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Using long-baseline interferometric gravitational waves detectors for high precision measures of the gravitational acceleration PDF

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Preview Using long-baseline interferometric gravitational waves detectors for high precision measures of the gravitational acceleration

Using long-baseline interferometri gravitational waves dete tors for high 7 pre ision measures of the gravitational 0 0 a eleration 2 n a Christian Corda J 5 February 7, 2008 2 3 v INFN - Sezione di Pisa and Università di Pisa, Via F. Buonarroti 2, I - 56127 6 PISA, Italy 4 0 1 1 E-mail address: hristian. ordaego-gw.it 6 0 / c q Abstra t - r g A derivation of the opti al axis lenght (cid:29)u tations due by tilts of the : mirrors of the Fabry-Perot avity of long-baseline interferometers for the v dete tion of gravitational waves in presen e of the gravitational (cid:28)eld of i X the earth is dis ussed. By omparing with the typi al tilt-indu ed noises it is shown that this potential signal, whi h is onsidered a weak sour e r a of noise, is negligible for the (cid:28)rst generation of gravitational waves inter- ferometers, but, in prin iple, this e(cid:27)e t ould be used for high pre ision measuresofthegravitationala elerationifadvan edproje tswilla hieve anhighsensitivity. Inthat asethepre isionofthemisure ouldbehigher thanthegravimeterrealizedbytheIstitutodiMetrologia(cid:16)GustavoColon- netti(cid:17). PACS numbers: 04.80.Nn, 04.80.-y, 04.25.Nx 1 Introdu tion Thedesignand onstru tionofanumberofsensitivedete torsforgravitational waves (GWs) is underway today. We have some laser interferometers like the VIRGO dete tor, being built in Cas ina, near Pisa by a joint Italian-Fren h 1 ollaboration,theGEO600dete tor,beingbuiltinHanover,Germanybyajoint Anglo-Germany ollaboration,thetwoLIGOdete tors,beingbuiltintheUnited States (one in Hanford, Washington and the other in Livingston,Louisiana)by ajointCalte h-Mit ollaboration,andtheTAMA300dete tor,beingbuiltnear Tokyo, Japan. We have many bar dete tors urrently in operation too, and several interferometers and bars that are in a phase of planning and proposal stages (for the urrent status of gravitational wavesexperiments see [1, 2℄). Theresultsofthesedete torswillhaveafundamentalimpa tonastrophysi s andgravitationphysi s. Therewillbelotsofexperimentaldatatobeanalyzed, andtheoristswillbefor edtointera twithlotsofexperimentsanddataanalysts to extra t the physi s from the data stream. Dete tors for GWs will also be important to verify that gravitational waves only hange distan es perpendi ular to their dire tion of propagation and to on(cid:28)rm or ruling out the physi al onsisten y of General Relativity or of any othertheoryofgravitation[3,4,5,6℄. Thisisbe auseinthe ontextofExtended TheoriesofGravityweknowthatsomedi(cid:27)eren esfrom GeneralRelativityand the others theories an be seen starting by the linearized theory of gravity [3, 4, 6℄. In this paper we dis uss on the possibility of an other use of long-baseline interferometri GWs dete tors, giving a derivation of the opti al axis lenght (cid:29)u tations due by tilts of the two mirros of the Fabry-Perot opti al avity in presen e of the gravitational (cid:28)eld of the earth. By omparing with the typi al tilt-indu ed noises it is shown that this potential signal, whi h is onsidered a weak sour e of noise [7, 8, 9℄, is negligible for the (cid:28)rst generation of GWs interferometers, but, in prin iple, this e(cid:27)e t ould be used for high pre ision measures of the gravitational a eleration if advan ed proje ts will a hieve an high sensitivity. In that ase the pre ision of the misure ould be higher than the one of the gravimeter of the Istituto di Metrologia (cid:16)Gustavo Colonnetti(cid:17) (IMGC) [10℄. 