Systematic errors for matched filtering of gravitational waves from inspiraling compact binaries

PHYSICAL REVIEW D, VOLUME 63, 082005

Systematic errors for matched filtering of gravitational waves from inspiraling compact binaries

Philippe CanitrotDARC, Observatoire de Paris, 92195 Meudon Cedex, France,

Fresnel, VIRGO, Observatoire de la Coˆte d’Azur, F-06304 Nice Cedex, France,and Laboratoire de l’Acce´lerateur Lineaire, B.P. 34 91898 Orsay Cedex, France

~Received 1 September 2000; published 27 March 2001!

We computed a thorough set of ambiguity functions with templates corresponding to various post-Newtonian approximations of the real signal. We study the detection of gravitational waves emitted by theinspiraling phase of compact binaries in order to study systematically the induced bias in the parameters. Thenoise spectrum is taken from the VIRGO interferometer, which has an effective frequency range larger than theone predicted for LIGO. We first confirm the results of previous authors that the Newtonian filter has a verylow capability of detection and that the 2 PN restricted wave form is good enough for detection in the case ofneutron stars binaries. Moreover, we also show that constant spins aligned with the orbital momentum have nosignificant effect in on-line selection. We point out that the maximization may lead to unphysical values ofparameters to compensate the systematic errors due to imperfectly modeled templates, so that one should usea wider range of variation of the mass ratio parametern5 reduced mass over the total mass~not restricted to0<n<1/4). We also demonstrate that the higher harmonics at one and three times the orbital frequency cannotalways be neglected for detection. The loss of signal to noise ratio amounts to 6% with 1.4 and 10 solar massesbinary in certain cases.

DOI: 10.1103/PhysRevD.63.082005 PACS number~s!: 04.80.Nn, 07.05.Kf, 97.80.2d

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I. INTRODUCTION

As previously shown in the literature, the most promisisources for gravitational waves detection by ground-bainterferometer such as VIRGO, the Laser InterferomeGravitational Wave Observatory~LIGO!, GEO, and TAMA@1–5#, are inspiraling compact binaries provided that thetrophysical rate of coalescences is sufficient@6,7#. Becausethey are the best modeled astrophysical sources, the opfiltering technique appears to be particularly adapted totect their signal in very noisy data@8–10#. Furthermore, de-tection with second generation detectors is predicted to ylow statistical errors and thus allow an accurate measuremof the parameters with the optimal filtering@11–13#.

As the exact solution in general relativity of the motiontwo compact objects is not known, templates are compuusing high post-Newtonian~PN! approximations. Howeverbecause the templates used are not exact representatiothe signal, the optimal filtering detection method will intrduce a loss of signal to noise ratio and systematic errorthe parameter estimates. The aim of this work is to systatically study the loss of signal to noise ratio and the bintroduced in the parameter estimates, given the expesensitivity for VIRGO. To measure the efficiency of templates, Apostolatos introduced the notion of ambiguity funtions and fitting factors@14,15#. The ambiguity function isthe ratio between the signal to noise ratio obtained withbest template in the family of imperfect PN templates, athe signal to noise ratio which would be obtained withperfectly matched template. The fitting factor~FF!, for agiven set of templates, is the maximum value of their amguity function. It allows to find the best parameters maximing the probability of detection and then to compute the bfor the real parameters of the signal. A practical criteriondetermine whether a template is good enough, is to havethan 10% event loss. This loss of events can be comp

0556-2821/2001/63~8!/082005~13!/$20.00 63 0820

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assuming binaries uniformly distributed in space and sithe signal strength is proportional to the inverse of the dtance, then 12FF3. Thus a 10% loss of events corresponto FF597%.

The bias of the parameters found in this work is vedifferent from the root mean square error, which dependsthe realizations of the noise and the statistical bound usemodel them. In Refs.@16,17# Monte Carlo simulations havebeen done. They show that the covariance matrix underemate by a factor of two the root mean square errors for sigto noise ratio less than 20. For the following computatiowe assume an infinite realization of the noise, i.e., an infinnumber of detectors, we have thus no statistical errors.are only interested in the validation of template families.

In Refs. @18–20# computations of fitting factors have aready been done taking into account the LIGO noise. Tcase of templates from Newtonian order to 2 PN approximtions has been examined. These authors find acceptablting factors for 1.5 and 2 PN templates family, even if tsignal has vanishing spins. They found low bias (<0.01%)in the chirp mass parameter and a bias less than 10% fototal mass. The results presented here are similar forfitting factors but different for the systematic errors. Furthmore we introduce new computations with 2.5 PN expasions and we discuss about the neglect of higher harmonIn addition the results differ from previous ones, becauseuse the VIRGO noise sensitivity curve, which has a widfrequency range, allowing more cycles to be measured.nally the method differs mainly in the use of different prameters.

