results for au:Szilagyi_B in:gr-qc

- Mar 22 2018 gr-qc arXiv:1803.07695v1In this paper, we test the performance of templates in detection and characterization of Spin-orbit resonant (SOR) binaries. We use precessing SEOBNRv3 waveforms as well as \it four numerical relativity (NR) waveforms to model GWs from SOR binaries and filter them through IMRPhenomD, SEOBNRv4 (non-precessing) and IMRPhenomPv2 (precessing) approximants. We find that IMRPhenomD and SEOBNRv4 recover only $\sim70\%$ of injections with fitting factor (FF) higher than 0.97 (or 90\% of injections with ${\rm FF} >0.9$).However, using the sky-maxed statistic, IMRPhenomPv2 performs magnificently better than their non-precessing counterparts with recovering $99\%$ of the injections with FFs higher than 0.97. Interestingly, injections with $\Delta \phi = 180^{\circ}$ have higher FFs ($\Delta \phi$ is the angle between the components of the black hole spins in the plane orthogonal to the orbital angular momentum) as compared to their $\Delta \phi =0^{\circ}$ and generic counterparts. This implies that we will have a slight observation bias towards $\Delta \phi=180^{\circ}$ SORs while using non-precessing templates for searches. All template approximants are able to recover most of the injected NR waveforms with FFs $>0.95$. For all the injections including NR, the error in estimating chirp mass remains below $<10\%$ with minimum error for $\Delta \phi = 180^{\circ}$ resonant binaries. The symmetric mass ratio can be estimated with errors below $15\%$. The effective spin parameter $\chi_{\rm eff}$ is measured with maximum absolute error of 0.13. The in-plane spin parameter $\chi_p$ 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 non-quadrupole modes but also better search strategies for precessing GW signals.
- May 30 2017 gr-qc arXiv:1705.09833v1We 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 simulations. This procedure bypasses approximations used in semi-analytical models for compact binary coalescence. In this work, we use only the full posterior parameter distribution for generic nonprecessing binaries, drawing inferences away from the set of NR simulations used, via interpolation of a single scalar quantity (the marginalized log-likelihood, $\ln {\cal L}$) 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 $l\le2$ as well as $l\le3$ harmonic modes. Using the $l\le3$ 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 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.
- A Numerical Relativity Waveform Surrogate Model for Generically Precessing Binary Black Hole MergersMay 22 2017 gr-qc arXiv:1705.07089v2A generic, non-eccentric binary black hole (BBH) system emits gravitational waves (GWs) that are completely described by 7 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 $7$ dimensions of the intrinsic non-eccentric 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 $\ell \leq 4$ modes, begins about $20$ orbits before merger, and can be evaluated in $\sim~50\,\mathrm{ms}$. 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 would otherwise require millions of numerical relativity waveforms, such as parameter inference and tests of general relativity with GW observations.
- Nov 24 2016 gr-qc astro-ph.HE arXiv:1611.07531v2Parameter 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 analyses 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 $\sim$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.
- Nov 14 2016 gr-qc arXiv:1611.03703v3We 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 towards 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 $\gtrsim 4$ and double spin $\gtrsim 0.8$ will be crucial to resolve 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.
- We present a time domain waveform model that describes the inspiral-merger-ringdown (IMR) of compact binary systems whose components are non-spinning, 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 non-linear 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 incorporate higher-order post-Newtonian corrections for the energy flux of quasi-circular binaries and gravitational self-force corrections to the binding energy of compact binaries. This enhanced inspiral evolution prescription is combined with an analytical prescription for the merger-ringdown evolution using a catalog of numerical relativity simulations. This IMR waveform model reproduces effective-one-body waveforms for systems with mass-ratios between 1 to 15 in the zero eccentricity limit. Using a set of eccentric numerical relativity simulations, not used during calibration, we show that our eccentric model accurately reproduces the features of eccentric compact binary coalescence throughout the merger. Using this model we show that the gravitational wave transients GW150914 and GW151226 can be effectively recovered with template banks of quasi-circular, spin-aligned waveforms if the eccentricity $e_0$ of these systems when they enter the aLIGO band at a gravitational wave frequency of 14 Hz satisfies $e_0^{\rm GW150914}\leq0.15$ and $e_0^{\rm GW151226}\leq0.1$.
- Jul 28 2016 gr-qc arXiv:1607.07962v1The 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 $S_{\rm BH}/M_{\rm BH}^2=0.99$.
