Gravitational Waveforms from Coalescing Binary Black Holes

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
CollaboratorsIts 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 Hahndol Code Hahndol??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

• 31 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-Differencingsee 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 OB128M, 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 Browns 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 (?BL) and time-dep. regular part. Treat ?BL
  analytically and evolve only the regular part.
• Use the following K-/Gamma-driver conditions for
  gauges. (BL factor)
• Problem is that, because of ?BL 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 t0.
• (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 t35M) (2) No BL factor (3) NoBL Shifting
  Shift

8
Hahndol Code

• The problem (b)
• In practice, we find that the stuff doesnt seem
  to spill over.
• Movie Head-on collision w/ L/M9 using
  NoBLShifting 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 ?4 (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 ? 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/M4.99, J/M20.779
• Re-Coulomb invariant ReC(horizon) -1/(8M2) for
  quiecent BHs. Movie Horizon at ReC -1/2
  (yellowish) at T0 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 Time11M,19M,24M where BHs are crossing the
  x-axis. (Note FMR boundaries are at 2M, 4M, etc.)
  

13
QC0 Waveforms

• Waveforms (Re L2, M2 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 L2, M2 mode) at r20M (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 r30M,
• Final J0.65

17
QC0 (Energy Conservation?)

• Calculate ADM Mass (Murchadha York, 1974)
• Energy conservation Minit-Mfinal EGW?
• r40M,50M, Solid represents M(t),
  Dashed M(t0)-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 (M1M20.5)
• Right Panel dE/dt (total energy loss 0.00040)

19
Head-On Collision

• Left Pane Waveforms in different resolutions.
  Proper separation 9M
• Right Panel convergence behavior of the
  waveforms.
• No apparent problems up to L11-12M. Promising
  for collision with large initial separation