Data Analysis for Coalescing Binariesdragonschool.roma2.infn.it/lectures/vicere-mondragone-2004.pdfN N−1 ∑ κ=0 1 2 Sn [κ]ei2πκ(a−b)/N (16) that is, R and the two-sidedspectrum

Data Analysis for Coalescing Binaries

Andrea Viceré

9th September 2004

Istituto di Fisica dell’Università di UrbinoVia S.Chiara 27, I-61029 Urbino, ITALY

e-mail: [email protected]

AbstractThe purpose of these lessons is to introduce the basic data analysis techniques for the

detection of GW emitted by stars in coalescence. We will focus on the “standard” approachbased on matched filtering, without attempting to cover the broader range of methods, de-vised to cope with possible limitations in the waveform knowledge.

Dragon School 2004 Data Analysis for Coalescing Binaries

An example of signal from a coalescence

In this example I have chosen a BH-BH event, in which the masses of the BH are set at m1 =10M�, m2 = 20M�, and the starting frequency of observation is at 16Hz. The last portion of theevent has been enlarged to show the merger and ringdown phases. However, it is the chirp portionof the signal that will be mostly dealt with during this lessons.

Dragon School 2004 Data Analysis for Coalescing Binaries 1

The signal phases

In order to plan for the data analysis we need to take into account the expected shape of thesignal: we can split it into three phases[6]

Inspiral giving a chirping signal, with a definite time-frequency relation[3, 5, 4, 1, 8]

Merger heavily dependent on the details of the collision, and on the equation of state of nuclearmatter for NS-NS coalescences

Ringdown composed by damped sinusoids, with relative amplitudes and phases hard to predict.

The CB detection relies presently mostly on the inspiral phase, although resummation methods[12,13] allow to predict (for binary black-holes) also the waveform in the merger phase, and connect it tothe ringdown phase.


The approximated inspiral signal

(h+

h×

)= A [ν(t)]2/3

(cosφ(t)cos(2θ) cos2 i+1

2 +sinφ(t)sin(2θ)cosi

sinφ(t)cos(2θ)cosi−cosφ(t)sin(2θ) cos2 i+12

)

In this approximation, only the funda-mental harmonic of the signal is consid-ered! i describes the inclination of thebinary orbit wrt the line of sight, while θis the wave polarization; α,δ locate thesource in the sky, while φ(t) is (twice)the orbital phase and ν(t) ≡ φ(t). Theamplitude A depends on the distanceand the masses of the stars.


The signal seen by the detector

The detector will see both polarizations and get a signal

s(t) = F+ (α, δ, t)h+ (t)+F× (α, δ, t)h× (t) (1)

which in principle depends on time also through the antenna patterns F+,×.

This dependence can be neglected for ground based interferometers, because the signal per-manence in the detection bandwidth is short: therefore

s(t) = A [ν(t)]2/3cos(φ(t)+φ0) (2)

is the signal at the output of the receiver, with some unknown constant amplitude A and an unknowninitial phase φ0, which depend both on the binary orbital parameters and on the detector orientation.

This dependence is immaterial, for a single detector analysis, but we will need it when dealingwith a network of interferometers.


Phase and signal at Newtonian order

The Newtonian approximation for the phase φ is inaccurate close to the ISCO, but allows tograsp the main signal characteristics: one has

φ(t) =16πν0τ(ν0)

11

[1−(

1− tτ(ν0)

)5/8]

; ν(t) = ν0

(1− t

τ(ν0)

)−3/8

, (3)

where ν0 is the signal frequency at t = 0 and τ(ν0) is the time remaining before the coalescence.

Albeit approximate, this formula shows that thesignal is a chirp with amplitude and frequencygrowing in time.Up to this order, all the physics is embodied in asingle parameter τ(ν0): this is no more true athigher orders, and the phase formula changes sig-nificantly.


Signal duration

It is instructive to notice that the time-frequency formula can be used to express the permanencetime tu− tl of the signal in a certain band [νl , νu] as

tu− tl = τ(ν0)

[(νl

ν0

)−8/3

−(

νu

ν0

)−8/3]

(4)

which means that extending the band toward smaller frequencies (νl → 0) leads to a rapid increase(faster than 1

ν2l) of the signal length.