This work can have a direct influence on template costruction and computational cost~see Owen andSathyaprakash for an estimation of number of templaneeded@21,22#!. The huge number of templates involves tneed for hierarchical search@23,24#. Damour, Iyer andSathyaprakash@25# have introduced Pade´ approximants for

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PHILIPPE CANITROT PHYSICAL REVIEW D 63 082005

inspiraling binaries that could have a better match for detion.

In Sec. II we introduce our notations, and briefly remithe definition of signal to noise ratio. We also presentapproximations used for these computations. In Sec. III,present the main numerical results including those obtaiwith signals taking into account spins. In Sec. IV, we discuthe use of the restricted wave form.

II. METHODS

In this section, we introduce all the notations and presthe approximations used in the computation. Masses arepressed in second. Dotted letters represent time derivati

A. Parameters of the gravitational signal

The scalar responseh of the interferometer to any gravitational signal is given by

h5F1h11F3h3 , ~2.1!

where F1 and F3 are the beam pattern functions, whicdepend on the relative orientation of the polarization planethe gravitational wave to the direction into the interferoeter:

F151

2~11cos2a!cos 2b cos 2j1cosa sin 2b sin 2j

~2.2!

F3521

2~11cos2a!cos 2bsin2j1cosa sin 2b cos 2j,

~2.3!

wherea, b andj are Euler angles. We now consider compact binary stars; in the inspiraling phase, the restricted wform @26# is given by

h1,3522cmn

d~mv!2/3H ~11ci

2!cos 2f for 1,

2cisin 2f for 3,~2.4!

where i is the inclination angle between the binary and tobserver, i.e., the angle between the normal to the orbplane and the perpendicular of the interferometer plane,ci5cosi, si5sin i. d is the luminosity distance andc thelight speed.m is the total mass,n the reduced mass over totmass andf the orbital phase,v5f.

In order to factorizeh, it is useful to introduce the variables

F5AF12 ~11ci

2!214F32 ci

2 <2, ~2.5!

and

cosZ5F1~11ci

2!

F, sinZ5

2F3ci

F.

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Then the scalar response of the interferometer for the grtational signal of inspiraling compact binaries, in time dmain, with use of the restricted wave form, reads

h~ t !524Amn~mv!2/3cos~2f2Z!, ~2.6!

whereA5(c/d)(F/2).In all our studies, we choose the maximal value forF, that

is F52. In @9#, Finn and in@28#, Hello show that the influ-ence of the angles in loss of signal to noise ratio is importathe mean value forF2 is 0.64 and maximum values forF areunlikely. 2/3 of the sources haveF,0.8. For the presenwork, the value ofF has no consequences. In@27#, the in-verse problem is solved. It requires a network of three detors and the authors expect an accuracy of 10% for thetance and 1024 sr for the position in the sky. In this papewe are not interested in the distance or angle measurem

B. Instantaneous frequency

In this section, we present the model for templates. Tanalysis was entirely carried out in the frequency domain

The computation of the phase is based on the balaequation

dE

dt52L, ~2.7!

whereE is the binding energy of the binary in the centermass frame, as computed using post-Newtonian approxitions andL is the total power luminosity emitted in all directions as gravitational waves. 2.5 post-Newtonian~PN! de-velopments@29–31# give for the mechanical energyE

E5EnewtX121

12~91n!x2

1

8 S 27219n11

3n2Cx21O~x3! D ,

~2.8!

where

Enewt521

2mn~mv!2/3

c5

G, ~2.9!

and for the power lost by gravitational wavesL

L5LnewtX12S 1247

3361

35

12n D x14px3/2

1S 244711

90721

9271

504n1

65

18n2D x2

2S 8191

6721

535

24n Dpx5/21O~x3!C, ~2.10!

where

Lnewt532

5n2~mv!5

c5

G. ~2.11!

In the previous expansionsx5(mv)2/3 is a small PN param-eter. From Eq.~2.7!, we obtain

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SYSTEMATIC ERRORS FOR MATCHED FILTERING OF . . . PHYSICAL REVIEW D 63 082005

1

v52

1

LdE

dv, ~2.12!

and we write

df

dv5

v

v, ~2.13!

so we integrate these two equations and obtain the phafas a function of the orbital pulsationv.

Let g( f ) be the Fourier transform of a functiong(t),

g~ f !5E2`

`

g~ t !ei2p f tdt. ~2.14!

We use the stationary phase approximation to computeFourier transform. This approximation is supposed tovalid up to 2 PN, but we assume the same validity at higorders. In@32# it is shown that this transformation is veraccurate at Newtonian order and sufficient for the practapplication of matched filtering. Then we note that the Forier frequencyf of the signal and the orbital pulsation arelated throughv5p f . In the following we express everything as functions of the Fourier frequencyf.

