35 research outputs found
Bondi-Sachs Energy-Momentum for the CMC Initial Value Problem
The constraints on the asymptotic behavior of the conformal factor and
conformal extrinsic curvature imposed by the initial value equations of general
relativity on constant mean extrinsic curvature (CMC) hypersurfaces are
analyzed in detail. We derive explicit formulas for the Bondi-Sachs energy and
momentum in terms of coefficients of asymptotic expansions on CMC hypersurfaces
near future null infinity. Precise numerical results for the Bondi-Sachs
energy, momentum, and angular momentum are used to interpret physically
Bowen-York solutions of the initial value equations on conformally flat CMC
hypersurfaces of the type obtained earlier by Buchman et al. [Phys. Rev. D
80:084024 (2009)].Comment: version to be published in Phys. Rev.
Black hole initial data on hyperboloidal slices
We generalize Bowen-York black hole initial data to hyperboloidal constant
mean curvature slices which extend to future null infinity. We solve this
initial value problem numerically for several cases, including unequal mass
binary black holes with spins and boosts. The singularity at null infinity in
the Hamiltonian constraint associated with a constant mean curvature
hypersurface does not pose any particular difficulties. The inner boundaries of
our slices are minimal surfaces. Trumpet configurations are explored both
analytically and numerically.Comment: version for publication in Phys. Rev.
Tetrad formalism for numerical relativity on conformally compactified constant mean curvature hypersurfaces
We present a new evolution system for Einstein's field equations which is
based on tetrad fields and conformally compactified hyperboloidal spatial
hypersurfaces which reach future null infinity. The boost freedom in the choice
of the tetrad is fixed by requiring that its timelike leg be orthogonal to the
foliation, which consists of constant mean curvature slices. The rotational
freedom in the tetrad is fixed by the 3D Nester gauge. With these conditions,
the field equations reduce naturally to a first-order constrained symmetric
hyperbolic evolution system which is coupled to elliptic equations for the
gauge variables. The conformally rescaled equations are given explicitly, and
their regularity at future null infinity is discussed. Our formulation is
potentially useful for high accuracy numerical modeling of gravitational
radiation emitted by inspiraling and merging black hole binaries and other
highly relativistic isolated systems.Comment: Corrected factor of 2 errors in Eqs. (A8) and (A9) and a few typos;
final versio
Improved outer boundary conditions for Einstein's field equations
In a recent article, we constructed a hierarchy B_L of outer boundary
conditions for Einstein's field equations with the property that, for a
spherical outer boundary, it is perfectly absorbing for linearized
gravitational radiation up to a given angular momentum number L. In this
article, we generalize B_2 so that it can be applied to fairly general
foliations of spacetime by space-like hypersurfaces and general outer boundary
shapes and further, we improve B_2 in two steps: (i) we give a local boundary
condition C_2 which is perfectly absorbing including first order contributions
in 2M/R of curvature corrections for quadrupolar waves (where M is the mass of
the spacetime and R is a typical radius of the outer boundary) and which
significantly reduces spurious reflections due to backscatter, and (ii) we give
a non-local boundary condition D_2 which is exact when first order corrections
in 2M/R for both curvature and backscatter are considered, for quadrupolar
radiation.Comment: accepted Class. Quant. Grav. numerical relativity special issue; 17
pages and 1 figur
Simulations of unequal-mass black hole binaries with spectral methods
This paper presents techniques and results for simulations of unequal-mass, nonspinning binary black holes with pseudospectral 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 nonspinning binary black holes 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
Implementation of Absorbing Boundary Conditions for the Einstein Equations
Based on a recent study of the linearized Bianchi equations by Buchman and Sarbach, we construct and implement a hierarchy of absorbing boundary conditions for the Einstein equations in generalized harmonic gauge. As a test problem, we demonstrate that we can evolve multipolar gravitational waves without any spurious reflections at linear order in perturbation theory
Boundary Conditions for the Einstein Evolution System
New boundary conditions are constructed and tested numerically for a general
first-order form of the Einstein evolution system. These conditions prevent
constraint violations from entering the computational domain through timelike
boundaries, allow the simulation of isolated systems by preventing physical
gravitational waves from entering the computational domain, and are designed to
be compatible with the fixed-gauge evolutions used here. These new boundary
conditions are shown to be effective in limiting the growth of constraints in
3D non-linear numerical evolutions of dynamical black-hole spacetimes.Comment: 21 pages, 12 figures, submitted to PR
Effective-one-body waveforms calibrated to numerical relativity simulations: coalescence of non-precessing, spinning, equal-mass black holes
