Simulations of Binary Black Simulations of Binary Black Hole CoalescenceHole Coalescence

Frans PretoriusFrans PretoriusPrinceton UniversityPrinceton University

University of Maryland University of Maryland Physics ColloquiumPhysics Colloquium

May 8, 2007 May 8, 2007

OutlineOutline• Why study binary black hole systems?Why study binary black hole systems?

– expected to be among the strongest and most promising sources of gravitational waves that could expected to be among the strongest and most promising sources of gravitational waves that could be observed by be observed by gravitational wave detectorsgravitational wave detectors

– understand the strong-field regime of general relativityunderstand the strong-field regime of general relativity

• Why do we need to Why do we need to simulatesimulate them? them?

– understanding the nature of the gravitational waves emitted during a merger event may be understanding the nature of the gravitational waves emitted during a merger event may be essentialessential for successful detection for successful detection

– the two-body problem in GR is the two-body problem in GR is unsolvedunsolved, and no analytic solution techniques (perturbative or , and no analytic solution techniques (perturbative or other) known that could be applied during the final stages of an inspiral and mergerother) known that could be applied during the final stages of an inspiral and merger

• MethodologyMethodology

– brief overview of numerical relativity, the difficulties in discretizing the field equations, and an brief overview of numerical relativity, the difficulties in discretizing the field equations, and an evolution scheme based on generalized harmonic coordinatesevolution scheme based on generalized harmonic coordinates

• Simulation resultsSimulation results

– highlights about what has been learnt about the merger process for astrophysically relevant initial highlights about what has been learnt about the merger process for astrophysically relevant initial data parametersdata parameters

– binaries in eccentric orbits constructed via scalar field collapsebinaries in eccentric orbits constructed via scalar field collapse

• evidence that, to a evidence that, to a certaincertain degree, some of the interesting phenomenology of test particle motion persists in the degree, some of the interesting phenomenology of test particle motion persists in the equal mass merger regime, including unstable equal mass merger regime, including unstable circularcircular orbits and corresponding zoom-whirl like behavior orbits and corresponding zoom-whirl like behavior

• speculative application to the high-energy scattering problemspeculative application to the high-energy scattering problem

Gravitational Waves in General Gravitational Waves in General RelativityRelativity

• Einstein’s theory of general relativity states that we Einstein’s theory of general relativity states that we live in a 4 dimensional, curved spacetimelive in a 4 dimensional, curved spacetime

– curvature is responsible for what we describe as the force curvature is responsible for what we describe as the force of gravity of gravity

– matter/energy is responsible for curvature in the geometry matter/energy is responsible for curvature in the geometry

– localized disturbances in the geometry propagate at the localized disturbances in the geometry propagate at the speed of light — speed of light — these are gravitational wavesthese are gravitational waves

– bulk motions of dense concentrations of matter/energy bulk motions of dense concentrations of matter/energy produce gravitational waves that may be strong enough to produce gravitational waves that may be strong enough to detectdetect

Weak field nature of gravitational Weak field nature of gravitational waveswaves

• Far from the source, the effect of a gravitational wave is to Far from the source, the effect of a gravitational wave is to cause distortions in the geometry transverse to the cause distortions in the geometry transverse to the direction of propagationdirection of propagation

• Two linearly independent polarizations (Two linearly independent polarizations (++ and and xx))

– schematic effect of a wave, traveling into the slide, on the schematic effect of a wave, traveling into the slide, on the distances between an initially circular ring of particles:distances between an initially circular ring of particles:

++

xx

timetime

The network of gravitational wave The network of gravitational wave detectorsdetectors

LIGO HanfordLIGO Hanford

LIGO LivingstonLIGO Livingston

ground based laser interferometersground based laser interferometersLIGO/VIRGO/GEO/TAMALIGO/VIRGO/GEO/TAMA

space-based laser interferometer (hopefully space-based laser interferometer (hopefully with get funded for a 201? Lauch)with get funded for a 201? Lauch)

LISALISA

ALLEGRO/NAUTILUS/AURIGA/…ALLEGRO/NAUTILUS/AURIGA/…resonant bar detectorsresonant bar detectors

ALLEGROALLEGROAURIGAAURIGA

Pulsar timing network, CMB anisotropyPulsar timing network, CMB anisotropy

The Crab nebula … a supernovae The Crab nebula … a supernovae remnant harboring a pulsar remnant harboring a pulsar

Segment of the CMB Segment of the CMB from WMAP from WMAP

source frequency (Hz)source frequency (Hz)

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relics from the big bang, inflationrelics from the big bang, inflation

exotic physics in the early universe: phase transitions, cosmic strings, domain walls, …exotic physics in the early universe: phase transitions, cosmic strings, domain walls, …

1-10 M1-10 M๏๏ BH/BH BH/BH

mergersmergers

NS/BH mergersNS/BH mergers

NS/NS mergersNS/NS mergers

pulsars, pulsars, supernovaesupernovae

EMR inspiralEMR inspiral

NS binariesNS binaries

WD binariesWD binaries

101022-10-1066 M M๏๏ BH/BH BH/BH

mergersmergers

>10>1066 M M๏๏ BH/BH mergersBH/BH mergers

CMB CMB anisotropyanisotropy

Pulsar timingPulsar timing LISALISA LIGO/…LIGO/…Bar Bar detectordetectorss

Overview of expected gravitational wave Overview of expected gravitational wave sourcessources

Binary black holes in the UniverseBinary black holes in the Universe• strong, though strong, though circumstantialcircumstantial evidence that evidence that

black holes are ubiquitous objects in the universeblack holes are ubiquitous objects in the universe

– supermassive black holes (supermassive black holes (101066 M M๏๏ - - 101099 M M๏๏) thought to ) thought to exist at the centers of most galaxiesexist at the centers of most galaxies

• high stellar velocities near the centers of galaxies, jets high stellar velocities near the centers of galaxies, jets in active galactic nuclei, x-ray emission, …in active galactic nuclei, x-ray emission, …

– more massive stars are expected to form BH’s at more massive stars are expected to form BH’s at the end of their livesthe end of their lives

