Numerical Simulations of Black Hole Spacetimeslindblom/Talks/LANL_5March2008.pdf · 2009-11-06 ·...

Numerical Simulations of Black Hole Spacetimes

Lee Lindblom

Theoretical AstrophysicsCalifornia Institute of Technology

Los Alamos Astrophysics Seminar5 March 2008

Lee Lindblom (Caltech) Numerical Black Hole Simulations LANL – 3/5/2008 1 / 27

Caltech-Cornell Numerical Relativity Collaboration

Group leaders: Lee Lindblom, Mark Scheel, and Harald Pfeifferat Caltech; Saul Teukolsky and Larry Kidder at Cornell.

Kidder Lindblom Pfeiffer Scheel Teukolsky

Caltech group: Michael Boyle, Jeandrew Brink, Luisa Buchman,Tony Chu, Michael Cohen, Lee Lindblom, Keith Matthews,Harald Pfeiffer, Mark Scheel, Bela Szilagyi, Kip Thorne.Cornell group: Matthew Duez, Francois Foucart, Lawrence Kidder,Francois Limousin, Geoffrey Lovelace, Abdul Mroue,Robert Owen, Nick Taylor, Saul Teukolsky.


Motivation: Gravitational Wave AstronomyRecent work in numerical relativity is aimed at providing modelwaveforms for gravitational wave (GW) astronomy (LIGO, etc.).

Binary black hole systems emit large amounts of GW as the holesinspiral and ultimately merge. These are expected to be amongthe strongest sources detectable by LIGO.Numerical waveforms may be useful in detection (to constructbetter data filters), and/or in modeling detected signals.


Motivation: Gravitational Wave AstronomyRecent work in numerical relativity is aimed at providing modelwaveforms for gravitational wave (GW) astronomy (LIGO, etc.).Binary black hole systems emit large amounts of GW as the holesinspiral and ultimately merge. These are expected to be amongthe strongest sources detectable by LIGO.

Numerical waveforms may be useful in detection (to constructbetter data filters), and/or in modeling detected signals.


Motivation: Gravitational Wave AstronomyRecent work in numerical relativity is aimed at providing modelwaveforms for gravitational wave (GW) astronomy (LIGO, etc.).Binary black hole systems emit large amounts of GW as the holesinspiral and ultimately merge. These are expected to be amongthe strongest sources detectable by LIGO.Numerical waveforms may be useful in detection (to constructbetter data filters), and/or in modeling detected signals.

0 0.1 0.2 0.3 0.4 0.5

-2×10-19

-1×10-19

0

1×10-19

2×10-19

t (sec.)

n(t)

h(t)


Motivation: Gravitational Wave AstronomyRecent work in numerical relativity is aimed at providing modelwaveforms for gravitational wave (GW) astronomy (LIGO, etc.).Binary black hole systems emit large amounts of GW as the holesinspiral and ultimately merge. These are expected to be amongthe strongest sources detectable by LIGO.Numerical waveforms may be useful in detection (to constructbetter data filters), and/or in modeling detected signals.

0 0.1 0.2 0.3 0.4 0.5

-2×10-19

-1×10-19

0

1×10-19

2×10-19

t (sec.)

n(t)

h(t)

0 0.1 0.2 0.3 0.4 0.5

-2×10-19

-1×10-19

0

1×10-19

2×10-19

t (sec.)

n(t)

h(t)

100 h(t)


Gravitational Wave Data AnalysisSignals s(t) are detected in the noisy LIGO data by projectingthem onto a template h(λ, t) using a noise-weighted inner product:

ρ(λ) = 2∫ ∞

0

s(f )h ∗(f , λ)

Sh(f )df

[∫ ∞0

h(f , λ)h ∗(f , λ)

Sh(f )df

]−1/2

.

The signal to noise ratio, ρ(λ), is maximized by adjusting thetemplate parameters λ.A detection occurs whenever a signal is present that matches asignal template with ρ(λ) > ρmin.