- Jul 20 2016 gr-qc astro-ph.HE arXiv:1607.05377v1In fall of 2015, the two LIGO detectors measured the gravitational wave signal GW150914, which originated from a pair of merging black holes. In the final 0.2 seconds (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. 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.
- We compare GW150914 directly to simulations of coalescing binary black holes in full general relativity, accounting for all the spin-weighted quadrupolar modes, and separately accounting for all the quadrupolar and octopolar modes. Consistent with the posterior distributions reported in LVC_PE[1] (at 90% confidence), we find the data are compatible with a wide range of nonprecessing and precessing simulations. Followup simulations performed using previously-estimated binary parameters most resemble the data. Comparisons including only the quadrupolar modes constrain the total redshifted mass Mz ∈[64 - 82M_⊙], mass ratio q = m2/m1 ∈[0.6,1], and effective aligned spin \chi_eff ∈[-0.3, 0.2], where \chi_eff = (S1/m1 + S2/m2) ⋅\hatL /M. Including both quadrupolar and octopolar modes, we find the mass ratio is even more tightly constrained. 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 \chi_eff are consistent with the data. Though correlated, the components' spins (both in magnitude and directions) are not significantly constrained by the data. 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 between 64.0 - 73.5M_⊙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 LVC_PE[1].
- May 17 2016 gr-qc arXiv:1605.04332v4We present a new approach for the Cauchy-characteristic extraction of gravitational radiation strain, news function, and the flux of the energy-momentum, supermomentum and angular momentum associated with the Bondi-Metzner-Sachs asymptotic symmetries. In Cauchy-characteristic extraction, a characteristic evolution code takes numerical data on an inner worldtube supplied by a Cauchy evolution code, and propagates it outwards to obtain the space-time metric in a neighborhood of null infinity. The metric is first determined in a scrambled form in terms of coordinates determined by the Cauchy formalism. In prior treatments, the waveform is first extracted from this metric and then transformed into an asymptotic inertial coordinate system. This procedure provides the physically proper description of the waveform and the radiated energy but it does not generalize to determine the flux of angular momentum or supermomentum. Here we formulate and implement a new approach which transforms the full metric into an asymptotic inertial frame and provides a uniform treatment of all the radiation fluxes associated with the asymptotic symmetries. Computations are performed and calibrated using the Spectral Einstein Code (SpEC).
- Apr 05 2016 gr-qc arXiv:1604.00782v2We present results on the inspiral, merger, and post-merger 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 $\approx 22$ 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 semi-analytical 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 post-merger 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.
- Feb 12 2016 gr-qc astro-ph.HE arXiv:1602.03840v2On 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 $36^{+5}_{-4} M_\odot$ and $29^{+4}_{-4} M_\odot$; 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 $410^{+160}_{-180}$ Mpc, corresponding to a redshift $0.09^{+0.03}_{-0.04}$ assuming standard cosmology. The source location is constrained to an annulus section of $610$ deg$^2$, primarily in the southern hemisphere. The binary merges into a black hole of $62^{+4}_{-4} M_\odot$ and spin $0.67^{+0.05}_{-0.07}$. This black hole is significantly more massive than any other inferred from electromagnetic observations in the stellar-mass regime.
- 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.
- Jan 21 2016 gr-qc arXiv:1601.05396v2Coalescing binary black holes are among the primary science targets for second generation ground-based gravitational wave (GW) detectors. Reliable GW 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 ($1\le m_1/m_2\le 3$), and samples independently both black hole spins up to 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 $\gtrsim 50M_\odot$). 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 $\rho$, except that SEOBNRv2's efficiency drops mildly for both black hole spins aligned with large magnitude. For parameter estimation, modeling inaccuracies of SEOBNRv2 are found to be smaller than systematic uncertainties for moderately strong GW events up to roughly $\rho\lesssim 15$. PhenomD's modeling errors are found to be smaller than SEOBNRv2's, and are generally irrelevant for $\rho\lesssim 20$. 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 GW models during parameter estimation for recovery of binary's masses and spins.
- Dec 22 2015 gr-qc arXiv:1512.06800v2We 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 to 3. The simulations cover on average 24 inspiral orbits, plus merger and ringdown, with low initial orbital eccentricities $e<10^{-4}$. 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 finite length of the numerical waveforms are of similar magnitude, with gravitational wave extraction errors somewhat 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.