• a few dozen candidate stellar mass black holes in x-ray a few dozen candidate stellar mass black holes in x-ray binary systems … companion too massive to be a binary systems … companion too massive to be a neutron starneutron star

VLA image of the galaxy NGC 326, with HST image VLA image of the galaxy NGC 326, with HST image of jets inset. CREDIT: NRAO/AUI, STScI (inset)of jets inset. CREDIT: NRAO/AUI, STScI (inset)

• detection of gravitational waves from BH detection of gravitational waves from BH mergers would provide mergers would provide directdirect evidence for black evidence for black holesholes, as well as give valuable information on , as well as give valuable information on stellar evolution theory and large scale structure stellar evolution theory and large scale structure formation and evolution in the universeformation and evolution in the universe

• this will also be an this will also be an unprecedentedunprecedented test of general test of general relativityrelativity, as the last stages of a merger takes , as the last stages of a merger takes place in the highly dynamical and non-linear place in the highly dynamical and non-linear strong-field regimestrong-field regime

Two merging galaxies in Abell 400. Credits: X-ray, NASA/CXC/ Two merging galaxies in Abell 400. Credits: X-ray, NASA/CXC/ AIfA/D.Hudson & T.Reiprich et al.; Radio: NRAO/VLA/NRL)AIfA/D.Hudson & T.Reiprich et al.; Radio: NRAO/VLA/NRL)

The two body problemThe two body problem• Newtonian gravity solution for the dynamics of two point-like Newtonian gravity solution for the dynamics of two point-like

masses in a bound orbit: motion along an ellipsemasses in a bound orbit: motion along an ellipse

• in general relativity there is no (analytic) solution … several in general relativity there is no (analytic) solution … several approximations with different realms of validityapproximations with different realms of validity

– test particle limittest particle limit

• geodesic motion of a particle about a black hole (i.e. self-gravity of particle geodesic motion of a particle about a black hole (i.e. self-gravity of particle is ignored)is ignored)

• already get some very interesting behavioralready get some very interesting behavior– perihelion precessionperihelion precession– unstable and chaotic orbitsunstable and chaotic orbits– ““zoom-whirl” behaviorzoom-whirl” behavior

– Post-Newtonian (PN) expansionsPost-Newtonian (PN) expansions

• self-gravity accounted for, though slow motion (relative to c) and weak self-gravity accounted for, though slow motion (relative to c) and weak gravitational fields assumedgravitational fields assumed

• begins to incorporate “radiation-reaction”; i.e. how the orbit decays via the begins to incorporate “radiation-reaction”; i.e. how the orbit decays via the emission of gravitational wavesemission of gravitational waves

– black hole (BH) perturbation theoryblack hole (BH) perturbation theory

• can be used to model the “ring-down” of the final BH that is formed in a can be used to model the “ring-down” of the final BH that is formed in a collisioncollision

• can also describe the radiation caused by a test particle in orbit about the can also describe the radiation caused by a test particle in orbit about the BHBH

• binary black hole mergersbinary black hole mergers– all the above assumptions break down close to the merger of all the above assumptions break down close to the merger of

comparable mass BHs: self gravity can’t be ignored, the comparable mass BHs: self gravity can’t be ignored, the gravitational fields are not weak, and the BHs are moving at gravitational fields are not weak, and the BHs are moving at sizeable fractions of the speed of lightsizeable fractions of the speed of light

From N. Cornish and J. Levin, CQG 20, 1649 (2003)From N. Cornish and J. Levin, CQG 20, 1649 (2003)

Numerical RelativityNumerical Relativity• Numerical relativity is concerned with solving the field equations of general relativityNumerical relativity is concerned with solving the field equations of general relativity

using computers. using computers.

• When written in terms of the spacetime metric, defined by the usual line elementWhen written in terms of the spacetime metric, defined by the usual line element

the field equations form a the field equations form a system of 10 coupled, non-linear, second order partial system of 10 coupled, non-linear, second order partial differential equations, each depending on the 4 spacetime coordinatesdifferential equations, each depending on the 4 spacetime coordinates

– it is this system of equations that we need to solve for the 10 metric elements (plus whatever it is this system of equations that we need to solve for the 10 metric elements (plus whatever matter we want to couple to gravity)matter we want to couple to gravity)

– for many problems this has turned out to be quite an undertaking, due in part to the for many problems this has turned out to be quite an undertaking, due in part to the mathematical complexity of the equations, and also the heavy computational resources mathematical complexity of the equations, and also the heavy computational resources required to solve themrequired to solve them

• The field equations may be complicated, but they are The field equations may be complicated, but they are thethe equations that we believe equations that we believe govern the structure of space and time (barring quantum effects and ignoring govern the structure of space and time (barring quantum effects and ignoring matter). That they can, in principle, be solved in many “real-universe” scenarios is a matter). That they can, in principle, be solved in many “real-universe” scenarios is a remarkable and unique situation in physics. remarkable and unique situation in physics.

πTG 8

dxdxgds 2

Minimal requirements for a formulation of the field equations Minimal requirements for a formulation of the field equations that that mightmight form the basis of a successful numerical integration form the basis of a successful numerical integration

schemescheme• Choose coordinates/system-of-variables that fix the character of the equationsChoose coordinates/system-of-variables that fix the character of the equations

– three common choicesthree common choices• free evolutionfree evolution — system of hyperbolic equations — system of hyperbolic equations• constrained evolutionconstrained evolution — system of hyperbolic and elliptic equations — system of hyperbolic and elliptic equations• characteristiccharacteristic or or null evolutionnull evolution — integration along the lightcones of the spacetime — integration along the lightcones of the spacetime

• For free evolution, need a system of equations that is well behaved off the For free evolution, need a system of equations that is well behaved off the ”constraint ”constraint manifold”manifold”

– analytically, if satisfied at the initial time the constraint equations of GR will be satisfied for all timeanalytically, if satisfied at the initial time the constraint equations of GR will be satisfied for all time

