Transcript Towards Gravitational Wave Astronomy

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
• 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
… Outline
• kinetic energy dominated mergers: relevant to superPlanck 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
•
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. Suen PRL 71, 2851 (1993) : Improved simulation of
head-on collision
•
B. Bruegmann Int. J. Mod. Phys. D8, 85 (1999) : First grazing collision of two black holes
•
mid 90’s-early 2000: Binary Black Hole Grand Challenge Alliance
•
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
–
Cornell,PSU,Syracuse,UT Austin,U Pitt, UIUC,UNC, Wash. U, NWU … head-on collisions, grazing collisions,
cauchy-characteristic matching, singularity excision
coordinates
•
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) 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
• 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!
LIGO/VIRGO/GEO/TAMA
ground based laser interferometers
LISA
space-based laser interferometer (hopefully
with get funded for a 20?? Launch)
LIGO Hanford
LIGO Livingston
ALLEGRO/NAUTILUS/AURIGA/…
resonant bar detectors
Pulsar timing network, CMB anisotropy
Segment of the CMB
from WMAP
AURIGA
ALLEGRO
The Crab nebula … a supernovae
remnant harboring a pulsar
Overview of expected gravitational wave sources
source “strength”
Pulsar timing
CMB
anisotropy
LISA
LIGO/…
Bar
detectors
>106 M๏ BH/BH mergers
102-106 M๏ BH/BH
mergers
EMR inspiral
1-10 M๏ BH/BH
mergers
NS/BH mergers
NS binaries
NS/NS mergers
WD binaries
pulsars,
supernovae
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
•
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
 tm   M
   
 tH   M 
 R 
 6 
 10 Rs 
4
– 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 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

1

2 2
M
 M   Rs 

11
kHz

 
R3
M

 R 
3/ 2
• 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 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 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)]:
QNM
M
 32 kHz 
2
 M
 QNM
 M
 20s
 M 


0.3
 1  0.63(1  j )

 1  j 0.45

0.3
 1  0.631  j 
j=a/Mf , the Kerr spin parameter of the black hole

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
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
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 pluspolarization
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
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 employ
the Effective One Body (EOB) approach
[A. Buonanno and T. Damour, et al.,
Phys.Rev.D59:084006,1999]. Example here
is for quasi-circular inspiral of non-spinning
BH’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
•
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. grqc/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, offcenter 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
m2,v2
b
m1,v1
• consider v~c. In general two, distinct end-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?
•
large extra dimension scenarios [N. Arkani-Hamed , S. Dimopoulos & G.R. Dvali,
•
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
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
–
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
•
ATLAS experiment at the LHC
One of the water tanks at the Pierre Auger Observatory
if a lot of gravitation radiation is produced during the
collision this could show up as a missing energy signal
at the LHC
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=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
•
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
• 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
• 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 : the Planck Luminosity regime
• The Planck Luminosity Lp is
c5
Lp   1052 W  1059 ergs/s
G
– 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
• 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
e
L
Lp
41  e 
• 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
T0 M0
(M=g M0 )
Re[Y4]
g2.9 collision example
T30 M0
(M=g M0 )
Re[Y4]
g2.9 collision example
T83 M0
(M=g M0 )
Re[Y4]
g2.9 collision example
T115M0
(M=g M0 )
Re[Y4]
g2.9 collision example
T137 M0
(M=g M0 )
Re[Y4]
g2.9 collision example
T156 M0
(M=g M0 )
Re[Y4]
g2.9 collision example
T202 M0
(M=g M0 )
Re[Y4]
g2.9 collision example
T252 M0
(M=g M0 )
Re[Y4]
g2.9 collision example
T292 M0
(M=g M0 )
Re[Y4]
g2.9 collision example
T405 M0
(M=g M0 )
Re[Y4]
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)
suggests the result will be ~ 14 %, a bit less than 1/2 the Penrose bound
Conclusions
• We are learning a lot about the
the nature of gravity through
numerical solution 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