Binary Black Hole Simulations

Home , Binary black hole, Numerical relativity

Harald Pfeiffer

California Institute of Technology

Mike Boyle, Duncan Brown, Lee Lindblom, Geoffrey Lovelace, Larry Kidder, Mark Scheel, Saul Teukolsky

APS April Meeting, Jacksonville, Apr 17, 2007

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 1 / 1 Gravitational wave detectors LISA (201x)

LIGO (2 sites)

VIRGO GEO 600

Among prime targets: Binary black hole systems

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 2 / 1 Stages of binary black hole evolution

Knowledge of waveform allows to

I Enhance detector sensitivity

I Test general relativity

I Extract information about source (→ astrophysics)

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 3 / 1 Tasks for Numerical relativity: – simulate “late” inspiral and merger. – determine what “late” means.

Tools for computing the waveform

2 Merger Inspiral – v c: perturbative expansion in v/c 1 (post-Newtonian expansion) Inspiral Ringdown – v/c large: Numerical relativity 0 Merger

– Numerical relativity -1 Ringdown – BH perturbation theory -300 -200 -100 0 100 t/m – Numerical relativity

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 4 / 1 Tools for computing the waveform

2 Merger Inspiral – v c: perturbative expansion in v/c 1 (post-Newtonian expansion) Inspiral Ringdown – v/c large: Numerical relativity 0 Merger

– Numerical relativity -1 Ringdown – BH perturbation theory -300 -200 -100 0 100 t/m – Numerical relativity

Tasks for Numerical relativity: – simulate “late” inspiral and merger. – determine what “late” means.

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 4 / 1 Since late 1990s: Groundwork and foundations

I New evolution systems

I AMR and spectral infrastructures

I Initial data

I Boundary conditions, gauge conditions

2005: The last pieces! a) Pretorius – constraint damping for generalized harmonic b) Goddard and Brownsville – Gauge conditions for moving punctures

Since 2005: The Golden Age of Numerical Relativity

History

1964 Hahn & Lindquist: Collisions of Wormholes 1970’s Smarr & Eppley: Head on collisions 1994-99 NSF Binary black hole grand challange

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 5 / 1 2005: The last pieces! a) Pretorius – constraint damping for generalized harmonic b) Goddard and Brownsville – Gauge conditions for moving punctures

Since 2005: The Golden Age of Numerical Relativity

History

1964 Hahn & Lindquist: Collisions of Wormholes 1970’s Smarr & Eppley: Head on collisions 1994-99 NSF Binary black hole grand challange

Since late 1990s: Groundwork and foundations

I New evolution systems

I AMR and spectral infrastructures

I Initial data

I Boundary conditions, gauge conditions

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 5 / 1 Since 2005: The Golden Age of Numerical Relativity

History

1964 Hahn & Lindquist: Collisions of Wormholes 1970’s Smarr & Eppley: Head on collisions 1994-99 NSF Binary black hole grand challange

Since late 1990s: Groundwork and foundations

I New evolution systems

I AMR and spectral infrastructures

I Initial data

I Boundary conditions, gauge conditions

2005: The last pieces! a) Pretorius – constraint damping for generalized harmonic b) Goddard and Brownsville – Gauge conditions for moving punctures

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 5 / 1 History

1964 Hahn & Lindquist: Collisions of Wormholes 1970’s Smarr & Eppley: Head on collisions 1994-99 NSF Binary black hole grand challange

Since late 1990s: Groundwork and foundations

I New evolution systems

I AMR and spectral infrastructures

I Initial data

I Boundary conditions, gauge conditions

2005: The last pieces! a) Pretorius – constraint damping for generalized harmonic b) Goddard and Brownsville – Gauge conditions for moving punctures

Since 2005: The Golden Age of Numerical Relativity

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 5 / 1 Computational approaches in the Golden Age Finite difference AMR

I Albert-Einstein Institut (Germany), Goddard, Jena (Germany), LSU, PSU, Gravitational wave ﬂux small Princeton, Rochester −5 I Impressive short inspirals with I E˙ /E ∼ 10 mergers I E˙ drives inspiral I Accurate long inspirals difﬁcult High accuracy required Multi-domain spectral methods I Absolute phase error δφ 1 I Cornell/Caltech Solutions are smooth I Impressive long inspiral simulations

