Gravitational Waveforms from Coalescing Binary Black Holes

Gravitational Waveforms Gravitational Waveforms from Coalescing Binary Black Holesfrom Coalescing Binary Black Holes

Dae-Il (Dale) Choi NASA Goddard Space Flight Center, MD, USA

Universities Space Research Association, USA

Supported by NASA ATP02-0043-0056 & NASA Advanced Supercomputing Project “Columbia”

Numerical Relativity 2005 Workshop NASA Goddard Space Flight Center, Greenbelt, MD, NOV 2, 2005

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Beyond Einstein: From the Big Bang to Black Holes

Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005

CollaboratorsCollaboratorsIt’s teamworkIt’s teamwork

Joan Centrella, John Baker (NASA/GSFC)

Jim van Meter, Michael Koppitz (National Research Council)

Breno Imbiriba, W. Darian Boggs, Stefan Mendez-Diez (University of Maryland)

Other collaborators

J. David Brown (North Carolina State Univ.)

David Fiske (DAC, formerly NASA/GSFC)

Kevin Olson (NASA/GSFC)

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OutlineOutline

Methodology: Hahndol Code [Hahndol= 한돌 =translation of “Ein-stein” into Korean]

Results: Inspiral merger from the ISCO (QC0)

Results: Head-on collision (if time allows)

Movie of the real part of Psi4

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Hahndol CodeHahndol Code

3+1 Numerical Relativity Code

– BSSN formalism following Imbiriba et al, PRD70, 124025 (2004), Alcubierre at al PRD67, 084023 (2003) except the new gauge conditions.

– Uses finite differencing (mixed 2nd and 4th order FD, Mesh-Adapted-Differencing–see posters for details), iterative Crank-Nicholson time integrator.

– Computational infrastructure based on PARAMESH (MacNiece, Olson) Scalability shown up to 864 CPUs with ~ 95% efficiency.

Mesh refinement

– Currently use fixed mesh structure with mesh boundaries at (2,4,8,16,32,64)M for QC0 runs.

– The innermost level contains the both black holes.

– For higher QC-sequence, AMR implementation being tested.

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Hahndol CodeHahndol CodeOuter boundary conditions

– Impose outgoing Sommerfeld conditions on all BSSN variables.

– But, basic strategy is to push OB far away so that OB does not contaminate regions of interests.

– With OB=128M, no harmful effects on the dynamics of black holes nor waveform extraction (QC0). If desired, OB can be put at 256M or beyond.

Initial data solver

– Uses multi-grid method on a non-uniform grid using Brown’s algorithm: Brown & Lowe, JCP 209, 582-598, 2005 (gr-qc/0411112).

– Generate QC ID by solving HCE using puncture method (Brandt & Bruegmann, 1997).

– Bowen-York prescription for the extrinsic curvature for binary black holes.

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Hahndol CodeHahndol CodeTraditional gauge conditions (AEI, etc.)

– Split conformal factor into time-indep. singular part (ΨBL) and time-dep. regular part. Treat ΨBL analytically and evolve only the regular part.

– Use the following K-/Gamma-driver conditions for gauges. (BL factor)

– Problem is that, because of ΨBL factor, black holes cannot move.

– Requires co-rotation shift. But it involves superluminal shift.

Alternative gauge conditions

– Do not split into singular/regular part. No BL factor. – Combined with the driver conditions, let the black holes move across the grid.

– Does this really work?

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Hahndol CodeHahndol CodeNot so fast! Two concerns.

– (a) Puncture memory effect: BHs move but still spiky errors at where the punctures were at t=0.

– (b) Messy stuff near the would-have-been puncture locations if they were moving.

The problem (a)

– Caused by the zero-speed mode in the Gamma driver shift condition

– Can be alleviated by “shifting shift”

[Movies] comparison bet. (1) Traditional (crashed at t=35M) (2) No BL factor (3) NoBL + Shifting Shift

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Hahndol CodeHahndol CodeThe problem (b)

– In practice, we find that the stuff doesn’t seem to “spill over”.

