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Numerical relativity: Triumphs and challenges of binary black hole simulations Mark A. Scheel Caltech SXS Collaboration: www.black-holes.org ROM-GR, Jun 06 2013 Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 1 / 37

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Page 1: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Numerical relativity: Triumphs and challenges ofbinary black hole simulations

Mark A. ScheelCaltech

SXS Collaboration: www.black-holes.org

ROM-GR, Jun 06 2013

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 1 / 37

Page 2: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Outline

1 Introduction: Gravitational-wave sources and numerical relativity2 Triumphs: Current capabilities of simulations3 Challenges

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 2 / 37

Page 3: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

1. Introduction

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 3 / 37

Page 4: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Black Holes

Black holes: most strongly gravitating objects in the Universe.Obey Einstein’s general relativistic field equations.Form from collapse of matter (stars, gas, . . . )Energy source for many astrophysical phenomena.

Black-hole binaries expected to occur in the Universe.

S

S

1

2

M

M

2

1

Orbit decays as energy lost to gravitationalradiation.

Eventually black holes collide, merge, andform a final black hole.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 4 / 37

Page 5: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Neutron Stars

Neutron star (NS): made of degenerate matter at nuclear density.Formed in supernovae.

NS/NS & BH/NS binaries produce gravitational radiation.

Need additional physics besides general relativityHydrodynamics (shocks, . . . )Microphysics (finite temperature, composition, . . . )Magnetic fieldsNeutrino transport

Many more parameters (but some less important for LIGO)Remainder of talk: concentrate on BH/BH

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 5 / 37

Page 6: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

LIGOLIGO: Laser Interferometer Gravitational-Wave Observatory

LIGO planned in 2 phases:Initial, 2005-2010.Advanced, ∼ 2015.

Advanced LIGO should detect waves from compact binaries.Similar detectors in Europe (Virgo), Japan (KAGRA)

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 6 / 37

Page 7: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Source modeling and LIGODetailed models of gravitational waves sources will help:

Detection of signalsMatched filtering technique requires waveform templates, greatlyimproves detection rate.

Parameter estimationCompare measured signal with model to learn about sources

Test general relativityMeasure populations/distributions/properties of BHs, NSsLearn about microphysics of dense matterMultimessenger astronomy. . .

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 7 / 37

Page 8: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

What do we know about black holes?

A handful of exact solutionsSchwarzschild (1916): Static black holeKerr (1963): Stationary, rotating black hole

M

J

χ = J/M2; |χ| ≤ 1

(units: G = c = 1)Global theorems (e.g. Hawking area theorem)

Perturbations about exact solutions

No way to solve dynamical strong-gravity problems until recently.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 8 / 37

Page 9: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Numerical relativity (NR)Write Einstein’s field equations as an initial value problem for gµν .

Gµν = 8πTµν ⇒

Constraints (like ∇ · B = 0)Evolution eqs. (like ∂tB = −∇× E)

Choose unconstrained data on an initial time sliceChoose gauge (=coordinate) conditions

Get yourself a computer cluster, andSolve constraints at t = 0

(For us: 4 (+1) coupled nonlinear 2nd-order elliptic PDEs)Use evolution eqs. to advance in time

(For us: 50 coupled nonlinear 1st-order hyperbolic PDEs)

First successful black-hole binary computation: Pretorius 2005Today several research groups worldwide have NR codes.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 9 / 37

Page 10: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Numerical relativity (NR)Write Einstein’s field equations as an initial value problem for gµν .

Gµν = 8πTµν ⇒

Constraints (like ∇ · B = 0)Evolution eqs. (like ∂tB = −∇× E)

Choose unconstrained data on an initial time sliceChoose gauge (=coordinate) conditions

Get yourself a computer cluster, andSolve constraints at t = 0

(For us: 4 (+1) coupled nonlinear 2nd-order elliptic PDEs)Use evolution eqs. to advance in time

(For us: 50 coupled nonlinear 1st-order hyperbolic PDEs)

First successful black-hole binary computation: Pretorius 2005Today several research groups worldwide have NR codes.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 9 / 37

Page 11: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Binary black hole waveforms

time

Inspiral

Ringdown

MergerS

S

1

2

M

M

2

1

Waveform divided into 3 parts:

Inspiral: BHs far apart, described by post-Newtonian (PN) theory.Merger: Nonlinear, need NR.Ringdown: Single BH, described by pert. theory or NR.

