Solucion numerica de las ecuaciones de Einstein:

Choques de agujeros negrosJose Antonio Gonzalez

IFM-UMSNH

25-Abril-2008

ENOAN 2008

• Introduction– Binary black hole problem

• Some ingredients– 3+1 decomposition

– Formulation of the equations

– Initial data

– Gauge

– Mesh refinement

– Boundary conditions

– Excision

– Diagnostic tools

• Applications• Conclusions

OverviewOverview

The big pictureThe big picture

Model GR (numerical relativity) PN Perturbation theory Non-GR?

External Physics Astrophysics Fundamental physics Cosmology

DetectorsPhysical System

describes

observe

test implications

Help d

etec

t

Numerical relativityNumerical relativity

-Two 10 solar mass black holes-Frequency ~ 100Hz-Distort the 4km mirror spacingby about 10^-18 m

3+1 decomposition3+1 decomposition

GR: “Space and time exist together as Spacetime’’

Numerical relativity: reverse this process!

ADM 3+1 decomposition Arnowitt, Deser, Misner (1962)York (1979)Choquet-Bruhat, York (1980)

3-metric ijlapse

shift

i

lapse, shift Gauge

Einstein equations 6 Evolution equations

4 Constraints

Constraints preserved under evolution!

ADM equationsADM equations

Evolution equations

ijijt KL 2)(

]2[)( KKKKRDDKL ijjm

imijjiijt

Constraints

02 ijijKKKR

0 KDKD iijj

Evolution

Solve constraints initially Evolve data Reconstruct spacetime Extract physics

Formulation of the equationsFormulation of the equations

ADM: unsuccessful; weakly hyperbolic!

BSSN (most popular) Shibata, Nakamura ‘95Baumgarte, Shapiro ‘99

Balance laws: Bona, Massó (H-code)

Many more:

Sarbach et.al.‘02;Gundlach, Martin-Garcia

Split degrees of freedom (similar to initial data split) Hyperbolicity

Generalized harmonic formulation

Harmonic gauge well-posed Wave equations for BBH-breakthrough

Choquet-Bruhat ‘62

Pretorius ‘05g

ADM-like family:Harmonic family:Control of constraints:

KST, NOR,… Z4 LSU, Caltech, Gundlach

Garfinkle ‘04

The BSSN formulationThe BSSN formulation

Initial dataInitial data

Two difficulties: Constraints, realistic data

York-Lichnerowicz split ijij ~4 KAK ijijij 3

1

Conformal transverse traceless Physical transverse traceless Thin sandwich

York, LichnerowiczO’Murchadha, YorkWilson, Mathews; York

Conformal flatness:

Spurious radiation does not seem problematic, but alternatives studied

Generalized analytic solutions:

Time symmetric, -holes:

Spin, linear momenta:Punctures

Brill-Lindquist, Misner (1960s)Bowen, York (1980)Brandt, Brügmann (1997)

Isotropic Schwarzschild

N

Excision data: Isolated Horizon condition on excision surface

Meudon group; Cook, Pfeiffer; Ansorg

Quasi-circularity: Effective potential method PN fit helical killing vector

GaugeGauge

Specific problem in GR: Coordinates constructed during evolutions

Highly non-trivial: Prescribe to avoid coordinate singularities

Einstein equations say nothing about , i, i

Maximal slicing, min.distortion shiftSmarr, York ‘78

Driver conditionsBalakrishnaet.al.’96

1+log, -driverAEI

~

Moving puncturesUTB, Goddard ‘06

Bona-Massó familyBona, Massó ‘95

Harmonic coordsChoquet-Bruhat‘62

Generalized harmonicGarfinkle ‘04Pretorius ‘05

Study singularity avoidanceAlcubierre ‘03

Analytic studies

gauge sources relation to i ,

Drive to stationarity

specialcase

specialcase

Mesh-refinement, boundariesMesh-refinement, boundaries

3 length scales: BHWave lengthWave zone M

M

M

100

10

1

Choptuik ’93 AMR, Critical phenomena

Stretch coordinates: Fish-eye Lazarus, AEI, UTB

FMR, Moving boxes: Berger-Oliger BAM Brügmann’96 Carpet Schnetter et.al.’03

AMR: Steer resolution via scalar Paramesh: MacNeice et.al.’00, Goddard modified Berger-Oliger: Pretorius, Choptuik ’05 SAMRAI

Refinement boundaries: reflections, stability Lehner, Liebling, Reula ‘05

Outer boundary conditionsOuter boundary conditions

Problems: Well-posedness of equations? Constraint violations?

BCs that satisfy constraints and/or well-posedness

Friedrich, Nagy ‘99 Calabrese, Lehner, Tiglio ‘02 Frittelli, Goméz ‘04 Sarbach, Tiglio ‘04 Kidder et.al.‘05, Lindblom et.al.‘06

Tested with success in BBH simulation: Lindblom et.al.‘06

Conformal, null-formulation: Untested in BBH simulations

Compactification in 3+1 Pretorius ‘05

Push boundaries “far out”, use Sommerfeld condition Used successfully by most groups; accuracy limits?

