On the background estimation by time slides in a network of gravitational wave detectors

HAL Id: in2p3-00394333

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Submitted on 25 Nov 2009

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network of gravitational wave detectors

Michal Was, Marie-Anne Bizouard, Violette Brisson, Fabien Cavalier, Michel

Davier, Patrice Hello, Nicolas Leroy, Florent Robinet, Vavoulidis Miltiadis

To cite this version:

network of gravitational wave dete tors

Mi haª W¡s, Marie-Anne Bizouard, Violette Brisson, Fabien

Cavalier, Mi hel Davier, Patri e Hello, Ni olas Leroy, Florent

Robinet and Miltiadis Vavoulidis.

LAL,UnivParis-Sud,CNRS/IN2P3,Orsay,Fran e.

E-mail: mwaslal.in2p3.fr

Abstra t. Time shiftingthe outputs of GravitationalWavedete tors operating in oin iden e is a onvenient way to estimate the ba kground in a sear h for short duration signals. However this pro edure is limited as in reasing indenitely the number of time shifts does not provide better estimates. We show that the false alarmrateestimationerrorsaturateswiththenumberoftimeshifts. Inparti ular,for dete torswithverydierenttriggerratesthiserrorsaturatesatalargevalue. Expli it omputationsaredonefor2dete tors,andfor3dete torswherethedete tionstatisti reliesonthelogi alOR ofthe oin iden esofthe3 ouplesinthenetwork.

1. Introdu tion

Kilometri interferometri Gravitational Wave (GW) dete tors su h as LIGO [1℄ or

Virgo [2℄ have been taking data with in reasing sensitivities over the past years

[3,4,5,6,7,8,9℄. Itisexpe tedthatshortdurationGWevents,e.g. theso- alledbursts

emitted by gravitational ollapses or the signals emitted by ompa t binary inspirals,

are very rare. Moreover the output of the dete tors is primarily (non Gaussian) noise,

andthisba kgroundnoiseisingeneralnotmodeled. ThisimpliesthatwithasingleGW

dete tor it is very di ult to estimate the ba kground event rate, and then to assess

the signi an e of some GW andidate.

On the ontrary when dealingwith a network of dete tors (thatmeans in pra ti e

at least two dete tors of the same lass), there is a onventional and simple way to

estimate the ba kground, that is the rate of oin ident events due to dete tor noise.

This onsistsof timeshiftingthe sear halgorithmoutputs(ortriggers)ofea hdete tor

withrespe ttotheother(s),bysomeunphysi aldelays,mu hlargerthanthelighttravel

time between the dete tors and mu h larger than the typi al duration of an expe ted

GW signal. The next step isthen to look for oin iden es between shifted triggersjust

as if the shifted streams were syn hronized. As we deal with a priori rare events, we

need toset inpra ti e lowfalse alarmrates inthe analysis. The question thenarises of

howmanytimeslidesare neededfor orre tlyestimatingthe ba kground andespe ially

its tails where rare (non-Gaussian noise) events lay. Note, that in pra ti e in burst

or binary inspiral sear hes, a hundred or more time slides are done [3, 6, 7℄, due in

parti ular to limited omputational resour es. Su h a limitation of ourse depends on

thedurationofthedierentdete torsdatastreamsandonthe omplexityof onsisten y

tests performed on oin ident triggers. For example, time slides omputation for one

year of data sampled at 16384 Hz (LIGO) or 20 kHz (Virgo) an rapidly be ome a

omputationalburden, espe iallywhen omputationallyintensive onsisten y tests like

the

χ

2

veto [10℄ are used.

Inthis paper,weshowthat thepre isiononthe ba kground estimation,usingtime

slides of trigger streams, is in fa t limited. Indeed the varian e of the false alarm rate

estimation does not indenitely de rease as the number of time slides in reases as we

wouldnaivelybelieve. Onthe ontrarythisvarian esaturatesatsomepoint,depending

on the trigger rates hosen in ea h individual dete tor and on the oin iden e time

window set for identifying oin ident events in the network of dete tors.

