Angular patterns of interferometric detectors and resonant bars for tensorial and scalar gravitational waves

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Angular patterns of interferometric detectors and

resonant bars for tensorial and scalar gravitational waves

Nicolas Arnaud, Monica Varvella

To cite this version:

O tober 2004

Angular patterns of interferometri dete tors and resonant bars for tensorial and s alar gravitational waves

Ni olas Arnaud, Moni a Varvella

Laboratoire de l'A élérateur linéaire

CNRS-IN2P3etUniversité deParis-Sud, Bât. 200,B.P.34,91898 OrsayCedex(Fran e)

Abstra t

Amongthemain hallengesofgravitationalwave(GW)dataanalysis,thenon-uniformity of urrent GW dete tor responses is a major on ern. Indeed, it strongly limits the e ien yofanynetworkofGWantennas,eventhoughsu h ongurationsaremandatory to separate trueGW signals from noiseu tuations. This arti le just aims at giving a omplete analyti al des riptionof GW dete tor antennapatterns. Most of its ontents anbefoundelsewhereintheliterature,butwethoughtusefulto olle tthemin asingle pla ewithadditionaldetails,inordertoprovideanunderstandingas ompleteaspossible of this essentialbasi feature of anyGW networkdata analysis. The twomain typesof GWdete torsinterferometersandresonantbarsare onsideredinthisarti le,aswell as both tensorial and s alarGW, the former predi tedby the General Relativity while thelatero ursin alternativetheoriesofgravity.

1 Introdu tion

Analyzing properly the data provided by GW dete tors is as di ult and omplex as the experimental work aiming at operatingthese instruments at their best sensitivities withthe highest duty y le. GW signals o ur at random times with waveforms a priori unknown,

whi h makes ompulsory the useof several ltering te hniques inparallel to try not to miss anyof theserare events.

Inaddition to these omputing hallenges (algorithm design, management of large omputer

farms,sele tion ofpotentiallyinteresting events...), anotherproblemmakestheanalysiseven more omplex: thespatialresponseof urrentGWdete torsisnotuniform. Indeed,itdepends ontherelative positionof theantenna withrespe tto thesour e lo ation [1℄. Therefore,the

GWamplitudesasso iatedtothesamesignal anbeverydierentindistantdete tors,whi h makes more di ult  and never 100% e ient  any network analysis using outputs from several antenna. Yet,su hmethods are ompulsory to validatea real event witha satisfying

onden e level seee.g. [2, 3,4 , 5, 6,7, 8, 9℄ as a single dete tor annot easily separate realGWfromrandom noise u tuations.

This arti le aims at summarizing the main features of the antenna patterns of urrent GW

dete tors: interferometers and resonant masses. In both ases, two types of GW are on-sidered: GWgenerated in theGeneral Relativity framework (tensorial GW) and s alarGW predi ted by alternative theories of gravity  see [10℄ and referen es therein for a review of

Todes ribetheee tofaGWonthespatialmetri ,letusstartwithits'properframe'dened in Figure1 for details. TheGWpropagates along thethird axis ofthe frame, whilethe two

rst are hosenaligned withtheindu ed perturbations,lo ated inthetransverse plane.

n

e

ψ

x

propa

e

₊

Figure 1: GW lo al frame in whi h the GW tensor an be simply written. The angle is an additional degree of freedom, taking into a ount rotations of the frame around the

sour e-Earth line.

In General Relativity, a GW is des ribed in the Transverse-Tra eless gauge by two time-dependent polarizations: h

+ et h



. In its proper frame, the tensor measuring the spatial

perturbation indu edbytheGWissimply:

H GR = 0  h + h  0 h  h + 0 0 0 0 1 A (1)

AddingaGWs alar omponentbto thetwo tensorialonessimplyrequiresto hange thetwo non-zero diagonal termsofthe tensor: h

+ !(h + +b)and h + !( h + +b). Onegets: H s alar = 0  h + +b h  0 h  h + +b 0 0 0 0 1 A (2)

Now, to ompute antennapatterns, one needs to move from theGWproperframe to Earth-based frames asso iated to GW dete tors. A rst degree of freedom is visible in Figure 1:

let~n propa

 thepolarizationangle around~n propa

.