As a rule of thumb, and neglecting the dependence on the upper frequency, it is worth remem-bering the following formula

duration≈ 34

(MM�

)−53( νl

40Hz

)−83sec (5)

where M� is the mass of the Sun.


Post-newtonian corrections

A PN2 form of the signal, in the Taylor approximation, is given by

s(t) ∝ [ν(t)]2/3cos[Φ(ν(t))+Φl ] (6)

with ν(t) the instantaneous frequency (a function of time) ; the phase Φ(ν) is a series

Φ(ν) =16π5

τ0νr

[(1−(

ννr

)−53)

+54

τ1

τ0

(1−(

ννr

)−1)

−2516

τ1.5

τ0

(1−(

ννr

)−23)

+52

τ2

τ0

(1−(

ννr

)−13)]

(7)

where νr is a reference frequency, chosen to compute the τ parameters.

The ν(t) is implicitly given by the relation

dνdt

=3νl

8τ0

(ννr

)113[

1− 3τ1

4τ0

(ννr

)23

+5τ1.5ν8τ0νr

− 12

(τ2

τ0− 9

8

(τ1

τ0

)2)(

ννr

)43]

(8)


which can be inverted to give

t− tr = τ0

[1−(

ννr

)−83]

+ τ1

[1−(

ννr

)−2]

(9)

−τ1.5

[1−(

ννr

)−53]

+ τ2

[1−(

ννr

)−43]

at the same order in the Post Newtonian expansion.

In turn, the parameters τ0,1,1.5,2 depend on the masses of the stars as

τ0 ≡ 5256π

ν−1r (πMνr)−5/3η−1

τ1 ≡ 5192π

ν−1r (πMνr)−1

(743336

+114

η)

η−1

τ1.5 ≡ 18

ν−1r (πMνr)−2/3η−1 (10)

τ2 ≡ 5128π

ν−1r (πMνr)−1/3η−1

(30586731016064

+54291008

η+617144

η2

)through M ≡m1+m2 and η≡ µ

M . The combination M ≡ µ3/5M2/5 = η3/5M is called chirp mass.


The data acquired by the detector

Any detector actually does not have access to the continuous time form s(t) of the signal, but toa discrete time expression s[n], obtained sampling the continuous signal with a sampling frequencyfs, that is every dt = 1

fsseconds.

It is useful then to look at the signal(s) as vector(s)

s≡ (s[0], s[1], . . . s[N−1]) (11)

where N is the length of the vector, corresponding to a duration

T = dt N =Nfs

(12)

of the time window.

The transformation from the continuous to the discrete form is delicate, for instance one canhave an aliasing effect in sampling the theoretical waveform at too low a sampling rate.


Signal and noise

The signal s, if present, will be mixed with the instrumental noise: the output of the interferometerwill be of the form

x = s+n (13)

where the noise n is assumed to be Gaussian, which means that

P(n) = N e−12n·R−1·n (14)

is the probability of a instance n of noise; R is the correlation matrix,

Rab≡ E [nanb] . (15)

For stationary noise, Rab depends only on the difference a−b, and one has

Rab = R

(a−b

fs

)=

fsN

N−1

∑κ=0

12

Sn [κ]ei2πκ(a−b)/N (16)

that is, R and the two-sided spectrum 1/2Sn are Fourier pairs.


The sensitivity h of a detector

To show a detector sensitivity it is customary to plot the quantity

h( f )≡√

Sn( f ) (17)

where Sn( f ) is the one-sided noise spectrum.


Example: signal in VIRGO noise

A chirp emitted by a pair m1 = 4M�, m2 = 5M�(SNR= 10) embedded in VIRGO noise.

Despite the relevant SNR value, no evidence of the signal can be grasped “by eye” in the data.


The detection problem

The output x of the detector is an unknown mixture of signal and noise, that is we have todistinguish among two hypotheses

H0 : no signal is present, that is the outputs are distributed as Gaussian noise

H1 : some signal is present.

In the technical language, H1 is a composite hypothesis, because the signal can come in manyshapes.

