find that nonlinear distortions to the complex amplitudes of some quasinormal modes can be as large as at the peak of the waveform. Furthermore, the change in the mass and spin results in a drift in the quasinormal mode frequencies, which for large amplitude perturbations causes the nonlinear waveform to rapidly dephase with respect to its linear counterpart. Surprisingly, despite these nonlinear effects creating significant deviations in the nonlinear waveform, we show that a linear quasinormal mode model still performs quite well from close to the peak amplitude onward. Comparing the quality of quasinormal mode fits for the linear and nonlinear waveforms, we show that the main obstruction to measuring high- overtones is the transient part of the waveform, already present at the linear level.
General relativistic force-free electrodynamics with a discontinuous Galerkin-finite difference hybrid method
Abstract
Relativistic plasmas around compact objects can sometimes be approximated as being force-free. In this limit, the plasma inertia is negligible and the overall dynamics is governed by global electric currents. We present a novel numerical approach for simulating such force-free plasmas, which allows for high accuracy in smooth regions as well as capturing dissipation in current sheets. Using a high-order accurate discontinuous Galerkin method augmented with a conservative finite-difference method, we demonstrate efficient global simulations of black hole and neutron star magnetospheres. In addition to a series of challenging test problems, we show that our approach can—depending on the physical properties of the system and the numerical implementation—be up to \(10x\) more efficient than conventional simulations, with a speedup of \(2-3x\) for most problems we consider in practice.
Optimizing post-Newtonian parameters and fixing the BMS frame for numerical-relativity waveform hybridizations
Abstract
Numerical relativity (NR) simulations of binary black holes provide precise waveforms, but are typically too computationally expensive to produce waveforms with enough orbits to cover the whole frequency band of gravitational-wave observatories. Accordingly, it is important to be able to hybridize NR waveforms with analytic, post- Newtonian (PN) waveforms, which are accurate during the early inspiral phase. We show that to build such hybrids, it is crucial to both fix the Bondi-Metzner-Sachs (BMS) frame of the NR waveforms to match that of PN theory, and optimize over the PN parameters to mitigate the error caused by the discrepancy between NR and PN parameters. We test such a hybridization procedure including all spin-weighted spherical harmonic modes with for , using 29 NR waveforms with mass ratios and spin magnitudes . We find that for spin-aligned systems, the PN and NR waveforms agree very well. The difference is limited by the small nonzero orbital eccentricity of the NR waveforms, or equivalently by the lack of eccentric terms in the PN waveforms. To maintain full accuracy of the simulations, the matching window for spin-aligned systems should be at least 5 orbits long and end at least 15 orbits before merger. For precessing systems, the errors are larger than for spin-aligned cases. The errors are likely limited by the absence of precession- related spin-spin PN terms. Using long NR waveforms, we find that there is no optimal choice of the matching window within this time span, because the hybridization result for precessing cases is always better if using earlier or longer matching windows. We provide the mean orbital frequency of the smallest acceptable matching window as a function of the target error between the PN and NR waveforms and the black hole spins.
Scalarization of isolated black holes in scalar Gauss-Bonnet theory in the fixing-the-equations approach
Abstract
One of the most promising avenues to perform numerical evolutions in theories beyond general relativity is the fixing-the-equations approach, a proposal in which new “driver” equations are added to the evolution equations in a way that allows for stable numerical evolutions. In this direction, we extend the numerical relativity code spectre to evolve a “fixed” version of scalar Gauss-Bonnet theory in the decoupling limit, a phenomenologically interesting theory that allows for hairy black hole solutions in vacuum. We focus on isolated black hole systems both with and without linear and angular momentum, and propose a new driver equation to improve the recovery of such stationary solutions. We demonstrate the effectiveness of the latter by numerically evolving black holes that undergo spontaneous scalarization using different driver equations. Finally, we evaluate the accuracy of the obtained solutions by comparing with the original unaltered theory.
Stability of hypermassive neutron stars with realistic rotation and entropy profiles
Abstract
Binary neutron star mergers produce massive, hot, rapidly differentially rotating neutron star remnants; electromagnetic and gravitational wave signals associated with the subsequent evolution depend on the stability of these remnants. Stability of relativistic stars has previously been studied for uniform rotation and for a class of differential rotation with monotonic angular velocity profiles. Stability of those equilibria to axisymmetric perturbations was found to respect a turning point criterion: along a constant angular momentum sequence, the onset of unstable stars is found at maximum density less than but close to the density of maximum mass. In this paper, we test this turning point criterion for nonmonotonic angular velocity profiles and nonisentropic entropy profiles, both chosen to more realistically model postmerger equilibria. Stability is assessed by evolving perturbed equilibria in 2D using the spectral einstein Code. We present tests of the code’s new capability for axisymmetric metric evolution. We confirm the turning point theorem and determine the region of our rotation law parameter space that provides highest maximum mass for a given angular momentum.
High angular momentum hot differentially rotating equilibrium star evolutions in conformally flat spacetime
Abstract
The conformal flatness approximation to the Einstein equations has been successfully used in many astrophysical applications such as initial data constructions and dynamical simulations. Although it has been shown that full general relativistic strongly differentially rotating equilibrium models deviate by at most a few percentage from their conformally flat counterparts, whether those conformally flat solutions remain stable has not been fully addressed. To further understand the limitations of the conformal flatness approximation, in this work, we construct spatially conformally flat hot hypermassive neutron stars with postmergerlike rotation laws, and perform conformally flat evolutions and analysis over dynamical timescales. We find that enforcing conformally flat spacetime could change the equilibrium of quasitoroidal models with high angular momentum for J≳9GM⊙²/c compared to fully general relativistic cases. In contrast, all the quasispherical models considered in this work remain stable even with high angular momentum J=9GM⊙²/c. Our investigation suggests that the quasispherical models are suitable initial data for long-lived hypermassive neutron star modeling in conformally flat spacetime.
Numerical relativity multimodal waveforms using absorbing boundary conditions
Class.Quant.Grav. 41, 175011 (2024)
Abstract
Errors due to imperfect boundary conditions in numerical relativity simulations of binary black holes (BBHs) can produce unphysical reflections of gravitational waves which compromise the accuracy of waveform predictions, especially for subdominant modes. A system of higher order absorbing boundary conditions which greatly reduces this problem was introduced in earlier work (Buchman and Sarbach 2006 Class. Quantum Grav. 23 6709). In this paper, we devise two new implementations of this boundary condition system in the Spectral Einstein Code (SpEC), and test them in both linear multipolar gravitational wave and inspiralling mass ratio 7:1 BBH simulations. One of our implementations in particular is shown to be extremely robust and to produce accuracy superior to the standard freezing-Ψ\(_{0}\) boundary condition usually used by SpEC.
Toward a self-consistent framework for measuring black hole ringdowns
Abstract
The ringdown portion of a binary black hole merger consists of a sum of modes, each containing an infinite number of tones that are exponentially damped sinusoids. In principle, these can be measured as gravitational-waves with observatories like LIGO/Virgo/KAGRA, however in practice it is unclear how many tones can be meaningfully resolved. We investigate the consistency and resolvability of the overtones of the quadrupolar ℓ=m=2 mode by starting at late times when the gravitational waveform is expected to be well approximated by the ℓmn=220 tone alone. We present a Bayesian inference framework to measure the tones in numerical relativity data. We measure tones at different start times, checking for consistency: we classify a tone as stably recovered if and only if the 95% credible intervals for amplitude and phase at time t overlap with the credible intervals at all subsequent times. We test a set of tones including the first four overtones of the fundamental mode and the 320 tone and find that the 220 and 221 tones can be measured consistently with the inclusion of additional overtones. The 222 tone measurements can be stabilized when we include the 223 tone, but only in a narrow time window, after which it is too weak to measure. The 223 tone recovery appears to be unstable, and does not become stable with the introduction of the 224 tone. We find that N=3 tones can be stably recovered simultaneously. However, when analyzing N≥4 tones, the amplitude of one tone is consistent with zero. Thus, within our framework, one can identify only N=3 tones with nonzero amplitude that are simultaneously stable.
Nonlinear effects in black hole ringdown from scattering experiments: Spin and initial data dependence of quadratic mode coupling
Abstract
We investigate quadratic quasinormal mode coupling in black hole spacetime through numerical experiments of single perturbed black holes using both numerical relativity and second-order black hole perturbation theory. Focusing on the dominant quadrupolar modes, we find good agreement (within ) between these approaches, with discrepancies attributed to truncation error and uncertainties from mode fitting. Our results align with earlier studies extracting the coupling coefficients from select binary black hole merger simulations, showing consistency for the same remnant spins. Notably, the coupling coefficient is insensitive to a diverse range of initial data, including configurations that led to a significant (up to 5%) increase in the remnant black hole mass. These findings present opportunities for testing the nonlinear dynamics of general relativity with ground-based gravitational wave observatories. Lastly, we provide evidence of a bifurcation in coupling coefficients between counterrotating and corotating quasinormal modes as black hole spin increases.
#2023
Black Hole Spectroscopy for Precessing Binary Black Hole Coalescences
Abstract
The spectroscopic study of black hole quasinormal modes in gravitational-wave ringdown observations is hindered by our ignorance of which modes should dominate astrophysical signals for different binary configurations, limiting tests of general relativity and astrophysics. In this work, we present a description of the quasinormal modes that are excited in the ringdowns of comparable mass, quasi-circular precessing binary black hole coalescences -- a key region of parameter space that has yet to be fully explored within the framework of black hole spectroscopy. We suggest that the remnant perturbation for precessing and non- precessing systems is approximately the same up to a rotation, which implies that the relative amplitudes of the quasinormal modes in both systems are also related by a rotation. We present evidence for this by analyzing an extensive catalog of numerical relativity simulations. Additional structure in the amplitudes is connected to the system's kick velocity and other asymmetries in the orbital dynamics. We find that the ringdowns of precessing systems need not be dominated by the \({(\ell,m)=(2,\pm 2)}\) quasinormal modes, and that instead the \((2,\pm 1)\) or \((2,0)\) quasinormal modes can dominate. Our results are consistent with a ringdown analysis of the LIGO-Virgo gravitational wave signal GW190521, and may also help in understanding phenomenological inspiral-merger-ringdown waveform model systematics.
Neutrino fast flavor oscillations with moments: Linear stability analysis and application to neutron star mergers
Abstract
Providing an accurate modeling of neutrino physics in dense astrophysical environments such as binary neutron star mergers presents a challenge for hydrodynamic simulations. Nevertheless, understanding how flavor transformation can occur and affect the dynamics, the mass ejection, and the nucleosynthesis will need to be achieved in the future. Computationally expensive, large-scale simulations frequently evolve the first classical angular moments of the neutrino distributions. By promoting these quantities to matrices in flavor space, we develop a linear stability analysis of fast flavor oscillations using only the first two “quantum” moments, which notably requires generalizing the classical closure relations that appropriately truncate the hierarchy of moment equations in order to treat quantum flavor coherence. After showing the efficiency of this method on a well-understood test situation, we perform a systematic search of the occurrence of fast flavor instabilities in a neutron star merger simulation. We discuss the successes and shortcomings of moment linear stability analysis, as this framework provides a time-efficient way to design and study better closure prescriptions in the future.
Fully relativistic three-dimensional Cauchy-characteristic matching for physical degrees of freedom
Abstract
A fully relativistic three-dimensional Cauchy-characteristic matching (CCM) algorithm is implemented for physical degrees of freedom in a numerical relativity code spectre. The method is free of approximations and can be applied to any physical system. We test the algorithm with various scenarios involving smooth data, including the propagation of Teukolsky waves within a flat background, the perturbation of a Kerr black hole with a Teukolsky wave, and the injection of a gravitational-wave pulse from the characteristic grid. Our investigations reveal no numerical instabilities in the simulations. In addition, the tests indicate that the CCM algorithm effectively directs characteristic information into the inner Cauchy system, yielding higher precision in waveforms and smaller violations of Bondi-gauge constraints, especially when the outer boundary of the Cauchy evolution is at a smaller radius.
Extending black-hole remnant surrogate models to extreme mass ratios
Abstract
Numerical-relativity surrogate models for both black-hole merger waveforms and remnants have emerged as important tools in gravitational-wave astronomy. While producing very accurate predictions, their applicability is limited to the region of the parameter space where numerical-relativity simulations are available and computationally feasible. Notably, this excludes extreme mass ratios. We present a machine-learning approach to extend the validity of existing and future numerical-relativity surrogate models toward the test-particle limit, targeting in particular the mass and spin of postmerger black-hole remnants. Our model is trained on both numerical-relativity simulations at comparable masses and analytical predictions at extreme mass ratios. We extend the gaussian-process-regression model NRSur7dq4Remnant, validate its performance via cross validation, and test its accuracy against additional numerical-relativity runs. Our fit, which we dub NRSur7dq4EmriRemnant, reaches an accuracy that is comparable to or higher than that of existing remnant models while providing robust predictions for arbitrary mass ratios.
Gravitational waves from binary neutron star mergers with a spectral equation of state
Abstract
In numerical simulations of binary neutron star systems, the equation of state of the dense neutron star matter is an important factor in determining both the physical realism and the numerical accuracy of the simulations. Some equations of state used in simulations are or smoother in the pressure/density relationship function, such as a polytropic equation of state, but may not have the flexibility to model stars or remnants of different masses while keeping their radii within known astrophysical constraints. Other equations of state, such as tabular or piecewise polytropic, may be flexible enough to model additional physics and multiple stars’ masses and radii within known constraints, but are not as smooth, resulting in additional numerical error. We will study in this paper a recently developed family of equation of state, using a spectral expansion with sufficient free parameters to allow for a larger flexibility than current polytropic equations of state, and with sufficient smoothness to reduce numerical errors compared to tabulated or piecewise polytropic equations of state. We perform simulations at three mass ratios with a common chirp mass, using two distinct spectral equations of state, and at multiple numerical resolutions. We evaluate the gravitational waves produced from these simulations, comparing the phase error between resolutions and equations of state, as well as with respect to analytical models. From our simulations, we estimate that the phase difference at the merger for binaries with a dimensionless weighted tidal deformability difference greater than can be captured by the spectral Einstein code for these equations of state.
Large-scale Evolution of Seconds-long Relativistic Jets from Black Hole–Neutron Star Mergers
Astrophys.J.Lett. 954, L21 (2023)
Abstract
We present the first numerical simulations that track the evolution of a black hole–neutron star (BH–NS) merger from premerger to r ≳ 10\(^{11}\) cm. The disk that forms after a merger of mass ratio q = 2 ejects massive disk winds (3–5 × 10\(^{−2}\)M\(_{⊙}\)). We introduce various postmerger magnetic configurations and find that initial poloidal fields lead to jet launching shortly after the merger. The jet maintains a constant power due to the constancy of the large- scale BH magnetic flux until the disk becomes magnetically arrested (MAD), where the jet power falls off as L\(_{j}\) ∼ t\(^{−2}\). All jets inevitably exhibit either excessive luminosity due to rapid MAD activation when the accretion rate is high or excessive duration due to delayed MAD activation compared to typical short gamma-ray bursts (sGRBs). This provides a natural explanation for long sGRBs such as GRB 211211A but also raises a fundamental challenge to our understanding of jet formation in binary mergers. One possible implication is the necessity of higher binary mass ratios or moderate BH spins to launch typical sGRB jets. For postmerger disks with a toroidal magnetic field, dynamo processes delay jet launching such that the jets break out of the disk winds after several seconds. We show for the first time that sGRB jets with initial magnetization σ\(_{0}\) > 100 retain significant magnetization (σ ≫ 1) at r > 10\(^{10}\) cm, emphasizing the importance of magnetic processes in the prompt emission. The jet–wind interaction leads to a power-law angular energy distribution by inflating an energetic cocoon whose emission is studied in a companion paper.
A positivity-preserving adaptive-order finite-difference scheme for GRMHD
Class.Quant.Grav. 40, 245014 (2023)
Abstract
We present an adaptive-order positivity-preserving conservative finite-difference scheme that allows a high-order solution away from shocks and discontinuities while guaranteeing positivity and robustness at discontinuities. This is achieved by monitoring the relative power in the highest mode of the reconstructed polynomial and reducing the order when the polynomial series no longer converges. Our approach is similar to the multidimensional optimal order detection strategy, but differs in several ways. The approach is a priori and so does not require retaking a time step. It can also readily be combined with positivity-preserving flux limiters that have gained significant traction in computational astrophysics and numerical relativity. This combination ultimately guarantees a physical solution both during reconstruction and time stepping. We demonstrate the capabilities of the method using a standard suite of very challenging 1d, 2d, and 3d general relativistic magnetohydrodynamics test problems.
Numerical relativity surrogate model with memory effects and post-Newtonian hybridization
Abstract
Numerical relativity simulations provide the most precise templates for the gravitational waves produced by binary black hole mergers. However, many of these simulations use an incomplete waveform extraction technique—extrapolation—that fails to capture important physics, such as gravitational memory effects. Cauchy-characteristic evolution (CCE), by contrast, is a much more physically accurate extraction procedure that fully evolves Einstein’s equations to future null infinity and accurately captures the expected physics. In this work, we present a new surrogate model, NRHybSur3dq8_CCE, built from CCE waveforms that have been mapped to the post-Newtonian (PN) BMS frame and then hybridized with PN and effective one-body (EOB) waveforms. This model is trained on 102 waveforms with mass ratios q≤8 and aligned spins χ1z,χ2z∈[-0.8,0.8]. The model spans the entire LIGO-Virgo-KAGRA (LVK) frequency band (with flow=20Hz) for total masses M≳2.25M⊙ and includes the ℓ≤4 and (ℓ,m)=(5,5) spin-weight -2 spherical harmonic modes, but not the (3, 1), (4, 2) or (4, 1) modes. We find that NRHybSur3dq8_CCE can accurately reproduce the training waveforms with mismatches ≲2×10-4 for total masses 2.25M⊙≤M≤300M⊙ and can, for a modest degree of extrapolation, capably model outside of its training region. Most importantly, unlike previous waveform models, the new surrogate model successfully captures memory effects.
Characterizing the Directionality of Gravitational Wave Emission from Matter Motions within Core-collapse Supernovae
Abstract
We analyze the directional dependence of the gravitational wave (GW) emission from 15 3D neutrino radiation hydrodynamic simulations of core-collapse supernovae (CCSNe). Using spin weighted spherical harmonics, we develop a new analytic technique to quantify the evolution of the distribution of GW emission over all angles. We construct a physics-informed toy model that can be used to approximate GW distributions for general ellipsoid-like systems, and use it to provide closed form expressions for the distribution of GWs for different CCSN phases. Using these toy models, we approximate the protoneutron star (PNS) dynamics during multiple CCSN stages and obtain similar GW distributions to simulation outputs. When considering all viewing angles, we apply this new technique to quantify the evolution of preferred directions of GW emission. For nonrotating cases, this dominant viewing angle drifts isotropically throughout the supernova, set by the dynamical timescale of the PNS. For rotating cases, during core bounce and the following tens of milliseconds, the strongest GW signal is observed along the equator. During the accretion phase, comparable—if not stronger—GW amplitudes are generated along the axis of rotation, which can be enhanced by the low T/∣W∣ instability. We show two dominant factors influencing the directionality of GW emission are the degree of initial rotation and explosion morphology. Lastly, looking forward, we note the sensitive interplay between GW detector site and supernova orientation, along with its effect on detecting individual polarization modes.
Numerical simulations of black hole-neutron star mergers in scalar-tensor gravity
Abstract
We present a numerical-relativity simulation of a black hole-neutron star merger in scalar-tensor (ST) gravity with binary parameters consistent with the gravitational wave event GW200115. In this exploratory simulation, we consider the Damour-Esposito-Farèse extension to Brans-Dicke theory, and maximize the effect of spontaneous scalarization by choosing a soft equation of state and ST theory parameters at the edge of known constraints. We extrapolate the gravitational waves, including tensor and scalar (breathing) modes, to future null-infinity. The numerical waveforms undergo wave cycles before the merger, and are in good agreement with predictions from post-Newtonian theory during the inspiral. We find the ST system evolves faster than its general-relativity (GR) counterpart due to dipole radiation, merging a full gravitational-wave cycle before the GR counterpart. This enables easy differentiation between the ST waveforms and GR in the context of parameter estimation. However, we find that dipole radiation’s effect may be partially degenerate with the NS tidal deformability during the late inspiral stage, and a full Bayesian analysis is necessary to fully understand the degeneracies between ST and binary parameters in GR.
Worldtube excision method for intermediate-mass-ratio inspirals: Scalar-field model in 3+1 dimensions
Abstract
Binary black hole simulations become increasingly more computationally expensive with smaller mass ratios, partly because of the longer evolution time, and partly because the lengthscale disparity dictates smaller time steps. The program initiated by Dhesi et al. [Phys. Rev. D 104, 124002 (2021)] explores a method for alleviating the scale disparity in simulations with mass ratios in the intermediate astrophysical range (\(10^{-4} \leq q \leq 10^{-2}\)), where purely perturbative methods may not be adequate. A region (“worldtube”) much larger than the small black hole is excised from the numerical domain, and replaced with an analytical model approximating a tidally deformed black hole. Here we apply this idea to a toy model of a scalar charge in a fixed circular geodesic orbit around a Schwarzschild black hole, solving for the massless Klein-Gordon field. This is a first implementation of the worldtube excision method in full \(3+1\) dimensions. We demonstrate the accuracy and efficiency of the method, and discuss the steps toward applying it for evolving orbits and, ultimately, in the binary black-hole scenario. Our implementation is publicly accessible in the spectre numerical relativity code.
Laying the foundation of the effective-one-body waveform models SEOBNRv5: Improved accuracy and efficiency for spinning nonprecessing binary black holes
Abstract
We present SEOBNRv5HM, a more accurate and faster inspiral-merger- ringdown gravitational waveform model for quasicircular, spinning, nonprecessing binary black holes within the effective-one-body (EOB) formalism. Compared to its predecessor, SEOBNRv4HM, the waveform model (i) incorporates recent high-order post-Newtonian results in the inspiral, with improved resummations, (ii) includes the gravitational modes , in addition to the (2,2), (3,3), (2,1), (4,4), (5,5) modes already implemented in SEOBNRv4HM, (iii) is calibrated to larger mass ratios and spins using a catalog of 442 numerical-relativity (NR) simulations and 13 additional waveforms from black-hole perturbation theory, and (iv) incorporates information from second-order gravitational self-force in the nonspinning modes and radiation-reaction force. Computing the unfaithfulness against NR simulations, we find that for the dominant (2,2) mode the maximum unfaithfulness in the total mass range is below for 90% of the cases (38% for SEOBNRv4HM). When including all modes up to we find 98% (49%) of the cases with unfaithfulness below (), while these numbers reduce to 88% (5%) when using SEOBNRv4HM. Furthermore, the model shows improved agreement with NR in other dynamical quantities (e.g., the angular momentum flux and binding energy), providing a powerful check of its physical robustness. We implemented the waveform model in a high-performance python package (pyseobnr), which leads to evaluation times faster than SEOBNRv4HM by a factor of 10 to 50, depending on the configuration, and provides the flexibility to easily include spin-precession and eccentric effects, thus making it the starting point for a new generation of EOBNR waveform models (SEOBNRv5) to be employed for upcoming observing runs of the LIGO- Virgo-KAGRA detectors.
Simulating neutron stars with a flexible enthalpy-based equation of state parametrization in spectre
Abstract
Numerical simulations of neutron star mergers represent an essential step toward interpreting the full complexity of multimessenger observations and constraining the properties of supranuclear matter. Currently, simulations are limited by an array of factors, including computational performance and input physics uncertainties, such as the neutron star equation of state. In this work, we expand the range of nuclear phenomenology efficiently available to simulations by introducing a new analytic parametrization of cold, beta- equilibrated matter that is based on the relativistic enthalpy. We show that the new enthalpy parametrization can capture a range of nuclear behavior, including strong phase transitions. We implement the enthalpy parametrization in the spectre code, simulate isolated neutron stars, and compare performance to the commonly used spectral and polytropic parametrizations. We find comparable computational performance for nuclear models that are well represented by either parametrization, such as simple hadronic equations of state. We show that the enthalpy parametrization further allows us to simulate more complicated hadronic models or models with phase transitions that are inaccessible to current parametrizations.
#2022
General relativistic simulations of collapsing binary neutron star mergers with Monte Carlo neutrino transport
Abstract
Recent gravitational wave observations of neutron-star-neutron-star and neutron-star-black-hole binaries appear to indicate that massive neutron stars may not be too uncommon in merging systems. These discoveries have led to an increased interest in the simulation of merging compact binaries involving massive stars. In this paper, we present a first set of evolution of massive neutron star binaries using Monte Carlo radiation transport for the evolution of neutrinos. We study a range of systems, from nearly symmetric binaries that collapse to a black hole before forming a disk or ejecting material, to more asymmetric binaries in which tidal disruption of the lower mass star leads to the production of more interesting postmerger remnants. For the latter type of systems, we additionally study the impact of viscosity on the properties of the outflows, and compare our results to two recent simulations of identical binaries performed with the whiskythc code. We find agreement on the black hole properties, disk mass, and mass and velocity of the outflows within expected numerical uncertainties, and some minor but noticeable differences in the evolution of the electron fraction when using a subgrid viscosity model, with viscosity playing a more minor role in our simulations. The method used to account for r-process heating in the determination of the outflow properties appears to have a larger impact on our result than those differences between numerical codes. We also use the simulation with the most ejected material to verify that our newly implemented Lagrangian tracers provide a reasonable sampling of the matter outflows as they leave the computational grid. We note that, given the lack of production of hot outflows in these mergers, the main role of neutrinos in these systems is to set the composition of the postmerger remnant. One of the main potential uses of our simulations is, thus, as improved initial conditions for longer evolutions of such remnants.
Eccentric binary black holes: Comparing numerical relativity and small mass-ratio perturbation theory
Abstract
The modeling of unequal mass binary black hole systems is of high importance to detect and estimate parameters from these systems. Numerical relativity (NR) is well suited to study systems with comparable component masses, m1∼m2, whereas small mass ratio (SMR) perturbation theory applies to binaries where q=m2/m1≪1. This work investigates the applicability for NR and SMR as a function of mass ratio for eccentric nonspinning binary black holes. We produce 52 NR simulations with mass ratios between 1:10 and 1:1 and initial eccentricities up to 0.8. From these we extract quantities like gravitational wave energy and angular momentum fluxes and periastron advance, and assess their accuracy. To facilitate comparison, we develop tools to map between NR and SMR inspiral evolutions of eccentric binary black holes. We derive post-Newtonian accurate relations between different definitions of eccentricity. Based on these analyses, we introduce a new definition of eccentricity based on the (2,2)-mode of the gravitational radiation, which reduces to the Newtonian definition of eccentricity in the Newtonian limit. From the comparison between NR simulations and SMR results, we quantify the unknown next-to-leading order SMR contributions to the gravitational energy and angular momentum fluxes, and periastron advance. We show that in the comparable mass regime these contributions are subdominant and higher order SMR contributions are negligible.
Nonlinearities in Black Hole Ringdowns
Phys.Rev.Lett. 130, 081402 (2023)
Abstract
The gravitational wave strain emitted by a perturbed black hole (BH) ringing down is typically modeled analytically using first-order BH perturbation theory. In this Letter, we show that second-order effects are necessary for modeling ringdowns from BH merger simulations. Focusing on the strain’s angular harmonic, we show the presence of a quadratic effect across a range of binary BH mass ratios that agrees with theoretical expectations. We find that the quadratic mode’s amplitude exhibits quadratic scaling with the fundamental mode—its parent mode. The nonlinear mode’s amplitude is comparable to or even larger than that of the linear mode. Therefore, correctly modeling the ringdown of higher harmonics—improving mode mismatches by up to 2 orders of magnitude—requires the inclusion of nonlinear effects.
Fixing the BMS frame of numerical relativity waveforms with BMS charges
Abstract
The Bondi-van der Burg-Metzner-Sachs (BMS) group, which uniquely describes the symmetries of asymptotic infinity and therefore of the gravitational waves that propagate there, has become increasingly important for accurate modeling of waveforms. In particular, waveform models, such as post-Newtonian (PN) expressions, numerical relativity (NR), and black hole perturbation theory, produce results that are in different BMS frames. Consequently, to build a model for the waveforms produced during the merging of compact objects, which ideally would be a hybridization of PN, NR, and black hole perturbation theory, one needs a fast and robust method for fixing the BMS freedoms. In this work, we present the first means of fixing the entire BMS freedom of NR waveforms to match the frame of either PN waveforms or black hole perturbation theory. We achieve this by finding the BMS transformations that change certain charges in a prescribed way—e.g., finding the center-of-mass transformation that maps the center-of-mass charge to a mean of zero. We find that this new method is 20 times faster, and more correct when mapping to the superrest frame, than previous methods that relied on optimization algorithms. Furthermore, in the course of developing this charge- based frame fixing method, we compute the PN expression for the Moreschi supermomentum to 3PN order without spins and 2PN order with spins. This Moreschi supermomentum is effectively equivalent to the energy flux or the null memory contribution at future null infinity . From this PN calculation, we also compute oscillatory ( modes) and spin-dependent memory terms that have not been identified previously or have been missing from strain expressions in the post-Newtonian literature.
Numerical-relativity surrogate modeling with nearly extremal black-hole spins
Class.Quant.Grav. 40, 055003 (2023)
Abstract
Numerical relativity (NR) simulations of binary black hole (BBH) systems provide the most accurate gravitational wave predictions, but at a high computational cost—especially when the black holes have nearly extremal spins (i.e. spins near the theoretical upper limit) or very unequal masses. Recently, the technique of reduced order modeling has enabled the construction of ‘surrogate models’ trained on an existing set of NR waveforms. Surrogate models enable the rapid computation of the gravitational waves emitted by BBHs. Typically these models are used for interpolation to compute gravitational waveforms for BBHs with mass ratios and spins within the bounds of the training set. Because simulations with nearly extremal spins are so technically challenging, surrogate models almost always rely on training sets with only moderate spins. In this paper, we explore how well surrogate models can extrapolate to nearly extremal spins when the training set only includes moderate spins. For simplicity, we focus on one-dimensional surrogate models trained on NR simulations of BBHs with equal masses and equal, aligned spins. We assess the performance of the surrogate models at higher spin magnitudes by calculating the mismatches between extrapolated surrogate model waveforms and NR waveforms, by calculating the differences between extrapolated and NR measurements of the remnant black-hole mass, and by testing how the surrogate model improves as the training set extends to higher spins. We find that while extrapolation in this one-dimensional case is viable for current detector sensitivities, surrogate models for next-generation detectors should use training sets that extend to nearly extremal spins.
Multipole moments on the common horizon in a binary-black-hole simulation
Abstract
We construct the covariantly defined multipole moments on the common horizon of an equal-mass, nonspinning, quasicircular binary-black- hole system. We see a strong correlation between these multipole moments and the gravitational waveform. We find that the multipole moments are well described by the fundamental quasinormal modes at sufficiently late times. For each nonzero multipole moment with , at least two fundamental quasinormal modes of different are detectable in the best model. These models provide faithful estimates of the true mass and spin of the remnant black hole. We also show that by including overtones, the mass multipole moment admits an excellent quasinormal-mode description at all times after the merger. This demonstrates the perhaps surprising power of perturbation theory near the merger.
Late-time post-merger modeling of a compact binary: effects of relativity, r-process heating, and treatment of transport
Class.Quant.Grav. 40, 085008 (2023)
Abstract
Detectable electromagnetic counterparts to gravitational waves from compact binary mergers can be produced by outflows from the black hole-accretion disk remnant during the first 10 s after the merger. Two-dimensional axisymmetric simulations with effective viscosity remain an efficient and informative way to model this late-time post-merger evolution. In addition to the inherent approximations of axisymmetry and modeling turbulent angular momentum transport by a viscosity, previous simulations often make other simplifications related to the treatment of the equation of state and turbulent transport effects. In this paper, we test the effect of these modeling choices. By evolving with the same viscosity the exact post-merger initial configuration previously evolved in Newtonian viscous hydrodynamics, we find that the Newtonian treatment provides a good estimate of the disk ejecta mass but underestimates the outflow velocity. We find that the inclusion of heavy nuclei causes a notable increase in ejecta mass. An approximate inclusion of r-process effects has a comparatively smaller effect, except for its designed effect on the composition. Diffusion of composition and entropy, modeling turbulent transport effects, has the overall effect of reducing ejecta mass and giving it a speed with lower average and more tightly-peaked distribution. Also, we find significant acceleration of outflow even at distances beyond 10 000 km, so that thermal wind velocities only asymptote beyond this radius and at higher values than often reported.
Quasinormal-mode filters: A new approach to analyze the gravitational-wave ringdown of binary black-hole mergers
Abstract
We propose two frequency-domain filters to analyze ringdown signals of binary black hole mergers. The first rational filter is constructed based on a set of (arbitrary) quasinormal modes (QNMs) of the remnant black holes, whereas the second full filter comes from the transmissivity of the remnant black holes. The two filters can remove corresponding QNMs from original time-domain ringdowns, while changing early inspiral signals in a trivial way—merely a time and phase shift. After filtering out dominant QNMs we can visualize the existence of various subdominant effects. For example, by applying our filters to a GW150914-like numerical relativity (NR) waveform, we find second-order effects in the (l=4,m=4),(l=5,m=4), and (l=5,m=5) harmonics; the spherical-spheroidal mixing mode in the (l=2,m=2) harmonic; and a mixing mode in the (l=2,m=1) harmonic due to a gravitational recoil. In another NR simulation where two component spins are antialigned with the orbital angular momentum we also find retrograde modes. The filters are sensitive to the remnant properties (i.e., mass and spin) and thus have a potential application to future data analyses and parameter estimations. We also investigate the stability of the full filter. Its connection to the instability of QNM spectra is discussed.
Success of the small mass-ratio approximation during the final orbits of binary black hole simulations
Abstract
Recent studies have shown the surprising effectiveness of the small mass-ratio approximation (SMR) in modeling the relativistic two-body problem even at comparable masses. Up to now this effectiveness has been demonstrated only during inspiral, before the binary transitions into plunge and merger. Here we examine the binding energy of nonspinning binary black hole simulations with mass ratios from to equal mass. We show for the first time that the binaries undergo a transition to plunge as predicted by analytic theory, and estimate the size of the transition region, which is gravitational wave cycles for equal mass binaries. By including transition, the SMR expansion of the binding energy is accurate until the last cycle of gravitational wave emission. This is true even for comparable mass binaries such as those observed by current gravitational wave detectors, where the transition often makes up much of the observed signal. Our work provides further evidence that the SMR approximation can be directly applied to current gravitational wave observations.
Surrogate model for gravitational wave signals from nonspinning, comparable-to large-mass-ratio black hole binaries built on black hole perturbation theory waveforms calibrated to numerical relativity
Abstract
We present a reduced-order surrogate model of gravitational waveforms from nonspinning binary black hole systems with comparable to large mass-ratio configurations. This surrogate model, BHPTNRSur1dq1e4, is trained on waveform data generated by point- particle black hole perturbation theory (ppBHPT) with mass ratios varying from 2.5 to 10,000. BHPTNRSur1dq1e4 extends an earlier waveform model, EMRISur1dq1e4, by using an updated transition-to- plunge model, covering longer durations up to 30,500m1 (where m1 is the mass of the primary black hole), includes several more spherical harmonic modes up to ℓ=10, and calibrates subdominant modes to numerical relativity (NR) data. In the comparable mass-ratio regime, including mass ratios as low as 2.5, the gravitational waveforms generated through ppBHPT agree surprisingly well with those from NR after this simple calibration step. We also compare our model to recent SXS and RIT NR simulations at mass ratios ranging from 15 to 32, and find the dominant quadrupolar modes agree to better than ≈10-3. We expect our model to be useful to study intermediate-mass-ratio binary systems in current and future gravitational-wave detectors.
Targeted large mass ratio numerical relativity surrogate waveform model for GW190814
Abstract
Gravitational wave observations of large mass ratio compact binary mergers like GW190814 highlight the need for reliable, high-accuracy waveform templates for such systems. We present nrhybsur2dq15, a new surrogate model trained on hybridized numerical relativity (NR) waveforms with mass ratios and aligned spins |χ1z|≤0.5 and χ2z=0. We target the parameter space of GW190814-like events as large mass ratio NR simulations are very expensive. The model includes the (2, 2), (2, 1), (3, 3), (4, 4), and (5, 5) spin-weighted spherical harmonic modes and spans the entire LIGO-Virgo bandwidth (with ) for total masses M≳9.5M⊙. nrhybsur2dq15 accurately reproduces the hybrid waveforms, with mismatches below ∼2×10-3 for total masses 10M⊙≤M≤300M⊙. This is at least an order of magnitude improvement over existing semianalytical models for GW190814-like systems. Finally, we reanalyze GW190814 with the new model and obtain source parameter constraints consistent with previous work.