The position ofthesour e inthe skyismonitored bythree angles: two re ordingitslo ation

in a xedframe, plus a phase taking into a ount the Earth proper rotational motion. The most onvenient hoi eistousethe 'equatorialframe'asitsthirddire tionisalignedwiththe Earth rotation axis; inthis frame, a sour e is labeled byits right as ension  (a 'longitude')

and its de lination Æ (a 'latitude'), both shownin Figure2.

z

δ

Point

Vernal

α

Source

x

Figure 2: Denition of the equatorial frame: the third dire tion of the frame 'z' is aligned

withtheEarthrotational axiswhile therst one, 'x',pointstoward thevernal point.

Dete tor positions on Earth are labeled by their latitudes l and their longitudes L (positive west-wardsby onvention). Fordete torshavingprivilegeddire tionslikeinterferometers and resonant masses,additional angles arerequired  seebelowand Figure3 forthedenition of

these lo al angles.

 For an interferometer, two angles are mandatory: the angle between thetwo arms  it will be shown that =90

Æ

is optimal  and a angle ,monitoring the lo al orienta-tion of the antenna on Earth. By onvention, is hosen to be the angle between the

lo al South-Northdire tion andthe interferometer armbise ting line, ounted ounter- lo kwise. Table1 summarizes these informations for therst generation of large-s ale interferometers.

 For resonant bars, one angle is su ient to des ribe the bar lo al orientation. In this ase, isdened astheanglebetween the South-Northlo aldire tionandthebar axis, ounted ounter lo kwiseagain. Table 2 ontains geographi al informations needed to

des ribethe resonant bars urrently operated.

Finally,thelo al hourangleH(t) ismandatoryto take into a ount thedete tormotionwith respe tto the elestialsphere dueto Earthproperrotation. For asour e ofrightas ension 

Greenwi h

withT

Greenwi h

(0)beingtheGreenwi hsiderealtimeat0hUTand1:002737915 Æ

=hour , asa sidereal day lastsapproximately 23 hoursand 56minutes.

ρ

Local South

Local North

χ / 2

γ

Local North

Local South

Interferometer

Resonant bar

Figure3: Des riptionof GWdete torlo al orientations.

Dete tor Latitude l ( Æ ) Longitude L ( Æ )  ( Æ ) ( Æ )

ACIGA[11 ℄ -31.4 -115.7 90.0 Not de ided

GEO600[12 ℄ 52.3 -9.8 94.3 158.8

LIGO Hanford[13 ℄ 46.5 119.4 90.0 261.8

LIGO Livingston[13 ℄ 30.6 90.8 90.0 333.0

TAMA300[14℄ 35.7 -139.5 90.0 315.0

Virgo [15 ℄ 43.6 -10.5 90.0 206.5

Table1: Interferometer lo ations onEarth

Dete tor Latitude l ( Æ ) Longitude L ( Æ ) ( Æ ) ALLEGRO[16 ℄ 30.5 -268.8 40.0 AURIGA[17℄ 45.4 -12.0 136.0 EXPLORER [18 ℄ 46.5 -6.2 141.0 NAUTILUS [19℄ 41.8 -12.7 136.0 NIOBE [20 ℄ -31.9 -115.8 0.0

A GWdete toris sensitive to alinear ombination h(t) ofthedierent GWpolarizations: h(t) = F + (t)h + (t) + F  (t)h  (t) | {z }

GeneralRelativitytensorialGW + F b (t)b(t) | {z } S alarGW (4)

The weightingfa torsF +

,F 

andF b

are alledantennapatternfun tions. To ompute them

analyti ally, one needs to transportthe GWperturbation tensor from theGW frame to the dete torlo alframe. Using theangles introdu ed intheprevious se tion,this transformation anbesplitintothreerotations: GWproperframe! elestialsphereframe!frame entered

on the dete tor ! dete tor lo al frame. If P is the full transformation matrix, the GW perturbation matrix M expressedinthedete torlo alframe is equalto:

M =

t

P  H  P (5)

The antenna pattern fun tions are nally omputed from theM matrix oe ients, using a formulawhi h depends onthe type ofGWdete tor onsidered.