After introdu ing the general denitions in se tion 2,we give expli it formulas for

the two-dete tor and the three-dete tor ase in respe tively Se . 3 and 4. In the latter

werestri tourselvestotheparti ularanalysiss hemewherewearelookingattheunion

(logi al OR) of the three ouples of dete tors. This is a tually the onguration of

interest, sin e it is more sensitive than simplysear hing for triple oin iden es(logi al

AND)[11℄. Inea h ase(2or3dete tors)we he k theanalyti alresultwithaMonte

Carlo simulation and nd ex ellent agreement. In se tion 5 these results are applied

2. Denitions

2.1. Poissonapproximation for trigger generation

Ba kground triggersinthe dete torsare due torareglit hes. Often these glit hes ome

ingroups,but most analysispipelines lustertheirtriggers, soea hglit hgroupresults

inonlyonenaltrigger. This lusteringpro edureisreasonableaslongasthe resulting

triggerrate ismu hlower thanthe inverse of the typi al lustering time length. In this

limit the lustered triggers are independent events. Thus, throughout this paper we

willassumethatea hdete tor produ esrandomba kgroundtriggers,whi hare Poisson

distributed intime.

2.2. Problem des ription

We look then at the oin iden e between two Poisson pro esses. The single

interferometer trigger rate will be noted

FA

1

,

FA

2

, ..., the oin iden e rate will be simplynoted

FA

. Wedenote by

FA(T )

f

the rate resultingfrom ounting the numberof oin iden e between two data streams, that are shifted by some time

T

. In parti ular for zero lag (

T = 0

) the measured rate is

FA(0)

f

. So the quantities with tildes are the experimentallymeasured rates, andthe quantities withouttildesare thea tual Poisson

distributionparameters. Thepurposeofthe paperistostudy thepropertiesofthetime

shiftingmethod,whi huses

FA =

c

1

R

P

R

k=1

FA(T

f

k

)

asanestimatorof

FA

,where

R

isthe number of time slides and

T

k

is the k

th

time slide stride.

2.3. Poissonpro ess model

To modela Poisson pro ess witheventrate

FA

1

we dis retize the datastream duration

T

withbinsof length

∆t

, thedis retizationtimes ale,e.g. eitherthedete tor sampling rateorthe lusteringtimes ale. Thus, forea hbinaneventispresentwithaprobability

p = FA

1

∆t ‡

.

To ease the al ulation we des ribe the Poisson pro ess realizations with a

ontinuous random variable. We take

x

uniformly distributed in the volume

[0, 1]

N

where

N =

T

∆t

is the number of samples, then ompare

x

k

(the k th

oordinate of

x

) with

p

. When

x

k

< p

there is an event in time bin

k

, otherwise there is none. Thus

x

hara terizes one realization of a Poisson pro ess, and it an be easily seen that the uniform distributionof

x

leads to aPoisson distributionof events.

2.4. Coin iden es

We hoosethesampling

∆t

tobeequaltotwi ethe oin iden etimewindow

τ

c

,inorder to simplify the modeling of the oin iden e between two pro esses. More pre isely, for

when thereisaneventinthe same timebin

k

forboth pro esses. Thisis dierentfrom the usualdenition,whereeventsare saidin oin iden ewhenthey areless thanatime

windowapart. Thisbinningtime oin iden ehasonaveragethesameee tsasdening

as oin ident events that are less than

±

1

2

∆t = ±τ

c

apart. The analyti al results are derivedusingthisnonstandarddenition,buttheyareinpre iseagreementwithMonte

Carlo simulationsthat are performedusing the usualdenition of time oin iden e.

3. The ase of two dete tors

3.1. Time slides between two dete tors

Let

x, y ∈ [0, 1]

N

be two realizations of Poisson pro esses with respe tively

p = FA

1

∆t

and

q = FA

2

∆t

. There is a oin ident event intime bin

k

when

x

k

< p

and

y

k

< q

. So the total numberof oin iden es forthis realizationis

N

X

k=1

1

(x

k

< p)

1

(y

k

< q)

(1) where

(

1

(a) = 1

if

a

is true 1

(a) = 0

if

a

isfalse (2)

Thusthe meannumberof oin iden es withouttime slides is asexpe ted

Mean =

Z

x

1

· · ·

Z

x

N

Z

y

1

· · ·

Z

y

N

N

X

k=1

1

(x

k

< p)

1

(y

k

< q)

d

x

1

· · ·

d

x

N

d

y

1

· · ·

d

y

N

|

{z

}

d

V

= Npq.