I Merger difﬁcult

Overview

Problem characteristics Multiple length scales

I Size of BH’s ∼ 1

I Separation ∼ 10

I Wavelength λ ∼ 100

I Wave extraction at several λ

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Computational approaches in the Golden Age Finite difference AMR

I Albert-Einstein Institut (Germany), Goddard, Jena (Germany), LSU, PSU, Princeton, Rochester

I Impressive short inspirals with mergers

I Accurate long inspirals difﬁcult High accuracy required Multi-domain spectral methods I Absolute phase error δφ 1 I Cornell/Caltech Solutions are smooth I Impressive long inspiral simulations

I Merger difﬁcult

Overview

Problem characteristics Multiple length scales

I Size of BH’s ∼ 1

I Separation ∼ 10

I Wavelength λ ∼ 100

I Wave extraction at several λ Gravitational wave ﬂux small −5 I E˙ /E ∼ 10

I E˙ drives inspiral

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Computational approaches in the Golden Age Finite difference AMR

I Albert-Einstein Institut (Germany), Goddard, Jena (Germany), LSU, PSU, Princeton, Rochester

I Impressive short inspirals with mergers

I Accurate long inspirals difﬁcult Multi-domain spectral methods

I Cornell/Caltech Solutions are smooth I Impressive long inspiral simulations

I Merger difﬁcult

Overview

Problem characteristics Multiple length scales

I Size of BH’s ∼ 1

I Separation ∼ 10

I Wavelength λ ∼ 100

I Wave extraction at several λ Gravitational wave ﬂux small −5 I E˙ /E ∼ 10

I E˙ drives inspiral High accuracy required

I Absolute phase error δφ 1

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Computational approaches in the Golden Age Finite difference AMR

I Albert-Einstein Institut (Germany), Goddard, Jena (Germany), LSU, PSU, Princeton, Rochester

I Impressive short inspirals with mergers

I Accurate long inspirals difﬁcult Multi-domain spectral methods

I Cornell/Caltech

I Impressive long inspiral simulations

I Merger difﬁcult

Overview

Problem characteristics Multiple length scales

I Size of BH’s ∼ 1

I Separation ∼ 10

I Wavelength λ ∼ 100

I Wave extraction at several λ Gravitational wave ﬂux small −5 I E˙ /E ∼ 10

I E˙ drives inspiral High accuracy required

I Absolute phase error δφ 1 Solutions are smooth

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Finite difference AMR

I Albert-Einstein Institut (Germany), Goddard, Jena (Germany), LSU, PSU, Princeton, Rochester

I Impressive short inspirals with mergers

I Accurate long inspirals difﬁcult Multi-domain spectral methods

I Cornell/Caltech

I Impressive long inspiral simulations

I Merger difﬁcult

Overview

Problem characteristics Computational approaches Multiple length scales in the Golden Age

I Size of BH’s ∼ 1

I Separation ∼ 10

I Wavelength λ ∼ 100

I Wave extraction at several λ Gravitational wave ﬂux small −5 I E˙ /E ∼ 10

I E˙ drives inspiral High accuracy required

I Absolute phase error δφ 1 Solutions are smooth

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Multi-domain spectral methods

I Cornell/Caltech

I Impressive long inspiral simulations

I Merger difﬁcult

Overview

Problem characteristics Computational approaches Multiple length scales in the Golden Age

I Size of BH’s ∼ 1 Finite difference AMR

I Separation ∼ 10 I Albert-Einstein Institut I Wavelength λ ∼ 100 (Germany), Goddard, Jena I Wave extraction at several λ (Germany), LSU, PSU, Gravitational wave ﬂux small Princeton, Rochester −5 I Impressive short inspirals with I E˙ /E ∼ 10 mergers I E˙ drives inspiral I Accurate long inspirals difﬁcult High accuracy required

I Absolute phase error δφ 1 Solutions are smooth

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Overview

Problem characteristics Computational approaches Multiple length scales in the Golden Age

I Size of BH’s ∼ 1 Finite difference AMR

I Merger difﬁcult

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Split space-time into space and time t+dt t Evolution equations ∂t gij = ... cf. Maxwell equations ∂t Kij = ... ~ ~ ∂t E = ∇ × B ~ ~ Constraints ∂t B = −∇ × E R[g ] + K 2 − K K ij = 0 ij ij ∇ · E~ = 0 ∇ K ij − gij K = 0 j ∇ · B~ = 0

