Black Holes Mergers

Frans PretoriusPrinceton University

UC Davis Physics Department Colloquium

March 15, 2010

Outline• Motivation: why explore black hole collisions?

– gravitational wave astronomy, and understanding the highly dynamical, strong-field regime of general relativity

• rest mass dominated mergers: of relevance to gravitational wave astronomy

– brief overview of the anatomy of black hole mergers in the universe − adiabatic inspiral

• kinetic energy dominated collisions: of relevance to putative scenarios of black hole formation at the LHC

– the black hole scattering problem

– aside: but what do black hole collisions have to do with particle collisions?

– recent results: may contain regimes where near 100% of the total energy of the system is radiated in gravitational waves (even for a deflection), “zoom-whirl” behavior is seen, and the remnant black hole in a merger is near-extremal

• Conclusions

Motivation: why study black hole collisions?

• gravitational wave astronomy

– almost overwhelming circumstantial evidence that black holes exist in our universe

– to obtain conclusive evidence, we need to “see” the black holes in the “light” they emit … gravitational waves. However, isolated single black holes do not radiate, so we need to look for binary mergers for the cleanest direct signature of the existence of black holes

– understanding the nature of the waves emitted in the process is important for detecting such events, and moreover will be crucial in deciphering the signals

• extracting the parameters of the binary

• obtain clues about the environment of the binary

• how accurately does Einstein’s theory describe the event?

The network of gravitational wave detectors

LIGO Hanford

LIGO Livingston

ground based laser interferometersLIGO/VIRGO/GEO/TAMA

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

LISA

ALLEGRO/NAUTILUS/AURIGA/…resonant bar detectors

ALLEGROAURIGA

Pulsar timing network, CMB anisotropy

The Crab nebula … a supernovae remnant harboring a pulsar

Segment of the CMB from WMAP

source frequency (Hz)

sou

rce

“str

en

gth

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10410-12 10-8 10-4 1

relics from the big bang, inflation

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

1-10 M๏ BH/BH mergers

NS/BH mergers

NS/NS mergers

pulsars, supernovae

EMR inspiral

NS binaries

WD binaries

102-106 M๏ BH/BH mergers

>106 M๏ BH/BH mergers

CMB anisotropy

Pulsar timing LISA LIGO/…Bar detectors

Overview of expected gravitational wave sources

Anatomy of a 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

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Anatomy of a 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 few 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 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

3.0

45.0

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120

)1(63.01kHz322

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QNM

QNM

Sample evolution --- Cook-Pfeiffer Quasi-circular initial data

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 infinity

If 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

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

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?

• 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 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

Kinetic energy dominated mergers: the black hole scattering problem

• consider v~c. In general, at least two, distinct end-states possible

• for b<b* one black hole, after a collision

• for b>b* two isolated black holes, after a deflection

bm1,v1

m2,v2

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 1019 GeV calculated from the fundamental constants measured on our 4-D Brane

• In the TeV range 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 & Fishler hep-th/9906038, Dimopoulos & Landsberg PRL 87 161602 (2001), Feng & Shapere, PRL 88 021303 (2002), …]

• 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

One of the water tanks at the Pierre Auger Observatory

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 purely classical, 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=2mc2g 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

Hoop Conjecture and Particle Collisions

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Hoop Conjecture and Particle Collisions

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Hoop Conjecture and Particle Collisions

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Hoop Conjecture and Particle Collisions

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v vBlack hole forms!

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”

Evidence to support this

• 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

• Another argument is that black hole formation results from a strong focusing of one particle from the passage of the near-shock wave geometry (as seen in its rest frame) of the other [Kaloper & Terning Int.J.Mod.Phys.D17:665-672,2008]

– purely geodesic argument, and gives correct order of magnitude for critical impact parameter

– though, considering an equal mass collision, if particle A is perturbed by the background geometry of particle B, one can flip the perspective and come to the same conclusion re. particle B being perturbed; furthermore, if a black hole does form, the geometry changes drastically relative to that of either particle’s. How then can one trust the test-body calculation?

– considering a solition, or star like model of a finite sized particle rather than representing it as a point, as the focusing starts to compress the star, won’t internal pressure forces counteract this?

» h

– Geodesics

– Pressure of soliutons

High speed soliton collision simulations

• Test this hypothesis by colliding self-gravitating solitons, boson stars in this case (M.W. Choptuik & FP arXiv:0908.1780 [gr-qc])

• Very computationally expensive to run high-g simulations, so need to start with a relatively compact boson star that will reach hoop-conjecture limits with reasonable g ’s.