- Sep 22 2015 gr-qc arXiv:1509.05782v3Gravitational 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 $\sim 15$ 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 ${\lambda}$.
- Aug 28 2015 gr-qc arXiv:1508.06986v1We 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 quasi-local angular momentum integrals. When constructing irrotational binary neutron stars, we find a very small residual dimensionless spin of $\sim 2\times 10^{-4}$. 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 $\sim 0.1\%$. The neutron stars show quasi-normal mode oscillations at an amplitude which increases with the rotation rate of the stars.
- Jul 02 2015 gr-qc arXiv:1507.00103v1Coalescing binaries of neutron stars (NS) and black holes (BH) 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 $10-30$ orbit numerical relativity (NR) merger simulations. In this paper we study the accuracy of these models using new NR simulations that span $36-88$ orbits, with mass-ratios and black hole spins $(q,\chi_{BH}) = (7, \pm 0.4), (7, \pm 0.6)$, and $(5, -0.9)$. 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 $\geq 0.5$ or $\leq -0.3$, with waveform mismatch accumulating during early inspiral; (ii) comparison with numerical waveforms indicate that this disagreement is due to phasing errors of SEOBNRv1, with SEOBNRv2 in good agreement with all of our simulations; (iii) Phenomenological waveforms disagree with SEOBNRv2 over most of the NSBH binary parameter space; (iv) comparison with NR waveforms shows that most of the model's dephasing accumulates near the frequency interval where it switches to a phenomenological phasing prescription; and finally (v) both SEOBNR and post-Newtonian (PN) models are effectual for NSBH systems, but PN waveforms will give a significant bias in parameter recovery. Our results suggest that future gravitational-wave detection searches and parameter estimation efforts targeted at NSBH systems with $q\lesssim 7$ and $\chi_\mathrm{BH} \approx [-0.9, +0.6]$ will benefit from using SEOBNRv2 templates. For larger black hole spins and/or binary mass-ratios, we recommend the models be further investigated as suitable NR simulations become available.
- Jun 05 2015 gr-qc arXiv:1506.01689v1Construction 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.
- Simulating a binary black hole (BBH) 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 non-spinning BBH 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 \em 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 ${}_{-2}Y_{\ell m}$ waveform modes resolved by the NR code up to $\ell=8.$ We compare our surrogate model to Effective One Body waveforms from $50$-$300 M_\odot$ for advanced LIGO detectors and find that the surrogate is always more faithful (by at least an order of magnitude in most cases).
- Feb 26 2015 gr-qc arXiv:1502.06987v2We 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).
- Feb 18 2015 gr-qc arXiv:1502.04953v1We 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 $45.5\,M_\odot$. 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.
- Feb 17 2015 astro-ph.HE gr-qc arXiv:1502.04146v1We 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 a 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 1.4 solar mass with a black hole of 7 solar mass and dimensionless spin a/M=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 the remnant is less neutron-rich. The spatial distribution of the neutrinos is significantly affected by relativistic effects. Over the short timescale evolved, we do not observe purely neutrino-driven outflows. However, a small amount of material (3e-4Msun) 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. Through r-process nucleosynthesis, that material should produce high-opacity lanthanides in the polar region, and could thus affect the lightcurve of radioactively powered electromagnetic transients. We also show that by the end of the simulation, while the bulk of the disk is neutron-rich, its outer layers have a higher electron fraction. As that material would be the first to be unbound by disk outflows on longer timescales, the changes in Ye experienced during the formation of the disk could have an impact on the nucleosynthesis outputs from neutrino-driven and viscously-driven outflows. [Abridged]
- Feb 09 2015 gr-qc arXiv:1502.01747v1Binary 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 $\sim 1^{\circ}$ with NR during the inspiral, increasing to $5^{\circ}$ 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.
- Jan 23 2015 gr-qc arXiv:1501.05358v1We present simulations of binary black holes mergers in which, after the common outer horizon has formed, the marginally outer trapped surfaces (MOTSs) corresponding to the individual black holes continue to approach and eventually penetrate each other. This has very interesting consequences according to recent results in the theory of MOTSs. Uniqueness and stability theorems imply that two MOTSs which touch with a common outer normal must be identical. This suggests a possible dramatic consequence of the collision between a small and large black hole. If the penetration were to continue to completion then the two MOTSs would have to coalesce, by some combination of the small one growing and the big one shrinking. Here we explore the relationship between theory and numerical simulations, in which a small black hole has halfway penetrated a large one.