Solving Einstein’s equations – basic idea

Task: Find space-time metric gab such that Rab[gab] = 0

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 7 / 1 Evolution equations ∂t gij = ... cf. Maxwell equations ∂t Kij = ... ~ ~ ∂t E = ∇ × B ~ ~ Constraints ∂t B = −∇ × E R[g ] + K 2 − K K ij = 0 ij ij ∇ · E~ = 0 ∇ K ij − gij K = 0 j ∇ · B~ = 0

Solving Einstein’s equations – basic idea

Task: Find space-time metric gab such that Rab[gab] = 0

Split space-time into space and time t+dt t

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 7 / 1 cf. Maxwell equations

~ ~ ∂t E = ∇ × B ~ ~ Constraints ∂t B = −∇ × E R[g ] + K 2 − K K ij = 0 ij ij ∇ · E~ = 0 ∇ K ij − gij K = 0 j ∇ · B~ = 0

Solving Einstein’s equations – basic idea

Task: Find space-time metric gab such that Rab[gab] = 0

Split space-time into space and time t+dt t Evolution equations

∂t gij = ...

∂t Kij = ...

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 7 / 1 cf. Maxwell equations

~ ~ ∂t E = ∇ × B ~ ~ ∂t B = −∇ × E ∇ · E~ = 0 ∇ · B~ = 0

Solving Einstein’s equations – basic idea

Task: Find space-time metric gab such that Rab[gab] = 0

Split space-time into space and time t+dt t Evolution equations

∂t gij = ...

∂t Kij = ...

Constraints 2 ij R[gij ] + K − Kij K = 0 ij ij ∇j K − g K = 0

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 7 / 1 Solving Einstein’s equations – basic idea

Task: Find space-time metric gab such that Rab[gab] = 0

Split space-time into space and time t+dt t Evolution equations ∂t gij = ... cf. Maxwell equations ∂t Kij = ... ~ ~ ∂t E = ∇ × B ~ ~ Constraints ∂t B = −∇ × E R[g ] + K 2 − K K ij = 0 ij ij ∇ · E~ = 0 ∇ K ij − gij K = 0 j ∇ · B~ = 0

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 7 / 1 a Harmonic coordinates x = 0:

gab = lower order terms.

b Generalized harmonic coordinates gabx ≡ Ha (Friedrich 1985, Pretorius 2005.)

b Constraint Ca ≡ Ha − gabx = 0. Constraint damping (Gundlach, et al., Pretorius, 2005) 1 1 0 = − g + ∇ C + γ t C − g tcC + l. o. 2 ab (a b) (a b) 2 ab c

∂t Ca ∼ −γCa.

Evolution equations

Einstein’s equations: 1 0 = R [g ] = − g + ∇ Γ +lower order terms Γ = −g x b. ab ab 2 ab (a b) a ab

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 8 / 1 b Generalized harmonic coordinates gabx ≡ Ha (Friedrich 1985, Pretorius 2005.)

b Constraint Ca ≡ Ha − gabx = 0. Constraint damping (Gundlach, et al., Pretorius, 2005) 1 1 0 = − g + ∇ C + γ t C − g tcC + l. o. 2 ab (a b) (a b) 2 ab c

∂t Ca ∼ −γCa.

Evolution equations

Einstein’s equations: 1 0 = R [g ] = − g + ∇ Γ +lower order terms Γ = −g x b. ab ab 2 ab (a b) a ab

a Harmonic coordinates x = 0:

gab = lower order terms.

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 8 / 1 b Constraint Ca ≡ Ha − gabx = 0. Constraint damping (Gundlach, et al., Pretorius, 2005) 1 1 0 = − g + ∇ C + γ t C − g tcC + l. o. 2 ab (a b) (a b) 2 ab c

∂t Ca ∼ −γCa.

Evolution equations

Einstein’s equations: 1 0 = R [g ] = − g + ∇ Γ +lower order terms Γ = −g x b. ab ab 2 ab (a b) a ab

a Harmonic coordinates x = 0:

gab = lower order terms.

b Generalized harmonic coordinates gabx ≡ Ha (Friedrich 1985, Pretorius 2005.)