• choose parameters to give a boson star with R/2M ~ 22

– thus, hoop-conjecture suggests a collision of two of these with =11g in the center of mass frame will be the marginal case

Case 1: free-fall collision from rest

• Here, gravity dominates the interaction, causing the boson stars to coalesce into a single, highly perturbed boson star (this case eventually collapses to form a black hole)

Symmetry axis

Both the color and height of

the surface represent the magnitude of

the scalar field. Scale M is the

total rest-mass of of the boson

stars

Case 2 : g = 2

• Here, though gravity strongly perturbs the boson stars, kinetic energy “wins” and causes them to pass through each other

– soliton-like interference pattern can be seen as the boson star matter interacts

– superposition of initial data, and subsequent truncation, cause some component of the field to move in the wrong direction; the truncation error part converges away with resolution, the initial data part lessens the further the initial separation

Case 3 : g = 4

• Here, the early matter interaction looks similar, but now the gravitational interaction of the kinetic energy of the solitons causes gravitational collapse and black hole formation

– NOTE: gauge than previous case: the coordinate spreading of the solitons before collision, and shrinking of the horizon afterwards, are just coordinate effects; also, different color scale

High speed particle collisions

• What is remarkable here is that both geodesic focusing arguments and the Penrose type-construction capture the “leading order” behavior in a process which seems to be highly dynamical and non-linear interaction

– There is no dynamics, or interaction in either treatment … the former looks at geodesics of Schwarzschild, the latter cutting and pasting two geometries together

– The only place where the Einstein equations entered was the static single particle model-geometries

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. Cardoso, E. Berti, J.A Gonzalez, T. Hinderer & N. Yunes (see also M. Shibata, H. Okawa & T. Yamamoto, PRD 78:101501, 2008)

– results obtain with U. Sperhake’s Lean code (BSSN with moving punctures)

• However, with application to the LHC in mind, such a project can only be pursued in some approximate manner

– unknown Planck-scale physics & structure of the extra dimensions

– 4D simulations “barely” feasible … generic 10(11?) dimensional simulations impossible in foreseeable future

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?

ergs/s10W10 59525

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cLp

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

pLe

eL

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head-on collision example, =2.9g

Re[Y4]

Head-on collisions• For the =2.9g example, roughly 8% of the total energy is emitted in gravitational

waves during the collision … this gives 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) suggests the result will be ~ 14 %, about 1/2 the Penrose bound

Grazing collisions• Things get much more interesting for small, yet non-zero impact parameters,

or grazing collisions

– luminosity and total radiated energy increase significantly, even in close-encounter scattering cases

– in merger examples, remnant black hole acquires a significant spin, close to extremal

– hypothesize in the large g limit, near a “critical” impact parameter, close to 100% of the energy of the system could be radiated in gravitational waves

• plots below are from a relatively modest g of 1.5, yet the luminosity is considerably larger than the extrapolated infinite boost head-on collision case (numerical errors ~ 5-15% in luminosty, ~3% in final spin)

Threshold behavior• One reason for this is the existence of a threshold in parameter space, dividing

two distinct solutions (i.e. cannot “smoothly deform” spacetime from one to the other)

– for small impact parameters the end-state is a single black hole, while for large impact parameters there are two black holes flying apart

• The following illustrates what could happen as one tunes to threshold, assuming smooth dependence of the trajectories as a function of b

• non-spinning case (so we have evolution in a plane), and only showing one of the BH trajectories for clarity

• solid blue (black) – merger (escape)

• dashed blue (black) – merger (escape) for values of b closer to threshold

Threshold behavior

• Continuing the fine tuning then, one would expect the trajectories to approach a circular “whirl”

• This is exactly what happens in the geodesic limit, and one gets so-called “zoom-whirl” orbits

– these orbits are intimately related to the existence of unstable spherical orbits about black holes

• Going away from the test-particle limit, such zoom-whirl behavior is seen (FP and D. Khurana, CQG. 24, S83, 2007 M.C. Washik et al., PRL 101:061102,2008, J. Healy et al., arXiv:0905.3914 , J. Healy et al., arXiv:0907.0671 ), though radiation reaction prevents the whirling behavior from being sustained indefinitely

• In theory the whirl phase could be sustained until all the “excess” kinetic energy 2m( -1g ) is radiated away, which in the large g case is an arbitrarily large fraction of the total energy of the system

Threshold behavior• Quadrupole physics does a very good job of describing the

energy emission during low-speed (equal mass) whirling

– applied to the high speed regime suggests a fraction p/40 ~ 8% of the total energy is radiated per orbit … perhaps can get on the order of a dozen or so orbits by fine-tuning the impact parameter?