Redshift factor and the small mass-ratio limit in binary black hole simulations
Abstract
We present a calculation of the Detweiler redshift factor in binary black hole simulations based on its relation to the surface gravity. The redshift factor has far-reaching applications in analytic approximations, gravitational self-force calculations, and conservative two-body dynamics. By specializing to nonspinning, quasicircular binaries with mass ratios ranging from mA/mB=1 to mA/mB=9.5 we are able to recover the leading small-mass-ratio (SMR) prediction with relative differences of order from simulations alone. The next-to-leading order term that we extract agrees with the SMR prediction arising from self-force calculations, with differences of a few percent. These deviations from the first-order conservative prediction are consistent with nonadiabatic effects that can be accommodated in an SMR expansion. This fact is also supported by a comparison to the conservative post-Newtonian prediction of the redshifts. For the individual redshifts, a reexpansion in terms of the symmetric mass ratio does not improve the convergence of the series. However we find that when looking at the sum of the redshift factors of both back holes, zA+zB, which is symmetric under the exchange of the masses, a reexpansion in ν accelerates its convergence. Our work provides further evidence of the surprising effectiveness of SMR approximations in modeling even comparable mass binary black holes.
Gravitational-wave echoes from numerical-relativity waveforms via spacetime construction near merging compact objects
Abstract
We propose a new approach toward reconstructing the late-time near- horizon geometry of merging binary black holes, and toward computing gravitational-wave echoes from exotic compact objects. A binary black-hole merger spacetime can be divided by a timelike hypersurface into a black-hole perturbation (BHP) region (in which the spacetime geometry can be approximated by homogeneous linear perturbations of the final Kerr black hole) and a nonlinear region. At late times, the boundary between the two regions is an infalling shell. The BHP region contains late-time gravitational waves emitted toward the future horizon, as well as those emitted toward future null infinity. In this region, by imposing no-ingoing-wave conditions at past null infinity and matching outgoing waves at future null infinity with waveforms computed from numerical relativity, we can obtain waves that travel toward the future horizon. In particular, the Newman-Penrose associated with the ingoing wave on the horizon is related to tidal deformations measured by fiducial observers floating above the horizon. We further determine the boundary of the BHP region on the future horizon by imposing that inside the BHP region can be faithfully represented by quasinormal modes. Using a physically motivated method to impose boundary conditions near the horizon and applying the so-called Boltzmann reflectivity, we compute the quasinormal modes of nonrotating exotic compact objects, as well as gravitational-wave echoes. We also investigate the detectability of these echoes in current and future detectors and prospects for parameter estimation.
Fully precessing higher-mode surrogate model of effective-one-body waveforms
Abstract
We present a surrogate model of SEOBNRv4PHM, a fully precessing time-domain effective-one-body waveform model including subdominant modes. We follow an approach similar to that used to build recent numerical relativity surrogate models. Our surrogate is 5000 M in duration, covers mass ratios up to and dimensionless spin magnitudes up to 0.8. Validating the surrogate against an independent test set, we find that the median mismatch error is less than , which is typically smaller than the modeling error of SEOBNRv4PHM itself. At high total mass, a few percent of configurations can exceed mismatches of if they are highly precessing and exceed a mass ratio of . This surrogate is nearly 2 orders of magnitude faster than the underlying time-domain SEOBNRv4PHM model and can be evaluated in . Bayesian inference analyses with SEOBNRv4PHM are typically very computationally demanding and can take from weeks to months to complete. The 2 order of magnitude speedup attained by our surrogate model enables practical parameter estimation analyses with this waveform family. This is crucial because Bayesian inference allows us to recover the masses and spins of binary black hole mergers given a model of the emitted gravitational waveform along with a description of the noise.
#2021
High-accuracy numerical models of Brownian thermal noise in thin mirror coatings
Class.Quant.Grav. 40, 025015 (2023)
Abstract
Brownian coating thermal noise in detector test masses is limiting the sensitivity of current gravitational-wave detectors on Earth. Therefore, accurate numerical models can inform the ongoing effort to minimize Brownian coating thermal noise in current and future gravitational-wave detectors. Such numerical models typically require significant computational resources and time, and often involve closed-source commercial codes. In contrast, open-source codes give complete visibility and control of the simulated physics, enable direct assessment of the numerical accuracy, and support the reproducibility of results. In this article, we use the open-source SpECTRE numerical relativity code and adopt a novel discontinuous Galerkin numerical method to model Brownian coating thermal noise. We demonstrate that SpECTRE achieves significantly higher accuracy than a previous approach at a fraction of the computational cost. Furthermore, we numerically model Brownian coating thermal noise in multiple sub-wavelength crystalline coating layers for the first time. Our new numerical method has the potential to enable fast exploration of realistic mirror configurations, and hence to guide the search for optimal mirror geometries, beam shapes and coating materials for gravitational-wave detectors.
A scalable elliptic solver with task-based parallelism for the SpECTRE numerical relativity code
Abstract
Elliptic partial differential equations must be solved numerically for many problems in numerical relativity, such as initial data for every simulation of merging black holes and neutron stars. Existing elliptic solvers can take multiple days to solve these problems at high resolution and when matter is involved, because they are either hard to parallelize or require a large amount of computational resources. Here we present a new solver for linear and nonlinear elliptic problems that is designed to scale with resolution and to parallelize on computing clusters. To achieve this we employ a discontinuous Galerkin discretization, an iterative multigrid- Schwarz preconditioned Newton-Krylov algorithm, and a task-based parallelism paradigm. To accelerate convergence of the elliptic solver we have developed novel subdomain-preconditioning techniques. We find that our multigrid-Schwarz preconditioned elliptic solves achieve iteration counts that are independent of resolution, and our task-based parallel programs scale over 200 million degrees of freedom to at least a few thousand cores. Our new code solves a classic initial data problem for binary black holes faster than the spectral code SpEC when distributed to only eight cores, and in a fraction of the time on more cores. It is publicly accessible in the next-generation SpECTRE numerical relativity code. Our results pave the way for highly parallel elliptic solves in numerical relativity and beyond.
High precision ringdown modeling: Multimode fits and BMS frames
Abstract
Quasinormal mode (QNM) modeling is an invaluable tool for characterizing remnant black holes, studying strong gravity, and testing general relativity. Only recently have QNM studies begun to focus on multimode fitting to numerical relativity strain waveforms. As gravitational wave observatories become even more sensitive they will be able to resolve higher-order modes. Consequently, multimode QNM fits will be critically important, and in turn require a more thorough treatment of the asymptotic frame at . The first main result of this work is a method for systematically fitting a QNM model containing many modes to a numerical waveform produced using Cauchy-characteristic extraction (CCE), a waveform extraction technique which is known to resolve memory effects. We choose the modes to model based on their power contribution to the residual between numerical and model waveforms. We show that the all-mode strain mismatch improves by a factor of ∼10⁵ when using multimode fitting as opposed to only fitting the (2,±2,n) modes. Our most significant result addresses a critical point that has been overlooked in the QNM literature: the importance of matching the Bondi-van der Burg- Metzner-Sachs (BMS) frame of the numerical waveform to that of the QNM model. We show that by mapping the numerical waveforms—which exhibit the memory effect—to a BMS frame known as the super rest frame, there is an improvement of in the all-mode strain mismatch compared to using a strain waveform whose BMS frame is not fixed. Furthermore, we find that by mapping CCE waveforms to the super rest frame, we can obtain all-mode mismatches that are, on average, a factor of better than using the publicly available extrapolated waveforms. We illustrate the effectiveness of these modeling enhancements by applying them to families of waveforms produced by numerical relativity and comparing our results to previous QNM studies.
SpECTRE Cauchy-characteristic evolution system for rapid, precise waveform extraction
Abstract
We give full details regarding the new Cauchy-characteristic evolution (CCE) system in spectre. The implementation is built to provide streamlined flexibility for either extracting waveforms during the process of a spectre binary compact object simulation or as a stand-alone module for extracting waveforms from worldtube data provided by another code base. Using our recently presented improved analytic formulation, the CCE system is free of pure-gauge logarithms that would spoil the spectral convergence of the scheme. It gracefully extracts all five Weyl scalars, in addition to the news and the strain. The spectre CCE system makes significant improvements on previous implementations in modularity, ease of use, and speed of computation.
Simulating magnetized neutron stars with discontinuous Galerkin methods
Abstract
Discontinuous Galerkin methods are popular because they can achieve high order where the solution is smooth, because they can capture shocks while needing only nearest-neighbor communication, and because they are relatively easy to formulate on complex meshes. We perform a detailed comparison of various limiting strategies presented in the literature applied to the equations of general relativistic magnetohydrodynamics. We compare the standard Lambda Pi^N limiter, the hierarchical limiter of Krivodonova, the simple WENO limiter, the HWENO limiter, and a discontinuous Galerkin-finite-difference hybrid method. The ultimate goal is to understand what limiting strategies are able to robustly simulate magnetized Tolman-Oppenheimer-Volkoff stars without any fine-tuning of parameters. Among the limiters explored here, the only limiting strategy we can endorse is a discontinuous Galerkin-finite-difference hybrid method.
A high-order shock capturing discontinuous Galerkin–finite difference hybrid method for GRMHD
Class.Quant.Grav. 39, 195001 (2022)
Abstract
We present a discontinuous Galerkin (DG)–finite difference (FD) hybrid scheme that allows high-order shock capturing with the DG method for general relativistic magnetohydrodynamics. The hybrid method is conceptually quite simple. An unlimited DG candidate solution is computed for the next time step. If the candidate solution is inadmissible, the time step is retaken using robust FD methods. Because of its a posteriori nature, the hybrid scheme inherits the best properties of both methods. It is high-order with exponential convergence in smooth regions, while robustly handling discontinuities. We give a detailed description of how we transfer the solution between the DG and FD solvers, and the troubled-cell indicators necessary to robustly handle slow-moving discontinuities and simulate magnetized neutron stars. We demonstrate the efficacy of the proposed method using a suite of standard and very challenging 1D, 2D, and 3D relativistic magnetohydrodynamics test problems. The hybrid scheme is designed from the ground up to efficiently simulate astrophysical problems such as the inspiral, coalescence, and merger of two neutron stars.
Efficient simulations of high-spin black holes with a new gauge
Abstract
We present a new choice of initial data for binary black hole simulations that significantly improves the efficiency of high-spin simulations. We use spherical Kerr-Schild coordinates, where the horizon of a rotating black hole is spherical, for each black hole. The superposed spherical Kerr-Schild initial data reduce the runtime by a factor of 2 compared to standard superposed Kerr-Schild for an intermediate resolution spin-0.99 binary-black-hole simulation. We also explore different variations of the gauge conditions imposed during the evolution, one of which produces an additional speed-up of 1.3.
Universal features of gravitational waves emitted by superkick binary black hole systems
Abstract
We use numerical relativity to study the merger and ringdown stages of “superkick” binary black hole systems (those with equal mass and antiparallel spins). We find a universal way to describe the mass and current quadrupole gravitational waves emitted by these systems during the merger and ringdown stage: (i) The time evolutions of these waves are insensitive to the progenitor’s parameters (spins) after being normalized by their own peak values. (ii) The peak values, which encode all the spin information of the progenitor, can be consistently fitted to formulas inspired by post-Newtonian theory. We find that the universal evolution of the mass quadrupole wave can be accurately modeled by the so-called Backwards One-Body (BOB) model. However, the BOB model, in its present form, leads to a lower waveform match and a significant parameter-estimation bias for the current quadrupole wave. We also decompose the ringdown signal into seven overtones, and study the dependence of mode amplitudes on the progenitor’s parameters. Such dependence is found to be insensitive to the overtone index (up to a scaling factor). Finally, we use the Fisher matrix technique to investigate how the ringdown waveform can be at least as important for parameter estimation as the inspiral stage. Assuming the Cosmic Explorer, we find the contribution of ringdown portion dominates as the total mass exceeds . For massive binary black hole (BBH) systems, the accuracy of parameter measurement is improved by incorporating the information of ringdown—the ringdown sector gives rise to a different parameter correlation from inspiral stage; hence, the overall parameter correlation is reduced in full signal.
Fixing the BMS frame of numerical relativity waveforms
Abstract
Understanding the Bondi-Metzner-Sachs (BMS) frame of the gravitational waves produced by numerical relativity is crucial for ensuring that analyses on such waveforms are performed properly. It is also important that models are built from waveforms in the same BMS frame. Up until now, however, the BMS frame of numerical waveforms has not been thoroughly examined, largely because the necessary tools have not existed. In this paper, we show how to analyze and map to a suitable BMS frame for numerical waveforms calculated with the Spectral Einstein Code (SpEC). However, the methods and tools that we present are general and can be applied to any numerical waveforms. We present an extensive study of 13 binary black hole systems that broadly span parameter space. From these simulations, we extract the strain and also the Weyl scalars using both SpECTRE’s Cauchy-characteristic extraction module and also the standard extrapolation procedure with a displacement memory correction applied during postprocessing. First, we show that the current center-of-mass correction used to map these waveforms to the center-of-mass frame is not as effective as previously thought. Consequently, we also develop an improved correction that utilizes asymptotic Poincaré charges instead of a Newtonian center-of-mass trajectory. Next, we map our waveforms to the post-Newtonian (PN) BMS frame using a PN strain waveform. This helps us find the unique BMS transformation that minimizes the norm of the difference between the numerical and PN strain waveforms during the early inspiral phase. We find that once the waveforms are mapped to the PN BMS frame, they can be hybridized with a PN strain waveform much more effectively than if one used any of the previous alignment schemes, which only utilize the Poincaré transformations.
Comparing Remnant Properties from Horizon Data and Asymptotic Data in Numerical Relativity
Abstract
We present a new study of remnant black hole properties from 13 binary black hole systems, numerically evolved using the Spectral Einstein Code. The mass, spin, and recoil velocity of each remnant were determined quasilocally from apparent horizon data and asymptotically from Bondi data (h,ψ4,ψ3,ψ2,ψ1) computed at future null infinity using SpECTRE’s Cauchy characteristic evolution. We compare these independent measurements of the remnant properties in the bulk and on the boundary of the spacetime, giving insight into how well asymptotic data are able to reproduce local properties of the remnant black hole in numerical relativity. We also discuss the theoretical framework for connecting horizon quantities to asymptotic quantities and how it relates to our results. This study recommends a simple improvement to the recoil velocities reported in the Simulating eXtreme Spacetimes waveform catalog, provides an improvement to future surrogate remnant models, and offers new analysis techniques for evaluating the physical accuracy of numerical simulations.
Implementation of Monte Carlo Transport in the General Relativistic SpEC Code
Abstract
Neutrino transport and neutrino−matter interactions are known to play an important role in the evolution of neutron star mergers and of their post-merger remnants. Neutrinos cool remnants, drive post- merger winds, and deposit energy in the low-density polar regions where relativistic jets may eventually form. Neutrinos also modify the composition of the ejected material, impacting the outcome of nucleosynthesis in merger outflows and the properties of the optical/infrared transients that they power (kilonovae). So far, merger simulations have largely relied on approximate treatments of the neutrinos (leakage, moments) that simplify the equations of radiation transport in a way that makes simulations more affordable but also introduces unquantifiable errors in the results. To improve on these methods, we recently published a first simulation of neutron star mergers using a low-cost Monte Carlo algorithm for neutrino radiation transport. Our transport code limits costs in optically thick regions by placing a hard ceiling on the value of the absorption opacity of the fluid, yet all approximations made within the code are designed to vanish in the limit of infinite numerical resolution. We provide here an in-depth description of this algorithm, of its implementation in the SpEC merger code, and of the expected impact of our approximations in optically thick regions. We argue that the last is a subdominant source of error at the accuracy reached by current simulations and for the interactions currently included in our code. We also provide tests of the most important features of this code.
Eccentric binary black hole surrogate models for the gravitational waveform and remnant properties: comparable mass, nonspinning case
Abstract
We develop new strategies to build numerical relativity surrogate models for eccentric binary black hole systems, which are expected to play an increasingly important role in current and future gravitational-wave detectors. We introduce a new surrogate waveform model, NRSur2dq1Ecc, using 47 nonspinning, equal-mass waveforms with eccentricities up to 0.2 when measured at a reference time of 5500M before merger. This is the first waveform model that is directly trained on eccentric numerical relativity simulations and does not require that the binary circularizes before merger. The model includes the (2,2), (3,2), and (4,4) spin-weighted spherical harmonic modes. We also build a final black hole model, NRSur2dq1EccRemnant, which models the mass, and spin of the remnant black hole. We show that our waveform model can accurately predict numerical relativity waveforms with mismatches ≈10-3, while the remnant model can recover the final mass and dimensionless spin with absolute errors smaller than ≈5×10-4M and ≈2×10-3 respectively. We demonstrate that the waveform model can also recover subtle effects like mode mixing in the ringdown signal without any special ad hoc modeling steps. Finally, we show that despite being trained only on equal-mass binaries, NRSur2dq1Ecc can be reasonably extended up to mass ratio q≈3 with mismatches ≃10-2 for eccentricities smaller than ∼0.05 as measured at a reference time of 2000M before merger. The methods developed here should prove useful in the building of future eccentric surrogate models over larger regions of the parameter space.
#2020
Up-down instability of binary black holes in numerical relativity
Abstract
Binary black holes with spins that are aligned with the orbital angular momentum do not precess. However, post-Newtonian calculations predict that "up-down" binaries, in which the spin of the heavier (lighter) black hole is aligned (antialigned) with the orbital angular momentum, are unstable when the spins are slightly perturbed from perfect alignment. This instability provides a possible mechanism for the formation of precessing binaries in environments where sources are preferentially formed with (anti) aligned spins. In this paper, we present the first full numerical relativity simulations capturing this instability. These simulations span \(\sim 100\) orbits and \(\sim 3\)-\(5\) precession cycles before merger, making them some of the longest numerical relativity simulations to date. Initialized with a small perturbation of \(1^{\circ}\)-\(10^{\circ}\), the instability causes a dramatic growth of the spin misalignments, which can reach \(\sim 90^{\circ}\) near merger. We show that this leaves a strong imprint on the subdominant modes of the gravitational wave signal, which can potentially be used to distinguish up-down binaries from other sources. Finally, we show that post-Newtonian and effective-one-body approximants are able to reproduce the unstable dynamics of up-down binaries extracted from numerical relativity.
Adding gravitational memory to waveform catalogs using BMS balance laws
Abstract
Accurate models of gravitational waves from merging binary black holes are crucial for detectors to measure events and extract new science. One important feature that is currently missing from the Simulating eXtreme Spacetimes (SXS) Collaboration’s catalog of waveforms for merging black holes, and other waveform catalogs, is the gravitational memory effect: a persistent, physical change to spacetime that is induced by the passage of transient radiation. We find, however, that by exploiting the Bondi-van der Burg-Metzner- Sachs (BMS) balance laws, which come from the extended BMS transformations, we can correct the strain waveforms in the SXS catalog to include the missing displacement memory. Our results show that these corrected waveforms satisfy the BMS balance laws to a much higher degree of accuracy. Furthermore, we find that these corrected strain waveforms coincide especially well with the waveforms obtained from Cauchy-characteristic extraction (CCE) that already exhibit memory effects. These corrected strain waveforms also evade the transient junk effects that are currently present in CCE waveforms. Last, we make our code for computing these contributions to the BMS balance laws and memory publicly available as a part of the python package sxs, thus enabling anyone to evaluate the expected memory effects and violation of the BMS balance laws.
Extending gravitational wave extraction using Weyl characteristic fields
Abstract
We present a detailed methodology for extracting the full set of Newman-Penrose Weyl scalars from numerically generated spacetimes without requiring a tetrad that is completely orthonormal or perfectly aligned to the principal null directions. We also describe how to implement an extrapolation technique for computing the Weyl scalars’ contribution at asymptotic null infinity in postprocessing. These methods have been used to produce Ψ4 and h waveforms for the Simulating eXtreme Spacetimes (SXS) waveform catalog and now have been expanded to produce the entire set of Weyl scalars. These new waveform quantities are critical for the future of gravitational wave astronomy in order to understand the finite-amplitude gauge differences that can occur in numerical waveforms. We also present a new analysis of the accuracy of waveforms produced by the Spectral Einstein Code. While ultimately we expect Cauchy characteristic extraction to yield more accurate waveforms, the extraction techniques described here are far easier to implement and have already proven to be a viable way to produce production-level waveforms that can meet the demands of current gravitational-wave detectors.
High-accuracy waveforms for black hole-neutron star systems with spinning black holes
Abstract
The availability of accurate numerical waveforms is an important requirement for the creation and calibration of reliable waveform models for gravitational wave astrophysics. For black hole-neutron star binaries (BHNS), very few accurate waveforms are however publicly available. Most recent models are calibrated to a large number of older simulations with good parameter space coverage for low-spin nonprecessing binaries but limited accuracy, and a much smaller number of longer, more recent simulations limited to nonspinning black holes. In this paper, we present long, accurate numerical waveforms for three new systems that include rapidly spinning black holes, and one precessing configuration. We study in detail the accuracy of the simulations, and in particular perform for the first time in the context of BHNS binaries a detailed comparison of waveform extrapolation methods to the results of Cauchy characteristic extraction. The new waveforms have <0.1 rad phase errors during inspiral, rising to ∼(0.2–0.4) rad errors at merger, and ≲1% error in their amplitude. We compute the faithfulness of recent analytical models to these numerical results for the late inspiral and merger phases covered by the numerical simulations, and find that models specifically designed for BHNS binaries perform well (faithfulness F>0.99) for binaries seen face on. For edge-on observations, particularly for precessing systems, disagreements between models and simulations increase, and models that include precession and/or higher-order modes start to perform better than BHNS models that currently lack these features.
Monte-Carlo neutrino transport in neutron star merger simulations
Astrophys.J.Lett. 902, L27 (2020)
Abstract
Gravitational waves and electromagnetic signals from merging neutron star binaries provide valuable information about the the properties of dense matter, the formation of heavy elements, and high-energy astrophysics. To fully leverage observations of these systems, we need numerical simulations that provide reliable predictions for the properties of the matter unbound in these mergers. An important limitation of current simulations is the use of approximate methods for neutrino transport that do not converge to a solution of the transport equations as numerical resolution increases, and thus have errors that are impossible to quantify. Here, we report on a first simulation of a binary neutron star merger that uses Monte-Carlo techniques to directly solve the transport equations in low-density regions. In high-density regions, we use approximations inspired by implicit Monte-Carlo to greatly reduce the cost of simulations, while only introducing errors quantifiable through more expensive convergence studies. We simulate an unequal mass neutron star binary merger up to 5 ms past merger, and report on the properties of the matter and neutrino outflows. Finally, we compare our results to the output of our best approximate “M1” transport scheme, demonstrating that an M1 scheme that carefully approximates the neutrino energy spectrum only leads to ∼10% uncertainty in the composition and velocity of the ejecta, and ∼20% uncertainty in the ν e and luminosities and energies. The most significant disagreement found between M1 and Monte-Carlo results is a factor of ∼2 difference in the luminosity of heavy-lepton neutrinos.
Comparison of momentum transport models for numerical relativity
Abstract
The main problems of nonvacuum numerical relativity, compact binary mergers and stellar collapse, involve hydromagnetic instabilities and turbulent flows, so that kinetic energy at small scales leads to mean effects at large scale that drive the secular evolution. Notable among these effects is momentum transport. We investigate two models of this transport effect, a relativistic Navier-Stokes system and a turbulent mean stress model, that are similar to all of the prescriptions that have been attempted to date for treating subgrid effects on binary neutron star mergers and their aftermath. Our investigation involves both stability analysis and numerical experimentation on star and disk systems. We also begin the investigation of the effects of particle and heat transport on postmerger simulations. We find that correct handling of turbulent heating is crucial for avoiding unphysical instabilities. Given such appropriate handling, the evolution of a differentially rotating star and the accretion rate of a disk are reassuringly insensitive to the choice of prescription. However, disk outflows can be sensitive to the choice of method, even for the same effective viscous strength. We also consider the effects of eddy diffusion in the evolution of an accretion disk and show that it can interestingly affect the composition of outflows.
Computation of displacement and spin gravitational memory in numerical relativity
Abstract
We present the first numerical relativity waveforms for binary black hole mergers produced using spectral methods that show both the displacement and the spin memory effects. Explicitly, we use the SXS (Simulating eXtreme Spacetimes) Collaboration’s spec code to run a Cauchy evolution of a binary black hole merger and then extract the gravitational wave strain using spectre’s version of a Cauchy- characteristic extraction. We find that we can accurately resolve the strain’s traditional m=0 memory modes and some of the m≠0 oscillatory memory modes that have previously only been theorized. We also perform a separate calculation of the memory using equations for the Bondi-Metzner-Sachs charges as well as the energy and angular momentum fluxes at asymptotic infinity. Our new calculation uses only the gravitational wave strain and two of the Weyl scalars at infinity. Also, this computation shows that the memory modes can be understood as a combination of a memory signal throughout the binary’s inspiral and merger phases, and a quasinormal mode signal near the ringdown phase. Additionally, we find that the magnetic memory, up to numerical error, is indeed zero as previously conjectured. Last, we find that signal-to-noise ratios of memory for LIGO, the Einstein Telescope, and the Laser Interferometer Space Antenna with these new waveforms and new memory calculation are larger than previous expectations based on post-Newtonian or minimal waveform models.
Axisymmetric hydrodynamics in numerical relativity using a multipatch method
Class.Quant.Grav. 37, 235010 (2020)
Abstract
We describe a method of implementing the axisymmetric evolution of general-relativistic hydrodynamics and magnetohydrodynamics through modification of a multipatch grid scheme. In order to ease the computational requirements required to evolve the post-merger phase of systems involving binary compact massive objects in numerical relativity, it is often beneficial to take advantage of these system’s tendency to rapidly settle into states that are nearly axisymmetric, allowing for 2D evolution of secular timescales. We implement this scheme in the spectral Einstein code and show the results of application of this method to four test systems including viscosity, magnetic fields, and neutrino radiation transport. Our results show that this method can be used to quickly allow already existing 3D infrastructure that makes use of local coordinate system transformations to be made to run in axisymmetric 2D with the flexible grid creation capabilities of multipatch methods. Our code tests include a simple model of a binary neutron star postmerger remnant, for which we confirm the formation of a massive torus which is a promising source of post-merger ejecta.
Aligned-spin neutron-star–black-hole waveform model based on the effective-one-body approach and numerical-relativity simulations
Abstract
After the discovery of gravitational waves from binary black holes (BBHs) and binary neutron stars (BNSs) with the LIGO and Virgo detectors, neutron-star black holes (NSBHs) are the natural next class of binary systems to be observed. In this work, we develop a waveform model for aligned-spin NSBHs combining a BBH baseline waveform (available in the effective-one-body approach) with a phenomenological description of tidal effects (extracted from numerical-relativity simulations) and correcting the amplitude during the late inspiral, merger and ringdown to account for the NS tidal disruption. In particular, we calibrate the amplitude corrections using NSBH waveforms obtained with the numerical- relativity spectral Einstein code (spec) and the sacra code. The model was calibrated using simulations with NS masses in the range 1.2–1.4 M⊙, tidal deformabilities up to 4200 (for a 1.2 M⊙ NS), and dimensionless BH spin magnitude up to 0.9. Based on the simulations used and on checking that sensible waveforms are produced, we recommend our model to be employed with a NS mass in the range 1–3 M⊙, tidal deformability 0–5000, and (dimensionless) BH spin magnitude up to 0.9. We also validate our model against two new, highly accurate NSBH waveforms with BH spin 0.9 and mass ratios 3 and 4, characterized by tidal disruption, produced with SpEC, and find very good agreement. Furthermore, we compute the unfaithfulness between waveforms from NSBH, BBH, and BNS systems, finding that it will be challenging for the Advanced LIGO-Virgo detector network at design sensitivity to distinguish different source classes. We perform a Bayesian parameter-estimation analysis on a synthetic numerical-relativity signal in zero noise to study parameter biases. Finally, we reanalyze GW170817, with the hypothesis that it is a NSBH. We do not find evidence to distinguish the BNS and NSBH hypotheses; however, the posterior for the mass ratio is shifted to less equal masses under the NSBH hypothesis.
Multipolar Effective-One-Body Waveforms for Precessing Binary Black Holes: Construction and Validation
Abstract
As gravitational-wave detectors become more sensitive and broaden their frequency bandwidth, we will access a greater variety of signals emitted by compact binary systems, shedding light on their astrophysical origin and environment. A key physical effect that can distinguish among different formation scenarios is the misalignment of the spins with the orbital angular momentum, causing the spins and the binary’s orbital plane to precess. To accurately model such precessing signals, especially when masses and spins vary in the wide astrophysical range, it is crucial to include multipoles beyond the dominant quadrupole. Here, we develop the first multipolar precessing waveform model in the effective-one-body (EOB) formalism for the entire coalescence stage (i.e., inspiral, merger and ringdown) of binary black holes: SEOBNRv4PHM. In the nonprecessing limit, the model reduces to SEOBNRv4HM, which was calibrated to numerical-relativity (NR) simulations, and waveforms from black-hole perturbation theory. We validate SEOBNRv4PHM by comparing it to the public catalog of 1405 precessing NR waveforms of the Simulating eXtreme Spacetimes (SXS) collaboration, and also to 118 SXS precessing NR waveforms, produced as part of this project, which span mass ratios 1-4 and (dimensionless) black-hole’s spins up to 0.9. We stress that SEOBNRv4PHM is not calibrated to NR simulations in the precessing sector. We compute the unfaithfulness against the 1523 SXS precessing NR waveforms, and find that, for 94% (57%) of the cases, the maximum value, in the total mass range 20−200 M⊙, is below 3% (1%). Those numbers change to 83% (20%) when using the inspiral-merger-ringdown, multipolar, precessing phenomenological model IMRPhenomPv3HM. We investigate the impact of such unfaithfulness values with two Bayesian, parameter-estimation studies on synthetic signals. We also compute the unfaithfulness between those waveform models as a function of the mass and spin parameters to identify in which part of the parameter space they differ the most. We validate them also against the multipolar, precessing NR surrogate model NRSur7dq4, and find that the SEOBNRv4PHM model outperforms IMRPhenomPv3HM.
#2019
Gravitational waveforms of binary neutron star inspirals using post-Newtonian tidal splicing
Abstract
The tidal deformations of neutron stars within an inspiraling compact binary alter the orbital dynamics, imprinting a signature on the gravitational wave signal. Modeling this signal could be done with numerical-relativity simulations, but these are too computationally expensive for many applications. Analytic post- Newtonian treatments are limited by unknown higher-order nontidal terms. This paper further builds upon the “tidal splicing” model in which post-Newtonian tidal terms are “spliced” onto numerical relativity simulations of black-hole binaries. We improve on previous treatments of tidal splicing by including spherical harmonic modes beyond the (2,2) mode, expanding the post-Newtonian expressions for tidal effects to 2.5 order, including dynamical tide corrections, and adding a partial treatment of the spin-tidal dynamics. Furthermore, instead of numerical relativity simulations, we use the spin-aligned binary black hole (BBH) surrogate model “NRHybSur3dq8” to provide the BBH waveforms that are input into the tidal slicing procedure. This allows us to construct spin-aligned, inspiraling TaylorT2 and TaylorT4 splicing waveform models that can be evaluated quickly. These models are tested against existing binary neutron star and black hole–neutron star simulations. We implement the TaylorT2 splicing model as an extension to “NRHybSur3dq8,” creating a model that we call “NRHybSur3dq8Tidal.”
Impact of subdominant modes on the interpretation of gravitational-wave signals from heavy binary black hole systems
Abstract
Over the past year, a handful of new gravitational wave models have been developed to include multiple harmonic modes thereby enabling for the first time fully Bayesian inference studies including higher modes to be performed. Using one recently developed numerical relativity surrogate model, NRHybSur3dq8, we investigate the importance of higher modes on parameter inference of coalescing massive binary black holes. We focus on examples relevant to the current three-detector network of observatories, with a detector- frame mass set to 120 M⊙ and with signal amplitude values that are consistent with plausible candidates for the next few observing runs. We show that for such systems the higher mode content will be important for interpreting coalescing binary black holes, reducing systematic bias, and computing properties of the remnant object. Even for comparable-mass binaries and at low signal amplitude, the omission of higher modes can influence posterior probability distributions. We discuss the impact of our results on source population inference and self-consistency tests of general relativity. Our work can be used to better understand asymmetric binary black hole merger events, such as GW190412. Higher modes are critical for such systems, and their omission usually produces substantial parameter biases.
Numerical relativity simulation of GW150914 beyond general relativity
Abstract
We produce the first astrophysically relevant numerical binary black hole gravitational waveform in a higher-curvature theory of gravity beyond general relativity. We simulate a system with parameters consistent with GW150914, the first LIGO detection, in order-reduced dynamical Chern-Simons gravity, a theory with motivations in string theory and loop quantum gravity. We present results for the leading- order corrections to the merger and ringdown waveforms, as well as the ringdown quasinormal mode spectrum. We estimate that such corrections may be discriminated in detections with signal to noise ratio ≳180–240, with the precise value depending on the dimension of the GR waveform family used in data analysis.
Spectral Cauchy-Characteristic Extraction of the Gravitational Wave News Function
Abstract
We present an improved spectral algorithm for Cauchy-characteristic extraction and characteristic evolution of gravitational waves in numerical relativity. The new algorithms improve spectral convergence both at the poles of the spherical-polar grid and at future null infinity, as well as increase the temporal resolution of the code. The key to the success of these algorithms is a new set of high-accuracy tests, which we present here. We demonstrate the accuracy of the code and compare with the existing pittnull implementation.
Smooth Equations of State for High-Accuracy Simulations of Neutron Star Binaries
Abstract
High-accuracy numerical simulations of merging neutron stars play an important role in testing and calibrating the waveform models used by gravitational wave observatories. Obtaining high-accuracy waveforms at a reasonable computational cost, however, remains a significant challenge. One issue is that high-order convergence of the solution requires the use of smooth evolution variables, while many of the equations of state used to model the neutron star matter have discontinuities, typically in the first derivative of the pressure. Spectral formulations of the equation of state have been proposed as a potential solution to this problem. Here, we report on the numerical implementation of spectral equations of state in the spectral Einstein code. We show that, in our code, spectral equations of state allow for high-accuracy simulations at a lower computational cost than commonly used “piecewise polytrope” equations state. We also demonstrate that not all spectral equations of state are equally useful: different choices for the low-density part of the equation of state can significantly impact the cost and accuracy of simulations. As a result, simulations of neutron star mergers present us with a trade-off between the cost of simulations and the physical realism of the chosen equation of state.
Unequal Mass Binary Neutron Star Simulations with Neutrino Transport: Ejecta and Neutrino Emission
Abstract
We present 12 new simulations of unequal mass neutron star mergers. The simulations are performed with the spec code, and utilize nuclear-theory-based equations of state and a two-moment gray neutrino transport scheme with an improved energy estimate based on evolving the number density. We model the neutron stars with the SFHo, LS220, and DD2 equations of state (EOS) and we study the neutrino and matter emission of all 12 models to search for robust trends between binary parameters and emission characteristics. We find that the total mass of the dynamical ejecta exceeds 0.01 M⊙ only for SFHo with weak dependence on the mass ratio across all models. We find that the ejecta have a broad electron fraction (Ye) distribution (≈0.06–0.48), with mean 0.2. Ye increases with neutrino irradiation over time, but decreases with increasing binary asymmetry. We also find that the models have ejecta with a broad asymptotic velocity distribution (≈0.05–0.7c). The average velocity lies in the range 0.2c-0.3c and decreases with binary asymmetry. Furthermore, we find that disk mass increases with binary asymmetry and stiffness of the EOS. The Ye of the disk increases with softness of the EOS. The strongest neutrino emission occurs for the models with soft EOS. For (anti) electron neutrinos we find no significant dependence of the magnitude or angular distribution or neutrino luminosity with mass ratio. The heavier neutrino species have a luminosity dependence on mass ratio but an angular distribution which does not change with mass ratio.
Numerical binary black hole collisions in dynamical Chern-Simons gravity
Abstract
We produce the first numerical relativity binary black hole gravitational waveforms in a higher-curvature theory beyond general relativity. In particular, we study head-on collisions of binary black holes in order-reduced dynamical Chern-Simons gravity. This is a precursor to producing beyond-general-relativity waveforms for inspiraling binary black hole systems that are useful for gravitational wave detection. Head-on collisions are interesting in their own right, however, as they cleanly probe the quasinormal mode spectrum of the final black hole. We thus compute the leading-order dynamical Chern-Simons modifications to the complex frequencies of the postmerger gravitational radiation. We consider equal-mass systems, with equal spins oriented along the axis of collision, resulting in remnant black holes with spin. We find modifications to the complex frequencies of the quasinormal mode spectrum that behave as a power law with the spin of the remnant, and that are not degenerate with the frequencies associated with a Kerr black hole of any mass and spin. We discuss these results in the context of testing general relativity with gravitational wave observations.
Surrogate models for precessing binary black hole simulations with unequal masses
Phys.Rev.Research. 1, 033015 (2019)
Abstract
Only numerical relativity simulations can capture the full complexities of binary black hole mergers. These simulations, however, are prohibitively expensive for direct data analysis applications such as parameter estimation. We present two new fast and accurate surrogate models for the outputs of these simulations: the first model, NRSur7dq4, predicts the gravitational waveform and the second model, \RemnantModel, predicts the properties of the remnant black hole. These models extend previous 7-dimensional, non- eccentric precessing models to higher mass ratios, and have been trained against 1528 simulations with mass ratios \(q\leq4\) and spin magnitudes \(\chi_1,\chi_2 \leq 0.8\), with generic spin directions. The waveform model, NRSur7dq4, which begins about 20 orbits before merger, includes all \(\ell \leq 4\) spin-weighted spherical harmonic modes, as well as the precession frame dynamics and spin evolution of the black holes. The final black hole model, \RemnantModel, models the mass, spin, and recoil kick velocity of the remnant black hole. In their training parameter range, both models are shown to be more accurate than existing models by at least an order of magnitude, with errors comparable to the estimated errors in the numerical relativity simulations. We also show that the surrogate models work well even when extrapolated outside their training parameter space range, up to mass ratios \(q=6\).