The Fourier transform of the signal at 2.5 PN is given

h~ f !521

4AA10p

3~p f !27/6M5/6exp„iF~ f !…,

~2.15!

whereM5mn3/5 is the so-called chirp mass.F( f ) is givenby the stationary phase approximation

F~ f !52p f tc22fc2p

41Z1

3

128P, ~2.16!

P5t0~p f !25/31t1~p f !211t3/2~p f !22/3

1t2~p f !21/31t5/2S 11 ln~p f !

C D , ~2.17!

t05M25/3,

t1520

9 S 743

3361

11

4n DM21n22/5,

t3/25216pM22/3n23/5,

t2510S 3058673

10160641

5429

1008n1

617

144n2DM21/3n24/5,

t5/2540

9 S 7729

6721

3

8n Dpn21,

wheretc andfc are respectively the time and phase at clescence,C a constant of integration. In the computation, wcan use a single parameter, namelyf t , which replaces22fc2p/41Z or 22fc2p/41Z2t5/2ln C at 2.5 PN.

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C. Matched filtering

The principles of matched filtering are well known~see@8–13# for a complete introduction!. We assume an infinitenumber of noise realizations in order to be in the ideal cfor the matched filtering. We write the detector outputo(t)as the sum of the noisen(t) and a signals(t). In the case ofthe interferometers(t) is given byh(t), defined in Eq.~2.1!.We introduce a time domain filterg(t). We denote byc(t)the correlation function:c(t)5*o(t8)g(t81t)dt8. The sig-

nal to noise ratio~SNR! is defined byr5 c/Ac22 c2, wherethe average—denoted by a bar—is over an infinite setidentical detectors.n(t) is modeled in the frequency domaiby its power spectral densitySh( f ). We use the one-sided

spectral density:n( f )n* ( f 8 )51/2d( f 2 f 8)Sh( f ), where *represents the complex conjugate.

It is convenient to define a scalar product as

^s,g&54 ReS E0

1` s* ~ f !g~ f !

Sh~ f !df D , ~2.18!

to have the signal to noise ratio~SNR! given by the simpleexpression

r5^s,g&

A^g,g&. ~2.19!

As a result of the theory of linear filters, the maximumSNR is obtained when the filter—modulo its norm—is equto the signal:rmax5A^s,s&. Because the template family wuse will not be an exact representation of the signal, it intduces a mismatch which is measured the ambiguity funcAF

AF5r~s,g!

r~s,s!5

^s,g&

A^g,g&^s,s&, ~2.20!

which yields

AF~DXi !5

E0

` f 27/3

Sh~ f !cos„Fs~ f !2Fg~ f !…df

E0

` f 27/3

Sh~ f !df

, ~2.21!

where the subscripts is relative to the signal andg to thetemplate;F is given by Eq.~2.16!. Xi are the different pa-rameters of the chirp,DXi5Xis2Xig . In this paper, we willsearch for the maximum ofAF over a template family for agiven typical signal. The member of a given family is chaacterized by the value of its parameters: the integrationrameterstc , f t and the mass parameterst i . The maximumvalue of the ambiguity function for a given set of parametwill be called the fitting factorFF. The parameters obtainethrough the maximization of the AF give the systemaerrors—bias—in the measured parameters. They differ frthe statistical errors which are found to be very low@11–13#.

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D. Noise spectrum

Figure 1 gives the expected sensitivity in the early staof VIRGO @33#.

We fit this curve with the following analytical formulas iwhich the low frequency cutofff c is set at 4 Hz~for filteredseismic noise!:

Sh~ f !55 `, f , f c ,4310236

f 51

3.6310243

f

14310246X11S f

500D2C Hz21, f . f c ,

~2.22!

where 4310236/ f 5 refers to the pendulum thermal nois3.6310243/ f to the mirrors thermal noise and 4310246@11( f /500)2# to the quantum noise. We do not take into acount the wire vibration noise~violin modes!. We write thepower spectral density asSh( f )5Sh0( f )S( f ) with Sh0( f )54310246 Hz21 and S( f )51010/ f 51900/f 111( f /500)2

~dimensionless!. The maximum sensitivity is at 483 Hz wit3.9310223 Hz21/2; see Fig. 2.

FIG. 2. Analytical fit for VIRGO noise.

FIG. 1. Expected sensitivity for VIRGO noise.

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For the computation of the ambiguity function we thtake for the low frequency cutf c54 Hz.

