b.s. sathyaprakash cardiff university
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Gravitational Waves from Binary Coalescences Looking for Needle in a haystack, Mondragone School, Rome September 7, 2004. B.S. Sathyaprakash Cardiff University. Gravitational Waves: Ripples in the Fabric of Spacetime. - PowerPoint PPT PresentationTRANSCRIPT
Gravitational WavesGravitational Wavesfrom Binary Coalescencesfrom Binary Coalescences
Looking for Needle in a haystack, Mondragone School, Rome
September 7, 2004
B.S. SathyaprakashCardiff University
Sept 7, 2004 2Gravitational Waves from Binaries
• In Newton’s law of gravity the gravitational potential is given by Poisson’s equation:
2(t, X)= 4G(t,X)
• In general relativity for weak gravitational fields, for which one can assume that background metric is nearly flat
g = + h where |h| << 1,• Einstein’s equations reduce to wave equations:
h = 8GT .• Gravitational waves are caused by asymmetric motion
and non-stationary fields• According to Einstein’s general relativity gravity is not
a force but a warping of spacetime: Gravitational waves are ripples in the curvature of spacetime that carry information about changing gravitational fields
Gravitational Waves:Ripples in the Fabric of Spacetime
Sept 7, 2004 3Gravitational Waves from Binaries
Gravitational Wave Observables
• Quadrupole Formula gives luminosity, amplitude and frequency of GW:– Luminosity
L = (Asymmetry) v10
– Luminosity is a strong function of velocity: A black hole binary source brightens up a million times in just a few minutes before merger
– Amplitude
h = (Asymmetry) (M/R) (M/r)– The amplitude gives strain
caused in space as the wave propagates
– Frequency
f = √– Dynamical frequency in the
system
Quasi-normal modes from a BH at 0.1-1 Gpc can generate detectable
amplitudes
Sept 7, 2004 4Gravitational Waves from Binaries
Period Decay in Hulse-Taylor BinaryIn 1974 Hulse and Taylor observed the first pulsar in a binary
• Two neutron stars in orbit– Each has mass 1.4 times the
mass of the Sun. Orbital period ~ 7.5 Hrs
– the stars are whirling around each other at ~ a thousandth the speed of light
• According to Einstein’s theory the binary should emit GW – Emission of the waves causes the
two stars to spiral towards each other and a decrease in the orbital period
– This decrease in period - about 10 micro seconds per year - is exactly as predicted by Einstein’s theory
• Eventually the binary will coalesce Eventually the binary will coalesce emitting a burst of GW that will be emitting a burst of GW that will be observable using instruments that observable using instruments that are currently being builtare currently being built
But that will take another 100 million
years
Sept 7, 2004 5Gravitational Waves from Binaries
Discovery of fastest binary pulsar
Burgay et al Nature 2003
• A brief history of pulsar discoveries– First pulsar, Carb PSR1919+21: Hewish and Bell 1967
– First binary pulsar PSR 1913+16: Hulse and Taylor 1974
– First millisecond pulsar PSR 1937+21 : Backer et al 1982
• Fastest known binary pulsar J0737-3039: Burgay et al 2003– In December 2003 Burgay et al discovered a new pulsar in a
binary J0737-3039 that is expected to open a new area of astrophysics/astronomy
– Strongly relativistic (period 2.5 Hrs), mildly eccentric (0.088), highly inclined (i > 87 deg)
– Faster than PSR 1913+16, J7037-3039 is the most relativistic neutron star binary
– Greatest periastron advance: d/dt 16.8 degrees per year (thought to be fully general relativistic) – indeed very large compared to relativistic part of Mercury’s perihelion advance of 42 sec per century
Sept 7, 2004 6Gravitational Waves from Binaries
Sept 7, 2004 7Gravitational Waves from Binaries
Discovery of the second pulsarLyne et al Science 2004
• Soon the companion was detected directly and confirmed to be a pulsar
• B has a spin period much larger: 2.5 s as opposed to 2.25 ms of A
Sept 7, 2004 8Gravitational Waves from Binaries
Masses of the component stars
• Six parameters, that are a function of the two masses, can be measured– (1) Periastron
advance, (2) gravitational red-shift, (3) mass ratio, shapiro time delay pulse (4) “range” and (5) “shape”, (6) orbital decay due to GW emission
• Masses are roughly 1.34 and 1.25 solar masses
Sept 7, 2004 9Gravitational Waves from Binaries
Nature of GW Observations
• Interferometric antennas are broadband detectors– Ground-based: 1-2 kHz
bandwidth around 100 Hz
– LISA: 0.1 Hz bandwidth around 1 millihertz
– Can observe different states of a source in the same detector and follow the phasing of the waves
– Should be possible to deduce the dynamics of the source from the phasing of the waves
1 102 103
104
10
10-
20
10-
25
10-20
Frequency Hz
Am
p.