For LIGO searches ρmin ≈ 8.



ρ(λ) = 2∫ ∞

0

s(f )h ∗(f , λ)

Sh(f )df

[∫ ∞0

h(f , λ)h ∗(f , λ)

Sh(f )df

]−1/2

.

The signal to noise ratio, ρ(λ), is maximized by adjusting thetemplate parameters λ.

A detection occurs whenever a signal is present that matches asignal template with ρ(λ) > ρmin.




ρ(λ) = 2∫ ∞

0

s(f )h ∗(f , λ)

Sh(f )df

[∫ ∞0

h(f , λ)h ∗(f , λ)

Sh(f )df

]−1/2

.

The signal to noise ratio, ρ(λ), is maximized by adjusting thetemplate parameters λ.A detection occurs whenever a signal is present that matches asignal template with ρ(λ) > ρmin.




ρ(λ) = 2∫ ∞

0

s(f )h ∗(f , λ)

Sh(f )df

[∫ ∞0

h(f , λ)h ∗(f , λ)

Sh(f )df

]−1/2

.

The signal to noise ratio, ρ(λ), is maximized by adjusting thetemplate parameters λ.A detection occurs whenever a signal is present that matches asignal template with ρ(λ) > ρmin. For LIGO searches ρmin ≈ 8.

0 0.1 0.2 0.3 0.4 0.5

-2×10-19

-1×10-19

0

1×10-19

2×10-19

t (sec.)

n(t)

h(t)

10 100 1000 1000010-24

10-23

10-22

10-21

f (Hz)

h(f)

S1/2(f)


Why Is Numerical Relativity So Difficult?

Dynamics of binary black hole problem is driven by delicateadjustments to orbit due to emission of gravitational waves.Very big computational problem:

Must evolve ∼ 50 dynamical fields (spacetime metric plus all firstderivatives).Must accurately resolve features on many scales from black holehorizons r ∼ GM/c2 to emitted waves r ∼ 100GM/c2.Many grid points are required & 106 even if points are locatedoptimally.

Most representations of the Einstein equations havemathematically ill-posed initial value problems.Constraint violating instabilities destroy stable numerical solutionsin many well-posed forms of the equations.

Unstable BBH Movie


Recent Progress in Numerical Relativity

Frans Pretorius performs first numerical BBH inspiral, merger andringdown calculations in the spring of 2005 using a “generalizedharmonic” formulation of the Einstein equations. Pretorius InspiralMovie

Groups at NASA GSFC and U. Texas–Brownsville simultaneouslyannounce similar BBH simulations in the fall of 2005 using verydifferent methods (BSSN–puncture).LSU/AEI collaboration obtains similar results in Dec. 2005.Penn State group begins the study of physical properties of BBHorbits in early 2006 by evolving unequal mass binaries andmeasuring the kick velocity using BSSN–puncture methods....


Outline of Remainder of Talk:

Technical issues:

Constraint Damping.Pseudo-Spectral Methods.Feedback Control Systems.

Science results:

Compare numerical waveforms with post-Newtonianapproximations.


Gauge and Constraints in ElectromagnetismThe usual representation of the vacuum Maxwell equations splitinto evolution equations and constraints:

∂t~E = ~∇× ~B, ∇ · ~E = 0,

∂t~B = −~∇× ~E , ∇ · ~B = 0.

These equations are often written in the more compact4-dimensional notation: ∇aFab = 0 and ∇[aFbc] = 0,where Fab has components ~E and ~B.

Maxwell’s equations are often re-expressed in terms of a vectorpotential Fab = ∇aAb −∇bAa :

∇a∇aAb −∇b∇aAa = 0.This form of Maxwell’s equations is manifestly hyperbolic as longas the gauge is chosen correctly, e.g., let ∇aAa = H(x , t), giving:

∇a∇aAb ≡(−∂2

t + ∂2x + ∂2

y + ∂2z

)Ab = ∇bH.