We present the first attempt at calibrating the effective-one-body (EOB)
model to accurate numerical-relativity simulations of spinning, non-precessing
black-hole binaries. Aligning the EOB and numerical waveforms at low frequency
over a time interval of 1000M, we first estimate the phase and amplitude errors
in the numerical waveforms and then minimize the difference between numerical
and EOB waveforms by calibrating a handful of EOB-adjustable parameters. In the
equal-mass, spin aligned case, we find that phase and fractional amplitude
differences between the numerical and EOB (2,2) mode can be reduced to 0.01
radians and 1%, respectively, over the entire inspiral waveforms. In the
equal-mass, spin anti-aligned case, these differences can be reduced to 0.13
radians and 1% during inspiral and plunge, and to 0.4 radians and 10% during
merger and ringdown. The waveform agreement is within numerical errors in the
spin aligned case while slightly over numerical errors in the spin anti-aligned
case. Using Enhanced LIGO and Advanced LIGO noise curves, we find that the
overlap between the EOB and the numerical (2,2) mode, maximized over the
initial phase and time of arrival, is larger than 0.999 for binaries with total
mass 30-200Ms. In addition to the leading (2,2) mode, we compare four
subleading modes. We find good amplitude and frequency agreements between the
EOB and numerical modes for both spin configurations considered, except for the
(3,2) mode in the spin anti-aligned case. We believe that the larger difference
in the (3,2) mode is due to the lack of knowledge of post-Newtonian spin
effects in the higher modes.Comment: 15 pages, 8 figures, typos fixed in Eqs.(7-10
Implementation of higher-order absorbing boundary conditions for the Einstein equations
We present an implementation of absorbing boundary conditions for the
Einstein equations based on the recent work of Buchman and Sarbach. In this
paper, we assume that spacetime may be linearized about Minkowski space close
to the outer boundary, which is taken to be a coordinate sphere. We reformulate
the boundary conditions as conditions on the gauge-invariant
Regge-Wheeler-Zerilli scalars. Higher-order radial derivatives are eliminated
by rewriting the boundary conditions as a system of ODEs for a set of auxiliary
variables intrinsic to the boundary. From these we construct boundary data for
a set of well-posed constraint-preserving boundary conditions for the Einstein
equations in a first-order generalized harmonic formulation. This construction
has direct applications to outer boundary conditions in simulations of isolated
systems (e.g., binary black holes) as well as to the problem of
Cauchy-perturbative matching. As a test problem for our numerical
implementation, we consider linearized multipolar gravitational waves in TT
gauge, with angular momentum numbers l=2 (Teukolsky waves), 3 and 4. We
demonstrate that the perfectly absorbing boundary condition B_L of order L=l
yields no spurious reflections to linear order in perturbation theory. This is
in contrast to the lower-order absorbing boundary conditions B_L with L<l,
which include the widely used freezing-Psi_0 boundary condition that imposes
the vanishing of the Newman-Penrose scalar Psi_0.Comment: 25 pages, 9 figures. Minor clarifications. Final version to appear in
Class. Quantum Grav
Inspiral-merger-ringdown multipolar waveforms of nonspinning black-hole binaries using the effective-one-body formalism
We calibrate an effective-one-body (EOB) model to numerical-relativity
simulations of mass ratios 1, 2, 3, 4, and 6, by maximizing phase and amplitude
agreement of the leading (2,2) mode and of the subleading modes (2,1), (3,3),
(4,4) and (5,5). Aligning the calibrated EOB waveforms and the numerical
waveforms at low frequency, the phase difference of the (2,2) mode between
model and numerical simulation remains below 0.1 rad throughout the evolution
for all mass ratios considered. The fractional amplitude difference at peak
amplitude of the (2,2) mode is 2% and grows to 12% during the ringdown. Using
the Advanced LIGO noise curve we study the effectualness and measurement
accuracy of the EOB model, and stress the relevance of modeling the
higher-order modes for parameter estimation. We find that the effectualness,
measured by the mismatch, between the EOB and numerical-relativity
polarizations which include only the (2,2) mode is smaller than 0.2% for
binaries with total mass 20-200 Msun and mass ratios 1, 2, 3, 4, and 6. When
numerical-relativity polarizations contain the strongest seven modes, and
stellar-mass black holes with masses less than 50Msun are considered, the
mismatch for mass ratio 6 (1) can be as high as 5% (0.2%) when only the EOB
(2,2) mode is included, and an upper bound of the mismatch is 0.5% (0.07%) when
all the four subleading EOB modes calibrated in this paper are taken into
account. For binaries with intermediate-mass black holes with masses greater
than 50Msun the mismatches are larger. We also determine for which
signal-to-noise ratios the EOB model developed here can be used to measure
binary parameters with systematic biases smaller than statistical errors due to
detector noise.Comment: 26 pages, 25 figures, published Phys. Rev. D versio