Sp
ec.
Hz-
1/2
Sept 7, 2004 10Gravitational Waves from Binaries
Nature of GW Observations (Cont.)
•GW antennas are fundamentally observers of strong fields and relativistic sources
h ~ (M/R) (M/r) ~ (M/R) v2
– At a given distance strong gravity sources have the highest amplitude
•Future antennas will observe a large number of sources at high red-shifts
Sept 7, 2004 11Gravitational Waves from Binaries
Span of Upcoming Ground-Based Antennas
M
Sept 7, 2004 12Gravitational Waves from Binaries
Span of LISA
Sept 7, 2004 13Gravitational Waves from Binaries
Chirping Binaries Are Standard Candles
• Compact binary sources are standard candles– Amplitude of the binary depends on distance to the
source d and chirpmass: 2/3 M
– If the source chirps, that is its frequency changes, during the course of observation then it is possible to measure its chirpmass
– Interferometers determine the amplitude of the waves and the chirpy nature of the wave helps to determine the chirpmass
– Thus, it is possible to determine the luminosity distance to a source
• However, it is not possible to measure the red-shift of a source from GW observations– Will need electromagnetic observations
Sept 7, 2004 14Gravitational Waves from Binaries
Binary Black Hole Waveforms – Current Status
• Post-Newtonian and post-Minkowskian approximations– Energy is known to order O (v 6) – Gravitational wave flux is known to order O (v 7) (but still one
unknown parameter)
• Improved dynamics by defining new energy and flux functions and their Pade approximants
– Works extremely well in the test mass limit where we know the exact answer and can compare the improved model with
– But how can we be sure that this also works in the comparable mass case
• Effective one-body approach– An improved Hamiltonian approach in which the two-body problem is
mapped on to the problem of a test body moving in an effective potential
– Can be extended to work beyond the last stable orbit and predict the waveform during the plunge phase until r =3M.
• Phenomenological models to extend beyond the post-Newtonian region
– A way of unifying different models under a single framework
Sept 7, 2004 15Gravitational Waves from Binaries
What do we know from PN expansion
• Gravitational wave flux– Transverse-traceless part of the metric
perturbation extracted at infinity
• Relativistic binding energy– Corrections to the Newtonian binding
energy of the system
• Use energy balance equation to determine the phasing– Rate of change of binding energy = GW
fluxd/dt = (d/dv) (dv/dE) (dE/dt)
Sept 7, 2004 16Gravitational Waves from Binaries
Probing inspiral, plunge and merger
Sept 7, 2004 17Gravitational Waves from Binaries
Now known up to
3.5 PN order
Binding energy:
Gravitational wave flux:
Post-Newtonian Expansions of GW Flux and Energy
Sept 7, 2004 18Gravitational Waves from Binaries
Why Invent Improved PN Waveforms?
Damour, Iyer, BSS 98, 00; Buonanno, Damour 98, 00; Damour, Jaranowski, Schaefer 99; Damour 01
• Standard post-Newtonian expansion is very slowly convergent
• Re-summation techniques are proven to be convergent and robust in the test mass limit
• There are no alternatives to deal with physics close to, and beyond, the last stable orbit (but rapid progress being made in NR)
• Effective one-body is approach is the latest
Sept 7, 2004 19Gravitational Waves from Binaries
P-Approximants
Construct analytically well-behaved new energy and flux functions (remove branch points in energy, induce a linear term and handle log term in flux):
2. Using Taylor expansions of new energy and flux construct Pade approximants which are consistent with the PN expansion
3. Work back and re-define the P-approximants of energy and flux functions
Sept 7, 2004 20Gravitational Waves from Binaries
Cauchy Convergence TableCompute overlaps <npN,mpN>
Standard pN-approximants(10,10) 3pN 4pN 5pN 6pN 7pN
3pN 0.87 0.69 0.96 0.77
4pN 0.61 0.79 0.68
5pN 0.69 0.92
6pN 0.76
7pN
(10,1.4) 3pN 4pN 5pN 6pN 7pN
3pN 0.64 0.68 0.56 0.72
4pN 0.56 0.45 0.60
5pN 0.92 0.96
6pN 0.89
7pN
(1.4,1.4) 3pN 4pN 5pN 6pN 7pN
3pN 0.63 0.82 0.95 0.92
4pN 0.54 0.60 0.58
5pN 0.88 0.92
6pN 0.99
7pN
Sept 7, 2004 21Gravitational Waves from Binaries
(1.4,1.4) 3pN 4pN 5pN 6pN 7pN
3pN 0.68 0.60 0.63 0.64
4pN 0.91 0.98 0.99
5pN 0.96 0.95
