Lawrence E. Kidder

High-accuracy comparison of numerical relativity simulations with post-Newtonian expansions

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.

A New generalized harmonic evolution system

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.

High-accuracy waveforms for binary black hole inspiral, merger, and ringdown

The first spectral numerical simulations of 16 orbits, merger, and ringdown of an equal-mass nonspinning 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 Mf/M=0.951 62±0.000 02, and the final black hole spin is Sf/Mf^2=0.686 46±0.000 04.

Effective-one-body model for black-hole binaries with generic mass ratios and spins

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 20 and 200M_⊙, the effective-one-body nonprecessing (dominant mode) waveforms have overlap 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. We also show that—without further calibration— the precessing effective-one-body (dominant mode) waveforms have overlap above 97% with two very long, strongly precessing numerical-relativity waveforms, when maximizing only on the initial phase and time.

Solving Einstein's equations with dual coordinate frames

A method is introduced for solving Einsteins 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 Einsteins 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.

A Multidomain spectral method for solving elliptic equations

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.

Extending the lifetime of 3D black hole computations with a new hyperbolic system of evolution equations

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 three-dimensional black hole to t ≃ 600 M – 1300 M, and we 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.

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

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

Catalog of 174 Binary Black Hole Simulations for Gravitational Wave Astronomy

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.

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

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