3.1 Tensorial GW

Let~n 1

and~n 2

beunitve torsalongtheinterferometer arms;theintera tion between theGW andtheantenna an be written [3℄:

h = 1 2 t ~ n 1
M
~n 1 t ~ n 2
M
~n 2
(6)

withM theGW perturbation tensor dened above in Eq. (5). After extensive al ulations, the nal form of the 'General Relativity' antenna pattern fun tions F

+ and F  is given by [3,8,21℄:  F + F   =sin  os2 sin2 sin2 os2  a b  (7) with a= 1 16

sin2 (3 os2l)(3 os2Æ) os2H 1

4

os2 sinl(3 os2Æ)sin2H

1

4

sin2 sin2l sin2Æ osH 1

2

os2 osl sin2Æ sinH

3 4 sin2 os 2 l os 2 Æ

b= os2 sinl sinÆ os2H + 1

4

sin2 (3 os2l) sinÆ sin2H

os2 osl osÆ osH + 1

2

are fa torized: the polarization angle simplyintrodu es a rotation of theantenna pattern, while theF oe ientsare learly maximalfor orthogonalarms. Theaand bfun tionshave

noparti ularphysi almeaning: theirdistributionsdierfromoneinterferometertotheother. On theother hand,the ombination

p a

2 +b

2

hasa unique distribution for all dete tors,see Se tion 4.

3.2 S alar GW

For a s alar GW, the antenna pattern fun tion is omputed by following exa tly the same

pro edure. Original al ulationsgive:

F b = sin 2 [K s sin2 + K os2 ℄ (8) with K s = sin 2 H os 2 Æ sin 2 l os 2 Æ os 2 H os 2 lsin 2 Æ + 1 2

sin2sin2Æ osH

K

= sinl os 2

Æ sin2H + osl sin2Æ sinH

Likefortensorial GW,sinisas alingfa torfortheinterferometerresponse, omingdire tly

from Eq. (6) and the lo al orientation angle dependen e is fa torized. But the most important feature hereis thatF

b

doesnot depend on thepolarization angle .

4 Interferometer antenna pattern interpretation

To understand the shapes of these antenna pattern fun tions, it is interesting to re-express them in aparti ular frame, shown inFigure4; itstwo rst axisare alongthe dete torarms, thus assumed to be perpendi ular, whi h is the ase for all interferometers apart GEO600

whi hexhibitsonlyasmalldeviationfromthisoptimalsituation seeTable1. These expres-sionsare simpleenough to explain themainfeaturesof theinterferometer patterns.

Using thespheri al angles introdu ed inFigure 4,one gets:

F + = 1 2 (1 + os 2

) os (2) os (2 ) ossin(2) sin(2 ) (9)

F  = 1 2 (1 + os 2

) os (2) sin(2 ) + os sin(2) os (2 ) (10)

F b = 1 2 sin 2  os (2) (11)

The expressions for F +

and F 

are well-known  for instan e, they an be found in Ref. [1℄  and present the same stru tures than Eq. (7). It is lear that F

+; and F b are very dierent. First, jF b

j 1=2, whi h means that at best 50% of thes alar GW amplitude an

be re overedinan interferometer. Thisfeaturestrongly limitsthepotential ofsu hantennas forthe dete tion of s alarsignals.

Ontheotherhand,F +

andF 

anrea h1;extremizingthemrequires os=1,i.e. aGW

perpendi ular tothe interferometer plane on theotherhand,for these dire tions, F b

e

_x

e

_y

e

_z

φ

θ

(polarisation )

ψ

Source direction

Interferometer

Arms

Figure4: Interferometerlo alframe: itstworstdire tionsarealignedwiththedete torarms (assumed to be perpendi ular) andspheri al oordinates(;) areintrodu ed.

