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Black Holes Mergers. Frans Pretorius Princeton University XXIèmes Rencontres de Blois June 25, 2009. Outline …. Numerical relativity and black hole mergers: brief history and current state of the field

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black holes mergers

Black Holes Mergers

Frans Pretorius

Princeton University

XXIèmesRencontres de Blois June 25, 2009

outline
Outline …
  • Numerical relativity and black hole mergers: brief history and current state of the field
  • Categorize the two-body problem in general relativity into two classes: rest mass dominated and kinetic energy dominated mergers
  • rest mass dominated mergers: relevant to gravitational wave astronomy
      • phases: “Newtonian”, inspiral quasi-circular inspiral, plunge/merger, ringdown
      • brief overview of a couple of the more interesting results recently uncovered by numerical simulations
        • “simplicity” of the merger waveform
        • large recoil velocities
outline1
… Outline
  • kinetic energy dominated mergers: relevant to super-Planck scale particle collisions
    • though what do black hole collisions have to do with particle collisions?
    • black hole scattering problem: either a deflection or a merger, the latter will be followed by a ringdown phase
    • either case could be accompanied by copious amounts of gravitational wave emission
    • may contain regimes where the luminosity approaches the Planck luminosity, and the remnant black hole in a merger is near-extremal
brief and incomplete history of the binary black hole problem in numerical relativity
Brief (and incomplete) history of the binary black hole problem in numerical relativity
  • Hahn and Lindquist, Ann. Phys. 19, 304 (1964)First simulation of “wormhole” initial data
  • L. Smarr, PhD Thesis (1977): First head-on collision simulation
  • P. Anninos, D. Hobill, E.Seidel, L. Smarr, W. SuenPRL 71, 2851 (1993): Improved simulation of head-on collision
  • B. BruegmannInt. J. Mod. Phys. D8, 85 (1999) : First grazing collision of two black holes
  • mid 90’s-early 2000: Binary Black Hole Grand Challenge Alliance
    • Cornell,PSU,Syracuse,UTAustin,U Pitt, UIUC,UNC, Wash. U, NWU … head-on collisions, grazing collisions, cauchy-characteristic matching, singularity excision
  • B. Bruegmann, W. Tichy, N. Jansen PRL 92, 211101 (2004): First full orbit of a quasi-circular binary
  • FP, PRL 95, 121101 (2005): First “complete” numerical solution of a non head-on merger event: orbit, coalescence, ringdown and gravitational wave extraction, using generalized harmonic coordinates
  • M. Campanelli, C. O. Lousto, P. Marronetti, Y. ZlochowerPRL 96, 111101, (2006);J. G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter PRL 96, 111102, (2006)obtained similar stable evolution using the “BSSN” formalism
    • 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 inside black holes, computational algorithms, initial data, extracting useful physical information from simulations, etc.
current state of the field
Current state of the field
  • Two quite different, stable methods of integrating the Einstein field equations for this problem
    • generalized harmonic with constraint damping, F.Pretorius, PRL 95, 121101 (2005)
      • Caltech/Cornell
      • LSU/LUI/BYU
      • PITT/AEI/LSU-CCT
      • Princeton
      • UMD
    • BSSN with “moving punctures” M. Campanelli, C. O. Lousto, P. Marronetti, Y. Zlochower PRL 96, 111101, (2006); J. G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter PRL 96, 111102, (2006)
      • Jena/FAU
      • LSU/AEI/UNAM
      • NASA Goddard
      • Pennstate
      • RIT
      • UIUC/Bowdoin
      • U. Sperhake
      • U.Tokyo/UWM
motivation to study astrophysical mergers gravitational wave observatories
Motivation to study astrophysical mergers:gravitational wave observatories!