Compact Binary Waveform Center-of-Mass Corrections
Abstract
We present a detailed study of the center-of-mass (c.m.) motion seen in simulations produced by the Simulating eXtreme Spacetimes (SXS) collaboration. We investigate potential physical sources for the large c.m. motion in binary black hole simulations and find that a significant fraction of the c.m. motion cannot be explained physically, thus concluding that it is largely a gauge effect. These large c.m. displacements cause mode mixing in the gravitational waveform, most easily recognized as amplitude oscillations caused by the dominant (2,±2) modes mixing into subdominant modes. This mixing does not diminish with increasing distance from the source; it is present even in asymptotic waveforms, regardless of the method of data extraction. We describe the current c.m.-correction method used by the SXS collaboration, which is based on counteracting the motion of the c.m. as measured by the trajectories of the apparent horizons in the simulations, and investigate potential methods to improve that correction to the waveform. We also present a complementary method for computing an optimal c.m. correction or evaluating any other c.m. transformation based solely on the asymptotic waveform data.
The SXS Collaboration catalog of binary black hole simulations
Class.Quant.Grav. 36, 195006 (2019)
Abstract
Accurate models of gravitational waves from merging black holes are necessary for detectors to observe as many events as possible while extracting the maximum science. Near the time of merger, the gravitational waves from merging black holes can be computed only using numerical relativity. In this paper, we present a major update of the Simulating eXtreme Spacetimes (SXS) Collaboration catalog of numerical simulations for merging black holes. The catalog contains 2018 distinct configurations (a factor of 11 increase compared to the 2013 SXS catalog), including 1426 spin-precessing configurations, with mass ratios between 1 and 10, and spin magnitudes up to 0.998. The median length of a waveform in the catalog is 39 cycles of the dominant gravitational-wave mode, with the shortest waveform containing 7.0 cycles and the longest 351.3 cycles. We discuss improvements such as correcting for moving centers of mass and extended coverage of the parameter space. We also present a thorough analysis of numerical errors, finding typical truncation errors corresponding to a waveform mismatch of ∼10−4. The simulations provide remnant masses and spins with uncertainties of 0.03% and 0.1% (90th percentile), about an order of magnitude better than analytical models for remnant properties. The full catalog is publicly available at www.black-holes.org/waveforms.
Numerical simulations of neutron star-black hole binaries in the near-equal-mass regime
Abstract
Simulations of neutron star–black hole (NSBH) binaries generally consider black holes with masses in the range (5–10)M⊙, where we expect to find most stellar mass black holes. The existence of lower mass black holes, however, cannot be theoretically ruled out. Low- mass black holes in binary systems with a neutron star companion could mimic neutron star–neutron star (NSNS) binaries, as they power similar gravitational waves and electromagnetic signals. To understand the differences and similarities between NSNS mergers and low-mass NSBH mergers, numerical simulations are required. Here, we perform a set of simulations of low-mass NSBH mergers, including systems compatible with GW170817. Our simulations use a composition and temperature dependent equation of state (DD2) and approximate neutrino transport, but no magnetic fields. We find that low-mass NSBH mergers produce remnant disks significantly less massive than previously expected, and consistent with the postmerger outflow mass inferred from GW170817 for a moderately asymmetric mass ratio. Whether postmerger disk outflows can also explain the inferred velocity and composition of that event’s ejecta is an open question that our merger simulations cannot answer at this point. The dynamical ejecta produced by systems compatible with GW170817 are negligible except if the mass ratio and black hole spin are at the edge of the allowed parameter space. The dynamical ejecta are cold, neutron-rich, and surprisingly slow for ejecta produced during the tidal disruption of a neutron star: v∼(0.1–0.15)c. We also find that the final mass of the remnant black hole is consistent with existing analytical predictions, while the final spin of that black hole is noticeably larger than expected—up to χBH=0.84 for our equal mass case.
Black Hole Ringdown: The Importance of Overtones
Abstract
It is possible to infer the mass and spin of the remnant black hole from binary black hole mergers by comparing the ringdown gravitational wave signal to results from studies of perturbed Kerr spacetimes. Typically, these studies are based on the fundamental quasinormal mode of the dominant ℓ=m=2 harmonic. By modeling the ringdown of accurate numerical relativity simulations, we find, in agreement with previous findings, that the fundamental mode alone is insufficient to recover the true underlying mass and spin, unless the analysis is started very late in the ringdown. Including higher overtones associated with this ℓ=m=2 harmonic resolves this issue and provides an unbiased estimate of the true remnant parameters. Further, including overtones allows for the modeling of the ringdown signal for all times beyond the peak strain amplitude, indicating that the linear quasinormal regime starts much sooner than previously expected. This result implies that the spacetime is well described as a linearly perturbed black hole with a fixed mass and spin as early as the peak. A model for the ringdown beginning at the peak strain amplitude can exploit the higher signal-to-noise ratio in detectors, reducing uncertainties in the extracted remnant quantities. These results should be taken into consideration when testing the no-hair theorem.
On the properties of the massive binary black hole merger GW170729
Abstract
We present a detailed investigation into the properties of GW170729, the gravitational wave with the most massive and distant source confirmed to date. We employ an extensive set of waveform models, including new improved models that incorporate the effect of higher- order waveform modes which are particularly important for massive systems. We find no indication of spin-precession, but the inclusion of higher-order modes in the models results in an improved estimate for the mass ratio of (0.3–0.8) at the 90% credible level. Our updated measurement excludes equal masses at that level. We also find that models with higher-order modes lead to the data being more consistent with a smaller effective spin, with the probability that the effective spin is greater than zero being reduced from 99% to 94%. The 90% credible interval for the effective spin parameter is now (-0.01-0.50). Additionally, the recovered signal-to-noise ratio increases by ∼0.3 units compared to analyses without higher-order modes; the overall Bayes factor in favor of the presence of higher- order modes in the data is 5.1∶1. We study the effect of common spin priors on the derived spin and mass measurements, and observe small shifts in the spins, while the masses remain unaffected. We argue that our conclusions are robust against systematic errors in the waveform models. We also compare the above waveform-based analysis which employs compact-binary waveform models to a more flexible wavelet- and chirplet-based analysis. We find consistency between the two, with overlaps of ∼0.9, typical of what is expected from simulations of signals similar to GW170729, confirming that the data are well-described by the existing waveform models. Finally, we study the possibility that the primary component of GW170729 was the remnant of a past merger of two black holes and find this scenario to be indistinguishable from the standard formation scenario.
Comparison of post-Newtonian mode amplitudes with numerical relativity simulations of binary black holes
Class.Quant.Grav. 37, 065006 (2020)
Abstract
Gravitational waves from the coalescence of two black holes carry the signature of the strong field dynamics of binary black holes. In this work we have used numerical relativity simulations and post- Newtonian theory to investigate this dynamics. Post-Newtonian theory is a low-velocity expansion that assumes the companion bodies to be point-particles, while numerical relativity treats black holes as extended objects with horizons and fully captures their dynamics. There is a priori no reason for the waveforms computed using these disparate methods to agree with each other, especially at late times when the black holes move close to the speed of light. We find, remarkably, that the leading order amplitudes in post-Newtonian theory agree well with the full general relativity solution for a large set of spherical harmonic modes, even in the most dynamical part of the binary evolution, with only some modes showing distinctly different behavior than that found by numerical relativity simulations. In particular, modes with spherical harmonic indices as well as are least modified from their dominant post- Newtonian behavior. Understanding the nature of these modes in terms of the post-Newtonian description will aid in formulating better models of the emitted waveforms in the strong field regime of the dynamics.
#2018
Surrogate model of hybridized numerical relativity binary black hole waveforms
Abstract
Numerical relativity (NR) simulations provide the most accurate binary black hole gravitational waveforms, but are prohibitively expensive for applications such as parameter estimation. Surrogate models of NR waveforms have been shown to be both fast and accurate. However, NR-based surrogate models are limited by the training waveforms’ length, which is typically about 20 orbits before merger. We remedy this by hybridizing the NR waveforms using both post- Newtonian and effective one-body waveforms for the early inspiral. We present NRHybSur3dq8, a surrogate model for hybridized nonprecessing numerical relativity waveforms, that is valid for the entire LIGO band (starting at 20 Hz) for stellar mass binaries with total masses as low as 2.25 M⊙. We include the ℓ≤4 and (5, 5) spin- weighted spherical harmonic modes but not the (4, 1) or (4, 0) modes. This model has been trained against hybridized waveforms based on 104 NR waveforms with mass ratios q≤8, and |χ1z|,|χ2z|≤0.8, where χ1z (χ2z) is the spin of the heavier (lighter) black hole in the direction of orbital angular momentum. The surrogate reproduces the hybrid waveforms accurately, with mismatches ≲3×10-4 over the mass range 2.25 M⊙≤M≤300 M⊙. At high masses (M≳40 M⊙), where the merger and ringdown are more prominent, we show roughly 2 orders of magnitude improvement over existing waveform models. We also show that the surrogate works well even when extrapolated outside its training parameter space range, including at spins as large as 0.998. Finally, we show that this model accurately reproduces the spheroidal-spherical mode mixing present in the NR ringdown signal.
Gravitational waveforms from spectral Einstein code simulations: Neutron star-neutron star and low-mass black hole-neutron star binaries
Abstract
Gravitational waveforms from numerical simulations are a critical tool to test and analytically calibrate the waveform models used to study the properties of merging compact objects. In this paper, we present a series of high-accuracy waveforms produced with the spectral Einstein code (SpEC) for systems involving at least one neutron star. We provide for the first time waveforms with subradian accuracy over more than twenty cycles for low-mass black hole- neutron star binaries, including binaries with nonspinning objects, and binaries with rapidly spinning neutron stars that maximize the impact on the gravitational wave signal of the near-resonant growth of the fundamental excitation mode of the neutron star (f-mode). We also provide for the first time with SpEC a high-accuracy neutron star-neutron star waveform. These waveforms are made publicly available as part of the SxS catalogue. We compare our results to analytical waveform models currently implemented in data analysis pipelines. For most simulations, the models lie outside of the predicted numerical errors in the last few orbits before merger, but do not show systematic deviations from the numerical results: comparing different models appears to provide reasonable estimates of the modeling errors. The sole exception is the equal-mass simulation using a rapidly counterrotating neutron star to maximize the impact of the excitation of the f-mode, for which all models perform poorly. This is however expected, as even the single model that takes f-mode excitation into account ignores the significant impact of the neutron star spin on the f-mode excitation frequency.
Imprints of r-process heating on fall-back accretion: distinguishing black hole–neutron star from double neutron star mergers
Mon.Not.Roy.Astron.Soc. 485, (2019)
Abstract
Mergers of compact binaries containing two neutron stars (NS–NS), or a neutron star and a stellar mass black hole (NS–BH), are likely progenitors of short-duration gamma-ray bursts (SGRBs). A fraction \({\gtrsim } 20{{\ \rm per\ cent}}\) of SGRBs is followed by temporally extended (≳minute-long), variable X-ray emission, attributed to ongoing activity of the central engine. One source of late-time engine activity is fall-back accretion of bound tidal ejecta; however, observed extended emission light curves do not track the naively anticipated, uninterrupted \(t^{−5/3}\) power-law decay, instead showing a lull or gap in emission typically lasting tens of seconds after the burst. Here, we re-examine the impact of heating due to rapid neutron capture (r-process) nucleosynthesis on the rate of the fall-back accretion, using ejecta properties extracted from numerical relativity simulations of NS–BH mergers. Heating by the r-process has its greatest impact on marginally bound matter, hence its relevance to late-time fall-back. Depending on the electron fraction of the ejecta and the mass of the remnant black hole, r-process heating can imprint a range of fall-back behaviour, ranging from temporal gaps of up to tens of seconds to complete late-time cut-off in the accretion rate. This behaviour is robust to realistic variations in the nuclear heating experienced by different parts of the ejecta. Central black holes with masses \({\lesssim } 3\, \mathrm{M}_{\odot }\) typically experience absolute cut-offs in the fall-back rate, while more massive \({\gtrsim } 6\!-\!8\, \mathrm{M}_{\odot }\) black holes instead show temporal gaps. We thus propose that SGRBs showing extended X-ray emission arise from NS–BH, rather than NS–NS, mergers. Our model implies an NS–BH merger detection rate by LIGO that, in steady state, is comparable to or greater than that of NS–NS mergers.
Evolving Metric Perturbations in dynamical Chern-Simons Gravity
Abstract
The stability of rotating black holes in dynamical Chern-Simons gravity (dCS) is an open question. To study this issue, we evolve the leading-order metric perturbation in order-reduced dynamical Chern-Simons gravity. The source is the leading-order dCS scalar field coupled to the spacetime curvature of a rotating black hole background. We use a well-posed, constraint-preserving scheme. We find that the leading-order metric perturbation numerically exhibits linear growth, but that the level of this growth converges to zero with numerical resolution. This analysis shows that spinning black holes in dCS gravity are numerically stable to leading-order perturbations in the metric.
The binary black hole explorer: on-the-fly visualizations of precessing binary black holes
Class.Quant.Grav. 36, 095007 (2019)
Abstract
Binary black hole mergers are of great interest to the astrophysics community, not least because of their promise to test general relativity in the highly dynamic, strong field regime. Detections of gravitational waves from these sources by LIGO and Virgo have garnered widespread media and public attention. Among these sources, precessing systems (with misaligned black-hole spin/orbital angular momentum) are of particular interest because of the rich dynamics they offer. However, these systems are, in turn, more complex compared to nonprecessing systems, making them harder to model or develop intuition about. Visualizations of numerical simulations of precessing systems provide a means to understand and gain insights about these systems. However, since these simulations are very expensive, they can only be performed at a small number of points in parameter space. We present binaryBHexp, a tool that makes use of surrogate models of numerical simulations to generate on-the-fly interactive visualizations of precessing binary black holes. These visualizations can be generated in a few seconds, and at any point in the 7-dimensional parameter space of the underlying surrogate models. With illustrative examples, we demonstrate how this tool can be used to learn about precessing binary black hole systems.
Gravitational waveforms for high spin and high mass-ratio binary black holes: A synergistic use of numerical-relativity codes
Abstract
Observation and characterization of gravitational waves from binary black holes requires accurate knowledge of the expected waveforms. The late inspiral and merger phase of the waveform is obtained through direct numerical integration of the full 3-dimensional Einstein equations. The Spectral Einstein Code (SpEC) utilizes a multi-domain pseudo-spectral method tightly adapted to the geometry of the black holes; it is computationally efficient and accurate, but—for high mass-ratios and large spins—sometimes requires manual fine-tuning for the merger-phase of binaries. The Einstein Toolkit (ET) employs finite difference methods and the moving puncture technique; it is less computationally efficient, but highly robust. For some mergers with high mass ratio and large spins, the efficient numerical algorithms used in SpEC have failed, whereas the simpler algorithms used in the ET were successful. Given the urgent need of testing the accuracy of waveform models currently used in LIGO and Virgo inference analyses for high mass ratios and spins, we present here a synergistic approach to numerical-relativity: We combine SpEC and ET waveforms into complete inspiral-merger-ringdown waveforms, taking advantage of the computational efficiency of the pseudo- spectral code during the inspiral, and the robustness of the finite- difference code at the merger. We validate our method against a case where complete waveforms from both codes are available, compute three new hybrid numerical-relativity waveforms, and compare them with analytical waveform models currently used in LIGO and Virgo science. All the waveforms and the hybridization code are publicly available.
Numerical black hole initial data and shadows in dynamical Chern–Simons gravity
Class.Quant.Grav. 36, 054001 (2019)
Abstract
We present a scheme for generating first-order metric perturbation initial data for an arbitrary background and source. We then apply this scheme to derive metric perturbations in order-reduced dynamical Chern–Simons gravity (dCS). In particular, we solve for metric perturbations on a black hole background that are sourced by a first-order dCS scalar field. This gives us the leading-order metric perturbation to the spacetime in dCS gravity. We then use these solutions to compute black hole shadows in the linearly perturbed spacetime by evolving null geodesics. We present a novel scheme to decompose the shape of the shadow into multipoles parametrized by the spin of the background black hole and the perturbation parameter . We find that we can differentiate the presence of a pure Kerr spacetime from a spacetime with a dCS perturbation using the shadow, allowing in part for a null- hypothesis test of general relativity. We then consider these results in the context of the event horizon telescope.
High-accuracy mass, spin, and recoil predictions of generic black-hole merger remnants
Phys.Rev.Lett. 122, 011101 (2019)
Abstract
We present accurate fits for the remnant properties of generically precessing binary black holes, trained on large banks of numerical- relativity simulations. We use Gaussian process regression to interpolate the remnant mass, spin, and recoil velocity in the seven-dimensional parameter space of precessing black-hole binaries with mass ratios q≤2, and spin magnitudes χ1, χ2≤0.8. For precessing systems, our errors in estimating the remnant mass, spin magnitude, and kick magnitude are lower than those of existing fitting formulae by at least an order of magnitude (improvement is also reported in the extrapolated region at high mass ratios and spins). In addition, we also model the remnant spin and kick directions. Being trained directly on precessing simulations, our fits are free from ambiguities regarding the initial frequency at which precessing quantities are defined. We also construct a model for remnant properties of aligned-spin systems with mass ratios q≤8, and spin magnitudes χ1, χ2≤0.8. As a byproduct, we also provide error estimates for all fitted quantities, which can be consistently incorporated into current and future gravitational-wave parameter- estimation analyses. Our model(s) are made publicly available through a fast and easy-to-use Python module called surfinBH.
Systematic effects from black hole-neutron star waveform model uncertainties on the neutron star equation of state
Abstract
We identify various contributors of systematic effects in the measurement of the neutron star (NS) tidal deformability and quantify their magnitude for several types of neutron star—black hole (NSBH) binaries. Gravitational waves from NSBH mergers contain information about the components’ masses and spins as well as the NS equation of state. Extracting this information requires comparison of the signal in noisy detector data with theoretical templates derived from some combination of post-Newtonian (PN) approximants, effective one-body (EOB) models, and numerical relativity (NR) simulations. The accuracy of these templates is limited by errors in the NR simulations, by the approximate nature of the PN/EOB waveforms, and by the hybridization procedure used to combine them. In this paper, we estimate the impact of these errors by constructing and comparing a set of PN-NR hybrid waveforms, for the first time with NR waveforms from two different codes, namely, SpEC and sacra, for such systems. We then attempt to recover the parameters of the binary using two non-precessing template approximants. As expected, these errors have negligible effect on detectability. Mass and spin estimates are moderately affected by systematic errors for near equal-mass binaries, while the recovered masses can be inaccurate at higher mass ratios. Large uncertainties are also found in the tidal deformability Λ, due to differences in PN base models used in hybridization, numerical relativity NR errors, and inherent limitations of the hybridization method. We find that systematic errors are too large for tidal effects to be accurately characterized for any realistic NS equation of state model. We conclude that NSBH waveform models must be significantly improved if they are to be useful for the extraction of NS equation of state information or even for distinguishing NSBH systems from binary black holes.
Comparison of binary black hole initial data sets
Abstract
We present improvements to the construction of binary black hole initial data used in the Spectral Einstein Code (SpEC). We introduce new boundary conditions for the extended conformal thin sandwich elliptic equations that enforce the excision surfaces to be slightly inside rather than on the apparent horizons, thus avoiding extrapolation into the black holes at the last stage of initial data construction. We find that this improves initial data constraint violations near and inside the apparent horizons by about 3 orders of magnitude. We construct several initial data sets that are intended to be astrophysically equivalent but use different free data, boundary conditions, and initial gauge conditions. These include free data chosen as a superposition of two black holes in time-independent horizon-penetrating harmonic and damped harmonic coordinates. We also implement initial data for which the initial gauge satisfies the harmonic and damped harmonic gauge conditions; this can be done independently of the free data, since this amounts to a choice of the time derivatives of the lapse and shift. We compare these initial data sets by evolving them. We show that the gravitational waveforms extracted during the evolution of these different initial data sets agree very well after excluding initial transients. However, we do find small differences between these waveforms, which we attribute to small differences in initial orbital eccentricity, and in initial BH masses and spins, resulting from the different choices of free data. Among the cases considered, we find that superposed harmonic initial data lead to significantly smaller transients, smaller variation in BH spins and masses during these transients, smaller constraint violations, and more computationally efficient evolutions. Finally, we study the impact of initial data choices on the construction of zero-eccentricity initial data.
Constraining the parameters of GW150914 and GW170104 with numerical relativity surrogates
Abstract
Gravitational-wave (GW) detectors have begun to observe coalescences of heavy black hole binaries (M≳50 M⊙) at a consistent pace for the past few years. Accurate models of gravitational waveforms are essential for unbiased and precise estimation of source parameters, such as masses and spins of component black holes. Recently developed surrogate models based on high-accuracy numerical relativity (NR) simulations provide ideal models for constraining physical parameters describing these heavy black hole merger events. In this paper, we first demonstrate the viability of these multi- modal surrogate models as reliable parameter estimation tools. We show that within a fully Bayesian framework, NR surrogates can help extract additional information from GW observations that is inaccessible to traditional models. We demonstrate this by analyzing a set of synthetic signals with NR surrogate templates and comparing against contemporary approximate models. We then consider the case of two of the earliest binary black holes detected by the LIGO observatories, GW150914 and GW170104. We reanalyze their data with the generically precessing NR-based surrogate model and freely provide the resulting posterior samples as supplemental material. We find that our refined analysis is able to extract information from sub-dominant GW harmonics in data, and therefore better resolve the degeneracy in measuring source luminosity distance and orbital inclination for both events. Our analysis estimates the sources of both events to be 20%–25% further away than was previously estimated. Our analysis also constrains their orbital orientations more tightly around face-on or face-off configurations than before. Additionally, for GW150914 we constrain the effective inspiral spin χeff more tightly around zero. This work is one of the first to unambiguously extract sub-dominant GW mode information from real events. It is also a first step toward eliminating the approximations used in semi-analytic waveform models from GW parameter estimation. It strongly motivates that NR surrogates be extended to cover more of the binary black hole parameter space.
Constructing a boosted, spinning black hole in the damped harmonic gauge
Abstract
The damped harmonic gauge is important for numerical relativity computations based on the generalized harmonic formulation of Einstein’s equations and is used to reduce coordinate distortions near binary black hole mergers. However, currently there is no prescription to construct quasiequilibrium binary black hole initial data in this gauge. Instead, initial data are typically constructed using a superposition of two boosted analytic single black hole solutions as free data in the solution of the constraint equations. Then, a smooth time-dependent gauge transformation is done early in the evolution to move into the damped harmonic gauge. Using this strategy to produce initial data in the damped harmonic gauge would require the solution of a single black hole in this gauge, which is not known analytically. In this work we construct a single boosted, spinning, equilibrium black hole in damped harmonic coordinates as a regular time-independent coordinate transformation from Kerr-Schild coordinates. To do this, we derive and solve a set of four coupled, nonlinear, elliptic equations for this transformation, with appropriate boundary conditions. This solution can now be used in the construction of damped harmonic initial data for binary black holes.
Distinguishing the nature of comparable-mass neutron star binary systems with multimessenger observations: GW170817 case study
Abstract
The discovery of GW170817 with gravitational waves (GWs) and electromagnetic (EM) radiation is prompting new questions in strong- gravity astrophysics. Importantly, it remains unknown whether the progenitor of the merger comprised two neutron stars (NSs) or a NS and a black hole (BH). Using new numerical-relativity simulations and incorporating modeling uncertainties, we produce novel GW and EM observables for NS-BH mergers with similar masses. A joint analysis of GW and EM measurements reveals that if GW170817 is a NS-BH merger, ≲40% of the binary parameters consistent with the GW data are compatible with EM observations.
Signatures of quark-hadron phase transitions in general-relativistic neutron-star mergers
Phys.Rev.Lett. 122, 061101 (2019)
Abstract
Merging binaries of neutron-stars are not only strong sources of gravitational waves, but also have the potential of revealing states of matter at densities and temperatures not accessible in laboratories. A crucial and long-standing question in this context is whether quarks are deconfined as a result of the dramatic increase in density and temperature following the merger. We present the first fully general-relativistic simulations of merging neutron- stars including quarks at finite temperatures that can be switched off consistently in the equation of state. Within our approach, we can determine clearly what signatures a quark-hadron phase transition would leave in the gravitational-wave signal. We show that if after the merger the conditions are met for a phase transition to take place at several times nuclear saturation density, they would lead to a postmerger signal considerably different from the one expected from the inspiral, that can only probe the hadronic part of the equations of state, and to an anticipated collapse of the merged object. We also show that the phase transition leads to a very hot and dense quark core that, when it collapses to a black hole, produces a ringdown signal different from the hadronic one. Finally, in analogy with what is done in heavy-ion collisions, we use the evolution of the temperature and density in the merger remnant to illustrate the properties of the phase transition in a QCD phase diagram.
Black hole-neutron star mergers using a survey of finite-temperature equations of state
Abstract
Each of the potential signals from a black hole–neutron star merger should contain an imprint of the neutron star equation of state: gravitational waves via its effect on tidal disruption, the kilonova via its effect on the ejecta, and the gamma-ray burst via its effect on the remnant disk. These effects have been studied by numerical simulations and quantified by semianalytic formulas. However, most of the simulations on which these formulas are based use equations of state without finite temperature and composition-dependent nuclear physics. In this paper, we simulate black hole–neutron star mergers varying both the neutron star mass and the equation of state, using three finite-temperature nuclear models of varying stiffness. Our simulations largely vindicate formulas for ejecta properties but do not find the expected dependence of disk mass on neutron star compaction. We track the early evolution of the accretion disk, largely driven by shocking and fallback inflow, and do find notable equation-of-state effects on the structure of this early-time, neutrino-bright disk.
Measuring the properties of nearly extremal black holes with gravitational waves
Abstract
Characterizing the properties of black holes is one of the most important science objectives for gravitational-wave observations. Astrophysical evidence suggests that black holes that are nearly extremal (i.e., spins near the theoretical upper limit) might exist and, thus, might be among the merging black holes observed with gravitational waves. In this paper, we explore how well current gravitational wave parameter estimation methods can measure the spins of rapidly spinning black holes in binaries. We simulate gravitational-wave signals using numerical-relativity waveforms for nearly-extremal, merging black holes. For simplicity, we confine our attention to binaries with spins parallel or antiparallel with the orbital angular momentum. We find that recovering the holes’ nearly extremal spins is challenging. When the spins are nearly extremal and parallel to each other, the resulting parameter estimates do recover spins that are large, though the recovered spin magnitudes are still significantly smaller than the true spin magnitudes. When the spins are nearly extremal and antiparallel to each other, the resulting parameter estimates recover the small effective spin but incorrectly estimate the individual spins as nearly zero. We study the effect of spin priors and argue that a commonly used prior (uniform in spin magnitude and direction) hinders unbiased recovery of large black-hole spins.
General-relativistic neutron star evolutions with the discontinuous Galerkin method
Abstract
Simulations of relativistic hydrodynamics often need both high accuracy and robust shock-handling properties. The discontinuous Galerkin method combines these features—a high order of convergence in regions where the solution is smooth and shock-capturing properties for regions where it is not—with geometric flexibility and is therefore well suited to solve the partial differential equations describing astrophysical scenarios. We present here evolutions of a general-relativistic neutron star with the discontinuous Galerkin method. In these simulations, we simultaneously evolve the spacetime geometry and the matter on the same computational grid, which we conform to the spherical geometry of the problem. To verify the correctness of our implementation, we perform standard convergence and shock tests. We then show results for evolving, in three dimensions, a Kerr black hole; a neutron star in the Cowling approximation (holding the spacetime metric fixed); and, finally, a neutron star where the spacetime and matter are both dynamical. The evolutions show long-term stability, good accuracy, and an improved rate of convergence versus a comparable-resolution finite-volume method.
Detection and characterization of spin-orbit resonances in the advanced gravitational wave detectors era
Abstract
Spin-orbit resonances have important astrophysical implications as the evolution and subsequent coalescence of supermassive black hole binaries in one of these configurations may lead to low recoil velocity of merger remnants. It has also been shown that black hole spins in comparable mass stellar-mass black hole binaries could preferentially lie in a resonant plane when their gravitational waves (GWs) enter the advanced LIGO frequency band [1]. Therefore, it is highly desirable to investigate the possibility of detection and subsequent characterization of such GW sources in the advanced detector era, which can, in turn, improve our perception of their high mass counterparts. The current detection pipelines involve only nonprecessing templates for compact binary searches whereas parameter estimation pipelines can afford to use approximate precessing templates. In this paper, we test the performance of these templates in detection and characterization of spin-orbit resonant binaries. We use fully precessing time-domain SEOBNRv3 waveforms as well as four numerical relativity (NR) waveforms to model GWs from spin-orbit resonant binaries and filter them through IMRPhenomD, SEOBNRv4 and IMRPhenomPv2 approximants. We find that the nonprecessing approximants IMRPhenomD and SEOBNRv4 recover only ∼70% of injections with fitting factor (FF) higher than 0.97 (or 90% of injections with FF>0.9). This loss in signal-to-noise ratio is mainly due to the missing physics in these approximants in terms of precession and nonquadrupole modes. However, if we use a new statistic, i.e., maximizing the matched filter output over the sky- location parameters as well, the precessing approximant IMRPhenomPv2 performs magnificently better than their nonprecessing counterparts with recovering 99% of the injections with FFs higher than 0.97. Interestingly, injections with Δϕ=180° have higher FFs (Δϕ is the angle between the components of the black hole spins in the plane orthogonal to the orbital angular momentum) as compared to their Δϕ=0° and generic counterparts. This is because Δϕ=180° binaries are not as strongly precessing as Δϕ=0° and generic binaries. This implies that we will have a slight observation bias towards Δϕ=180° and away from Δϕ=0° resonant binaries while using nonprecessing templates for searches. Moreover, all template approximants are able to recover most of the injected NR waveforms with FFs >0.95. For all the injections including NR, the systematic error in estimating chirp mass remains below <10% with minimum error for Δϕ=180° resonant binaries. The symmetric mass-ratio can be estimated with errors below 15%. The effective spin parameter χeff is measured with maximum absolute error of 0.13. The in-plane spin parameter χp is mostly underestimated indicating that a precessing signal will be recovered as a relatively less precessing signal. Based on our findings, we conclude that we not only need improvements in waveform models towards precession and nonquadrupole modes but also better search strategies for precessing GW signals.
Critical behavior in 3D gravitational collapse of massless scalar fields
Abstract
We present results from a study of critical behavior in 3D gravitational collapse with no symmetry assumptions. The source of the gravitational field is a massless scalar field. This is a well- studied case for spherically symmetric gravitational collapse, allowing us to understand the reliability and accuracy of the simulations. We study both supercritical and subcritical evolutions to see if one provides more accurate results than the other. We find that even for nonspherical initial data with 35% of the power in the ℓ=2 spherical harmonic, the critical solution is the same as in spherical symmetry.
#2017
On choosing the start time of binary black hole ringdowns
Abstract
The final stage of a binary black hole merger is ringdown, in which the system is described by a Kerr black hole with quasinormal mode perturbations. It is far from straightforward to identify the time at which the ringdown begins. Yet determining this time is important for precision tests of the general theory of relativity that compare an observed signal with quasinormal mode descriptions of the ringdown, such as tests of the no-hair theorem. We present an algorithmic method to analyze the choice of ringdown start time in the observed waveform. This method is based on determining how close the strong field is to a Kerr black hole (Kerrness). Using numerical relativity simulations, we characterize the Kerrness of the strong- field region close to the black hole using a set of local, gauge- invariant geometric and algebraic conditions that measure local isometry to Kerr. We produce a map that associates each time in the gravitational waveform with a value of each of these Kerrness measures; this map is produced by following outgoing null characteristics from the strong and near-field regions to the wave zone. We perform this analysis on a numerical relativity simulation with parameters consistent with GW150914—the first gravitational- wave detection. We find that the choice of ringdown start time of 3 ms after merger used in the GW150914 study [B. P. Abbott (Virgo Collaboration and LIGO Scientific Collaboration), Phys. Rev. Lett. 116, 221101 (2016).PRLTAO0031-900710.1103/PhysRevLett.116.221101] to test general relativity corresponds to a high dimensionless perturbation amplitude of ∼7.5×10-3 in the strong-field region. This suggests that in higher signal-to-noise detections, one would need to start analyzing the signal at a later time for studies that depend on the validity of black hole perturbation theory.
Evolution of the Magnetized, Neutrino-Cooled Accretion Disk in the Aftermath of a Black Hole Neutron Star Binary Merger
Abstract
Black hole–torus systems from compact binary mergers are possible engines for gamma-ray bursts (GRBs). During the early evolution of the postmerger remnant, the state of the torus is determined by a combination of neutrino cooling and magnetically driven heating processes, so realistic models must include both effects. In this paper, we study the postmerger evolution of a magnetized black hole–neutron star binary system using the Spectral Einstein Code (SpEC) from an initial postmerger state provided by previous numerical relativity simulations. We use a finite-temperature nuclear equation of state and incorporate neutrino effects in a leakage approximation. To achieve the needed accuracy, we introduce improvements to SpEC’s implementation of general-relativistic magnetohydrodynamics (MHD), including the use of cubed-sphere multipatch grids and an improved method for dealing with supersonic accretion flows where primitive variable recovery is difficult. We find that a seed magnetic field triggers a sustained source of heating, but its thermal effects are largely cancelled by the accretion and spreading of the torus from MHD-related angular momentum transport. The neutrino luminosity peaks at the start of the simulation, and then drops significantly over the first 20 ms but in roughly the same way for magnetized and nonmagnetized disks. The heating rate and disk’s luminosity decrease much more slowly thereafter. These features of the evolution are insensitive to grid structure and resolution, formulation of the MHD equations, and seed field strength, although turbulent effects are not fully converged.
Eccentric binary black hole inspiral-merger-ringdown gravitational waveform model from numerical relativity and post-Newtonian theory
Abstract
We present a prescription for computing gravitational waveforms for the inspiral, merger and ringdown of nonspinning moderately eccentric binary black hole systems. The inspiral waveform is computed using the post-Newtonian expansion and the merger waveform is computed by interpolating a small number of quasicircular NR waveforms. The use of circular merger waveforms is possible because binaries with moderate eccentricity circularize in the last few cycles before the merger, which we demonstrate up to mass ratio q=m1/m2=3. The complete model is calibrated to 23 numerical relativity (NR) simulations starting ≈20 cycles before the merger with eccentricities eref≤0.1 and mass ratios q≤3, where eref is the eccentricity ≈7 cycles before the merger. The NR waveforms are long enough that they start below 30 Hz (10 Hz) for BBH systems with total mass M≥80 M⊙ (230 M⊙). We find that, for the sensitivity of advanced LIGO at the time of its first observing run, the eccentric model has a faithfulness with NR of over 97% for systems with total mass M≥85M⊙ across the parameter space (eref≤0.1, q≤3). For systems with total mass M≥70M⊙, the faithfulness is over 97% for eref≲0.05 and q≤3. The NR waveforms and the Mathematica code for the model are publicly available.
Fundamental frequencies and resonances from eccentric and precessing binary black hole inspirals
Classical and Quantum Gravity 34, 124001 (2017)
Abstract
Binary black holes which are both eccentric and undergo precession remain unexplored in numerical simulations. We present simulations of such systems which cover about 50 orbits at comparatively high mass ratios 5 and 7. The configurations correspond to the generic motion of a nonspinning body in a Kerr spacetime, and are chosen to study the transition from finite mass-ratio inspirals to point particle motion in Kerr. We develop techniques to extract analogs of the three fundamental frequencies of Kerr geodesics, compare our frequencies to those of Kerr, and show that the differences are consistent with self-force corrections entering at first order in mass ratio. This analysis also locates orbital resonances where the ratios of our frequencies take rational values. At the considered mass ratios, the binaries pass through resonances in one to two resonant cycles, and we find no discernible effects on the orbital evolution. We also compute the decay of eccentricity during the inspiral and find good agreement with the leading order post- Newtonian prediction.
Parameter estimation method that directly compares gravitational wave observations to numerical relativity
Abstract
We present and assess a Bayesian method to interpret gravitational wave signals from binary black holes. Our method directly compares gravitational wave data to numerical relativity (NR) simulations. In this study, we present a detailed investigation of the systematic and statistical parameter estimation errors of this method. This procedure bypasses approximations used in semianalytical models for compact binary coalescence. In this work, we use the full posterior parameter distribution for only generic nonprecessing binaries, drawing inferences away from the set of NR simulations used, via interpolation of a single scalar quantity (the marginalized log likelihood, lnL) evaluated by comparing data to nonprecessing binary black hole simulations. We also compare the data to generic simulations, and discuss the effectiveness of this procedure for generic sources. We specifically assess the impact of higher order modes, repeating our interpretation with both l≤2 as well as l≤3 harmonic modes. Using the l≤3 higher modes, we gain more information from the signal and can better constrain the parameters of the gravitational wave signal. We assess and quantify several sources of systematic error that our procedure could introduce, including simulation resolution and duration; most are negligible. We show through examples that our method can recover the parameters for equal mass, zero spin, GW150914-like, and unequal mass, precessing spin sources. Our study of this new parameter estimation method demonstrates that we can quantify and understand the systematic and statistical error. This method allows us to use higher order modes from numerical relativity simulations to better constrain the black hole binary parameters.