(3)

To onsider a number

R

of time slides we take a set of

R

ir ularpermutations of

[[1, N]]

. Time-sliding a ve tor

x

by the ir ularpermutation

π

transforms the ve tor

x

into the ve torof oordinates

x

π(k)

. Thenthe meannumberof oin iden es is simply

Mean =

Z

x

1

· · ·

Z

x

N

Z

y

1

· · ·

Z

y

N

1

R

X

π

X

k

1

(x

k

< p)

1

(y

π(k)

< q)

d

V = Npq,

(4)

thus there isnobias resultingfromthe use of time slides.

3.2. Computation of the varian e

In order to have an estimate of the statisti al error, we ompute the varian e with

R

time slides. The se ondmomentis

10

0

10

1

10

2

10

3

10

4

10

−2

10

−1

10

0

10

1

10

2

number of time slides

variance of the number of coincident events

analytical formula

Monte Carlo

Figure1. Thesolidline istheanalyti alformula(7)of thevarian eandthedashed lineistheMonteCarlovarian easafun tionofthenumberoftimesslides. TheMonte Carlohasbeenperformedwith

FA

1

= 0.7 Hz, FA

2

= 0.8 Hz, τ

c

= 1 ms,

500trialsand a

T = 10

4

_s

datastreamlength.

We an then ex hange integrals and sums. To ompute the integrals we distinguish

two ases: when

k 6= l

the integrals on

x

k

and

x

l

are independent, and the integration over

x

1

, . . . , x

N

givesa

p

2

ontribution;otherwise theintegrationgivesa

p

ontribution. Analogously for the

y

variables we obtain

q

or

q

2

depending on whether

π

1

(k) = π

2

(l)

ornot.

The omputation of this integral, detailedinAppendix A, yields

Var = Npq



1

R

+ p + q +

pq − (p + q)

R

− 2pq



(6)

≃ Npq



1

R

+ p + q



,

(7)

where the lastlineis anapproximation inthe limit

p, q,

1

R

≪ 1

, whi h is reasonableas far asGW analysis is on erned.

3.3. Interpretation

Ea h term in equation (7) an be interpreted. The

1

R

is what we would expe t if we onsidered

R

independentPoisson pro ess realizationsinsteadof

R

timeslides. The

of the event probability

p

from a single realization of a Poisson pro ess with a mean numberofevents

Np

is

p = p+δp

b

,where

δp

istherandomstatisti alerrorwithvarian e

hδp

2

_{i =}

p

N

. This yieldsthe mean rate of oin iden es

Mean

= N b

pb

q = N (p + δp) (q + δq) ≃ Npq + Npδq + Nqδp

(8) whi h orresponds to a varian e of

hN

2

_p

2

_δq

2

_{+ Nq}

2

_δp

2

_{i = Npq(p + q)}

, be ause

δp

and

δq

are independent errors. Thus, whenusing onlyonerealization forthesingle dete tor triggers, we have astatisti al error on the single dete tor pro ess rate. This statisti al

erroris systemati allypropagatedtothe oin iden erateof ea htimeslide, that yields

theextratermsinthevarian eas omparedtoindependentpro essrealizations. One an

seethatthisextratermisimportantwhen

1

R

< max(p, q)

;for aseswherethe oin ident false alarm rate is maintainedxed (

pq

onstant), the ee t is most noti eable when

p

and

q

are very dierent.

This gives anestimateofthe varian e ofthe numberof oin ident events inadata

stream of length

T

. After onverting to the estimation of the oin iden e false alarm rate we obtain

Mean

FA

c

=

Mean

T

= FA

1

FA

2

∆t,

(9)

Var

FA

c

=

Var

T

2

≃ FA

1

FA

2

∆t

T



1

R

+ FA

1

∆t + FA

2

∆t



.