6pN 1.00
7pN
(10,10) 3pN 4pN 5pN 6pN 7pN
3pN 0.68 0.66 0.74 0.75
4pN 0.99 0.94 0.94
5pN 0.90 0.90
6pN 1.00
7pN
(10,1.4) 3pN 4pN 5pN 6pN 7pN
3pN 0.41 0.39 0.40 0.40
4pN 0.91 0.99 0.99
5pN 0.94 0.93
6pN 1.00
7pN
Cauchy Convergence TableCompute overlaps <npN,mpN>
P-approximants
Sept 7, 2004 22Gravitational Waves from Binaries
Exact GW flux - Kerr Case Shibata 96
a=0.0, 0.25, 0.5, 0.75, 0.95
Sept 7, 2004 23Gravitational Waves from Binaries
Post-Newtonian flux - Kerr case
Tagoshi, Shibata, Tanaka, Sasaki Phys Rev D54, 1429, 1996
a=0.0, 0.25, 0.5, 0.75, 0.95
Sept 7, 2004 24Gravitational Waves from Binaries
P-approximant flux - Kerr casePorter 01
a = 0.0, 0.25, 0.5, 0.75, 0.95
Sept 7, 2004 25Gravitational Waves from Binaries
P-approximant flux - Kerr casePorter 01
a=0.0, 0.25, 0.5, 0.75, 0.95
Sept 7, 2004 26Gravitational Waves from Binaries
Effective One-Body ApproachBuonanno and Damour 98
• Map the two-body problem onto an effective one-body problem, i.e. the motion of a test particle in some effective external metric
• In the absence of RR the effective metric will be a static, spherically symmetric deformation of the Schwarzschild geometry (symmetric mass ratio being the deformation parameter)
• It is a particular non-perturbative method for re-summing the post-Newtonian expansion of the equations-of-motion
• Condense essential information about dynamics in just one function - a radial potential:
A(r=M/u) = 1-2u+2u3 +a4()u4 + …• Dynamics very reliable up to r=6M• EOB allows the computation of the orbit beyond
ISCO, up to r ~ 2.8M - the plunge phase
Sept 7, 2004 27Gravitational Waves from Binaries
Effective One-Body in Summary
• The dynamics of a compact binary driven by radiation reaction governed by Damour-Deruelle equations
Acceleration = [Conservative part] + RR
• At second post-Newtonian approximation
a=[A0+c-2A2+c-4 A4] + c-5AReac
• Conservative dynamics can be reduced to dynamics of relative coordinates, H(q,p)
• Starting from H(q,p), compute the effective metric
Sept 7, 2004 28Gravitational Waves from Binaries
is the Hamiltonian
is the Hamiltonian
The equations motion
Sept 7, 2004 29Gravitational Waves from Binaries
EOB gives both inspiral and merger
Drawn here separately only to show transition
Sept 7, 2004 30Gravitational Waves from Binaries
EOB signal in frequency domain
Damour, Iyer and Sathyaprakash 00
EOB Signals are wide-band
Sept 7, 2004 31Gravitational Waves from Binaries
Phenomenological Waveforms – detection template family
• Using the stationary phase approximation one can compute the Fourier transform of a binary black hole chirp which has the form
h(f) = h0 f -7/6 exp [i k f (k-5)/3]
• Where are the related to the masses and can only take certain values for physical systems
• Buonanno, Chen and Vallisneri (2002) introduced, by hand, amplitude corrections and proposed that be allowed to take non-physical values and frequencies extended beyond their natural cutoff points at the last stable orbit
• Such models, though unrealistic, seem to cover all the known families of post-Newtonian and improved models– Such DTFs have also been extended to the spinning case
where they seem to greatly reduce the number of free parameters required in a search
Sept 7, 2004 32Gravitational Waves from Binaries
Summary on Waveforms
• PN theory is now known to a reliably high order in post-Newtonian theory– O(v7)
• Resummed approaches are (1) convergent (in Cauchy sense), (2) robust (wrt variation of parameters), (3) faithful (in parameter estimation) and (4) effectual (in detecting true general relativistic signal)
• EOB approach gives a better evolution up to ISCO most likely reliable for all - including BH-BH - binary inspirals
• Detection template families (DTF) are an efficient way of exploring a larger physical space than what is indicated by various approximations
Sept 7, 2004 33Gravitational Waves from Binaries
Gravitational capture and testing uniqueness of black hole
spacetimesRyan; Finn and Thorne
Babak and Glampedakis 03
Sept 7, 2004 34Gravitational Waves from Binaries
Weighing the Graviton
• If gravitons are massive then their velocity will depend on their frequency via some dispersion relation
• Black hole binaries emit a chirping signal whose frequency evolution will be modulated as it traverses across from the source to the detector