Numerical binary black hole mergers in dynamical Chern-Simons gravity: Scalar field
Abstract
Testing general relativity in the nonlinear, dynamical, strong-field regime of gravity is one of the major goals of gravitational wave astrophysics. Performing precision tests of general relativity (GR) requires numerical inspiral, merger, and ringdown waveforms for binary black hole (BBH) systems in theories beyond GR. Currently, GR and scalar-tensor gravity are the only theories amenable to numerical simulations. In this article, we present a well-posed perturbation scheme for numerically integrating beyond-GR theories that have a continuous limit to GR. We demonstrate this scheme by simulating BBH mergers in dynamical Chern-Simons gravity (dCS), to linear order in the perturbation parameter. We present mode waveforms and energy fluxes of the dCS pseudoscalar field from our numerical simulations. We find good agreement with analytic predictions at early times, including the absence of pseudoscalar dipole radiation. We discover new phenomenology only accessible through numerics: a burst of dipole radiation during merger. We also quantify the self-consistency of the perturbation scheme. Finally, we estimate bounds that GR-consistent LIGO detections could place on the new dCS length scale, approximately ℓ≲O(10) km.
Numerical relativity waveform surrogate model for generically precessing binary black hole mergers
Abstract
A generic, noneccentric binary black hole (BBH) system emits gravitational waves (GWs) that are completely described by seven intrinsic parameters: the black hole spin vectors and the ratio of their masses. Simulating a BBH coalescence by solving Einstein’s equations numerically is computationally expensive, requiring days to months of computing resources for a single set of parameter values. Since theoretical predictions of the GWs are often needed for many different source parameters, a fast and accurate model is essential. We present the first surrogate model for GWs from the coalescence of BBHs including all seven dimensions of the intrinsic noneccentric parameter space. The surrogate model, which we call NRSur7dq2, is built from the results of 744 numerical relativity simulations. NRSur7dq2 covers spin magnitudes up to 0.8 and mass ratios up to 2, includes all ℓ≤4 modes, begins about 20 orbits before merger, and can be evaluated in ∼50 ms. We find the largest NRSur7dq2 errors to be comparable to the largest errors in the numerical relativity simulations, and more than an order of magnitude smaller than the errors of other waveform models. Our model, and more broadly the methods developed here, will enable studies that were not previously possible when using highly accurate waveforms, such as parameter inference and tests of general relativity with GW observations.
Effects of waveform model systematics on the interpretation of GW150914
Classical and Quantum Gravity 34, 104002 (2017)
Abstract
Parameter estimates of GW150914 were obtained using Bayesian inference, based on three semi-analytic waveform models for binary black hole coalescences. These waveform models differ from each other in their treatment of black hole spins, and all three models make some simplifying assumptions, notably to neglect sub-dominant waveform harmonic modes and orbital eccentricity. Furthermore, while the models are calibrated to agree with waveforms obtained by full numerical solutions of Einstein’s equations, any such calibration is accurate only to some non-zero tolerance and is limited by the accuracy of the underlying phenomenology, availability, quality, and parameter-space coverage of numerical simulations. This paper complements the original analyses of GW150914 with an investigation of the effects of possible systematic errors in the waveform models on estimates of its source parameters. To test for systematic errors we repeat the original Bayesian analysis on mock signals from numerical simulations of a series of binary configurations with parameters similar to those found for GW150914. Overall, we find no evidence for a systematic bias relative to the statistical error of the original parameter recovery of GW150914 due to modeling approximations or modeling inaccuracies. However, parameter biases are found to occur for some configurations disfavored by the data of GW150914: for binaries inclined edge-on to the detector over a small range of choices of polarization angles, and also for eccentricities greater than ∼0.05. For signals with higher signal-to-noise ratio than GW150914, or in other regions of the binary parameter space (lower masses, larger mass ratios, or higher spins), we expect that systematic errors in current waveform models may impact gravitational-wave measurements, making more accurate models desirable for future observations.
A Surrogate model of gravitational waveforms from numerical relativity simulations of precessing binary black hole mergers
Physical Review D 95, 104023 (2017)
Abstract
We present the first surrogate model for gravitational waveforms from the coalescence of precessing binary black holes. We call this surrogate model NRSur4d2s. Our methodology significantly extends recently introduced reduced-order and surrogate modeling techniques, and is capable of directly modeling numerical relativity waveforms without introducing phenomenological assumptions or approximations to general relativity. Motivated by GW150914, LIGO’s first detection of gravitational waves from merging black holes, the model is built from a set of 276 numerical relativity (NR) simulations with mass ratios q≤2, dimensionless spin magnitudes up to 0.8, and the restriction that the initial spin of the smaller black hole lies along the axis of orbital angular momentum. It produces waveforms which begin ∼30 gravitational wave cycles before merger and continue through ringdown, and which contain the effects of precession as well as all ℓ∈{2,3} spin-weighted spherical-harmonic modes. We perform cross-validation studies to compare the model to NR waveforms not used to build the model and find a better agreement within the parameter range of the model than other, state-of-the-art precessing waveform models, with typical mismatches of 10-3. We also construct a frequency domain surrogate model (called NRSur4d2s_FDROM) which can be evaluated in 50 ms and is suitable for performing parameter estimation studies on gravitational wave detections similar to GW150914.
The influence of neutrinos on r-process nucleosynthesis in the ejecta of black hole-neutron star mergers
Mon. Not. Roy. Astron. Soc. 464, 3907 (2017)
Abstract
During the merger of a black hole and a neutron star, baryonic mass can become unbound from the system. Because the ejected material is extremely neutron-rich, the r-process rapidly synthesizes heavy nuclides as the material expands and cools. In this work, we map general relativistic models of black hole–neutron star mergers into a Newtonian smoothed particle hydrodynamics (SPH) code and follow the evolution of the thermodynamics and morphology of the ejecta until the outflows become homologous. We investigate how the subsequent evolution depends on our mapping procedure and find that the results are robust. Using thermodynamic histories from the SPH particles, we then calculate the expected nucleosynthesis in these outflows while varying the level of neutrino irradiation coming from the post-merger accretion disc. We find that the ejected material robustly produces r-process nucleosynthesis even for unrealistically high neutrino luminosities, due to the rapid velocities of the outflow. None the less, we find that neutrinos can have an impact on the detailed pattern of the r-process nucleosynthesis. Electron neutrinos are captured by neutrons to produce protons while neutron capture is occurring. The produced protons rapidly form low-mass seed nuclei for the r-process. These low-mass seeds are eventually incorporated into the first r-process peak at A ∼ 78. We consider the mechanism of this process in detail and discuss if it can impact galactic chemical evolution of the first peak r-process nuclei.
Dynamical ejecta from precessing neutron star-black hole mergers with a hot, nuclear-theory based equation of state
Classical and Quantum Gravity 34, 044002 (2017)
Abstract
Neutron star-black hole binaries are among the strongest sources of gravitational waves detectable by current observatories. They can also power bright electromagnetic signals (gamma-ray bursts, kilonovae), and may be a significant source of production of r-process nuclei. A misalignment of the black hole spin with respect to the orbital angular momentum leads to precession of that spin and of the orbital plane, and has a significant effect on the properties of the post-merger remnant and of the material ejected by the merger. We present a first set of simulations of precessing neutron star-black hole mergers using a hot, composition dependent, nuclear- theory based equation of state (DD2). We show that the mass of the remnant and of the dynamical ejecta are broadly consistent with the result of simulations using simpler equations of state, while differences arise when considering the dynamics of the merger and the velocity of the ejecta. We show that the latter can easily be understood from assumptions about the composition of low-density, cold material in the different equations of state, and propose an updated estimate for the ejecta velocity which takes those effects into account. We also present an updated mesh-refinement algorithm which allows us to improve the numerical resolution used to evolve neutron star-black hole mergers.
Improved effective-one-body model of spinning, nonprecessing binary black holes for the era of gravitational-wave astrophysics with advanced detectors
Physical Review D 95, 044028 (2017)
Abstract
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.
Complete waveform model for compact binaries on eccentric orbits
Physical Review D 95, 024038 (2017)
Abstract
We present a time domain waveform model that describes the inspiral, merger and ringdown of compact binary systems whose components are nonspinning, and which evolve on orbits with low to moderate eccentricity. The inspiral evolution is described using third-order post-Newtonian equations both for the equations of motion of the binary, and its far-zone radiation field. This latter component also includes instantaneous, tails and tails-of-tails contributions, and a contribution due to nonlinear memory. This framework reduces to the post-Newtonian approximant TaylorT4 at third post-Newtonian order in the zero-eccentricity limit. To improve phase accuracy, we also incorporate higher-order post-Newtonian corrections for the energy flux of quasicircular binaries and gravitational self-force corrections to the binding energy of compact binaries. This enhanced prescription for the inspiral evolution is combined with a fully analytical prescription for the merger-ringdown evolution constructed using a catalog of numerical relativity simulations. We show that this inspiral-merger-ringdown waveform model reproduces the effective-one-body model of Ref. [Y. Pan , Phys. Rev. D 89, 061501 (2014).PRVDAQ1550-799810.1103/PhysRevD.89.061501] for quasicircular black hole binaries with mass ratios between 1 to 15 in the zero-eccentricity limit over a wide range of the parameter space under consideration. Using a set of eccentric numerical relativity simulations, not used during calibration, we show that our new eccentric model reproduces the true features of eccentric compact binary coalescence throughout merger. We use this model to show that the gravitational-wave transients GW150914 and GW151226 can be effectively recovered with template banks of quasicircular, spin-aligned waveforms if the eccentricity e0 of these systems when they enter the aLIGO band at a gravitational-wave frequency of 14 Hz satisfies e0GW150914≤0.15 and e0GW151226≤0.1. We also find that varying the spin combinations of the quasicircular, spin-aligned template waveforms does not improve the recovery of nonspinning, eccentric signals when e0≥0.1. This suggests that these two signal manifolds are predominantly orthogonal.
#2016
Modeling the source of GW150914 with targeted numerical-relativity simulations
Classical and Quantum Gravity 33, 244002 (2016)
Abstract
In fall of 2015, the two LIGO detectors measured the gravitational wave signal GW150914, which originated from a pair of merging black holes (Abbott et al Virgo, LIGO Scientific 2016 Phys. Rev. Lett. 116 061102). In the final 0.2 s (about 8 gravitational-wave cycles) before the amplitude reached its maximum, the observed signal swept up in amplitude and frequency, from 35 Hz to 150 Hz. The theoretical gravitational-wave signal for merging black holes, as predicted by general relativity, can be computed only by full numerical relativity, because analytic approximations fail near the time of merger. Moreover, the nearly-equal masses, moderate spins, and small number of orbits of GW150914 are especially straightforward and efficient to simulate with modern numerical-relativity codes. In this paper, we report the modeling of GW150914 with numerical- relativity simulations, using black-hole masses and spins consistent with those inferred from LIGO’s measurement (Abbott et al LIGO Scientific Collaboration, Virgo Collaboration 2016 Phys. Rev. Lett. 116 241102). In particular, we employ two independent numerical- relativity codes that use completely different analytical and numerical methods to model the same merging black holes and to compute the emitted gravitational waveform, we find excellent agreement between the waveforms produced by the two independent codes. These results demonstrate the validity, impact, and potential of current and future studies using rapid-response, targeted numerical-relativity simulations for better understanding gravitational-wave observations.
Impact of an improved neutrino energy estimate on outflows in neutron star merger simulations
Physical Review D 94, 123016 (2016)
Abstract
Binary neutron star mergers are promising sources of gravitational waves for ground-based detectors such as Advanced LIGO. Neutron-rich material ejected by these mergers may also be the main source of r-process elements in the Universe, while radioactive decays in the ejecta can power bright electromagnetic postmerger signals. Neutrino-matter interactions play a critical role in the evolution of the composition of the ejected material, which significantly impacts the outcome of nucleosynthesis and the properties of the associated electromagnetic signal. In this work, we present a simulation of a binary neutron star merger using an improved method for estimating the average neutrino energies in our energy- integrated neutrino transport scheme. These energy estimates are obtained by evolving the neutrino number density in addition to the neutrino energy and flux densities. We show that significant changes are observed in the composition of the polar ejecta when comparing our new results with earlier simulations in which the neutrino spectrum was assumed to be the same everywhere in optically thin regions. In particular, we find that material ejected in the polar regions is less neutron rich than previously estimated. Our new estimates of the composition of the polar ejecta make it more likely that the color and time scale of the electromagnetic signal depend on the orientation of the binary with respect to an observer’s line of sight. These results also indicate that important observable properties of neutron star mergers are sensitive to the neutrino energy spectrum, and may need to be studied through simulations including a more accurate, energy-dependent neutrino transport scheme.
Redshift Factor and the First Law of Binary Black Hole Mechanics in Numerical Simulations
Physical Review Letters 117, 191101 (2016)
Abstract
The redshift factor z is an invariant quantity of fundamental interest in post-Newtonian and self-force descriptions of compact binaries. It connects different approximation schemes, and plays a central role in the first law of binary black hole mechanics, which links local quantities to asymptotic measures of energy and angular momentum in these systems. Through this law, the redshift factor is conjectured to have a close relation to the surface gravity of the event horizons of black holes in circular orbits. We propose and implement a novel method for extracting the redshift factor on apparent horizons in numerical simulations of quasicircular binary inspirals. Our results confirm the conjectured relationship between z and the surface gravity of the holes and that the first law holds to a remarkable degree for binary inspirals. The redshift factor enables tests of analytic predictions for z in spacetimes where the binary is only approximately circular, giving a new connection between analytic approximations and numerical simulations.
Initial data for black hole-neutron star binaries, with rotating stars
Classical and Quantum Gravity 33, 225012 (2016)
Abstract
The coalescence of a neutron star with a black hole is a primary science target of ground-based gravitational wave detectors. Constraining or measuring the neutron star spin directly from gravitational wave observations requires knowledge of the dependence of the emission properties of these systems on the neutron star spin. This paper lays foundations for this task, by developing a numerical method to construct initial data for black hole–neutron star binaries with arbitrary spin on the neutron star. We demonstrate the robustness of the code by constructing initial-data sets in large regions of the parameter space. In addition to varying the neutron star spin-magnitude and spin-direction, we also explore neutron star compactness, mass-ratio, black hole spin, and black hole spin-direction. Specifically, we are able to construct initial data sets with neutron stars spinning near centrifugal break-up, and with black hole spins as large as \({S}_{\mathrm{BH}}/{M}_{\mathrm{BH}}^{2}=0.99\).
Directly comparing GW150914 with numerical solutions of Einstein's equations for binary black hole coalescence
Physical Review D 94, 064035 (2016)
Abstract
We compare GW150914 directly to simulations of coalescing binary black holes in full general relativity, including several performed specifically to reproduce this event. Our calculations go beyond existing semianalytic models, because for all simulations—including sources with two independent, precessing spins—we perform comparisons which account for all the spin-weighted quadrupolar modes, and separately which account for all the quadrupolar and octopolar modes. Consistent with the posterior distributions reported by Abbott et al. [Phys. Rev. Lett. 116, 241102 (2016)] (at the 90% credible level), we find the data are compatible with a wide range of nonprecessing and precessing simulations. Follow-up simulations performed using previously estimated binary parameters most resemble the data, even when all quadrupolar and octopolar modes are included. Comparisons including only the quadrupolar modes constrain the total redshifted mass Mz∈[64 M⊙-82 M⊙], mass ratio 1/q=m2/m1∈[0.6,1], and effective aligned spin χeff∈[-0.3,0.2], where χeff=(S1/m1+S2/m2)·L^/M. Including both quadrupolar and octopolar modes, we find the mass ratio is even more tightly constrained. Even accounting for precession, simulations with extreme mass ratios and effective spins are highly inconsistent with the data, at any mass. Several nonprecessing and precessing simulations with similar mass ratio and χeff are consistent with the data. Though correlated, the components’ spins (both in magnitude and directions) are not significantly constrained by the data: the data is consistent with simulations with component spin magnitudes a1,2 up to at least 0.8, with random orientations. Further detailed follow-up calculations are needed to determine if the data contain a weak imprint from transverse (precessing) spins. For nonprecessing binaries, interpolating between simulations, we reconstruct a posterior distribution consistent with previous results. The final black hole’s redshifted mass is consistent with Mf,z in the range 64.0 M⊙-73.5 M⊙ and the final black hole’s dimensionless spin parameter is consistent with af=0.62–0.73. As our approach invokes no intermediate approximations to general relativity and can strongly reject binaries whose radiation is inconsistent with the data, our analysis provides a valuable complement to Abbott et al. [Phys. Rev. Lett. 116, 241102 (2016)].
How should spin-weighted spherical functions be defined?
Journal of Mathematical Physics 57, 092504 (2016)
Abstract
Spin-weighted spherical functions provide a useful tool for analyzing tensor-valued functions on the sphere. A tensor field can be decomposed into complex-valued functions by taking contractions with tangent vectors on the sphere and the normal to the sphere. These component functions are usually presented as functions on the sphere itself, but this requires an implicit choice of distinguished tangent vectors with which to contract. Thus, we may more accurately say that spin-weighted spherical functions are functions of both a point on the sphere and a choice of frame in the tangent space at that point. The distinction becomes extremely important when transforming the coordinates in which these functions are expressed, because the implicit choice of frame will also transform. Here, it is proposed that spin-weighted spherical functions should be treated as functions on the spin group. This approach more cleanly reflects the geometry involved, and allows for a more elegant description of the behavior of spin-weighted functions. In this form, the spin- weighted spherical harmonics have simple expressions as elements of the Wigner \(\mathfrak{D}\) representations, and transformations under rotation are simple. Two variants of the angular-momentum operator are defined directly in terms of the spin group; one is the standard angular-momentum operator \(\mathbf{L}\), while the other is shown to be related to the spin-raising operator \(\eth\). Computer code is also included, providing an explicit implementation of the spin- weighted spherical harmonics in this form.
Toroidal horizons in binary black hole mergers
Physical Review D 94, 064009 (2016)
Abstract
We find the first binary black hole event horizon with a toroidal topology. It has been predicted that generically the event horizons of merging black holes should briefly have a toroidal topology. However, such a phase has never been seen in numerical simulations. Instead, in all previous simulations, the topology of the event horizon transitions directly from two spheres during the inspiral to a single sphere as the black holes merge. We find a coordinate transformation to a foliation of spacelike hypersurfaces that “cut a hole” through the event horizon surface, resulting in a toroidal event horizon, thus reconciling the numerical work with theoretical expectations. The demonstration requires extremely high numerical precision, which is made possible by a new event horizon code described in a companion paper. A torus could potentially provide a mechanism for violating topological censorship. However, these toroidal event horizons satisfy topological censorship by construction, because we can always trivially apply the inverse coordinate transformation to remove the topological feature.
Parallel adaptive event horizon finder for numerical relativity
Physical Review D 94, 064008 (2016)
Abstract
With Advanced LIGO detecting the gravitational waves emitted from a pair of merging black holes in late 2015, we have a new perspective into the strong field regime of binary black hole systems. Event horizons are the defining features of such black hole spacetimes. We introduce a new code for locating event horizons in numerical simulations based on a Delaunay triangulation on a topological sphere. The code can automatically refine arbitrary regions of the event horizon surface to find and explore features such as the hole in a toroidal event horizon, as discussed in our companion paper. We also investigate various ways of integrating the geodesic equation and find evolution equations that can be integrated efficiently with high accuracy.
SpECTRE: A task-based discontinuous Galerkin code for relativistic astrophysics
J.Comput.Phys. 335, 7061 (2017)
Abstract
We introduce a new relativistic astrophysics code, SpECTRE, that combines a discontinuous Galerkin method with a task-based parallelism model. SpECTRE's goal is to achieve more accurate solutions for challenging relativistic astrophysics problems such as core-collapse supernovae and binary neutron star mergers. The robustness of the discontinuous Galerkin method allows for the use of high-resolution shock capturing methods in regions where (relativistic) shocks are found, while exploiting high-order accuracy in smooth regions. A task-based parallelism model allows efficient use of the largest supercomputers for problems with a heterogeneous workload over disparate spatial and temporal scales. We argue that the locality and algorithmic structure of discontinuous Galerkin methods will exhibit good scalability within a task-based parallelism framework. We demonstrate the code on a wide variety of challenging benchmark problems in (non)-relativistic (magneto)-hydrodynamics. We demonstrate the code's scalability including its strong scaling on the NCSA Blue Waters supercomputer up to the machine's full capacity of nodes using threads.
On the accuracy and precision of numerical waveforms: Effect of waveform extraction methodology
Classical and Quantum Gravity 33, 165001 (2016)
Abstract
We present a new set of 95 numerical relativity simulations of non- precessing binary black holes (BBHs). The simulations sample comprehensively both black-hole spins up to spin magnitude of 0.9, and cover mass ratios 1–3. The simulations cover on average 24 inspiral orbits, plus merger and ringdown, with low initial orbital eccentricities \(e\lt {10}^{-4}\). A subset of the simulations extends the coverage of non-spinning BBHs up to mass ratio q = 10. Gravitational waveforms at asymptotic infinity are computed with two independent techniques: extrapolation and Cauchy characteristic extraction. An error analysis based on noise-weighted inner products is performed. We find that numerical truncation error, error due to gravitational wave extraction, and errors due to the Fourier transformation of signals with finite length of the numerical waveforms are of similar magnitude, with gravitational wave extraction errors dominating at noise-weighted mismatches of \(\sim 3\times {10}^{-4}\). This set of waveforms will serve to validate and improve aligned-spin waveform models for gravitational wave science.
Properties of the Binary Black Hole Merger GW150914
Physical Review Letters 116, 241102 (2016)
Abstract
On September 14, 2015, the Laser Interferometer Gravitational-Wave Observatory (LIGO) detected a gravitational-wave transient (GW150914); we characterize the properties of the source and its parameters. The data around the time of the event were analyzed coherently across the LIGO network using a suite of accurate waveform models that describe gravitational waves from a compact binary system in general relativity. GW150914 was produced by a nearly equal mass binary black hole of masses 36-4+5M⊙ and 29-4+4M⊙; for each parameter we report the median value and the range of the 90% credible interval. The dimensionless spin magnitude of the more massive black hole is bound to be <0.7 (at 90% probability). The luminosity distance to the source is 410-180+160 Mpc, corresponding to a redshift 0.09-0.04+0.03 assuming standard cosmology. The source location is constrained to an annulus section of 610 deg2, primarily in the southern hemisphere. The binary merges into a black hole of mass 62-4+4M⊙ and spin 0.67-0.07+0.05. This black hole is significantly more massive than any other inferred from electromagnetic observations in the stellar-mass regime.
GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence
Physical Review Letters 116, 241103 (2016)
Abstract
We report the observation of a gravitational-wave signal produced by the coalescence of two stellar-mass black holes. The signal, GW151226, was observed by the twin detectors of the Laser Interferometer Gravitational-Wave Observatory (LIGO) on December 26, 2015 at 03:38:53 UTC. The signal was initially identified within 70 s by an online matched-filter search targeting binary coalescences. Subsequent off-line analyses recovered GW151226 with a network signal-to-noise ratio of 13 and a significance greater than 5σ. The signal persisted in the LIGO frequency band for approximately 1 s, increasing in frequency and amplitude over about 55 cycles from 35 to 450 Hz, and reached a peak gravitational strain of 3.4+0.7−0.9×10−22. The inferred source-frame initial black hole masses are 14.2+8.3−3.7M⊙ and 7.5+2.3−2.3M⊙, and the final black hole mass is 20.8+6.1−1.7M⊙. We find that at least one of the component black holes has spin greater than 0.2. This source is located at a luminosity distance of 440+180−190 Mpc corresponding to a redshift of 0.09+0.03−0.04. All uncertainties define a 90% credible interval. This second gravitational-wave observation provides improved constraints on stellar populations and on deviations from general relativity.
Tests of General Relativity with GW150914
Physical Review Letters 116, 221101 (2016)
Abstract
The LIGO detection of GW150914 provides an unprecedented opportunity to study the two-body motion of a compact-object binary in the large-velocity, highly nonlinear regime, and to witness the final merger of the binary and the excitation of uniquely relativistic modes of the gravitational field. We carry out several investigations to determine whether GW150914 is consistent with a binary black-hole merger in general relativity. We find that the final remnant’s mass and spin, as determined from the low-frequency (inspiral) and high-frequency (postinspiral) phases of the signal, are mutually consistent with the binary black-hole solution in general relativity. Furthermore, the data following the peak of GW150914 are consistent with the least-damped quasinormal mode inferred from the mass and spin of the remnant black hole. By using waveform models that allow for parametrized general-relativity violations during the inspiral and merger phases, we perform quantitative tests on the gravitational-wave phase in the dynamical regime and we determine the first empirical bounds on several high- order post-Newtonian coefficients. We constrain the graviton Compton wavelength, assuming that gravitons are dispersed in vacuum in the same way as particles with mass, obtaining a 90%-confidence lower bound of 1013 km. In conclusion, within our statistical uncertainties, we find no evidence for violations of general relativity in the genuinely strong-field regime of gravity.
Simulations of inspiraling and merging double neutron stars using the Spectral Einstein Code
Physical Review D 93, 124062 (2016)
Abstract
We present results on the inspiral, merger, and postmerger evolution of a neutron star-neutron star (NSNS) system. Our results are obtained using the hybrid pseudospectral-finite volume Spectral Einstein Code (SpEC). To test our numerical methods, we evolve an equal-mass system for ≈22 orbits before merger. This waveform is the longest waveform obtained from fully general-relativistic simulations for NSNSs to date. Such long (and accurate) numerical waveforms are required to further improve semianalytical models used in gravitational wave data analysis, for example, the effective one body models. We discuss in detail the improvements to SpEC’s ability to simulate NSNS mergers, in particular mesh refined grids to better resolve the merger and postmerger phases. We provide a set of consistency checks and compare our results to NSNS merger simulations with the independent bam code. We find agreement between them, which increases confidence in results obtained with either code. This work paves the way for future studies using long waveforms and more complex microphysical descriptions of neutron star matter in SpEC.
Accuracy of binary black hole waveform models for aligned-spin binaries
Physical Review D 93, 104050 (2016)
Abstract
Coalescing binary black holes are among the primary science targets for second generation ground-based gravitational wave detectors. Reliable gravitational waveform models are central to detection of such systems and subsequent parameter estimation. This paper performs a comprehensive analysis of the accuracy of recent waveform models for binary black holes with aligned spins, utilizing a new set of 84 high-accuracy numerical relativity simulations. Our analysis covers comparable mass binaries (mass-ratio 1≤q≤3), and samples independently both black hole spins up to a dimensionless spin magnitude of 0.9 for equal-mass binaries and 0.85 for unequal mass binaries. Furthermore, we focus on the high-mass regime (total mass ≳50M⊙). The two most recent waveform models considered (PhenomD and SEOBNRv2) both perform very well for signal detection, losing less than 0.5% of the recoverable signal-to-noise ratio ρ, except that SEOBNRv2’s efficiency drops slightly for both black hole spins aligned at large magnitude. For parameter estimation, modeling inaccuracies of the SEOBNRv2 model are found to be smaller than systematic uncertainties for moderately strong GW events up to roughly ρ≲15. PhenomD’s modeling errors are found to be smaller than SEOBNRv2’s, and are generally irrelevant for ρ≲20. Both models’ accuracy deteriorates with increased mass ratio, and when at least one black hole spin is large and aligned. The SEOBNRv2 model shows a pronounced disagreement with the numerical relativity simulation in the merger phase, for unequal masses and simultaneously both black hole spins very large and aligned. Two older waveform models (PhenomC and SEOBNRv1) are found to be distinctly less accurate than the more recent PhenomD and SEOBNRv2 models. Finally, we quantify the bias expected from all four waveform models during parameter estimation for several recovered binary parameters: chirp mass, mass ratio, and effective spin.
Effects of Neutron-Star Dynamic Tides on Gravitational Waveforms within the Effective-One-Body Approach
Physical Review Letters 116, 181101 (2016)
Abstract
Extracting the unique information on ultradense nuclear matter from the gravitational waves emitted by merging neutron-star binaries requires robust theoretical models of the signal. We develop a novel effective-one-body waveform model that includes, for the first time, dynamic (instead of only adiabatic) tides of the neutron star as well as the merger signal for neutron-star–black-hole binaries. We demonstrate the importance of the dynamic tides by comparing our model against new numerical-relativity simulations of nonspinning neutron-star–black-hole binaries spanning more than 24 gravitational-wave cycles, and to other existing numerical simulations for double neutron-star systems. Furthermore, we derive an effective description that makes explicit the dependence of matter effects on two key parameters: tidal deformability and fundamental oscillation frequency.
The integration of angular velocity
Adv.Appl.Clifford Algebras 27, (2017)
Abstract
A common problem in physics and engineering is determination of the orientation of an object given its angular velocity. When the direction of the angular velocity changes in time, this is a nontrivial problem involving coupled differential equations. Several possible approaches are examined, along with various improvements over previous efforts. These are then evaluated numerically by comparison to a complicated but analytically known rotation that is motivated by the important astrophysical problem of precessing black-hole binaries. It is shown that a straightforward solution directly using quaternions is most efficient and accurate, and that the norm of the quaternion is irrelevant. Integration of the generator of the rotation can also be made roughly as efficient as integration of the rotation. Both methods will typically be twice as efficient as naive vector- or matrix-based methods. Implementation by means of standard general-purpose numerical integrators is stable and efficient, so that such problems can be readily solved as part of a larger system of differential equations. Possible generalization to integration in other Lie groups is also discussed.
Transformations of asymptotic gravitational-wave data
Physical Review D 93, 084031 (2016)
Abstract
Gravitational-wave data is gauge dependent. While we can restrict the class of gauges in which such data may be expressed, there will still be an infinite-dimensional group of transformations allowed while remaining in this class, and almost as many different—though physically equivalent—waveforms as there are transformations. This paper presents a method for calculating the effects of the most important transformation group, the Bondi-Metzner-Sachs (BMS) group, consisting of rotations, boosts, and supertranslations (which include time and space translations as special cases). To a reasonable approximation, these transformations result in simple coupling between the modes in a spin-weighted spherical-harmonic decomposition of the waveform. It is shown that waveforms from simulated compact binaries in the publicly available SXS waveform catalog contain unmodeled effects due to displacement and drift of the center of mass, accounting for mode mixing at typical levels of 1%. However, these effects can be mitigated by measuring the average motion of the system’s center of mass for a portion of the inspiral, and applying the opposite transformation to the waveform data. More generally, controlling the BMS transformations will be necessary to eliminate the gauge ambiguity inherent in gravitational-wave data for both numerical and analytical waveforms. Open-source code implementing BMS transformations of waveforms is supplied along with this paper in the supplemental materials.
Observation of Gravitational Waves from a Binary Black Hole Merger
Physical Review Letters 116, 061102 (2016)
Abstract
On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational- wave strain of 1.0×10−21. It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and a false alarm rate estimated to be less than 1 event per 203 000 years, equivalent to a significance greater than 5.1σ. The source lies at a luminosity distance of 410+160−180 Mpc corresponding to a redshift z=0.09+0.03−0.04. In the source frame, the initial black hole masses are 36+5−4M⊙ and 29+4−4M⊙, and the final black hole mass is 62+4−4M⊙, with 3.0+0.5−0.5M⊙c2 radiated in gravitational waves. All uncertainties define 90% credible intervals. These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.
Low mass binary neutron star mergers : gravitational waves and neutrino emission
Physical Review D 93, 044019 (2016)
Abstract
Neutron star mergers are among the most promising sources of gravitational waves for advanced ground-based detectors. These mergers are also expected to power bright electromagnetic signals, in the form of short gamma-ray bursts, infrared/optical transients powered by r-process nucleosynthesis in neutron-rich material ejected by the merger, and radio emission from the interaction of that ejecta with the interstellar medium. Simulations of these mergers with fully general relativistic codes are critical to understand the merger and postmerger gravitational wave signals and their neutrinos and electromagnetic counterparts. In this paper, we employ the Spectral Einstein Code to simulate the merger of low mass neutron star binaries (two 1.2M⊙ neutron stars) for a set of three nuclear-theory-based, finite temperature equations of state. We show that the frequency peaks of the postmerger gravitational wave signal are in good agreement with predictions obtained from recent simulations using a simpler treatment of gravity. We find, however, that only the fundamental mode of the remnant is excited for long periods of time: emission at the secondary peaks is damped on a millisecond time scale in the simulated binaries. For such low mass systems, the remnant is a massive neutron star which, depending on the equation of state, is either permanently stable or long lived (i.e. rapid uniform rotation is sufficient to prevent its collapse). We observe strong excitations of l=2, m=2 modes, both in the massive neutron star and in the form of hot, shocked tidal arms in the surrounding accretion torus. We estimate the neutrino emission of the remnant using a neutrino leakage scheme and, in one case, compare these results with a gray two-moment neutrino transport scheme. We confirm the complex geometry of the neutrino emission, also observed in previous simulations with neutrino leakage, and show explicitly the presence of important differences in the neutrino luminosity, disk composition, and outflow properties between the neutrino leakage and transport schemes.
Gravitational waveforms for neutron star binaries from binary black hole simulations
Physical Review D 93, 044064 (2016)
Abstract
Gravitational waves from binary neutron star (BNS) and black hole/neutron star (BHNS) inspirals are primary sources for detection by the Advanced Laser Interferometer Gravitational-Wave Observatory. The tidal forces acting on the neutron stars induce changes in the phase evolution of the gravitational waveform, and these changes can be used to constrain the nuclear equation of state. Current methods of generating BNS and BHNS waveforms rely on either computationally challenging full 3D hydrodynamical simulations or approximate analytic solutions. We introduce a new method for computing inspiral waveforms for BNS/BHNS systems by adding the post-Newtonian (PN) tidal effects to full numerical simulations of binary black holes (BBHs), effectively replacing the nontidal terms in the PN expansion with BBH results. Comparing a waveform generated with this method against a full hydrodynamical simulation of a BNS inspiral yields a phase difference of <1 radian over ∼15 orbits. The numerical phase accuracy required of BNS simulations to measure the accuracy of the method we present here is estimated as a function of the tidal deformability parameter λ.
#2015
Binary neutron stars with arbitrary spins in numerical relativity
Physical Review D 92, 124012 (2015)
Abstract
We present a code to construct initial data for binary neutron star systems in which the stars are rotating. Our code, based on a formalism developed by Tichy, allows for arbitrary rotation axes of the neutron stars and is able to achieve rotation rates near rotational breakup. We compute the neutron star angular momentum through quasilocal angular momentum integrals. When constructing irrotational binary neutron stars, we find a very small residual dimensionless spin of ∼2×10-4. Evolutions of rotating neutron star binaries show that the magnitude of the stars’ angular momentum is conserved, and that the spin and orbit precession of the stars is well described by post-Newtonian approximation. We demonstrate that orbital eccentricity of the binary neutron stars can be controlled to ∼0.1%. The neutron stars show quasinormal mode oscillations at an amplitude which increases with the rotation rate of the stars.
Improvements to the construction of binary black hole initial data
Classical and Quantum Gravity 32, 245010 (2015)
Abstract
Construction of binary black hole initial data is a prerequisite for numerical evolutions of binary black holes. This paper reports improvements to the binary black hole initial data solver in the Spectral Einstein Code, to allow robust construction of initial data for mass-ratio above 10:1, and for dimensionless black hole spins above 0.9, while improving efficiency for lower mass-ratios and spins. We implement a more flexible domain decomposition, adaptive mesh refinement and an updated method for choosing free parameters. We also introduce a new method to control and eliminate residual linear momentum in initial data for precessing systems, and demonstrate that it eliminates gravitational mode mixing during the evolution. Finally, the new code is applied to construct initial data for hyperbolic scattering and for binaries with very small separation.
Gauge invariant spectral Cauchy characteristic extraction
Classical and Quantum Gravity 32, 235018 (2015)
Abstract
We present gauge invariant spectral Cauchy characteristic extraction. We compare gravitational waveforms extracted from a head-on black hole merger simulated in two different gauges by two different codes. We show rapid convergence, demonstrating both gauge invariance of the extraction algorithm and consistency between the legacy Pitt null code and the much faster spectral Einstein code (SpEC).
Comparing post-Newtonian and numerical relativity precession dynamics
Physical Review D 92, 104028 (2015)
Abstract
Binary black-hole systems are expected to be important sources of gravitational waves for upcoming gravitational-wave detectors. If the spins are not colinear with each other or with the orbital angular momentum, these systems exhibit complicated precession dynamics that are imprinted on the gravitational waveform. We develop a new procedure to match the precession dynamics computed by post-Newtonian (PN) theory to those of numerical binary black-hole simulations in full general relativity. For numerical relativity (NR) simulations lasting approximately two precession cycles, we find that the PN and NR predictions for the directions of the orbital angular momentum and the spins agree to better than ∼1° with NR during the inspiral, increasing to 5° near merger. Nutation of the orbital plane on the orbital time scale agrees well between NR and PN, whereas nutation of the spin direction shows qualitatively different behavior in PN and NR. We also examine how the PN equations for precession and orbital-phase evolution converge with PN order, and we quantify the impact of various choices for handling partially known PN terms.