- Dec 05 2014 gr-qc arXiv:1412.1803v1Astrophysical 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 more robustly and more efficiently. We use these methods to simulate an unequal-mass, precessing binary black hole coalescence, where the larger black hole has $S/m^2=0.99$. We also use these methods to simulate a non-precessing binary black hole 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).
- Nov 27 2014 gr-qc arXiv:1411.7297v2The spin angular momentum $S$ of an isolated Kerr black hole is bounded by the surface area $A$ of its apparent horizon: $8\pi S \le A$, 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 $8\pi S < A$ is satisfied in all cases but is very close to equality on the common apparent horizon at the instant it first appears. We also 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), we construct quasiequilibrium binary-black-hole initial data with "overspun" marginally trapped surfaces with $8\pi S > A$ and for which our lower bound on their Booth-Fairhurst extremality exceeds unity. These superextremal surfaces are always surrounded by marginally outer trapped surfaces (i.e., by apparent horizons) with $8\pi S<A$. 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. (Abstract abbreviated.)
- Jun 30 2014 gr-qc arXiv:1406.7029v2We present a spectral algorithm for solving the full nonlinear vacuum Einstein field equations in the Bondi framework. Developed within the Spectral Einstein Code (SpEC), we demonstrate spectral characteristic evolution as a technical precursor to Cauchy Characteristic Extraction (CCE), a rigorous method for obtaining gauge-invariant gravitational waveforms from existing and future astrophysical simulations. We demonstrate the new algorithm's 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.
- May 16 2014 gr-qc arXiv:1405.3693v1As 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 (\tt 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.
- May 12 2014 astro-ph.HE gr-qc arXiv:1405.2144v1Dynamical 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.
- May 07 2014 astro-ph.HE gr-qc arXiv:1405.1121v2We present a first exploration of the results of neutron star-black hole mergers using black hole masses in the most likely range of $7M_\odot-10M_\odot$, 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 ($\chi_{\rm BH}\gtrsim 0.7$), we show that the merger consistently produces large amounts of cool ($T\lesssim 1\,{\rm MeV}$), unbound, neutron-rich material ($M_{\rm ej}\sim 0.05M_\odot-0.20M_\odot$). A comparable amount of bound matter is initially divided between a hot disk ($T_{\rm max}\sim 15\,{\rm MeV}$) with typical neutrino luminosity $L_\nu\sim 10^{53}\,{\rm erg/s}$, and a cooler tidal tail. After a short period of rapid protonization of the disk lasting $\sim 10\,{\rm ms}$, 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 $M_{\rm BH}\sim 7M_\odot-10M_\odot$ 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.
- Jan 29 2014 gr-qc arXiv:1401.7038v1Many relevant applications in gravitational wave physics share a significant common problem: the seven-dimensional parameter space of gravitational waveforms from precessing compact binary inspirals and coalescences is large enough to prohibit covering the space of waveforms with sufficient density. We find that by using the reduced basis method together with a parametrization of waveforms based on their phase and precession, we can construct ultra-compact yet high-accuracy representations of this large space. As a demonstration, we show that less than $100$ judiciously chosen precessing inspiral waveforms are needed for $200$ cycles, mass ratios from $1$ to $10$ and spin magnitudes $\le 0.9$. In fact, using only the first $10$ reduced basis waveforms yields a maximum mismatch of $0.016$ over the whole range of considered parameters. We test whether the parameters selected from the inspiral regime result in an accurate reduced basis when including merger and ringdown; we find that this is indeed the case in the context of a non-precessing effective-one-body model. This evidence suggests that as few as $\sim 100$ numerical simulations of binary black hole coalescences may accurately represent the seven-dimensional parameter space of precession waveforms for the considered ranges.
- Jan 07 2014 gr-qc astro-ph.CO arXiv:1401.0939v1The 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]
- Dec 04 2013 gr-qc arXiv:1312.0701v2This paper develops a method for solving Einstein's equation numerically on multi-cube 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 non-linear numerical results to the analytical perturbation solutions, which are also derived here.
- Nov 12 2013 gr-qc arXiv:1311.2544v1Gravitational 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.
- Nov 12 2013 gr-qc arXiv:1311.2565v1The 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.
- Gravitational waves (GW) from coalescing stellar-mass black hole binaries (BBH) 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 BBH are constructed from intermediary analytical waveform models which are calibrated against numerical relativity simulations and which can be aluated 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 non-spinning 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 non-spinning 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.