Idea: Match NR simulation to PN, just before PN becomes inaccurate.

PN: perturbative expansion in powers of v/c

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 10 / 37

Page 12: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

2. Capabilities of BBHSimulations

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 11 / 37

Page 13: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

NR CodesAbout a dozen in existence

Numerical Methods:

BSSN Generalized Harmonic

Finite Differencing Spectral

Treatment of Singularities:

Formulation of Equations:

Moving Punctures Excision

Most codes

Princeton, AEI Harmonic Code SXS Collaboration (SpEC)

Comparing different codes improves confidence in results.Most examples I will show will be from SpEC.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 12 / 37

Page 14: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

NR CodesAbout a dozen in existence

Numerical Methods:

BSSN Generalized Harmonic

Finite Differencing Spectral

Treatment of Singularities:

Formulation of Equations:

Moving Punctures Excision

Most codes

Princeton, AEI Harmonic Code SXS Collaboration (SpEC)

Comparing different codes improves confidence in results.Most examples I will show will be from SpEC.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 12 / 37

Page 15: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

NR CodesAbout a dozen in existence

Numerical Methods:

BSSN Generalized Harmonic

Finite Differencing Spectral

Treatment of Singularities:

Formulation of Equations:

Moving Punctures Excision

Most codes

Princeton, AEI Harmonic Code

SXS Collaboration (SpEC)

Comparing different codes improves confidence in results.Most examples I will show will be from SpEC.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 12 / 37

Page 16: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

NR CodesAbout a dozen in existence

Numerical Methods:

BSSN Generalized Harmonic

Finite Differencing Spectral

Treatment of Singularities:

Formulation of Equations:

Moving Punctures Excision

Most codes

Princeton, AEI Harmonic Code SXS Collaboration (SpEC)

Comparing different codes improves confidence in results.Most examples I will show will be from SpEC.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 12 / 37

Page 17: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

NR CodesAbout a dozen in existence

Numerical Methods:

BSSN Generalized Harmonic

Finite Differencing Spectral

Treatment of Singularities:

Formulation of Equations:

Moving Punctures Excision

Most codes

Princeton, AEI Harmonic Code SXS Collaboration (SpEC)

Comparing different codes improves confidence in results.Most examples I will show will be from SpEC.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 12 / 37

Page 18: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

SpEC - Spectral Einstein Codehttp://www.black-holes.org/SpEC.html

Parallel computer code developed at Caltech, Cornell, CITA (Toronto),Washington State, UC Fullerton, plus several contributors at otherinstitutions.

Solves nonlinear Einstein equations in 3+1 dimensions.Handles dynamical black holes.Relativistic Hydrodynamics.

Over 50 researchers have contributed to SpEC.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 13 / 37

Page 19: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Brief Description of SpEC GR ModuleSpectral methods

Solve equations on finite spatial regions called subdomains.

f (x , y , z) =∑LMN`mn f`mnT`(x)Tm(y)Tn(z)

f (r , θ, φ) =∑LMN`mn f`mnTn(r)Y`m(θ, φ)

Choose spectral basis functions based on subdomain shapes.Exponential convergence for smooth problems. High accuracy.