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 8 / 1 Constraint damping (Gundlach, et al., Pretorius, 2005) 1 1 0 = − g + ∇ C + γ t C − g tcC + l. o. 2 ab (a b) (a b) 2 ab c

∂t Ca ∼ −γCa.

Evolution equations

Einstein’s equations: 1 0 = R [g ] = − g + ∇ Γ +lower order terms Γ = −g x b. ab ab 2 ab (a b) a ab

a Harmonic coordinates x = 0:

gab = lower order terms.

b Generalized harmonic coordinates gabx ≡ Ha (Friedrich 1985, Pretorius 2005.)

b Constraint Ca ≡ Ha − gabx = 0.

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 8 / 1 Evolution equations

Einstein’s equations: 1 0 = R [g ] = − g + ∇ Γ +lower order terms Γ = −g x b. ab ab 2 ab (a b) a ab

a Harmonic coordinates x = 0:

gab = lower order terms.

b Generalized harmonic coordinates gabx ≡ Ha (Friedrich 1985, Pretorius 2005.)

b Constraint Ca ≡ Ha − gabx = 0. Constraint damping (Gundlach, et al., Pretorius, 2005) 1 1 0 = − g + ∇ C + γ t C − g tcC + l. o. 2 ab (a b) (a b) 2 ab c

∂t Ca ∼ −γCa.

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 8 / 1 Derivatives known analytically (N) N−1 du (x) X dφk (x) = u˜ . dx k dx k=0

Evolve u(xi ) by method of lines h k i ∂t u(xi ) = F − A(u) ∂k u . x=xi

Spectral Evolution code

Rewrite as ﬁrst order symmetric hyperbolic system (Lindblom et al. 2005)

k ∂t u + A(u) ∂k u = F(u).

Approximate solution by truncated series N−1 (N) X u(x, t) ≈ u (x, t) ≡ u˜k (t)Φk (x), k=0 with associated collocation points xi .

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 9 / 1 Evolve u(xi ) by method of lines h k i ∂t u(xi ) = F − A(u) ∂k u . x=xi

Spectral Evolution code

Rewrite as ﬁrst order symmetric hyperbolic system (Lindblom et al. 2005)

k ∂t u + A(u) ∂k u = F(u).

Approximate solution by truncated series N−1 (N) X u(x, t) ≈ u (x, t) ≡ u˜k (t)Φk (x), k=0 with associated collocation points xi . Derivatives known analytically (N) N−1 du (x) X dφk (x) = u˜ . dx k dx k=0

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 9 / 1 Spectral Evolution code

Rewrite as ﬁrst order symmetric hyperbolic system (Lindblom et al. 2005)

k ∂t u + A(u) ∂k u = F(u).

Evolve u(xi ) by method of lines h k i ∂t u(xi ) = F − A(u) ∂k u . x=xi

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 9 / 1 t Horizon

Outside Horizon

x All modes propagate inside light cone

I Excise interior of BH

I No boundary condition needed

Black hole singularity excision

Boundary conditions

I Find characteristic ﬁelds & speeds

I Impose BCs on incoming ﬁelds only

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 10 / 1 Black hole singularity excision

t Horizon

Boundary conditions Outside I Find characteristic ﬁelds & speeds Horizon I Impose BCs on incoming ﬁelds only

x All modes propagate inside light cone

I Excise interior of BH

I No boundary condition needed

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 10 / 1 Black hole singularity excision

t Horizon

Boundary conditions Outside I Find characteristic ﬁelds & speeds Horizon I Impose BCs on incoming ﬁelds only

x All modes propagate inside light cone

I Excise interior of BH

I No boundary condition needed

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 10 / 1 Outer shells have ﬁxed angular resolution. Cost increases linearly with radius of outer boundary.

Domain-decomposition

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 11 / 1 Outer shells have ﬁxed angular resolution. Cost increases linearly with radius of outer boundary.

Domain-decomposition

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 11 / 1 Outer shells have ﬁxed angular resolution. Cost increases linearly with radius of outer boundary.

Domain-decomposition

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 11 / 1 Outer shells have ﬁxed angular resolution. Cost increases linearly with radius of outer boundary.