- Sep 17 2013 gr-qc arXiv:1309.3605v1We 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.
- Sep 05 2013 gr-qc arXiv:1309.1141v1At 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.
- Sep 04 2013 gr-qc arXiv:1309.0544v3We compute the periastron advance using the effective-one-body formalism for binary black holes moving on quasi-circular 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 (PN) corrections linear in the spin of the particle. In fact, even when only the leading spin PN 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.
- Sep 04 2013 gr-qc arXiv:1309.0541v2We study the general relativistic periastron advance in spinning black hole binaries on quasi-circular orbits, with spins aligned or anti-aligned 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 non-spinning particle orbiting a Kerr black hole of mass M and spin S = -0.5 M^2, 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.
- Jul 30 2013 gr-qc astro-ph.CO arXiv:1307.7685v1We 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 non spinning 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 dM< 5 10^-4M_BH and their spins within da< 10^-3M_BH. 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 non-disrupting 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 ~10Msun, at least until the advent of gravitational wave detectors with a sensitivity comparable to that of the proposed Einstein Telescope.
- Jul 24 2013 gr-qc arXiv:1307.6232v1We 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.
- Jul 22 2013 gr-qc arXiv:1307.5307v3The 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 ~100-200 solar masses, 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.
- May 28 2013 gr-qc arXiv:1305.5991v2The 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 anti-aligned with the orbital angular momentum. The initial spin magnitudes have $|\chi_i| \lesssim 0.95$ and are more concentrated in the aligned direction because of the greater astrophysical interest of this case. We combine this data with five previously reported simulations of the same configuration, but with different spin magnitudes, including the highest spin simulated to date, $\chi_i \approx 0.97$. This data set is sufficiently accurate to enable us to offer improved analytic fitting formulae for the final spin and for the energy radiated by gravitational waves as a function of initial spin. The improved fitting formulae 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.
- Apr 23 2013 gr-qc arXiv:1304.6077v3This paper 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.
- Apr 12 2013 astro-ph.HE gr-qc arXiv:1304.3384v3Neutrino 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 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/M^2=0.9) system with the K_0=220 MeV Lattimer-Swesty equation of state. 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 (i) to make the disk much denser and more compact, (ii) to cause the average electron fraction Y_e of the disk to rise to about 0.2 and then gradually decrease again, and (iii) to gradually cool the disk. The disk is initially hot (T~6 MeV) and luminous in neutrinos (L_\nu~10^54 erg s^-1), but the neutrino luminosity decreases by an order of magnitude over 50 ms of post-merger evolution.
- Feb 27 2013 gr-qc astro-ph.HE arXiv:1302.6297v2Black 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 $S/M^2$=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 $S/M^2$=0.97, the highest simulated to date. We find that the black hole's large spin leads to the most massive accretion disk 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 disk appears to be remarkably stable. We also find that the high black-hole spin persists until shortly before the time of merger; afterwards, both merger and accretion spin down the black hole.
- Dec 20 2012 gr-qc astro-ph.HE arXiv:1212.4810v2Black 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 post-merger remnant are very sensitive to the parameters of the binary. In this paper, we study the impact of the radius of the neutron star and the alignment of the black hole spin for systems within the range of mass ratio currently deemed most likely for field binaries (M_BH ~ 7 M_NS) and for black hole spins large enough for the neutron star to disrupt (J/M^2=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 0.3M_NS to 0.1M_NS for changes of only 2 km in the NS radius; (ii) 0.01-0.05M_sun of unbound material can be ejected with kinetic energy >10^51 ergs, a significant increase compared to low mass ratio, low spin binaries. This ejecta could power detectable optical and radio afterglows. (iii) Only a small fraction (<3%) 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. (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 (v>300 km/s) can be given to the final black hole as a result of a precessing BHNS merger, when the disruption of the neutron star occurs just outside or within the innermost stable spherical orbit.
- Nov 27 2012 gr-qc arXiv:1211.6079v2Simulations 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.
- Oct 11 2012 gr-qc astro-ph.HE arXiv:1210.3007v2This 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 non-spinning 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 3.5PN, hybrids for higher-mass-ratio systems would require NR waveforms with many more gravitational-wave (GW) cycles to guarantee no adverse impact on parameter estimation. Furthermore, we investigate the potential improvement in hybrid waveforms that can be expected from 4th order post-Newtonian waveforms, and find that knowledge of this 4th post-Newtonian order would significantly improve the accuracy of hybrid waveforms.
