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Gravitational Waveforms from Coalescing Binary Black Holes

Gravitational Waveforms from 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”

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Gravitational Waveforms from Coalescing Binary Black Holes

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  1. Gravitational Waveforms from 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

  2. CollaboratorsIt’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)

  3. Outline • Methodology:HahndolCode [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

  4. Hahndol 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.

  5. Hahndol Code • Outer 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.

  6. Hahndol Code • Traditional 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?

  7. Hahndol Code • Not 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

  8. Hahndol Code • The 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).

  9. Hahndol 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.]

  10. Evolution 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

  11. 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.)

  12. 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.)

  13. QC0 Waveforms • Waveforms (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)

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

  15. QC0 (dE/dt, dJz/dt) • Energy & angular momentum loss due to GW • dE/dt, dJz/dt

  16. QC0 (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

  17. 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!

  18. Head-On Collision • Left 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)

  19. Left 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 Head-On Collision

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