BBH initial data - Astrophysics...2 Formalism & Numerics H. Pfeiﬀer, NumRel 2005, Goddard Space...

1

Quasi-equilibrium binary black hole initial data

Harald P. Pfeiffer

California Institute of Technology

Collaborators: Greg Cook, Larry Kidder, Mark Scheel,

Saul Teukolsky, James York

Numerical Relativity 2005, Goddard Space Flight Center, Nov 4, 2005

Outline:

1. Formalism & Numerics

2. Non-uniqueness in conformal thin sandwich

3. Properties of the constructed ID sets

4. Public initial data repository

H. Pfeiffer, NumRel 2005, Goddard Space Flight Center, Nov 4 2005

2

Formalism & Numerics


3Quasi-equilibrium method

Basic idea:Approx. time-independence in corotating frame

Approx. helical Killing vector

(both concepts essentially equivalent,

both useful depending on context)

History:

• Wilson & Matthews 1985: Binary neutron stars

• Gourgoulhon, Grandclement & Bonazzola, 2002a,bBBH ID with inner boundary conditions

basically right, but various deficiencies

• Cook & HP, 2002, 2003, 2004 (especially Cook & Pfeiffer, PRD 70, 104106, 2004)General quasi-equilibrium method with isolated horizon BCs


4

Quasi-equilibrium method (the easy pieces)

• Time-independence in corotating frame

⇒ vanishing time derivatives


4




• Extented conformal thin sandwich formalism

∂tg̃ij = 0 = ∂tK∇̃2ψ −

1

8R̃ψ −

1

12K

2ψ

4+

1

8ÃijÃ

ij=0

∇̃j

ψ6

2NLβij

!−

2

3ψ

6∇̃iK−∇̃j

ψ6

2Nũij

!=0

∇̃2`Nψ´−Nψ

„1

8R̃+

5

12K

2ψ

4+

7

8ÃijÃ

ij«

=

−ψ5(∂t − βk∂k)K


4




• Extented conformal thin sandwich formalism

∂tg̃ij = 0 = ∂tK∇̃2ψ −

1

8R̃ψ −

1

12K

2ψ

4+

1

8ÃijÃ

ij=0

∇̃j

ψ6

2NLβij

!−

2

3ψ

6∇̃iK−∇̃j

ψ6

2Nũij

!=0

∇̃2`Nψ´−Nψ

„1

8R̃+

5

12K

2ψ

4+

7

8ÃijÃ

ij«

=

−ψ5(∂t − βk∂k)K• Boundary conditions at infinity

ψ = 1

βi= (~Ωorbital × ~r)

i

N = 1

• New contribution: inner boundary conditions (next slides)


5Quasi-equilibrium excision boundary conditions

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Σ

kµ

S

• Excise topological spheres S

• Require1. S be apparent horizons2. The AH’s remain stationary in evolution

3. Shear of kµ vanishes (isolated horizon)

⇒ Lkθ = 0 ⇒ AH moves along kµ and MAH initially constant







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Σ

kµ

S





• Rewrite in conformal variables ⇒ BC’s on ψ and βi







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Σ

kµ

S






• General spin possible (→ Greg Cook’s talk)







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Σ

kµ

S






• General spin possible (→ Greg Cook’s talk)

• One still must specify...1. Conformal metric g̃ij2. Shape of excision surfaces S3. Mean curvature K

4. Lapse boundary condition


6Spectral elliptic solver (HP, Kidder, Scheel & Teukolsky, 2003)

Expand solution in basis-functions & solve for expansion-coefficients




Smooth solutions ⇒ exponential convergence

30 45 60 75 90N

10-8

10-6

10-4

10-2

||H||2||M||2|δEADM||δMK||δJz|





• Superior accuracy: Numerical errors � physical effects

30 45 60 75 90N

10-8

10-6

10-4

10-2






• Superior accuracy: Numerical errors � physical effects• Superior efficiency: Large parameter studies

30 45 60 75 90N

10-8

10-6

10-4

10-2


HP, Kidder, Scheel, Teukolsky 2003





• Superior accuracy: Numerical errors � physical effects• Superior efficiency: Large parameter studies• Domain decomposition: Nontrivial topologies & Multiple length-scales

30 45 60 75 90N

10-8

10-6

10-4

10-2


HP, Kidder, Scheel, Teukolsky 2003


7

Non-uniqueness


8Extended conformal thin sandwich equations

0.01 0.1 1A~

0.01

0.1

1

10

100

ADM energy

Standa

rd CT

S

Exten

ded CT

S

g̃ij = δij + Ãhij

∂tg̃ij = ÃḣijK = ∂tK = 0

(perturbed flat space w/o inner b’dries)

HP & York, 2005



0.01 0.1 1A~

0.01

0.1

1

10

100

ADM energy

Standa

rd CT

S

Exten

ded CT

S

0.01 0.1 1A~

0.01

0.1

1

10

100

ADM energy

Standa

rd CT

S

Extented CTS, too!

Exten

ded CT

S




HP & York, 2005



0.01 0.1 1A~

0.01

0.1

1

10

100

ADM energy

Standa

rd CT

S

Exten

ded CT

S

0.01 0.1 1A~

0.01

0.1

1

10

100

ADM energy

Standa

rd CT

S

Extented CTS, too!