(10)

Toverifythese results,aMonteCarlosimulationhas been performed. The Poisson

pro essesare reatedasdes ribedinse tion2,usingasamplingrateof

16384 Hz

,thena simple oin iden e test witha windowof

τ

c

= 1 ms

isapplied. The time shiftsare done byaddinganintegernumberofse ondstoalleventsandapplyingamodulo

T

operation. The formulahas been testedusing 500 realizations of

T = 10

4

_s

long Poisson pro esses,

and using between 1 and

10

4

time slides for ea h realization. Figure 1 shows that the

analyti alformula(7)andtheMonteCarloagreewellforanynumberoftimeslides,and

that the varian e starts saturating when a few hundred time slides are used. We an

see thattheidenti ationof thesamplingtimeandthe oin iden etimewindowhas no

onsequen e on the result, the hoi e between binning and windowing the oin iden es

is ahigher order ee t.

3.4. Straightforward extensions of the model

In real data analysis, there are times when one of the dete tors does not take s ien e

quality data for te hni al reasons. Thus, the data set is divided into disjointsegments,

and the ba kground estimation is often done by ir ular time slides on ea h segment

separately. Afterwards the results from all the segments are ombined to get the

ba kground false alarm estimation. The omputation dis ussed above extends to this

ase with minimal hanges. The ir ular permutations have to be hanged to ir ular

Another aveat is that for real data analysis the oin iden e pro edure is often

more ompli ated. Some of those ompli ations are event onsisten y tests, e.g. do the

two oin ident events have a similar frequen y? We an model this by adding some

parameter

f

distributed uniformlyin

[0, 1]

atta hed toea hevent, and thenrequesting a oin iden ein the parameter

f

.

For this model the results willbe the same as those above, up to a fa tor of order

1. Indeed, instead of applying a window of size

∆t

to our events, we are now working in a2 dimensional(for instan e time-frequen y) spa e and using are tangular window

in this 2D parameter spa e. The pro edure in both ases is the same  applying D

dimensional re tangular windows to events distributed uniformly in a D dimensional

spa e up tothe dimension of the spa e.

4. The ase of three dete tors

4.1. Time slides between three dete tors

In the ase of three dete tors one natural extension is to ask for events that are seen

by at least two dete tors, whi h means look for oin iden e between two dete tors for

ea h dete tor pair, but ounting the oin iden es between three dete tors only on e.

This OR strategy in a interferometer network has been shown to be more e ient

than a dire t three fold oin iden e strategy (AND strategy) [11℄. For time slides,

when shifting the events of the se ond dete tor with some permutation

π

,we alsoshift the events of the third dete tor by the same amount but inthe opposite dire tionwith

π

−1

. To write ompa t equations we abbreviate

X =

1

(x

k

< p)

,

Y =

1

(y

π(k)

< q)

,

Z =

1

(z

π

−1

(k)

< r)

, d

V =

d

x

1

· · ·

d

x

N

d

y

1

· · ·

d

y

N

d

z

1

· · ·

d

z

N

, where

r = FA

3

∆t

is the eventprobabilityperbinofthethirddete torandtheve tor

z

des ribesitsrealizations. Thus, the meannumberof oin iden es inthe framework des ribed inse tion 3.1is

Mean =

Z

· · ·

Z

₁

R

X

π

X

k

[XY + Y Z + XZ − 2XY Z]

d

V = N [pq + pr + qr − 2pqr] .

(11)

4.2. Computation of the varian e

The se ond moment an be written ompa tlyas

M

2

=

Z

· · ·

Z

₁

R

2

X

π

1

X

π

2

X

k

X

l

[XY X

′

Y

′

+ XZX

′

Z

′

+ Y ZY

′

Z

′

+ 4XY ZX

′

Y

′

Z

′

+ 2XY X

′

Z

′

+ 2XY Y

′

Z

′

+2XZY

′

_Z

′

_{− 4XY X}

′

_Y

′

_Z

′

_{− 4XZX}

′

_Y

′

_Z

′

_{− 4Y ZX}

′

_Y

′

_Z

′

_]

d

V,

(12)

where the

′

10

0

10

1

10

2

10

3

10

−1

10

0

10

1

10

2

number of time slides

variance of the number of coincident events

analytical formula

Monte Carlo

Figure2. Thesolidlineistheanalyti alformula(14)ofthevarian eandthedotted lineistheMonteCarlovarian easafun tionofthenumberoftimeslides. TheMonte Carlohas beenperformed with

FA

1

= 0.04 Hz, FA

2

= 0.08 Hz, FA

3

= 0.16 Hz, τ

_c

=

31 ms,

500trialsanda

T = 10

4

_s

datastream length.