∂t~E = ~∇× ~B, ∇ · ~E = 0,

∂t~B = −~∇× ~E , ∇ · ~B = 0.

These equations are often written in the more compact4-dimensional notation: ∇aFab = 0 and ∇[aFbc] = 0,where Fab has components ~E and ~B.Maxwell’s equations are often re-expressed in terms of a vectorpotential Fab = ∇aAb −∇bAa :

∇a∇aAb −∇b∇aAa = 0.

This form of Maxwell’s equations is manifestly hyperbolic as longas the gauge is chosen correctly, e.g., let ∇aAa = H(x , t), giving:

∇a∇aAb ≡(−∂2

t + ∂2x + ∂2

y + ∂2z

)Ab = ∇bH.



∂t~E = ~∇× ~B, ∇ · ~E = 0,

∂t~B = −~∇× ~E , ∇ · ~B = 0.

These equations are often written in the more compact4-dimensional notation: ∇aFab = 0 and ∇[aFbc] = 0,where Fab has components ~E and ~B.Maxwell’s equations are often re-expressed in terms of a vectorpotential Fab = ∇aAb −∇bAa :

∇a∇aAb −∇b∇aAa = 0.This form of Maxwell’s equations is manifestly hyperbolic as longas the gauge is chosen correctly, e.g., let ∇aAa = H(x , t), giving:

∇a∇aAb ≡(−∂2

t + ∂2x + ∂2

y + ∂2z

)Ab = ∇bH.


Constraint Damping

Where are the constraints: ∇a∇aAb = ∇bH?

Gauge condition becomes a constraint: 0 = C ≡ ∇aAa − H .

Maxwell’s equations imply that this constraint is preserved:

∇a∇a C = 0.

Modify evolution equations by adding multiples of the constraints:

∇a∇aAb = ∇bH+γ0tb C = ∇bH+γ0tb (∇aAa − H).

These changes also affect the constraint evolution equation,

∇a∇a C−γ0tb∇b C = 0,

so constraint violations are damped when γ0 > 0.


Constraint Damping

Where are the constraints: ∇a∇aAb = ∇bH?Gauge condition becomes a constraint: 0 = C ≡ ∇aAa − H .

Maxwell’s equations imply that this constraint is preserved:

∇a∇a C = 0.

Modify evolution equations by adding multiples of the constraints:

∇a∇aAb = ∇bH+γ0tb C = ∇bH+γ0tb (∇aAa − H).

These changes also affect the constraint evolution equation,

∇a∇a C−γ0tb∇b C = 0,

so constraint violations are damped when γ0 > 0.


Constraint Damped Einstein System“Generalized Harmonic” form of Einstein’s equations haveproperties similar to Maxwell’s equations:

Gauge (coordinate) conditions are imposed by specifying thedivergence of the spacetime metric: ∂agab = Hb + ...Evolution equations become manifestly hyperbolic: gab = ...Gauge conditions become constraints.Constraint damping terms can be added which make numericalevolutions stable.

0 100 20010-10

10-8

10-6

10-4

10-2

t/M

|| C ||

With Constraint Damping

WithoutConstraintDamping


Numerical Solution of Evolution Equations∂tu = F (u, ∂xu, x , t).

Choose a grid of spatial points, xn.

x n−1 x n x n+1

Evaluate the function u on this grid: un(t) = u(xn, t).

Approximate the spatial derivatives at the grid points∂xu(xn) =

∑k Dn kuk .

Evaluate F at the grid points xn in terms of the uk : F (uk , xn, t).

Solve the coupled system of ordinary differential equations,

dun(t)

dt= F [uk (t), xn, t ],

using standard numerical methods (e.g. Runge-Kutta).



Choose a grid of spatial points, xn.Evaluate the function u on this grid: un(t) = u(xn, t).

x n−1 x n x n+1

un−1 un un+1


∑k Dn kuk .



dun(t)

dt= F [uk (t), xn, t ],

using standard numerical methods (e.g. Runge-Kutta).



