Tracking the precession of compact binaries from their gravitational-wave signal

We present a simple method to track the precession of a black-hole-binary system, using only information from the gravitational-wave (GW) signal. Our method consists of locating the frame from which the magnitude of the $(\ell=2,|m|=2)$ modes is maximized, which we denote the "quadrupole-aligned" frame. We demonstrate the efficacy of this method when applied to waveforms from numerical simulations. In the test case of an equal-mass nonspinning binary, our method locates the direction of the orbital angular momentum to within $(\Delta \theta, \Delta \phi) = (0.05^{\circ},0.2^{\circ})$. We then apply the method to a $q = M_2/M_1 = 3$ binary that exhibits significant precession. In general a spinning binary's orbital angular momentum $\mathbf{L}$ is \emph{not} orthogonal to the orbital plane. Evidence that our method locates the direction of $\mathbf{L}$ rather than the normal of the orbital plane is provided by comparison with post-Newtonian (PN) results. Also, we observe that it accurately reproduces similar higher-mode amplitudes to a comparable non-spinning (and therefore non-precessing) binary, and that the frequency of the $(\ell=2,|m|=2)$ modes is consistent with the "total frequency" of the binary's motion. The simple form of the quadrupole-aligned waveform will be useful in attempts to analytically model the inspiral-merger-ringdown (IMR) signal of precessing binaries, and in standardizing the representation of waveforms for studies of accuracy and consistency of source modelling efforts, both numerical and analytical.Comment: 11 pages, 12 figures, 1 tabl

Length requirements for numerical-relativity waveforms

One way to produce complete inspiral-merger-ringdown gravitational waveforms from black-hole-binary systems is to connect post-Newtonian (PN) and numerical-relativity (NR) results to create "hybrid" waveforms. Hybrid waveforms are central to the construction of some phenomenological models for GW search templates, and for tests of GW search pipelines. The dominant error source in hybrid waveforms arises from the PN contribution, and can be reduced by increasing the number of NR GW cycles that are included in the hybrid. Hybrid waveforms are considered sufficiently accurate for GW detection if their mismatch error is below 3% (i.e., a fitting factor about 0.97). We address the question of the length requirements of NR waveforms such that the final hybrid waveforms meet this requirement, considering nonspinning binaries with q = M_2/M_1 \in [1,4] and equal-mass binaries with \chi = S_i/M_i^2 \in [-0.5,0.5]. We conclude that for the cases we study simulations must contain between three (in the equal-mass nonspinning case) and ten (the \chi = 0.5 case) orbits before merger, but there is also evidence that these are the regions of parameter space for which the least number of cycles will be needed.Comment: Corrected some typo

Dramatic slowing of compositional relaxations in the approach to the glass transition for a bimodal colloidal suspension

Molecular dynamics simulation was used to study a model colloidal suspension with two species of slightly different sized colloidal particles in an explicit solvent. In this work we calculated the four interdiffusion coefficients for the ternary system, which were then used to calculate the decay coefficients D± of the two independent diffusive modes. We found that the slower D- decay mode, which is associated with the system's ability to undergo compositional changes, was responsible for the long-time decay in the intermediate scattering function. We also found that a decrease in D- to negligible values at a packing fraction of Φg=0.592 resulted in an extreme slow-down in the long-time decay of the intermediate scattering function often associated with the glass transition. Above Φg, the system formed a long-lived metastable state that did not relax to its equilibrium crystal state within the simulation time window. We concluded that the inhibition of crystallization was caused by the inability of the quenched fluid to undergo the compositional changes needed for the formation of the equilibrium crystal

Rotational tuning of interaction in metamaterials

We experimentally observe the tuning of metamaterials through the relative rotation of the elements about their common axis. In contrast to previous results we observe a crossing of resonances, where the symmetric and anti-symmetric modes become degenerate. We associate this effect with an interplay between the magnetic and electric near-field interactions and verify this by calculations based on the interaction energy between resonators

Dynamics of a model colloidal suspension from dilute to freezing

Molecular dynamics simulation was used to study a model colloidal suspension at a range of packing fractions from the dilute limit up to the freezing point. This study builds on previous work by the authors which modeled the colloidal particles with a hard core surrounded by a Weeks-Chandler-Anderson potential with modified interaction parameters, and included an explicit solvent. In this work, we study dynamical properties of the model by first calculating the velocity autocorrelation function, the self-diffusion coefficient, and the mutual diffusion coefficient. We also perform detailed calculations of the colloidal particle intermediate scattering function to study the change in dynamics leading up to the freezing point, and to determine whether the current model can be used to interpret light scattering experiments. We then perform a multiexponential analysis on the intermediate scattering function results and find that the data are fitted well by the sum of two exponentials, which is in line with previous analysis of experimental colloidal suspensions. The amplitudes and decay coefficients of the two modes are determined over a large range of wave vectors at packing fractions leading up to the freezing point. We found that the maximum wave vector at which macroscopic diffusive behavior was observed decreased as the packing fraction increased, and a simple extrapolation shows the maximum wave vector going to zero at the melting point. Lastly, the ratio of the two decay coefficients is compared to the scaling law proposed by Segre and Pusey. It was found that the ratio was not constant, but instead was wave vector dependent

