post newtonian theory and data analysis of...

Post Newtonian theory and data analysis of gravitational waves from inspiraling

compact binaries

Hideyuki Tagoshi

December 7 - 11, 2009 at APCTP Seoul in Korea

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Plan of talk 1.  Post-Newtonian approximation from a user’s point of view 1.1 Post-Newtonian approximation: Basics 1.2 How to treat the retarded integrals 1.3 Equations of motion 1.4 Current status

2.  Post-Newtonian approximation from black hole perturbation theory

3.  Data analysis of post-Newtonian wave form from inspiraling compact binaries

4.  Science from post-Newtonian wave form

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Introduction

Gravitational wave generation formalism

Analytical, semi-analytical method Post-Newtonian approximation, Black hole perturbation method, ......

Full numerical method Numerical Relativity

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Post-Newtonian approximation

•  Post-Newtonian theory has very long history. It’s history is as long as the general relativity itself.

Einstein (1915) Perihelion shift of mercury Lorentz, Droste (1917), Einstein,Infeld,Hoffmann(1938), Fock (1959), Chandrasekhar et al. (1965-),...

PN approximation for compact binaries

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NS or BH binaries are most promising sources for ground Based interferometers

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•  Post-Newtonian (PN) approximation is “week field” and “slow motion” approximation expansion expansion G: Newton’s gravitational constant v: typical speed of the system c: speed of light

week field approximation = post-Minkowski (PM) approximation

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Post-Minkowski and Post-Newton

Post-Minkowski (PM) expansion

dimension of G imply PM expansion is done with M : mass scale of the system L: length scale of the system

1/c expansion is partially done. in the equation of motion,

Newton

1PN

2PN

3PN

1PM 2PM 3PM 4PM PM

PN

Bel et al. (1981)

........

How to call the post-Newtonian order

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There are some discrepancies between literatures about how to call the PN order. In the case of compact binaries and radiation from them, most widely used way is as follows.

• Equations of motion Newtonian order = 0PN PN order: if the term is higher than Newtonian, it is called (n/2)PN.

Radiation reaction appears first at 2.5PN order.

• In the wave formulas, or in the wave luminosity formulas, PN order is the order relative to the lowest order term even if the radiation formulas is already at higher order in

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PN approximation Einstein eqs.

harmonic condition

Einstein eqs. are written in harmonic coordinate as

is at least quadratic in This equation matches non-linear expansion in the form

:Minkowski metric

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PN approximation No-incoming radiation condition is imposed at past null infinity of Minkowski background. This is done by using the retarded green function.


Non-linear iteration

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n-th order

Near zone 1/c expansion

Far zone

Gravitational wave

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A difficulty in PN expansion The retarded integral contains integral of non-linear term:

does not have compact support, it extends to infinity.

Since PN expansion is done in (1/c)^n series, we expand formally with respect to 1/c, we have something like

This gives divergent integral at larger n. Epstein-Wagoner (1975) Wagoner-Will (1976) 1PN wave

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PN approximation is near zone approximation

• Post-Newtonian expansion is valid only in near zone A: physical quantity

In PN approximation, we assume L: length scale T: time scale

However, in the far zone (wave zone), frequency of wave

wave length of wave

Therefore,

PN expansion can not be done formally. (Note however, after we compute the wave in the wave zone, we can expand it with respect to (v/c) if the wave formulas contain (v/c)^n. )

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How to treat the retarded integral Blanchet-Damour “Matched asymptotic expansion” In near zone, one expands with 1/c,

and solve them iteratively by using “generalized Poisson integral”

: Laurant expand [] in terms of B, and take the “finite part” at B=0. (B:complex number) analytic continuation method

(finite)

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How to treat the retarded integral Near zone solution is not enough, since it is not possible to impose the correct boundary condition. To do it, matching to the far zone solution is necessary.

At far zone, Blanchet-Damour use “Multipolar-post-Minkowski expansion” post-Minkowski solution (Gn expansion) + multipole expansion

Then, we match two solutions, near zone sol. and far zone sol. and determine the undetermined homogeneous solution in the near zone solution. The matching is done with a help of multipole expansion using STF (symmetric trace free tensor).

Poujade-Blanchet (2002) have shown that this scheme can be done, in principle, upto arbitrary order of PN expansion.