– numerically the constraints can only be satisfied to within the truncation error of the numerical numerically the constraints can only be satisfied to within the truncation error of the numerical scheme, hence we do not want a formulation that is “unstable” when the evolution proceeds scheme, hence we do not want a formulation that is “unstable” when the evolution proceeds slightly off the constraint manifoldslightly off the constraint manifold

• Need well behaved coordinates (or gauges) that do not develop pathologies when the Need well behaved coordinates (or gauges) that do not develop pathologies when the spacetime is evolvedspacetime is evolved

– typically need dynamical coordinate conditions that can adapt to unfolding features of the typically need dynamical coordinate conditions that can adapt to unfolding features of the spacetimespacetime

• Boundary conditions also historically a source of headaches Boundary conditions also historically a source of headaches

– naive BC’s don’t preserve the constraint nor are representative of the physicsnaive BC’s don’t preserve the constraint nor are representative of the physics– fancy BC’s can preserve the constraints, but again miss the physicsfancy BC’s can preserve the constraints, but again miss the physics– solution … compactify to infinity solution … compactify to infinity

• Geometric singularities in black hole spacetimes need to be dealt with Geometric singularities in black hole spacetimes need to be dealt with

Generalized Harmonic Evolution SchemeGeneralized Harmonic Evolution Scheme

• Einstein equations in Einstein equations in generalized harmonicgeneralized harmonic form with form with constraint dampingconstraint damping::

where the generalized harmonic constraints are ,where the generalized harmonic constraints are , is a constant, is a constant, nn is a unit time-like vector is a unit time-like vector

and and are the Christoffel symbols are the Christoffel symbols

• Need Need gauge evolution equationsgauge evolution equations to close the system; use the following with to close the system; use the following with and and nn constants, and constants, and is the so-called lapse function: is the so-called lapse function:

• Matter stress energy supplied by a massless scalar field Matter stress energy supplied by a massless scalar field : : 0

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Computational issues in solving numerical Computational issues in solving numerical solution of the field equationssolution of the field equations

• Each equation contains tens to hundreds of individual terms, requiring Each equation contains tens to hundreds of individual terms, requiring on the order of on the order of several thousand floating point operations several thousand floating point operations per grid per grid point point with any evolution scheme.with any evolution scheme.

• Problems of interest often have Problems of interest often have several orders of magnitude of several orders of magnitude of relevant physical length scalesrelevant physical length scales that need to be well resolved. In an that need to be well resolved. In an equal mass binary black hole merger for example:equal mass binary black hole merger for example:

• radius of each black hole radius of each black hole R~2MR~2M• orbital radius orbital radius ~ 20M (~ 20M (which is also the dominant wavelength of radiation emitted)which is also the dominant wavelength of radiation emitted)• outer boundary outer boundary ~ 200M~ 200M, as the waves must be measured in the weak-field , as the waves must be measured in the weak-field

regime to coincide with what detectors will seeregime to coincide with what detectors will see

– Can solve these problems with a combination of hardware technology — Can solve these problems with a combination of hardware technology — supercomputers — and software algorithms, in particular adaptive mesh supercomputers — and software algorithms, in particular adaptive mesh refinement (AMR)refinement (AMR)

• vast majority of numerical relativity codes today use finite difference techniques vast majority of numerical relativity codes today use finite difference techniques (predominantly 2(predominantly 2ndnd to 4 to 4thth order), notable exception is the Caltech/Cornell pseudo- order), notable exception is the Caltech/Cornell pseudo-spectral code spectral code

• How to deal with the true geometric singularities that exist inside all How to deal with the true geometric singularities that exist inside all black holes?black holes?

• excisionexcision

Brief (and incomplete) history of the binary Brief (and incomplete) history of the binary black hole problem in numerical relativityblack hole problem in numerical relativity

• L. Smarr, L. Smarr, PhD Thesis (1977)PhD Thesis (1977) : First head-on collision simulation : First head-on collision simulation

• P. Anninos, D. Hobill, E.Seidel, L. Smarr, W. Suen P. Anninos, D. Hobill, E.Seidel, L. Smarr, W. Suen PRL 71, 2851 (1993)PRL 71, 2851 (1993) : Improved : Improved simulation of head-on collisionsimulation of head-on collision

• B. Bruegmann B. Bruegmann Int. J. Mod. Phys. D8, 85 (1999)Int. J. Mod. Phys. D8, 85 (1999) : First grazing collision of two black holes : First grazing collision of two black holes

• mid 90’s-early 2000: Binary Black Hole Grand Challenge Alliancemid 90’s-early 2000: Binary Black Hole Grand Challenge Alliance– Cornell,PSU,Syracuse,UT Austin,U Pitt, UIUC,UNC, Wash. U, NWU … head-on collisions, Cornell,PSU,Syracuse,UT Austin,U Pitt, UIUC,UNC, Wash. U, NWU … head-on collisions,

grazing collisions, cauchy-characteristic matching, singularity excisiongrazing collisions, cauchy-characteristic matching, singularity excision

• B. Bruegmann, W. Tichy, N. Jansen B. Bruegmann, W. Tichy, N. Jansen PRL 92, 211101 (2004)PRL 92, 211101 (2004) : First full orbit of a quasi- : First full orbit of a quasi-circular binarycircular binary

• FP, FP, PRL 95, 121101 (2005)PRL 95, 121101 (2005) : First “complete” simulation of a non head-on merger event: : First “complete” simulation of a non head-on merger event: orbit, coalescence, ringdown and gravitational wave extractionorbit, coalescence, ringdown and gravitational wave extraction

• M. Campanelli, C. O. Lousto, P. Marronetti, Y. Zlochower M. Campanelli, C. O. Lousto, P. Marronetti, Y. Zlochower PRL 96, 111101, (2006)PRL 96, 111101, (2006);; J. G. J. G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter PRL 96, 111102, (2006)PRL 96, 111102, (2006) … several … several other groups have now repeated these results: PSU, Jena, AEI, LSU, Caltech/Cornellother groups have now repeated these results: PSU, Jena, AEI, LSU, Caltech/Cornell

– note that to go from “a to b” here has required a tremendous amount of research in understanding the note that to go from “a to b” here has required a tremendous amount of research in understanding the mathematical structure of the field equations, stable discretization schemes, dealing with geometric singularities mathematical structure of the field equations, stable discretization schemes, dealing with geometric singularities inside black holes, computational algorithms, initial data, extracting useful physical information from simulations, inside black holes, computational algorithms, initial data, extracting useful physical information from simulations, etc. etc.