LIGO/VIRGO/GEO/TAMA

LISA

ground based laser interferometers

space-based laser interferometer (hopefully with get funded for a 20?? Launch)

LIGO Hanford

LIGO Livingston

ALLEGRO/NAUTILUS/AURIGA/…

Pulsar timing network, CMB anisotropy

resonant bar detectors

Segment of the CMB from WMAP

AURIGA

The Crab nebula … a supernovae remnant harboring a pulsar

ALLEGRO

slide7

Overview of expected gravitational wave sources

Pulsar timing

LISA

LIGO/…

Bar detectors

CMB anisotropy

>106 M๏ BH/BH mergers

102-106 M๏ BH/BH mergers

source “strength”

1-10 M๏ BH/BH mergers

NS/BH mergers

NS/NS mergers

pulsars, supernovae

EMR inspiral

NS binaries

WD binaries

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

relics from the big bang, inflation

10-12

10-8

10-4

1

104

source frequency (Hz)

anatomy of a rest mass dominated merger
Anatomy of a Rest Mass Dominated Merger
  • In the conventional scenario of a black hole merger in the universe, one can break down the evolution into 4 stages: Newtonian, inspiral, plunge/merger and ringdown
  • Newtonian
    • in isolation, radiation reaction will cause two black holes of mass M in a circular orbit with initial separation R to merge within a time tm relative to the Hubble time tH
    • label the phase of the orbit Newtonian when the separation is such that the binary will take longer than the age of the universe to merge, for then to be of relevance to gravitational wave detection, other “Newtonian” processes need to operate, e.g. dynamical friction, n-body encounters, gas-drag, etc. For e.g.,
      • two solar mass black holes need to be within 1 million Schwarzschild radii ~ 3 million km
      • two 109 solar mass black holes need to be within 6 thousand Schwarzschild radii ~ 1 parsec
anatomy of a rest mass dominated merger1
Anatomy of a Rest Mass Dominated Merger
  • inspiral  quasi-circular inspiral (QSI)
    • In the inspiral phase, energy loss through gravitational wave emission is the dominate mechanism forcing the black holes closer together
    • to get an idea for the dominant timescale during inspiral, for equal mass, circular binaries the Keplarian orbital frequency offers a good approximation until very close to merger
      • the dominant gravitational wave frequency is twice this
    • Post-Newtonian techniques provide an accurate description of certain aspects of the process until remarkably close to merger
    • if the initial pericenter of the orbit is sufficiently large, the orbit will loose its eccentricity long before merger [Peters & Matthews, Phys.Rev. 131 (1963)] and become quasi-circular
anatomy of a rest mass dominated merger2
Anatomy of a Rest Mass Dominated Merger
  • plunge/merger
    • this is the time in the merger when the two event horizons coalesce into one
      • we know the two black holes must merge into one if cosmic censorship holds (and no indications of a failure yet in any merger simulations)
    • full numerical solution of the field equations are required to solve for the geometry of spacetime in this stage
      • Only within the last 3 years, following a couple of breakthroughs, has numerical relativity been able to complete the picture by filling in the details of the final, non-perturbative phase of the merger
      • At present two known stable formulations of the field equations, generalized harmonic[FP, PRL 95, 121101 (2005)], and BSSN with moving punctures[M. Campanelli, C. O. Lousto, P. Marronetti, Y. Zlochower PRL 96, 111101, (2006); J. G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter PRL 96, 111102, (2006)]
    • in all cases studied to date, this stage is exceedingly short, leaving its imprint in on the order of 1-2 gravitational wave cycles, at roughly twice the final orbital frequency
anatomy of a rest mass dominated merger3
Anatomy of a Rest Mass Dominated Merger
  • ringdown
    • in the final phase of the merger, the remnant black hole “looses all its hair”, settling down to a Kerr black hole
    • one possible definition for when plunge/merger ends and ringdown begins, is when the spacetime can adequately be described as a Kerr black hole perturbed by a set of quasi-normal modes (QNM)
    • the ringdown portion of the waveform will be dominated by the fundamental harmonic of the quadrupole QNM, with characteristic frequency and decay time [Echeverria, PRD 34, 384 (1986)]:j=a/Mf , the Kerr spin parameter of the black hole
sample evolution cook pfeiffer quasi circular initial data
Sample evolution --- Cook-Pfeiffer Quasi-circular initial data

A. Buonanno, G.B. Cook and F.P.; Phys.Rev.D75:124018,2007

  • This animation shows the lapse function in the orbital plane.The lapse function represents the relative time dilation between a hypothetical observer at the given location on the grid, and an observer situated very far from the system --- the redder the color, the slower local clocks are running relative to clocks at infinityIf this were in “real-time” it would correspond to the merger of two ~5000 solar mass black holes
  • Initial black holes are close to non-spinning Schwarzschild black holes; final black hole is a Kerr a black hole with spin parameter ~0.7, and ~4% of the total initial rest-mass of the system is emitted in gravitational waves
slide13