Choose a grid of spatial points, xn.Evaluate the function u on this grid: un(t) = u(xn, t).

x n−1 x n x n+1

un−1 un un+1


∑k Dn kuk .



dun(t)

dt= F [uk (t), xn, t ],

using standard numerical methods (e.g. Runge-Kutta).Lee Lindblom (Caltech) Numerical Black Hole Simulations LANL – 3/5/2008 11 / 27

Basic Numerical Methods

Different numerical methods use different ways of choosing thegrid points xn, and different expressions for the spatial derivatives

∂xu(un) =∑

k Dn kuk .

Most numerical groups use finite difference methods:Uniformly spaced grids: xn − xn−1 = ∆x = constant.Use Taylor expansions,

un−1 = u(xn−∆x) = u(xn)− ∂xu(xn)∆x + ∂2x u(xn)∆x2/2 +O(∆x3),

un+1 = u(xn +∆x) = u(xn) + ∂xu(xn)∆x + ∂2x u(xn)∆x2/2 +O(∆x3),

(1)to obtain the needed expressions for ∂xu:

∂xu(xn) =un+1 − un−1

2∆x+O(∆x2).

Grid spacing decreases as the number of grid points N increases,∆x ∼ 1/N . Errors in finite difference methods scale as N−p.


Basic Numerical Methods II

A few groups (including ours) use pseudo-spectral methods.

Represent functions as finite sums: u(x , t) =∑N−1

k=0 uk (t)eikx .Choose grid points xn to allow exact (and efficient) inversion of theseries: uk (t) =

∑N−1n=0 wn u(xn, t)e−ikxn .

Obtain derivative formulas by differentiating the series:∂xu(xn, t) =

∑N−1k=0 uk (t)∂xeikxn =

∑N−1m=0 Dn m u(xm, t).

Errors in spectral methods are dominated by the size of uN .Estimate the errors (for Fourier series of smooth functions):

uN =1

2π

∫ π

−πu(x)e−iNxdx =

12π

(−iN

)∫ π

−π

du(x)

dxe−iNxdx

=1

2π

(−iN

)p ∫ π

−π

dpu(x)

dxp e−iNxdx ≤ 1Np max

∣∣∣∣dpu(x)

dxp

∣∣∣∣ .Errors in spectral methods decrease faster than any power of N.











uN =1

2π

∫ π


12π

(−iN

)∫ π

−π

du(x)

dxe−iNxdx

=1

2π

(−iN

)p ∫ π

−π

dpu(x)


∣∣∣∣dpu(x)

dxp

∣∣∣∣ .

Errors in spectral methods decrease faster than any power of N.











uN =1

2π

∫ π


12π

(−iN

)∫ π

−π

du(x)

dxe−iNxdx

=1

2π

(−iN

)p ∫ π

−π

dpu(x)


∣∣∣∣dpu(x)

dxp

∣∣∣∣ .Errors in spectral methods decrease faster than any power of N.


Comparing Different Numerical Methods

x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

U(x

,t)

0 π/2 π 3π/2 2π

N=200

t=0.0

x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

U(x

,t)0 π/2 π 3π/2 2π

N=200

t=100.0

x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

U(x

,t)

0 π/2 π 3π/2 2π

N=200

t=200.0

Wave propagation with second-order finite difference method:

Wave propagation with spectral method:

Figures from Hesthaven, Gottlieb, & Gottlieb (2007).Lee Lindblom (Caltech) Numerical Black Hole Simulations LANL – 3/5/2008 14 / 27

Comparing Different Numerical Methods

x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

U(x

,t)

0 π/2 π 3π/2 2π

N=10

t=0.0

x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

U(x

,t)

0 π/2 π 3π/2 2π

N=10

t=100.0

x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

U(x

,t)

0 π/2 π 3π/2 2π

N=10

t=200.0

x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

U(x

,t)