Accuracy and precision of gravitational-wave models of inspiraling neutron star-black hole binaries with spin: Comparison with matter-free numerical relativity in the low-frequency regime
Physical Review D 92, 102001 (2015)
Abstract
Coalescing binaries of neutron stars and black holes are one of the most important sources of gravitational waves for the upcoming network of ground-based detectors. Detection and extraction of astrophysical information from gravitational-wave signals requires accurate waveform models. The effective-one-body and other phenomenological models interpolate between analytic results and numerical relativity simulations, that typically span O(10) orbits before coalescence. In this paper we study the faithfulness of these models for neutron star-black hole binaries. We investigate their accuracy using new numerical relativity (NR) simulations that span 36–88 orbits, with mass ratios q and black hole spins χBH of (q,χBH)=(7,±0.4),(7,±0.6), and (5,-0.9). These simulations were performed treating the neutron star as a low-mass black hole, ignoring its matter effects. We find that (i) the recently published SEOBNRv1 and SEOBNRv2 models of the effective-one-body family disagree with each other (mismatches of a few percent) for black hole spins χBH≥0.5 or χBH≤-0.3, with waveform mismatch accumulating during early inspiral; (ii) comparison with numerical waveforms indicates that this disagreement is due to phasing errors of SEOBNRv1, with SEOBNRv2 in good agreement with all of our simulations; (iii) phenomenological waveforms agree with SEOBNRv2 only for comparable-mass low-spin binaries, with overlaps below 0.7 elsewhere in the neutron star-black hole binary parameter space; (iv) comparison with numerical waveforms shows that most of this model’s dephasing accumulates near the frequency interval where it switches to a phenomenological phasing prescription; and finally (v) both SEOBNR and post-Newtonian models are effectual for neutron star-black hole systems, but post-Newtonian waveforms will give a significant bias in parameter recovery. Our results suggest that future gravitational-wave detection searches and parameter estimation efforts would benefit from using SEOBNRv2 waveform templates when focused on neutron star-black hole systems with q≲7 and χBH≈[-0.9,+0.6]. For larger black hole spins and/or binary mass ratios, we recommend the models be further investigated as NR simulations in that region of the parameter space become available.
Approaching the Post-Newtonian Regime with Numerical Relativity: A Compact-Object Binary Simulation Spanning 350 Gravitational-Wave Cycles
Physical Review Letters 115, 031102 (2015)
Abstract
We present the first numerical-relativity simulation of a compact- object binary whose gravitational waveform is long enough to cover the entire frequency band of advanced gravitational-wave detectors, such as LIGO, Virgo, and KAGRA, for mass ratio 7 and total mass as low as 45.5M⊙. We find that effective-one-body models, either uncalibrated or calibrated against substantially shorter numerical- relativity waveforms at smaller mass ratios, reproduce our new waveform remarkably well, with a negligible loss in detection rate due to modeling error. In contrast, post-Newtonian inspiral waveforms and existing calibrated phenomenological inspiral-merger- ringdown waveforms display greater disagreement with our new simulation. The disagreement varies substantially depending on the specific post-Newtonian approximant used.
Post-merger evolution of a neutron star-black hole binary with neutrino transport
Physical Review D 91, 124021 (2015)
Abstract
We present a first simulation of the post-merger evolution of a black hole-neutron star binary in full general relativity using an energy-integrated general-relativistic truncated moment formalism for neutrino transport. We describe our implementation of the moment formalism and important tests of our code, before studying the formation phase of an accretion disk after a black hole-neutron star merger. We use as initial data an existing general-relativistic simulation of the merger of a neutron star of mass 1.4M⊙ with a black hole of mass 7M⊙ and dimensionless spin χBH=0.8. Comparing with a simpler leakage scheme for the treatment of the neutrinos, we find noticeable differences in the neutron-to-proton ratio in and around the disk, and in the neutrino luminosity. We find that the electron neutrino luminosity is much lower in the transport simulations, and that both the disk and the disk outflows are less neutron rich. The spatial distribution of the neutrinos is significantly affected by relativistic effects, due to large velocities and curvature in the regions of strongest emission. Over the short time scale evolved, we do not observe purely neutrino- driven outflows. However, a small amount of material (3×10-4M⊙) is ejected in the polar region during the circularization of the disk. Most of that material is ejected early in the formation of the disk, and is fairly neutron rich (electron fraction Ye∼0.15–0.25). Through r-process nucleosynthesis, that material should produce high-opacity lanthanides in the polar region, and could thus affect the light curve of radioactively powered electromagnetic transients. We also show that by the end of the simulation, while the bulk of the disk remains neutron rich (Ye∼0.15–0.2 and decreasing), its outer layers have a higher electron fraction: 10% of the remaining mass has Ye>0.3. As that material would be the first to be unbound by disk outflows on longer time scales, and as composition evolution is slower at later times, the changes in Ye experienced during the formation phase of the disk could have an impact on nucleosynthesis outputs from neutrino-driven and viscously driven outflows. Finally, we find that the effective viscosity due to momentum transport by neutrinos is unlikely to have a strong effect on the growth of the magnetorotational instability in the post-merger accretion disk.
Improved methods for simulating nearly extremal binary black holes
Classical and Quantum Gravity 32, 105009 (2015)
Abstract
Astrophysical black holes could be nearly extremal (that is, rotating nearly as fast as possible), therefore, nearly extremal black holes could be among the binaries that current and future gravitational-wave observatories will detect. Predicting the gravitational waves emitted by merging black holes requires numerical-relativity simulations, but these simulations are especially challenging when one or both holes have mass m and spin S exceeding the Bowen–York limit of \(S/{{m}^{2}}=0.93\). We present improved methods that enable us to simulate merging, nearly extremal black holes (i.e., black holes with \(S/{{m}^{2}}\gt 0.93\)) more robustly and more efficiently. We use these methods to simulate an unequal-mass, precessing binary black hole (BBH) coalescence, where the larger black hole has \(S/{{m}^{2}}=0.99\). We also use these methods to simulate a non-precessing BBH coalescence, where both black holes have \(S/{{m}^{2}}=0.994\), nearly reaching the Novikov–Thorne upper bound for holes spun up by thin accretion disks. We demonstrate numerical convergence and estimate the numerical errors of the waveforms, we compare numerical waveforms from our simulations with post-Newtonian and effective-one-body waveforms, we compare the evolution of the black hole masses and spins with analytic predictions, and we explore the effect of increasing spin magnitude on the orbital dynamics (the so-called ‘orbital hangup’ effect).
Nearly extremal apparent horizons in simulations of merging black holes
Classical and Quantum Gravity 32, 065007 (2015)
Abstract
The spin angular momentum S of an isolated Kerr black hole is bounded by the surface area A of its apparent horizon: \(8\pi S\leqslant A\), with equality for extremal black holes. In this paper, we explore the extremality of individual and common apparent horizons for merging, rapidly spinning binary black holes. We consider simulations of merging black holes with equal masses M and initial spin angular momenta aligned with the orbital angular momentum, including new simulations with spin magnitudes up to \(S/{{M}^{2}}=0.994\). We measure the area and (using approximate Killing vectors) the spin on the individual and common apparent horizons, finding that the inequality \(8\pi S\lt A\) is satisfied in all cases but is very close to equality on the common apparent horizon at the instant it first appears. We also evaluate the Booth–Fairhurst extremality, whose value for a given apparent horizon depends on the scaling of the horizon’s null normal vectors. In particular, we introduce a gauge-invariant lower bound on the extremality by computing the smallest value that Booth and Fairhurst’s extremality parameter can take for any scaling. Using this lower bound, we conclude that the common horizons are at least moderately close to extremal just after they appear. Finally, following Lovelace et al (2008 Phys. Rev. D 78 084017), we construct quasiequilibrium binary-black hole initial data with ‘overspun’ marginally trapped surfaces with \(8\pi S\gt A\). We show that the overspun surfaces are indeed superextremal: our lower bound on their Booth–Fairhurst extremality exceeds unity. However, we confirm that these superextremal surfaces are always surrounded by marginally outer trapped surfaces (i.e., by apparent horizons) with \(8\pi S\lt A\). The extremality lower bound on the enclosing apparent horizon is always less than unity but can exceed the value for an extremal Kerr black hole.
What does a binary black hole merger look like?
Classical and Quantum Gravity 32, 065002 (2015)
Abstract
We present a method of calculating the strong-field gravitational lensing caused by many analytic and numerical spacetimes. We use this procedure to calculate the distortion caused by isolated black holes (BHs) and by numerically evolved BH binaries. We produce both demonstrative images illustrating details of the spatial distortion and realistic images of collections of stars taking both lensing amplification and redshift into account. On large scales the lensing from inspiraling binaries resembles that of single BHs, but on small scales the resulting images show complex and in some cases self- similar structure across different angular scales.
Fast and accurate prediction of numerical relativity waveforms from binary black hole mergers using surrogate models
Physical Review Letters 115, 121102 (2015)
Abstract
Simulating a binary black hole coalescence by solving Einstein’s equations is computationally expensive, requiring days to months of supercomputing time. Using reduced order modeling techniques, we construct an accurate surrogate model, which is evaluated in a millisecond to a second, for numerical relativity (NR) waveforms from nonspinning binary black hole coalescences with mass ratios in [1, 10] and durations corresponding to about 15 orbits before merger. We assess the model’s uncertainty and show that our modeling strategy predicts NR waveforms not used for the surrogate’s training with errors nearly as small as the numerical error of the NR code. Our model includes all spherical-harmonic Yℓm−2 waveform modes resolved by the NR code up to ℓ=8. We compare our surrogate model to effective one body waveforms from 50M⊙ to 300M⊙ for advanced LIGO detectors and find that the surrogate is always more faithful (by at least an order of magnitude in most cases).
Spectral characteristic evolution: a new algorithm for gravitational wave propagation
Classical and Quantum Gravity 32, 025008 (2015)
Abstract
We present a spectral algorithm for solving the full nonlinear vacuum Einstein field equations in the Bondi framework. Developed within the Spectral Einstein Code, we demonstrate spectral characteristic evolution as a technical precursor to Cauchy characteristic extraction, a rigorous method for obtaining gauge- invariant gravitational waveforms from existing and future astrophysical simulations. We demonstrate the new algorithmʼs stability, convergence, and agreement with existing evolution methods. We explain how an innovative spectral approach enables a two orders of magnitude improvement in computational efficiency.
#2014
Magnetic effects on the low-T /|W | instability in differentially rotating neutron stars
Physical Review D 90, 104014 (2014)
Abstract
Dynamical instabilities in protoneutron stars may produce gravitational waves whose observation could shed light on the physics of core-collapse supernovae. When born with sufficient differential rotation, these stars are susceptible to a shear instability (the "low-T/|W| instability"), but such rotation can also amplify magnetic fields to strengths where they have a considerable impact on the dynamics of the stellar matter. Using a new magnetohydrodynamics module for the Spectral Einstein Code, we have simulated a differentially-rotating neutron star in full 3D to study the effects of magnetic fields on this instability. Though strong toroidal fields were predicted to suppress the low-T/|W| instability, we find that they do so only in a small range of field strengths. Below 4e13 G, poloidal seed fields do not wind up fast enough to have an effect before the instability saturates, while above 5e14 G, magnetic instabilities can actually amplify a global quadrupole mode (this threshold may be even lower in reality, as small-scale magnetic instabilities remain difficult to resolve numerically). Thus, the prospects for observing gravitational waves from such systems are not in fact diminished over most of the magnetic parameter space. Additionally, we report that the detailed development of the low-T/|W| instability, including its growth rate, depends strongly on the particular numerical methods used. The high-order methods we employ suggest that growth might be considerably slower than found in some previous simulations.
Accretion disks around binary black holes of unequal mass: General relativistic MHD simulations of postdecoupling and merger
Physical Review D 90, 104030 (2014)
Abstract
We report results from simulations in general relativity of magnetized disks accreting onto merging black hole binaries, starting from relaxed disk initial data. The simulations feature an effective, rapid radiative cooling scheme as a limiting case of future treatments with radiative transfer. Here we evolve the systems after binary-disk decoupling through inspiral and merger, and analyze the dependence on the binary mass ratio with q≡mbh/MBH=1,1/2, and 1/4. We find that the luminosity associated with local cooling is larger than the luminosity associated with matter kinetic outflows, while the electromagnetic (Poynting) luminosity associated with bulk transport of magnetic field energy is the smallest. The cooling luminosity around merger is only marginally smaller than that of a single, non-spinning black hole. Incipient jets are launched independently of the mass ratio, while the same initial disk accreting on a single non-spinning black hole does not lead to a jet, as expected. For all mass ratios we see a transient behavior in the collimated, magnetized outflows lasting 2−5(M/108M⊙)days after merger: the outflows become increasingly magnetically dominated and accelerated to higher velocities, boosting the Poynting luminosity. These sudden changes can alter the electromagnetic emission across the jet and potentially help distinguish mergers of black holes in AGNs from single accreting black holes based on jet morphology alone.
Gravitational-wave modes from precessing black-hole binaries
Abstract
Gravitational waves from precessing black-hole binaries exhibit features that are absent in nonprecessing systems. The most prominent of these is a parity-violating asymmetry that beams energy and linear momentum preferentially along or opposite to the orbital angular momentum, leading to recoil of the binary. The asymmetry will appear as amplitude and phase modulations at the orbital frequency. For strongly precessing systems, it accounts for at least 3% amplitude modulation for binaries in the sensitivity band of ground-based gravitational-wave detectors, and can exceed 50% for massive systems. Such asymmetric features are also clearly visible when the waves are decomposed into modes of spin-weighted spherical harmonics, and are inherent in the waves themselves---rather than resulting from residual eccentricity in numerical simulations, or from mode-mixing due to precession. In particular, there is generically no instantaneous frame for which the mode decomposition will have any symmetry. We introduce a method to simplify the expressions for waveforms given in analytical relativity, which can be used to combine existing high-order waveforms for nonprecessing systems with expressions for the precessing contributions, leading to improved accuracy and a unified treatment of precessing and nonprecessing binaries. Using this method, it is possible to clarify the nature and the origins of the asymmetries and show the effects of asymmetry on recoils more clearly. We present post-Newtonian (PN) expressions for the waveform modes that include these terms, complete to the relative 2PN level in spin (proportional to \(v^4/c^4\) times a certain combination of the spins). Comparing the results of those expressions to numerical results, we find good qualitative agreement. We also demonstrate how these expressions can be used to efficiently calculate waveforms for gravitational-wave astronomy.
Neutron star-black hole mergers with a nuclear equation of state and neutrino cooling: Dependence in the binary parameters
Physical Review D 90, 024026 (2014)
Abstract
We present a first exploration of the results of neutron star-black hole mergers using black hole masses in the most likely range of 7M⊙−10M⊙, a neutrino leakage scheme, and a modeling of the neutron star material through a finite-temperature nuclear-theory based equation of state. In the range of black hole spins in which the neutron star is tidally disrupted (χBH≳0.7), we show that the merger consistently produces large amounts of cool (T≲1MeV), unbound, neutron-rich material (Mej∼0.05M⊙−0.20M⊙). A comparable amount of bound matter is initially divided between a hot disk (Tmax∼15MeV) with typical neutrino luminosity Lν∼1053erg/s, and a cooler tidal tail. After a short period of rapid protonization of the disk lasting ∼10ms, the accretion disk cools down under the combined effects of the fall-back of cool material from the tail, continued accretion of the hottest material onto the black hole, and neutrino emission. As the temperature decreases, the disk progressively becomes more neutron-rich, with dimmer neutrino emission. This cooling process should stop once the viscous heating in the disk (not included in our simulations) balances the cooling. These mergers of neutron star-black hole binaries with black hole masses MBH∼7M⊙−10M⊙ and black hole spins high enough for the neutron star to disrupt provide promising candidates for the production of short gamma-ray bursts, of bright infrared post-merger signals due to the radioactive decay of unbound material, and of large amounts of r-process nuclei.
The NINJA-2 project: detecting and characterizing gravitational waveforms modelled using numerical binary black hole simulations
Classical and Quantum Gravity 31, 115004 (2014)
Abstract
The Numerical INJection Analysis (NINJA) project is a collaborative effort between members of the numerical relativity and gravitational-wave astrophysics communities. The purpose of NINJA is to study the ability to detect gravitational waves emitted from merging binary black holes and recover their parameters with next- generation gravitational-wave observatories. We report here on the results of the second NINJA project, NINJA-2, which employs 60 complete binary black hole hybrid waveforms consisting of a numerical portion modelling the late inspiral, merger, and ringdown stitched to a post-Newtonian portion modelling the early inspiral. In a "blind injection challenge" similar to that conducted in recent LIGO and Virgo science runs, we added 7 hybrid waveforms to two months of data recolored to predictions of Advanced LIGO and Advanced Virgo sensitivity curves during their first observing runs. The resulting data was analyzed by gravitational-wave detection algorithms and 6 of the waveforms were recovered with false alarm rates smaller than 1 in a thousand years. Parameter estimation algorithms were run on each of these waveforms to explore the ability to constrain the masses, component angular momenta and sky position of these waveforms. We also perform a large-scale monte- carlo study to assess the ability to recover each of the 60 hybrid waveforms with early Advanced LIGO and Advanced Virgo sensitivity curves. Our results predict that early Advanced LIGO and Advanced Virgo will have a volume-weighted average sensitive distance of 300Mpc (1Gpc) for \(10M_{\odot}+10M_{\odot}\) (\(50M_{\odot}+50M_{\odot}\)) binary black hole coalescences. We demonstrate that neglecting the component angular momenta in the waveform models used in matched-filtering will result in a reduction in sensitivity for systems with large component angular momenta. [Abstract abridged for ArXiv, full version in PDF]
Key elements of robustness in binary black hole evolutions using spectral methods
International Journal of Modern Physics D 23, 30014 (2014)
Abstract
As a network of advanced-era gravitational wave detectors is nearing its design sensitivity, efficient and accurate waveform modeling becomes more and more relevant. Understanding of the nature of the signal being sought can have an order unity effect on the event rates seen in these instruments. The paper provides a description of key elements of the Spectral Einstein Code (SpEC), with details of our spectral adaptive mesh refinement (AMR) algorithm that has been optimized for binary black hole (BBH) evolutions. We expect that the gravitational waveform catalog produced by our code will have a central importance in both the detection and parameter estimation of gravitational waves in these instruments.
Inspiral-merger-ringdown waveforms of spinning, precessing black-hole binaries in the effective-one-body formalism
Physical Review D 89, 084006 (2014)
Abstract
We describe a general procedure to generate spinning, precessing waveforms that include inspiral, merger and ringdown stages in the effective-one-body (EOB) approach. The procedure uses a precessing frame in which precession-induced amplitude and phase modulations are minimized, and an inertial frame, aligned with the spin of the final black hole, in which we carry out the matching of the inspiral-plunge to merger-ringdown waveforms. As a first application, we build spinning, precessing EOB waveforms for the gravitational modes l=2 such that in the nonprecessing limit those waveforms agree with the EOB waveforms recently calibrated to numerical-relativity waveforms. Without recalibrating the EOB model, we then compare EOB and post-Newtonian precessing waveforms to two numerical-relativity waveforms produced by the Caltech-Cornell-CITA collaboration. The numerical waveforms are strongly precessing and have 35 and 65 gravitational-wave cycles. We find a remarkable agreement between EOB and numerical-relativity precessing waveforms and spins' evolutions. The phase difference is ~ 0.2 rad at merger, while the mismatches, computed using the advanced-LIGO noise spectral density, are below 2% when maximizing only on the time and phase at coalescence and on the polarization angle.
Decline of the current quadrupole moment during the merger phase of binary black hole coalescence
Abstract
Utilizing the tools of tendex and vortex, we study the highly dynamic plunge and merger phases of several π -symmetric binary black hole coalescences. In particular, we observe a decline of the strength of the current quadrupole moment compared to that of the mass quadrupole moment during the merger phase, contrary to a naive estimate according to the dependence of these moments on the separation between the black holes. We further show that this decline of the current quadrupole moment is achieved through the remnants of the two individual spins becoming nearly aligned or anti-aligned with the total angular momentum. We also speculate on the ability to achieve a consistency between the electric and magnetic parity quasinormal modes.
Effective-one-body model for black-hole binaries with generic mass ratios and spins
Physical Review D 89, 061502 (2014)
Abstract
Gravitational waves emitted by black-hole binary systems have the highest signal-to-noise ratio in LIGO and Virgo detectors when black-hole spins are aligned with the orbital angular momentum and extremal. For such systems, we extend the effective-one-body inspiral-merger-ringdown waveforms to generic mass ratios and spins calibrating them to 38 numerical-relativity nonprecessing waveforms produced by the SXS Collaboration. The numerical-relativity simulations span mass ratios from 1 to 8, spin magnitudes up to 98% of extremality, and last for 40 to 60 gravitational-wave cycles. When the total mass of the binary is between 20Msun and 200Msun, the effective-one-body nonprecessing (dominant mode) waveforms have overlaps above 99% (using the advanced-LIGO design noise spectral density) with all of the 38 nonprecessing numerical waveforms, when maximizing only on initial phase and time. This implies a negligible loss in event rate due to modeling. Moreover, without further calibration, we show that the precessing effective-one-body (dominant mode) waveforms have overlaps above 97% with two very long, strongly precessing numerical-relativity waveforms, when maximizing only on the initial phase and time.
Stability of nonspinning effective-one-body model in approximating two-body dynamics and gravitational-wave emission
Physical Review D 89, 061501 (2014)
Abstract
The detection of gravitational waves and the extraction of physical information from them requires the prediction of accurate waveforms to be used in template banks. For that purpose, the accuracy of effective-one-body (EOB) waveforms has been improved over the last years by calibrating them to numerical-relativity (NR) waveforms. So far, the calibration has employed a handful of NR waveforms with a total length of ~30 cycles, the length being limited by the computational cost of NR simulations. Here we address the outstanding problem of the stability of the EOB calibration with respect to the length of NR waveforms. Performing calibration studies against NR waveforms of nonspinning black-hole binaries with mass ratios 1, 1.5, 5, and 8, and with a total length of ~60 cycles, we find that EOB waveforms calibrated against either 30 or 60 cycles will be indistinguishable by the advanced detectors LIGO and Virgo when the signal-to-noise ratio (SNR) is below 110. When extrapolating to a very large number of cycles, using very conservative assumptions, we can conclude that state-of-the-art nonspinning EOB waveforms of any length are sufficiently accurate for parameter estimation with advanced detectors when the SNR is below 20, the mass ratio is below 5 and total mass is above 20 Msun. The results are not conclusive for the entire parameter space because of current NR errors.
Accretion disks around binary black holes of unequal mass: General relativistic magnetohydrodynamic simulations near decoupling
Physical Review D 89, 064060 (2014)
Abstract
We report on simulations in general relativity of magnetized disks onto black hole binaries. We vary the binary mass ratio from 1:1 to 1:10 and evolve the systems when they orbit near the binary-disk decoupling radius. We compare (surface) density profiles, accretion rates (relative to a single, non-spinning black hole), variability, effective α-stress levels and luminosities as functions of the mass ratio. We treat the disks in two limiting regimes: rapid radiative cooling and no radiative cooling. The magnetic field lines clearly reveal jets emerging from both black hole horizons and merging into one common jet at large distances. The magnetic fields give rise to much stronger shock heating than the pure hydrodynamic flows, completely alter the disk structure, and boost accretion rates and luminosities. Accretion streams near the horizons are among the densest structures; in fact, the 1:10 no-cooling evolution results in a refilling of the cavity. The typical effective temperature in the bulk of the disk is ∼105(M/108M⊙)−1/4(L/Ledd)1/4K yielding characteristic thermal frequencies ∼1015(M/108M⊙)−1/4(L/Ledd)1/4(1+z)−1Hz. These systems are thus promising targets for many extragalactic optical surveys, such as LSST, WFIRST, and PanSTARRS.
Including realistic tidal deformations in binary black-hole initial data
Physical Review D 89, 064062 (2014)
Abstract
A shortcoming of current binary black-hole initial data is the generation of spurious gravitational radiation, so-called junk radiation, when they are evolved. This problem is a consequence of an oversimplified modeling of the binary's physics in the initial data. Since junk radiation is not astrophysically realistic, it contaminates the actual waveforms of interest and poses a numerical nuisance. The work here presents a further step towards mitigating and understanding the origin of this issue, by incorporating post-Newtonian results in the construction of constraint-satisfying binary black-hole initial data. Here we focus on including realistic tidal deformations of the black holes in the initial data, by building on the method of superposing suitably chosen black hole metrics to compute the conformal data. We describe the details of our initial data for an equal-mass and nonspinning binary, compute the subsequent relaxation of horizon quantities in evolutions, and quantify the amount of junk radiation that is generated. These results are contrasted with those obtained with the most common choice of conformally flat (CF) initial data, as well as superposed Kerr-Schild (SKS) initial data. We find that when realistic tidal deformations are included, the early transients in the horizon geometries are significantly reduced, along with smaller deviations in the relaxed black hole masses and spins from their starting values. Likewise, the junk radiation content in the l=2 modes is reduced by a factor of ∼1.7 relative to CF initial data, but only by a factor of ∼1.2 relative to SKS initial data. More prominently, the junk radiation content in the 3≤l≤8 modes is reduced by a factor of ∼5 relative to CF initial data, and by a factor of ∼2.4 relative to SKS initial data.
Solving Einstein's equation numerically on manifolds with arbitrary spatial topologies
Physical Review D 89, 044044 (2014)
Abstract
This paper develops a method for solving Einstein's equation numerically on multicube representations of manifolds with arbitrary spatial topologies. This method is designed to provide a set of flexible, easy to use computational procedures that make it possible to explore the never before studied properties of solutions to Einstein's equation on manifolds with arbitrary toplogical structures. A new covariant, first-order symmetric-hyperbolic representation of Einstein's equation is developed for this purpose, along with the needed boundary conditions at the interfaces between adjoining cubic regions. Numerical tests are presented that demonstrate the long-term numerical stability of this method for evolutions of a complicated, time-dependent solution of Einstein's equation coupled to a complex scalar field on a manifold with spatial topology \({S}^{3}\). The accuracy of these numerical test solutions is evaluated by performing convergence studies and by comparing the full nonlinear numerical results to the analytical perturbation solutions, which are also derived here.
Template banks for binary black hole searches with numerical relativity waveforms
Physical Review D 89, 042002 (2014)
Abstract
Gravitational waves from coalescing stellar-mass black hole binaries (BBHs) are expected to be detected by the Advanced Laser Interferometer gravitational-wave observatory and Advanced Virgo. Detection searches operate by matched filtering the detector data using a bank of waveform templates. Traditionally, template banks for BBHs are constructed from intermediary analytical waveform models which are calibrated against numerical relativity simulations and which can be evaluated for any choice of BBH parameters. This paper explores an alternative to the traditional approach, namely, the construction of template banks directly from numerical BBH simulations. Using nonspinning BBH systems as an example, we demonstrate which regions of the mass-parameter plane can be covered with existing numerical BBH waveforms. We estimate the required number and required length of BBH simulations to cover the entire nonspinning BBH parameter plane up to mass ratio 10, thus illustrating that our approach can be used to guide parameter placement of future numerical simulations. We derive error bounds which are independent of analytical waveform models; therefore, our formalism can be used to independently test the accuracy of such waveform models. The resulting template banks are suitable for advanced LIGO searches.
Initial data for high-compactness black hole-neutron star binaries
Class.Quant.Grav. 33, 105009 (2016)
Abstract
For highly compact neutron stars, constructing numerical initial data for black hole-neutron star binary evolutions is very difficult. We describe improvements to an earlier method that enable it to handle these more challenging cases. We examine the case of a 6:1 mass ratio system in inspiral close to merger, where the star is governed by a polytropic \(\Gamma=2\), an SLy, or an LS220 equation of state. In particular, we are able to obtain a solution with a realistic LS220 equation of state for a star with compactness 0.26 and mass 1.98 \(M_\odot\), which is representative of the highest reliably determined neutron star masses. For the SLy equation of state, we can obtain solutions with a comparable compactness of 0.25, while for a family of polytropic equations of state, we obtain solutions with compactness up to 0.21, the largest compactness that is stable in this family. These compactness values are significantly higher than any previously published results. We find that improvements in adapting the computational domain to the neutron star surface and in accounting for the center of mass drift of the system are the key ingredients allowing us to obtain these solutions.
#2013
Comparing gravitational waveform extrapolation to Cauchy-characteristic extraction in binary black hole simulations
Phys. Rev. D 88, 124010 (2013)
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.
Catalog of 174 Binary Black Hole Simulations for Gravitational Wave Astronomy
Physical Review Letters 111, 241104 (2013)
Abstract
This Letter presents a publicly available catalog of 174 numerical binary black hole simulations following up to 35 orbits. The catalog includes 91 precessing binaries, mass ratios up to 8∶1, orbital eccentricities from a few percent to 10-5, black hole spins up to 98% of the theoretical maximum, and radiated energies up to 11.1% of the initial mass. We establish remarkably good agreement with post- Newtonian precession of orbital and spin directions for two new precessing simulations, and we discuss other applications of this catalog. Formidable challenges remain: e.g., precession complicates the connection of numerical and approximate analytical waveforms, and vast regions of the parameter space remain unexplored.
Periastron advance in spinning black hole binaries: Gravitational self-force from numerical relativity
Physical Review D 88, 124027 (2013)
Abstract
We study the general relativistic periastron advance in spinning black hole binaries on quasicircular orbits, with spins aligned or antialigned with the orbital angular momentum, using numerical- relativity simulations, the post-Newtonian approximation, and black hole perturbation theory. By imposing a symmetry by exchange of the bodies’ labels, we devise an improved version of the perturbative result and use it as the leading term of a new type of expansion in powers of the symmetric mass ratio. This allows us to measure, for the first time, the gravitational self-force effect on the periastron advance of a nonspinning particle orbiting a Kerr black hole of mass M and spin S=-0.5M2, down to separations of order 9M. Comparing the predictions of our improved perturbative expansion with the exact results from numerical simulations of equal-mass and equal-spin binaries, we find a remarkable agreement over a wide range of spins and orbital separations.
Joint approach for reducing eccentricity and spurious gravitational radiation in binary black hole initial data construction
Phys. Rev. D 88, 084033 (2013)
Abstract
At the beginning of binary black hole simulations, there is a pulse of spurious radiation (or junk radiation) resulting from the initial data not matching astrophysical quasi-equilibrium inspiral exactly. One traditionally waits for the junk radiation to exit the computational domain before taking physical readings, at the expense of throwing away a segment of the evolution, and with the hope that junk radiation exits cleanly. We argue that this hope does not necessarily pan out as junk radiation could excite long-lived constraint violation. Another complication with the initial data is that it contains orbital eccentricity that needs to be removed, usually by evolving the early part of the inspiral multiple times with gradually improved input parameters. We show that this procedure is also adversely impacted by junk radiation. In this paper, we do not attempt to eliminate junk radiation directly, but instead tackle the much simpler problem of ameliorating its long-lasting effects. We report on the success of a method that achieves this goal by combining the removal of junk radiation and eccentricity into a single "joint-elimination" procedure. This approach has the following benefits: (1) We do not have to contend with the influence of junk radiation on eccentricity measurements for later iterations of the eccentricity reduction procedure. (2) We re-enforce constraints periodically by invoking the initial data solver, removing the constraint violation excited by junk radiation previously. (3) The wasted simulation segment associated with the junk radiation's evolution is absorbed into the eccentricity reduction iterations. Furthermore, (1) and (2) together allow us to carry out our joint-elimination procedure at low resolution, even when the subsequent "production run" is intended as a high resolution simulation.
Precession-tracking coordinates for simulations of compact-object binaries
Physical Review D 88, 084031 (2013)
Abstract
Binary black hole simulations with black hole excision using spectral methods require a coordinate transformation into a corotating coordinate system where the black holes are essentially at rest. This paper presents and discusses two coordinate transformations that are applicable to precessing binary systems, one based on Euler angles, the other on quaternions. Both approaches are found to work well for binaries with moderate precession, i.e., for cases where the orientation of the orbital plane changes by ≪90°. For strong precession, performance of the Euler-angle parametrization deteriorates, eventually failing for a 90° change in orientation because of singularities in the parametrization (“gimbal lock”). In contrast, the quaternion representation is invariant under an overall rotation and handles any orientation of the orbital plane as well as the Euler-angle technique handles nonprecessing binaries.
Periastron advance in spinning black hole binaries: comparing effective-one-body and numerical relativity
Physical Review D 88, 084005 (2013)
Abstract
We compute the periastron advance using the effective-one-body formalism for binary black holes moving on quasicircular orbits and having spins collinear with the orbital angular momentum. We compare the predictions with the periastron advance recently computed in accurate numerical-relativity simulations and find remarkable agreement for a wide range of spins and mass ratios. These results do not use any numerical-relativity calibration of the effective- one-body model, and stem from two key ingredients in the effective- one-body Hamiltonian: (i) the mapping of the two-body dynamics of spinning particles onto the dynamics of an effective spinning particle in a (deformed) Kerr spacetime, fully symmetrized with respect to the two-body masses and spins, and (ii) the resummation, in the test-particle limit, of all post-Newtonian corrections linear in the spin of the particle. In fact, even when only the leading spin post-Newtonian corrections are included in the effective-one- body spinning Hamiltonian but all the test-particle corrections linear in the spin of the particle are resummed we find very good agreement with the numerical results (within the numerical error for equal-mass binaries and discrepancies of at most 1% for larger mass ratios). Furthermore, we specialize to the extreme mass-ratio limit and derive, using the equations of motion in the gravitational skeleton approach, analytical expressions for the periastron advance, the meridional Lense-Thirring precession and spin precession frequency in the case of a spinning particle on a nearly circular equatorial orbit in Kerr spacetime, including also terms quadratic in the spin.
Black Hole-Neutron Star Mergers with a Hot Nuclear Equation of State: Outflow and Neutrino-cooled Disk for a Low-mass, High-spin Case
Abstract
Neutrino emission significantly affects the evolution of the accretion tori formed in black hole-neutron star mergers. It removes energy from the disk, alters its composition, and provides a potential power source for a gamma-ray burst. To study these effects, simulations in general relativity with a hot microphysical equation of state (EOS) and neutrino feedback are needed. We present the first such simulation, using a neutrino leakage scheme for cooling to capture the most essential effects and considering a moderate mass (1.4 M☉ neutron star, 5.6 M☉ black hole), high-spin (black hole J/M2 = 0.9) system with the K0 = 220 MeV Lattimer-Swesty EOS. We find that about 0.08 M☉ of nuclear matter is ejected from the system, while another 0.3 M☉ forms a hot, compact accretion disk. The primary effects of the escaping neutrinos are (1) to make the disk much denser and more compact, (2) to cause the average electron fraction Yeof the disk to rise to about 0.2 and then gradually decrease again, and (3) to gradually cool the disk. The disk is initially hot (T ~ 6 MeV) and luminous in neutrinos (Lν ~ 1054 erg s–1), but the neutrino luminosity decreases by an order of magnitude over 50 ms of post-merger evolution.
Final spin and radiated energy in numerical simulations of binary black holes with equal masses and equal, aligned or antialigned spins
Physical Review D 88, 064014 (2013)
Abstract
The behavior of merging black holes (including the emitted gravitational waves and the properties of the remnant) can currently be computed only by numerical simulations. This paper introduces ten numerical relativity simulations of binary black holes with equal masses and equal spins aligned or antialigned with the orbital angular momentum. The initial spin magnitudes have |χi|≲0.95 and are more concentrated in the aligned direction because of the greater astrophysical interest of this case. We combine these data with five previously reported simulations of the same configuration, but with different spin magnitudes, including the highest spin simulated to date, χi≈0.97. This data set is sufficiently accurate to enable us to offer improved analytic fitting formulas for the final spin and for the energy radiated by gravitational waves as a function of initial spin. The improved fitting formulas can help to improve our understanding of the properties of binary black hole merger remnants and can be used to enhance future approximate waveforms for gravitational wave searches, such as effective-one-body waveforms.
First direct comparison of nondisrupting neutron star-black hole and binary black hole merger simulations
Physical Review D 88, 064017 (2013)
Abstract
We present the first direct comparison of numerical simulations of neutron star-black hole and black hole-black hole mergers in full general relativity. We focus on a configuration with nonspinning objects and within the most likely range of mass ratio for neutron star-black hole systems (q=6). In this region of the parameter space, the neutron star is not tidally disrupted prior to merger, and we show that the two types of mergers appear remarkably similar. The effect of the presence of a neutron star on the gravitational wave signal is not only undetectable by the next generation of gravitational wave detectors, but also too small to be measured in the numerical simulations: even the plunge, merger and ringdown signals appear in perfect agreement for both types of binaries. The characteristics of the post-merger remnants are equally similar, with the masses of the final black holes agreeing within δMBH<5×10-4MBH and their dimensionless spins within δχBH<10-3. The rate of periastron advance in the mixed binary agrees with previously published binary black hole results, and we use the inspiral waveforms to place constraints on the accuracy of our numerical simulations independent of algorithmic choices made for each type of binary. Overall, our results indicate that nondisrupting neutron star-black hole mergers are exceptionally well modeled by black hole-black hole mergers, and that given the absence of mass ejection, accretion disk formation, or differences in the gravitational wave signals, only electromagnetic precursors could prove the presence of a neutron star in low-spin systems of total mass ∼10M⊙, at least until the advent of gravitational wave detectors with a sensitivity comparable to that of the proposed Einstein Telescope.