2 Using laser interferometers like gravimeters The oupling of angular tilts of the avity mirrorswith interferometer's imper- fe tions(likenonperfe t enteringoftheopti alaxisonthetwomirrorrotation enters, interferometers asymmetries and input laser geometry (cid:29)u tations) has been long studied like a parti ular noise sour e [11, 12, 13℄. In next Se tions the e(cid:27)e t of the oupling of mirror tilts with gravitational earth (cid:28)eld will be analyzed, (cid:28)rst like a sour e of noise for GWs interferometers, omparing with other greater noises asso iated to mirror tilts (that however do not limit the sensitivity of the interferometer) and showing that in the (cid:28)rst generation of in- terferometersthis noise is negligible, then the possibility of using this e(cid:27)e t for highpre isionmeasuresofthegravitationala elerationwithadvan edproje ts will be dis ussed. Thepossibilityofusinginterferometerslikehighpre isiongravimetersiswell knownin the s ienti(cid:28) ommunity. A tually falling orner ube interferometers 2 Y Photon Photon X Figure1: asimpleMi helsoninterferometer: photons anbe laun hedfrom the beam-splitter to be boun ed ba k by the mirror arebeingusedforabsolutemeasuresofthegravitationala eleration,beingthe best absolute instruments for this type of misure, be ause they are based on (cid:28)rst prin iple measures and require no alibration [10, 14℄. In parti ular the gravimeter reali1ze0d−8bmyst−h2e IMGC measures the gravitational a eleration with a sensitivity of [10℄. The instrument, based on symmetri motion, uses one mirror whi h is free falling in a va uum 1h0a−m9bmer. To dete t the (cid:29)ight path, a laser interferometer with a sensitivity of is used [10℄. In next Se tions of this paper it will be shown that advan ed interferometers for the dete tion of GWs ould in prin iple a hieve a better sensitivity for measures of the gravitational a eleration. 3 E(cid:27)e t of tilts in the frame of the lo al observer Be ause we are in a laboratory enviroment on earth, we typi ally use the oor- dinate system in whi h the spa e-time is lo ally (cid:29)at (see refs. [6, 15, 16, 17℄) and the distan e between any two points is given simply by the di(cid:27)eren e in their oordinates in the sense of Newtonian physi s. In this frame, alled the frame of the lo al observer, gravitational signals manifest themself by exerting tidal for es on the masses (the mirror and the beam-splitter in the ase of a simple Mi helson interferometer, see Figure 1, the input and re(cid:29)e ting mirrors for a Fabry-Perot avity, like the Virgo interferometer, see Figure 2). A detailed analysis of the frame of the lo al observer is given in ref. [15℄, se t. 13.6. Here we remember only the more important features of this frame: x0 the time oordinate is the proper time of the observerO; spatial axes are entered in O; in the spe ial ase of zero a eleration and zero rotation the spatial oor- x j dinates are the proper distan es along the axes and the frame of the lo al observerredu es to a lo al Lorentz frame: in this ase the line element reads 3 Ffffffffffffffffffff Figure 2: scheme of the Virgo interferometer 4 Figure 2: S heme of the Virgo interferometer C A O B Teta AB = R Figure 3: the imput and far mirrors of a Fabry - Perot avity, it is , OB =L BC =d OC =L+a , and ds2 =( dx0)2+δ dxidxj +O(dxj 2)dxαdxβ; ij − | | (1) the e(cid:27)e t of weak gravitational signals on test masses is des ribed by the equation for geodesi deviation in this frame x¨i = Ri xk, − 0k0 (2) Ri e where we have alled 0k0 the linearized Riemann tensor [15℄. L x Let us onsider a Feabry - Perot avity of lenght in the dire tion (i.e. x y we are assuming that the arms of our interferometer are in the and axes), R>L havingthe input mirror(cid:29)atand thefarmirror onvexofradius (likethe Virgo interferometer, see Figures 2, 3). We are interested in verti al tilts generated by the earth gravitational (cid:28)eld z (i.e. in the dire tion), thus, by putting the origin of our oordinatesystem in θ the input mirror (Figure 3) we all the tilts of the se ond mirror around the horizontal axis from the ideal position orresponding to the ase of avity axis lying on line joining the mirror enter of rotation. The geometri al e(cid:27)e ts on 2 the avityarea(cid:28)rst orderdispla ementof the opti alaxison the mirror that d l we all , and a se ond order variation of the lenght of the avity (ref. [11℄ and Figure 3) given by 5 d=(L+a)θ (3) and 2 l=L+dθ =L+(L+a)θ . (4) The more general interval in the frame of a lo al observer is [6, 15, 16, 17℄ 2 2 2 2 2 ds =g00dt +dx +dy +dz , (5) G=1 c=1 ~=1 (i.e. in this paper we work with , and ) where g00 = [1+2V(z)]. − (6) Equation(6)istheformulawhi hgivesthe orresponden ebetweenNewto- V(z) nian theory and General Relativity and is the Newtonian potential of the earth [6, 15, 16, 17℄. x Let us onsider the interval for photons propagating along the -axis in absen e of gravitational tilts [6, 15, 16℄ 2 2 2 ds =g00dt +dx . (7) y y We re all that the dire tion an be negle ted be ause the absen e of the dependen einthemetri (5)impliesthatphotonmomentuminthisdire tionis z onserved. We also emphasize that photon momentum in the dire tion is not z onserved, for the dependen e in eqs. (6) and (5) (see ref. [17℄ for details). x Thus photonslaun hed in the axiswill de(cid:29)e t outofthis axis. Butthis e(cid:27)e t z an be negle ted be ause the photon de(cid:29)e tion intdozt2he dire tion will be at most of (cid:28)rst order [6, 16℄. Then, to (cid:28)rst order, the term an be negle ted. ds=0 The ondition for a null traje tory ( ) gives the oordinate velo ity of the photons dx 2 2 v ( ) =1+2V(z =0), ≡ dt (8) whi h is a onvenient quantity for al ulations of the photon propagation time between the the two the mirrors[6, 16℄. To (cid:28)rst order the oordinate velo ity is approximated by v [1+V(z =0)], ≈± (9) + with and − for the forward and return trip respe tively. In absen e of xm1 =0 gravitationaltiltsthe oordinatesoftheinputmirror andofthese ond xm2 =T =L mirror donot hangeunderthein(cid:29)uen eofthegravitational(cid:28)eld of the earth, thus we an (cid:28)nd the duration of the forward trip of the light as L dx T01 = Z0 v (10) 6 T =L where isthetransittimeofthephotoninabsen eofanygravitational (cid:28)eld, i.e. in a Minkowskian spa etime. To (cid:28)rst order eq. (10) an be rewritten as T T01 T V(z =0)dx=T(1 V(z =0)). ≈ −Z0 − (11) In analogous way the propagation time in the return trip is 0 T02 =T V(z =0)( dx)=T(1 V(z =0))=T01. −Z − − (12) T T01 T02 Thesumof and giveustheround-triptimeforphotonstravelingbetween theboundaries. Thenweobtainthedeviationofthisround-triptime(distan