(This differs from widely used finite-difference methods)

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 14 / 37

Page 20: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Brief Description of SpEC GR ModuleExcision

To avoid singularities, we excise the interiors of BHs.

excision boundary(slightly inside horizon)

(This differs from another widely used approach called “moving punctures”)

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 15 / 37

Page 21: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Aside: Apparent horizons

Event horizon (EH)Boundary of region where photons can escape to infinity.Nonlocal

Apparent horizon (AH)Smooth closed surface of zero null expansion. ∇µkµ = 0Local

Space

k

s

Time

grey=EH; red,green=AH

Theorem: if an AH exists,it cannot be outside an EH.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 16 / 37

Page 22: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Brief Description of SpEC GR ModuleHandling Merger

Eventually, common horizon forms around both BHs.Regrid onto new grid with only one excised region.Continue evolution on new grid, until final BH settles down.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 17 / 37

Page 23: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Current BBH capabilities: spins

Straightforward initial data construction limited to spins χ <∼ 0.93.Even χ = 0.93 is only 60% of possible Erot

What are spins of real black holes?

Highly uncertainAccretion models: χ ∼ 0.95Some observations suggestχ > 0.98

Spin parameter: χ = J/M2, |χ| ≤ 1

0 0.2 0.4 0.6 0.8 1Spin parameter χ

0

0.2

0.4

0.6

0.8

1

Ero

t / E

rot,

(χ=

1)

Rotational energy / rot. energy if χ=1

Spin 0.95

Spin 0.97

Adapted from Lovelace, MAS, Szilagyi PRD83:024010,2011

Want to explore large spins.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 18 / 37

Page 24: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Current BBH capabilities: spins

Spins ∼ 0.97.

Lovelace, Boyle, MAS, Szilágyi, CQG 29:045003 (2012)

Movie: Geoffrey LovelaceMark A. Scheel (Caltech) Numerical relativity Jun 06 2013 19 / 37

Page 25: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Current BBH capabilities: mass ratios

Largest to date is 100:1 (2 orbits), Lousto & Zlochower 2011

Large mass ratios are difficult:

Time scale of orbit ∼ M1 +M2Size of time step ∼ Msmall

Want to explore all mass ratios.Extreme mass ratio: pert. theory.

SXS Collaboration: Mass ratio 8:1Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 20 / 37

Page 26: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Current BBH capabilities: Precession

Simulation and Movie: Nick Taylor, SXS Collaboration

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 21 / 37

Page 27: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Current BBH capabilities: Precession

Color = Vorticity(a measure of spin)

Mass ratio 6Large hole spin ∼ 0.91Small hole spin ∼ 0.3

Movie: Robert McGehee and Alex Streicher, SXS Collaboration

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 22 / 37

Page 28: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Current BBH capabilities: Accuracy

3000 6000 9000

t/M

10-4

10-2

δφ

10240 10320

-0.06

0

0.06

rh/M

h+

hx

θ=π/2, φ=0

L1L2

L3L4

L5

L4

L5

Mroue et. al., SXS Collaboration, arXiv:1304.6077

Mass ratio=3Large hole spin=0.5Small hole spin=0

31 orbitsPrecession

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 23 / 37

Page 29: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

New BBH capabilities: simulation catalogs

0009 0032 0038 0041 0043 0044 0046 0054 0057 0059 0062 0109 0111 0122 0147 0151 0154