Domain-decomposition

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 11 / 1 Domain-decomposition

Outer shells have ﬁxed angular resolution. Cost increases linearly with radius of outer boundary.

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 11 / 1 t

Horizon

Outside Horizon

Map between “moving” and “inertial” coordinates:

~xinertial = a(t)R(t)~xmoving

R(t) rotation, a(t) radial scaling.

Moving black holes – Dual frame method

Scheel et al., 2006 t

Horizon

Outside Horizon

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 12 / 1 t

Horizon

Outside Horizon

Map between “moving” and “inertial” coordinates:

~xinertial = a(t)R(t)~xmoving

R(t) rotation, a(t) radial scaling.

Moving black holes – Dual frame method

Scheel et al., 2006 t

Horizon

Outside Horizon

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 12 / 1 t

Horizon

Outside Horizon

Map between “moving” and “inertial” coordinates:

~xinertial = a(t)R(t)~xmoving

R(t) rotation, a(t) radial scaling.

Moving black holes – Dual frame method

Scheel et al., 2006 t

Horizon

Outside Horizon

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 12 / 1 Map between “moving” and “inertial” coordinates:

~xinertial = a(t)R(t)~xmoving

R(t) rotation, a(t) radial scaling.

Moving black holes – Dual frame method

Scheel et al., 2006 t t

Horizon Horizon

Outside Outside Horizon Horizon

x x

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 12 / 1 Map between “moving” and “inertial” coordinates:

~xinertial = a(t)R(t)~xmoving

R(t) rotation, a(t) radial scaling.

Moving black holes – Dual frame method

Scheel et al., 2006 t t

Horizon Horizon

Outside Outside Horizon Horizon

x x

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 12 / 1 Moving black holes – Dual frame method

Scheel et al., 2006 t t

Horizon Horizon

Outside Outside Horizon Horizon

x x

Map between “moving” and “inertial” coordinates:

~xinertial = a(t)R(t)~xmoving

R(t) rotation, a(t) radial scaling.

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 12 / 1 Dynamic feedback control

δϕ C y Example: Control of φ C Measure BH location Cx (t), Cy (t) x

Update φ periodically s.t. Cy → 0: φ τ -Control in action ( =5M1) 3 d φ ... 3 3 (t) = φ , k∆T ≤t <(k + 1)∆T d φ/dt dt 3 k 2e-05

... » 1 C 3 d „ C « 3 d 2 „ C «– 0 φ = − y + y + y k τ 3 τ 2 τ 2 Cx dt Cx dt Cx t=k-2e-05∆T

-4e-05 Cy/Cx

0 10 20 30 40 time

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 13 / 1 I Consider characteristics of constraint evolution system

I set incoming constraint-modes to zero ⇒ BCs on some fundamental ﬁelds

Must be transparent to gravitational waves.

I Consider Newman-Penrose scalars – Ψ0 ≡ 0 ⇒ BCs on some fundamental ﬁelds -3 0 Lindblom et al., 2006 10 10 -1 10 -2 Should keep coordinates well-behaved 10 -6 -3 10 – Sommerfeld BC on gauge modes 10 〈 -4Ψ 〉 Rinne et al. 2007 10R 4 -5 -9 10 10 -6 10 -7 10 -8 -12 10 10 0 100 200 300 -9 10 0 200t/M 400 600 time

Outer boundary conditions

Must prevent inﬂux of constraint violations

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 14 / 1 Must be transparent to gravitational waves.

Outer boundary conditions

Must prevent inﬂux of constraint violations

I Consider characteristics of constraint evolution system

I set incoming constraint-modes to zero ⇒ BCs on some fundamental ﬁelds

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 14 / 1 0 10 -1 10 -2 Should keep coordinates well-behaved 10 -3 – Sommerfeld BC on gauge modes 10 -4 Rinne et al. 2007 10 -5 10 -6 10 -7 10 -8 10 -9 10 0 200 400 600 time

Outer boundary conditions

Must prevent inﬂux of constraint violations

I Consider characteristics of constraint evolution system

I set incoming constraint-modes to zero ⇒ BCs on some fundamental ﬁelds

Must be transparent to gravitational waves.