Gravitational waves from the simulation

A depiction of the gravitational waves emitted in the orbital plane of the binary. Shown is the real component of the Newman Penrose scalar y4, which in the wave zone is proportional to the second time derivative of the usual plus-polarization

The plus-component of the wave from the same simulation, measured on the axis normal to the orbital plane

what does the merger wave represent
What does the merger wave represent?
  • Scale the system to two 10 solar mass (~2x1031 kg) BHs
    • radius of each black hole in the binary is ~ 30km
    • radius of final black hole is ~ 60km
    • distance from the final black hole where the wave was measured ~ 1500km
    • frequency of the wave ~ 200Hz (early inspiral) - 800Hz (ring-down)
what does the merger wave represent1
What does the merger wave represent?
  • fractional oscillatory “distortion” in space induced by the wave transverse to the direction of propagation has a maximum amplitude DL/L~ 3x10-3
      • a 2m tall person will get stretched/squeezed by ~ 6 mm as the wave passes
      • LIGO’s arm length would change by ~ 12m. Wave amplitude decays like 1/distance from source; e.g. at 10Mpc the change in arms ~ 5x10-17m (1/20 the radius of a proton, which is well within the ballpark of what LIGO is trying to measure!!)
  • despite the seemingly small amplitude for the wave, the energy it carries is enormous — around 1030 kg c2 ~ 1047 J ~ 1054 ergs
      • peak luminosity is about 1/100th the Planck luminosity of 1059ergs/s !!
      • luminosity of the sun ~ 1033ergs/s, a bright supernova or milky-way type galaxy ~ 1042 ergs/s
      • if all the energy reaching LIGO from the 10Mpc event could directly be converted to sound waves, it would have an intensity level of ~ 80dB
highlights of recent results simplicity of merger waveform
Highlights of recent results: simplicity of merger waveform
  • the “non-linear” phase of the merger is surprisingly short
  • great boon for data analysis, as this suggests an efficient LIGO template bank could be compiled by stitching together quick-to-calculate perturbative waveforms, guided by a handful of numerical waveforms
  • to-date, some of the best examples employthe Effective One Body (EOB) approach [A. Buonanno and T. Damour, et al., Phys.Rev.D59:084006,1999]. Example hereis for quasi-circular inspiral of non-spinningBH’s [A. Buonanno et al., Phys.Rev.D76:104049,2007]
    • EOB inspiral connected to 3 leading QNMs
    • added “pseudo” 4PN term to EOB model, with coefficient determined by a best-fit match to a set of numerical results
    • used simulation results for final spin and black hole mass to fix the QNM frequencies and decay constants

4:1 mass ratio example

highlights of recent results large recoil velocities
Highlights of recent results: large recoil velocities
  • significant recoil can be imparted to the remnant black hole due to asymmetric beaming of radiation during the merger, up to 4000km/s in some cases
    • Herrmann et al., gr-qc/0701143; Koppitz et al., gr-qc/0701163; Campanelli et al. gr-qc/0701164 & gr-qc/0702133, Gonzalez et al, arXiv:gr-qc/0702052, Tichy & Marronetti, arXiv:gr-qc/0703075v1
  • there are far reaching consequences to this, some that could be detected via electromagnetic observations, in particular for supermassive black hole mergers
    • offset or double galactic nuclei, displaced active galactic nuclei, wiggling jets, enlarged cores, lopsided cores, x-ray afterglows, feedback trails, off-center flares from tidally disrupted stars, hypervelocity stars, a population of galaxies without supermassive black holes, etc.
      • Merritt et al., ApJ. 607 (2004) L9-L12; Milosavljevic & Phinney, ApJ 622, L93(2005); Gualandris & Merrit arXiv:0708.0771 & , arXiv:0708.3083; Lippai et al. arXiv:0801.0739; Kornreich & Lovelace, arXiv:0802.2058; Devecchi et al. arXiv:0805.2609; Komossa & Merrit arXiv:0807.0223 & arXiv:0811.1037; Fujita arXiv:0808.1726 & arXiv:0810.1520
    • a 2650km/s recoiling black hole could explain the emission line spectra from quasar SDSSJ092712.65+294344.0 [S. Komossa et al., ApJ.678:L81,2008]
kinetic energy dominated mergers the black hole scattering problem
Kinetic energy dominated mergers: the black hole scattering problem