0 π/2 π 3π/2 2π

N=200

t=0.0

x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

U(x

,t)0 π/2 π 3π/2 2π

N=200

t=100.0

x

0.0

0.5

1.0

1.5

2.0

2.5

3.0

U(x

,t)

0 π/2 π 3π/2 2π

N=200

t=200.0

Wave propagation with second-order finite difference method:

Wave propagation with spectral method:

Figures from Hesthaven, Gottlieb, & Gottlieb (2007).Lee Lindblom (Caltech) Numerical Black Hole Simulations LANL – 3/5/2008 14 / 27

Moving Black Holes

Black hole interior is not in causal contact with exterior.Interior is removed, introducing an excision boundary.

Numerical grid must be moved when black holes move too far.Problems:

Difficult to get smooth extrapolation at trailing edge of horizon.Causality trouble at leading edge of horizon.

x

Horizon

HorizonOutside

t

Solution:Choose coordinates that smoothlytrack the motions of the centersof the black holes.

x

Horizon

HorizonOutside

t


Moving Black Holes

Black hole interior is not in causal contact with exterior.Interior is removed, introducing an excision boundary.Numerical grid must be moved when black holes move too far.

Problems:


x

Horizon

HorizonOutside

t


x

Horizon

HorizonOutside

t


Moving Black Holes

Black hole interior is not in causal contact with exterior.Interior is removed, introducing an excision boundary.Numerical grid must be moved when black holes move too far.Problems:

Difficult to get smooth extrapolation at trailing edge of horizon.

Causality trouble at leading edge of horizon.

x

Horizon

HorizonOutside

tSolution:Choose coordinates that smoothlytrack the motions of the centersof the black holes.

x

Horizon

HorizonOutside

t


Moving Black Holes

Black hole interior is not in causal contact with exterior.Interior is removed, introducing an excision boundary.Numerical grid must be moved when black holes move too far.Problems:


x

Horizon

HorizonOutside

t


x

Horizon

HorizonOutside

t


Horizon Tracking Coordinates

Coordinates must be used that track the motions of the holes.A coordinate transformation from “inertial” coordinates, (x , y , z), to“co-moving” coordinates (x , y , z), consisting of a rotation followedby an expansion, x

yz

= e a(t)

cosϕ(t) − sinϕ(t) 0sinϕ(t) cosϕ(t) 0

0 0 1

xyz

,

is general enough to keep the holes fixed in co-movingcoordinates for suitably chosen functions a(t) and ϕ(t).Since the motions of the holes are not known a priori, thefunctions a(t) and ϕ(t) must be chosen dynamically andadaptively as the system evolves.


Horizon Tracking Coordinates II

x

δϕ y

c

c

Measure the co-moving centers of the holes: xc(t) and yc(t), orequivalently

Qx (t) =xc(t)− xc(0)

xc(0),

Qy (t) =yc(t)xc(t)

.

Choose the map parameters a(t) and ϕ(t) to keep Qx (t) andQy (t) small.Changing the map parameters by the small amounts, δa and δϕ,results in associated small changes in δQx and δQy :

δQx = −δa, δQy = −δϕ.


Horizon Tracking Coordinates IIIMeasure the quantities Q y (t), dQ y (t)/dt , d 2Q y (t)/dt2, and set

d 3ϕ

dt3 = λ3Q y + 3λ2 dQ y

dt+ 3λ

d 2Q y

dt2 = −d 3Q y

dt3 .

The solutions to this “closed-loop” equation for Q y have the formQ y (t) = (At2 + Bt + C)e−λt , so Q y always decreases as t →∞.

0 1000 2000 3000 4000-2×10-4

-1×10-4

0

1×10-4

2×10-4

Qy =

t / MADM

Qx =

yc(t)

xc(t)

xc(t) - x

c(0)

xc(0)

This works! This simple rotationplus expansion map allows usto evolve binary black holesto just before merger.More complicated maps that controlthe shapes of the horizons allow usto simulate the merger and ringdownas well.



d 3ϕ


dt+ 3λ

d 2Q y

dt2 = −d 3Q y

dt3 .