The omputation of this integral, detailedinAppendix B, yields

M

2

=

N

R

n

(pq + pr + qr − 2pqr)

+(R − 1) [pq(p + q + pq) + pr(p + r + pr) + qr(q + r + qr) + 6pqr − 4pqr(p + q + r)]

+ [(R − 1)(N − 3) + (N − 1)] (pq + pr + qr − 2pqr)

2

o

,

(13) and an beapproximatedin the limit

p, q, r,

1

R

≪ 1

by

Var ≃ N(pq + pr + qr)



1

R

+ p + q + r +

3pqr

pq + pr + qr



.

(14) 4.3. Interpretation

Similarly to se tion 3.3the extra terms in equation (14) an be explained through the

errorin theestimation ofthe single dete tor event rate. Usingthe same notationsasin

se tion 3.3 the mean oin iden e numberis

Usingtheindependen e ofestimationerrors andre allingthat

hδp

2

_{i =}

p

N

we obtainthe varian e of this mean value

Var

= N

2



_hδp

2

_{i(q + r)}

2

_{+ hδq}

2

_{i(p + r)}

2

_{+ hδr}

2

_{i(p + q)}

2



= N [(pq + pr + qr)(p + q + r) + 3pqr] ,

(16)

that orresponds to the extra terms in equation(14).

After onverting to the estimation of the false alarm rate weobtain

Mean

_FA

c

≃ (FA

1

FA

2

+ FA

1

FA

3

+ FA

2

FA

3

)∆t,

(17)

Var

FA

c

≃ (FA

1

FA

2

+ FA

1

FA

3

+ FA

2

FA

3

)

∆t

T



1

R

+ FA

1

∆t + FA

2

∆t + FA

3

∆t +

3FA

1

FA

2

FA

3

FA

1

FA

2

+ FA

1

FA

3

+ FA

2

FA

3

∆t



.

(18)

To he k the 3 dete tor results we performed a Monte Carlo similar tothe one of

the2dete tor ase(seese tion3.3). Theonlydieren eisthenumberofdete tors, and

we hoose a dierent oin iden e window:

τ

c

= 31 ms §

. To he k that the assumption of equal and opposite time slides does not inuen e the result, inthe Monte Carlo the

data in the se ond dete tor are shifted by

T

k

and in the third dete tor by

3T

k

. Figure 2 shows that the Monte Carlo and the 3 dete tor OR formula(14) agree reallywell.

4.4. The ase of

D

dete tors

Forthe sake of ompletenesswe an generalize the interpretationdone inse tion 3.3to

the ase of

D

dete torsinthe AND onguration. This generalizationofequation (8) to

D

dete tors yieldsa varian eon the number of oin iden es

Var ≃ N

D

Y

i=1

p

i

1

R

+

D

X

i=1

Y

j6=i

p

j

!

,

(19)

where

p

i

is the probability for dete tor

i

tohave anevent ina given time bin.

The interpretation an also be generalized in the OR ase, that is oin iden e

betweenanypairofdete tors,althoughthe omputationismore umbersomeasdetailed

in Appendix C and yields

Var ≃ N

"

X

i<j

p

i

p

j

!

1

R

+

D

X

i=1

p

i

!

+

1

2

X

i6=j, j6=k, k6=i

p

i

p

j

p

k

#

,

(20)

where

p

i

is the event probabilityperbin in the i th

dete tor.

5. Dis ussion

We nally dis uss the onsequen es of the above results on GW data analysis. To be

able to put numbers intothe equationswe willlook at a du ialGW data taking run.

We hoose the run properties tobe:

•

a durationof

T = 10

7

_s

, that isroughly 4 months

•

twodete torswithalighttraveltimeseparationof

25 ms

,andweusethesametime asthe oin iden ewindow, sothat

∆t = 50 msk

,assumingperfe t timinga ura y of trigger generators.

•

a desired oin iden e false alarmrate of

10

−8

_Hz

, i.e. one event every three years.

Wewilllookattwospe ial asesofsingledete torthreshold hoi e. Onesymmetri

ase, wherethresholds areset so thatthe single dete tortrigger rateinea h dete toris

roughlythe same. Oneasymmetri ase, whereinoneof the dete torsthere isonlyone

trigger. This asymmetri aseisextreme butinstru tive,be ausetuningthe thresholds

to obtain the best sensitivity often yields asymmetri trigger rates between dierent

dete tors.