- Aug 04 2012 gr-qc arXiv:1208.0630v1We 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.
- Jun 15 2012 gr-qc arXiv:1206.2943v3Initial 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 \emphnot 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
- Jun 15 2012 gr-qc arXiv:1206.3015v2This 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.
- Jan 26 2012 gr-qc arXiv:1201.5319v1The 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 63 hybrid waveforms consisting of a numerical portion modelling the late inspiral, merger, and ringdown stitched to a post-Newtonian portion modelling 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 $(\ell,m)=(2,2)$ 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.
- Nov 08 2011 gr-qc astro-ph.HE arXiv:1111.1677v2General 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.
- Oct 12 2011 gr-qc astro-ph.HE arXiv:1110.2229v1Motivated 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.
- Nov 19 2010 gr-qc arXiv:1011.4223v3We develop and calibrate a characteristic waveform extraction tool whose major improvements and corrections of prior versions allow satisfaction of the accuracy standards required for advanced LIGO data analysis. The extraction tool uses a characteristic evolution code to propagate numerical data on an inner worldtube supplied by a 3+1 Cauchy evolution to obtain the gravitational waveform at null infinity. With the new extraction tool, high accuracy and convergence of the numerical error can be demonstrated for an inspiral and merger of mass M binary black holes even for an extraction worldtube radius as small as R = 20M. The tool provides a means for unambiguous comparison between waveforms generated by evolution codes based upon different formulations of the Einstein equations and based upon different numerical approximations.
- Oct 15 2010 gr-qc astro-ph.HE arXiv:1010.2777v3Astrophysically 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.
- The accurate modeling of gravitational radiation is a key issue for gravitational wave astronomy. As simulation codes reach higher accuracy, systematic errors inherent in current numerical relativity wave-extraction methods become evident, and may lead to a wrong astrophysical interpretation of the data. In this paper, we give a detailed description of the Cauchy-characteristic extraction technique applied to binary black hole inspiral and merger evolutions to obtain gravitational waveforms that are defined unambiguously, that is, at future null infinity. By this method we remove finite-radius approximations and the need to extrapolate data from the near zone. Further, we demonstrate that the method is free of gauge effects and thus is affected only by numerical error. Various consistency checks reveal that energy and angular momentum are conserved to high precision and agree very well with extrapolated data. In addition, we revisit the computation of the gravitational recoil and find that finite radius extrapolation very well approximates the result at $\scri$. However, the (non-convergent) systematic differences to extrapolated data are of the same order of magnitude as the (convergent) discretisation error of the Cauchy evolution hence highlighting the need for correct wave-extraction.
- Sep 22 2009 gr-qc arXiv:0909.3557v1Several 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.
- Jul 16 2009 gr-qc arXiv:0907.2637v3Gravitational radiation is properly defined only at future null infinity ($\scri$), but in practice it is estimated from data calculated at a finite radius. We have used characteristic extraction to calculate gravitational radiation at $\scri$ for the inspiral and merger of two equal mass non-spinning black holes. Thus we have determined the first unambiguous merger waveforms for this problem. The implementation is general purpose, and can be applied to calculate the gravitational radiation, at $\scri$, given data at a finite radius calculated in another computation.
- May 01 2009 gr-qc arXiv:0904.4873v2A new gauge driver is introduced for the generalized harmonic (GH) 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. 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 are demonstrated through numerical tests, which impose a new damped-wave gauge condition on the evolutions of single black-hole spacetimes.
- Aug 07 2008 gr-qc arXiv:0808.0861v2We develop, test and compare new numerical and geometrical methods for improving the accuracy of extracting waveforms using characteristic evolution. The new numerical method involves use of circular boundaries to the stereographic grid patches which cover the spherical cross-sections of the outgoing null cones. We show how an angular version of numerical dissipation can be introduced into the characteristic code to damp the high frequency error arising form the irregular way the circular patch boundary cuts through the grid. The new geometric method involves use of the Weyl tensor component $\Psi_4$ to extract the waveform as opposed to the original approach via the Bondi news function. We develop the necessary analytic and computational formula to compute the $O(1/r)$ radiative part of $\Psi_4$ in terms of a conformally compactified treatment of null infinity. These methods are compared and calibrated in test problems based upon linearized waves.
