[PPT] - Numerical relativity: Triumphs and challenges of binary black hole PowerPoint Presentation

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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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Outline

1

Introduction: Gravitational-wave sources and numerical relativity

2

Triumphs: Current capabilities of simulations

3

Challenges

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

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1. Introduction

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

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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 gravitational radiation. Eventually black holes collide, merge, and form a final black hole.

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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 relativity

Hydrodynamics (shocks, . . . ) Microphysics (finite temperature, composition, . . . ) Magnetic fields Neutrino transport

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

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LIGO

LIGO: 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)

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Source modeling and LIGO

Detailed models of gravitational waves sources will help: Detection of signals

Matched filtering technique requires waveform templates, greatly improves detection rate.

Parameter estimation

Compare measured signal with model to learn about sources

Test general relativity Measure populations/distributions/properties of BHs, NSs Learn about microphysics of dense matter Multimessenger astronomy . . .

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What do we know about black holes?

A handful of exact solutions

Schwarzschild (1916): Static black hole Kerr (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

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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 slice Choose gauge (=coordinate) conditions Get yourself a computer cluster, and Solve 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)

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

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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 slice Choose gauge (=coordinate) conditions Get yourself a computer cluster, and Solve 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 2005 Today several research groups worldwide have NR codes.

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

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Binary black hole waveforms

time

Inspiral Ringdown Merger

S 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

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2. Capabilities of BBH

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.

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SpEC - Spectral Einstein Code

http://www.black-holes.org/SpEC.html

Parallel computer code developed at Caltech, Cornell, CITA (Toronto), Washington State, UC Fullerton, plus several contributors at other institutions. Solves nonlinear Einstein equations in 3+1 dimensions. Handles dynamical black holes. Relativistic Hydrodynamics. Over 50 researchers have contributed to SpEC.

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Brief Description of SpEC GR Module

Spectral methods

Solve equations on finite spatial regions called subdomains.

f(x, y, z) = LMN

ℓmn fℓmnTℓ(x)Tm(y)Tn(z)

f(r, θ, φ) = LMN

ℓmn fℓmnTn(r)Yℓm(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)

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Brief Description of SpEC GR Module

Excision

To avoid singularities, we excise the interiors of BHs.

excision boundary (slightly inside horizon) (This differs from another widely used approach called “moving punctures”)

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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 = 0 Local

Space k s Time

grey=EH; red,green=AH

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

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Brief Description of SpEC GR Module

Handling 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. ⇒

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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 uncertain Accretion models: χ ∼ 0.95 Some observations suggest χ > 0.98 Spin parameter: χ = J/M2, |χ| ≤ 1

0.2 0.4 0.6 0.8 1

Spin parameter χ

0.2 0.4 0.6 0.8 1

Erot / Erot,(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.

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Current BBH capabilities: spins

Spins ∼ 0.97.

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

Movie: Geoffrey Lovelace

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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 + M2 Size of time step ∼ Msmall Want to explore all mass ratios. Extreme mass ratio: pert. theory. SXS Collaboration: Mass ratio 8:1

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Current BBH capabilities: Precession

Simulation and Movie: Nick Taylor, SXS Collaboration

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Current BBH capabilities: Precession

Color = Vorticity (a measure of spin) Mass ratio 6 Large hole spin ∼ 0.91 Small hole spin ∼ 0.3

Movie: Robert McGehee and Alex Streicher, SXS Collaboration

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Current BBH capabilities: Accuracy

3000 6000 9000

t/M

10

4

10

2

δφ

10240 10320

0.06

0.06

rh/M

h+ hx θ=π/2, φ=0 L1 L2 L3 L4 L5 L4 L5

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

Mass ratio=3 Large hole spin=0.5 Small hole spin=0 31 orbits Precession

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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

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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) collaboration

Goal:

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

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

Georgia Tech

191 simulations (Pekowsky et al, arXiv:1304.3176)

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Applications of BBH simulations

Gravitational-wave studies Calibrate and test analytic waveform models Inject into LIGO data analysis pipelines Construct NR-only template banks . . .

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Applications of BBH simulations

Astrophysics Construct formulae for remnant properties Study gravitational recoil Examine precession effects Examine effects of eccentricity . . .

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Applications of BBH simulations

Fundamental relativity Study critical phenomena Study black holes in higher dimensions Examine relativistic head-on collisions Study topology and behavior of event horizon during merger Understand dynamics of strong gravity and wave generation . . .

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3. Challenges

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Challenge: parameter space coverage

χ

A

0.0 0.2 0.4 0.6 0.8 1.0 χB 0.0 0.2 0.4 0.6 0.8 1.0 η 0.08 0.12 0.16 0.20 0.24 SXS Collaboration catalog, Mroue et. al. arXiv:1304.6077

η = mAmB/(mA + mB)2 Red/blue arrows = Initial spin directions Spins up to 0.97 Mass ratios up to 8 Very sparse coverage!

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

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Challenge: parameter space coverage

0.0 0.2 0.4 0.6 0.8 1.0

χA

0.00 0.10 0.20 0.25

η

1 1 0.0 0.2 0.4 0.6 0.8 1.0

χB

0.00 0.10 0.20 0.25

η

1 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.

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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

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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

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Challenge: Initial data

Astrophysical initial data Initial data should be ’snapshot’ of inspiral from t = −∞ Tidal distortion, initial gravitational radiation are not correct. ⇒ “Junk radiation” spoils beginning of simulation. ⇒ Masses & spins relax during junk epoch.

500 1000 1500 2000

t/m

0.001

0.001 r M ψ4

Eccentricity Cannot a priori choose initial data to get desired eccentricity. Can produce small eccentricity via iterative scheme: expensive.

200 400 600 800 1000

t/m

1×10

6

2×10

6

3×10

6

4×10

6

m

2Ω

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:1210.2958

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Challenge: Computational expense

Example runs from SpEC code (SXS collaboration): Mass Ratio Spin A Spin B N orbits Run time (CPU-h) 1 16 8k 1 0.95 0.95 25 200k 3 0.7 0.3 26 34k 6 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).

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Challenge: Computational expense

NR too computationally expensive for

Covering 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.

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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)

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 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=χ2=+0.43655

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

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Summary

Numerical Relativity now becoming mature, especially for BBH 7D parameter space for BBH, more for NS/NS, NS/BH Challenges:

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

In the future

Improvements 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