In actual computation, upto 3.5PN order has been done.

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How to treat the retarded integral

Will et al. : Direct Integration of Relaxed Einstein equations(DIRE)

PN expansion direct integration

In principle, there is no divergence.

Will-Wiseman (1996) 2PN wave form. (2PN order beyond lowest quadrupole radiation)

Pati-Will (2002) 2.5PN equations of motion + 3.5PN order. 3PN equations of motion : not yet.

DIRE techniques are also used by Itoh-Futamase (‘03,’04,’09) in deriving the 3PN and 3.5PN equations of motion.

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PN equations of motion

How to represent the stars (or BHs)

1. Point particle (delta function) method

Since field variables diverges at the position of the particles, regularization procedure is necessary.

Damour(1981): Reitz kernel method (analytic continuation)

Hadamard regularization: Blanchet, Faye, Ponsot (1998) : 2.5PN EOM : no ambiguity Jaranowski, Schafer, Damour (1998-2001) : 3PN EOM Blanchet, Faye (2000-2001) :3PN EOM ambiguity appear

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Damour,Jaranowski,Schafer (2001) ADM Hamiltonian Blanchet,Damour,Esposito-Farese (2004) Harmonic coordinate

Dimensional regularization

Space-time dimension is set D=d+1 in the intermediate formulas. Then set

Blanchet,Damour,Esposito-Farese,Iyer (2005) 3PN energy flux formulas was completed.

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Other method: •  Fluid ball approach Kopejkin(1985)

Pati, Will (2002)

•  Strong field point particle limit +Surface integral approach

Itoh, Futamase, Asada (2001) 2.5PN Itoh, Futamase (2003, 2004) 3PN Itoh (2009) 3.5PN

In general relativity, there is not point particle physically. If the size of body R shrinks, it becomes a black hole. We also need to shrink M and keep M/R constant to express the strong field compact objects. These scaling can matches the post-Newtonian scaling, and thus can be introduced in PN theory (Futamase (’87)).

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Spinning binaries’ EOM Spinning binaries’ EOM are derived by using the spinning particles’ energy momentum tensor given as

1.5PN spin-orbit terms, 2PN spin-spin terms have been well-known for a long time. Tagoshi, Ohashi, Owen (2001) 2.5PN SO terms. Faye,Blanchet,Buonanno (2006) 2.5PN SO terms, 1PN spin-precession Damour,Jaranowski,Schafer (2008) 2.5PN SO terms in Hamiltonian

Some higher order spin-spin (SS) terms, Porto,Rothstein (2008)

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PN approximation of binary: history Equations of motion 1PN many works: Lorentz-Droste, Eddinton-Clark, EIH, Fock, Papapetrou, ... 2PN Japanese group (Ohta,Okamura,Kimura,Hiida) (1973-74) Discovery of PSR 1913 + 16 (1974) 2.5PN Damour-Deruelle (1981), Schafer(1985), Kopejkin(1985) R&D of Laser interferometer (‘90) TAMA DT1 (‘98), LIGO S1 (‘02) Cutler et al. (1993) “The last three minutes” : necessity of higher order PN 2.5PN revisited Blanchet, Faye, Ponsot (1998), Itoh, Futamase, Asada (2001) Pati, Will (2002) 3PN (Hadamard regularization): Jaranowski, Schafer, Damour (1998-2001) Blanchet, Faye (2000-2001) 3PN (Dimensional regularization): Damour,Jaranowski,Schafer (2001) ADM Hamiltonian Blanchet,Damour,Esposito-Farese (2004) Harmonic coordinate 3PN (Strong field point particle limit, surface integral approach) Itoh, Futamase (2003-2004)

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Post-Newtonian equations of motion: status

In the relative coordinate,

by ~1980

Radiation reaction

Radiation reaction

Damour,Jaranowski,Schafer Blanchet,Damour,Esposito-Farese (2004) Itoh,Futamase (‘03, ‘04)

Pati,Will (‘02) Nissanke,Blanchet (‘05) Itoh(‘09)

spin-orbit, spin-spin terms

Tagoshi, Ohashi, Owen (‘01) Faye,Blanchet,Buonanno (‘06) Damour,Jaranowski,Schafer (‘08)

conservative, radiation reaction from tail (‘??)

by ~1980

1PN higher order SO

Appendix: Pole in dimensional regularization

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there remains a pole part in the EOM. However, such pole can be renormalized into some shifts of the “bare” world-lines by

In dimensional regularization,

Blanchet, Detweiler, Le Tiec, Whiting (arXiv:0910.0207) compare the 3PN result and self-force computed with the black hole perturbation by using a gauge invariant quantity.