III. NUMERICAL COMPUTATIONS

A. Methods

For our computations we consider three generic sittions: two neutron stars of 1.4m( @neutron-star–neutron-sta~NS-NS!#, m( being the solar mass, one neutron star1.4m( and a black hole of 10m( @neutron-star–black-hole~NS-BH!# and two black holes of 10m( ~BH-BH!. Themaximum SNR is given by

rmax51

2A10

3p22/3

c

d

M5/6

ASh0

Af 7/3, f 7/35Ef c

f l f 27/3

S~ f !df ,

wheref l is a high frequency cutoff, corresponding to the lastable orbit in the Schwarzschild metric, and given byf l5(63/2pm)21. Numerically we have rmax

57.806(10Mpc/d)(M/m()5/6Af 7/3104. In our three casesat 10 Mpc rmax(NS-NS!518.5, rmax(NS-BH!536.6 andrmax(BH-BH!580.5.

We can note that 67% of the value off 7/3 is between 20Hz and 200 Hz. So most of the contribution to SNR is in thrange. For the high frequency cut, it is not necessary tofurther than 1 kHz because we have 98% of the value off 7/3with f l51 kHz.

So we try to find fitting factors, when using families otemplates of PN orders lower than the PN order of the sigWe would like to stress that because the integrals becovery oscillatory, we need to use a very accurate integraalgorithm to compute the ambiguity function. We have usDO1AJF of the NAG© library.

We know@8# that it is sufficient to use only two values fothe parameterf t :Df t50 andDf t5p/2. Indeed the ambi-guity function AF writes with AF05AF(Df t50) andAF15AF(Df t5p/2),

AF5AF0cos~Df t!1AF1sin~Df t!

5AAF021AF1

2cos~Df t2u!,

and tanu5AF1 /AF0. So one only needs to maximizAAF0

21AF12, andDf t5u.

TABLE I. Results obtained with 1 PN order signal and Newtoian templates. FF is the fitting factor, PV512FF3 the loss ofevents percentage andr the signal to noise ratio.

Binary FF~%! PV~%! r

Mg

Ms~%!

Df t(rad) Dtc(ms)

NS-NS 36 95 6.7 95 20.33 28NS-BH 54 84 19.7 89 21.11 9BH-BH 85 39 68.9 87 1.22 8

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B. Results

In this section, we investigate the detection of a sigemitted by a compact binary system involving two nospinning objects, by means of a family of PN templates c

FIG. 3. Fitting factor in percentage~gray scale on the left! for 1PN signal and Newtonian filter. Horizontal axis is relative errort0. Vertical axis is relative error for time of coalescence. Maximzation over phase at coalescence is done. Top: two neutron scenter: neutron star and black hole; bottom: two black holes.

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responding also to non-spinning objects. We evaluatefitting factor with signal calculated from Newtonian up to 2PN order. We have also considered an artifact signal calated at 2.5 PN order and completed to 5.5 PN order in

rs;

FIG. 4. Fitting factor in percentage~gray scale on the left! for 2PN signal and 1.5 PN filter. Horizontal axis is relative error fort0.Vertical axis is relative error for time of coalescence. Maximizatiover phase at coalescence and over time at coalescence isTop: two neutron stars; center: neutron star and black hole; botttwo black holes.

5-5

, PVl


TABLE II. Results obtained with 2 PN order signal and 1.5 PN order filter. FF is the fitting factor512FF3 the loss of events percentage andr the signal to noise ratio.ng is found out far from the physicarange, which explains the imaginary masses.


Mg

Ms~%!

Dn

ns Df t~rad!

m1

m(

/m2

m( Dtc(ms)

NS-NS 98.84 3.4 18 100.02 20.22 21.48 1.246 i0.58 0.42NS-BH 99.70 0.9 36 100.05 20.25 20.79 1.6/8.36 1.20BH-BH 99.96 0.1 80 100.23 20.31 21.41 8.56 i4.7 1.00

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limit n→0 ~see further for more details!. We introduce thequantity PV512 FF3, representing the percentage of losignals coming from uniformly distributed sources in spaFirst results were obtained with 1 PN signal and Newtontemplates, see Table I and Fig. 3. In the Newtonian cathere is only one parameter mass, the chirp mass whicequivalent tot0. The stepsDtc50.1 ms andDt0 /t0s5131023 numerically show an accuracy of 1% for FF. Itimportant to notice that we cannot neglect the parametertc .With Dtc50, the FF is reduced by 20%. Table I demostrates that Newtonian filters must be rejected.