0030 0033 0034 0045 0048 0049 0050 0051 0052 0053 0156 0159 0167

00106 0064 0104 0114 0115 0116 0117 0118 0119 0120 0121 0123 0165

0040 0124 0125 0126 0127 0128 0129 0130 0131 0132 0133 0134 0135

0042 0137 0138 0139 0140 0141 0142 0144 0145 0146 0148 0149 0162

0006 0008 0010 0012 0017 0019 0021 0023 0024 0163 0164

0027 0031 0047 0055 0063 0075 0076 0079 0080 0081 0168

0082 0092 0094 0095 0096 0097 0108 0110 0112 0113 0169

0011 0013 0020 0022 0136 0143 0150 0152 0155 0161

0025 0026 0028 0029 0039 0060 0077 0078 0102 0171

0014 0037 0056 0061 0153 0157 0158 0160 0166

0001 0005 0007 0015 0074 0098 0100

0018 0035 0058 0066 0067 0068 0070

0065 0071 0072 0073 0086 0093 0099

0004 0036 0069 0084 0085 0103 0170

0002 0003 0016 0083 0087 0105

2000 4000 6000 8000 10000 12000

0088

2000 4000 6000 8000 10000 12000

0089

2000 4000 6000 8000 10000 12000

0090

2000 4000 6000 8000 10000 12000

0091

2000 4000 6000 8000 10000 12000

0101

2000 4000 6000 8000

0107

171 simulations, all with inspiral, merger, and ringdown

Mroue, MAS, Szilagyi, Pfeiffer, Boyle, Hemberger, Kidder, Lovelace, Ossokine,Taylor, Zenginoglu, Buchman, Chu, Giesler, Owen, Teukolsky, arXiv:1304.6077

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 24 / 37

Page 30: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

New BBH capabilities: simulation catalogs

Other catalogs:NRAR (Numerical Relativity/Analytical Relativity) project

Goal: Improve analytic waveform models using NR simulations.9 NR codes.25 simulations in first round; in preparation.

NINJA (Numerical INJection Analysis) collaborationGoal:

Add numerical waveforms into LIGO/Virgo detector noiseTest how well detection pipelines can detect/identify them

8 NR groups.56 hybridized waveforms, CQG 29, 124001 (2012)

Georgia Tech191 simulations (Pekowsky et al, arXiv:1304.3176)

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 25 / 37

Page 31: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Applications of BBH simulations

Gravitational-wave studiesCalibrate and test analytic waveform modelsInject into LIGO data analysis pipelinesConstruct NR-only template banks. . .

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 26 / 37

Page 32: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Applications of BBH simulations

AstrophysicsConstruct formulae for remnant propertiesStudy gravitational recoilExamine precession effectsExamine effects of eccentricity. . .

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 27 / 37

Page 33: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Applications of BBH simulations

Fundamental relativityStudy critical phenomenaStudy black holes in higher dimensionsExamine relativistic head-on collisionsStudy topology and behavior of event horizon during mergerUnderstand dynamics of strong gravity and wave generation. . .

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 28 / 37

Page 34: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

3. Challenges

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 29 / 37

Page 35: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Challenge: parameter space coverage

A

0.00.2

0.40.6

0.81.0

B

0.0

0.20.4

0.60.8

1.0

0.08

0.12

0.16

0.20

0.24

0.0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3.0

SXS Collaboration catalog, Mroue et. al. arXiv:1304.6077

η = mAmB/(mA + mB)2

Red/blue arrows =Initial spin directionsSpins up to 0.97Mass ratios up to 8

Very sparse coverage!

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 30 / 37

Page 36: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Challenge: parameter space coverage

0.0 0.2 0.4 0.6 0.8 1.0χA

0.00

0.10

0.20

0.25

η

1

0

1

0.0 0.2 0.4 0.6 0.8 1.0χB

0.00

0.10

0.20

0.25

η

1

0

1

η = mAmB/(mA + mB)2

SXS Collaboration catalog, Mroue et. al. arXiv:1304.6077

Dual challenge:Parameter space is large, 7D.Extreme parameter values are difficult.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 31 / 37

Page 37: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Challenge: many orbits

Waveform visible to LIGO includes hundreds of binary orbits.NR simulates many fewer orbits (most to date is 34).Solution: Use PN for inspiral, NR afterwards.

PN NR

time

Problem: Where is the matching point?Equal mass, no spin: can match 10 orbits before merger.’BH/NS’ parameters: (mass ratio ∼ 7, moderate spins),PN is still inaccurate dozens of orbits before merger.

Dual challenge:NR needs to simulate more orbits.We don’t know how many orbits are needed! (but are testing this)

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 32 / 37

Page 38: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Challenge: many orbits

Waveform visible to LIGO includes hundreds of binary orbits.NR simulates many fewer orbits (most to date is 34).Solution: Use PN for inspiral, NR afterwards.