I Consider Newman-Penrose scalars – Ψ0 ≡ 0 ⇒ BCs on some fundamental ﬁelds -3 Lindblom et al., 2006 10

-6 10 〈 Ψ 〉 R 4 -9 10

-12 10 0 100 200 300 t/M

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 14 / 1 -3 0 10 10 -1 10 -2 10 -6 -3 10 10 〈 -4Ψ 〉 10R 4 -5 -9 10 10 -6 10 -7 10 -8 -12 10 10 0 100 200 300 -9 10 0 200t/M 400 600 time

Outer boundary conditions

Must prevent inﬂux of constraint violations

I Consider characteristics of constraint evolution system

I set incoming constraint-modes to zero ⇒ BCs on some fundamental ﬁelds

Must be transparent to gravitational waves.

I Consider Newman-Penrose scalars – Ψ0 ≡ 0 ⇒ BCs on some fundamental ﬁelds 0 Lindblom et al., 2006 10 -1 10 -2 Should keep coordinates well-behaved 10 -3 – Sommerfeld BC on gauge modes 10 -4 Rinne et al. 2007 10 -5 10 -6 10 -7 10 -8 10 -9 10 0 200 400 600 time

Outer boundary conditions

Must prevent inﬂux of constraint violations

I Consider characteristics of constraint evolution system

I set incoming constraint-modes to zero ⇒ BCs on some fundamental ﬁelds

Must be transparent to gravitational waves.

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 14 / 1 Allow for nonzero initial radial velocities of BHs. Tune radial velocities (and Ω0) to reduce orbital eccentricity.

17.6 0.001 ds/dt Proper separation s/MADM Quasi-circular Quasi-circular (e~0.01) 1st iteration -4 1st iteration (e~7x10 ) 0 2nd iteration -5 17.2 2nd iteration (e~5x10 )

-0.001

16.8 -0.002

16.4 -0.003

-0.004 0 500 1000 0 500 1000 t/M ADM t/MADM

Improve initial data through evolutions HP et al., 2007 Circular orbits ⇒ eccentric inspiral.

17.6 0.001 ds/dt Proper separation s/MADM Quasi-circular Quasi-circular 0 17.2

-0.001

16.8 -0.002

16.4 -0.003

-0.004 0 500 1000 0 500 1000 t/M ADM t/MADM

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 15 / 1 17.6 0.001 ds/dt Proper separation s/MADM Quasi-circular Quasi-circular (e~0.01) 1st iteration -4 1st iteration (e~7x10 ) 0 2nd iteration -5 17.2 2nd iteration (e~5x10 )

-0.001

16.8 -0.002

16.4 -0.003

-0.004 0 500 1000 0 500 1000 t/M ADM t/MADM

Improve initial data through evolutions HP et al., 2007 Circular orbits ⇒ eccentric inspiral. Allow for nonzero initial radial velocities of BHs. Tune radial velocities (and Ω0) to reduce orbital eccentricity.

17.6 0.001 ds/dt Proper separation s/MADM Quasi-circular Quasi-circular 1st iteration -4 1st iteration (e~7x10 ) 0 17.2

-0.001

16.8 -0.002

16.4 -0.003

-0.004 0 500 1000 0 500 1000 t/M ADM t/MADM

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 15 / 1 17.6 0.001 ds/dt Proper separation s/MADM Quasi-circular Quasi-circular 1st iteration -4 1st iteration (e~7x10 ) 0 17.2

-0.001

16.8 -0.002

16.4 -0.003

-0.004 0 500 1000 0 500 1000 t/M ADM t/MADM

17.6 0.001 ds/dt Proper separation s/MADM Quasi-circular Quasi-circular (e~0.01) 1st iteration -4 1st iteration (e~7x10 ) 0 2nd iteration -5 17.2 2nd iteration (e~5x10 )

-0.001

16.8 -0.002

16.4 -0.003

-0.004 0 500 1000 0 500 1000 t/M ADM t/MADM

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 15 / 1 Eccentricity reduced

y 0

-5

-10

-10 -5 0 5 10 x

Orbital trajectory

Quasi-circular

y 0

-5

-10

-10 -5 0 5 10 x

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 16 / 1 Orbital trajectory

Quasi-circular Eccentricity reduced

10 10

5 5

y 0 y 0

-5 -5

-10 -10

-10 -5 0 5 10 -10 -5 0 5 10 x x

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 16 / 1 −7 Mirr initially increases by 6 · 10 – “Junk” radiation falling into BH −7 |δMirr| < 10 in the next 3.5 orbits – Limit on tidal heating