A standard way to proceed is to split the problem in two: first one assumes that there is somesignal, and finds which is the most probable one.

Then, one assigns a degree of significance to the supposed signal presence.


The maximum likelihood procedure

Given the output x, the probability of it is

P(x) = P(x|H0)P(H0)+P(x|H1)P(H1) ; (18)

using the Bayes ruleP(H1|x)P(x) = P(x|H1)P(H1) (19)

one can invert the relation among probabilities and obtain

P(H1|x) =Λ(x)

Λ(x)+P(H0)/P(H1): (20)

where Λ(x) = P(x|H1)P(x|H0)

is the likelihood ratio.

Even if we do not know the a priori probabilities P(H0,1), we know that the a posteriori probabilityof H1 is maximum when Λ(x) is maximum. In other words, the signal which maximizes the likelihoodis the most probable one, assuming that H1 is true.


The likelihood expression

Assume that the signal has a certain form s(t) = Acos(φ(t)+φ0), what is the associated like-lihood?

Obviously the combination x−s is distributed as Gaussian noise, hence

P(x|H1,s) = P(x−s|H0) = N e−12(x−s)·R−1·(x−s) (21)

and the likelihood ratio (for a given signal s) is given by

Λ(x|s) = e−12s·R−1·s+s·R−1·x. (22)

We have to recall that we should search the maximum of Λ varying the signal parameters: in partic-ular, we can analytically maximize over the amplitude A and the initial phase φ0.

It is now convenient to introduce a notational abbreviation:

〈a, b〉 ≡ a·R−1 ·b (23)

which is a sort of scalar product among signals.


Amplitude maximization

It is immediate to show that the most probable amplitude is given by

AMLE =〈s, x〉〈s, s〉

(24)

where s(t) = cos(φ(t)+φ0). Hence

maxA

Λ(x|s) = exp

(12〈s, x〉2

〈s, s〉

); (25)

in other words, one has to define the normalized template

v =s√〈s, s〉

(26)

and look for the maximum of the expression |〈v, x〉|: this is the famous matched filtering procedure,invented by Wiener.


Phase maximization

The template v depends on the initial phase φ0: we can maximize the likelihood (or its logarithm)over the possible phases.

This can be done definings= s0cos(φ0)+s1sin(φ0) ; (27)

where s0 ≡ Acos(φ(t)) , s1 ≡−Asin(φ(t)); we have therefore

〈s, x〉2

〈s, s〉=

(〈s0, x〉+ 〈s1, x〉 tanφ0)2

〈s0, s0〉+ tan2φ0〈s1, s1〉+2tanφ0〈s0, s1〉(28)

which can be maximized over tanφ0 obtaining

maxA,φ0

2lnΛ =〈s0,x〉2〈s0,s0〉

+ 〈s1,x〉2〈s1,s1〉

−2〈s0,s1〉〈s0,x〉〈s1,x〉〈s0,s0〉〈s1,s1〉

1− 〈s0,s1〉2〈s0,s0〉〈s1,s1〉

(29)

notice that we could not assume 〈s0, s1〉= 0 (why?).


Orthogonal templates

The non-orthogonality of the two basis signals s0, s1 is a nuisance: however we are free tochoose new signals sp, sq which are truly in quadrature, by choosing appropriate angles φp, φq whichrender the product

〈sp, sq〉= 〈s0cosφp+s1sinφp, s0cosφq+s1sinφq〉= 0; (30)

in this new basis, the matching procedure is simplified and one gets

maxA,φ0

2lnΛ =〈sp, x〉2

〈sp, sp〉+〈sq, x〉2

〈sq, sq〉(31)

that is, adsorbing the normalization in the templates vp, vq

maxA,φ0

2lnΛ = 〈vp, x〉2+ 〈vq, x〉2 (32)

the usual sum in quadrature of the outputs of the two matched filters.


The scalar product in the frequency domain

It is worth noticing that the scalar product 〈vp, x〉 assumes a simple form in the frequencydomain: remembering that R is the Fourier transform of the noise spectrum, one has

〈vp, x〉= 2Z +∞

0

vp( f ) x∗ ( f )+ v∗p( f ) x( f )Sn( f )

d f (33)

where we have adopted a continuous frequency notation, although this expression should be in-tended in discrete frequency and discrete time.