Current state of the field Current state of the field

• Two quite different, stable methods of integrating the Einstein field Two quite different, stable methods of integrating the Einstein field equations for this problemequations for this problem

– generalized harmonic coordinates with constraint dampinggeneralized harmonic coordinates with constraint damping, , F.Pretorius, F.Pretorius, PRL 95, 121101 (2005)PRL 95, 121101 (2005)

• Caltech/Cornell, Caltech/Cornell, L. Lindblom et al., Class.Quant.Grav. 23 (2006) S447-S462L. Lindblom et al., Class.Quant.Grav. 23 (2006) S447-S462

• PITT/AEI/LSU, PITT/AEI/LSU, B. Szilagyi et al., gr-qc/0612150B. Szilagyi et al., gr-qc/0612150

– BSSN with “moving punctures”,BSSN with “moving punctures”, M. Campanelli, C. O. Lousto, P. Marronetti, Y. M. Campanelli, C. O. Lousto, P. Marronetti, Y. Zlochower PRL 96, 111101, (2006); J. G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Zlochower PRL 96, 111101, (2006); J. G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter PRL 96, 111102, (2006)Meter PRL 96, 111102, (2006)

• Pennstate, Pennstate, F. Herrmann et al., gr-gc/0601026F. Herrmann et al., gr-gc/0601026

• Jena/FAU, Jena/FAU, J. A. Gonzalez et al., gr-gc/06010154J. A. Gonzalez et al., gr-gc/06010154

• LSU/AEI/UNAM, LSU/AEI/UNAM, J. Thornburg et al., gr-gc/0701038J. Thornburg et al., gr-gc/0701038

• U.Tokyo/UWM, U.Tokyo/UWM, M. Shibata et al, astro-ph/0611522M. Shibata et al, astro-ph/0611522

• U. Sperhake, gr-qc/0606079U. Sperhake, gr-qc/0606079

Evolution of Cook-Pfeiffer Quasi-Evolution of Cook-Pfeiffer Quasi-circular BBH Merger Initial datacircular BBH Merger Initial data

This animation shows the This animation shows the lapse functionlapse function in the in the orbital plane.orbital plane.

The lapse function The lapse function represents the relative represents the relative time dilation between a time dilation between a hypothetical observer at hypothetical observer at the given location on the the given location on the grid, and an observer grid, and an observer situated very far from the situated very far from the system --- the redder the system --- the redder the color, the slower local color, the slower local clocks are running relative clocks are running relative to clocks at infinityto clocks at infinity

If this were in “real-time” If this were in “real-time” it would correspond to the it would correspond to the merger of two ~5000 merger of two ~5000 solar mass black holessolar mass black holes

Initial black holes are Initial black holes are close to non-spinning close to non-spinning Schwarzschild black holes; Schwarzschild black holes; final black hole is Kerr a final black hole is Kerr a black hole with spin black hole with spin parameter parameter ~0.7~0.7

A. Buonanno, G.B. Cook and F.P.; A. Buonanno, G.B. Cook and F.P.; gr-qc/0610122gr-qc/0610122

Gravitational waves from the Gravitational waves from the simulationsimulation

A depiction of the A depiction of the gravitational waves gravitational waves emitted in the orbital emitted in the orbital plane of the binary. plane of the binary. Shown is the real Shown is the real component of the component of the Newman Penrose scalar Newman Penrose scalar , which in the wave , which in the wave zone is proportional to zone is proportional to the second time the second time derivative of the usual derivative of the usual plus-polarizationplus-polarization

What does the merger wave represent? What does the merger wave represent? • Scale the system to two Scale the system to two 10 solar mass (~10 solar mass (~ 2x102x103131 kg) BHs kg) BHs

– radius of each black hole in the binary is ~ radius of each black hole in the binary is ~ 30km30km

– radius of final black hole is ~ radius of final black hole is ~ 60km60km

– distance from the final black hole where the wave was measured ~distance from the final black hole where the wave was measured ~ 1500km 1500km

– frequency of the wave ~frequency of the wave ~ 200Hz (early inspiral) - 800Hz (ring-down) 200Hz (early inspiral) - 800Hz (ring-down)

– fractional oscillatory “distortion” in space induced by the wave transverse to fractional oscillatory “distortion” in space induced by the wave transverse to the direction of propagation has a the direction of propagation has a maximummaximum amplitude amplitude DL/LDL/L ~ 3x10~ 3x10-3-3

• a 2m tall person will get stretched/squeezed by ~ a 2m tall person will get stretched/squeezed by ~ 6 mm6 mm as the wave passes as the wave passes

• LIGO’s arm length would change by ~ LIGO’s arm length would change by ~ 12m12m. Wave amplitude decays like . Wave amplitude decays like 1/distance from source; e.g. at 10Mpc the change in arms ~ 1/distance from source; e.g. at 10Mpc the change in arms ~ 5x105x10-17-17m m (1/20 the (1/20 the radius of a proton, which is well within the ballpark of what LIGO is trying to radius of a proton, which is well within the ballpark of what LIGO is trying to measure!!)measure!!)

