# Presentation: 099887

##  Calculating Gravitational Wave Signatures from Black Hole Binary Coalescence

**Joan Centrella**
***Laboratory for High Energy Astrophysics***
***NASA/GSFC***

*The Astrophysics of GW Sources Workshop                                April 24-26, 2003*

## Final Coalescence of BH binary.....

- Dominant energy loss mechanism is GW emission
- Coalescence time for binary of total mass *M* and separation *a* (point masses, circular orbits):
- For binary of total mass *M* = 2 x 106 M **** to coalesce
- 	within  tH ~ 1010 years
- 		 separation *a* < *a*max ~ 2.53 x 104 *M* ~ 2.4 x 10-3 pc
- GW detectors will observe the stages of this coalescence, typically ~ 103 orbits
- LISA: will observe this for ~ 1 year
- Ground-based detectors: will observe stellar BH binaries for ~ 1 minute

## Final coalescence proceeds in 3 stages . . .

- GW produced in all three phases of this evolution . . .
- Waveforms and dynamics scale with BH masses and spins
-   source modeling applicable to stellar,
- 	intermediate mass, & massive BHs...

*(graphic courtesy of Kip Thorne)*

***measure masses and spins of binary BHs***

***detect normal modes of ringdown to identify final Kerr BH***

* ****strong-field spacetime dynamics, spin flips and couplings...***

## Focus on the merger stage...

- Inspiral lasts until separation *a* ~ 3R**Schw** = 6*M*
*	**	**** *
- BHs leave quasi-static orbits and plunge together
- Expect ~ several cycles of gravitational radiation from merger
**“****burst” waveform, observable  by LISA for ~ minutes – hours**
**knowledge of merger waveform important to enhance detectability in ground-based GW observatories....**
- Merger can be phenomenologically rich
**effects of spin: spin-spin and spin-orbit couplings, spin flips**
**possible ejection of final BH for *****M*****1**** ****≠ *****M*****2****  **
**test of General Relativity in the dynamical, nonlinear regime**
- Strong, highly nonlinear, dynamical gravitational fields
- Requires numerical solution of the full Einstein
- 	equations in 3 spatial dimensions + time...

## Using numerical relativity to evolve BH mergers...

- Einstein equations: **(*****a,b = 0,1,2,3 – spacetime indices)***
- “3+1”  split spacetime into 3-D spatial slices + time
- Kinematical conditions: freely specificable gauge choices
**lapse function ****α****: measures proper time (****α**** dt) between slices**
**shift vector *****β*****i****: allows moving of spatial coordinates as you evolve from slice to slice**
- Metric becomes **(*****i,j = 1,2,3 – spatial indices)***
- Choice of *α*, *β****i*** of critical to successful spacetime          evolutions....

## Einstein equations split into 2 sets....

- Constraint equations set conditions on spacelike slices
- Hamiltonian constraint:
- Momentum constraints:
- Constraints also serve as initial value equations
- Evolution equations in ADM *(Arnowit, Deser, Misner)* form: 1st order in time, 2nd order in space
- 3-metric:
- extrinsic curvature:

## Initial data for BH mergers....

**Goal:** initial data for BHs representing the ***realistic astrophysical state*** of a binary that has spiralled in by the emission of gravitational radiation
- Need to solve 4 nonlinear constraint equations on initial slice
- Most techniques use a conformal decomposition *(York, et al...)*
- Split basic variables into ***freely specifiable*** and ***constrained***
- Choose the freely specifiable pieces
- Constraints determine the remaining variables              	  guaranteed solution to initial value equations
- physical 3-metric **γ****ij **:			*ψ* is conformal factor
- split extrinsic curvature into trace and trace-free parts:
- Various approaches further decompose ***A*****ij**....

## How to be sure the initial data is astrophysically valid?

- Initial data sets generally contain gravitational radiation due to the selection of free variables and the solution process
**these have largely been chosen to simplify the solution and decouple the constraint equations**
- Most current data sets for BH binaries near the last stable orbit contain (likely unphysical) gravitational radiation ~ few percent of the total system mass *(Pfeiffer, Cook, Teukolsky 2002)*
***This is comparable to the total GW energy expected for the merger process itself !***
- Adopt a more physical approach....
**use binary parameters determined by PN expansion near the end of the inspiral to inform the selection of free variables**
**challenging: the metric and extrinsic curvature that emerge from the solution process can be changed significantly from the PN input**
**recent work by Tichy, et al. shows promise....**

## Evolving BH binary mergers...

- Need to evolve BH binary for ~ few orbits near
- 	the last stable orbit (LSO) at the end of the inspiral,        through merger and ringdown...and extract the GW signature
- Orbital period near LSO is *P* ~ 70*M *(possibly larger...)
*	********** ******need total evolution of ~ few hundred M at minimun***
***Challenges:***
**choice of formalism ******** critical for long term stability of evolution**
**how to represent the BHs on a computational grid**
**choice of gauge: slicing and conditions on spatial coordinates**
**multiple spatial and temporal scales ******** adaptive mesh refinement**
**boundary conditions**
**choice of numerical methods (finite differencing, spectral methods)**
**role of the constraint equations in evolution schemes**
**parallelism and efficiency of computer code.....**