Symmetri dete tor ase Inthis asewehavethe singledete tortriggerrate

FA

1

=

FA

2

= FA

s

=

q

FA

∆t

≃ 4.5 × 10

−4

Hz

, whi h gives using equation(10) the fra tional error of the false alarm estimation

σ

FA

FA

≃ 3.2



1

R

+ 4.5 × 10

−5



1

2

p = q = 2.25 × 10

−5

≃ 0.32

for

R = 100

≃ 0.02

for

R → ∞.

So for 100 time slides we get a typi al error of 30% in the false alarm estimation,

and the errorsaturates at2% for

R & 20000

.

Asymmetri dete tor ase In this extreme ase the singledete tor triggerrates are

FA

1

=

_T

1

= 10

−7

Hz

and

FA

2

=

FA

FA

1

∆t

= 2 Hz

, whi h gives using equation (10) the

fra tional error of the falsealarm estimation

σ

FA

FA

≃ 3.2



1

R

+ 0.1



1

2

p = 5 × 10

−9

, q = 0.1

≃ 1.05

for

R = 100

≃ 1

for

R → ∞.

So the error saturates at100%, and this saturation is a hieved for

R & 10

.

Those two examples show that the maximal number of useful time slides and the

false alarm estimation pre isionstrongly depends on the relative properties of the two

k

Asnotedin se tion2.4, oin identtriggersaredenedaslessthat

±

1

dete tors. In parti ular when there are mu hmore triggers inone dete tor than in the

other,the ba kground anbebadly estimated andin reasing the number oftime slides

does not solve the issue.

6. Con lusions

Wehave studied the statisti alerror in the ba kground estimation of event-based GW

data analysis when using the time slide method. Under the assumption of stationary

noise we analyti ally omputed this error in both the two-dete tor and three-dete tor

ase, and found ex ellent agreement with Monte Carlosimulations.

The important resulting onsequen es are: the pre ision on the ba kground

estimation saturates as a fun tion of the numberof time slides, this saturation is most

relevantfordete torswithaverydierenttriggerrate wheretheba kgroundestimation

pre ision an bepoorfor any numberof time slides.

Let us note that the time slide method an be used in other situations than

GW data ba kground estimation. For example it an be used to estimate the rate

of a idental oin iden es between a GW hannel and an environmental hannel in a

GW interferometer; or in any experiment where oin iden es between two (or more)

trigger generators are looked for. The results of this paper an be straightforwardly

extended tosu h anexperiment.

Another limitation, the non stationarity of the data, has not been investigated in

this paper. Data non stationarity is a well known issue in GW data analysis [12℄. In

the ontext of the time slides method it raises the question whether the time shifted

data are still representative of the zero lag data, when large time shifts are used. It

involves both the problemof the measure of the level of data non stationarity,and the

estimation of the error it indu es on the ba kground estimation. Further work on this

issue willbethe subje tof a future paper.

Appendix A. Two-dete tor integral

To ompute the integral

M

2

=

Z

x

1

· · ·

Z

x

N

Z

y

1

· · ·

Z

y

N

"

1

R

X

π

X

k

1

(x

k

< p)

1

(y

π(k)

< q)

#

2

=

Z

· · ·

Z

1

R

2

X

π

1

X

π

2

X

k

X

l

1

(x

k

< p)

1

(x

l

< p)

1

(y

π

1

(k)

< q)

1

(y

π

2

(l)

< q),

(A.1)

we put the sums outside the integrals. When

k 6= l

, the integrals on

x

k

and

x

l

are independent, and the integration over

x

1

, . . . , x

N

gives a

p

2

ontribution. Otherwise

the integration gives a

p

ontribution. Analogously for the

y

variables we get

q

2

Thus weget fourtypes of integrals

integral

×

numberof su h integrals

k = l,π

₂

−1

◦ π

1

(k) = l

1

R

2

pq × NR

(A.2a)

k 6= l,π

₂

−1

◦ π

1

(k) = l

1

R

2

p

2

_{q × NR(R − 1)}

(A.2b)

k = l,π

₂

−1

◦ π

1

(k) 6= l

1

R

2

pq

2

_{× NR(R − 1)}

(A.2 )

k 6= l,π

₂

−1

◦ π

1

(k) 6= l

1

R

2

p

2

_q

2

_{× N [R(R − 1)(N − 2) + R(N − 1)]}

(A.2d)

Hereweusedthatthe ompositionoftwo ir ularpermutationisa ir ularpermutation,

and that the only ir ularpermutation with axed point isthe identity.