• However, an interesting phenomena seems to be happening in the high-speed limit where a huge enhancement relative to the quadrupole estimate occurs

– E.g. already with the modest value g=1.5 case we get ~20%/whirl, and an early look with g=2.9 gives ~35% (and we’re still far from threshold)

– in an approximate effective one body description, the reason appears to be that in the high-speed limit the orbital whirl frequency coincides with the dominant quasi-normal oscillation frequency of the effective black hole, so there is a resonant emission of gravitational waves [Berti et al., arXiv:1003.0812v1[gr-qc]]

– suggests in this regime one can obtain exceedingly high luminosities with off-center collisions, without fine-tuning the impact parameter

Threshold behavior• Note also, for the high speed

scattering problem, the threshold impact parameter where one gets whirling will be less than the critical impact parameter resulting in a single final black hole end-state

– for impact parameters slightly larger than the whirl-threshold, sufficient energy is lost during the whirl phase that the two black holes become a bound system, and will merge at some point in the future

– maximum total energy radiated, and largest final remnant spin, seems to occur for impact parameters at the whirl-threshold From g=1.5

example

Scatter example, =1.5g

Re[Y4]

Whirl, then scatter , =1.5g

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Whirl, then merger, =1.5g

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Conclusions• Over the past few years numerical solution of the Einstein

field equations has filled in many gaps in our knowledge of the black hole merger problem– one surprising (unsurprising) and interesting (reassuring) aspect of many of

the new solutions is how much of the phenomenology can be explained in terms of existing perturbative or approximate models, even in regimes where one would think they should not apply

• The next few years will probably bring several new interesting solutions– just beginning to scratch the surface of the kinetic energy dominated regime

– with application to the LHC in mind, if the 4D examples are any indication, existing estimates based on trapped surface calculations and geodesic properties of higher dimensional calculations give decent estimates of the cross section of black hole formation, however they vastly underestimate the “lost energy” signal due to gravitational wave emission for most impact parameters

– need to begin including the effects of higher dimensions to confirm this

The infinite boost limit• Give an arbitrary, static, charge-free, spherically symmetric soliton of

gravitational rest mass m a Lorentz boost of g. Take the limit g, m0 such that the energy E= mg is constant. In the limit, the solution is given by the Aichelburg-Sexl spacetime [GRG 2, 303 (1971)] (originally derived as the infinite boost limit of a Schwarzschild black hole)

• the spacetime is a plane-fronted gravitational “shock”-wave, with Minkowski spacetime on either side of the shock

• Collide two such spacetimes together … even though the solution to the future of the collision is not known, trapped surfaces can be found at the moment of collision [Penrose 1974, Eardley & Giddings PRD 66, 044011 (2002), Yoshino & Rychkov, PRD D71, 104028 (2005), … ]

2GE/c2

v=c v=c

No characteristic width; black hole always forms in head-on collision

spacetime diagram of collision

?

Minkowski

Minkowski

Minkowski

The infinite boost limit• The presence of trapped surfaces thus gives an argument in

favor of the hypothesis of black hole formation in ultrarelativstic collisions, however

– this is a singular limit, and only in the most trivial sense does the solution provide a good approximation to the geometry of a finite-g soliton (i.e. it’s a good approximation when you’re far enough away that the soliton spacetime is close to Minkowksi)

• going to the infinite boost limit, the algebraic type of the Weyl tensor changes from type D (two distinct eigenvectors) to type N (1 distinct eigenvector), and the spacetime ceases to be asymptotically flat

– when two finite-gamma solitions collide, the non-trivial spacetime dynamics that will (or will not) cause black hole formation will unfold precisely in the regime where the Aichelberg-Sexl solution is not a good approximation

– simulations of boson star collisions [D. Choi et al; K. Lai, PhD Thesis 2004, C. Palenzuela et al Phys.Rev.D75:064005,2007] suggest that gravity becomes weaker in the interaction as the initial velocity is increased

Anatomy of a 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

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