# Presentation: 510287

## 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**

## CollaboratorsIt’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)

## Outline

- 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

## 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.

## 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.

## 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?

## 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

## 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).

## 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.]

## 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

## 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.)

## 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.)

## 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)

## QC0 (Waveforms)

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

## QC0 (dE/dt, dJz/dt)

- Energy & angular momentum loss due to GW
- dE/dt, dJz/dt

## 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

| 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%) |

## 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!

## 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)

## Head-On Collision

- 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