In the following computations, filters are characterizedfour parameters:tc , f t and two ‘‘mass’’ parameters necesary to compute thet i . m1 andm2, the two stars masses, anot good parameters, because one needs more computafor the t i , and the space of parameters cannot be tiled uformly, see@21,22#. t0 is a good parameter; we can useconstant step to fill the space and it describes the most pacycles in the signal. So this parameter must be as clospossible to the signal parameter value. For the second mparameter, we could chooset1, in order to facilitate compu-tations of the AF, and to have an uniform tiling of the spaof parameters. But for higher PN templates, we cannotrectly obtain the nextt i . Mohanty@24# and following Owenand Sathyaprakash@22# find convenient to uset3/2. But forquestions of generalities, we do not use this one becawhen we introduce spins, see Sec. III C, the othert i are notany more analytically computable. So for the second pareter, we can choose the reduced mass orn. Becausen ap-pears in the definition oft i , this parameter seems to be thmost convenient. But with this choice, the phase in thecannot be written any longer in terms ofDt i f

i .Computations have been made with signals of orders

PN, 2 PN, 2.5 PN, 2PN and templates of orders 1 PN,PN, 2.5 PN, 1 PN, respectively. The ambiguity functio

08200

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have been numerically constructed with an absolute numcal accuracy of 0.1%. So we choose steps of 1023 forDt0 /t0s and 1022 for Dn/ns . The step forDtc depends onthe total mass: from 0.1 ms to 5 ms.

In Table II and Fig. 4, results are presented for a signa2 PN and filters at 1.5 PN. We have seen that the resultsinsufficient for templates at 1 PN, FF is of the order fro50% to 60%. FF with 2.5 PN signal and 2 PN templateslower than FF of Table II~10 to 15 % lower!. This result isnot surprising, because as remarked by Poisson@34#, 2.5 PNis a less good approximation of the signal than 2 PN forcomputation of the flux in the test mass limit. 3 and 3.5 Porders should be better approximations.

We see in Table II that the loss of signal become negible and 1.5 PN templates are good filters for detection oPN signal. We have a relatively low bias for the chirp mawhich was expected becauset0 provides the maximum number of cycles. SoDt0 has to be the closest to zero. On thcontrary the others parametersDtc andDn get all the effectof the addition of the 2 PN terms. Therefore, they areclose from zero. For the same reason~number of cycles!, theFF is better and thet0 bias is bigger with higher total massThe main difference between our results and those prously presented comes from the fact that we do not obtaincorresponding to the physical region of the parameters. WDn<0, the FF was only of 30 to 50 %. These results adifferent from those made with the advanced LIGO sensiity, maybe because for LIGO the maximum contributionsignal to noise ratio is around 100 Hz and measurescycles, leading to a lower bias. As a consequenceVIRGO, the number of filters must be increased becausevolume of the covered parameter space is bigger. To findcorrect value ofn, an off-line analysis with more parameteis needed.

ting
TABLE III. Signal 2.5 PN1 corrections 5.5 PN in test mass approximation, 2 PN filter. FF is the fitfactor, PV512FF3 the loss of events percentage andr the signal to noise ratio.

Mg

Ms~%!

Dn

ns Df t(rad)

m1

m(

/m2

m( Dtc(ms)

NS-NS 93.13 19 17 99.87 0.30 0.5 0.77/2.7 24.6NS-BH 95.10 14 34 99.10 0.49 21.17 0.98/16.1 216.8BH-BH 99.89 0.3 80 99.37 0.41 1.19 4.9/22.5 225.2

1 / 100 m( 98.2 5.3 17 95.95 0.63 20.78 0.64/176 2640

5-6

istime atlar mass


FIG. 5. Fitting factor in percentage~gray scale on the left! for signal 2.5 PN1 5.5 PN corrections and 2 PN filter. Horizontal axisrelative error fort0. Vertical axis is relative error for time of coalescence. Maximization over phase at coalescence and overcoalescence is done. Left up: two neutron stars; left down: neutron star and black hole; right up: two black holes; right down: 1 soand 100 solar masses.

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In order to test the efficiency of 2 PN templates, we costruct an artifact signal, composed of the 2.5 PN signwhere we add the terms of 3 to 5.5 PN in the test mapproximation. This terms have been calculated by TanTagoshi and Sasaki@35,36#. See Appendix A for the expression of the signal. The results are presented in Table IIIFig. 5. We find acceptable fitting factors, but with importasystematic errors in the masses.

In this case, computations were also made with a binof 1 and 100m( , where the approximationn→0 is nearlyvalid. For this maximization we need a step of 1024 forDt0 /t0s and 1024 for Dn/ns . We find Dtc with a step of0.01 ms. The FF is not very good and the bias, even forchirp mass, are important. This is not surprising becausecannot provide a sufficient number of cycles and the fquency range of integration is small: from 4 to 44 Hz.