This expression allows to maximize over the possible shifts in time of the signal: we just noticethat a time shift ∆t of the template corresponds to a phase factor ei2π f ∆t in its Fourier transform:hence we define

〈vp, x〉(∆t) = 2Z +∞

0

[vp( f ) x∗ ( f )

Sn( f )ei2π f ∆t +c.c.

]d f (34)

which is the inverse Fourier transform ofvp( f )x∗( f )

Sn( f ) , whose maximum over ∆t locates in the data themost likely signal.


The matched filtering statistics

The matched filter applied to pure noise has simple statistical properties:

E [〈a, n〉〈b, n〉] = 4ℜZ Z ∞

0a( f ) b∗ ( f ′)

E [n( f ′) n∗ ( f )]Sn( f )Sn( f ′)

d f d f′

= 4ℜZ ∞

0

a( f ) b∗ ( f )Sn( f )

d f = 〈a, b〉 . (35)

Given a signal in noise, we have therefore

SNR≡ E [〈s, s+n〉]√E[〈s, n〉2

] =√〈s, s〉 . (36)

Working with normalized templates v = s√〈s,s〉

, this means that

E [〈v, s+n〉] = SNR (37)

and 〈v, n〉 is a Gaussian variable with zero mean and unit variance!


SNR distribution

It is worth recalling that we actually use two templates in quadrature, and build the variableρ2 ≡ 〈vp, s+n〉2+ 〈vq, s+n〉2.

In absence of the signal, the two variables 〈vp, n〉 ; 〈vq, n〉 are Gaussian, and uncorrelated:

E [〈vp, n〉〈vq, n〉] = E [〈vp, vq〉] = 0 (38)

The statistic is a Rayleigh variable (a 2 DOF χ2),

with the distribution Pχ22(ρ) = ρe−

ρ2

2 as in figure.In presence of a signal, one would have a non-central χ2.


An example of “detection”

Given the distribution of the statistic ρ, it is clear that detection means assessing whether a“peak” in the output of the correlator ρ2(∆t)

ρ2(∆t)≡ [〈vp, x〉(∆t)]2+[〈vq, x〉(∆t)]2 (39)

is statistically significant.

One expects a time series distributed asPχ2

2and a signal should manifest itself as

a sufficiently high peak above the noise,which corresponds, given our normal-izations, to a sufficiently high SNR.


The Stationary Phase Approximation

It is possible to exploit the approximate identity[8]Zg(t)eiφ(t)dt ≈

∣∣∣∣ 2πiφ′′ (t0)

∣∣∣∣g(t0)eiφ(t0) (40)

where t0 is the stationary point for the phase φ(t), to obtain an analytical form of the Fourier transformof the signal.

Approximately

s( f ) ∝ f−7/6expi[π

4+Φc−2π f tc+Ψ( f )

](41)

where

Ψ( f ) =−2πνr

[3τo

5

(f

νr

)−53

+ τ1

(f

νr

)−1

− 3τ1.5

2

(f

νr

)−23

+3τ2

(f

νr

)−13]

(42)

is the expression of the phase and Φc, tc phase and time at a reference point.


A simplified expression for the SNR

It is worth noting that, as long as the SPA is valid, the expression for the SNR becomes verysimple[6] (for a source optimally oriented wrt the detector):

SNR =

√4

Z |s( f )|2

Sn( f )d f =

GNM 5/6

r (cπ)2/3

(56

)1/2√Z

f−73

Sn( f )d f

= 1.56×10−19

[MM�

]5/6[Mpcr

]√Zf−

73

Sn( f )d f . (43)

This is why in the literature one frequently finds the function f−7/6 plotted against the sensitivityh( f )≡

√Sn( f ); but only the integral has a meaning in the detection.

Substituting the sensitivities for the different instruments, one can exploit the above formula tocompare them as detectors of chirps, (exercise!): and one can choose the frequency limits to imposeon the integral.