The details of the ombinatori s are as follows.

• k = l, π

₂

−1

◦ π

1

(k) = l

: There are

N

dierent

k

values. For ea h of them there is only one

l

that isequal to it. Here

π

−1

2

◦ π

1

is a ir ularpermutationwith a xed point, so it is the identity. There are

R

dierent

π

1

, and for ea h of them only

π

2

= π

1

gives the identity.

• k 6= l, π

₂

−1

◦ π

1

(k) = l

: There are

N

dierent

k

values. For every pair

π

1

6= π

2

we get

π

−1

1

◦ π

2

(k) 6= k

. And the hoi e of this pair determines uniquely an

l

that is not equal to

k

. Thereare

R(R − 1)

su hpairs.

• k = l, π

₂

−1

◦ π

1

(k) 6= l

: There are

N

dierent

k

values. The value of

l

is determined by the equality

k = l

. And there are

R(R − 1)

pairs of

π

1

, π

2

su h that

π

−1

1

◦ π

2

(k) 6= k

.

• k 6= l, π

₂

−1

◦ π

1

(k) 6= l

: There are

N

dierent

k

values. In the ase where

π

1

6= π

2

, we need that

l 6= k

and

l 6= π

−1

2

◦ π

1

(k)

,there are

N − 2

su h

l

. In the ase where

π

1

= π

2

we get

k = π

−1

2

◦ π

1

(k)

,so there is only one inequality on

l

, and there are

N − 1

possible

l

.

By summingthe 4 terms aboveand subtra ting

Mean

Appendix B. Three-dete tor integral

Wewant to ompute the integral

M

2

=

Z

· · ·

Z

₁

R

2

X

π

1

X

π

2

X

k

X

l

[XY X

′

_Y

′

_{+ XZX}

′

_Z

′

_{+ Y ZY}

′

_Z

′

_{+ 4XY ZX}

′

_Y

′

_Z

′

_{+ 2XY X}

′

_Z

′

_{+ 2XY Y}

′

_Z

′

+2XZY

′

Z

′

− 4XY X

′

Y

′

Z

′

− 4XZX

′

Y

′

Z

′

− 4Y ZX

′

Y

′

Z

′

] ,

(B.1)

where the

′

denotes whether the hidden variablesare

π

1

, k

or

π

2

, l

. Similarly toAppendix A wehave here eightkind of integrals.

X

Y

Z

number of su h integrals

k = l, π

−1

₂

◦ π

1

(k) = l, π

2

◦ π

−1

1

(k) = l,

NR

k = l, π

−1

₂

◦ π

1

(k) = l, π

2

◦ π

−1

1

(k) 6= l,

0

k = l, π

−1

₂

◦ π

1

(k) 6= l, π

2

◦ π

−1

1

(k) = l,

0

k = l, π

−1

₂

◦ π

1

(k) 6= l, π

2

◦ π

−1

1

(k) 6= l,

NR(R − 1)

k 6= l, π

−1

₂

◦ π

1

(k) = l, π

2

◦ π

−1

1

(k) = l,

0

k 6= l, π

−1

₂

◦ π

1

(k) = l, π

2

◦ π

−1

1

(k) 6= l,

NR(R − 1)

k 6= l, π

−1

₂

◦ π

1

(k) 6= l, π

2

◦ π

−1

1

(k) = l,

NR(R − 1)

k 6= l, π

−1

2

◦ π

1

(k) 6= l, π

2

◦ π

−1

1

(k) 6= l, NR [(R − 1)(N − 3) + (N − 1)]

In these ombinatori omputationswe needtoassume thatalltranslations aresmaller

than

N/4

, to ensure that

π

−1

2

◦ π

1

◦ π

2

−1

◦ π

1

(k) = k ⇒ π

1

= π

2

. This assumption is reallyreasonable, and the result would not be signi antly dierentwithout it.