– despite the seemingly small amplitude for the wave, the energy it carries is despite the seemingly small amplitude for the wave, the energy it carries is enormous — around enormous — around 10103030 kg c kg c22 ~ 10 ~ 104747 J ~ 10 J ~ 105454 ergs ergs

• peak luminosity is about 1/100peak luminosity is about 1/100thth the Planck luminosity of 10 the Planck luminosity of 105959ergs/s !!ergs/s !!• luminosity of the sun ~ 10luminosity of the sun ~ 103333ergs/s, a bright supernova or milky-way type galaxy ~ ergs/s, a bright supernova or milky-way type galaxy ~

10104242 ergs/s ergs/s

Summary of equal mass quasi-circular merger Summary of equal mass quasi-circular merger resultsresults

• Remarkable simplicity in the waveformRemarkable simplicity in the waveform

– ““plunge/merger” phase very short, ~ 10-20M plunge/merger” phase very short, ~ 10-20M – waveform dominated by the quadrupole modewaveform dominated by the quadrupole mode– despite some initial eccentricity, the inspiral phase can be well described as a quasi-despite some initial eccentricity, the inspiral phase can be well described as a quasi-

circular inspiral driven by quadrupole GW emissioncircular inspiral driven by quadrupole GW emission– close to 4% of the total initial energy of the system is released in gravitational waves; close to 4% of the total initial energy of the system is released in gravitational waves;

roughly ½ of this in the final couple of orbits, and ½ during the merger/ringdown phase.roughly ½ of this in the final couple of orbits, and ½ during the merger/ringdown phase.

• Though the merger phase is short, it is characterized by a steep rise in the Though the merger phase is short, it is characterized by a steep rise in the frequency of the waveformfrequency of the waveform

– spans a broad range of frequencies, and depending upon the black hole masses could be spans a broad range of frequencies, and depending upon the black hole masses could be the dominant contribution to the LIGO/LISA signal the dominant contribution to the LIGO/LISA signal

• Pre & post merger well approximated by perturbative methodsPre & post merger well approximated by perturbative methods

– Post-Newtonian for the inspiralPost-Newtonian for the inspiral– Black hole perturbation theory for the ring down phaseBlack hole perturbation theory for the ring down phase

• Given the short time between the inspiral and ringdown phases, seems Given the short time between the inspiral and ringdown phases, seems reasonable that it might be possible to construct an analytic mode of the entire reasonable that it might be possible to construct an analytic mode of the entire merger eventmerger event

– will need to understand the non-linear excitation of quasi-normal modes (QNM), and will need to understand the non-linear excitation of quasi-normal modes (QNM), and probably need higher order PN methods, as naïve constructions quite sensitive to the probably need higher order PN methods, as naïve constructions quite sensitive to the time of the matchtime of the match

Beyond equal mass, non-spinning Beyond equal mass, non-spinning mergersmergers

• Detailed studies underway by several groups --- a couple of highlights:Detailed studies underway by several groups --- a couple of highlights:

– ““kick” velocity of remnant black hole due to asymmetric beaming of kick” velocity of remnant black hole due to asymmetric beaming of radiationradiation

• up to 175km/s for non-spinning, unequal mass components [up to 175km/s for non-spinning, unequal mass components [F. Herrmann et al., F. Herrmann et al., gr-qc/0601026; J.G. Baker et al. astro-ph/0603204; J.A. Gonzalez et al., gr-qc/0610154gr-qc/0601026; J.G. Baker et al. astro-ph/0603204; J.A. Gonzalez et al., gr-qc/0610154]]

• typical values for spinning black holes of 100’s km/s, but can be as large as typical values for spinning black holes of 100’s km/s, but can be as large as 4000km/s for equal mass black holes with spins vectors anti-aligned and in the 4000km/s for equal mass black holes with spins vectors anti-aligned and in the orbital plane [orbital plane [F. Herrmann et al., gr-qc/0701143; M. Koppitz et al., gr-qc/0701163; M. Campanelli et al. F. Herrmann et al., gr-qc/0701143; M. Koppitz et al., gr-qc/0701163; M. Campanelli et al. gr-qc/0701164gr-qc/0701164 & gr-qc/gr-qc/0702133 & gr-qc/gr-qc/0702133]]

– uniform sampling over spin vector orientations and mass ratios for two a=.9 black holes uniform sampling over spin vector orientations and mass ratios for two a=.9 black holes with m1/m2 between 1 and 10 suggested only around 2% parameter space has kicks with m1/m2 between 1 and 10 suggested only around 2% parameter space has kicks larger than 1000km/s, and 10% larger than 500km/s [larger than 1000km/s, and 10% larger than 500km/s [J. Schnittman & A. Buonanno, J. Schnittman & A. Buonanno, astro-ph/0702641astro-ph/0702641]]

– astrophysical population is most likely highly non-uniform, e.g. torques from accreting gas astrophysical population is most likely highly non-uniform, e.g. torques from accreting gas in supermassive merger scenarios tend to align the spin and orbital angular momenta, in supermassive merger scenarios tend to align the spin and orbital angular momenta, which will result in more modest kick velocities <~200km/s [which will result in more modest kick velocities <~200km/s [T.Bogdanovic et al, T.Bogdanovic et al, astro-ph/0703054astro-ph/0703054]]

– strong spin-orbit coupling effects near merger that can cause significant strong spin-orbit coupling effects near merger that can cause significant precession of the orbital plane and orientation of the spins, as well as precession of the orbital plane and orientation of the spins, as well as enhance (reduce) the gravitational wave energy for spins aligned (anti-enhance (reduce) the gravitational wave energy for spins aligned (anti-aligned) with the orbital angular momentum aligned) with the orbital angular momentum [[M. Campanelli, et al., PRD 74, 041051(2006) &M. Campanelli, et al., PRD 74, 041051(2006) & gr-qc/0612076gr-qc/0612076 ]]

Scalar field collapse driven binariesScalar field collapse driven binaries

• Look at equal mass mergersLook at equal mass mergers

– initial scalar field pulses separated a coordinate (proper) distance initial scalar field pulses separated a coordinate (proper) distance 8.9M8.9M ((10.8M10.8M ) on the x-axis, one boosted with ) on the x-axis, one boosted with boost parameterboost parameter kk in the in the +y+y direction, the other with direction, the other with kk in the in the -y-y direction direction

• note, resultant black hole velocities are related to, but not equal to note, resultant black hole velocities are related to, but not equal to kk

• To find interesting orbital dynamics, tune the parameter To find interesting orbital dynamics, tune the parameter kk to get to get as many orbits as possibleas many orbits as possible