Massive disc formation in the tidal disruption of a neutron star by a nearly extremal black hole
Classical and Quantum Gravity 30, 135004 (2013)
Abstract
Black hole–neutron star (BHNS) binaries are important sources of gravitational waves for second-generation interferometers, and BHNS mergers are also a proposed engine for short, hard gamma-ray bursts. The behavior of both the spacetime (and thus the emitted gravitational waves) and the neutron-star matter in a BHNS merger depend strongly and nonlinearly on the black hole's spin. While there is a significant possibility that astrophysical black holes could have spins that are nearly extremal (i.e. near the theoretical maximum), to date fully relativistic simulations of BHNS binaries have included black-hole spins only up to S/M2 = 0.9, which corresponds to the black hole having approximately half as much rotational energy as possible, given the black hole's mass. In this paper, we present a new simulation of a BHNS binary with a mass ratio q = 3 and black-hole spin S/M2 = 0.97, the highest simulated to date. We find that the black hole's large spin leads to the most massive accretion disc and the largest tidal tail outflow of any fully relativistic BHNS simulations to date, even exceeding the results implied by extrapolating results from simulations with lower black-hole spin. The disc appears to be remarkably stable. We also find that the high black-hole spin persists until shortly before the time of merger; afterward, both merger and accretion spin down the black hole.
Dynamical excision boundaries in spectral evolutions of binary black hole spacetimes
Classical and Quantum Gravity 30, 115001 (2013)
Abstract
Simulations of binary black hole systems using the Spectral Einstein Code (SpEC) are done on a computational domain that excises the regions inside the black holes. It is imperative that the excision boundaries are outflow boundaries with respect to the hyperbolic evolution equations used in the simulation. We employ a time- dependent mapping between the fixed computational frame and the inertial frame through which the black holes move. The time- dependent parameters of the mapping are adjusted throughout the simulation by a feedback control system in order to follow the motion of the black holes, to adjust the shape and size of the excision surfaces so that they remain outflow boundaries, and to prevent large distortions of the grid. We describe in detail the mappings and control systems that we use. We show how these techniques have been essential in the evolution of binary black hole systems with extreme configurations, such as large spin magnitudes and high mass ratios, especially during the merger, when apparent horizons are highly distorted and the computational domain becomes compressed. The techniques introduced here may be useful in other applications of partial differential equations that involve time- dependent mappings.
Black-hole-neutron-star mergers at realistic mass ratios: Equation of state and spin orientation effects
Physical Review D 87, 084006 (2013)
Abstract
Black-hole–neutron-star mergers resulting in the disruption of the neutron star and the formation of an accretion disk and/or the ejection of unbound material are prime candidates for the joint detection of gravitational-wave and electromagnetic signals when the next generation of gravitational-wave detectors comes online. However, the disruption of the neutron star and the properties of the postmerger remnant are very sensitive to the parameters of the binary (mass ratio, black-hole spin, neutron star radius). In this paper, we study the impact of the radius of the neutron star and the alignment of the black-hole spin on black-hole–neutron-star mergers within the range of mass ratio currently deemed most likely for field binaries (MBH∼7MNS) and for black-hole spins large enough for the neutron star to disrupt (JBH/MBH2=0.9). We find that (i) In this regime, the merger is particularly sensitive to the radius of the neutron star, with remnant masses varying from 0.3MNS to 0.1MNS for changes of only 2 km in the NS radius; (ii) 0.01M⊙–0.05M⊙ of unbound material can be ejected with kinetic energy ≳1051 ergs, a significant increase compared to low mass ratio, low spin binaries. This ejecta could power detectable postmerger optical and radio afterglows. (iii) Only a small fraction of the Advanced LIGO events in this parameter range have gravitational-wave signals which could offer constraints on the equation of state of the neutron star (at best ∼3% of the events for a single detector at design sensitivity). (iv) A misaligned black-hole spin works against disk formation, with less neutron-star material remaining outside of the black hole after merger, and a larger fraction of that material remaining in the tidal tail instead of the forming accretion disk. (v) Large kicks vkick≳300 km/s can be given to the final black hole as a result of a precessing black-hole–neutron-star merger, when the disruption of the neutron star occurs just outside or within the innermost stable spherical orbit.
Suitability of hybrid gravitational waveforms for unequal-mass binaries
Physical Review D 87, 024009 (2013)
Abstract
This article studies sufficient accuracy criteria of hybrid post- Newtonian (PN) and numerical relativity (NR) waveforms for parameter estimation of strong binary black-hole sources in second-generation ground-based gravitational-wave detectors. We investigate equal-mass nonspinning binaries with a new 33-orbit NR waveform, as well as unequal-mass binaries with mass ratios 2, 3, 4 and 6. For equal masses, the 33-orbit NR waveform allows us to recover previous results and to extend the analysis toward matching at lower frequencies. For unequal masses, the errors between different PN approximants increase with mass ratio. Thus, at 3.5 PN, hybrids for higher-mass-ratio systems would require NR waveforms with many more gravitational-wave cycles to guarantee no adverse impact on parameter estimation. Furthermore, we investigate the potential improvement in hybrid waveforms that can be expected from fourth- order post-Newtonian waveforms and find that knowledge of this fourth post-Newtonian order would significantly improve the accuracy of hybrid waveforms.
Error-analysis and comparison to analytical models of numerical waveforms produced by the NRAR Collaboration
Classical and Quantum Gravity 31, 025012 (2013)
Abstract
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 binary's 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.
#2012
Black-hole-neutron-star mergers: Disk mass predictions
Physical Review D 86, 124007 (2012)
Abstract
Determining the final result of black hole-neutron star mergers, and in particular the amount of matter remaining outside the black hole at late times and its properties, has been one of the main motivations behind the numerical simulation of these systems. Black hole-neutron star binaries are amongst the most likely progenitors of short gamma-ray bursts --- as long as massive (probably a few percents of a solar mass), hot accretion disks are formed around the black hole. Whether this actually happens strongly depends on the physical characteristics of the system, and in particular on the mass ratio, the spin of the black hole, and the radius of the neutron star. We present here a simple two-parameter model, fitted to existing numerical results, for the determination of the mass remaining outside the black hole a few milliseconds after a black hole-neutron star merger (i.e. the combined mass of the accretion disk, the tidal tail, and the potential ejecta). This model predicts the remnant mass within a few percents of the mass of the neutron star, at least for remnant masses up to 20% of the neutron star mass. Results across the range of parameters deemed to be the most likely astrophysically are presented here. We find that, for 10 solar mass black holes, massive disks are only possible for large neutron stars (R>12km), or quasi-extremal black hole spins (a/M>0.9). We also use our model to discuss how the equation of state of the neutron star affects the final remnant, and the strong influence that this can have on the rate of short gamma-ray bursts produced by black hole-neutron star mergers.
Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes. III. Quasinormal pulsations of Schwarzschild and Kerr black holes
Phys. Rev. D 86, 104028 (2012)
Abstract
In recent papers, we and colleagues have introduced a way to visualize the full vacuum Riemann curvature tensor using frame-drag vortex lines and their vorticities, and tidal tendex lines and their tendicities. We have also introduced the concepts of horizon vortexes and tendexes and 3-D vortexes and tendexes (regions where vorticities or tendicities are large). Using these concepts, we discover a number of previously unknown features of quasinormal modes of Schwarzschild and Kerr black holes. These modes can be classified by mode indexes \((n,l,m)\), and parity, which can be electric \([(-1)^l]\) or magnetic \([(-1)^{l+1}]\). Among our discoveries are these: (i) There is a near duality between modes of the same \((n,l,m)\): a duality in which the tendex and vortex structures of electric-parity modes are interchanged with the vortex and tendex structures (respectively) of magnetic-parity modes. (ii) This near duality is perfect for the modes' complex eigenfrequencies (which are well known to be identical) and perfect on the horizon; it is slightly broken in the equatorial plane of a non-spinning hole, and the breaking becomes greater out of the equatorial plane, and greater as the hole is spun up; but even out of the plane for fast-spinning holes, the duality is surprisingly good. (iii) Electric-parity modes can be regarded as generated by 3-D tendexes that stick radially out of the horizon. As these "longitudinal," near-zone tendexes rotate or oscillate, they generate longitudinal-transverse near-zone vortexes and tendexes, and outgoing and ingoing gravitational waves. The ingoing waves act back on the longitudinal tendexes, driving them to slide off the horizon, which results in decay of the mode's strength. (iv) By duality, magnetic-parity modes are driven in this same manner by longitudinal, near-zone vortexes that stick out of the horizon. [Abstract abridged.]
Precessing Binary Black Holes Simulations: Quasicircular Initial Data
Abstract
In numerical evolutions of binary black holes (BBH) it is desirable to easily control the orbital eccentricity of the BBH, and the number of orbits completed by the binary through merger. This paper presents fitting formulae that allow to choose initial-data parameters for generic precessing BBH resulting in an orbital eccentricity \(\sim 10^{-4}\), and that allow to predict the number of orbits to merger. We further demonstrate how these fits can be used to choose initial-data parameters of desired non-zero eccentricity. For both usage scenarios, no costly exploratory BBH evolutions are necessary, but both usage scenarios retain the freedom to refine the fitted parameters further based on the results of BBH evolutions. The presented fitting formulas are based on 729 BBH configurations which are iteratively reduced to eccentricity \(\lesssim 10^{-4}\), covering mass-ratios between 1 and 8 and spin-magnitude up to 0.5. 101 of these configurations are evolved through the BBH inspiral phase.
Geometrically motivated coordinate system for exploring spacetime dynamics in numerical-relativity simulations using a quasi-Kinnersley tetrad
Physical Review D 86, 084020 (2012)
Abstract
We investigate the suitability and properties of a quasi-Kinnersley tetrad and a geometrically motivated coordinate system as tools for quantifying both strong-field and wave-zone effects in numerical relativity (NR) simulations. We fix the radial and latitudinal coordinate degrees of freedom of the metric, using the Coulomb potential associated with the quasi-Kinnersley transverse frame. These coordinates are invariants of the spacetime and can be used to unambiguously fix the outstanding spin-boost freedom associated with the quasi-Kinnersley frame (resulting in a preferred quasi- Kinnersley tetrad (QKT)). In the limit of small perturbations about a Kerr spacetime, these coordinates and QKT reduce to Boyer- Lindquist coordinates and the Kinnersley tetrad, irrespective of the simulation gauge choice. We explore the properties of this construction both analytically and numerically, and we gain insights regarding the propagation of radiation described by a super-Poynting vector. We also quantify in detail the peeling properties of the chosen tetrad and gauge. We argue that these choices are particularly well suited for a rapidly converging wave-extraction algorithm as the extraction location approaches infinity, and we explore numerically the extent to which this property remains applicable on the interior of a computational domain. Using a number of additional tests, we verify that the prescription behaves as required in the appropriate limits regardless of simulation gauge. We explore the behavior of the geometrically motivated coordinate system in dynamical binary-black-hole NR mergers, and find them useful for visualizing features in NR simulations such as the spurious 'junk' radiation. Finally, we carefully scrutinize the head-on collision of two black holes and, for example, the way in which the extracted waveform changes as it moves through the computational domain.
Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes. II. Stationary black holes
Physical Review D 86, 084049 (2012)
Abstract
When one splits spacetime into space plus time, the Weyl curvature tensor (which equals the Riemann tensor in vacuum) splits into two spatial, symmetric, traceless tensors: the tidal field \(E\), which produces tidal forces, and the frame-drag field \(B\), which produces differential frame dragging. In recent papers, we and colleagues have introduced ways to visualize these two fields: tidal tendex lines (integral curves of the three eigenvector fields of \(E\)) and their tendicities (eigenvalues of these eigenvector fields)/ and the corresponding entities for the frame-drag field: frame-drag vortex lines and their vorticities. These entities fully characterize the vacuum Riemann tensor. In this paper, we compute and depict the tendex and vortex lines, and their tendicities and vorticities, outside the horizons of stationary (Schwarzschild and Kerr) black holes/ and we introduce and depict the black holes' horizon tendicity and vorticity (the normal-normal components of \(E\) and \(B\) on the horizon). For Schwarzschild and Kerr black holes, the horizon tendicity is proportional to the horizon's intrinsic scalar curvature, and the horizon vorticity is proportional to an extrinsic scalar curvature. We show that, for horizon-penetrating time slices, all these entities (\(E\), \(B\), the tendex lines and vortex lines, the lines' tendicities and vorticities, and the horizon tendicities and vorticities) are affected only weakly by changes of slicing and changes of spatial coordinates, within those slicing and coordinate choices that are commonly used for black holes. [Abstract is abbreviated.]
Are different approaches to constructing initial data for binary black hole simulations of the same astrophysical situation equivalent?
Physical Review D 86, 084054 (2012)
Abstract
Initial data for numerical evolutions of binary-black holes have been dominated by 'conformally flat' (CF) data (i.e., initial data where the conformal background metric is chosen to be flat) because they are easy to construct. However, CF initial data cannot simulate nearly extremal spins, while more complicated 'conformally curved' initial data (i.e., initial data in which the background metric is \emph{not} explicitly chosen to be flat), such as initial data where the spatial metric is chosen to be proportional to a weighted superposition of two Kerr-Schild (SKS) black holes can. Here we establish the consistency between the astrophysical results of these two initial data schemes for nonspinning binary systems. We evolve the inspiral, merger, and ringdown of two equal-mass, nonspinning black holes using SKS initial data and compare with an analogous simulation using CF initial data. We find that the resultant gravitational-waveform phases agree to within \(\delta \phi \lesssim 10^{-2}\) radians and the amplitudes agree to within \(\delta A/A \lesssim 5 \times 10^{-3}\), which are within the numerical errors of the simulations. Furthermore, we find that the final mass and spin of the remnant black hole agree to one part in \(10^{5}\).
Simulations of unequal-mass black hole binaries with spectral methods
Physical Review D 86, 084033 (2012)
Abstract
This paper presents techniques and results for simulations of unequal mass, non-spinning black hole binaries with pseudo-spectral methods. Specifically, we develop an efficient root-finding procedure to ensure the black hole initial data have the desired masses and spins, we extend the dual coordinate frame method and eccentricity removal to asymmetric binaries. Furthermore, we describe techniques to simulate mergers of unequal mass black holes. The second part of the paper presents numerical simulations of non- spinning black hole binaries with mass ratios 2, 3, 4 and 6, covering between 15 and 22 orbits, merger and ringdown. We discuss the accuracy of these simulations, the evolution of the (initially zero) black hole spins, and the remnant black hole properties.
Prototype effective-one-body model for nonprecessing spinning inspiral-merger-ringdown waveforms
Physical Review D 86, 024011 (2012)
Abstract
We first use five non-spinning and two mildly spinning (chi_i \simeq -0.44, +0.44) numerical-relativity waveforms of black-hole binaries and calibrate an effective-one-body (EOB) model for non-precessing spinning binaries, notably its dynamics and the dominant (2,2) gravitational-wave mode. Then, we combine the above results with recent outcomes of small-mass-ratio simulations produced by the Teukolsky equation and build a prototype EOB model for detection purposes, which is capable of generating inspiral-merger-ringdown waveforms for non-precessing spinning black-hole binaries with any mass ratio and individual black-hole spins -1 \leq chi_i \lesssim 0.7. We compare the prototype EOB model to two equal-mass highly spinning numerical-relativity waveforms of black holes with spins chi_i = -0.95, +0.97, which were not available at the time the EOB model was calibrated. In the case of Advanced LIGO we find that the mismatch between prototype-EOB and numerical-relativity waveforms is always smaller than 0.003 for total mass 20-200 M_\odot, the mismatch being computed by maximizing only over the initial phase and time. To successfully generate merger waveforms for individual black-hole spins chi_i \gtrsim 0.7, the prototype-EOB model needs to be improved by (i) better modeling the plunge dynamics and (ii) including higher-order PN spin terms in the gravitational-wave modes and radiation-reaction force.
Numerical simulations of compact object binaries
Classical and Quantum Gravity 29, 124004 (2012)
Abstract
Coalescing compact object binaries consisting of black holes and/or Neutron stars are a prime target for ground-based gravitational wave detectors. This article reviews the status of numerical simulations of these systems, with an emphasis on recent progress.
The NINJA-2 catalog of hybrid post-Newtonian/numerical-relativity waveforms for non-precessing black-hole binaries
Classical and Quantum Gravity 29, 124001 (2012)
Abstract
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 and parameter- estimation algorithms using numerically generated waveforms and to foster closer collaboration between the numerical-relativity and data-analysis communities. The first NINJA project used only a small number of injections of short numerical-relativity waveforms, which limited its ability to draw quantitative conclusions. The goal of the NINJA-2 project is to overcome these limitations with long post- Newtonian—numerical-relativity hybrid waveforms, large numbers of injections and the use of real detector data. We report on the submission requirements for the NINJA-2 project and the construction of the waveform catalog. Eight numerical-relativity groups have contributed 56 hybrid waveforms consisting of a numerical portion modeling the late inspiral, merger and ringdown stitched to a post- Newtonian portion modeling the early inspiral. We summarize the techniques used by each group in constructing their submissions. We also report on the procedures used to validate these submissions, including examination in the time and frequency domains and comparisons of waveforms from different groups against each other. These procedures have so far considered only the (ℓ, m) = (2, 2) mode. Based on these studies, we judge that the hybrid waveforms are suitable for NINJA-2 studies. We note some of the plans for these investigations.
High-accuracy gravitational waveforms for binary black hole mergers with nearly extremal spins
Classical and Quantum Gravity 29, 045003 (2012)
Abstract
Motivated by the possibility of observing gravitational waves from merging black holes whose spins are nearly extremal (i.e., 1 in dimensionless units), we present numerical waveforms from simulations of merging black holes with the highest spins simulated to date: (1) a 25.5-orbit inspiral, merger, and ringdown of two holes with equal masses and spins of magnitude 0.97 aligned with the orbital angular momentum/ and (2) a previously reported 12.5-orbit inspiral, merger, and ringdown of two holes with equal masses and spins of magnitude 0.95 anti-aligned with the orbital angular momentum. First, we consider the horizon mass and spin evolution of the new aligned-spin simulation. During the inspiral, the horizon area and spin evolve in remarkably close agreement with Alvi's analytic predictions, and the remnant hole's final spin agrees reasonably well with several analytic predictions. We also find that the total energy emitted by a real astrophysical system with these parameters---almost all of which is radiated during the time included in this simulation---would be 10.952% of the initial mass at infinite separation. Second, we consider the gravitational waveforms for both simulations. After estimating their uncertainties, we compare the waveforms to several post-Newtonian approximants, finding significant disagreement well before merger, although the phase of the TaylorT4 approximant happens to agree remarkably well with the numerical prediction in the aligned-spin case. We find that the post-Newtonian waveforms have sufficient uncertainty that hybridized waveforms will require far longer numerical simulations (in the absence of improved post-Newtonian waveforms) for accurate parameter estimation of low-mass binary systems.
Black hole-neutron star mergers for \(10M_\odot\) black holes
Physical Review D 85, 044015 (2012)
Abstract
General relativistic simulations of black hole-neutron star mergers have currently been limited to low-mass black holes (less than 7 solar mass), even though population synthesis models indicate that a majority of mergers might involve more massive black holes (10 solar mass or more). We present the first general relativistic simulations of black hole-neutron star mergers with 10 solar mass black holes. For massive black holes, the tidal forces acting on the neutron star are usually too weak to disrupt the star before it reaches the innermost stable circular orbit of the black hole. Varying the spin of the black hole in the range a/M = 0.5-0.9, we find that mergers result in the disruption of the star and the formation of a massive accretion disk only for large spins a/M>0.7-0.9. From these results, we obtain updated constraints on the ability of BHNS mergers to be the progenitors of short gamma-ray bursts as a function of the mass and spin of the black hole. We also discuss the dependence of the gravitational wave signal on the black hole parameters, and provide waveforms and spectra from simulations beginning 7-8 orbits before merger.
Toroidal horizons in binary black hole inspirals
Physical Review D 85, 024031 (2012)
Abstract
We examine the structure of the event horizon for numerical simulations of two black holes that begin in a quasicircular orbit, inspiral, and finally merge. We find that the spatial cross section of the merged event horizon has spherical topology (to the limit of our resolution), despite the expectation that generic binary black hole mergers in the absence of symmetries should result in an event horizon that briefly has a toroidal cross section. Using insight gained from our numerical simulations, we investigate how the choice of time slicing affects both the spatial cross section of the event horizon and the locus of points at which generators of the event horizon cross. To ensure the robustness of our conclusions, our results are checked at multiple numerical resolutions. 3D visualization data for these resolutions are available for public access online. We find that the structure of the horizon generators in our simulations is consistent with expectations, and the lack of toroidal horizons in our simulations is due to our choice of time slicing.
#2011
Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes: General theory and weak-gravity applications
Phys. Rev. D 84, 124014 (2011)
Abstract
When one splits spacetime into space plus time, the Weyl curvature tensor (vacuum Riemann tensor) gets split into two spatial, symmetric, and trace-free (STF) tensors: (i) the Weyl tensor's so-called "electric" part or tidal field, and (ii) the Weyl tensor's so-called "magnetic" part or frame-drag field. Being STF, the tidal field and frame-drag field each have three orthogonal eigenvector fields which can be depicted by their integral curves. We call the integral curves of the tidal field's eigenvectors tendex lines, we call each tendex line's eigenvalue its tendicity, and we give the name tendex to a collection of tendex lines with large tendicity. The analogous quantities for the frame-drag field are vortex lines, their vorticities, and vortexes. We build up physical intuition into these concepts by applying them to a variety of weak-gravity phenomena: a spinning, gravitating point particle, two such particles side by side, a plane gravitational wave, a point particle with a dynamical current-quadrupole moment or dynamical mass-quadrupole moment, and a slow-motion binary system made of nonspinning point particles. [Abstract is abbreviated; full abstract also mentions additional results.]
Geometric approach to the precession of compact binaries
Phys. Rev. D 84, 124011 (2011)
Abstract
We discuss a geometrical method to define a preferred reference frame for precessing binary systems and the gravitational waves they emit. This minimal-rotation frame is aligned with the angular-momentum axis and fixes the rotation about that axis up to a constant angle, resulting in an essentially invariant frame. Gravitational waveforms decomposed in this frame are similarly invariant under rotations of the inertial frame and exhibit relatively smoothly varying phase. By contrast, earlier prescriptions for radiation-aligned frames induce extraneous features in the gravitational-wave phase which depend on the orientation of the inertial frame, leading to fluctuations in the frequency that may compound to many gravitational-wave cycles. We explore a simplified description of post-Newtonian approximations for precessing systems using the minimal-rotation frame, and describe the construction of analytical/numerical hybrid waveforms for such systems.
Inspiral-merger-ringdown multipolar waveforms of nonspinning black-hole binaries using the effective-one-body formalism
Physical Review D 84, 124052 (2011)
Abstract
We calibrate an effective-one-body (EOB) model to numerical- relativity simulations of mass ratios 1, 2, 3, 4, and 6, by maximizing phase and amplitude agreement of the leading (2,2) mode and of the subleading modes (2,1), (3,3), (4,4) and (5,5). Aligning the calibrated EOB waveforms and the numerical waveforms at low frequency, the phase difference of the (2,2) mode between model and numerical simulation remains below 0.1 rad throughout the evolution for all mass ratios considered. The fractional amplitude difference at peak amplitude of the (2,2) mode is 2% and grows to 12% during the ringdown. Using the Advanced LIGO noise curve we study the effectualness and measurement accuracy of the EOB model, and stress the relevance of modeling the higher-order modes for parameter estimation. We find that the effectualness, measured by the mismatch, between the EOB and numerical-relativity polarizations which include only the (2,2) mode is smaller than 0.2% for binaries with total mass 20-200 Msun and mass ratios 1, 2, 3, 4, and 6. When numerical-relativity polarizations contain the strongest seven modes, and stellar-mass black holes with masses less than 50Msun are considered, the mismatch for mass ratio 6 (1) can be as high as 5% (0.2%) when only the EOB (2,2) mode is included, and an upper bound of the mismatch is 0.5% (0.07%) when all the four subleading EOB modes calibrated in this paper are taken into account. For binaries with intermediate-mass black holes with masses greater than 50Msun the mismatches are larger. We also determine for which signal-to- noise ratios the EOB model developed here can be used to measure binary parameters with systematic biases smaller than statistical errors due to detector noise.
Implicit-explicit evolution of single black holes
Physical Review D 84, 084023 (2011)
Abstract
Numerical simulations of binary black holes---an important predictive tool for the detection of gravitational waves---are computationally expensive, especially for binaries with high mass ratios or with rapidly spinning constituent holes. Existing codes for evolving binary black holes rely on explicit timestepping methods, for which the timestep size is limited by the smallest spatial scale through the Courant-Friedrichs-Lewy condition. Binary inspiral typically involves spatial scales (the spatial resolution required by a small or rapidly spinning hole) which are orders of magnitude smaller than the relevant (orbital, precession, and radiation-reaction) timescales characterizing the inspiral. Therefore, in explicit evolutions of binary black holes, the timestep size is typically orders of magnitude smaller than the relevant physical timescales. Implicit timestepping methods allow for larger timesteps, and they often reduce the total computational cost (without significant loss of accuracy) for problems dominated by spatial rather than temporal error, such as for binary-black-hole inspiral in corotating coordinates. However, fully implicit methods can be difficult to implement for nonlinear evolution systems like the Einstein equations. Therefore, in this paper we explore implicit-explicit (IMEX) methods and use them for the first time to evolve black-hole spacetimes. Specifically, as a first step toward IMEX evolution of a full binary-black-hole spacetime, we develop an IMEX algorithm for the generalized harmonic formulation of the Einstein equations and use this algorithm to evolve stationary and perturbed single-black-hole spacetimes. Numerical experiments explore the stability and computational efficiency of our method.
Periastron Advance in Black-Hole Binaries
Physical Review Letters 107, 141101 (2011)
Abstract
The general relativistic (Mercury-type) periastron advance is calculated here for the first time with exquisite precision in full general relativity. We use accurate numerical relativity simulations of spinless black hole binaries with mass ratios 1/8 < m1/m2 < 1 and compare with the predictions of several analytic approximation schemes. We find the effective-one-body model to be remarkably accurate, and, surprisingly, so also the predictions of self-force theory [replacing m1/m2 --> m1m2/(m1+m2)^2]. Our results can inform a universal analytic model of the two-body dynamics, crucial for ongoing and future gravitational-wave searches.
Suitability of post-Newtonian/numerical-relativity hybrid waveforms for gravitational wave detectors
Classical and Quantum Gravity 28, 134002 (2011)
Abstract
This article presents a study of the sufficient accuracy of post- Newtonian and numerical relativity waveforms for the most demanding usage case: parameter estimation of strong sources in advanced gravitational wave detectors. For black hole binaries, these detectors require accurate waveform models which can be constructed by fusing an analytical post-Newtonian inspiral waveform with a numerical relativity merger-ringdown waveform. We perform a comprehensive analysis of errors that enter such "hybrid waveforms". We find that the post-Newtonian waveform must be aligned with the numerical relativity waveform to exquisite accuracy, about 1/100 of a gravitational wave cycle. Phase errors in the inspiral phase of the numerical relativity simulation must be controlled to less than about 0.1rad. (These numbers apply to moderately optimistic estimates about the number of GW sources; exceptionally strong signals require even smaller errors.) The dominant source of error arises from the inaccuracy of the investigated post-Newtonian Taylor-approximants. Using our error criterium, even at 3.5-th post- Newtonian order, hybridization has to be performed significantly before the start of the longest currently available numerical waveforms which cover 30 gravitational wave cycles. The current investigation is limited to the equal-mass, zero-spin case and does not take into account calibration errors of the gravitational wave detectors.
Horizon dynamics of distorted rotating black holes
Physical Review D 83, 104018 (2011)
Abstract
We present numerical simulations of a Kerr black hole perturbed by a pulse of ingoing gravitational radiation. For strong perturbations we find up to five concentric marginally outer trapped surfaces. These trapped surfaces appear and disappear in pairs, so that the total number of such surfaces at any given time is odd. The world tubes traced out by the marginally outer trapped surfaces are found to be spacelike during the highly dynamical regime, approaching a null hypersurface at early and late times. We analyze the structure of these marginally trapped tubes in the context of the dynamical horizon formalism, computing the expansion of outgoing and incoming null geodesics, as well as evaluating the dynamical horizon flux law and the angular momentum flux law. Finally, we compute the event horizon. The event horizon is well-behaved and approaches the apparent horizon before and after the highly dynamical regime. No new generators enter the event horizon during the simulation.
Reducing orbital eccentricity of precessing black-hole binaries
Physical Review D 83, 104034 (2011)
Abstract
Building initial conditions for generic binary black-hole evolutions without initial spurious eccentricity remains a challenge for numerical-relativity simulations. This problem can be overcome by applying an eccentricity-removal procedure which consists in evolving the binary for a couple of orbits, estimating the eccentricity, and then correcting the initial conditions. The presence of spins can complicate this procedure. As predicted by post-Newtonian theory, spin-spin interactions and precession prevent the binary from moving along an adiabatic sequence of spherical orbits, inducing oscillations in the radial separation and in the orbital frequency. However, spin-induced oscillations occur at approximately twice the orbital frequency, therefore they can be distinguished from the initial spurious eccentricity, which occurs at approximately the orbital frequency. We develop a new removal procedure based on the derivative of the orbital frequency and find that it is successful in reducing the eccentricity measured in the orbital frequency to less than 0.0001 when moderate spins are present. We test this new procedure using numerical-relativity simulations of binary black holes with mass ratios 1.5 and 3, spin magnitude 0.5 and various spin orientations. The numerical simulations exhibit spin-induced oscillations in the dynamics at approximately twice the orbital frequency. Oscillations of similar frequency are also visible in the gravitational-wave phase and frequency of the dominant mode.
Frame-Dragging Vortexes and Tidal Tendexes Attached to Colliding Black Holes: Visualizing the Curvature of Spacetime
Physical Review Letters 106, 151101 (2011)
Abstract
When one splits spacetime into space plus time, the spacetime curvature (Weyl tensor) gets split into an "electric" part E_{jk} that describes tidal gravity and a "magnetic" part B_{jk} that describes differential dragging of inertial frames. We introduce tools for visualizing B_{jk} (frame-drag vortex lines, their vorticity, and vortexes) and E_{jk} (tidal tendex lines, their tendicity, and tendexes), and also visualizations of a black-hole horizon's (scalar) vorticity and tendicity. We use these tools to elucidate the nonlinear dynamics of curved spacetime in merging black-hole binaries.
Hyperboloidal layers for hyperbolic equations on unbounded domains
Journal of Computational Physics 230, 2286 (2011)
Abstract
We show how to solve hyperbolic equations numerically on unbounded domains by compactification, thereby avoiding the introduction of an artificial outer boundary. The essential ingredient is a suitable transformation of the time coordinate in combination with spatial compactification. We construct a new layer method based on this idea, called the hyperboloidal layer. The method is demonstrated on numerical tests including the one dimensional Maxwell equations using finite differences and the three dimensional wave equation with and without nonlinear source terms using spectral techniques.
Simulating merging binary black holes with nearly extremal spins
Physical Review D 83, 024010 (2011)
Abstract
Astrophysically realistic black holes may have spins that are nearly extremal (i.e., close to 1 in dimensionless units). Numerical simulations of binary black holes are important tools both for calibrating analytical templates for gravitational-wave detection and for exploring the nonlinear dynamics of curved spacetime. However, all previous simulations of binary-black-hole inspiral, merger, and ringdown have been limited by an apparently insurmountable barrier: the merging holes' spins could not exceed 0.93, which is still a long way from the maximum possible value in terms of the physical effects of the spin. In this paper, we surpass this limit for the first time, opening the way to explore numerically the behavior of merging, nearly extremal black holes. Specifically, using an improved initial-data method suitable for binary black holes with nearly extremal spins, we simulate the inspiral (through 12.5 orbits), merger and ringdown of two equal- mass black holes with equal spins of magnitude 0.95 antialigned with the orbital angular momentum.
Black hole-neutron star mergers: Effects of the orientation of the black hole spin
Physical Review D 83, 024005 (2011)
Abstract
The spin of black holes in black hole-neutron star (BHNS) binaries can have a strong influence on the merger dynamics and the postmerger state/ a wide variety of spin magnitudes and orientations are expected to occur in nature. In this paper, we report the first simulations in full general relativity of BHNS mergers with misaligned black hole spin. We vary the spin magnitude from a/m=0 to a/m=0.9 for aligned cases, and we vary the misalignment angle from 0 to 80 degrees for a/m=0.5. We restrict our study to 3:1 mass ratio systems and use a simple Gamma-law equation of state. We find that the misalignment angle has a strong effect on the mass of the postmerger accretion disk, but only for angles greater than ~ 40 degrees. Although the disk mass varies significantly with spin magnitude and misalignment angle, we find that all disks have very similar lifetimes ~ 100ms. Their thermal and rotational profiles are also very similar. For a misaligned merger, the disk is tilted with respect to the final black hole's spin axis. This will cause the disk to precess, but on a timescale longer than the accretion time. In all cases, we find promising setups for gamma-ray burst production: the disks are hot, thick, and hyperaccreting, and a baryon-clear region exists above the black hole.
#2010
Measuring orbital eccentricity and periastron advance in quasicircular black hole simulations
Physical Review D 82, 124016 (2010)
Abstract
We compare different methods of computing the orbital eccentricity of quasi-circular binary black hole systems using the orbital variables and gravitational wave phase and frequency. For eccentricities of about a per cent, most methods work satisfactorily. For small eccentricity, however, the gravitational wave phase allows a particularly clean and reliable measurement of the eccentricity. Furthermore, we measure the decay of the orbital eccentricity during the inspiral and find reasonable agreement with post-Newtonian results. Finally, we measure the periastron advance of non-spinning binary black holes, and we compare them to post- Newtonian approximations. With the low uncertainty in the measurement of the periastron advance, we positively detect deviations between fully numerical simulations and post-Newtonian calculations.
Spectral methods for the wave equation in second-order form
Physical Review D 82, 024037 (2010)
Abstract
Current spectral simulations of Einstein's equations require writing the equations in first-order form, potentially introducing instabilities and inefficiencies. We present a new penalty method for pseudo-spectral evolutions of second order in space wave equations. The penalties are constructed as functions of Legendre polynomials and are added to the equations of motion everywhere, not only on the boundaries. Using energy methods, we prove semi-discrete stability of the new method for the scalar wave equation in flat space and show how it can be applied to the scalar wave on a curved background. Numerical results demonstrating stability and convergence for multi-domain second-order scalar wave evolutions are also presented. This work provides a foundation for treating Einstein's equations directly in second-order form by spectral methods.
Degeneracy measures for the algebraic classification of numerical spacetimes
Phys. Rev. D 81, 124042 (2010)
Abstract
We study the issue of algebraic classification of the Weyl curvature tensor, with a particular focus on numerical relativity simulations. The spacetimes of interest in this context, binary black hole mergers, and the ringdowns that follow them, present subtleties in that they are generically, strictly speaking, type I, but in many regions approximately, in some sense, type D. To provide meaning to any claims of “approximate” Petrov class, one must define a measure of degeneracy on the space of null rays at a point. We will investigate such a measure, used recently to argue that certain binary black hole merger simulations ring down to the Kerr geometry, after hanging up for some time in Petrov type II. In particular, we argue that this hangup in Petrov type II is an artefact of the particular measure being used, and that a geometrically better-motivated measure shows a black hole merger produced by our group settling directly to Petrov type D.
Hyperboloidal evolution of test fields in three spatial dimensions
Physical Review D 81, 124010 (2010)
Abstract
We present the numerical implementation of a clean solution to the outer boundary and radiation extraction problems within the 3+1 formalism for hyperbolic partial differential equations on a given background. Our approach is based on compactification at null infinity in hyperboloidal scri fixing coordinates. We report numerical tests for the particular example of a scalar wave equation on Minkowski and Schwarzschild backgrounds. We address issues related to the implementation of the hyperboloidal approach for the Einstein equations, such as nonlinear source functions, matching, and evaluation of formally singular terms at null infinity.
Equation of state effects in black hole--neutron star mergers
Classical and Quantum Gravity 27, 114106 (2010)
Abstract
The merger dynamics of a black hole-neutron star (BHNS) binary is influenced by the neutron star equation of state (EoS) through the latter's effect on the neutron star's radius and on the character of the mass transfer onto the black hole. We study these effects by simulating a number of BHNS binaries in full general relativity using a mixed pseudospectral/finite difference code. We consider several models of the neutron star matter EoS, including Gamma=2 and Gamma=2.75 polytropes and the nuclear-theory based Shen EoS. For models using the Shen EoS, we consider two limits for the evolution of the composition: source-free advection and instantaneous beta- equilibrium. To focus on EoS effects, we fix the mass ratio to 3:1 and the initial aligned black hole spin to a/m=0.5 for all models. We confirm earlier studies which found that more compact stars create a stronger gravitational wave signal but a smaller postmerger accretion disk. We also vary the EoS while holding the compaction fixed. All mergers are qualitatively similar, but we find signatures of the EoS in the waveform and in the tail and disk structures.