C. Constant spins aligned with the orbital angularmomentum

In this section, we now take spins into account. We intduce the spin-orbit term at 1.5 PN and spin-spin term a

08200

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PN. For classical neutron stars these contributions are negible, but they could be important for black holes. From@37#,we can compute the change in the Fourier transform ofsignal. To get an analytical expression using again thetionary phase approximation, we need to assume consspin terms. Time-varying spins have been studied by Aptolatos@38#. He focused on the need to introduce somepropriate parameters to deal with spinning systems, and csequently increase the computing needs beyond the precapabilities.

Only t3/2 andt2 take new values, which read

t3/254~24p1b!M22/3n23/5,

t2510S 3058673

10160641

5429

1008n1

617

144n22sDM21/3n24/5.

L being a unit vector directed along the orbital angumomentum, andS1 and S2 the spins of the two bodies wdefinexi5Si /mi , the spin-orbit term

5-7

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TABLE IV. Signal 2 PN, parallel maximally rotating spins, 2 PN filter. FF is the fitting factor, PV512FF3 the loss of events percentage andr the signal to noise ratio.ng is found out far from the physicarange, which explains the imaginary masses.


Mg

Ms~%!

Dn

ns Df t(rad)

m1

m(

/m2

m( Dtc(ms)

NS-NS 99.04 2.8 18 99.96 23.35 0.32 0.586 i1.06 20.30NS-BH 99.60 1.2 36 99.65 211.4 20.61 1.256 i2.61 20.75BH-BH 99.98 0.06 80 99.80 23.12 20.88 4.266 i7.54 20.60

e

N

pwbis

oryes

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teth

al,

p-

b51

12 (i

S 113mi

2

m2175n D L•xi ,

and the spin-spin term

s5n

48~2247x1•x21721L•x1L•x2!.

With spins parallel to the orbital angular momentum, whave simplyL•xi5x i and x1•x25x1x2. Then consideringa maximally rotating Kerr black hole,x i51, we obtain themaximum values forb and s: bmax5

11312 2 19

3 n, smax5 79

8 n.We made computations with signal at 1.5 P

order1 spins and 1.5 PN templates, signals at 2 PN1 spinsand 1.5 PN templates and signals at 2 PN1 spins and 2 PNtemplates. The results are roughly the same as thosesented in Table IV and Fig. 6. The loss of events is very loless than 3%. The bias in the chirp mass are negligiblestill bigger than the root mean square error. But the mmatch in n is very large: we find an error from 300 t1100 % inn. In this last case, the ambiguity function, is veflat as a function ofDn/ns , and a step of 0.5 would bsufficient to have an accuracy of 0.2% for the FF. In all thecases, the parametern is found very far from the physicarange. Ifn stays in the physical region, FF will be betwee30 to 40 %. But for an on-line analysis, these templashould be sufficient, if an appropriate range is chosen forparameters.

08200

re-,ut-

e

se

IV. HARMONICS

From @26#, we introduce the next harmonics in the signwhich are corrections to the amplitude in 1/c, with c beingthe speed of light. The new scalar responseh of the interfer-ometer is given by

h~ t !524c

dmn~mv!2/3FF

2cos~2f2Z!1~mv!1/3

dm

m

sin i

8

3XF1

2cos~f2Z1!2

9F

2cos~3f2Z!CG , ~4.1!

with

F15AF12 ~51ci

2!2136F32 ci

2 <6, ~4.2!

and cosZ15F1(51ci2)/F1, sinZ156F3ci /F1. We have

udmu/m5A124n. Again, using the stationary phase aproximation, the new Fourier transform ofh is

h~ f !521

4AA10p

3~p f !27/6M5/6HexpF i S 2j~p f !

2p

41ZD G1~mp f !1/3A124n

sin i

8

3F221/3F1

FexpF i S j~2p f !2

p

41Z1D G

29S 2

3D 21/3

expH i F3jS 2

3p f D2

p

41ZG J GJ, ~4.3!

r.
TABLE V. Results obtained with 2 PN signal with 3 harmonics and 2 PN restricted wave form filte~a!For any value ofa, b andj. ~b! a5p/4 andb5j5p/8. FF is the fitting factor, PV512FF3 the loss ofevents percentage andr the signal to noise ratio.

Mg

Ms~%!

Dn

ns Df t(rad)

m1

m(

/m2

m( Dtc(ms)

i 5p/2 ~a!

NS-BH 93.7 17.7 18.2 1006631023 06531024 0.004 1.4/1061023 060.20.1/10 m( 96.6 9.6 5 1006631023 06531024 2631025 0.1/1061023 060.20.5/20 m( 91.8 22.6 12.5 1006631023 06531024 1.531023 0.5/2061023 060.2

i 5p/4 ~b!