- Feb 25 2008 gr-qc arXiv:0802.3341v2We present a set of well-posed constraint-preserving boundary conditions for a first-order in time, second-order in space, harmonic formulation of the Einstein equations. The boundary conditions are tested using robust stability, linear and nonlinear waves, and are found to be both less reflective and constraint preserving than standard Sommerfeld-type boundary conditions.
- Sep 25 2007 gr-qc arXiv:0709.3559v3We discuss results that have been obtained from the implementation of the initial round of testbeds for numerical relativity which was proposed in the first paper of the Apples with Apples Alliance. We present benchmark results for various codes which provide templates for analyzing the testbeds and to draw conclusions about various features of the codes. This allows us to sharpen the initial test specifications, design a new test and add theoretical insight.
- Binary black-hole systems with spins aligned with the orbital angular momentum are of special interest as they may be the preferred end-state of the inspiral of generic supermassive binary black-hole systems. In view of this, we have computed the inspiral and merger of a large set of binary systems of equal-mass black holes with spins aligned with the orbital angular momentum but otherwise arbitrary. By least-square fitting the results of these simulations we have constructed two "spin diagrams" which provide straightforward information about the recoil velocity |v_kick| and the final black-hole spin a_fin in terms of the dimensionless spins a_1 and a_2 of the two initial black holes. Overall they suggest a maximum recoil velocity of |v_kick|=441.94 km/s, and minimum and maximum final spins a_fin=0.3471 and a_fin=0.9591, respectively.
- Binary black-hole systems with spins aligned with the orbital angular momentum are of special interest, as studies indicate that this configuration is preferred in nature. If the spins of the two bodies differ, there can be a prominent beaming of the gravitational radiation during the late plunge, causing a recoil of the final merged black hole. We perform an accurate and systematic study of recoil velocities from a sequence of equal-mass black holes whose spins are aligned with the orbital angular momentum, and whose individual spins range from a = +0.584 to -0.584. In this way we extend and refine the results of a previous study and arrive at a consistent maximum recoil of 448 +- 5 km/s for anti-aligned models as well as to a phenomenological expression for the recoil velocity as a function of spin ratio. This relation highlights a nonlinear behavior, not predicted by the PN estimates, and can be readily employed in astrophysical studies on the evolution of binary black holes in massive galaxies. An essential result of our analysis is the identification of different stages in the waveform, including a transient due to lack of an initial linear momentum in the initial data. Furthermore we are able to identify a pair of terms which are largely responsible for the kick, indicating that an accurate computation can be obtained from modes up to l=3. Finally, we provide accurate measures of the radiated energy and angular momentum, finding these to increase linearly with the spin ratio, and derive simple expressions for the final spin and the radiated angular momentum which can be easily implemented in N-body simulations of compact stellar systems. Our code is calibrated with strict convergence tests and we verify the correctness of our measurements by using multiple independent methods whenever possible.
- Dec 26 2006 gr-qc arXiv:gr-qc/0612150v2We describe an explicit in time, finite-difference code designed to simulate black holes by using the excision method. The code is based upon the harmonic formulation of the Einstein equations and incorporates several features regarding the well-posedness and numerical stability of the initial-boundary problem for the quasilinear wave equation. After a discussion of the equations solved and of the techniques employed, we present a series of testbeds carried out to validate the code. Such tests range from the evolution of isolated black holes to the head-on collision of two black holes and then to a binary black hole inspiral and merger. Besides assessing the accuracy of the code, the inspiral and merger test has revealed that individual apparent horizons can touch and even intersect. This novel feature in the dynamics of the marginally trapped surfaces is unexpected but consistent with theorems on the properties of apparent horizons.
- Oct 06 2006 gr-qc arXiv:gr-qc/0610019v1The characteristic approach to numerical relativity is a useful tool in evolving gravitational systems. In the past this has been implemented using two patches of stereographic angular coordinates. In other applications, a six-patch angular coordinate system has proved effective. Here we investigate the use of a six-patch system in characteristic numerical relativity, by comparing an existing two-patch implementation (using second-order finite differencing throughout) with a new six-patch implementation (using either second- or fourth-order finite differencing for the angular derivatives). We compare these different codes by monitoring the Einstein constraint equations, numerically evaluated independently from the evolution. We find that, compared to the (second-order) two-patch code at equivalent resolutions, the errors of the second-order six-patch code are smaller by a factor of about 2, and the errors of the fourth-order six-patch code are smaller by a factor of nearly 50.