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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
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
1. Introduction
Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 3 / 37
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
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
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
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
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
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
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
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
2. Capabilities of BBHSimulations
Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 11 / 37
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
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
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
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
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
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
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
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
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
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
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
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
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
Current BBH capabilities: Precession
Simulation and Movie: Nick Taylor, SXS Collaboration
Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 21 / 37
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
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
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
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
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
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
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
3. Challenges
Mark A. Scheel (Caltech) Numerical relativity Jun 06 2013 29 / 37
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
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
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
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
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
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
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
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
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