One analso notethatF +

andF 

arezero infourdire tionsbelongingto theinterferometer

plane,denedby( os2=0): alongthearmbise tinglineandintheperpendi ulardire tion. AsF

+ andF



depend onthepolarization angle ,theyhavea zeromeaninanydire tion of the sky. Therefore, still following Ref. [8℄, one introdu es the -independent quantity F to

quantify the strengthofthe interferometer pattern ina given dire tion:

F = s F 2 + + F 2  2 (12)

Thesquaresensureanon- an ellation ofterms;whilethesquarerootredu es F toaquantity

homogeneoustoaGWamplitude themeaningfulquantityforwhat on ernsGWdete tion. Finally, thefa tor 1=

p

2 is simply a onvention 'averaging' thetwo squared terms, validated

belowwhen interpretations ofF will be given. FromEq. (7),itis learthat q F 2 + + F 2  ,the

normof theve tor(F +

;F 

), isa -independent quantity. Indeed,one has:

q F 2 + + F 2  = jsinj 2 p a 2 + b 2 (13)

Theprevious equation explains whythe distributionof thequantity p a 2 +b 2 hasa physi al

meaning. In the following, thetwo quantities F (ranging from 0 to 1= p

2 for perpendi ular arms)andjF

b

jwillbeusedto ompare tensorialands alarantennapatterns. Themeanvalue ofF isnothingbutthe ommon RMSoftheantennapatternsF

+ andF



,whi hhave 0-mean

A onvenient wayto visualizetheantennapattern ofGWdete tors istousetwo-dimensional densityplots,theskymaps[3 ,8 ℄. Anydire tionintheskyislo atedbya oupleof oordinates

(; os)or(;sinÆ)dependingonthe hosenframe 1

andtheantennapatternamplitudeis representedbya olor ode. Asarstexample,Figure5showsF andjF

b

jintheinterferometer

lo alframe (; os). All featuresmentioned intheprevious se tionare learly visible.

Figure 5: Antenna pattern omparison inthe interferometer lo al frame (; os): top plot, tensorial pattern; bottom plot, s alar pattern. F has four zeros in the interferometer plane

(in oming wave along the arm bise ting line) and is maximal for normal in iden e. On the other hand,jF

b

jismaximalfor GW oming alongone armand nullfor GWperpendi ular to the antenna.

In this parti ular frame, the antenna patterns look 'repetitive'. To see more 'attra tive'

pi tures, one an re ompute the same diagrams in a general frame, independent from any interferometer. This hange modiesthe pattern shapes, but not their hara teristi s. Therefore, now labeling the sour e position by the ouple (;sinÆ), theVirgo antenna

pat-1

terns  for tensorial and s alar GW  are shown in Figure 6. The two rosses show the dire tions perpendi ular to the interferometer plane,representedbythe urved dashedline.

Figure 6: GW antenna pattern (;sinÆ) for the Virgo dete tor. The two marks show the dire tions perpendi ular to the dete tor plane, represented by the dashed line. In this way, one an he kthatthetwodiagramspresenttheexpe tedfeatures,alreadyvisibleinFigure5.

To on ludeoninterferometerpatternskymaps,Figures7and8 omparetheantennapatterns

of thesixrst generationinterferometers, for tensorial and s alarGWrespe tively. For both plots, thelo alorientation ofACIGAhasbeen hoseninorderto optimizeits ontribution to afull-sizednetwork in ludingallinterferometers, assumedto have identi alsensitivities. This

orrespondsto ACIGA 0 Æ modulo 90 Æ

dueto the -dependen eof allantennapatterns. A last wayto ompare tensorial and s alarantenna patternsis to ompute their probability distributions, assuming auniform distribution of sour esinthesky.

Results areshown inFigure 9: both distributions have zeromean; theF b

plotsharply peaks

around zero while the F +;

distribution  identi al for F +

and F 

asexpe ted is approxi-mately at between -0.5 and0.5, beforede reasing ontheedges oftheplot.

2

To omputesu h antennapatterns, the origin of the angle  is hosenarbitrarily  indeed, at agiven sidereal time,Earth propermotion leads to ahorizon tal shift of the patternby a phaset +T

straight lines rossea h otherinthetwo dire tionsperpendi ular totheinterferometer plane  forwhi hthe antennaresponseisoptimalwhilethe urved dashedline showsthelo ation

of theinterferometer planeinwhi h arelo atedthefour blind dire tionsfor su h dete tors.