Effective band of the instrument

The expression given for the scalar product

〈a, b〉= 2Z +∞

0

a( f ) b∗ ( f )+ a( f )∗ b( f )Sn( f )

d f (44)

needs not to be evaluated over all possible frequencies: at low and high frequencies the noise Sn( f )is large and the contribution to the integral is small.

To be more quantitative, let us consider the ratio

〈s, s〉[νl ,νu]

〈s, s〉[0,∞]=

R νuνl

|s( f )|2Sn( f ) d fR ∞

0|s( f )|2Sn( f ) d f

≈R νu

νl

f−7/3

Sn( f ) d fR ∞0

f−7/3

Sn( f ) d f(45)

for an arbitrary signal s: this is the fraction of SNR2 that templates limited to the interval [νl , νu] canrecover. Actually, νu is set by the limits of the validity of the theoretical waveforms: but νl is morerelevant in practice.


Choice of the lower frequency cutoff

The template length has a rapid dependence on the lower frequency cutoff, which is thereforeof practical importance: we show some prospective noise curves for Virgo, and the correspondingSNR fraction, as a function of νl

the figure shows that a 40Hz cutoff is more than adequate to recover more than 95% of theSNR, at least for a “standard” Virgo.


The search for the maximum likelihood

For the amplitude and the initial phase we have been able to perform the maximization analyti-cally, and for the relative shift between the signal and the template we have exploited the translationproperties of the Fourier transform.

It is unfortunately impossible to maximize analytically over the remaining parameters, and inparticular over the masses of the stars: in order to determine the most likely signal, we need to testseveral templates against the data.

At first sight, given that the masses are a continuous parameter, this procedure appears tough:however we shall see that the templates don’t possess a perfect discriminating capacity amongdifferent signals, and this means that they are sensible, to a lesser extent, also to slightly mismatchedsignals.

The detection becomes therefore of a different nature from the classical Wiener problem: wedeal with filters that cannot be exactly matched to the signal.


The mismatched filter

Suppose that a template is applied to a signal with different physical parameters: masses, timeof coalescence tc, phase at the coalescence Φc: what is the result of the scalar product?

In other words, what is the superposition integral among templates with different parameters?

Apart a normalization,

〈q(Φc, tc,θ), q(Φ′c, t

′c,θ

′)〉 ∝Z +∞

0d f f−

73cos(∆Φc−2π f ∆tc+∆Ψ)

Sn( f ): (46)

regarding one of the templates as the signal, and maximizing over the phase of the other

max∆Φ0

〈q, q′〉 ∝∣∣∣∣Z +∞

0d f f−

73

expi (2π f ∆tc+∆Ψ)Sn( f )

∣∣∣∣ . (47)

This formula is very instructive at the Newtonian approximation: in that case one has

∆Ψ( f ) =−6π5

νr

(f

νr

)−53

∆τ0 ; (48)

the integrand suggests that a difference ∆τ0 can be in part compensated when maximizing over ∆tc.

The expression max∆Φ0,∆tc 〈q, q′〉 is sometimes called match among templates.


The ambiguity function

As a function of the remaining physical parameters, the match is called ambiguity function,because it answers to the question: how sensitive are we to a change of signal parameters?

Or equivalently, how ambiguous is the output of the match, as long as the determination of thesignal parameters is concerned?

Let us take a Newtonian signal and the scalar product in the plane [∆τ0+∆tc,∆τ0−∆tc]

along the direction ∆τ0 + ∆tc = 0 the scalar product remains almost at the maximum (unequalscales!). To some extent, this property remains true at higher orders in the PN expansion.


A metric in the signal space

Close to the maximum of the ambiguity function one can make a quadratic expansion, anddefine[9]

max∆Φc

〈q, q′〉= 1− γi j (λλλ)∆λi∆λ j (49)

where the matrix γγγ is defined as

γi j (λλλ) =−12

[∂2

∂λi∂λ j

(max∆Φl

〈q, q′〉)]

∆λ=0

. (50)

Given γγγ, and assuming that tc is the λ0 parameter, one easily finds

∆tc(optimal) =− 1γ00

∑i=1

γ0i∆λi (51)

and thereforemax

∆Φc,∆tc〈q, q′〉= 1−gi j (θ)∆θi∆θ j (52)

where θθθ are the remaining parameters, and

gi j (θθθ) = γi j (θθθ)−γ0i(θθθ)γ0 j(θθθ)

γ00(θθθ)(53)

is the metric projected on the ∆tc = ∆tc(optimal) variety.