PN NR

time

Problem: Where is the matching point?Equal mass, no spin: can match 10 orbits before merger.’BH/NS’ parameters: (mass ratio ∼ 7, moderate spins),PN is still inaccurate dozens of orbits before merger.

Dual challenge:NR needs to simulate more orbits.We don’t know how many orbits are needed! (but are testing this)

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 32 / 37

Page 39: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Challenge: Initial dataAstrophysical initial data

Initial data should be ’snapshot’ of inspiral from t = −∞Tidal distortion, initial gravitational radiation are not correct.

⇒ “Junk radiation” spoilsbeginning of simulation.

⇒ Masses & spins relaxduring junk epoch.

500 1000 1500 2000

t/m

-0.001

0

0.001

r M

ψ4

EccentricityCannot a priori choose initial datato get desired eccentricity.Can produce small eccentricity viaiterative scheme: expensive.

0 200 400 600 800 1000

t/m

0

1×10-6

2×10-6

3×10-6

4×10-6

m2Ω

Iter 0: eΩ

~0.018

Iter 1: eΩ

~0.0037

Iter 2: eΩ

~0.0009

Iter 3: eΩ

~0.0003

Iter 4: eΩ

~0.00015

.

Mroue & Pfeiffer, arXiv:1210.2958Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 33 / 37

Page 40: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Challenge: Computational expense

Example runs from SpEC code (SXS collaboration):

Mass Ratio Spin A Spin B N orbits Run time (CPU-h)1 0 0 16 8k1 0.95 0.95 25 200k3 0.7 0.3 26 34k6 0.91 0.3 6.5 38k

Best efficiency: few (∼ 50) cores, run many simulations at once.Days to months wallclock time, depending on parameters.Recent improvement by a factor of ∼ 5 for SpEC (Bela Szilagyi).

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 34 / 37

Page 41: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Challenge: Computational expense

NR too computationally expensive forCovering 7D parameter space by random sampling.’Live’ NR during data analysis.

Parameter space, expense worse when including neutron stars.

What to do?

Build analytical models (“EOB”, “PhenomC”) w/ unknown coefs.⇒ Determine coefficients by fitting to numerical simulations.

Reduced order modeling.

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 35 / 37

Page 42: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Analytical waveform models

“Effective One Body” model fitted to SpEC waveforms.

No spin: “EOBNRv2”Pan et. al. PRD 84:124052 (2011)

1000 2000 3000 4000

-0.02

0.00

0.02

NR Re(h33) R/M

EOB Re(h33) R/M

4800 4860 4920

-0.05

0.00

0.05

1000 2000 3000 4000(t - r

*)M

-0.10

-0.05

0.00

0.05

0.10∆φh (rad)

∆A / A

4800 4860 4920(t - r

*)/M

-0.2

0.0

0.2

0.4

0.6

q=6 (3,3)

Spins, no precession:“SEOBNRv1”Taracchini ea., PRD 86:024011 (2012)

0 1000 2000 3000-0.4

-0.2

0.0

0.2

0.4

NR Re(h22

)

EOB Re(h22

)

3300 3350 3400-0.4

-0.2

0.0

0.2

0.4

0 1000 2000 3000(t - r

*)/M

-0.1

0.0

0.1

0.2

0.3∆φ

∆Α/Α

3300 3350 3400(t - r

*)/M

-0.1

0.0

0.1

0.2

0.3

q=1, χ1=χ

2=+0.43655

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 36 / 37

Page 43: Numerical relativity: Triumphs and challenges of binary ...rom-gr/slides/Mark_Scheel.pdf · Numerical relativity (NR) Write Einstein’s field equations as an initial value problem

Summary

Numerical Relativity now becoming mature, especially for BBH7D parameter space for BBH, more for NS/NS, NS/BHChallenges:

Simulating enough binary orbits.Difficult corners of parameter space.How to choose ’important’ points in parameter space.Computational expense.

In the futureImprovements in efficiency, accuracy, parameter coverage.Include more physics for simulations with matter.Reduced Order Modeling

Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 37 / 37