Binary black holes on a budget: Simulations using workstations

Binary black hole simulations have traditionally been computationally very expensive: current simulations are performed in supercomputers involving dozens if not hundreds of processors, thus systematic studies of the parameter space of binary black hole encounters still seem prohibitive with current technology. Here we show how the multi-layered refinement level code BAM can be used on dual processor workstations to simulate certain binary black hole systems. BAM, based on the moving punctures method, provides grid structures composed of boxes of increasing resolution near the center of the grid. In the case of binaries, the highest resolution boxes are placed around each black hole and they track them in their orbits until the final merger when a single set of levels surrounds the black hole remnant. This is particularly useful when simulating spinning black holes since the gravitational fields gradients are larger. We present simulations of binaries with equal mass black holes with spins parallel to the binary axis and intrinsic magnitude of S/m^2= 0.75. Our results compare favorably to those of previous simulations of this particular system. We show that the moving punctures method produces stable simulations at maximum spatial resolutions up to M/160 and for durations of up to the equivalent of 20 orbital periods.Comment: 20 pages, 8 figures. Final version, to appear in a special issue of Class. Quantum Grav. based on the New Frontiers in Numerical Relativity Conference, Golm, July 200

Molecular dynamics simulation study of the static and dynamic properties of a model colloidal suspension with explicit solvent

Molecular dynamics simulation was used to study a colloidal suspension with explicit solvent to determine how inclusion of the solvent affects the structure and dynamics of the system. The solute was modelled as a hard-core particle enclosed in a Weeks-Chandler-Andersen (WCA) potential shell, while the solvent was modelled as a simple WCA fluid. We found that when the solute-solvent interaction included a hard core equal to half of the solute hard-core diameter, large depletion effects arose, leading to an effective attraction and large deviations from hard-sphere structure for the colloidal component. It was found that these effects could be eliminated by reducing the hard-core distance parameter in the solute-solvent interaction, thus allowing the solvent to penetrate closer to the colloidal particles. Three different values for the solute-solvent hard-core parameter were systematically studied by comparing the static structure factor and radial distribution function to the predictions of the Percus-Yevick theory for hard spheres. When the optimal value of the solute-solvent hard-core interaction parameter was found, this model was then used to study the dynamical behaviour of the colloidal suspension. This was done by first measuring the velocity autocorrelation function (VACF) over a large range of packing fractions. We found that this model predicted the sign of the long-time tail in the VACF in agreement with experimental values, something that single component hard-sphere systems have failed to do. The intermediate scattering functions at low wavevector were briefly studied to determine their behaviour in a dilute system. It was found that they could be modelled using a simple diffusion equation with a wavevector independent diffusion coefficient, making this model an excellent analogue of experimentally studied hard-sphere colloids

Bowen-York trumpet data and black-hole simulations

The most popular method to construct initial data for black-hole-binary simulations is the puncture method, in which compactified wormholes are given linear and angular momentum via the Bowen-York extrinsic curvature. When these data are evolved, they quickly approach a ``trumpet'' topology, suggesting that it would be preferable to use data that are in trumpet form from the outset. To achieve this, we extend the puncture method to allow the construction of Bowen-York trumpets, including an outline of an existence and uniqueness proof of the solutions. We construct boosted, spinning and binary Bowen-York puncture trumpets using a single-domain pseudospectral elliptic solver, and evolve the binary data and compare with standard wormhole-data results. We also show that for boosted trumpets the black-hole mass can be prescribed {\it a priori}, without recourse to the iterative procedure that is necessary for wormhole data.Comment: 15 pages, 14 figures. Published versio

BSSN in Spherical Symmetry

The BSSN (Baumgarte-Shapiro-Shibata-Nakamura) formulation of the Einstein evolution equations is written in spherical symmetry. These equations can be used to address a number of technical and conceptual issues in numerical relativity in the context of a single Schwarzschild black hole. One of the benefits of spherical symmetry is that the numerical grid points can be tracked on a Kruskal--Szekeres diagram. Boundary conditions suitable for puncture evolution of a Schwarzschild black hole are presented. Several results are shown for puncture evolution using a fourth--order finite difference implementation of the equations.Comment: This is the final version to be published in CQG. It contains much more information and detail than the original versio

Testing the validity of the single-spin approximation in inspiral-merger-ringdown waveforms

Gravitational-wave signals from black-hole binaries with nonprecessing spins are described by four parameters—each black hole’s mass and spin. It has been shown that the dominant spin effects can be modeled by a single spin parameter, leading to the development of several three-parameter waveform models. Previous studies indicate that these models should be adequate for gravitational-wave detection. In this paper we focus on the systematic biases that would result from using them to estimate binary parameters, and consider a one-parameter family of configurations at mass ratio 4 and for one choice of effective single spin. We find that for low-mass binaries within that family of configurations, where the observable waveform is dominated by the inspiral, the systematic bias in all physical parameters is smaller than the parameter uncertainty due to degeneracies between the mass ratio and the spins, at least up to signal-to-noise ratios (SNRs) of 50. For higher-mass binaries, where the merger and ringdown make a greater contribution to the observed signal, the bias in the mass ratio is comparable to its uncertainty at SNRs of only ∼30, and the bias in the measurement of the total spin is larger than the uncertainty defined by the 90% confidence region even at an SNR of only 10. Although this bias may be mitigated in future models by a better choice of single-effective-spin parameter, these results suggest that it may be possible to accurately measure both black-hole spins in intermediate-mass binaries