In the gauge invariant quantity, the pole in the dimensional regularization disappear and the 3PN result and the self-force result agree each other.

Lecture 2

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Plan of talk 1.  Post-Newtonian approximation from a user’s point of view 1.1 Post-Newtonian approximation: Basics 1.2 How to treat the retarded integrals 1.3 Equations of motion 1.4 Current status

2.  Post-Newtonian approximation from black hole perturbation theory

3.  Data analysis of post-Newtonian wave form from inspiraling compact binaries

4.  Science from post-Newtonian wave form

• Higher harmonics

When , this formula agree with that derived by using the BH perturbation theory

Gravitational wave luminosity formula:status

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Gravitational wave luminosity in circular orbit case

:Euler constant

PN, BH perturbation, Numerical Relativity

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Blanchet et al. arXiv:0910.0207 or

bita

l sep

arat

ion

Log(Mass ratio)

Black hole perturbation theory

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M

µ We consider the linear perturbation of black hole by assuming

Background metric is not Minkowski but black hole metric

Thus, this approach is not restricted to the slow motion sources.

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Black hole perturbation theory Linear perturbation of Kerr BH is formulated for the Weyl scalar . ・The basic equation is a 2nd order partial differential equations. ・It is usually expanded Fourier-harmonically, and is reduced to a 2nd order ordinary differential equation. ・In order to solve it, we need to use numerical computation in general.

Teukolsky equation:

BH perturbation theory Schwarzschild case(no source term)

Bardeen-Press-Teukolsky eq. (perturbation of Weyl Ψ4/0 )

Regge-Wheeler eq. (metric perturbation(odd parity))

Zerrilli equation (metric perturbation(even parity))

Chandrasekhar transformation

With source term (Sasaki,Nakamura, Phys.Lett.A,87,85(1981))

Bardeen-Press-Teukolsky eq. (perturbation of Weyl Ψ4/0 )

Regge-Wheeler eq. (with modfied source term)

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BH perturbation theory

Kerr BH

Teukolsky eq. (perturbation of Weyl Ψ4/0 ) Sasaki-Nakamura eq.

Sasaki,Nakamura, Phys.Lett.A,89,68 (1982) Sasaki,Nakamura, Prog.Theor.Phys., 67,1788 (1982)

If we set Kerr spin parameter=0, this reduces to Regge-Wheeler eq.

Regge-Wheeler like eq.

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Black hole perturbation theory Linear perturbation of Kerr BH is formulated for the Weyl scalar . ・The basic equation is a 2nd order partial differential equations. ・It is usually expanded Fourier-harmonically, and is reduced to a 2nd order ordinary differential equation. ・In order to solve it, we need to use numerical computation in general.

Teukolsky equation:

Black hole perturbation theory What is quite different from PN approximation is the Green function. Green function include the effect of the curvature of background BH space time.

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In order to impose the no-incoming radiation condition, we prepare two kinds of homogeneous solution of the Teukosky equation.


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and then, we construct the Green function, and the solution.

PN expansion of BH perturbation eqs.

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Any of equations of BH perturbation must be evaluated Numerically in order to derive exact value.

However, we can consider the approximate solution by expanding them in terms of G and/or 1/c.

Post-Minkowskian, or post-Newtonian approximation of the solutions of these equations can be derived analytically, or semi-analytically.

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• Sasaki, Nakamura，Oohara，Kojima (~1984)

Initial results Progress, Supplement 90 (1987)

• Sasaki，Nakamura + Shibata，Tanaka，Tagoshi (1994-1997) Higher order post-Newtonian term computation with Regge-Wheeler, or Sasaki-Nakamura eq.

Blanchet-Damour-Iyer Will-Wiseman

Schwarzshild 5.5PN order (Tanaka,Tagoshi,Sasaki) Kerr 　　　　　　　 4PN order (Tagoshi,Shibata,Tanaka,Sasaki)

• Mano-Suzuki-Takasugi (Analytic representation of the solutions of the Teukolsky equation) Greatly simplifies the PN computation. Also useful for numerical computation.