Exten

ded CT

S

0.01 0.1 1A~

0.01

0.1

1

10

100

ADM energy

Standa

rd CT

S

Extented CTS, too!

Exten

ded CT

S

apparenthorizon




Apparent horizons exist for small Ã!

HP & York, 2005


9

Properties of QE-ID sets


10Corotating BBH solutions

Arbitrary choices: Conformal flatness, S=sphere. Gauge choices: K = 0, ∂n(Nψ) = 0.

30 45 60 75 90N

10-8

10-6

10-4

10-2


Exponential convergence Lapse positive through horizon


11Sequences of quasi-circular orbits & ISCO

3.6 4 4.4 4.8J/µm

-0.06

-0.05

-0.04

-0.03

-0.02

E b/µ

CO: MS - d(αψ)/dr = 0CO: MS - αψ = 1/2CO: MS - d(αψ)/dr = (αψ)/2r

3.39 3.42

-0.065

-0.06

0.08 0.12 0.16mΩ0

-0.022

-0.02

-0.018

-0.016

-0.014

E b /

m GGB ’02

Cook&Pfeiffer ’04 (3 data points)

2,3 PN (standard)1,2,3 PN (EOB)

Cook ’94

Cook&Pfeiffer ’04 (3 data points)

PN (EOB)PN (standard)


12Towards evolving these ID

• ISCO and other diagnostics very promising

• But, ID only up to AH, whereas evolution codes excise inside AH

• Extrapolate data inward to 0.75rAH

• Constraints violated for r < rAH

0.6 0.8 1 21.4r

10-10

10-8

10-6

10-4

10-2Hamiltonian Constraint

Low resolution "Lev0"

high resolution "Lev5"Extrapolate

• The next slides highlight aspects ofevolution which are relevant to ID


13Evolution with fixed gauge – horizon motion

Same data as in Mark Scheel’s talk – separation 10.

0 1 2 3 4 5t/MAH

-0.001

-0.0005

0

0.0005

0.001

min(rAH) - rinitial

max(rAH) - rinitial

Initially at rest, no transient

N and βi are excellent initial gauge


13Evolution with fixed gauge – horizon motion

Same data as in Mark Scheel’s talk – separation 10.

0 1 2 3 4 5t/MAH

-0.001

-0.0005

0

0.0005

0.001

min(rAH) - rinitial

max(rAH) - rinitial

Initially at rest, no transient

0 5 10 15 20 25t/MAH

-0.04

-0.02

0

0.02

0.04

0.06

min(rAH ) - r

initial

max(r A

H) -

r initial

Horizon crosses excision bdry

On longer time-scales, AH deforms

N and βi are excellent initial gauge


14Apparent horizon mass

0 10 20 30 40 50t / MAH

1

1.00002

1.00004

1.00006

MAH / MAH(t=0)(3 different resolutions)


15Not all is well – Tidal distortions

Tidal distortions not captured correctly with current choices for g̃ij and S— Work in progress —


16

Public ID repository


17Initial data repository

• http://www.tapir.caltech.edu/~harald/PublicID• Equal mass BBHs in corotation• Two choices for Lapse-BC – Eq. (59a) or (59b) from Cook&HP, 2004


http://www.tapir.caltech.edu/~harald/PublicID



Concentrate on Lapse-BC (59a) for uniformity





Concentrate on Lapse-BC (59a) for uniformity

10 20 30 40coordinate separation d

1

235

10

10 20 30 40

0.01

0.02

0.03

0.05

0.1

mΩ

orbits un

til ISCO

(1.5PN)

PretoriusBruegmann et al 2004

Available separations



18Using the public QE-BBH initial data


The web-site contains:

• Data sets, containing gij, Kij, N, βi in Cartesian components

• Library to interpolate the data to any desired point (x, y, z)(as long as it is inside the covered computational domain)

• Example executable and example data-set(Schwarzschild in Kerr-Schild coordinates)



19Summary

• Framework for BBH initial data in a kinematical setting (helical Killing vector)

• Advantages:1. Agreement with PN

2. N > 0, AH initially constant, MAH exceedingly constant

• Tidal distortions not yet captured

• Data sets publicly availablehttp://www.tapir.caltech.edu/~harald/PublicID

1. Compute waveforms!

2. Compare and validate evolution codes on the same initial data



20Contents of a data set

1. The data in several resolutions (Lev2, ... Lev5), each in its own subdirectory

2. The file Convergence listing errors for each resolution:

#....N Nor-Linf Nor-L2 Ham-Linf Ham-L2 Mom-Linf Mom-L232.184 0.2280 0.03185 0.0339 0.00202 0.00552 0.000217

21Interpolation Library – suggestions welcome!

• Library libSpECLibraryID.a (compiled with gcc 3.4.3 on RHE 9)

• Header file PublicID.hpp:#include

void ReadData(const double Omega); // import from disk

void InterpolateData(const vector& x,const vector& y,const vector& z,vector& gxx, ... , vector& gzz,vector& Kxx, ... , vector& Kzz,vector& Betax, ... , vector& Betaz,vector& N);

void ReleaseData(); // free memory

• Test-executable InterpolateExample.cpp:

g++ InterpolateExample.cpp libSpECLibraryID.a -lblas


BBH initial data - Astrophysics...2 Formalism & Numerics H. Pfeiﬀer, NumRel 2005, Goddard Space...

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