Effective-one-body waveforms calibrated to numerical relativity simulations: Coalescence of nonprecessing, spinning, equal-mass black holes
Phys. Rev. D 81, 084041 (2010)
Abstract
We present the first attempt at calibrating the effective-one-body (EOB) model to accurate numerical-relativity simulations of spinning, non-precessing black-hole binaries. Aligning the EOB and numerical waveforms at low frequency over a time interval of 1000M, we first estimate the phase and amplitude errors in the numerical waveforms and then minimize the difference between numerical and EOB waveforms by calibrating a handful of EOB-adjustable parameters. In the equal-mass, spin aligned case, we find that phase and fractional amplitude differences between the numerical and EOB (2,2) mode can be reduced to 0.01 radians and 1%, respectively, over the entire inspiral waveforms. In the equal-mass, spin anti-aligned case, these differences can be reduced to 0.13 radians and 1% during inspiral and plunge, and to 0.4 radians and 10% during merger and ringdown. The waveform agreement is within numerical errors in the spin aligned case while slightly over numerical errors in the spin anti-aligned case. Using Enhanced LIGO and Advanced LIGO noise curves, we find that the overlap between the EOB and the numerical (2,2) mode, maximized over the initial phase and time of arrival, is larger than 0.999 for binaries with total mass 30M⊙-200M⊙. In addition to the leading (2,2) mode, we compare four subleading modes. We find good amplitude and frequency agreements between the EOB and numerical modes for both spin configurations considered, except for the (3,2) mode in the spin anti-aligned case. We believe that the larger difference in the (3,2) mode is due to the lack of knowledge of post-Newtonian spin effects in the higher modes.
#2009
High accuracy simulations of black hole binaries: spins anti-aligned with the orbital angular momentum
Phys. Rev. D 80, 124051 (2009)
Abstract
High-accuracy binary black hole simulations are presented for black holes with spins anti-aligned with the orbital angular momentum. The particular case studied represents an equal-mass binary with spins of equal magnitude S/m2=0.437 57±0.000 01. The system has initial orbital eccentricity ∼4×10-5, and is evolved through 10.6 orbits plus merger and ringdown. The remnant mass and spin are Mf=(0.961 109±0.000 003)M and Sf/Mf2=0.547 81±0.000 01, respectively, where M is the mass during early inspiral. The gravitational waveforms have accumulated numerical phase errors of ≲0.1 radians without any time or phase shifts, and ≲0.01 radians when the waveforms are aligned with suitable time and phase shifts. The waveform is extrapolated to infinity using a procedure accurate to ≲0.01 radians in phase, and the extrapolated waveform differs by up to 0.13 radians in phase and about 1% in amplitude from the waveform extracted at finite radius r=350M. The simulations employ different choices for the constraint damping parameters in the wave zone; this greatly reduces the effects of junk radiation, allowing the extraction of a clean gravitational wave signal even very early in the simulation.
Simulations of binary black hole mergers using spectral methods
Physical Review D 80, 124010 (2009)
Abstract
Several improvements in numerical methods and gauge choice are presented that make it possible now to perform simulations of the merger and ringdown phases of "generic" binary black-hole evolutions using the pseudo-spectral evolution code SpEC. These improvements include the use of a new damped-wave gauge condition, a new grid structure with appropriate filtering that improves stability, and better adaptivity in conforming the grid structures to the shapes and sizes of the black holes. Simulations illustrating the success of these new methods are presented for a variety of binary black- hole systems. These include fairly ``generic'' systems with unequal masses (up to 2:1 mass ratios), and spins (with magnitudes up to 0.4 M^2) pointing in various directions.
Extrapolating gravitational-wave data from numerical simulations
Physical Review D 80, 124045 (2009)
Abstract
Two complementary techniques are developed for obtaining the asymptotic form of gravitational-wave data at large radii from numerical simulations, in the form of easily implemented algorithms. It is shown that, without extrapolation, near-field effects produce errors in extracted waveforms that can significantly affect LIGO data analysis. The extrapolation techniques are discussed in the context of Newman-Penrose data applied to extrapolation of waveforms from an equal-mass, nonspinning black-hole binary simulation. The results of the two methods are shown to agree within error estimates. The various benefits and deficiencies of the methods are discussed.
Black hole initial data on hyperboloidal slices
Phys. Rev. D 80, 084024 (2009)
Abstract
We generalize Bowen-York black hole initial data to hyperboloidal constant mean curvature slices which extend to future null infinity. We solve this initial value problem numerically for several cases, including unequal mass binary black holes with spins and boosts. The singularity at null infinity in the Hamiltonian constraint associated with a constant mean curvature hypersurface does not pose any particular difficulties. The inner boundaries of our slices are minimal surfaces. Trumpet configurations are explored both analytically and numerically.
Final remnant of binary black hole mergers: Multipolar analysis
Phys. Rev. D 80, 084012 (2009)
Abstract
Methods are presented to define and compute source multipoles of dynamical horizons in numerical relativity codes, extending previous work in the isolated and dynamical horizon formalisms to allow for horizons that are not axisymmetric. These methods are then applied to a binary black hole merger simulation, providing evidence that the final remnant is a Kerr black hole, both through the (spatially) gauge-invariant recovery of the geometry of the apparent horizon, and through a detailed extraction of quasinormal ringing modes directly from the strong-field region.
Improved gauge driver for the generalized harmonic Einstein system
Physical Review D 80, 084019 (2009)
Abstract
A new gauge driver is introduced for the GH (generalized harmonic) representation of Einstein's equation. This new driver allows a rather general class of gauge conditions to be implemented in a way that maintains the hyperbolicity of the combined evolution system. This driver is more stable and effective, and unlike previous drivers, allows stable evolutions using the dual-frame evolution technique. Some useful new gauge conditions are also introduced, in which the lapse and the spatial coordinates satisfy damped-wave equations. Appropriate boundary conditions for this new gauge driver are constructed, and a new boundary condition for the 'gauge' components of the spacetime metric in the GH Einstein system is introduced. The stability and effectiveness of this new gauge driver and the new damped-wave coordinate conditions are demonstrated through numerical tests using evolutions of single black-hole spacetimes with large amplitude gauge perturbations.
Testing gravitational-wave searches with numerical relativity waveforms: results from the first Numerical INJection Analysis (NINJA) project
Classical and Quantum Gravity 26, 165008 (2009)
Abstract
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.
Momentum flow in black-hole binaries: II. Numerical simulations of equal-mass, head-on mergers with antiparallel spins.
Abstract
Research on extracting science from binary-black-hole (BBH) simulations has often adopted a "scattering matrix" perspective: given the binary's initial parameters, what are the final hole's parameters and the emitted gravitational waveform? In contrast, we are using BBH simulations to explore the nonlinear dynamics of curved spacetime. Focusing on the head-on plunge, merger, and ringdown of a BBH with transverse, antiparallel spins, we explore numerically the momentum flow between the holes and the surrounding spacetime. We use the Landau-Lifshitz field-theory-in-flat-spacetime formulation of general relativity to define and compute the density of field energy and field momentum outside horizons and the energy and momentum contained within horizons, and we define the effective velocity of each apparent and event horizon as the ratio of its enclosed momentum to its enclosed mass-energy. We find surprisingly good agreement between the horizons' effective and coordinate velocities. To investigate the gauge dependence of our results, we compare pseudospectral and moving-puncture evolutions of physically similar initial data; although spectral and puncture simulations use different gauge conditions, we find remarkably good agreement for our results in these two cases. We also compare our simulations with the post-Newtonian trajectories and near-field energy- momentum. [Abstract abbreviated; full abstract also mentions additional results.]
Orbiting binary black hole evolutions with a multipatch high order finite-difference approach
Physical Review D 80, 024027 (2009)
Abstract
We present numerical simulations of orbiting black holes for around twelve cycles, using a high-order multipatch approach. Unlike some other approaches, the computational speed scales almost perfectly for thousands of processors. Multipatch methods are an alternative to AMR (adaptive mesh refinement), with benefits of simplicity and better scaling for improving the resolution in the wave zone. The results presented here pave the way for multipatch evolutions of black hole-neutron star and neutron star-neutron star binaries, where high resolution grids are needed to resolve details of the matter flow.
Effective-one-body waveforms calibrated to numerical relativity simulations: Coalescence of nonspinning, equal-mass black holes
Phys. Rev. D 79, 124028 (2009)
Abstract
We calibrate the effective-one-body (EOB) model to an accurate numerical simulation of an equal-mass, non-spinning binary black-hole coalescence produced by the Caltech-Cornell collaboration. Aligning the EOB and numerical waveforms at low frequency over a time interval of ~1000M, and taking into account the uncertainties in the numerical simulation, we investigate the significance and degeneracy of the EOB adjustable parameters during inspiral, plunge and merger, and determine the minimum number of EOB adjustable parameters that achieves phase and amplitude agreements on the order of the numerical error. We find that phase and fractional amplitude differences between the numerical and EOB values of the dominant gravitational wave mode h22 can be reduced to 0.02 radians and 2%, respectively, until a time 26 M before merger, and to 0.1 radians and 10%, at a time 16M after merger (during ringdown), respectively. Using LIGO, Enhanced LIGO and Advanced LIGO noise curves, we find that the overlap between the EOB and the numerical h22, maximized only over the initial phase and time of arrival, is larger than 0.999 for equal-mass binary black holes with total mass 30-150 M⊙. In addition to the leading gravitational mode (2,2), we compare the dominant subleading modes (4,4) and (3,2) and find phase and amplitude differences on the order of the numerical error. We also determine the mass-ratio dependence of one of the EOB adjustable parameters by fitting to numerical inspiral waveforms for black-hole binaries with mass ratios 2:1 and 3:1. These results improve and extend recent successful attempts aimed at providing gravitational-wave data analysts the best analytical EOB model capable of interpolating accurate numerical simulations.
Reducing spurious gravitational radiation in binary-black-hole simulations by using conformally curved initial data
Class. Quantum Grav. 26, 114002 (2009)
Abstract
At early times in numerical evolutions of binary black holes, current simulations contain an initial burst of spurious gravitational radiation (also called "junk radiation") which is not astrophysically realistic. The spurious radiation is a consequence of how the binary-black- hole initial data are constructed: the initial data are typically assumed to be conformally flat. In this paper, I adopt a curved conformal metric that is a superposition of two boosted, non- spinning black holes that are approximately 15 orbits from merger. I compare junk radiation of the superposed-boosted-Schwarzschild (SBS) initial data with the junk of corresponding conformally flat, maximally sliced (CFMS) initial data. The SBS junk is smaller in amplitude than the CFMS junk, with the junk's leading-order spectral modes typically being reduced by a factor of order two or more.
Comparison of high-accuracy numerical simulations of black-hole binaries with stationary-phase post-Newtonian template waveforms for initial and advanced LIGO
Classical and Quantum Gravity 26, 114006 (2009)
Abstract
We study the effectiveness of stationary-phase approximated post- Newtonian waveforms currently used by ground-based gravitational- wave detectors to search for the coalescence of binary black holes by comparing them to an accurate waveform obtained from numerical simulation of an equal-mass non-spinning binary black hole inspiral, merger and ringdown. We perform this study for the Initial- and Advanced-LIGO detectors. We find that overlaps between the templates and signal can be improved by integrating the match filter to higher frequencies than used currently. We propose simple analytic frequency cutoffs for both Initial and Advanced LIGO, which achieve nearly optimal matches, and can easily be extended to unequal-mass, spinning systems. We also find that templates that include terms in the phase evolution up to 3.5 pN order are nearly always better, and rarely significantly worse, than 2.0 pN templates currently in use. For Initial LIGO we recommend a strategy using templates that include a recently introduced pseudo-4.0 pN term in the low-mass (\(M \leq 35 M_\odot\)) region, and 3.5 pN templates allowing unphysical values of the symmetric reduced mass \(\eta\) above this. This strategy always achieves overlaps within 0.3% of the optimum, for the data used here. For Advanced LIGO we recommend a strategy using 3.5 pN templates up to \(M=12 M_\odot\), 2.0 pN templates up to \(M=21 M_\odot\), pseudo-4.0 pN templates up to \(65 M_\odot\), and 3.5 pN templates with unphysical \(\eta\) for higher masses. This strategy always achieves overlaps within 0.7% of the optimum for Advanced LIGO.
Status of NINJA: the Numerical INJection Analysis project
Classical and Quantum Gravity 26, 114008 (2009)
Abstract
The 2008 NRDA conference introduced the Numerical INJection Analysis project (NINJA), a new collaborative effort between the numerical relativity community and the data analysis community. NINJA focuses on modeling and searching for gravitational wave signatures from the coalescence of binary system of compact objects. We review the scope of this collaboration and the components of the first NINJA project, where numerical relativity groups shared waveforms and data analysis teams applied various techniques to detect them when embedded in colored Gaussian noise.
Samurai project: Verifying the consistency of black-hole-binary waveforms for gravitational-wave detection
Physical Review D 79, 084025 (2009)
Abstract
We quantify the consistency of numerical-relativity black-hole- binary waveforms for use in gravitational-wave (GW) searches with current and planned ground-based detectors. We compare previously published results for the \((\ell=2,| m | =2)\) mode of the gravitational waves from an equal-mass nonspinning binary, calculated by five numerical codes. We focus on the 1000M (about six orbits, or 12 GW cycles) before the peak of the GW amplitude and the subsequent ringdown. We find that the phase and amplitude agree within each code's uncertainty estimates. The mismatch between the \((\ell=2,| m| =2)\) modes is better than \(10^{-3}\) for binary masses above \(60 M_{\odot}\) with respect to the Enhanced LIGO detector noise curve, and for masses above \(180 M_{\odot}\) with respect to Advanced LIGO, Virgo and Advanced Virgo. Between the waveforms with the best agreement, the mismatch is below \(2 \times 10^{-4}\). We find that the waveforms would be indistinguishable in all ground- based detectors (and for the masses we consider) if detected with a signal-to-noise ratio of less than \(\approx25\).
Implementation of higher-order absorbing boundary conditions for the Einstein equations
Class. Quantum Grav. 26 (2009) 075009
Abstract
We present an implementation of absorbing boundary conditions for the Einstein equations based on the recent work of Buchman and Sarbach. In this paper, we assume that spacetime may be linearized about Minkowski space close to the outer boundary, which is taken to be a coordinate sphere. We reformulate the boundary conditions as conditions on the gauge-invariant Regge–Wheeler–Zerilli scalars. Higher-order radial derivatives are eliminated by rewriting the boundary conditions as a system of ODEs for a set of auxiliary variables intrinsic to the boundary. From these we construct boundary data for a set of well-posed constraint-preserving boundary conditions for the Einstein equations in a first-order generalized harmonic formulation. This construction has direct applications to outer boundary conditions in simulations of isolated systems (e.g., binary black holes) as well as to the problem of Cauchyperturbative matching. As a test problem for our numerical implementation, we consider linearized multipolar gravitational waves in TT gauge, with angular momentum numbers \(\ell = 2\) (Teukolsky waves), 3 and 4. We demonstrate that the perfectly absorbing boundary condition \(\mathcal{B}_L\) of order \(L = \ell\) yields no spurious reflections to linear order in perturbation theory. This is in contrast to the lower-order absorbing boundary conditions \(\mathcal{B}_L\) with \(L < \ell\), which include the widely used freezing-\(\Psi_0\) boundary condition that imposes the vanishing of the Newman–Penrose scalar \(\Psi_0\).
Revisiting event horizon finders
Classical and Quantum Gravity 26, 035005 (2009)
Abstract
Event horizons are the defining physical features of black hole spacetimes, and are of considerable interest in studying black hole dynamics. Here, we reconsider three techniques to localise event horizons in numerical spacetimes: integrating geodesics, integrating a surface, and integrating a level-set of surfaces over a volume. We implement the first two techniques and find that straightforward integration of geodesics backward in time to be most robust. We find that the exponential rate of approach of a null surface towards the event horizon of a spinning black hole equals the surface gravity of the black hole. In head-on mergers we are able to track quasi-normal ringing of the merged black hole through seven oscillations, covering a dynamic range of about 10^5. Both at late times (when the final black hole has settled down) and at early times (before the merger), the apparent horizon is found to be an excellent approximation of the event horizon. In the head-on binary black hole merger, only {\em some} of the future null generators of the horizon are found to start from past null infinity/ the others approach the event horizons of the individual black holes at times far before merger.
High-accuracy waveforms for binary black hole inspiral, merger, and ringdown
Physical Review D 79, 024003 (2009)
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.
IMEX Evolution of Scalar Fields on Curved Backgrounds
Communications in Computational Physics 6, 1063 (2009)
Abstract
Inspiral of binary black holes occurs over a time-scale of many orbits, far longer than the dynamical time-scale of the individual black holes. Explicit evolutions of a binary system therefore require excessively many time steps to capture interesting dynamics. We present a strategy to overcome the Courant-Friedrichs-Lewy condition in such evolutions, one relying on modern implicit- explicit ODE solvers and multidomain spectral methods for elliptic equations. Our analysis considers the model problem of a forced scalar field propagating on a generic curved background. Nevertheless, we encounter and address a number of issues pertinent to the binary black hole problem in full general relativity. Specializing to the Schwarzschild geometry in Kerr-Schild coordinates, we document the results of several numerical experiments testing our strategy.
#2008
Binary-black-hole initial data with nearly extremal spins
Abstract
There is a significant possibility that astrophysical black holes with nearly-extremal spins exist. Numerical simulations of such systems require suitable initial data. In this paper, we examine three methods of constructing binary-black-hole initial data, focusing on their ability to generate black holes with nearly-extremal spins: (i) Bowen-York initial data, including standard puncture data (based on conformal flatness and Bowen-York extrinsic curvature), (ii) standard quasi-equilibrium initial data (based on the extended-conformal- thin-sandwich equations, conformal flatness, and maximal slicing), and (iii) quasi- equilibrium data based on the superposition of Kerr-Schild metrics. We find that the two conformally-flat methods (i) and (ii) perform similarly, with spins up to about 0.99 obtainable at the initial time. However, in an evolution, we expect the spin to quickly relax to a significantly smaller value around 0.93 as the initial geometry relaxes. For quasi- equilibrium superposed Kerr-Schild (SKS) data [method (iii)], we construct initial data with initial spins as large as 0.9997. We evolve SKS data sets with spins of 0.93 and 0.97 and find that the spin drops by only a few parts in 104 during the initial relaxation; therefore, we expect that SKS initial data will allow evolutions of binary black holes with relaxed spins above 0.99. [Abstract abbreviated; full abstract also mentions several secondary results.]
Evolving black hole-neutron star binaries in general relativity using pseudospectral and finite difference methods
Physical Review D 78, 104015 (2008)
Abstract
We present a code for solving the coupled Einstein-hydrodynamics equations to evolve relativistic, self-gravitating fluids. The Einstein field equations are solved in generalized harmonic coordinates on one grid using pseudospectral methods, while the fluids are evolved on another grid using shock-capturing finite difference or finite volume techniques. We show that the code accurately evolves equilibrium stars and accretion flows. Then we simulate an equal-mass nonspinning black hole-neutron star binary, evolving through the final four orbits of inspiral, through the merger, to the final stationary black hole. The gravitational waveform can be reliably extracted from the simulation.
Ineffectiveness of Padé resummation techniques in post-Newtonian approximations
Abstract
We test the resummation techniques used in developing Padé and Effective One Body (EOB) waveforms for gravitational wave detection. Convergence tests show that Padé approximants of the gravitational wave energy flux do not accelerate the convergence of the standard Taylor approximants even in the test mass limit, and there is no reason why Padé transformations should help in estimating parameters better in data analysis. Moreover, adding a pole to the flux seems unnecessary in the construction of these Padé-approximated flux formulas. Padé approximants may be useful in suggesting the form of fitting formulas. We compare a 15-orbit numerical waveform of the Caltech-Cornell group to the suggested Padé waveforms of Damour et al. in the equal mass, nonspinning quasi-circular case. The comparison suggests that the Padé waveforms do not agree better with the numerical waveform than the standard Taylor based waveforms. Based on this result, we design a simple EOB model by modifiying the ET EOB model of Buonanno et al., using the Taylor series of the flux with an unknown parameter at the fourth post-Newtonian order that we fit for. This simple EOB model generates a waveform having a phase difference of only 0.002 radians with the numerical waveform, much smaller than 0.04 radians the phase uncertainty in the numerical data itself. An EOB Hamiltonian can make use of a Padé transformation in its construction, but this is the only place Padé transformations seem useful.
High-accuracy numerical simulation of black-hole binaries: Computation of the gravitational-wave energy flux and comparisons with post-Newtonian approximants
Physical Review D 78, 104020 (2008)
Abstract
Expressions for the gravitational wave (GW) energy flux and center-of-mass energy of a compact binary are integral building blocks of post-Newtonian (PN) waveforms. In this paper, we compute the GW energy flux and GW frequency derivative from a highly accurate numerical simulation of an equal-mass, non-spinning black hole binary. We also estimate the (derivative of the) center-of-mass energy from the simulation by assuming energy balance. We compare these quantities with the predictions of various PN approximants (adiabatic Taylor and Pade models; non-adiabatic effective-one-body (EOB) models). We find that Pade summation of the energy flux does not accelerate the convergence of the flux series; nevertheless, the Pade flux is markedly closer to the numerical result for the whole range of the simulation (about 30 GW cycles). Taylor and Pade models overestimate the increase in flux and frequency derivative close to merger, whereas EOB models reproduce more faithfully the shape of and are closer to the numerical flux, frequency derivative and derivative of energy. We also compare the GW phase of the numerical simulation with Pade and EOB models. Matching numerical and untuned 3.5 PN order waveforms, we find that the phase difference accumulated until \(M \omega = 0.1\) is -0.12 radians for Pade approximants, and 0.50 (-0.28) radians for an EOB approximant with Keplerian (non-Keplerian) flux. We fit free parameters within the EOB models to minimize the phase difference, and discover degeneracies among these parameters. By tuning pseudo 4PN order coefficients in the radial potential or in the flux, or, if present, the location of the pole in the flux, we find that the accumulated phase difference can be reduced - if desired - to much less than the estimated numerical phase error (0.04 radians).
Initial data for black hole-neutron star binaries: a flexible, high-accuracy spectral method
Phys. Rev. D 77, 124051 (2008)
Abstract
We present a new numerical scheme to solve the initial value problem for black hole-neutron star binaries. This method takes advantage of the flexibility and fast convergence of a multidomain spectral representation of the initial data to construct high-accuracy solutions at a relatively low computational cost. We provide convergence tests of the method for both isolated neutron stars and irrotational binaries. In the second case, we show that we can resolve the small inconsistencies that are part of the quasi-equilibrium formulation, and that these inconsistencies are significantly smaller than observed in previous works. The possibility of generating a wide variety of initial data is also demonstrated through two new configurations inspired by results from binary black holes. First, we show that choosing a modified Kerr-Schild conformal metric instead of a flat conformal metric allows for the construction of quasi-equilibrium binaries with a spinning black hole. Second, we construct binaries in low-eccentricity orbits, which are a better approximation to astrophysical binaries than quasi-equilibrium systems.
#2007
High-accuracy comparison of numerical relativity simulations with post-Newtonian expansions
Phys. Rev. D 76, 124038 (2007)
Abstract
Numerical simulations of 15 orbits of an equal-mass binary black-hole system are presented. Gravitational waveforms from these simulations, covering more than 30 cycles and ending about 1.5 cycles before merger, are compared with those from quasicircular zero-spin post-Newtonian (PN) formulae. The cumulative phase uncertainty of these comparisons is about 0.05 radians, dominated by effects arising from the small residual spins of the black holes and the small residual orbital eccentricity in the simulations. Matching numerical results to PN waveforms early in the run yields excellent agreement (within 0.05 radians) over the first ~15 cycles, thus validating the numerical simulation and establishing a regime where PN theory is accurate. In the last 15 cycles to merger, however, generic time-domain Taylor approximants build up phase differences of several radians. But, apparently by coincidence, one specific post-Newtonian approximant, TaylorT4 at 3.5PN order, agrees much better with the numerical simulations, with accumulated phase differences of less than 0.05 radians over the 30-cycle waveform. Gravitational-wave amplitude comparisons are also done between numerical simulations and post-Newtonian, and the agreement depends on the post-Newtonian order of the amplitude expansion: the amplitude difference is about 6%–7% for zeroth order and becomes smaller for increasing order. A newly derived 3.0PN amplitude correction improves agreement significantly (<1% amplitude difference throughout most of the run, increasing to 4% near merger) over the previously known 2.5PN amplitude terms.
High accuracy simulations of Kerr tails: coordinate dependence and higher multipoles
Class. Quant. Grav. 25, 105022 (2008)
Abstract
We investigate the late time behavior of a scalar field on a fixed Kerr background using a 2+1 dimensional pseudospectral evolution code. We compare evolutions of pure axisymmetric multipoles in both Kerr-Schild and Boyer-Lindquist coordinates. We find that the late-time power-law decay rate depends upon the slicing of the background, confirming previous theoretical predictions for those decay rates. The accuracy of the numerical evolutions is sufficient to decide unambiguously between competing claims in the literature.
Gauge Drivers for the Generalized Harmonic Einstein Equations
Abstract
The generalized harmonic representation of Einstein's equation is manifestly hyperbolic for a large class of gauge conditions. Unfortunately most of the useful gauges developed over the past several decades by the numerical relativity community are incompatible with the hyperbolicity of the equations in this form. This paper presents a new method of imposing gauge conditions that preserves hyperbolicity for a much wider class of conditions, including as special cases many of the standard ones used in numerical relativity: e.g., K-freezing, Gamma-freezing, Bona-Masso slicing, conformal Gamma-drivers, etc. Analytical and numerical results are presented which test the stability and the effectiveness of this new gauge driver evolution system.
Using Full Information When Computing Modes of Post-Newtonian Waveforms From Inspiralling Compact Binaries in Circular Orbit
Phys. Rev. D 77, 044016 (2008)
Abstract
The increasing sophistication and accuracy of numerical simulations of compact binaries (especially binary black holes) presents the opportunity to test the regime in which post-Newtonian (PN) predictions for the emitted gravitational waves are accurate. In order to confront numerical results with those of post-Newtonian theory, it is convenient to compare multipolar decompositions of the two waveforms. It is pointed out here that the individual modes can be computed to higher post-Newtonian order by examining the radiative multipole moments of the system, rather than by decomposing the 2.5PN polarization waveforms. In particular, the dominant (l = 2, m = 2) mode can be computed to 3PN order. Individual modes are computed to as high a post-Newtonian order as possible given previous post-Newtonian results.
Estimating the final spin of a binary black hole coalescence
Phys. Rev. D 77, 026004 (2008)
Abstract
We present a straightforward approach for estimating the final black hole spin of a binary black hole coalescence with arbitrary initial masses and spins. Making some simple assumptions, we estimate the final angular momentum to be the sum of the individual spins plus the orbital angular momentum of a test particle orbiting at the last stable orbit around a Kerr black hole with a spin parameter of the final black hole. The formula we obtain is able to reproduce with reasonable accuracy the results from available numerical simulations, but, more importantly, it can be used to investigate what configurations might give rise to interesting dynamics. In particular, we discuss scenarios which might give rise to a "flip" in the direction of the total angular momentum of the system. By studying the dependence of the final spin upon the mass ratio and initial spins we find that our simple approach suggests that it is not possible to spin-up a black hole to extremal values through merger scenarios irrespective of the mass ratio of the objects involved.
Outer boundary conditions for Einstein's field equations in harmonic coordinates
Abstract
We analyze Einstein's vacuum field equations in generalized harmonic coordinates on a compact spatial domain with boundaries. We specify a class of boundary conditions which is constraint-preserving and sufficiently general to include recent proposals for reducing the amount of spurious reflections of gravitational radiation. In particular, our class comprises the boundary conditions recently proposed by Kreiss and Winicour, a geometric modification thereof, the freezing-\(\Psi_0\) boundary condition and the hierarchy of absorbing boundary conditions introduced by Buchman and Sarbach. Using the recent technique developed by Kreiss and Winicour based on an appropriate reduction to a pseudo-differential first order system, we prove well posedness of the resulting initial-boundary value problem in the frozen coefficient approximation. In view of the theory of pseudo-differential operators it is expected that the full nonlinear problem is also well posed. Furthermore, we implement some of our boundary conditions numerically and study their effectiveness in a test problem consisting of a perturbed Schwarzschild black hole.
Radiation reaction in the 2.5PN waveform from inspiralling binaries in circular orbits
Class. Quant. Grav. 24, 5307 (2007)
Abstract
In this Comment we compute the contributions of the radiation reaction force in the 2.5 post-Newtonian (PN) gravitational wave polarizations for compact binaries in circular orbits. (i) We point out and correct an inconsistency in the derivation of Arun, Blanchet, Iyer, and Qusailah. (ii) We prove that all contributions from radiation reaction in the 2.5PN waveform are actually negligible since they can be absorbed into a modification of the orbital phase at the 5PN order.
Testing outer boundary treatments for the Einstein equations
Class. Quantum Grav. 24 (2007) 4053--4078
Abstract
Various methods of treating outer boundaries in numerical relativity are compared using a simple test problem: a Schwarzschild black hole with an outgoing gravitational wave perturbation. Numerical solutions computed using different boundary treatments are compared to a `reference' numerical solution obtained by placing the outer boundary at a very large radius. For each boundary treatment, the full solutions including constraint violations and extracted gravitational waves are compared to those of the reference solution, thereby assessing the reflections caused by the artificial boundary. These tests use a first-order generalized harmonic formulation of the Einstein equations. Constraint-preserving boundary conditions for this system are reviewed, and an improved boundary condition on the gauge degrees of freedom is presented. Alternate boundary conditions evaluated here include freezing the incoming characteristic fields, Sommerfeld boundary conditions, and the constraint-preserving boundary conditions of Kreiss and Winicour. Rather different approaches to boundary treatments, such as sponge layers and spatial compactification, are also tested. Overall the best treatment found here combines boundary conditions that preserve the constraints, freeze the Newman-Penrose scalar \(\Psi_0\), and control gauge reflections.
Constraint Damping in First-Order Evolution Systems for Numerical Relativity
Abstract
A new constraint suppressing formulation of the Einstein evolution equations is presented, generalizing the five-parameter first-order system due to Kidder, Scheel and Teukolsky (KST). The auxiliary fields, introduced to make the KST system first-order, are given modified evolution equations designed to drive constraint violations toward zero. The algebraic structure of the new system is investigated, showing that the modifications preserve the hyperbolicity of the fundamental and constraint evolution equations. The evolution of the constraints for pertubations of flat spacetime is completely analyzed, and all finite-wavelength constraint modes are shown to decay exponentially when certain adjustable parameters satisfy appropriate inequalities. Numerical simulations of a single Schwarzschild black hole are presented, demonstrating the effectiveness of the new constraint-damping modifications.
Evolving relativistic fluid spacetimes using pseudospectral methods and finite differencing
Abstract
We present a new code for solving the coupled Einstein-hydrodynamics equations to evolve relativistic, self-gravitating fluids. The Einstein field equations are solved on one grid using pseudospectral methods, while the fluids are evolved on another grid by finite differencing. We discuss implementation details, such as the communication between the grids and the treatment of stellar surfaces, and present code tests.
Reducing Orbital Eccentricity in Binary Black Hole Simulations
Class. Quant. Grav. 24, S59 (2007)
Abstract
Binary black hole simulations starting from quasi-circular (i.e., zero radial velocity) initial data have orbits with small but non-zero orbital eccentricities. In this paper the quasi-equilibrium initial-data method is extended to allow non-zero radial velocities to be specified in binary black hole initial data. New low-eccentricity initial data are obtained by adjusting the orbital frequency and radial velocities to minimize the orbital eccentricity, and the resulting (\(\sim 5\) orbit) evolutions are compared with those of quasi-circular initial data. Evolutions of the quasi-circular data clearly show eccentric orbits, with eccentricity that decays over time. The precise decay rate depends on the definition of eccentricity; if defined in terms of variations in the orbital frequency, the decay rate agrees well with the prediction of Peters (1964). The gravitational waveforms, which contain \(\sim 8\) cycles in the dominant \(l=m=2\) mode, are largely unaffected by the eccentricity of the quasi-circular initial data. The overlap between the dominant mode in the quasi-circular evolution and the same mode in the low-eccentricity evolution is about 0.99.
The Einstein constraints: uniqueness and non-uniqueness in the conformal thin sandwich approach
Abstract
We study the appearance of multiple solutions to certain decompositions of Einstein's constraint equations. Pfeiffer and York recently reported the existence of two branches of solutions for identical background data in the extended conformal thin-sandwich decomposition. We show that the Hamiltonian constraint alone, when expressed in a certain way, admits two branches of solutions with properties very similar to those found by Pfeiffer and York. We construct these two branches analytically for a constant-density star in spherical symmetry, but argue that this behavior is more general. In the case of the Hamiltonian constraint this non-uniqueness is well known to be related to the sign of one particular term, and we argue that the extended conformal thin-sandwich equations contain a similar term that causes the breakdown of uniqueness.
Numerical implementation of isolated horizon boundary conditions
Abstract
We study the numerical implementation of a set of boundary conditions derived from the isolated horizon formalism, and which characterize a black hole whose horizon is in quasi-equilibrium. More precisely, we enforce these geometrical prescriptions as inner boundary conditions on an excised sphere, in the numerical resolution of the Conformal Thin Sandwich equations. As main results, we firstly establish the consistency of including in the set of boundary conditions a "constant surface gravity" prescription, interpretable as a lapse boundary condition, and secondly we assess how the prescriptions presented recently by Dain et al. for guaranteeing the well-posedness of the Conformal Transverse Traceless equations with quasi-equilibrium horizon conditions extend to the Conformal Thin Sandwich elliptic system. As a consequence of the latter analysis, we discuss the freedom of prescribing the expansion associated with the ingoing null normal at the horizon.
Testing the Accuracy and Stability of Spectral Methods in Numerical Relativity
Abstract
The accuracy and stability of the Caltech-Cornell pseudospectral code is evaluated using the KST representation of the Einstein evolution equations. The basic "Mexico City Tests" widely adopted by the numerical relativity community are adapted here for codes based on spectral methods. Exponential convergence of the spectral code is established, apparently limited only by numerical roundoff error. A general expression for the growth of errors due to finite machine precision is derived, and it is shown that this limit is achieved here for the linear plane-wave test. All of these tests are found to be stable, except for simulations of high amplitude gauge waves with nontrivial shift.
#2006
Towards absorbing outer boundaries in General Relativity
Class. Quant. Grav. 23, 6709-6744 (2006)
Abstract
We construct exact solutions to the Bianchi equations on a flat spacetime background. When the constraints are satisfied, these solutions represent in- and outgoing linearized gravitational radiation. We then consider the Bianchi equations on a subset of flat spacetime of the form \([0,T] \times B_R\), where \(B_R\) is a ball of radius \(R\), and analyze different kinds of boundary conditions on \(\partial B_R\). Our main results are: i) We give an explicit analytic example showing that boundary conditions obtained from freezing the incoming characteristic fields to their initial values are not compatible with the constraints. ii) With the help of the exact solutions constructed, we determine the amount of artificial reflection of gravitational radiation from constraint-preserving boundary conditions which freeze the Weyl scalar \(\Psi_0\) to its initial value. For monochromatic radiation with wave number k and arbitrary angular momentum number \(\ell \geq 2\), the amount of reflection decays as \(1/(kR)^4\) for large \(kR\). iii) For each \(L \geq 2\), we construct new local constraint-preserving boundary conditions which perfectly absorb linearized radiation with \(\ell \leq L\). (iv) We generalize our analysis to a weakly curved background of mass \(M\), and compute first order corrections in M/R to the reflection coefficients for quadrupolar odd-parity radiation. For our new boundary condition with \(L=2\), the reflection coefficient is smaller than the one for the freezing \(\Psi_0\) boundary condition by a factor of \(M/R\) for \(kR > 1.04\). Implications of these results for numerical simulations of binary black holes on finite domains are discussed.
Solving Einstein's Equations With Dual Coordinate Frames
Physical Review D, 74 104006 (2006)
Abstract
A method is introduced for solving Einstein's equations using two distinct coordinate systems. The coordinate basis vectors associated with one system are used to project out components of the metric and other fields, in analogy with the way fields are projected onto an orthonormal tetrad basis. These field components are then determined as functions of a second independent coordinate system. The transformation to the second coordinate system can be thought of as a mapping from the original "inertial" coordinate system to the computational domain. This dual-coordinate method is used to perform stable numerical evolutions of a black-hole spacetime using the generalized harmonic form of Einstein's equations in coordinates that rotate with respect to the inertial frame at infinity; such evolutions are found to be generically unstable using a single rotating coordinate frame. The dual-coordinate method is also used here to evolve binary black-hole spacetimes for several orbits. The great flexibility of this method allows comoving coordinates to be adjusted with a feedback control system that keeps the excision boundaries of the holes within their respective apparent horizons.
Self-Renormalization of the Classical Quasilocal Energy
Phys. Rev. D 75, 084026 (2007)
Abstract
Pointlike objects cause many of the divergences that afflict physical theories. For instance, the gravitational binding energy of a point particle in Newtonian mechanics is infinite. In general relativity, the analog of a point particle is a black hole and the notion of binding energy must be replaced by quasilocal energy. The quasilocal energy (QLE) derived by York, and elaborated by Brown and York, is finite outside the horizon but it was not considered how to evaluate it inside the horizon. We present a prescription for finding the QLE inside a horizon, and show that it is finite at the singularity for a variety of types of black hole. The energy is typically concentrated just inside the horizon, not at the central singularity.