The ambiguity function at work

The matrix g behaves like a metric in the space of signals, and this analogy is quite fruitful,because it allows to apply the methods of differential geometry to the problem of covering in aoptimal way the parameter space.

We can readily verify that a quadratic approximation is sensible, at least for values of the matchclose to 1:

to this end, we display in the figure the contoursof equal match around a certain point in the pa-rameter space, assumed bi-dimensional and pa-rameterized with the ∆τ0, ∆τ1 distances: we haveelliptic contours

We further notice that the axes of the ellipses are parallel to the eigenvectors of g, and the axeslengths are inversely proportional to the square root of the corresponding eigenvalues.


A grid detection strategy

Given the non-zero sensitivity of a template to slightly mismatched signals, a possible strategyis to cover the parameter space with a grid of templates, spaced so as to guarantee that any signalwill be recovered with an efficiency above a certain level.

This level, usually called the minimal match, or MM, tells us how dense the templates shouldbe.

The strategy requires to:

• define the limits of the parameter space

• tile it with templates, in the most efficient way.

Notice that this strategy introduces a bias in the detection: events corresponding to the nodes of thegrid are more likely to be detected. This fact, and the correlation between filter outputs, need to betaken into account when assigning confidence levels.


Example: a 2D parameter space

When parameterizing the signal space, it is convenient to select variables which make the metricas constant as possible: good variables are for instance the θ parameters considered by Owen andSathyaprakash[11], θ1 ≡ 2πνrτ0, θ2 ≡ 2πνrτ1.5.

As an example, we show how it lookslike a parameter space for stars in acertain mass range m1,2∈ [mmin, mmax].Please note the logarithmic axes:smaller masses lead to substantiallylonger signal duration, and larger pa-rameter space.

Covering this space with “ellipses”, how many templates will we need?


An application of the metric: the number of templates

To make a estimate, set up a regular rectangular lattice of templates spaced by a proper distanceds. The worst case is a signal at the center of an hypercube, at distance gi j ∆θi∆θ j = N

(ds2

)2from

any corner.

We imposeN(ds/2)2 ≤ 1−MM (54)

because we want to recover at least a fraction MM of the signal;

∆V = dsN ≤(

2√

(1−MM)/N)N

(55)

is the condition on the volume. It follows that

ntemplates ≥Rparameterspace dNθ

√detg(

2√

(1−MM)/N)N , (56)

the ratio of the volume available to the “fiducial” volume of each template[9].


Parameter estimation

Once a signal is detected, how accurately can its parameters be determined?

In other words, given that a signal was present in the data, and its parameters are extractedusing the maximum-likelihood procedure, what is the variance of the estimate?

The Cramer-Rao bound allows to set a lower limit to this variance[7].

We start from the probability distribution P(θθθ|x) of the parameters, conditioned by having re-ceived the data x, and define the Fisher information matrix

Γ≡−E

[∂2 lnP(θθθ|x)

∂θi∂θ j

]: (57)

let us assume that it is positive definite: then it is possible to prove that

varθi ≥[Γ−1(θθθ)

]ii, (58)

a limit which the MLE is not guaranteed to attain.

For the case at hand, we know that

− lnP(θθθ|x) =12〈x−s(θθθ) |x−s(θθθ)〉 (59)


where we include also the amplitude among the parameters, hence we get

Γi j = E

[12

⟨∂2s(θθθ)∂θi∂θ j

, s(θθθ)−x⟩

+⟨

∂s(θθθ)∂θi

,∂s(θθθ)∂θ j

⟩]=

⟨∂s(θθθ)

∂θi,

∂s(θθθ)∂θ j

⟩(60)

where the first term is zero (s−x is zero-mean Gaussian noise).

The detailed form of Γ and its inverse are complicated, but one result is apparent: Γ is propor-tional to the square of the signal amplitude1 that is to the SNR2; therefore stddevθi ∝ 1

SNR.

