Computing and analyzing gravitational radiation in black hole simulations using a new multi-block approach to numerical relativity

Numerical simulations of Kerr black holes are presented and the excitation of quasinormal modes is studied in detail. Issues concerning the extraction of gravitational waves from numerical space-times and analyzing them in a systematic way are discussed. A new multi-block infrastructure for solving first order symmetric hyperbolic time dependent partial differential equations is developed and implemented in a way that stability is guaranteed for arbitrary high order accurate numerical schemes. Multi-block methods make use of several coordinate patches to cover a computational domain. This provides efficient, flexible and very accurate numerical schemes. Using this code, three dimensional simulations of perturbed Kerr black holes are carried out. While the quasinormal frequencies for such sources are well known, until now little attention has been payed to the relative excitation strength of different modes. If an actual perturbed Kerr black hole emits two distinct quasinormal modes that are strong enough to be detected by gravitational wave observatories, these two modes can be used to test the Kerr nature of the source. This would provide a strong test of the so called no hair theorem of general relativity. A systematic method for analyzing ringdown waveforms is proposed. The so called time shift problem, an ambiguity in the definition of excitation amplitudes, is identified and it is shown that this problem can be avoided by looking at appropriately chosen relative mode amplitudes. Rotational mode coupling, the relative excitation strength of co- and counter rotating modes and overtones for slowly and rapidly spinning Kerr black holes are studied. A method for extracting waves from numerical space-times which generalizes one of the standard methods based on the Regge-Wheeler-Zerilli perturbation formalism is presented. Applying this to evolutions of single perturbed Schwarzschild black holes, the accuracy of the new method is compared to the standard approach and it is found that the errors resulting from the former are one to several orders of magnitude below the ones from the latter. It is demonstrated that even at large extraction radii (r=80M), the standard extraction approach produces errors that are dominantly of systematic nature and not due to numerical inaccuracies

A multi-block infrastructure for three-dimensional time-dependent numerical relativity

We describe a generic infrastructure for time evolution simulations in numerical relativity using multiple grid patches. After a motivation of this approach, we discuss the relative advantages of global and patch-local tensor bases. We describe both our multi-patch infrastructure and our time evolution scheme, and comment on adaptive time integrators and parallelisation. We also describe various patch system topologies that provide spherical outer and/or multiple inner boundaries. We employ penalty inter-patch boundary conditions, and we demonstrate the stability and accuracy of our three-dimensional implementation. We solve both a scalar wave equation on a stationary rotating black hole background and the full Einstein equations. For the scalar wave equation, we compare the effects of global and patch-local tensor bases, different finite differencing operators, and the effect of artificial dissipation onto stability and accuracy. We show that multi-patch systems can directly compete with the so-called fixed mesh refinement approach; however, one can also combine both. For the Einstein equations, we show that using multiple grid patches with penalty boundary conditions leads to a robustly stable system. We also show long-term stable and accurate evolutions of a one-dimensional non-linear gauge wave. Finally, we evolve weak gravitational waves in three dimensions and extract accurate waveforms, taking advantage of the spherical shape of our grid lines.Comment: 18 pages. Some clarifications added, figure layout improve

Optimized high-order derivative and dissipation operators satisfying summation by parts, and applications in three-dimensional multi-block evolutions

We construct optimized high-order finite differencing operators which satisfy summation by parts. Since these operators are not uniquely defined, we consider several optimization criteria: minimizing the bandwidth, the truncation error on the boundary points, the spectral radius, or a combination of these. We examine in detail a set of operators that are up to tenth order accurate in the interior, and we surprisingly find that a combination of these optimizations can improve the operators\u27 spectral radius and accuracy by orders of magnitude in certain cases. We also construct high-order dissipation operators that are compatible with these new finite difference operators and which are semi-definite with respect to the appropriate summation by parts scalar product. We test the stability and accuracy of these new difference and dissipation operators by evolving a three-dimensional scalar wave equation on a spherical domain consisting of seven blocks, each discretized with a structured grid, and connected through penalty boundary conditions. In particular, we find that the constructed dissipation operators are effective in suppressing instabilities that are sometimes otherwise present in the restricted full norm case. © Springer Science+Business Media, LLC 2007

