Gravitational Waves from Binary Coalescences Gravitational Waves from Binary Coalescences Looking for Needle in a haystack, Mondragone School, Rome September

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


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


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


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



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


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


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


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


Span of Upcoming Ground-Based Antennas

M


Span of LISA


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


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


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)


Probing inspiral, plunge and merger


Now known up to

3.5 PN order

Binding energy:

Gravitational wave flux:

Post-Newtonian Expansions of GW Flux and Energy


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


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


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


(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


Exact GW flux - Kerr Case Shibata 96

a=0.0, 0.25, 0.5, 0.75, 0.95


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


P-approximant flux - Kerr casePorter 01

a = 0.0, 0.25, 0.5, 0.75, 0.95


P-approximant flux - Kerr casePorter 01

a=0.0, 0.25, 0.5, 0.75, 0.95


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


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


is the Hamiltonian

is the Hamiltonian

The equations motion


EOB gives both inspiral and merger

Drawn here separately only to show transition


EOB signal in frequency domain

Damour, Iyer and Sathyaprakash 00

EOB Signals are wide-band


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


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


Gravitational capture and testing uniqueness of black hole

spacetimesRyan; Finn and Thorne

Babak and Glampedakis 03


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


Strong field tests of general relativity

Blanchet and Schaefer 95, Blanchet and Sathyaprakash 96

Gravitational wave tails







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


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


Binary Neutron Star Simulation


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


Black Hole Mergers: Exploring the

Nature of Spacetime Warpage

Thorne

AEI and NCSA


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


Binary Sources in LISA

Galaxy mergers

Galactic Binaries

Capture orbits


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


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.


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.


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

Documents

Gravitational Waves from Binary Coalescences Gravitational Waves from Binary Coalescences Looking for Needle in a haystack, Mondragone School, Rome September