– [Movie: Head-on collision w/ L/M~9 using NoBL+Shifting Shift] shows a good convergence of HC from 3 runs with different resolutions.

– Note, with the traditional gauge, HC too large and non-convergent.

For all the cases we considered, this new gauge conditions allow us to obtain convergent results (constraints, waveforms).

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Hahndol CodeHahndol Code

Wave extraction

– Compute the Newman-Penrose Weyl scalar Ψ4 (a gauge invariant measure)

where C is weyl tensor and (l,n,m,mbar) is a tetrad.

– Analyze its harmonic decomposition using a novel technique due to Misner (Misner 2004; Fiske 2005).

– Compute waveforms r ~ 20M, 30M, 40M and 50M.

Coulomb scalar χ [Beetle, et al, PRD72, 024013 (2005); Burko, Baumgarte & Beetle, gr-qc/0505028.]

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Evolution of Quasi Circular Initial DataEvolution of Quasi Circular Initial Data

QC-sequence (Minimization of effective potential, Cook 1994)

QC0, L/M=4.99, J/M^2=0.779

Re-Coulomb invariant: ReC(horizon) = -1/(8M2) for quiecent BHs. [Movie: Horizon at ReC~ -1/2 (yellowish) at T=0; Horizon at ReC~-1/8 (blue edge) late times.]

4M 180M

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QC0 (BH source region)QC0 (BH source region)Comparison of Re (Coulomb) scalar for three different resolutions: M/16, M/32, M/48 runs. [Only in this movie, time label is in terms of (M/2)]

(In this talk, different runs are labeled by the resolution in the finest resolution grid.)

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QC0 (BH source region)QC0 (BH source region)Convergence of HC near black holes along x-axis from M/24 (Dashed) and M/32 (Solid) runs. Data from Time=11M,19M,24M where BHs are crossing the x-axis. (Note FMR boundaries are at 2M, 4M, etc.)

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QC0 WaveformsQC0 WaveformsWaveforms (Re L=2, M=2 mode) from three runs, M/16, M/24, M/32 extracted at rextract =20M (Solid), 40M(Dashed). Plotted are (r x Psi4).

Good O(1/r) propagation behavior; M/24, M/32 are very close.

Comparison with Lazarus I--Baker et al, PRD 65,124012 (2002)

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QC0 (Waveforms)QC0 (Waveforms)Convergence of waveforms (real and imaginary parts of L=2, M=2 mode) at r=20M (upper panels), and 40M (lower panels).

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QC0 (dE/dt, dJz/dt)QC0 (dE/dt, dJz/dt)Energy & angular momentum loss due to GWdE/dt, dJz/dt

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QC0 QC0 (Energy and Angular momentum)(Energy and Angular momentum)

Total E and Total Jz loss (plotted for three resolutions and for 4 different extraction radii)

At r=30M,

Final J~0.65

Resolution E Jz

M/16 0.0494 -0.200 (26%)

M/24 0.0325 -0.133 (17%)

M/32 0.0315 -0.132 (17%)

Lazarus I 0.025 -0.093 (12%)

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QC0 (Energy Conservation?)QC0 (Energy Conservation?)Calculate ADM Mass (Murchadha & York, 1974)

Energy conservation: Minit-Mfinal= EGW?

r=40M,50M, Solid represents M(t), Dashed M(t=0)-EGW(t).

Minit-Mfinal= EGW!

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Head-On CollisionHead-On CollisionLeft Panel: Waveforms extracted at rextract = 20M, 30M, 40M, 50M

– Colored lines show O(1/r) propagation fall-off behavior (M1=M2=0.5)

Right Panel: dE/dt (total energy loss ~ 0.00040)

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Head-On CollisionHead-On CollisionLeft Pane: Waveforms in different resolutions. Proper separation ~9M

Right Panel: convergence behavior of the waveforms.

No apparent problems up to L~11-12M. Promising for collision with large initial separation

Gravitational Waveforms from Coalescing Binary Black Holes

Documents

dynamics of black holes

big bang

hahndol code hahndol

hahndol code3

nasagsfckevin olson

new gauge conditions

bl factorproblem

gammadriver conditions