The final spin from the coalescence of aligned-spin black-hole binaries

Determining the final spin of a black-hole (BH) binary is a question of key importance in astrophysics. Modelling this quantity in general is made difficult by the fact that it depends on the 7-dimensional space of parameters characterizing the two initial black holes. However, in special cases, when symmetries can be exploited, the description can become simpler. For black-hole binaries with unequal masses but with equal spins which are aligned with the orbital angular momentum, we show that the use of recent simulations and basic but exact constraints derived from the extreme mass-ratio limit allow to model this quantity with a simple analytic expression. Despite the simple dependence, the expression models very accurately all of the available estimates, with errors of a couple of percent at most. We also discuss how to use the fit to predict when a Schwarzschild BH is produced by the merger of two spinning BHs, when the total angular momentum of the spacetime ``flips'' sign, or under what conditions the final BH is ``spun-up'' by the merger. Finally, suggest an extension of the fit to include unequal-spin binaries, thus potentially providing a complete description of the final spin from the coalescence of generic black-hole binaries with spins aligned to the orbital angular momentum.Comment: Version matching the published one; small changes throughout to fit space constraints; corrects error in vii) about spin-up/dow

Spin Diagrams for Equal-Mass Black-Hole Binaries with Aligned Spins

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.Comment: 4 pages, 3 figs; small changes matching published versio

Gravitational-wave detectability of equal-mass black-hole binaries with aligned spins

Binary black-hole systems with spins aligned or anti-aligned to the orbital angular momentum provide the natural ground to start detailed studies of the influence of strong-field spin effects on gravitational wave observations of coalescing binaries. Furthermore, such systems 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 parallel to the orbital angular momentum but otherwise arbitrary. Our attention is particularly focused on the gravitational-wave emission so as to quantify how much spin effects contribute to the signal-to-noise ratio, to the horizon distances, and to the relative event rates for the representative ranges in masses and detectors. As expected, the signal-to-noise ratio increases with the projection of the total black hole spin in the direction of the orbital momentum. We find that equal-spin binaries with maximum spin aligned with the orbital angular momentum are more than "three times as loud" as the corresponding binaries with anti-aligned spins, thus corresponding to event rates up to 30 times larger. We also consider the waveform mismatch between the different spinning configurations and find that, within our numerical accuracy, binaries with opposite spins S_1=-S_2 cannot be distinguished whereas binaries with spin S_1=S_2 have clearly distinct gravitational-wave emissions. Finally, we derive a simple expression for the energy radiated in gravitational waves and find that the binaries always have efficiencies E_rad/M > 3.6%, which can become as large as E_rad/M = 10% for maximally spinning binaries with spins aligned with the orbital angular momentum.Comment: 18 pages, 11 figures, matches published versio

Faithful Effective-One-Body waveforms of equal-mass coalescing black-hole binaries

We continue the program of constructing, within the Effective-One-Body (EOB) approach, high-accuracy analytic waveforms describing the signal emitted by inspiralling and coalescing black hole binaries. Here, we compare a recently derived, resummed 3 PN-accurate EOB quadrupolar waveform to the results of a numerical simulation of the inspiral and merger of an equal-mass black hole binary. We find a remarkable agreement, both in phase and in amplitude, with a maximal dephasing which can be reduced below $\pm 0.005$ gravitational-wave (GW) cycles over 12 GW cycles corresponding to the end of the inspiral, the plunge, the merger and the beginning of the ringdown. This level of agreement is shown for two different values of the effective 4 PN parameter a_5, and for corresponding, appropriately "flexed" values of the radiation-reaction resummation parameter v_pole. In addition, our resummed EOB amplitude agrees to better than the 1% level with the numerical-relativity one up to the late inspiral. These results, together with other recent work on the EOB-numerical-relativity comparison, confirm the ability of the EOB formalism to faithfully capture the general relativistic waveforms.Comment: 13 pages, 3 figures. Small changes. Version published in Phys. Rev.

Systolic and Hyper-Systolic Algorithms for the Gravitational N-Body Problem, with an Application to Brownian Motion

A systolic algorithm rhythmically computes and passes data through a network of processors. We investigate the performance of systolic algorithms for implementing the gravitational N-body problem on distributed-memory computers. Systolic algorithms minimize memory requirements by distributing the particles between processors. We show that the performance of systolic routines can be greatly enhanced by the use of non-blocking communication, which allows particle coordinates to be communicated at the same time that force calculations are being carried out. Hyper-systolic algorithms reduce the communication complexity at the expense of increased memory demands. As an example of an application requiring large N, we use the systolic algorithm to carry out direct-summation simulations using 10^6 particles of the Brownian motion of the supermassive black hole at the center of the Milky Way galaxy. We predict a 3D random velocity of 0.4 km/s for the black hole.Comment: 33 pages, 10 postscript figure

Recoil velocities from equal-mass binary black-hole mergers: a systematic investigation of spin-orbit aligned configurations

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.Comment: 24 pages, 15 figures, 5 table