e) 2T from its unperturbed value as δ(2T)=2T0 2T = 2TV(z =0)). − − (13) Nowletussupposethat,in presen eofgravitationaltilts, these ondmirror z 2 has a weak motion in the dire tion. In this ase the mirror rotates by an θ amount whi h orresponds to a parallel displa ement of the opti al axis by the quantity δ(z) (T +a)θ =d ≡ (14) (see (cid:28)gure 3). Thus, to (cid:28)rst order, we an write for the duration of the forward trip T ∂V(z) T∗1 T ( V(z)dx=T[1 (V(0)+ δ(z))]. ≈ − Z0 − ∂z (15) δT1 = T∗1 T0 In this way the time di(cid:27)eren e − between the tilted and un- perturbed opti al axes an be written as δT1 =Tg(T +a)θ, (16) where ∂V(z) g ≡− ∂z (17) is the gravitational a eleration. The duration of the return trip is omputed as 0 ∂V(z) T∗2 T ( V(z))( dx)=T[1 (V(0)+ δ(z)))]=T∗1. ≈ − Z − − ∂z (18) T δT2 =T∗2 T0 Thus,thetimedi(cid:27)eren e − betweenthetiltedandunperturbed opti al axes for the return trip is the same for the forward trip δT2 =Tg(T +a)θ (19) andwehaveforthetotalvariationoftheroundtrippropertimeofaphoton 7 δT =δT1+δT2 =2Tg(T +a)θ =2Tgd (20) Wehavetore allthatexperimentalists omputethee(cid:27)e tthatthisvariation oftheroundtrippropertimegeneratesinthevariationofthephaseofthelight of the laser in the avity [9, 14℄, thus δΦ=Ω0δT =2Ω0Tgd, (21) is the signal whi h an be in prin iple omputed with long-baseline inter- Ωfe0ro=me2tπrfi 0 gravift0at=ion10a1l5wHazves dete tors and due to gravitational tilts, where and in the Virgo interferometer. For the other arm of the interferometer in whi h the mirrors are lo ated y along the axis of our oordinate system the analysis is parallel to the one above. L = T = 3Km In the ase of the Virgo interferometer we have 10−18amnd, in prin iple, we an ompute d1i0s−ta9nra eds (times) with a sensitivity of and angles with a sensitivity of [7, 8℄. In next se tion, by omparing with the typi al tilt-indu ed noises, it will be shown that pre ision measures of the gravitational a eleration are1in0−p9rmins −ip2le possible with long baseline interfer- ometers with a sensitivity of that is an order of magnitude higher than the gravimeter realized by the IMGC. 4 Typi al error tilt noise in the Fabry-Perot av- ity of laser interferometers The e(cid:27)e t of oupling of mirror tilts with the gravitational earth (cid:28)eld is a (cid:28)rst order e(cid:27)e t, while, as it is well known [7, 8℄, the lenght variation of a Fabry- Perot axis when mirrors rotate is a se ond order e(cid:27)e t. For an high pre ision alignmentoftheaxis, thegravitational oupling anover ometheothere(cid:27)e ts, thusit ouldgeneratenoise,or,fromanotherpointofview,su hane(cid:27)e t ould be put in eviden e and measurethe gravitationala eleration. It isθalso10k−n9orwand that, in prin iple,asmallhigh-freque iestiltofangularamplitude ≃ an be dete t [7, 8, 9, 13℄. But it has to be emphasized that, at least for the (cid:28)rst generation interferometers, the alignment of the a1v0it−y9rwaidll not be perfe t, and the angular sensitivity will probabily grater than . This is be ausetheappli ationofeq. (4)togravitationalwavesinterferometersgivesan intuitivesimpli(cid:28) ation: interferometerslikeVirgopotentiallydete twaveswi h 10Hz f 10KHz a frequen y whi h falls in the frequen y-range ≤ ≤ , [6, 7, 8℄ whiletypi alspe trumofmirrorrotationsisdominedbystati orslowlyvaryng 1Hz omponentsatfrequen ieswellbelow . IntherangeofinterestfortheGWs dete tion the Fourier transform of eq. (4) is simply [12℄ ˜l(f) d˜ǫ(f), ≈ (22) d˜ where an be approximated with