Transcript Towards Gravitational Wave Astronomy

Black Holes Mergers
Frans Pretorius
Princeton 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), “zoomwhirl” behavior is seen, and the remnant black hole in a merger is nearextremal
• 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/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 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 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)]:
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
Kinetic energy dominated mergers: the black hole
scattering problem
m2,v2
b
m1,v1
• 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
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=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
W
2Ggm/c2
1
g
2
1 v
1
c2
Hoop Conjecture and Particle Collisions
v
v
W
2Ggm/c2
1
g
2
1 v
2
c2
Hoop Conjecture and Particle Collisions
v
v
W
2Ggm/c2
1
g
2
1 v
5
c2
Hoop Conjecture and Particle Collisions
v
Black hole forms!
W
2Ggm/c2
1
g
2
1 v
 10
c2
v
Evidence to support this
• From the classical perspective, evidence to support this would be
solutions to the field equations demonstrating that weakly selfgravitating 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?
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
g11 in the center of mass frame will be the marginal case
Case 1: free-fall collision from rest
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
•
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)
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 typeconstruction 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
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
head-on collision example, g2.9
Re[Y4]
Head-on collisions
•
For the g2.9 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 socalled “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(g1) 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 /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, g1.5
Re[Y4]
Whirl, then scatter , g1.5
Re[Y4]
Whirl, then merger, g1.5
Re[Y4]
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, m0 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), … ]
v=c
v=c
?
2GE/c2
Minkowski
Minkowski
Minkowski
No characteristic width; black hole
always forms in head-on collision
spacetime diagram
of collision
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
 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