0 1000 2000 3000 4000-2×10-4

-1×10-4

0

1×10-4

2×10-4

Qy =

t / MADM

Qx =

yc(t)

xc(t)

xc(t) - x

c(0)

xc(0)

This works! This simple rotationplus expansion map allows usto evolve binary black holesto just before merger.

More complicated maps that controlthe shapes of the horizons allow usto simulate the merger and ringdownas well.



d 3ϕ


dt+ 3λ

d 2Q y

dt2 = −d 3Q y

dt3 .


0 1000 2000 3000 4000-2×10-4

-1×10-4

0

1×10-4

2×10-4

Qy =

t / MADM

Qx =

yc(t)

xc(t)

xc(t) - x

c(0)

xc(0)

This works! This simple rotationplus expansion map allows usto evolve binary black holesto just before merger.More complicated maps that controlthe shapes of the horizons allow usto simulate the merger and ringdownas well.


Caltech/Cornell Spectral Einstein Code (SpEC):Multi-domain pseudo-spectral method.

Constraint damped “generalized harmonic” Einstein equations:gab = Fab(g, ∂g).

Constraint-preserving, physical and gauge boundary conditions.


Evolving Binary Black Hole SpacetimesWe can now evolve BBH spacetimes with excellent accuracy andefficiency through many orbits plus merger plus ringdown.

-4 0 4

-4

0

4

x/m

y/m

-2 -1 0 1 2x/m

-2

-1

0

1

2

y/m

t=3936.3mt=3937.5mt=3938.7mt=3940.36mMerged BH

Head-on Merger MovieLee Lindblom (Caltech) Numerical Black Hole Simulations LANL – 3/5/2008 20 / 27

Numerical Gravitational WaveformsWe can now compute high precision gravitational waveforms forequal mass non-spinning BBH systems.

4100 4200t/m

0 1000 2000 3000 4000t/m

-0.4

-0.2

0

0.2

0.4

r R

e Ψ

4 22 / m

ω2

BBH Inspiral-Merger MovieLee Lindblom (Caltech) Numerical Black Hole Simulations LANL – 3/5/2008 21 / 27

Post-Newtonian Gravitational Waveforms

Until recently the only way to compute the gravitational waveformspredited by general relativity was through approximations.

The post-Newtonian approximation is an expansion of Einstein’sequations appropriate for weak gravitational fields and slowmoving sources.Post-Newtonian waveforms are very accurate for widely separatedbinary systems, but fail when the black holes get too close.All current compact binary searches on LIGO use PN basedwaveform templates.When do PN waveforms fail?



Until recently the only way to compute the gravitational waveformspredited by general relativity was through approximations.The post-Newtonian approximation is an expansion of Einstein’sequations appropriate for weak gravitational fields and slowmoving sources.Post-Newtonian waveforms are very accurate for widely separatedbinary systems, but fail when the black holes get too close.

All current compact binary searches on LIGO use PN basedwaveform templates.When do PN waveforms fail?



Until recently the only way to compute the gravitational waveformspredited by general relativity was through approximations.The post-Newtonian approximation is an expansion of Einstein’sequations appropriate for weak gravitational fields and slowmoving sources.Post-Newtonian waveforms are very accurate for widely separatedbinary systems, but fail when the black holes get too close.All current compact binary searches on LIGO use PN basedwaveform templates.

When do PN waveforms fail?



Until recently the only way to compute the gravitational waveformspredited by general relativity was through approximations.The post-Newtonian approximation is an expansion of Einstein’sequations appropriate for weak gravitational fields and slowmoving sources.Post-Newtonian waveforms are very accurate for widely separatedbinary systems, but fail when the black holes get too close.All current compact binary searches on LIGO use PN basedwaveform templates.When do PN waveforms fail?