6 Resonant bar pattern

Thesamepro edure anbeappliedtoresonantbarsto omputetheirantennapatterns. Dueto thebaraxisymmetri alshape,thethreefun tionsF

+ ,F

 andF

b

arefoundtobeproportional to sin 2 ( pol ),where  pol

isthepolaranglewithrespe tto thebaraxis. Therefore,one ansee

immediately thatdire tionsperpendi ular tothebar arestronglyfavored, whiletheresonant massis blindalong itsaxis. Indeed,let us dene two angularfun tions:

A bar

= ossinl sinH + sin osH (14)

B bar

tions perpendi ular to the dete tor plane areblind, while optimal responses are found along the arms,thus shiftedfrom45

Æ

withrespe tto theblinddire tionsfor tensorial GW.

In term ofthese variables,thebar pattern fun tionsarethefollowing:

F + =  A 2 bar B 2 bar  os2 + 2A bar B bar sin2 (16) F  = 2A bar B bar os2  A 2 bar B 2 bar  sin2 (17) ) F = A 2 bar + B 2 bar p 2 (18) F b = A 2 bar + B 2 bar = p 2F = sin 2 ( pol ) (19)

Thepreviousequationsshowthatforresonant bars,theknowledgeofF indeedproportional to thesine squaredofthepolar angle issu ient to des ribeall antenna patterns,both for

GR GW pattern distribution

Scalar GW pattern distribution

Figure9: GeneralRelativityands alarGWpattern distributions,assuming auniform distri-bution of sour es in thesky:  and uniform in [ ;℄ and sinÆ uniform in [ 1;1℄. Both distributionshavezeromeanandtheirRMSequaltothemeanvaluesofF arerespe tively

1= p

5(tensorial GW)and1= p

15 (s alar GW).Verti al s alesarearbitrary.

independentfromthe polarizationangle . Yet,F b

isnot bounded by0:5 but anrea h1 in aseof optimal orientation of thesour e: at equal sensitivities,bars appear more suitable

thaninterferometers forthe dete tionof s alarGW.

Aspredi ted by Eq. (19), the distribution of F hasthe same shape as thedistribution of a sinesquaredwhose osineisuniformbetween -1and1seeFigure10. Itpeaksatitsmaximal

value whi h isrea hed ona great ir le inthesky.

Finally, Figure 11 ompares the antenna patterns of the ve bars urrently running. Con-trary to interferometers  f. Figures 7 or 8 , the bars are almost aligned to optimize the

performan es ofthenetwork theyform; therefore,theirantennapatternsareverysimilar. As expe tedfromthedependen eonsin

2 ( 

pol

)previouslymentioned,abar isoptimallysensitive to GWperpendi ulartoitsaxisandblindalongit. Thesepatternsarevalidbothfortensorial

Resonant bar GW pattern distribution

Figure 10: Distribution of F for a resonant bar, assuming a uniform distribution of sour es in the sky. As expe ted, it has the same shape as the distribution of a sine squared whose osine is drawn uniformly between -1and 1. Its mean value whi h is also theRMS of the

zero-mean fun tionsF + and F   isequal to p 2=3. 7 Con lusion

The antenna patterns of interferometers and resonant bars have been analyti ally omputed inthisarti le,bothfor tensorialands alarGW.Theyare learlynonuniform onthe elestial sphere and in addition they depend on the types of dete tor su h as on the GW radiation onsidered. Interferometer antenna patternsaremore ompli atedthan resonant bar shapes,

whi h exhibitan axisymmetry. Thelatterdete tors appearmoresuitable tosear hfor s alar GWwaves, provided thatthey anrea h thesame sensitivitiesthan interferometers andthat the GWemission ismostlylo atedin theresonant bar sensitivity frequen yrange.

Inthefuture, this situationmayevolvewiththeintrodu tion ofspheri al resonant dete tors. Indeed, thesehave various advantages withrespe tto resonantbars seee.g. [22 ℄ : in

par-ti ular, theirresponseisindependent fromthesour elo ationand fromtheGWpolarization! Inaddition,usingasingledete tor,one anmergeinformation omingfromthevequadrupole modesandfromthemonopolemodeofthespheretofullyre onstru ttheGWtensorandthe

resonant bars are almost parallel. One an also note that they have their better sensitivity for GWperpendi ulartotheir axisand thattheyareblindforGWparallel toit. Theseplots

are valid both for tensorial and s alarGW.

A knowledgment

TheauthorswouldliketothanktheOrsayVirgogroupfortheirsupportandfruitful omments.

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