・Poisson, Cutler et al. (1993)

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€

x = (MΩϕ )1/3

(Tanaka,Tagoshi,Sasaki (‘96))

Black hole perturbation theory Schwarzschild, circular orbit

Luminosity of GW

5.5PN order

: These terms are already derived by using the BH perturbation theory

Gravitational wave luminosity formula:status

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PN formulas of Gravitational wave luminosity in circular orbit case

:Euler constant

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How far PN expansion needed? It is not easy to derive conclusive answer, since we do not know exact results. One way is to use the results from black hole perturbation (e.g., extreme mass ratio case), since exact value can be computed numerically.

Phase difference of chirp signals from 10Hz to coalescence

(n/2)PN

Data analysis of post-Newtonian wave form from inspiraling compact binaries

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Post-Newtonian wave form

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Restricted wave form (RWF) Lowest order in amplitude, and only twice the orbital phase are taken in the phase formula.

Detector response

:Detector response function

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Arun,Blanchet,Iyer,Qusailah (2004) 2.5PN wave form

Amplitude and phase modulation from higher harmonics mode are included.

However, since full wave form (FWF) is very complicated, only restricted wave form (RWF) has been used in the most data analysis in the past.

This is justified as a search strategy in a sense that the quadrupole mode is dominant in phasing formula. Further, matched filtering is not so sensitive to the amplitude modulation.

Full wave form (FWF)


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Restricted wave form (RWF) Lowest order in amplitude, and only twice the orbital phase are taken in the phase formula.

Detector response


Data analysis method

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Detector’s response


Let us consider restricted wave form (RWF)

We assume the detector’s noise is Gaussian. From Patrick Brady’s Lecture 2, we find the log- likelihood ratio is given as

(x: detector’s data)


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We take maximum of ln(λ) with respect to amplitude A and phase δc and we obtain

This is the formula of the matched filtering of inspiral signal. Among parameters describing signal,

only three have to be searched.

Multiple detectors case

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This maximum likelihood method (matched filtering) can be extended to the case of multiple detectors. Detection method based on the maximum likelihood in the case of multiple detectors is often called “coherent detection strategy” which is different from “coincident detection strategy”.

Pai, Dhurahdhar, Bose (2001), Finn (2001)

In the network coherent detection strategy, we need to search for the sky position, . The number of parameters needed to be searched is 5, . Miximization over other 4 parameters, , can be taken analytically.

Pai, Dhurahdhar, Bose (2001)


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There are some works which compare “coherent” and “coincident” strategies

Finn (2001) sine wave Arnaud et al. (2003) Gaussian shape wave Mukhopadhyay et al. (2006) inspiral wave Tagoshi et al. (2007) inspiral wave Mukhopadhyay et al. (2009) inspiral wave

All of these works have shown that in the case of Gaussian noise, coherent method works better than the coincident method as a detection method.

The coherent strategy is formulated only for RWF (Pai et al (‘01)). The coherent strategy for FWF is not worked out yet.

Science from PN wave form

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Let us consider the gravitational waves from coalescence of super massive black holes (SMBH) binaries, and its detection by LISA.

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Wave form with higher harmonics

Observed spectrum in LISA Arun et al. (07)

Due to interference between various harmonics, these complicated structure appear.

Full PN waveform is schematically written as

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With FWF, LISA will be able to detect 10^8 Msolar SMBH at 10Gpc with SNR=10. RWF does not have sensitivity for such mass

Arun et al. (07)

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More importantly, FWF improve the parameter estimation accuracy in general

Especially, angular position accuracy improved

possibility to localize the source


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Parameter estimation accuracy with FWF

Number of cluster of galaxies in the error circle can be less than 1. Then, we can identify the cluster of galaxy where SMBH exists, and determine the redshift of it. Distance-redshift relation gives constraint to the dark energy equation of state.

References

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Post-Newtonian approximation L.Blanchet, Living Reviews, lrr-2006-4 L.Blanchet, arXiv:0907.3596

Post-Newtonian equations of motion Futamase, Itoh, lrr-2007-2

Black hole perturbation, analytic approach Sasaki, Tagoshi, lrr-2003-6 Mino et al. Prog. Theor. Phys. Suppl. 128, 1, (1997)

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