– in the limit as in the limit as kk goes to 0, get head-on collisions goes to 0, get head-on collisions– in the large in the large kk limit, black holes are deflected but fly apart limit, black holes are deflected but fly apart

• Generically these black hole binaries will have some eccentricity Generically these black hole binaries will have some eccentricity (not easy to define given how close they are initially), and so (not easy to define given how close they are initially), and so arguably of less astrophysical significancearguably of less astrophysical significance

– want to explore the non-linear interaction of BH’s in full general want to explore the non-linear interaction of BH’s in full general relativity relativity

Scalar field Scalar field rr, compactified (code) , compactified (code) coordinatescoordinates

)2/tan(),2/tan(),2/tan( zzyyxx

Sample OrbitSample Orbit

Lapse function Lapse function , orbital plane, orbital plane

Lapse and Gravitational WavesLapse and Gravitational Waves6/8h resolution, v=0.21909 merger example6/8h resolution, v=0.21909 merger example

Real component of the Newman-Penrose Real component of the Newman-Penrose scalar scalar 44( times rM), orbital plane( times rM), orbital plane

The threshold of “immediate” mergerThe threshold of “immediate” merger

• Tuning the parameter Tuning the parameter kk between merger between merger and deflection, one approaches a very and deflection, one approaches a very interesting dynamical regimeinteresting dynamical regime

• the black holes move into a state of the black holes move into a state of near-circular evolution before either near-circular evolution before either merging (merging (k<k*k<k*), or moving apart ), or moving apart again (again (k<k*k<k*)---at least temporarily)---at least temporarily

• for for kk near near k*k* , there is exponential , there is exponential sensitivity of the resultant evolution sensitivity of the resultant evolution to the initial conditions. In fact, the to the initial conditions. In fact, the number of orbits number of orbits nn spent in this spent in this phase scales approximately likephase scales approximately like

where for this particular set of initial where for this particular set of initial conditions conditions

• In all cases, to within numerical error In all cases, to within numerical error the spin parameter of the final black the spin parameter of the final black hole for the cases that merge is hole for the cases that merge is a~0.7a~0.7

*kken

The threshold of “immediate” mergerThe threshold of “immediate” merger• The binary is still radiating significant amounts of energy (on the order of 1-1.5% The binary is still radiating significant amounts of energy (on the order of 1-1.5% per per

orbitorbit), yet the black holes do not spiral in), yet the black holes do not spiral in

• Technical comment: the coordinates seem to be very well adapted to the physics of the Technical comment: the coordinates seem to be very well adapted to the physics of the situation, as the situation, as the coordinatecoordinate motion plugged verbatim into the quadrupole formula for two motion plugged verbatim into the quadrupole formula for two point masses gives a very good approximated to the actual numerical waveform point masses gives a very good approximated to the actual numerical waveform measured in the far-field regime of the simulationmeasured in the far-field regime of the simulation

• taking these coordinates at face-value, they imply the binaries are orbiting well taking these coordinates at face-value, they imply the binaries are orbiting well within the inner-most stable circular orbit of the equivalent Kerr spacetime of the within the inner-most stable circular orbit of the equivalent Kerr spacetime of the final black holefinal black hole

Kerr equatorial geodesic Kerr equatorial geodesic analogue analogue

work with work with D. Khurana, D. Khurana,

gr-qc/0702084 gr-qc/0702084

• We can play the same fine-tuning game with equatorial geodesics on a black hole backgroundWe can play the same fine-tuning game with equatorial geodesics on a black hole background

– here, we tune between capture or escape of the geodesichere, we tune between capture or escape of the geodesic

– regardlessregardless of the initial conditions, at threshold one tunes to one of the unstable circular orbits of the initial conditions, at threshold one tunes to one of the unstable circular orbits of the of the Kerr geometry (Kerr geometry (for equatorial geodesics)for equatorial geodesics)

• I.e. I.e. anyany smooth, one parameter family of geodesics that has the property that at one extreme of the smooth, one parameter family of geodesics that has the property that at one extreme of the parameter the geodesic falls into the black hole, while at the other extreme it escapes, exhibits this behaviorparameter the geodesic falls into the black hole, while at the other extreme it escapes, exhibits this behavior

un-bound orbit exampleun-bound orbit example bound orbit example; the threshold orbit in this bound orbit example; the threshold orbit in this case is sometimes referred to as a homoclinic case is sometimes referred to as a homoclinic

orbitorbit

Kerr geodesic analogue cont. Kerr geodesic analogue cont.

• Can quantify the unstable behavior by calculating the Lyapunov Can quantify the unstable behavior by calculating the Lyapunov exponentexponentof the orbit (or in this case calling it an “instability” of the orbit (or in this case calling it an “instability” exponent may be more accurate)exponent may be more accurate)

– easy way: measure easy way: measure n(k-k*)n(k-k*) , and find the slope , and find the slope of of nn versus versus –ln|k-k*|–ln|k-k*| … … ..

– easier(?) way: do a perturbation theory calculation (following easier(?) way: do a perturbation theory calculation (following N. Cornish and J. Levin, CQG 20, 1649 (2003)N. Cornish and J. Levin, CQG 20, 1649 (2003)) …) …

12/3

2220

22

2/122

022

2

//

/21,2

,244

32

mamrm

rmarRmrar

mrmarRm

rr

Comparison of Comparison of • The dashed black lines are the The dashed black lines are the

previous formula evaluated for previous formula evaluated for various ranges of a & rvarious ranges of a & r

• The colored dots are from The colored dots are from calculations finding the capture-calculations finding the capture-threshold using numerical threshold using numerical integration of geodesicsintegration of geodesics

• Technical note: analytic Technical note: analytic expression derived using expression derived using Boyer-Lindquist Boyer-Lindquist coordinates, numerical coordinates, numerical integration done in Kerr-integration done in Kerr-Schild coordinates, though Schild coordinates, though neither expect nor see neither expect nor see significant differencessignificant differences

• The red-dashed ellipse is the The red-dashed ellipse is the single “dot” from the full single “dot” from the full numerical experiment numerical experiment performed. The size of the performed. The size of the ellipse is an indication of the ellipse is an indication of the numerical uncertainty, though numerical uncertainty, though the trend suggests that r is the trend suggests that r is slowly decreasing with the slowly decreasing with the approach to threshold, so the approach to threshold, so the ellipse may move a bit to the ellipse may move a bit to the left if one could tune closer to left if one could tune closer to thresholdthreshold

• interestingly, the final spin interestingly, the final spin parameter of the black hole parameter of the black hole that forms in the merger that forms in the merger case is ~0.7case is ~0.7

How far can this go in the non-linear How far can this go in the non-linear case?case?

