1. B.S. Sathyaprakash Cardiff University
Gravitational Waves from Binary Coalescences Looking for Needle in a haystack, Mondragone School, Rome September 7, 2004
B.S. Sathyaprakash
Cardiff University

2. Gravitational Waves: Ripples in the Fabric of Spacetime
In Newton’s law of gravity the gravitational potential is given by Poisson’s equation:
s2F(t, X)= 4pGr(t,X)
In general relativity for weak gravitational fields, for which one can assume that background metric is nearly flat
gab = hab + hab where |hab| << 1,
Einstein’s equations reduce to wave equations:
hab = 8pGTab .
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
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3. 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 = √r
Dynamical frequency in the system
Quasi-normal modes from a BH at Gpc can generate detectable amplitudes
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4. But that will take another 100 million years
Period Decay in Hulse-Taylor Binary In 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 emitting a burst of GW that will be observable using instruments that are currently being built
But that will take another 100 million years
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5. Discovery of fastest binary pulsar Burgay et al Nature 2003
A brief history of pulsar discoveries
First pulsar, Carb PSR : Hewish and Bell 1967
First binary pulsar PSR : Hulse and Taylor 1974
First millisecond pulsar PSR : Backer et al 1982
Fastest known binary pulsar J : Burgay et al 2003
In December 2003 Burgay et al discovered a new pulsar in a binary J 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 , J is the most relativistic neutron star binary
Greatest periastron advance: dw/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
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6. Gravitational Waves from Binaries
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7. Discovery of the second pulsar Lyne 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
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8. 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
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9. 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
Frequency Hz
Amp. Spec. Hz-1/2
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10. 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
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11. Span of Upcoming Ground-Based Antennas
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12. Span of LISA
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13. Chirping Binaries Are Standard Candles
Compact binary sources are standard candles
Amplitude of the binary depends on distance to the source d and chirpmass: h 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
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14. 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
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15. 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 flux
dw/dt = (dw/dv) (dv/dE) (dE/dt)
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16. Probing inspiral, plunge and merger
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17. Post-Newtonian Expansions of GW Flux and Energy
Gravitational wave flux:
Now known up to
3.5 PN order
Binding energy:
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18. Why Invent Improved PN Waveforms
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
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19. 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):
Using Taylor expansions of new energy and flux construct Pade approximants which are consistent with the PN expansion
Work back and re-define the P-approximants of energy and flux functions
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20. Cauchy Convergence Table Compute overlaps <npN,mpN>
Standard pN-approximants
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21. Cauchy Convergence Table Compute overlaps <npN,mpN>
P-approximants
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22. Exact GW flux - Kerr Case Shibata 96
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23. Post-Newtonian flux - Kerr case Tagoshi, Shibata, Tanaka, Sasaki Phys Rev D54, 1429, 1996
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24. P-approximant flux - Kerr case Porter 01
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25. P-approximant flux - Kerr case Porter 01
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26. Effective One-Body Approach Buonanno 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+2h u3 +a4(h)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
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27. 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
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28. is the Hamiltonian is the Hamiltonian The equations motion
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29. EOB gives both inspiral and merger Drawn here separately only to show transition
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30. EOB signal in frequency domain Damour, Iyer and Sathyaprakash 00
EOB Signals are wide-band
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31. 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 Syk f (k-5)/3]
Where y 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 y 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
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32. 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
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33. Gravitational capture and testing uniqueness of black hole spacetimes
Ryan; Finn and Thorne
Babak and Glampedakis 03
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34. Weighing the Graviton
Cliff Will
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
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35. Strong field tests of general relativity
Blanchet and Schaefer 95, Blanchet and Sathyaprakash 96
Gravitational wave tails
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36. Gravitational Waves from Binaries
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37. Gravitational Waves from Binaries
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38. Gravitational Waves from Binaries
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39. Gravitational Waves from Binaries
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40. Gravitational Waves from Binaries
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41. 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, m, h
Masses are over-determined
Observe the different phases of the dynamics using different template families
Inspiral, merger, quasi-normal modes
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42. Neutron Star Binary Inspiral
NS-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
~10 min
~10,000 cycles
~3 sec
~1000 cycles
20 Mpc
300 Mpc
Signal shape very well known
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43. Binary Neutron Star Simulation
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44. Neutron Star-Black Hole Inspiral and Neutron Star Tidal Disruption
NS/BH Binaries
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
43 Mpc
650 Mpc
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45. Black Hole Mergers: Exploring the Nature of Spacetime Warpage
AEI and NCSA
Thorne
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46. 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
100 Mpc inspiral
z=0.4 inspiral
Signal shape poorly known
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47. Binary Sources in LISA Galactic Binaries Galaxy mergers Capture orbits
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48. Merger of Supermassive Black Holes
Cutler and Vecchio
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
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49. 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.
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50. 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.
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51. 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 [Wm (1+z)3 + WL (1+z)3(1+w)] z
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