## Evolving BHs...effects of the choice of formalism

- Original ADM unstable for binary BH evolutions after ~ 13 *M*
**exponentially growing unstable modes **
**exact cause of instability is not yet known**
- Conformal ADM formalisms: much better stability properties
**BSSN: Baumgarte, Shapiro, Shibata, Nakamura**
**variations being developed to “tune” the system of equations and allow longer-lived evolutions**
**currently, can evolve a single BH for > several x 10****3**** M (*****Yo, et al.)***
**binary black hole evolutions now possible for ~ 100 M or longer *****(Alcubierre, et al.)***
**note: longevity also depends on gauge choices....**
- Hyperbolic formulations: “mathematically desirable”
**fully first order set of equations**
**introduce relatively large set of auxiliary variables**
**stability of single BH runs > few x ****10****3**** M **
**area of active research.... **

## Evolving BHs...choice of formalism

**BSSN or conformal ADM formalisms – have much improved stability properties though with a larger set of evolution equations:**
**plus additional constraint: **

## Evolving BHs...how to represent the BHs

- How to handle the singularity....
**horizon at r=2M**
**asymptotically flat as r ******** ****∞**
**map within horizon as r ******** 0 to an                                                                       **
-     **inner asymptotically flat region**
- “Puncture” method:
**conformal factor *****ψ***** ****contains 1/r terms at the centers of the BHs**
**factor these out and treat them analytically ******** no singular terms for initial data or evolution**
**works for one or more BHs**

## Grazing collision of BHs (Alcubierre, et al.)

- Puncture evolution (*β**i* = 0)
*M*1 = 1.5 *M*2
- BHs have general spins,
- 	momenta
- Evolve for t = 35 *M*
- Formation of common apparent
- 	horizon
- Extract *l* = *m* = 2 GW mode

## Evolving BHs...how to represent the BHs

- Excision: cut out a region containing singularity
***Event horizon*****: boundary between the events which emit light rays to ****∞ and those which do not                                                                    ******** requires knowledge of entire spacetime**
***Apparent horizon*****: useful for evolutions**
**outermost 2-surface in spatial slice whose **
**    ****outgoing null geodesics have zero expansion**
**is always located within an event horizon **
**    ******** ****it is safe to excise w/in AH**
**Need to set boundary conditions at excision**
**When an excised BH is moved,**
**     ****zones that were previously within**
**	****excised region are now on the**
**	****numerical grid**
**need methods to populate these points**
**	****with data **

## Moving a BH with excision... (Shoemaker, et al.)

- Move excised BH using coordinate transformation of stationary BH enabled by shift *β*i
- Cubical, spherical excision regions
- Stable for time t ~ 100 *M*

## Evolving BHs...choice of lapse and shift

- Lapse function: governs slicing of spacetime
***α***** = 1 ******** slices crash into singularity**
**“****singularity avoiding” slices wrap up**
- 	**around the singularity, but...**
**stretching of slices ******** large gradients**
- Shift vector: governs movement of
- 	spatial coords as spacetime evolves
***β*****i**** = 0 ******** grid points fall into BH region**
**new conditions for shift vector**
- 	**allow shifts that counter slice stretching**
**	****and enable longer evolutions**
- Application of singularity avoiding slices and
- 	new shift conditions  evolve punctures for > 1000 *M* (single BH) and > few x 100 *M* (binary BHs) ***(Alcubierre, et al.)***

## Long term evolution of single BHs... (Yo, et al.)

- Evolve a single BH, with and without rotation
- Further modification of conformal ADM formulation
- use new lapse and shift from Alcubierre, et al.
- Cubical, spherical excision regions
- Stable for times t > 1000 *M*,* *for BHs with *J/M* ≤ 0.9 *M*

## BH mergers...computational challenges

- Strong, dynamical gravitational fields
- Multiple scales:
*M*1 ≠ *M*2
- λGW ~ 10 – 100 Lsource
- need large enough grid to
- 	extract waves, and for
- 	outer BCs
*use of adaptive mesh refinement (AMR)*
- 	*(This all needs to be done in a parallel computing environment...)*

***Paramesh (Macneice, et al.)***

## Evolving GWs with mesh refinement...

***Evolving GWs with mesh refinement******...***

- Evolve source-free GW across grid
- Paramesh for mesh refinement
- With AMR: track the wave      
***(New, et al.)***
- With fixed mesh refinement:
**cross mesh refinement boundaries**
**higher order interpolation to remove spurious reflections at interface boundaries *****(Choi, et al.)***

##  Lazarus: the first astrophysical BH merger waveforms . . .	(Baker, Campanelli, Lousto, Takahashi)

**Coalescence of 2 equal mass nonrotating BHs **
**Start simulation near last stable circular orbit**
**Use 3-D Einstein solver (Cactus) to evolve merger until BH perturbation theory becomes applicable**
**Continue evolving numerically using BH perturbation theory through ringdown ******** final Kerr BH**
**~ 3% of total energy emitted as GW**
**~12% of total J emitted as GW**
**final Kerr BH has spin ~ 0.7S****max**
**Time measured in terms of mass:**
****** ****time in *****M***** = 5 x 10****-6**** sec (M/M********)**
*graphics courtesy of J. Baker (GSFC)*

## BH binary merger is expected to be phenomenologically rich....

**Effects of spin: spin-spin and spin-orbit couplings, spin flips...**
**Mergers of spinning BHs can cause dramatic changes in orientation of BH’s spin axis ******** sudden flip in direction of associated jet**
**Can identify the winged or X-type radio sources with galaxies in which a merger   has occurred (Merritt & Ekers)**
**Possible ejection of final BH for **
**	****M****1**** ****≠ M****2****  **
**Tests of General Relativity in the dynamical, strongly nonlinear regime...**
***Significant challenges remain... ***
***Numerical relativity has ***
***	******made real progress in modeling binary BH mergers.....***
***	******	******	******	************ ******stay tuned!***