• By including an additional parameter in matched filtering one could measure the mass of the graviton – LIGO, and especially LISA, should improve the
current limits on the mass of the graviton by several orders of magnitude
Cliff Will
Sept 7, 2004 35Gravitational Waves from Binaries
Strong field tests of general relativity
Blanchet and Schaefer 95, Blanchet and Sathyaprakash 96
Gravitational wave tails
Sept 7, 2004 36Gravitational Waves from Binaries
Sept 7, 2004 37Gravitational Waves from Binaries
Sept 7, 2004 38Gravitational Waves from Binaries
Sept 7, 2004 39Gravitational Waves from Binaries
Sept 7, 2004 40Gravitational Waves from Binaries
Sept 7, 2004 41Gravitational Waves from Binaries
How To Test Non-Linear Gravity
• Construct and use in GW searches models of the dynamics of sources under the influence of strong gravity, e.g. binary black hole sources:– Post-Newtonian (PN) approximations
– Improvements constructed from PN approximations
– Semi-analytical methods
– Numerical relativity predictions
• If PN expansion is known to a sufficiently high order employ more parameters than the number of independent parameters, e.g. M, , – Masses are over-determined
• Observe the different phases of the dynamics using different template families– Inspiral, merger, quasi-normal modes
Sept 7, 2004 42Gravitational Waves from Binaries
Neutron Star Binary InspiralNS-NS coalescence event rates
(V Kalogera, et al)
– Initial interferometers
• Range: 20 Mpc
• 1 per 40 yrs to 1 per 2 yrs
– Advanced interferometers
• Range: 300Mpc
• few per yr to several per day
– The discovery of a new binary pulsar have increased the rate upwards by an order of magnitude
Signal shape very well known
300 Mpc
~10 min
~10,000 cycles
20 Mpc~3 sec
~1000 cycles
Sept 7, 2004 43Gravitational Waves from Binaries
Binary Neutron Star Simulation
Sept 7, 2004 44Gravitational Waves from Binaries
NS/BH Binaries
43 Mpc
Neutron Star-Black Hole Inspiral and Neutron Star Tidal Disruption
650 Mpc
NS-BH Event rates
– Based on Population Synthesis
• Initial interferometers
– Range: 43 Mpc
– 1/1000 yrs to 1per yr
• Advanced interferometers
– Range: 650 Mpc
– 2 per yr to several per day
Sept 7, 2004 45Gravitational Waves from Binaries
Black Hole Mergers: Exploring the
Nature of Spacetime Warpage
Thorne
AEI and NCSA
Sept 7, 2004 46Gravitational Waves from Binaries
Black Hole Mergers: Event Rates
BH-BH event rates
– population synthesis
• Initial IFO– Range: 100 Mpc
– 1 in 100 yrs to several per yr
• Advanced IFO– Range: z=0.4
– 4 per month to 20 per day
• BH-BH rate is greater than NS-NS rate
NS/BH Binaries
Signal shape poorly known
z=0.4 inspiral
100 Mpc inspiral
Sept 7, 2004 47Gravitational Waves from Binaries
Binary Sources in LISA
Galaxy mergers
Galactic Binaries
Capture orbits
Sept 7, 2004 48Gravitational Waves from Binaries
Merger of Supermassive Black Holes
The high S/N at early times enables LISA to predict the time and position of the
coalescence event, allowing the event to be observed simultaneously by other
telescopes.
NGC6240, Hasinger et al
Cutler and Vecchio
Sept 7, 2004 49Gravitational Waves from Binaries
Binary Coalescences in EGO
• At frequencies > kHz detect normal modes of NS and measure the equation of state of matter at high densities and temperatures
• Probe the high red-shift Universe for black hole and NS mergers
• Resolve the origin of gamma-ray bursts and the expansion rate at red-shifts z ~ 2.
Sept 7, 2004 50Gravitational Waves from Binaries
Binary Black holes in Big Bang Observer
• Identify signals from every merging NS and stellar-mass black hole in the Universe and thereby determine rate of expansion of the Universe as a function of time and provide insights into dark energy
• Pinpoint radiation from the formation or merger of intermediate mass black holes believed to form from the first massive stars born in our Universe.
Sept 7, 2004 51Gravitational Waves from Binaries
Cosmology with Binary Coalescences
• Binary inspiral signals are standard candles:
|h| = [M(1+z)]5/6 f 2/3 (t) /dL
– Amplitude, redshift determines the luminosity distance
• Luminosity-redshift relation determines the cosmological model
dL(z) = (1+z) ∫ H-1(z’) dz’
H2(z) = H02 [m (1+z)3 + (1+z)3(1+w)]
0
z