• System is losing energy, and quite rapidly, so there must be a limit to the System is losing energy, and quite rapidly, so there must be a limit to the number of orbits we can getnumber of orbits we can get

• Hawking’s area theoremHawking’s area theorem: assume cosmic censorship and “reasonable” : assume cosmic censorship and “reasonable” forms of matter, then net area of all black holes in the universe can forms of matter, then net area of all black holes in the universe can notnot decrease with timedecrease with time

– the area of a single, isolated black hole is: the area of a single, isolated black hole is:

– initially, we have two non-rotating initially, we have two non-rotating (J=0)(J=0) black holes, each with mass black holes, each with mass M/2M/2::

– maximum energy that can be extracted from the system is if the final black hole maximum energy that can be extracted from the system is if the final black hole is also non-rotating:is also non-rotating:

in otherwords, the maximum energy that can be lost is a factor in otherwords, the maximum energy that can be lost is a factor 1-1/√2 ~ 29%1-1/√2 ~ 29%

– If the trend in the simulations continues, and the final If the trend in the simulations continues, and the final J~0.7MJ~0.7M22, we still get close , we still get close to to 24%24% energy that could be radiated energy that could be radiated

• the simulations show around 1-1.5% energy is lost per whirl, so we may get close to 15-the simulations show around 1-1.5% energy is lost per whirl, so we may get close to 15-30 orbits at the threshold of this fine-tuning process!30 orbits at the threshold of this fine-tuning process!

4

22 118

M

JMA

22 816 MMA ff

28 MAi

Can we go even further?Can we go even further?• The preceding back-of-the-envelope calculation assumed the The preceding back-of-the-envelope calculation assumed the

energy in the system was dominated by the rest mass of the energy in the system was dominated by the rest mass of the black holesblack holes

• What about the black hole scattering problem?What about the black hole scattering problem?

– give the black holes sizeable boosts, such that the net energy of give the black holes sizeable boosts, such that the net energy of the system is dominated by the kinetic energy of the black holesthe system is dominated by the kinetic energy of the black holes

– set up initial conditions to have a one-parameter family of solutions set up initial conditions to have a one-parameter family of solutions that smoothly interpolate between coalescence and scatter that smoothly interpolate between coalescence and scatter

• ““natural” choice is the impact parameternatural” choice is the impact parameter

– it is it is plausibleplausible that at threshold, that at threshold, allall of the kinetic energy is of the kinetic energy is converted to gravitational radiation (think of what happens to a converted to gravitational radiation (think of what happens to a “failed” merger, and what the resultant orbit “failed” merger, and what the resultant orbit mustmust look like in the look like in the limit)limit)

• this can be an this can be an arbitrarily large fraction of the total energyarbitrarily large fraction of the total energy of the system of the system (scale the rest mass to zero as the boost goes to 1) (scale the rest mass to zero as the boost goes to 1)

An application to the LHC?An application to the LHC?• The Large Hadron Collider (LHC) is a particle accelerator currently The Large Hadron Collider (LHC) is a particle accelerator currently

under construction near Lake Geneva, Switzerlandunder construction near Lake Geneva, Switzerland

– it will be able to collide beams of protons with center of mass it will be able to collide beams of protons with center of mass energies up to energies up to 14 TeV14 TeV

• In recent years the idea of In recent years the idea of large extra dimensionslarge extra dimensions have become have become popular [popular [N. Arkani-Hamed , S. Dimopoulos & G.R. Dvali, Phys.Lett.B429:263-272; L. Randall & N. Arkani-Hamed , S. Dimopoulos & G.R. Dvali, Phys.Lett.B429:263-272; L. Randall & R. Sundrum [Phys.Rev.Lett.83:3370-3373R. Sundrum [Phys.Rev.Lett.83:3370-3373]]

– we (ordinary particles) live on a 4-dimensional we (ordinary particles) live on a 4-dimensional branebrane of a of a higher dimensional spacetimehigher dimensional spacetime

• ““large” extra dimensions are sub-mm in size, but large large” extra dimensions are sub-mm in size, but large compared to the 4D compared to the 4D Planck lengthPlanck length of of 1010-33-33 cm cm

– gravity propagates in all dimensionsgravity propagates in all dimensions

• The 4D The 4D Planck Energy,Planck Energy, where we expect quantum where we expect quantum gravity effects to become important, is gravity effects to become important, is 10101919 GeV GeV; ; however the presence of extra dimensions can change however the presence of extra dimensions can change the “true” Planck energy the “true” Planck energy

• A Planck scale in the TeV range is preferred as this A Planck scale in the TeV range is preferred as this solves the hierarchy problemsolves the hierarchy problem

• current experiments rule out Planck energies <~ current experiments rule out Planck energies <~ 1TeV1TeV

• Collisions of particles with super-Planck energies in Collisions of particles with super-Planck energies in these scenarios would cause black holes to be these scenarios would cause black holes to be produced at the LHC!produced at the LHC!