NS-BH 96.9 9 16 1006631023 06531024 0.002 1.4/1061023 060.2

5-8

nal

n,

hat

ics

the

otal

the

es.atictheesthescil-rate

orxi-co


with

j~u!5utc2fc1C~u!,

FIG. 6. Fitting factor in percentage~gray scale on the left! for 2PN signal1 spins and 2 PN filter. Horizontal axis is relative errfor t0. Vertical axis is relative error for time of coalescence. Mamization over phase at coalescence and over time at coalescendone. Top: two neutron stars; center: neutron star and black hbottom: two black holes.

08200

C~u!53

256~t0u25/31t1u211t3/2u

22/3

1t2u21/3!.

We want to compute the new maximal value of the sigto noise ratio h,h& in order to compare it to the SNRr rest.obtained with the restricted waveform. In this computatiotwo new integrals depending on the noise appear:

f 25E0

` f 22

S~ f !df , f 5/35E

0

` f 25/3

S~ f !df . ~4.4!

^h,h& is given in Appendix B. To compare it tor rest. , inde-pendently of the position of the binaries, we can notice t

3<F1

F<5,

and we obtain

u^h,h&2r rest.2 u

r rest.2

<2f 2

f 7/3eS 11e

1

2

f 5/3

f 2D , ~4.5!

where

s5221/3F1

F19S 2

3D 21/3

, ~4.6!

e5ssin i

8A124nm1/3. ~4.7!

In the case of the VIRGO noise we havef 2 / f 7/3'5.5 andf 5/3/ f 2'6. We consider that we can neglect add harmonif this leads to less than 10% loss of events, so

u^h,h&2r rest.2 u

r rest.2

<331022. ~4.8!

To obtain the value given by Eq.~4.8! it is sufficient tosatisfy the conditione<331023 ~we have neglected theterm in e2).

For maximal values for the angles functions we obtainnumerical condition

dm

m S m

m(D 1/3

<0.1. ~4.9!

In the case of neutron star binaries, if we suppose the tmassm<4.5m( , it is sufficient to havedm<6%m to ne-glect harmonics, which can be assumed to be realistic incase of neutrons stars.

We now look at the case of stars of different massResults are presented in Table V and Fig. 7. No systemerrors are introduced when neglecting the harmonics infilter, which shows that the restricted wave form filter donot match at all the next harmonics. We can notice thatphase in the next harmonics are in this case, the most olatory possible. Besides the results of Table V demonst

e isle;

5-9

istime at


FIG. 7. Fitting factor in percentage~gray scale on the left! for signal 2 PN1 2 harmonics and 2 PN restricted filter. Horizontal axisrelative error fort0. Vertical axis is relative error for time of coalescence. Maximization over phase at coalescence and overcoalescence is done. Left up:i 5p/2 neutron star and black hole; left down:i 5p/2, 0.1 and 10 solar masses; right up:i 5p/2, 0.5 and 20solar masses; right down:i 5p/4, a5p/4 andb5j5p/8, neutron star and black hole.

vasen

inas

elinbenths

sslru

th

e

assicalanot,m-ins

onesfor

. Anm-ss.an-er-ctedthe

edO

that harmonics cannot be anymore neglected for certainues of the angles since it leads to more than 10% losevents.i 5p/2 is not a singular value, so it must be takinto account. The other case,i 5p/4 is a typical value for theangles. The resulting fitting factor is what is expectedmost cases. Similar results were obtained in the LIGO csee@39#.

V. CONCLUSION

In this paper, we have discussed which template modshould be taken for the matched filter search of an inspirabinary signal in the VIRGO experiment, i.e., what shouldnecessary to include in the post-Newtonian developmeWe have made exhaustive numerical computations ofambiguity function to estimate which class of wave formare good enough for an on-line analysis, when the clastemplates are not exactly similar to the signal. As previoupublished, we reject the Newtonian filter and find that filteusing the restricted wave form at 2 PN order should be sficient. We did not use the same parametrization as oauthors as we find more convenient to use parametersrectly derived from the chirp mass and the ratio of reduc

08200

l-of

e,

lsg

ts.e

ofysf-erdi-d

mass to total mass. We find that we obtain the chirp mwith a good accuracy. It may be needed to use unphysvalues for n to compensate for neglected post-Newtoniterms and in order to have the best fitting factors. If nfitting factors can be worse than 40%. Furthermore, we deonstrate that the maximum values for constant parallel spcan be omitted if we use values forDn very different fromthe real ones. Other numerical computations should be dfor varying spins following the work done by Apostolato@38#. As a consequence of the use of nonphysical valuesn, we have an increased volume for the parameter spaceoff-line analysis with template depending on more paraeters will be needed to find the correct values for the maWe also demonstrate that other harmonics in the signal cnot be neglected, in the case of binaries with stars of diffent masses, in certain cases. To conclude, the restriwaveform at 2 PN order is a good template model foron-line search of inspiral compact binaries.