Multi-patch approach: Efficiency AEI (Cactus): Thornburg et.al.: excision, Char.Code LSU (below), Austin (below), Cornell-Caltech (below)

Black hole excisionBlack hole excision

Cosmic censorship: Causal disconnection of region inside AH

Unruh ’84 cited in Thornburg ‘87

Grand Challenge: Causal differencing

“Simple Excision” Alcubierre, Brügmann ‘01

Dynamic “moving” excision Pitt-PSU-Texas PSU-Maya Pretorius

combined with Dual coordinate frame Caltech-Cornell

Mathematical properties: Wealth of literature

Diagnostic toolsDiagnostic tools

A computer just gives numbers! These are gauge dependent! Convert to physical information…

ADM mass, momentum Arnowitt, Deser, Misner ‘62

Bondi mass, News function (Characteristic approach)

Gravitational Waves Zerilli-Moncrief formalism Newman-Penrose scalar hih tttt4

Black hole quantities: mass, momentum, spin, area,…

Apparent Horizon Alcubierre, Gundlach (Cactus) Schnetter ‘03 Thornburg ‘03 (AHFinderDirect) Pretorius

Event horizon Diener ‘03 Isolated, Dynamic Hor. Ashtekar, Krishnan ’03

Ashtekar et.al. Dreyer et.al. ’02

2004

2007

How far we are?How far we are?

Spinning holes: The orbital hang-upSpinning holes: The orbital hang-up

Spins alligned with inspiral delayed, largerL

radrad JE ,

Spins anti-alligned with inspiral fast smallerL

radrad JE ,

No extreme Kerr holes produced

Gravitational recoilGravitational recoil

Anisotropic emission of GW carries away linear momentum recoil of remaining system

Merger of galaxies

Inspiral and merger of black holes Recoil of merged hole Displacement, Ejection?

Astrophysical relevanceAstrophysical relevance

BH inspiral kick possible ejection of BH from host

Escape velocities: globular clusters dSph dE large galaxies Merritt et al.’04

km/s 30km/s 10020 km/s 300100

km/s 1000

Non-spinning binariesNon-spinning binaries

Emerging picture: Kicks unlikely to exceed a few km/s 100

Numerical relativity allows accurate estimates Campanelli ’05 Herrmann et al.’06 Baker et al.’06

Close limit calculations Sopuerta et al.’06 a,b Upper and lower bounds

Including eccentricity increases kick

for small eccentricities

)1(kick ev

EOB approximation: account for deviations from Kepler law Damour & Gopakumar ‘06

Non-spinning binariesNon-spinning binaries

Systematic parameter study Gonzalez et al.’06

Moving puncture method BAM code Nested boxes, resolutions Extract calculate linear momentum Vary mass ratio:

150,000 CPU hours

/40mh s4

16.0...25.0

,4:1...1:1

q

Higher order PN Blanchet et al.’05

Non-spinning binaries: Maximal kickNon-spinning binaries: Maximal kick

km/s 117.175 005.0195.0 Maximal kick: at

Recoil of spinning binariesRecoil of spinning binaries

Kidder ’95: PN study including recoil of spinning holes

= “unequal mass” + “spin(-orbit)”

Penn State ‘07: Spin-orbit term larger

extrapolated:

8.0,...,2.0m

a

km/s 475v

AEI ’07:

extrapolated:

6.01 m

a6.0,...,0.02

m

a

km/s 440v

Recoil of spinning binariesRecoil of spinning binaries

UTB-Rochester

maximum predicted:km/s 454v

km/s 1300v

NASA Goddard:

km/s 52

km/s 3015

long

trans

v

v Spin effect

Unequal-mass effect

PN predictions remarkably robustFitting formulas

Discretization error: km/s 43v

Trajectories:

Getting even larger kicksGetting even larger kicks

Dependence on Extraction radiusDependence on Extraction radius

Error fall-off: km/s 120v

Reducing eccentricityReducing eccentricity

Data analysis and PN comparisonsData analysis and PN comparisons

Thick red line NR waveformsDashed black ‘best matched’ 3.5 PN waveformsThin green Hybrid waveforms

Since it is expensive to generate an entire physical bank of templates using numerical simulations, it is better to construct a phenomenological bank –unequal mass, non spinning black holes-

EccentricityEccentricity

IMRI’s: Motivation• Stellar mass black holes M~1-10 Msun

• Intermediate mass bh’s M~102-4 Msun

• Supermassive bh’s M~106-9 Msun

Why IMRIs and EMRIs are interesting?•Astrophysics•Data Analysis and gravitational waves detection:

Gravitational waves emited during the merger of stellar-mass black holes into a IMBHs will lie in the frequencies of Advanced LIGO (Brown et al. 2007)

•Tests of General Relativity•Comparison with PN and perturbation theory

•Numerical simulations are expensive

•How many orbits are required?

Data analysis 10? 100?

•How far we need to go in mass ratios?

Compare with PN!

1:100? 1:1000???

Hopefully not!

Mass ratio 1:10

–Resolution:

[η]= 1/M•Problems:

• M1 = 0.25 , M2 = 2.5 , M = M1+M2

• D = 19.25 = 7M

• q = M1/M2 = 10 , η = q/(1+q)2 = 0.0826

Parameters:

–Gauge:

Kick

Fitchett (MNRAS 203 1049,1983) Gonzalez et al. (PRL 98 091101, 2007)

V~62 km/s

Radiated energy ΔE/M=0.580192 η2

ΔE/M~0.004018

Berti et al. (2007)

Final spin

Damour and Nagar (2007)

aaF/MF~0.2602

ERAD = 0.011001

l=2 75.62%

l=3 16.36%

l=4 4.96%

l=5 1.74%

Energy distributionEnergy distribution

Conclusions• After a lot of work and effort….it seems to

work!

• It is over? No way!– It is necessary to improve accuracy– Now it is possible to do physics –the original

purpose of everything-– Data analysis– Parameter estimations

• Matter