1If we choose as amplitude parameter lnA !


Network data analysis

Things get slightly more complicated if we are interested in exploiting several detectors, forinstance building a LIGO-Virgo network, or even considering the three LIGOs.

We have two possible approaches to the detection problem:

coincidence analysis: it amounts to build independent lists of candidates for each detector, esti-mate their parameters, and then compare the lists in order to find compatible events, in terms ofrelative amplitudes, arrival times, physical parameters.

Coherent analysis: it consists (for instance) in maximizing the combined likelihood for the pres-ence of a specified signal in the different data streams, and amounts to a vector matched filtering.

It should be clear that the approaches are very different: to say the least, in the first case data neednot to be exchanged. It is also evident that under the assumption of Gaussian noise the coherentanalysis can be made optimal : however, this will not help against non-Gaussian events.


The interferometer network

It is worth recalling positions and orientations of the different ITFs under construction:

the distribution of the ITFs on the northern hemisphere suggests that a network analysis mightgive a better coverage of the sky. This is to some extent true, as we will show later.


Coherent Network DA

The basis of the coherent analysis is very simple[20]: if noises are independent, then the prob-ability of observing data xI on the different detectors (labeled by I) is simply the product of theprobabilities:

P(x1,x2, . . . |s) = ∏I

P(xI|s) (61)

hence the log-likelihood ratio is the sum of the individual LLR and one as

lnΛNW = ∑I

lnΛI = 〈s, x〉NW− 12〈s, s〉NW (62)

where we have introduced vectors of data and templates:

x(t) = (x1(t) ,x2(t) , . . .) . (63)

It is easy to show then the network template s = (s1(t) ,s2(t) , . . .) allows to recover the maximumSNR and generalizes the matched filtering to the network case.


Reference frames

Two issues are crucial: the delays due to the different detector locations, and the differentresponse of the detectors to the two polarizations of the wave.


Signal description

A convenient way[17, 20] to work with network signals is to use a tensor representation: define

w(t) =12

[(h+(t)+ ih×(t))eR+(h+(t)− ih×(t))eL] (64)

as the wave tensor, where h×,h+ are the two wave polarizations, and

eL,R≡12

(eX± i eY)⊗ (eX± i eY) (65)

are helicity states of the wave, in terms of versors eX,Y in the “wave frame”. A detector L is describedby the tensor

dL = n(L)1⊗n(L)1−n(L)2⊗n(L)2 (66)

where n(L)1,2 are versors oriented along the arms of the detector. The signal at each detector is

sL(t) = tr [w(t− τL (α, δ)) ·dL] (67)

where the delay τL at instrument L depends on the source direction, specified by the right ascensionα and the declination δ, in the network frame.

A basis for such tensors exists, the so-called Symmetric Trace Free tensors of rank 2. Therotations of these tensors, and therefore the transformations among frames are described by theGel’fand functions[18, 19], allowing to refer every expression to a common reference, for instancethe network frame.


How does the coherent analysis work?

✘ For each intrinsic parameter of the source (mass, spin, ...) and for each detector I one has twocorrelators CI

0(t) , CIπ/2(t), resulting from the Wiener filtering with the two templates in quadra-

ture.

✘ The network statistic is built combining the correlators with a matrix depending on the sourcedirection

SNR2 = pIJ (φ,θ)[CI

o(t− τI (φ,θ))CJo (t− τJ (φ,θ))+(0⇒ π/2)

]where τi (φ,θ) is the detector dependent delay to be used to take into account the different “timeof arrival” of the signal at each detector location.

✘ Finding an event means scanning not just time t but also the direction φ,θ; this increases thecomputational cost, also because the ambiguity function of the network is narrower than the oneof any single detector[20].


How some of the matrices pIJ look like

Here are a few diagonal entries

Left: pH4K,H4K Center: pL4K,L4K Right: pVirgo,Virgo

These matrices give an idea of how the detectors have a different sensitivity for different direc-tions in the sky.


Example: the network “energy”

The SNR2 available to the network depends now on the direction of the source, as well as onits orbital parameters. It is interesting to average over inclination and polarization, and plot it againstthe sky direction.