• can “detect” black holes by observing energy loss can “detect” black holes by observing energy loss (from gravitational radiation or newly formed black (from gravitational radiation or newly formed black holes escaping the detector) and/or measuring the holes escaping the detector) and/or measuring the particles that should be produced as the black holes particles that should be produced as the black holes decay via Hawking radiationdecay via Hawking radiation

The black hole scattering problem The black hole scattering problem

• Consider the high speed collision of two black holes with impact parameter Consider the high speed collision of two black holes with impact parameter bb

– good approximation to the collision of two partons if energy is beyond the Planck regimegood approximation to the collision of two partons if energy is beyond the Planck regime

– for sufficiently high velocities charge and spin of the parton will be irrelevant (though both will for sufficiently high velocities charge and spin of the parton will be irrelevant (though both will probably be important at LHC energies) probably be important at LHC energies)

• threshold of immediate merger threshold of immediate merger mustmust exist exist

• if similar scaling behavior is seen as with geodesics and full simulations of the equal mass/low if similar scaling behavior is seen as with geodesics and full simulations of the equal mass/low velocity regime in general, can use the geodesic analogue to obtain an approximate idea of the cross velocity regime in general, can use the geodesic analogue to obtain an approximate idea of the cross section and energy loss to radiation vs. impact parameter … Ingredients:section and energy loss to radiation vs. impact parameter … Ingredients:

– map geodesic motion on a Kerr back ground with map geodesic motion on a Kerr back ground with (M,a)(M,a) to the scattering problem with total initial to the scattering problem with total initial energy energy E=ME=M and angular momentum and angular momentum aa of the black hole that’s formed near threshold of the black hole that’s formed near threshold

– find find andand b* b* using geodesic motionusing geodesic motion

– assume a constant fraction assume a constant fraction of the remaining energy of the system is radiated per orbit near of the remaining energy of the system is radiated per orbit near threshold (estimate using quadrupole formulathreshold (estimate using quadrupole formula))

– Integrate near-threshold scaling relation to find Integrate near-threshold scaling relation to find E(b)E(b) with the above parameters and the following with the above parameters and the following “boundary” conditions: “boundary” conditions: E(0), E(b*)E(0), E(b*) and and E(infinity)

• E(b*)E(b*) must be ~ 1 in kinetic energy dominated regime must be ~ 1 in kinetic energy dominated regime• E(infinity)=0E(infinity)=0• E(0)E(0) … need some other input, either perturbative calculations, or full numerical simulations. … need some other input, either perturbative calculations, or full numerical simulations.

The black hole scattering problem The black hole scattering problem

• What value of the Kerr spin parameter to use?What value of the Kerr spin parameter to use?

– in the ultra-relativistic limit the geodesic asymptotes to the light-ring at in the ultra-relativistic limit the geodesic asymptotes to the light-ring at thresholdthreshold

– it also seems “natural” that in this limit the final spin of the black hole at it also seems “natural” that in this limit the final spin of the black hole at threshold is threshold is a=1a=1. This is consistent with simple estimates of energy/angular . This is consistent with simple estimates of energy/angular momentum radiated momentum radiated

• quadrupole physics gives the following for the relative rates at which energy vs. quadrupole physics gives the following for the relative rates at which energy vs. angular momentum is radiated in a circular orbit with orbital frequency angular momentum is radiated in a circular orbit with orbital frequency ::

• for the scattering problem with the same impact parameter as a threshold geodesic on for the scattering problem with the same impact parameter as a threshold geodesic on an extremal Kerr background, the initial an extremal Kerr background, the initial J/EJ/E22=1=1. The Boyer-Lindquist value of . The Boyer-Lindquist value of EE is ½ for is ½ for a geodesic on the light ring of an extremal Kerr BH, in that regime a geodesic on the light ring of an extremal Kerr BH, in that regime d(J/ Ed(J/ E22)=0)=0

• But now we have a bit of a dilemma, as the extremal Kerr background has But now we have a bit of a dilemma, as the extremal Kerr background has no no unstableunstable circular geodesics, and hence circular geodesics, and hence tends to infinity in this limit tends to infinity in this limit

– will use will use aa close to but not exactly 1 to find out what close to but not exactly 1 to find out what E(b)E(b) might look like might look like

2

2

211)/(

E

J

Edn

dE

Edn

EJd

Sample energy radiated vs. impact parameter curves Sample energy radiated vs. impact parameter curves (normalized) (normalized)

• An estimate of E(0) from An estimate of E(0) from Cardoso et al., Class.Quant.Grav. 22 Cardoso et al., Class.Quant.Grav. 22 (2005) L61-R84(2005) L61-R84

• Cross section for black hole Cross section for black hole formation (b<~1) at the LHC formation (b<~1) at the LHC would thus be would thus be 22EE22, though , though notice that a significant amount notice that a significant amount of energy could be lost to of energy could be lost to gravitational waves even if gravitational waves even if black holes are not formed black holes are not formed (b>1). Suggests effective cross (b>1). Suggests effective cross section signaling strong section signaling strong gravitational interaction could gravitational interaction could be several times larger than be several times larger than thisthis

• dE/dn ~ dE/dn ~ /40/40 in this limit, so expect all the in this limit, so expect all the energy to be radiated away in around a dozen energy to be radiated away in around a dozen orbits. orbits.

ConclusionsConclusions

• the next few of decades are going to be a very exciting time for the next few of decades are going to be a very exciting time for gravitational physicsgravitational physics

– numerical simulations are finally beginning to reveal the fascinating numerical simulations are finally beginning to reveal the fascinating landscape of binary coalescence with Einstein’s theory of general relativity landscape of binary coalescence with Einstein’s theory of general relativity

• most of parameter space still left to exploremost of parameter space still left to explore

– the “extreme” regions, though perhaps not astrophysically relevant, will be the most the “extreme” regions, though perhaps not astrophysically relevant, will be the most challenging to simulate, and may reveal some of the more interesting aspects of the challenging to simulate, and may reveal some of the more interesting aspects of the theorytheory

– gravitational wave detectors should allow us to see the universe in gravitational wave detectors should allow us to see the universe in gravitational radiation for the first timegravitational radiation for the first time

• even if we only see what we expect to see we can learn a lot about the universe, even if we only see what we expect to see we can learn a lot about the universe, though history tells us that each time a new window into the universe has been though history tells us that each time a new window into the universe has been opened, surprising things have been discoveredopened, surprising things have been discovered

• if we don’t see anything, something is “broken” … unless it’s the detectors even if we don’t see anything, something is “broken” … unless it’s the detectors even that will be a remarkable discovery that will be a remarkable discovery