ACKNOWLEDGMENTS

I thank Luc Blanchet and Jean-Yves Vinet who directthis work. I express gratitude toward people of the VIRG

5-10

th


group at Observatoire de la Coˆte d’Azur in Nice and at LALin Orsay for useful discussions and information aboutVIRGO Project. This work is supported by CNRS.

APPENDIX A: 5.5 POST-NEWTONIAN PHASE TERM INTEST MASS APPROXIMATION

From @36#, where v5(mp f )1/3, E5Enewte(v), L5Lnewtp(v), Enewt given by Eq.~2.9! andLnewt given by Eq.~2.11!, one has

08200

e e~v !51

v2 S 122v2

~123v2!1/221D ,

dE

df5

dEnewt

dfq~v !,

~A1!

and

q~v !5126v2

~123v2!23/2

5123

2v22

81

8v42

675

16v62

19845

128v82

137781

256v10.

~A2!

p~v !5121247

336v214pv32

44711

9072v42

8191

672pv51S 6643739519

698544001

16

3p22

3424

105ln 22

1712

105~g1 ln v ! D v6

216285

504pv71S 2

323105549467

31783752002

1369

126p21

39931

294ln 22

47385

1568ln 31

232597

4410~g1 ln v ! D v8

1S 265978667519

745113600p2

13696

105p ln 22

6848

105p~g1 ln v ! D v9

1S 22500861660823683

28319323032002

424223

6804p22

83217611

1122660ln 21

47385

196ln 31

916628467

7858620~g1 ln v ! D v10

1S 8399309750401

101708006400p1

8521283

17640p ln 22

142155

784p ln 31

177293

1176p~g1 ln v ! D v11, ~A3!

whereg50.577216 . . . is theEuler constant.The restricted wave form is still given by Eq.~2.15! where formally the phaseF( f ) is

F~ f !5f t12p f tc12p f Ep f c

p f

t8~F !dF2Ep f c

p f

2pFt8~F !dF, t8~ f !521

LdE

df. ~A4!

In 5.5 PN order, we obtain

F~ f !5f t12p f tc13

128n21v25L, ~A5!

where

L511S 3715

7561

55

9n D v2216pv31a2v41b2.5v

5ln v1~a31b3ln v !v61a3.5v71~a41b4ln v1c4ln2v !v8

1~a4.51b4.5ln v !v91~a51b5ln v !v101~a5.51b5.5ln v !v11, ~A6!

a2515293365

5080321

27145

504n1

3085

72n2

b2.55S 38645

25215n Dp

a3511583231236531

46942156802

640

3p22

6848

21g2

13696

21ln 21O~n!

5-11


b3526848

21

a3.5577096675

254016p1O~n!

a452550713843998885153

8304255306547202

90490

567p22

36812

189g2

1011020

3969ln 22

26325

196ln 31O~n!

b4522550713843998885153

2768085102182401

90490

189p21

36812

63g1

1011020

1323ln 21

78975

196ln 31O~n!

c4518406

631O~n!

a4.55S 105344279473163

187768627202

640

3p22

13696

21g2

27392

21ln 21O~n! Dp

b4.55213696

21p

a5521433006523295407126559

1263066979958784001

578223115

3048192p21

6470582647

27505170g1

53992839431

55010340ln 2

25512455

21952ln 31O~n!

b556470582647

275051701O~n!

a5.55S 1857541407236594411

2768085102182402

94390

567p22

3558011

7938g2

862549

1134ln 22

26325

196ln 31O~n! Dp

b5.5523558011

7938p1O~n!. ~A7!

APPENDIX B: FIRST CORRECTIONS TO THE AMPLITUDE

If we want to compute the maximum SNR with harmonics, we obtain

^h,h&5r rest.2 F11~124n!S sin i

8 D 2

~a12b!m2/3f 5/3

f 7/31A124n

sin i

82~d1e!m1/3

f 2

f 7/3G ~B1!

a5222/3F1

2

F2181S 2

3D 22/3

b52221/3F1

F9S 2

3D 21/3E0

` f 25/3

S~ f !df cosX3CS 2

3p f D2C~2p f !22fc1Z2Z1C/ f 5/3

d5221/3F1

F E0

` f 22

S~ f !df cos„C~2p f !22C~p f !1fc1Z12Z…/ f 2

e529S 2

3D 21/3E0

` f 22

S~ f !df cosS 3CS 2

3p f D22C~p f !2fcD / f 2 .

082005-12

R.

E.

V

. D

s.

ta

ev.

.

lez,


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Documents

Systematic errors for matched filtering of gravitational waves from inspiraling compact binaries