Left: LIGO network. Right: Virgo alone


The global network

If all the ITFs are included in the network, obviously the most sensible ones should account forthe greatest part of the SNR

and according to the design sensitivities, this role should be played by the LIGOs (note : seismicwall not shown).


The “energy” available to the global network

We can compare the network made of the three LIGOs (shown at left) with a global networkincluding also Virgo, GEO and TAMA (at right)

we see that the other detectors (with Virgo playing the major role) manage to render the sensi-tivity more spherical, but not fully.

For a better sky coverage, and provided that the current detectors are able to reach the designsensitivities, a second European large scale detector would greatly help.


Comparing with the coincidence analysis

In order to compare coherent and coincidence strategies, it is worth recalling that the SNR2

seen by the individual detectors and by the network obey to different statistics

✔ On a single detector the SNR2 is a χ2 with 2 DOF, hence if ξ is a threshold

PFA(ξ) = e−ξ; PDET (ξ, Esig) =Z ∞

ξe−E−EsigI0

(2√

E ∗Esig

)dE

✔ On the network, the corresponding quantity is a χ2 with 4 DOF, hence

PFA(ξ) = (1+ξ)e−ξ; PDET (ξ, Esig) =Z ∞

ξ

√E

Esige−E−EsigI1

(2√

E ∗Esig

)dE

Therefore interpretation of the SNR clearly depends on the kind of statistic: it is not meaningful to saythat “the network recovers more SNR”, we have to refer to PDET, PFA for a meaningful comparison.


Rules for the comparison

✔ Set a false alarm rate of the network as a whole (1 event/year)

✔ Generate events with random direction ϑ,ϕ and source parameters ε,ψ, but the same networkSNR. This means turning the response peanut into a sphere, to compare the strategies in a wayindependent from the source direction/polarization.

✔ Set false alarm rates RFA on the individual detectors, and rules to combine the events that leadto the same overall RFA as the “coherent network”.

✔ Compute the SNR seen by each detector, hence local detection probabilities PDET for eachsampled direction/polarization.

✔ Combine with various strategies (OR, AND); obtain the average PDET

✔ Compare with the coherent case, and vary the SNR available to the network.


Coherent vs AND with 2 detectors

✔ Two detectors must flag the event. Tune local RFA for a fair comparison.

✔ In average, at larger SNR the AND(2) gets close to the coherent case.

✔ The minimum is always zero: there exist directions/polarizations that only one detector is sensi-tive to!


Coherent vs AND with 3 detectors

✔ Now at least three detectors need to be above the local threshold.

✔ The result is slightly worse, but not qualitatively different: there exist blind directions/polarizations.

In presence of non-gaussian noises, a coincidence analysis will be necessary; such plots help toquantify how much we lose in this way[21].


Conclusions

We touched just few of the issues involved in the DA for CB: in particular

• no mention was done of sub-optimal methods, like the time-frequency transforms. They shouldbe less sensitive to errors in the filters, but will probably have a smaller PDET.

• We neglected the hierarchical methods: at the price of a slightly reduced detection probability,they will reduce significantly the computational burden[10].

• We didn’t mention the existence of improved templates, like those based on the Padéapproximants[16] and on the effective one-body approach[12, 13, 15], which allow to go closerto the coalescence: however, they don’t change the essentials of the detection strategy.

Despite the limitations, I just hope that this lessons provide a useful introduction to a vast literature.

References

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[12] A.Buonanno and T.Damour, Phys.Rev. D 59, 084006 (1999).

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[14] T.Damour, P.Jaranowski and G. Schäfer, Phys. Rev. D 62, 084011 (2000).

[15] T.Cokelaer, P.Jaranowski and J-Y. Vinet, Waveforms from binary black hole coalescences and3PN templates generation, VIR-NOT-OCA-1390-176 (2001).

[16] T. Damour, B. R. Iyer, and B. S. Sathyaprakash, Phys. Rev. D 57, 885 (1998).

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[21] A.Viceré, Network analysis for coalescing binaries: coherent vs coincidence based strategies,Class.Quantum Grav. (proceedings of GWDAW-8), to appear.


Documents

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