m2,v2

b

m1,v1

  • consider v~c. In general two, distinctend-states possible
    • for b<b* one black hole, after a collision
    • for b>b* two isolated black holes, after a deflection
  • because there are two distinct end-states, there must be some kind of threshold behavior approaching the critical impact parameter b*
motivation black hole formation at the lhc and in the atmosphere
Motivation: Black hole formation at the LHC and in the atmosphere?
  • large extra dimension scenarios [N. Arkani-Hamed , S. Dimopoulos & G.R. Dvali, PLB429:263-272; L. Randall & R. Sundrum, PRL.83:3370-3373] suggest the true Planck scale can be very different from what then would be an effective 4-dimensional Planck scale of 1019GeV calculated from the fundamental constants measured on our 4-D Brane
  • In the TeVrange is a “natural” choice to solve the hierarchy problem
  • Implications of this are that super-TeV particle collisions would probe the quantum gravity regime
    • collisions sufficiently above the Planck scale are expected to be dominated by the gravitational interaction, and arguments suggest that black hole formation will be the most likely result of the two-particle scattering event [Banks & Fishlerhep-th/9906038, Dimopoulos & Landsberg PRL 87 161602 (2001), Feng & Shapere, PRL 88 021303 (2002), …]

One of the water tanks at the Pierre Auger Observatory

  • current experiments rule out a Planck scale of <~ 1TeV
  • The LHC should reach center-of-mass energies of ~ 10 TeV
  • cosmic rays can have even higher energies than this, and so in both cases black hole formation could be expected
  • these black holes will be small and decay rapidly via Hawking radiation, which is the most promising route to detection
    • if a lot of gravitation radiation is produced during the collision this could show up as a missing energy signal at the LHC

ATLAS experiment at the LHC

but do super planck scale particle collisions form black holes
But do super-Planck scale particle collisions form black holes?
  • The argument that the ultra-relativistic collision of two particles should form a black hole is purelyclassical, and is essentially based on Thorne’s hoop conjecture
    • (4D) if an amount of matter/energy E is compacted to within a sphere of radius R=2GE/c4 corresponding to the Schwarzschild radius of a black hole of mass M=E/c2, a black hole will form
    • applied to the head-on collision of two “particles” each with rest mass m, characteristic size W, and center-of-mass frame Lorentz gamma factors g, this says a black hole will form if the Schwarzschild radius corresponding to the total energy E=2mg is greater that W
    • the quantum physics comes in when we say that the particle’s size is given by its de Broglie wavelength W= hc/E, from which one gets the Planck energy Ep=(hc5/G)1/2
evidence to support this
Evidence to support this
  • From the classical perspective, evidence to support this would be solutions to the field equations demonstrating that weakly self-gravitating objects, when boosted toward each other with large velocities so that the net mass of the space time (in the center of mass frame) is dominated by the kinetic energy, generically form a black hole when the interaction occurs within a region smaller than the Schwarzschild radius of the spacetime
    • generic: the outcome would have to be independent of the particular details of the structure and non-gravitational interactions between the particles, if the classical picture is to have any bearing on the problem
    • interaction: the non-linear interaction of the gravitational kinetic energy of the boosted particles will be key in determining what happens
      • consider the trivial counter-examples to the hoop conjecture applied to a single particle boosted to ultra-relativistic velocities, or a white hole “explosion”
      • What is oft quoted as evidence comes from studies of the infinite boost limit of BH collisions [Penrose 1974, Eardley & Giddings PRD 66, 044011 (2002)]
        • however, it is not obvious that this describes large-but-finite collisions: it is a singular limit, the gravitational field changes character from Petrov type D (Coloumb-like) to type N (pure gravitational wave) and is nowhere a good approximation to a boosted massive particle geometry, the spacetime is no longer asymptotically flat, … etc
high speed soliton collision simulations
High speed soliton collision simulations
  • Try to test this hypothesis by colliding self-gravitating solitons, boson stars in this case (work with M.W.Choptuik)
  • The follow use initial conditions where the Thorne estimates says g>11
    • get BH formation at g>3 already!