Adaptive Mesh Refinement for Coupled Elliptic-Hyperbolic Systems
J. Comput. Phys. 218, 246-274 (2006)
Abstract
We present a modification to the Berger and Oliger adaptive mesh refinement algorithm designed to solve systems of coupled, non-linear, hyperbolic and elliptic partial differential equations. Such systems typically arise during constrained evolution of the field equations of general relativity. The novel aspect of this algorithm is a technique of "extrapolation and delayed solution" used to deal with the non-local nature of the solution of the elliptic equations, driven by dynamical sources, within the usual Berger and Oliger time-stepping framework. We show empirical results demonstrating the effectiveness of this technique in axisymmetric gravitational collapse simulations. We also describe several other details of the code, including truncation error estimation using a self-shadow hierarchy, and the refinement-boundary interpolation operators that are used to help suppress spurious high-frequency solution components ("noise").
Approximate initial data for binary black holes
Abstract
We construct approximate analytical solutions to the constraint equations of general relativity for binary black holes of arbitrary mass ratio in quasicircular orbit. We adopt the puncture method of Brandt and Bruegmann to solve the constraint equations in the transverse-traceless decomposition and consider perturbations of Schwarzschild black holes caused by boosts and the presence of a binary companion. A superposition of these two perturbations then yields approximate, but fully analytic binary black hole initial data that are accurate to first order in the inverse of the binary separation and the square of the black holes' momenta. Even close to the innermost stable circular orbit, the perturbative treatment introduces errors that are remarkably small and only somewhat larger than the errors caused by the underlying assumptions of puncture data.
Circular orbits and spin in black-hole initial data
Abstract
The construction of initial data for black-hole binaries usually involves the choice of free parameters that define the spins of the black holes and essentially the eccentricity of the orbit. Such parameters must be chosen carefully to yield initial data with the desired physical properties. In this paper, we examine these choices in detail for the quasiequilibrium method coupled to apparent-horizon/quasiequilibrium boundary conditions. First, we compare two independent criteria for choosing the orbital frequency, the "Komar-mass condition" and the "effective-potential method," and find excellent agreement. Second, we implement quasi-local measures of the spin of the individual holes, calibrate these with corotating binaries, and revisit the construction of non-spinning black hole binaries. Higher-order effects, beyond those considered in earlier work, turn out to be important. Without those, supposedly non-spinning black holes have appreciable quasi-local spin; furthermore, the Komar-mass condition and effective potential method agree only when these higher-order effects are taken into account. We compute a new sequence of quasi-circular orbits for non-spinning black-hole binaries, and determine the innermost stable circular orbit of this sequence.
A minimization problem for the lapse and the initial-boundary value problem for Einstein's field equations
Class. Quant. Grav. 23, S477-S504 (2006)
Abstract
We discuss the initial-boundary value problem of General Relativity. Previous considerations for a toy model problem in electrodynamics motivate the introduction of a variational principle for the lapse with several attractive properties. In particular, it is argued that the resulting elliptic gauge condition for the lapse together with a suitable condition for the shift and constraint-preserving boundary conditions controlling the Weyl scalar \(\Psi_0\) are expected to yield a well posed initial-boundary value problem for metric formulations of Einstein's field equations which are commonly used in numerical relativity. To present a simple and explicit example we consider the 3+1 decomposition introduced by York of the field equations on a cubic domain with two periodic directions and prove in the weak field limit that our gauge condition for the lapse and our boundary conditions lead to a well posed problem. The method discussed here is quite general and should also yield well posed problems for different ways of writing the evolution equations, including first order symmetric hyperbolic or mixed first-order second-order formulations. Well posed initial-boundary value formulations for the linearization about arbitrary stationary configurations will be presented elsewhere.
A New Generalized Harmonic Evolution System
Class. Quant. Grav. 23, S447-S462 (2006)
Abstract
A new representation of the Einstein evolution equations is presented that is first order, linearly degenerate, and symmetric hyperbolic. This new system uses the generalized harmonic method to specify the coordinates, and exponentially suppresses all small short-wavelength constraint violations. Physical and constraint-preserving boundary conditions are derived for this system, and numerical tests that demonstrate the effectiveness of the constraint suppression properties and the constraint-preserving boundary conditions are presented.
Stable radiation-controlling boundary conditions for the generalized harmonic Einstein equations
Class. Quantum Grav. 23(22), 6275-6300 (2006)
Abstract
This paper is concerned with the initial-boundary value problem for the Einstein equations in a first-order generalized harmonic formulation. We impose boundary conditions that preserve the constraints and control the incoming gravitational radiation by prescribing data for the incoming fields of the Weyl tensor. High-frequency perturbations about any given spacetime (including a shift vector with subluminal normal component) are analyzed using the Fourier-Laplace technique. We show that the system is boundary-stable. In addition, we develop a criterion that can be used to detect weak instabilities with polynomial time dependence, and we show that our system does not suffer from such instabilities. A numerical robust stability test supports our claim that the initial-boundary value problem is most likely to be well-posed even if nonzero initial and source data are included.
New Minimal Distortion Shift Gauge
Abstract
Based on the recent understanding of the role of the densitized lapse function in Einstein's equations and of the proper way to pose the thin sandwich problem, a slight readjustment of the minimal distortion shift gauge in the 3+1 approach to the dynamics of general relativity allows this shift vector to serve as the vector potential for the longitudinal part of the extrinsic curvature tensor in the new approach to the initial value problem, thus extending the initial value decomposition of gravitational variables to play a role in the evolution as well. The new shift vector globally minimizes the changes in the conformal 3-metric with respect to the spacetime measure rather than the spatial measure on the time coordinate hypersurfaces, as the old shift vector did.
Recent analytical and numerical techniques applied to the Einstein equations
Lect.Notes Phys. 692, 223-249 (2006)
Abstract
Combining deeper insight of Einstein's equations with sophisticated numerical techniques promises the ability to construct accurate numerical implementations of these equations. We illustrate this in two examples, the numerical evolution of "bubble" and single black hole spacetimes. The former is chosen to demonstrate how accurate numerical solutions can answer open questions and even reveal unexpected phenomena. The latter illustrates some of the difficulties encountered in three-dimensional black hole simulations, and presents some possible remedies.
#2005
Multi-block simulations in general relativity: high order discretizations, numerical stability, and applications
Class.Quant.Grav. 22, 5283-5322 (2005)
Abstract
The need to smoothly cover a computational domain of interest generically requires the adoption of several grids. To solve the problem of interest under this grid-structure one must ensure the suitable transfer of information among the different grids involved. In this work we discuss a technique that allows one to construct finite difference schemes of arbitrary high order which are guaranteed to satisfy linear numerical and strict stability. The technique relies on the use of difference operators satisfying summation by parts and penalty techniques to transfer information between the grids. This allows the derivation of semidiscrete energy estimates for problems admitting such estimates at the continuum. We analyze several aspects of this technique when used in conjuction with high order schemes and illustrate its use in one, two and three dimensional numerical relativity model problems with non-trivial topologies, including truly spherical black hole excision.
Schwarzschild Tests of the Wahlquist-Estabrook-Buchman-Bardeen Tetrad Formulation for Numerical Relativity
Abstract
A first order symmetric hyperbolic tetrad formulation of the Einstein equations developed by Estabrook and Wahlquist and put into a form suitable for numerical relativity by Buchman and Bardeen (the WEBB formulation) is adapted to explicit spherical symmetry and tested for accuracy and stability in the evolution of spherically symmetric black holes (the Schwarzschild geometry). The lapse and shift which specify the evolution of the coordinates relative to the tetrad congruence are reset at frequent time intervals to keep the constant-time hypersurfaces nearly orthogonal to the tetrad congruence and the spatial coordinate satisfying a kind of minimal rate of strain condition. By arranging through initial conditions that the constant-time hypersurfaces are asymptotically hyperbolic, we simplify the boundary value problem and improve stability of the evolution. Results are obtained for both tetrad gauges ("Nester" and "Lorentz") of the WEBB formalism using finite difference numerical methods. We are able to obtain stable unconstrained evolution with the Nester gauge for certain initial conditions, but not with the Lorentz gauge.
Boundary conditions for Einstein's field equations: Analytical and numerical analysis
J. Hyperbolic Differential Eqs. 2, 839-883 (2005)
Abstract
Outer boundary conditions for strongly and symmetric hyperbolic formulations of 3D Einstein's field equations with a live gauge condition are discussed. The boundary conditions have the property that they ensure constraint propagation and control in a sense made precise in this article the physical degrees of freedom at the boundary. We use Fourier-Laplace transformation techniques to find necessary conditions for the well posedness of the resulting initial-boundary value problem and integrate the resulting three-dimensional nonlinear equations using a finite-differencing code. We obtain a set of constraint-preserving boundary conditions which pass a robust numerical stability test. We explicitly compare these new boundary conditions to standard, maximally dissipative ones through Brill wave evolutions. Our numerical results explicitly show that in the latter case the constraint variables, describing the violation of the constraints, do not converge to zero when resolution is increased while for the new boundary conditions, the constraint variables do decrease as resolution is increased. As an application, we inject pulses of "gravitational radiation" through the boundaries of an initially flat spacetime domain, with enough amplitude to generate strong fields and induce large curvature scalars, showing that our boundary conditions are robust enough to handle nonlinear dynamics. We expect our boundary conditions to be useful for improving the accuracy and stability of current binary black hole and binary neutron star simulations, for a successful implementation of characteristic or perturbative matching techniques, and other applications. We also discuss limitations of our approach and possible future directions.
Einstein Bianchi equations with sources
Abstract
In this article, dedicated to one of the best specialist of the FOSH systems, we couple the Bianchi equations with the equations satisfied by the dynamical acceleration of a charged fluid and the derivatives of the associated Maxwell field.
The geometry of a naked singularity created by standing waves near a Schwarzschild horizon, and its application to the binary black hole problem
Abstract
The most promising way to compute the gravitational waves emitted by binary black holes (BBHs) in their last dozen orbits, where post-Newtonian techniques fail, is a quasistationary approximation introduced by Detweiler and being pursued by Price and others. In this approximation the outgoing gravitational waves at infinity and downgoing gravitational waves at the holes' horizons are replaced by standing waves so as to guarantee that the spacetime has a helical Killing vector field. Because the horizon generators will not, in general, be tidally locked to the holes' orbital motion, the standing waves will destroy the horizons, converting the black holes into naked singularities that resemble black holes down to near the horizon radius. This paper uses a spherically symmetric, scalar-field model problem to explore in detail the following BBH issues: (i) The destruction of a horizon by the standing waves. (ii) The accuracy with which the resulting naked singularity resembles a black hole. (iii) The conversion of the standing-wave spacetime (with a destroyed horizon) into a spacetime with downgoing waves by the addition of a "radiation-reaction field". (iv) The accuracy with which the resulting downgoing waves agree with the downgoing waves of a true black-hole spacetime (with horizon). The model problem used to study these issues consists of a Schwarzschild black hole endowed with spherical standing waves of a scalar field. It is found that the spacetime metric of the singular, standing-wave spacetime, and its radiation-reaction-field-constructed downgoing waves are quite close to those for a Schwarzschild black hole with downgoing waves -- sufficiently close to make the BBH quasistationary approximation look promising for non-tidally-locked black holes.
Evolution of Binary Black Hole Spacetimes
Phys.Rev.Lett. 95, 121101 (2005)
Abstract
We describe early success in the evolution of binary black hole spacetimes with a numerical code based on a generalization of harmonic coordinates. Indications are that with sufficient resolution this scheme is capable of evolving binary systems for enough time to extract information about the orbit, merger and gravitational waves emitted during the event. As an example we show results from the evolution of a binary composed of two equal mass, non-spinning black holes, through a single plunge-orbit, merger and ring down. The resultant black hole is estimated to be a Kerr black hole with angular momentum parameter a~0.70. At present, lack of resolution far from the binary prevents an accurate estimate of the energy emitted, though a rough calculation suggests on the order of 5% of the initial rest mass of the system is radiated as gravitational waves during the final orbit and ringdown.
Axisymmetric Numerical Relativity
PhD thesis, University of Cambridge
Abstract
This thesis is concerned with formulations of the Einstein equations in axisymmetric spacetimes which are suitable for numerical evolutions. We develop two evolution systems based on the (2+1)+1 formalism. The first is a (partially) constrained scheme with elliptic gauge conditions arising from maximal slicing and conformal flatness. The second is a strongly hyperbolic first-order formulation obtained by combining the (2+1)+1 formalism with the Z4 formalism. A careful study of the behaviour of regular axisymmetric tensor fields enables us to cast the equations in a form that is well-behaved on the axis. Further topics include (non)uniqueness of solutions to the elliptic equations arising in constrained schemes, and comparisons between various boundary conditions used in numerical relativity. The numerical implementation is applied to adaptive evolutions of nonlinear Brill waves, including twist.
Uniqueness and Non-uniqueness in the Einstein Constraints
Phys.Rev.Lett. 95, 091101 (2005)
Abstract
The conformal thin sandwich (CTS) equations are a set of four of the Einstein equations, which generalize the Laplace-Poisson equation of Newton's theory. We examine numerically solutions of the CTS equations describing perturbed Minkowski space, and find only one solution. However, we find two distinct solutions, one even containing a black hole, when the lapse is determined by a fifth elliptic equation through specification of the mean curvature. While the relationship of the two systems and their solutions is a fundamental property of general relativity, this fairly simple example of an elliptic system with non-unique solutions is also of broader interest.
A model problem for the initial-boundary value formulation of Einstein's field equations
J. Hyperbolic Differential Eqs. 2, 397-435 (2005)
Abstract
In many numerical implementations of the Cauchy formulation of Einstein's field equations one encounters artificial boundaries which raises the issue of specifying boundary conditions. Such conditions have to be chosen carefully. In particular, they should be compatible with the constraints, yield a well posed initial-boundary value formulation and incorporate some physically desirable properties like, for instance, minimizing reflections of gravitational radiation. Motivated by the problem in General Relativity, we analyze a model problem, consisting of a formulation of Maxwell's equations on a spatially compact region of spacetime with timelike boundaries. The form in which the equations are written is such that their structure is very similar to the Einstein-Christoffel symmetric hyperbolic formulations of Einstein's field equations. For this model problem, we specify a family of Sommerfeld-type constraint-preserving boundary conditions and show that the resulting initial-boundary value formulations are well posed. We expect that these results can be generalized to the Einstein-Christoffel formulations of General Relativity, at least in the case of linearizations about a stationary background.
The initial value problem in numerical relativity
J. Hyperbolic Differential Eqs., 2, 497-520 (2005)
Abstract
The conformal method for constructing initial data for Einstein's equations is presented in both the Hamiltonian and Lagrangian picture (extrinsic curvature decomposition and conformal thin sandwich formalism, respectively), and advantages due to the recent introduction of a weight-function in the extrinsic curvature decomposition are discussed. I then describe recent progress in numerical techniques to solve the resulting elliptic equations, and explore innovative approaches toward the construction of astrophysically realistic initial data for binary black hole simulations.
The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method
Phys. Rev. D 71, 104017 (2005)
Abstract
The periodic standing wave (PSW) method for the binary inspiral of black holes and neutron stars computes exact numerical solutions for periodic standing wave spacetimes and then extracts approximate solutions of the physical problem, with outgoing waves. The method requires solution of a boundary value problem with a mixed (hyperbolic and elliptic) character. We present here a new numerical method for such problems, based on three innovations: (i) a coordinate system adapted to the geometry of the problem, (ii) an expansion in multipole moments of these coordinates and a filtering out of higher moments, and (iii) the replacement of the continuum multipole moments with their analogs for a discrete grid. We illustrate the efficiency and accuracy of this method with nonlinear scalar model problems. Finally, we take advantage of the ability of this method to handle highly nonlinear models to demonstrate that the outgoing approximations extracted from the standing wave solutions are highly accurate even in the presence of strong nonlinearities.
A strongly hyperbolic and regular reduction of Einstein's equations for axisymmetric spacetimes
Class. Quantum Grav. 22(6) 1143-1166 (2005)
Abstract
This paper is concerned exclusively with axisymmetric spacetimes. We want to develop reductions of Einstein's equations which are suitable for numerical evolutions. We first make a Kaluza-Klein type dimensional reduction followed by an ADM reduction on the Lorentzian 3-space, the (2+1)+1 formalism. We include also the Z4 extension of Einstein's equations adapted to this formalism. Our gauge choice is based on a generalized harmonic gauge condition. We consider vacuum and perfect fluid sources. We use these ingredients to construct a strongly hyperbolic first-order evolution system and exhibit its characteristic structure. This enables us to construct constraint-preserving stable outer boundary conditions. We use cylindrical polar coordinates and so we provide a careful discussion of the coordinate singularity on axis. By choosing our dependent variables appropriately we are able to produce an evolution system in which each and every term is manifestly regular on axis.
Boundary Conditions for the Einstein Evolution System
Abstract
New boundary conditions are constructed and tested numerically for a general first-order form of the Einstein evolution system. These conditions prevent constraint violations from entering the computational domain through timelike boundaries, allow the simulation of isolated systems by preventing physical gravitational waves from entering the computational domain, and are designed to be compatible with the fixed-gauge evolutions used here. These new boundary conditions are shown to be effective in limiting the growth of constraints in 3D nonlinear numerical evolutions of dynamical black-hole spacetimes.
A numerical examination of an evolving black string horizon
Phys. Rev. D 71, 064009 (2005)
Abstract
We use the numerical solution describing the evolution of a perturbed black string presented in Choptuik et al. (2003) to elucidate the intrinsic behavior of the horizon. It is found that by the end of the simulation, the affine parameter on the horizon has become very large and the expansion and shear of the horizon in turn very small. This suggests the possibility that the horizon might pinch off in infinite affine parameter.
The intermediate problem for binary black hole inspiral and the periodic standing wave approximation
Abstract
In calculations of the inspiral of binary black holes an intermediate approximation is needed that can bridge the post-Newtonian methods of the early inspiral and the numerical relativity computations of the final plunge. We describe here the periodic standing wave approximation: A numerical solution is found to the problem of a periodic rotating binary with helically symmetric standing wave fields, and from this solution an approximation is extracted for the physically relevant problem of inspiral with outgoing waves. The approximation underlying this approach has been recently confirmed with innovative numerical methods applied to nonlinear model problems.
Numerical Relativity Using a Generalized Harmonic Decomposition
Class.Quant.Grav. 22, 425-452 (2005)
Abstract
A new numerical scheme to solve the Einstein field equations based upon the generalized harmonic decomposition of the Ricci tensor is introduced. The source functions driving the wave equations that define generalized harmonic coordinates are treated as independent functions, and encode the coordinate freedom of solutions. Techniques are discussed to impose particular gauge conditions through a specification of the source functions. A 3D, free evolution, finite difference code implementing this system of equations with a scalar field matter source is described. The second-order-in-space-and-time partial differential equations are discretized directly without the use first order auxiliary terms, limiting the number of independent functions to fifteen--ten metric quantities, four source functions and the scalar field. This also limits the number of constraint equations, which can only be enforced to within truncation error in a numerical free evolution, to four. The coordinate system is compactified to spatial infinity in order to impose physically motivated, constraint-preserving outer boundary conditions. A variant of the Cartoon method for efficiently simulating axisymmetric spacetimes with a Cartesian code is described that does not use interpolation, and is easier to incorporate into existing adaptive mesh refinement packages. Preliminary test simulations of vacuum black hole evolution and black hole formation via scalar field collapse are described, suggesting that this method may be useful for studying many spacetimes of interest.
Initial data for Einstein's equations with superposed gravitational waves
Abstract
A method is presented to construct initial data for Einstein's equations as a superposition of a gravitational wave perturbation on an arbitrary stationary background spacetime. The method combines the conformal thin sandwich formalism with linear gravitational waves, and allows detailed control over characteristics of the superposed gravitational wave like shape, location and propagation direction. It is furthermore fully covariant with respect to spatial coordinate changes and allows for very large amplitude of the gravitational wave.
#2004
Nonlinear Development of the Secular Bar-mode Instability in Rotating Neutron Stars
Astrophys.J. 617, 490-499 (2004)
Abstract
We have modelled the nonlinear development of the secular bar-mode instability that is driven by gravitational radiation-reaction (GRR) forces in rotating neutron stars. In the absence of any competing viscous effects, an initially uniformly rotating, axisymmetric n=1/2 polytropic star with a ratio of rotational to gravitational potential energy T/|W| = 0.181 is driven by GRR forces to a bar-like structure, as predicted by linear theory. The pattern frequency of the bar slows to nearly zero, that is, the bar becomes almost stationary as viewed from an inertial frame of reference as GRR removes energy and angular momentum from the star. In this "Dedekind-like" state, rotational energy is stored as motion of the fluid in highly noncircular orbits inside the bar. However, in less than 10 dynamical times after its formation, the bar loses its initially coherent structure as the ordered flow inside the bar is disrupted by what appears to be a purely hydrodynamical, short-wavelength, "shearing" type instability. The gravitational waveforms generated by such an event are determined, and an estimate of the detectability of these waves is presented.
Velocities and Momenta in an Extended Elliptic Form of the Initial Value Conditions
Nuovo Cim. B119, 823-837 (2004)
Abstract
The complete form of the constraints following from their conformal structure is extended so as to include constant mean curvature and other mean curvature foliations. This step is demonstrated using the momentum phase space approach. This approach yields equations of exactly the same form as the extended conformal thin sandwich approach. In solving the equations, it is never necessary actually to perform a tensor decomposition.
Excision boundary conditions for black hole initial data
Abstract
We define and extensively test a set of boundary conditions that can be applied at black hole excision surfaces when the Hamiltonian and momentum constraints of general relativity are solved within the conformal thin-sandwich formalism. These boundary conditions have been designed to result in black holes that are in quasiequilibrium and are completely general in the sense that they can be applied with any conformal three-geometry and slicing condition. Furthermore, we show that they retain precisely the freedom to specify an arbitrary spin on each black hole. Interestingly, we have been unable to find a boundary condition on the lapse that can be derived from a quasiequilibrium condition. Rather, we find evidence that the lapse boundary condition is part of the initial temporal gauge choice. To test these boundary conditions, we have extensively explored the case of a single black hole and the case of a binary system of equal-mass black holes, including the computation of quasi-circular orbits and the determination of the inner-most stable circular orbit. Our tests show that the boundary conditions work well.
Optimal Constraint Projection for Hyperbolic Evolution Systems
Phys. Rev. D 70, 084017 (2004)
Abstract
Techniques are developed for projecting the solutions of symmetric hyperbolic evolution systems onto the constraint submanifold (the constraint-satisfying subset of the dynamical field space). These optimal projections map a field configuration to the "nearest" configuration in the constraint submanifold, where distances between configurations are measured with the natural metric on the space of dynamical fields. The construction and use of these projections is illustrated for a new representation of the scalar field equation that exhibits both bulk and boundary generated constraint violations. Numerical simulations on a black-hole background show that bulk constraint violations cannot be controlled by constraint-preserving boundary conditions alone, but are effectively controlled by constraint projection. Simulations also show that constraint violations entering through boundaries cannot be controlled by constraint projection alone, but are controlled by constraint-preserving boundary conditions. Numerical solutions to the pathological scalar field system are shown to converge to solutions of a standard representation of the scalar field equation when constraint projection and constraint-preserving boundary conditions are used together.
Critical Collapse of a Complex Scalar Field with Angular Momentum
Phys.Rev.Lett. 93, 131101 (2004)
Abstract
We report a new critical solution found at the threshold of axisymmetric gravitational collapse of a complex scalar field with angular momentum. To carry angular momentum the scalar field cannot be axisymmetric; however, its azimuthal dependence is defined so that the resulting stress energy tensor and spacetime metric are axisymmetric. The critical solution found is non-spherical, discretely self-similar with an echoing exponent of 0.42 (± 4%), and exhibits a scaling exponent of 0.11 (± 10%) in near critical collapse. Our simulations suggest that the solution is universal (within the imposed symmetry class), modulo a family-dependent constant phase in the complex plane.
Adaptive Mesh Refinement for Characteristic Codes
J.Comput.Phys. 198, 10-34 (2004)
Abstract
The use of adaptive mesh refinement (AMR) techniques is crucial for accurate and efficient simulation of higher dimensional spacetimes. In this work we develop an adaptive algorithm tailored to the integration of finite difference discretizations of wave-like equations using characteristic coordinates. We demonstrate the algorithm by constructing a code implementing the Einstein-Klein-Gordon system of equations in spherical symmetry. We discuss how the algorithm can trivially be generalized to higher dimensional systems, and suggest a method that can be used to parallelize a characteristic code.
Controlling the Growth of Constraints in Hyperbolic Evolution Systems
Abstract
Motivated by the need to control the exponential growth of constraint violations in numerical solutions of the Einstein evolution equations, two methods are studied here for controlling this growth in general hyperbolic evolution systems. The first method adjusts the evolution equations dynamically, by adding multiples of the constraints, in a way designed to minimize this growth. The second method imposes special constraint preserving boundary conditions on the incoming components of the dynamical fields. The efficacy of these methods is tested by using them to control the growth of constraints in fully dynamical 3D numerical solutions of a particular representation of the Maxwell equations that is subject to constraint violations. The constraint preserving boundary conditions are found to be much more effective than active constraint control in the case of this Maxwell system.
The Initial Value Problem Using Metric and Extrinsic Curvature
Abstract
The initial value problem is introduced after a thorough review of the essential geometry. The initial value equations are put into elliptic form using both conformal transformations and a treatment of the extrinsic curvature introduced recently. This use of the metric and the extrinsic curvature is manifestly equivalent to the author's conformal thin sandwich formulation. Therefore, the reformulation of the constraints as an elliptic system by use of conformal techniques is complete.
3D simulations of linearized scalar fields in Kerr spacetime
Abstract
We investigate the behavior of a dynamical scalar field on a fixed Kerr background in Kerr-Schild coordinates using a 3+1 dimensional spectral evolution code, and we measure the power-law tail decay that occurs at late times. We compare evolutions of initial data proportional to \(f(r) Y_{\ell m}(\theta,\phi)\) where \(Y_{\ell m}\) is a spherical harmonic and \((r,\theta,\phi)\) are Kerr-Schild coordinates, to that of initial data proportional to \(f(r_{BL}) Y_{\ell m}(\theta_{BL},\phi)\), where \((r_{BL},\theta_{BL})\) are Boyer-Lindquist coordinates. We find that although these two cases are initially almost identical, the evolution can be quite different at intermediate times; however, at late times the power-law decay rates are equal.
#2003
Critical Collapse of the Massless Scalar Field in Axisymmetry
Abstract
We present results from a numerical study of critical gravitational collapse of axisymmetric distributions of massless scalar field energy. We find threshold behavior that can be described by the spherically symmetric critical solution with axisymmetric perturbations. However, we see indications of a growing, non-spherical mode about the spherically symmetric critical solution. The effect of this instability is that the small asymmetry present in what would otherwise be a spherically symmetric self-similar solution grows. This growth continues until a bifurcation occurs and two distinct regions form on the axis, each resembling the spherically symmetric self-similar solution. The existence of a non-spherical unstable mode is in conflict with previous perturbative results, and we therefore discuss whether such a mode exists in the continuum limit, or whether we are instead seeing a marginally stable mode that is rendered unstable by numerical approximation.
Initial data for black hole evolutions
Ph.D. thesis, Cornell University, 2003
Abstract
We discuss the initial value problem of general relativity in its recently unified Lagrangian and Hamiltonian pictures and present a multi-domain pseudo-spectral collocation method to solve the resulting coupled nonlinear partial differential equations. Using this code, we explore several approaches to construct initial data sets containing one or two black holes: We compute quasi-circular orbits for spinning equal mass black holes and unequal mass (nonspinning) black holes using the effective potential method with Bowen-York extrinsic curvature. We compare initial data sets resulting from different decompositions, and from different choices of the conformal metric with each other. Furthermore, we use the quasi-equilibrium method to construct initial data for single black holes and for binary black holes in quasi-circular orbits. We investigate these binary black hole data sets and examine the limits of large mass-ratio and wide separation. Finally, we propose a new method for constructing spacetimes with superposed gravitational waves of possibly very large amplitude.
Towards the Final Fate of an Unstable Black String
Abstract
Black strings, one class of higher dimensional analogues of black holes, were shown to be unstable to long wavelength perturbations by Gregory and Laflamme in 1992, via a linear analysis. We revisit the problem through numerical solution of the full equations of motion, and focus on trying to determine the end-state of a perturbed, unstable black string. Our preliminary results show that such a spacetime tends towards a solution resembling a sequence of spherical black holes connected by thin black strings, at least at intermediate times. However, our code fails then, primarily due to large gradients that develop in metric functions, as the coordinate system we use is not well adapted to the nature of the unfolding solution. We are thus unable to determine how close the solution we see is to the final end-state, though we do observe rich dynamical behavior of the system in the intermediate stages.
Dynamical Gauge Conditions for the Einstein Evolution Equations
Abstract
The Einstein evolution equations have been written in a number of symmetric hyperbolic forms when the gauge fields--the densitized lapse and the shift--are taken to be fixed functions of the coordinates. Extended systems of evolution equations are constructed here by adding the gauge degrees of freedom to the set of dynamical fields, thus forming symmetric hyperbolic systems for the combined evolution of the gravitational and the gauge fields. The associated characteristic speeds can be made causal (i.e. less than or equal to the speed of light) by adjusting 14 free parameters in these new systems. And 21 additional free parameters are available, for example to optimize the stability of numerical evolutions. The gauge evolution equations in these systems are generalizations of the "K-driver" and "Gamma-driver" conditions that have been used with some success in numerical black hole evolutions.
A multidomain spectral method for solving elliptic equations
Comput.Phys.Commun. 152, 253-273 (2003)
Abstract
We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three distinct features. First, the combined problem of solving the PDE, satisfying the boundary conditions, and matching between different subdomains is cast into one set of equations readily accessible to standard linear and nonlinear solvers. Second, touching as well as overlapping subdomains are supported; both rectangular blocks with Chebyshev basis functions as well as spherical shells with an expansion in spherical harmonics are implemented. Third, the code is very flexible: The domain decomposition as well as the distribution of collocation points in each domain can be chosen at run time, and the solver is easily adaptable to new PDEs. The code has been used to solve the equations of the initial value problem of general relativity and should be useful in many other problems. We compare the new method to finite difference codes and find it superior in both runtime and accuracy, at least for the smooth problems considered here.
An Axisymmetric Gravitational Collapse Code
Class.Quant.Grav. 20, 1857-1878 (2003)
Abstract
We present a new numerical code designed to solve the Einstein field equations for axisymmetric spacetimes. The long term goal of this project is to construct a code that will be capable of studying many problems of interest in axisymmetry, including gravitational collapse, critical phenomena, investigations of cosmic censorship, and head-on black hole collisions. Our objective here is to detail the (2+1)+1 formalism we use to arrive at the corresponding system of equations and the numerical methods we use to solve them. We are able to obtain stable evolution, despite the singular nature of the coordinate system on the axis, by enforcing appropriate regularity conditions on all variables and by adding numerical dissipation to hyperbolic equations.
Extrinsic Curvature and the Einstein Constraints
Abstract
The Einstein initial-value equations in the extrinsic curvature (Hamiltonian) representation and conformal thin sandwich (Lagrangian) representation are brought into complete conformity by the use of a decomposition of symmetric tensors which involves a weight function. In stationary spacetimes, there is a natural choice of the weight function such that the transverse traceless part of the extrinsic curvature (or canonical momentum) vanishes.
#2002
Toward stable 3D numerical evolutions of black-hole spacetimes
Phys. Rev. D 66, 124005 (2002)
Abstract
Three dimensional (3D) numerical evolutions of static black holes with excision are presented. These evolutions extend to about 8000M, where M is the mass of the black hole. This degree of stability is achieved by using growth-rate estimates to guide the fine tuning of the parameters in a multiparameter family of symmetric hyperbolic representations of the Einstein evolution equations. These evolutions were performed using a fixed gauge in order to separate the intrinsic stability of the evolution equations from the effects of stability-enhancing gauge choices.
Energy Norms and the Stability of the Einstein Evolution Equations
Abstract
The Einstein evolution equations may be written in a variety of equivalent analytical forms, but numerical solutions of these different formulations display a wide range of growth rates for constraint violations. For symmetric hyperbolic formulations of the equations, an exact expression for the growth rate is derived using an energy norm. This expression agrees with the growth rate determined by numerical solution of the equations. An approximate method for estimating the growth rate is also derived. This estimate can be evaluated algebraically from the initial data, and is shown to exhibit qualitatively the same dependence as the numerically-determined rate on the parameters that specify the formulation of the equations. This simple rate estimate therefore provides a useful tool for finding the most well-behaved forms of the evolution equations.
First-order symmetrizable hyperbolic formulations of Einstein's equations including lapse and shift as dynamical fields
Class.Quant.Grav. 19, 5153-5162 (2002)
Abstract
First-order hyperbolic systems are promising as a basis for numerical integration of Einstein's equations. In previous work, the lapse and shift have typically not been considered part of the hyperbolic system and have been prescribed independently. This can be expensive computationally, especially if the prescription involves solving elliptic equations. Therefore, including the lapse and shift in the hyperbolic system could be advantageous for numerical work. In this paper, two first-order symmetrizable hyperbolic systems are presented that include the lapse and shift as dynamical fields and have only physical characteristic speeds.
Comparing initial-data sets for binary black holes
Abstract
We compare the results of constructing binary black hole initial data with three different decompositions of the constraint equations of general relativity. For each decomposition we compute the initial data using a superposition of two Kerr-Schild black holes to fix the freely specifiable data. We find that these initial-data sets differ significantly, with the ADM energy varying by as much as 5% of the total mass. We find that all initial-data sets currently used for evolutions might contain unphysical gravitational radiation of the order of several percent of the total mass. This is comparable to the amount of gravitational-wave energy observed during the evolved collision. More astrophysically realistic initial data will require more careful choices of the freely specifiable data and boundary conditions for both the metric and extrinsic curvature. However, we find that the choice of extrinsic curvature affects the resulting data sets more strongly than the choice of conformal metric.
Bianchi - Euler system for relativistice fluids and Bel - Robinson type energy
Comptes Rendus Mathematique, Volume 335, Number 8, 15 October 2002, pp. 711-716(6)
Abstract
We write a first order symmetric hyperbolic system coupling the Riemann with the dynamical acceleration of a relativistic fluid. W determine the associated, coupled, Bel - Robinson energy, and the integral equality that it satisfies.
Numerical evolutions of nonlinear r-modes in neutron stars
Abstract
Nonlinear evolution of the gravitational radiation (GR) driven instability in the r-modes of neutron stars is studied by full numerical 3D hydrodynamical simulations. The growth of the r-mode instability is found to be limited by the formation of shocks and breaking waves when the dimensionless amplitude of the mode grows to about three in value. This maximum mode amplitude is shown by numerical tests to be rather insensitive to the strength of the GR driving force. Upper limits on the strengths of possible nonlinear mode--mode coupling are inferred. Previously unpublished details of the numerical techniques used are presented, and the results of numerous calibration runs are discussed.
#2001
Extending the lifetime of 3D black hole computations with a new hyperbolic system of evolution equations
Abstract
We present a new many-parameter family of hyperbolic representations of Einstein's equations, which we obtain by a straightforward generalization of previously known systems. We solve the resulting evolution equations numerically for a Schwarzschild black hole in three spatial dimensions, and find that the stability of the simulation is strongly dependent on the form of the equations (i.e. the choice of parameters of the hyperbolic system), independent of the numerics. For an appropriate range of parameters we can evolve a single 3D black hole to \(t \simeq 600 M\) -- \(1300 M\), and are apparently limited by constraint-violating solutions of the evolution equations. We expect that our method should result in comparable times for evolutions of a binary black hole system.
Non-Linear Evolution of the r-Modes in Neutron Stars
Phys.Rev.Lett. 86, 1152-1155 (2001)
Abstract
The evolution of a neutron-star r-mode driven unstable by gravitational radiation (GR) is studied here using numerical solutions of the full non-linear fluid equations. The amplitude of the mode grows to order unity before strong shocks develop which quickly damp the mode. In this simulation the star loses about 40% of its initial angular momentum and 50% of its rotational kinetic energy before the mode is damped. The non-linear evolution causes the fluid to develop strong differential rotation which is concentrated near the surface and especially near the poles of the star.
#2000
Quasi-circular Orbits for Spinning Binary Black Holes
Abstract
Using an effective potential method we examine binary black holes where the individual holes carry spin. We trace out sequences of quasi-circular orbits and locate the innermost stable circular orbit as a function of spin. At large separations, the sequences of quasi-circular orbits match well with post-Newtonian expansions, although a clear signature of the simplifying assumption of conformal flatness is seen. The position of the ISCO is found to be strongly dependent on the magnitude of the spin on each black hole. At close separations of the holes, the effective potential method breaks down. In all cases where an ISCO could be determined, we found that an apparent horizon encompassing both holes forms for separations well inside the ISCO. Nevertheless, we argue that the formation of a common horizon is still associated with the breakdown of the effective potential method.
Black hole evolution by spectral methods
Abstract
Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that prohibit long-term evolution. Some of these instabilities may be due to the numerical method used, traditionally finite differencing. In this paper, we explore the use of a pseudospectral collocation (PSC) method for the evolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of Einstein's equations. We demonstrate that our PSC method is able to evolve a spherically symmetric black hole spacetime forever without enforcing constraints, even if we add dynamics via a Klein-Gordon scalar field. We find that, in contrast to finite-differencing methods, black hole excision is a trivial operation using PSC applied to a hyperbolic formulation of Einstein's equations. We discuss the extension of this method to three spatial dimensions.