Post-Newtonian Gravitational WaveformsTaylorT1

1 Rewrite energy-balance equation

−dEbinary

dΩ

dΩ

dt=

dEGW

dt⇒ dΩ

dt= − dEGW/dt

dEbinary/dΩ

2 Substitute Taylor series on right-hand side

dΩ

dt= −Ω10/3 (A0 + . . .+ AnΩn/3)

Ω−1/3(B0 + . . .+ BnΩn/3

)3 Numerically integrate once to find Ω4 Numerically integrate once more to find Φ

5 Numerically integrate once more to find Φ

TaylorT2, TaylorT3, . . .




−dEbinary

dΩ

dΩ

dt=

dEGW

dt⇒ dΩ

dt= − dEGW/dt

dEbinary/dΩ


dΩ

dt= −Ω10/3 (A0 + . . .+ AnΩn/3)

Ω−1/3(B0 + . . .+ BnΩn/3

)3 Re-expand right-hand side as a Taylor series, and truncate

dΩ

dt= −Ω11/3

(C0 + . . .+ CnΩn/3


TaylorT2, TaylorT3, . . .




−dEbinary

dΩ

dΩ

dt=

dEGW

dt⇒ dΩ

dt= − dEGW/dt

dEbinary/dΩ


dΩ

dt= −Ω10/3 (A0 + . . .+ AnΩn/3)

Ω−1/3(B0 + . . .+ BnΩn/3

)3 Re-expand right-hand side as a Taylor series, and truncate

dΩ

dt= −Ω11/3

(C0 + . . .+ CnΩn/3


TaylorT2, TaylorT3, . . .Lee Lindblom (Caltech) Numerical Black Hole Simulations LANL – 3/5/2008 23 / 27

Comparing Various Order PN with NR WaveformComparison of the numerical gravitational wave phase withpredictions of various post-Newtonian orders.

0 1000 2000 3000 4000

-0.2

0

0.2

0.4

0.6

0 1000 2000-0.02

0

0.02

0.04

∆φ

t/(M1+M2)

2PN

2.5PN

3.5PN

3PN

Lev5

0093h - 0093g

PN-phase comparison andnumerical error budget (May 2007)

Extrapolation Error


Comparing Various PN Methods

-4000 -3000 -2000 -1000 0t / M

-4

-3

-2

-1

0

1

Phas

e di

ffere

nce

(radi

ans)

02468101214Orbits before coalescence

TaylorT1 - TaylorT4

TaylorT3 - TaylorT4

TaylorT2 - TaylorT4


Comparing Various PN Methods with NR Waveform

0 1000 2000 3000 4000t / M

-0.1

-0.05

0

0.05

0.1

Frac

tiona

l am

plitu

de d

iffer

ence 48121620242832

GW cycles before merger

TaylorT3 - NR

TaylorT1 - NR

TaylorT4 - NR

TaylorT2 - NR


Comparing Various PN Methods with NR Waveform

0 1000 2000 3000 4000t / M

-1

-0.5

0

0.5

1Ph

ase

diffe

renc

e (ra

dian

s)48121620242832

GW cycles before merger

TaylorT3 - NR

TaylorT1 - NR

TaylorT4 - NR

TaylorT2 - NR


Summary

Advances in understanding the Einsteinequations provide new formulationssuitable for numerical evolutions:hyperbolic formulations with constraintdamping and well posed initial-boundaryvalue problems.High accuracy multi-orbit binary black holesimulations are now routine (but not yetcheap).Numerical waveforms suitable for LIGOdata analysis are starting to be generated.Interesting non-linear dynamics of binaryblack hole mergers are beginning to beinvestigated.


Numerical Simulations of Black Hole Spacetimeslindblom/Talks/LANL_5March2008.pdf · 2009-11-06 ·...

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Transcript of Numerical Simulations of Black Hole Spacetimeslindblom/Talks/LANL_5March2008.pdf · 2009-11-06 ·...