scalar field magnitude, g~3

g=4

high speed black hole collisions
High Speed Black Hole Collisions
  • If these soliton collisions are confirming the expected generic behavior of high energy particle collisions, this implies we can use any model of a particle to study the classical gravitational signatures of super-Planck scale collisions, including black holes!
  • Will describe some on-going studies of such scenarios with U. Sperhake, V. Cardose, E. Berti, J.A Gonzalez, T. Hinderer & N. Yunes (see also M. Shibata, H. Okawa & T. Yamamoto, PRD 78:101501, 2008)
  • However, with application to the LHC in mind, such a project can only be pursued in some approximate manner
    • unknown Planck-scale physics
    • unknown structure of the extra dimensions
    • 4D simulations “barely” feasible … generic 10(11?) dimensional simulations impossible in foreseeable future
aside t he planck luminosity regime
Aside : the Planck Luminosity regime
  • The Planck Luminosity Lp is
    • notice that Planck’s constant does not enter, hence this is a regime that is ostensibly described by classical general relativity
    • Lp may be a limiting luminosity for “reasonable” processes, and trying to reach it may reveal some of the more interesting aspects of the theory
      • would super-Planck luminosities be associated with violations of cosmic censorship?
the planck luminosity regime
The Planck Luminosity regime
  • Suppose a single black hole is formed during the collision, and a fraction e of the total energy of the system 2gmc2 (equal mass) is released as gravitational waves. Estimate that the shortest timescale on which the energy can be released is the light-crossing time of the remnant black hole ~2R/c = 4G(1-e)M/c3. This gives
  • Implies that a very efficient (>80%) prompt energy release mechanism will result in super-Planck luminosity, however
    • we know in the low-speed regime e is small (fraction of a percent)
    • if the limiting solution is a collision of Aichelburg-Sexl shock waves, Penrose found trapped surfaces at the moment of collision implying e<29%; seems to be consistent with simulation results (next slides)
    • expect to get much larger emission rates in grazing collisions
g 2 9 collision example
g=2.9 collision example

T=0 M0

(M=gM0 )

Re[Y4]

g 2 9 collision example1
g=2.9 collision example

T=30 M0

(M=gM0 )

Re[Y4]

g 2 9 collision example2
g=2.9 collision example

T=83 M0

(M=gM0 )

Re[Y4]

g 2 9 collision example3
g=2.9 collision example

T=115M0

(M=gM0 )

Re[Y4]

g 2 9 collision example4
g=2.9 collision example

T=137 M0

(M=gM0 )

Re[Y4]

g 2 9 collision example5
g=2.9 collision example

T=156 M0

(M=gM0 )

Re[Y4]

g 2 9 collision example6
g=2.9 collision example

T=202 M0

(M=gM0 )

Re[Y4]

g 2 9 collision example7
g=2.9 collision example

T=252 M0

(M=gM0 )

Re[Y4]

g 2 9 collision example8
g=2.9 collision example

T=292 M0

(M=gM0 )

Re[Y4]

g 2 9 collision example9
g=2.9 collision example

T=405 M0

(M=gM0 )

Re[Y4]

g 2 9 collision example10
g=2.9 collision example
  • roughly 8% of the total energy is emitted in gravitational waves during the collision … peak luminosity ~0.3% of Lp
    • about a factor of 6 less than the order-of-magnitude estimate (the relevant time scale seems to be given by the dominant quasi-normal mode frequency of the final black hole, the period of which is a few times larger that the light-crossing time of the black hole)
  • spectrum is reasonably flat (left) below a cut-off given by the QNM frequencies of the black hole, a result predicted by the “zero-frequency-limit” (ZFL) approximation
  • extrapolation to infinite boost limit using a curve motivated by the ZFL (right, red line) suggeststhe result will be ~ 14 %, a bit less than 1/2 the Penrose bound
conclusions
Conclusions
  • We are learning a lot about thethe nature of gravity through numericalsolution of the field equations
  • Black hole collisions are a fascinating probe of strong-field general relativity, though it is remarkable how “simple” some of the these highly non-linear, dynamical scenarios are turning out to be
    • merger phase and waveforms from quasi-circular inspiral
    • in the ultra-relativistic (UR) limit that the leading order behavior of the dynamics of the collision is captured by the cut-and-paste Penrose construction
  • Future work
    • barely scratched the surface of the UR limit – things are becoming very interesting in non head-on collisions : much higher luminosities, nearly extremely rotating BH’s are formed, onset of “zoom-whirl” behavior