The nal result is

M

2

=

N

R

n

(pq + pr + qr − 2pqr)

+(R − 1) [pq(p + q + pq) + pr(p + r + pr) + qr(q + r + qr) + 6pqr − 4pqr(p + q + r)]

+ [(R − 1)(N − 3) + (N − 1)] (pq + pr + qr − 2pqr)

2

o

_,

(B.2)

Appendix C. OR ase for

D

dete tors

Usingthe same heuristi as inse tion 3.3and 4.3we ompute the varian e of the time

slide estimation method for

D

dete tors in the OR ase. This heuristi yielded the sameresultsastheexa t omputationforthe2and3dete tor ase,thuswemayexpe t

it tostay true in the general ase.

As in equation(14), the varian eis the sum of the normal Poisson varian e

The estimateof the meanrate is

Mean = N

"

_D

X

j=1

j−1

X

i=1

(p

i

+ δp

i

) (p

j

+ δp

j

)

#

(C.2)

≃ N







D

X

j=1

j−1

X

i=1

p

i

p

j

+

D

X

j=1

δp

j







D

X

i=1

i6=j

p

i











 ,

(C.3)

whi hleads to avarian edue to multiplereuse of the data (assuming

hδp

2

i

i =

p

i

N

)

Var

Slides

/N =

D

X

j=1

p

j







D

X

i=1

i6=j

p

i













D

X

k=1

k6=j

p

k







(C.4)

=







D

X

j=1

D

X

i=1

i6=j

p

i

p

j







D

X

k=1

p

k

!

−

D

X

j=1

p

2

_j

D

X

i=1

i6=j

p

i

(C.5)

=

D

X

j=1

j−1

X

i=1

p

i

p

j

!

_D

X

k=1

p

k

!

+

1

2

D

X

j=1

D

X

i=1

i6=j

p

i

p

j





p

j

+ p

i

+

D

X

k=1

k6=i, k6=j

p

k







−

D

X

j=1

D

X

i=1

i6=j

p

i

p

2

j

(C.6)

=

D

X

j=1

j−1

X

i=1

p

i

p

j

!

_D

X

k=1

p

k

!

+

1

2

D

X

j=1

D

X

i=1

i6=j

D

X

k=1

k6=i, k6=j

p

i

p

j

p

k

.

(C.7)

This generalformula(C.7) is orre tlygivingba k the extra termsinequations(7)and

(14) for respe tivelythe 2and 3dete tor ase.

Referen es

[1℄ B. P. Abbott et al. LIGO: the laser interferometer gravitational-wave observatory. Rep. Prog. Phys.,72(7):076901,2009.

[2℄ F.A erneseetal. StatusofVirgo. Class.QuantumGrav.,25(11):114045,2008.

[3℄ B.P.Abbottetal. Sear hforgravitational-waveburstsinLIGOdatafromthefourths ien erun. Class.QuantumGrav.,24(22):5343,2007.

[4℄ B.P.Abbott etal. Sear hforgravitational-waveburstsin therstyearofthefthLIGOs ien e run. a eptedin Phys. Rev.D, arXiv/0905.0020

[5℄ B. P. Abbott et al. Sear h for High Frequen yGravitational Wave Bursts in the First Calendar YearofLIGO'sFifth S ien eRun. a eptedin Phys. Rev.D, arXiv/0904.4910

[6℄ B.P.Abbott et al. Sear hforgravitationalwavesfrombinaryinspirals inS3and S4LIGOdata. Phys. Rev. D,77(6):062002,2008

[8℄ B. P. Abbott et al. Sear h for gravitationalwavesfrom low mass ompa t binary oales en e in 186daysofLIGO'sfths ien erun Phys. Rev. D,80(4):047101,2009.

[9℄ F.A erneseet al. Gravitationalwaveburst sear hin theVirgo C7data. Class. QuantumGrav., 26(8):085009,2009.

[10℄ B. Allen.

χ

2

time-frequen y dis riminator for gravitational wave dete tion. Phys. Rev. D, 71(6):062001,2005.

[11℄ F.Beauvilleetal. A omparisonofmethodsforgravitationalwaveburstsear hesfromLIGOand Virgo. Class. QuantumGrav.,25:045002,2008.