Irreducible Mass 1.001 Mirr(t) / Mirr(t=0)

1.000

3 N~43 0.999 3 N~48 3 N~53 3 N~57 3 0.998 N~62 3 N~67

0 1000 2000 3000 4000 time

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 17 / 1 −7 Mirr initially increases by 6 · 10 – “Junk” radiation falling into BH −7 |δMirr| < 10 in the next 3.5 orbits – Limit on tidal heating

Irreducible Mass

M (t) / M (t=0) 1.00002 irr irr

1.00000

3 0.99998 N~43 3 N~48 3 N~53 0.99996 3 N~57 3 N~62 3 0.99994 N~67

0 1000 2000 3000 4000 time

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 17 / 1 Irreducible Mass -6 1×10 M (t) / M (t=0) M (t)/M (t=0) - 1 1.00002 irr irr irr irr 3 -7 N~57 8×10 1.00000 -7 × 3 6 10 N~67 3 0.99998 N~43 -7 3 4×10 N~48 3 3 N~62 N~53 0.99996 3 -7 N~57 2×10 3 N~62 3 0.99994 N~67 0

0 1000 2000 3000 4000 1 10 100 1000 time time + 1

−7 Mirr initially increases by 6 · 10 – “Junk” radiation falling into BH −7 |δMirr| < 10 in the next 3.5 orbits – Limit on tidal heating

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 17 / 1 Movie

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 18 / 1 Waveforms

0.004

0.002

-0.002

-0.004 0 1000 2000 3000 4000 t/MADM

-19 2×10

-19 1×10

-19 -1×10

-19 -2×10

0 0.1 0.2 0.3 0.4 t [sec]

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 19 / 1 Gravitational wave phase

-50

3 R=100M, N~62 φ 3 -100 R=150M, N~62 3 R=200M, N~62 3 Extrapolated, N~62 3 -150 N~43 3 N~48 3 N~53 3 -200 N~58

0 1000 2000 3000 4000 t / (m1+m2)

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 20 / 1 Truncation error ∼ 0.03rad Extraction radius ∼ 0.1rad Outer boundary ∼ 0.07rad

2PN deviates throughout run ≥ 3PN=NR for t<2000M Validation of NR PN 6= NR at late times

Comparison with post-Newtonian (GW-phase)

0.6

0.4 ∆φ 0.2

-0.2

0 1000 2000 3000 4000

t/(M1+M2) Numerical error budget

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 21 / 1 Extraction radius ∼ 0.1rad Outer boundary ∼ 0.07rad

2PN deviates throughout run ≥ 3PN=NR for t<2000M Validation of NR PN 6= NR at late times

Comparison with post-Newtonian (GW-phase)

0.6 φ φ N1 - N5 0.4 ∆φ φ φ N3 - N5 0.2

0 φ φ N4 - N5 φ - φ -0.2 N2 N5

0 1000 2000 3000 4000

t/(M1+M2) Numerical error budget Truncation error ∼ 0.03rad

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 21 / 1 Outer boundary ∼ 0.07rad

2PN deviates throughout run ≥ 3PN=NR for t<2000M Validation of NR PN 6= NR at late times

Comparison with post-Newtonian (GW-phase)

0.6 R=100M - R=infty R=150M - R=infty R=200M - R=infty 0.4 ∆φ 0.2

-0.2

0 1000 2000 3000 4000

t/(M1+M2) Numerical error budget Truncation error ∼ 0.03rad Extraction radius ∼ 0.1rad

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 21 / 1 2PN deviates throughout run ≥ 3PN=NR for t<2000M Validation of NR PN 6= NR at late times

Comparison with post-Newtonian (GW-phase)

0.6 Rbdry=200M - Rbdry=400M

Rbdry=400M - Rbdry=700M 0.4 ∆φ 0.2

-0.2

0 1000 2000 3000 4000

t/(M1+M2) Numerical error budget Truncation error ∼ 0.03rad Extraction radius ∼ 0.1rad Outer boundary ∼ 0.07rad

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 21 / 1 2PN deviates throughout run ≥ 3PN=NR for t<2000M Validation of NR PN 6= NR at late times

