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## B.S. Sathyaprakash Cardiff University

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**Gravitational Wavesfrom Binary CoalescencesLooking for**Needle in a haystack, Mondragone School, Rome September 7, 2004 B.S. Sathyaprakash Cardiff University**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 Gravitational Waves from Binaries**Quasi-normal modes from a BH at 0.1-1 Gpc can generate**detectable amplitudes 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 timesin 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 Gravitational 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 emitting a burst of GW that will be observable using instruments that are currently being built But that will take another 100 million years Gravitational Waves from Binaries**Discovery of fastest binary pulsarBurgay 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: 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 Gravitational 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 Gravitational 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 Gravitational Waves from Binaries**10-20**10-20 Amp. Spec. Hz-1/2 10-25 1 10 102 103 104 Frequency Hz 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 Gravitational 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 Gravitational Waves from Binaries**Span of Upcoming Ground-Based Antennas**M Gravitational Waves from Binaries**Span of LISA**Gravitational 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: 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 Gravitational 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 Gravitational 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 flux dw/dt = (dw/dv)(dv/dE) (dE/dt) Gravitational Waves from Binaries**Probing inspiral, plunge and merger**Gravitational Waves from Binaries**Post-Newtonian Expansions of GW Flux and Energy**Gravitational wave flux: Now known up to 3.5 PN order Binding energy: Gravitational 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 Gravitational 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): • 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 Gravitational Waves from Binaries**Cauchy Convergence TableCompute overlaps <npN,mpN>**Standard pN-approximants Gravitational Waves from Binaries**Cauchy Convergence TableCompute overlaps <npN,mpN>**P-approximants Gravitational Waves from Binaries**Exact GW flux - Kerr Case Shibata 96**a=0.0, 0.25, 0.5, 0.75, 0.95 Gravitational Waves from Binaries**Post-Newtonian flux - Kerr caseTagoshi, Shibata, Tanaka,**Sasaki Phys Rev D54, 1429, 1996 a=0.0, 0.25, 0.5, 0.75, 0.95 Gravitational Waves from Binaries**P-approximant flux - Kerr casePorter 01**a = 0.0, 0.25, 0.5, 0.75, 0.95 Gravitational Waves from Binaries**P-approximant flux - Kerr casePorter 01**a=0.0, 0.25, 0.5, 0.75, 0.95 Gravitational 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+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 Gravitational 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 Gravitational Waves from Binaries**is the Hamiltonian**is the Hamiltonian The equations motion Gravitational Waves from Binaries**EOB gives both inspiral and mergerDrawn here separately only**to show transition Gravitational Waves from Binaries**EOB signal in frequency domainDamour, Iyer and Sathyaprakash**00 EOB Signals are wide-band Gravitational 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) = h0f -7/6 exp [iSyk 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 Gravitational 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 Gravitational Waves from Binaries**Babak and Glampedakis 03**Gravitational capture and testing uniqueness of black hole spacetimes Ryan; Finn and Thorne Gravitational Waves from Binaries**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 Gravitational Waves from Binaries**Gravitational wave tails**Blanchet and Schaefer 95, Blanchet and Sathyaprakash 96 Strong field tests of general relativity Gravitational 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, m, h • Masses are over-determined • Observe the different phases of the dynamics using different template families • Inspiral, merger, quasi-normal modes Gravitational Waves from Binaries**~10 min**~3 sec 20 Mpc ~1000 cycles ~10,000 cycles 300 Mpc Signal shape very well known 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 Gravitational Waves from Binaries**Binary Neutron Star Simulation**Gravitational Waves from Binaries**NS/BH Binaries**43 Mpc 650 Mpc Neutron Star-Black Hole Inspiral and Neutron Star Tidal Disruption 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 Gravitational Waves from Binaries**AEI and NCSA**Thorne Black Hole Mergers: Exploring theNature of Spacetime Warpage Gravitational Waves from Binaries**z=0.4 inspiral**100 Mpc inspiral Signal shape poorly known Black Hole Mergers: Event Rates NS/BH Binaries 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 Gravitational Waves from Binaries**Galaxy mergers**Galactic Binaries Capture orbits Binary Sources in LISA Gravitational Waves from Binaries**Cutler and Vecchio**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 Gravitational 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. Gravitational 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. Gravitational Waves from Binaries