Comparison with post-Newtonian (GW-phase)

0.6

0.4 ∆φ 0.2

-0.2

0 1000 2000 3000 4000

t/(M1+M2) Numerical error budget Truncation error ∼ 0.03rad Extraction radius ∼ 0.1rad Outer boundary ∼ 0.07rad

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 21 / 1 ≥ 3PN=NR for t<2000M Validation of NR PN 6= NR at late times

Comparison with post-Newtonian (GW-phase)

0.6 2PN

0.4 ∆φ 0.2

-0.2

0 1000 2000 3000 4000

t/(M1+M2) Numerical error budget Truncation error ∼ 0.03rad Extraction radius ∼ 0.1rad Outer boundary ∼ 0.07rad

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 21 / 1 2PN deviates throughout run ≥ 3PN=NR for t<2000M Validation of NR PN 6= NR at late times

Comparison with post-Newtonian (GW-phase)

0.6 2PN

0.4 ∆φ 0.2

0 2.5PN

-0.2

0 1000 2000 3000 4000

t/(M1+M2) Numerical error budget Truncation error ∼ 0.03rad Extraction radius ∼ 0.1rad Outer boundary ∼ 0.07rad

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 21 / 1 2PN deviates throughout run ≥ 3PN=NR for t<2000M Validation of NR PN 6= NR at late times

Comparison with post-Newtonian (GW-phase)

0.6 2PN

0.4 ∆φ 3PN 0.2

0 2.5PN

-0.2

0 1000 2000 3000 4000

t/(M1+M2) Numerical error budget Truncation error ∼ 0.03rad Extraction radius ∼ 0.1rad Outer boundary ∼ 0.07rad

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 21 / 1 2PN deviates throughout run PN 6= NR at late times

Comparison with post-Newtonian (GW-phase)

0.6 2PN

0.4 ∆φ 3.5PN 3PN 0.2

0 2.5PN

-0.2

0 1000 2000 3000 4000

t/(M1+M2) Numerical error budget Truncation error ∼ 0.03rad Extraction radius ∼ 0.1rad Outer boundary ∼ 0.07rad

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 21 / 1 2PN deviates throughout run ≥ 3PN=NR for t<2000M Validation of NR PN 6= NR at late times

Comparison with post-Newtonian (GW-phase)

0.02 0.6 2PN 0.01

0.4 0 ∆φ 3.5PN -0.01 3PN 0.2 0 1000 2000

0 2.5PN

-0.2

0 1000 2000 3000 4000

t/(M1+M2) Numerical error budget Truncation error ∼ 0.03rad Extraction radius ∼ 0.1rad Outer boundary ∼ 0.07rad

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 21 / 1 2PN deviates throughout run ≥ 3PN=NR for t<2000M Validation of NR Comparison with post-Newtonian (GW-phase)

3 0.6 2PN 2

0.4 1 ∆φ 0 3.5PN 3PN 0.2 2500 3000 3500

0 2.5PN

-0.2

0 1000 2000 3000 4000

t/(M1+M2) Numerical error budget Truncation error ∼ 0.03rad Extraction radius ∼ 0.1rad Outer boundary ∼ 0.07rad

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 21 / 1 2PN deviates throughout run ≥ 3PN=NR for t<2000M Validation of NR PN 6= NR at late times Testing LIGO detection templates

In collaboration with D. Brown Good overlap!

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 22 / 1 Parameter estimation

Masses in simulation: 10.07Msun. Recovered 10.05Msun.

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 23 / 1 Head-on Merger

Evolve to common horizon, regrid, continue.

Apparent Horizons Event horizon (M. Cohen, in prep.)

Orbiting BHs have reached separation 2.2M (common horizon forms at ∼ 2M.)

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 24 / 1 Summary

Numerical relativity is in its golden age

Spectral methods achieve stunning accuracy

Comparison to PN in progress

I Equal-mass non-spinning BHs – Full agreement NR - PN up to ∼ 15 cycles before merger – Sufﬁcient accuracy to identify deviations at all available PN-orders

Future

I Merger!

I PN comparisons for spinning, non-equal mass binaries – Blanchet: PN converges exceptionally fast for q – For q > 1, Tinspiral ∝ q

Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 25 / 1