the root mean square of long - term d (cid:29)u tations of the arm lenght de(cid:28)ned by eq. (3). Equation (22) allows a 8 dire t omparision of the noise due to oupling of small, high frequen y mirror rotations with the gravitational (cid:28)eld or with the slowly varing mirror angle dp˜osit1i0o−n.5Indeeed the estimed long term (cid:29)u tation of the armθ is1o0f−t5hreaodrder of ≈ meters,whi h orrespondstoanangularamplitude ≃ −9 ,while θ 10 rad the gravitational oupling generates an angular amplitude ≃ . Thus the ouplingof mirrortilts with slowlyvaryngangleposition is the predominat e(cid:27)e t. In order to performe a (cid:28)rst measurement of the gravitational a eleration with long baseline interferometers with a sensitivity higher than the one of the gravimeterrθealiz1e0d−b9yratdhe IMGC, the sensitivity should be improved at a level better than ≃d˜ 10−9 whi h orresponds to a a misure of the (cid:16)gravitational g ouplingarm(cid:17) ≈ . This anbe made in prin ipe rememberingthat long baseline interferometers re y le the power by using a parti ular s heme that is shown in (cid:28)gure 2 (see also refs. [7, 8, 18℄). The lenght of the equivalent avity, whi h is onstituited by the re y ling mirror (PR in (cid:28)gure two) and the Mi helson interferometer, is monitored and maintened on risonan e by a suitablefeedba ksǫystem. Itispossibleaddingasmallhigd˜h-frequen ysinusoidal a c ˜lmodulatonoftilts whilealow-frequen ymisalignment ispresent,thelenght c of the re y ling avity will be a(cid:27)e ted by this geometri al oupling ˜l =d˜ǫ , c c a (23) while thee(cid:27)e t ofthe tilts due to the ouplingwithearth (cid:28)eld will be negli- giblebe ausetheshortlenghtofthe avityitself. Inthisway,atthemodulation ǫ a df˜requen y of , the signal will be useful for a good feedba k to redu e the arm c . Re olling1t0h−a1t8mth/e√sHenzsivity to re y ling avitylenght (cid:29)u tations is extimed of the o1rHdezr [7, 8, 18℄ and the bǫandwi1d0t−h8orafdthe feedba k an be a ˜labou1t0−10m, with a modulation amplitude of ≃ a redu tion up to c ≃ anbe possible. Then, usingthe(cid:28)rst generationparameters(even g if not in the1(cid:28)0r−s9t op1e0r−a1t0iomnsi−n2terferometers) in prin iple a measure of with a pre isionof − is possible, improvingthe one of the gravimeter realizedby the IMGC. Inadvan ed proje tsoflong-b1a0s−el1i9nme/in√teHrfzerometersfor thedete tionofGWsabetteǫrsen1s0it−iv7irtaydoftheorder isexpe ted, a thus,usingamodulationof ≃ (whi hdoesnotlowertheinterferom- g ete1r0p−e1r1form10a−n1 2ems)sw−i2ll be in prin iple possible a measure of with a pre ision of − . 5 Con lusion remarks Thepossibilityofanotheruseoflaserinterferometersforthedete tionofgravi- tationalwaveswithaderivationoftheopti alaxislenght(cid:29)u tationsduebytilts of the mirrors of the Fabry-Perot avity in presen e of the gravitational (cid:28)eld of the earth has been dis ussed. By omparing with the typi al tilt-indu ed noisesithasshownthatthis potentialsignal,whi his onsideredaweaksour e of noise, is negligibleforthe (cid:28)rst generationof gravitationalwavesinterferome- ters,but,inprin iple,thise(cid:27)e t ouldbeusedforhighpre isionmeasuresofthe 9 gravitational a eleration if advan ed proje ts will a hieve an high sensitivity. In that ase the pre ision of the misure ould be higher than the gravimeter realized by t1h0e−1I1stitu10to−1d2imMs−e2trologia (cid:16)Gustavo Colonnetti(cid:17) with a potential pre ision of − . A knowledgements Iwouldliketo thankSalvatoreCapozziello,MariaFeli iaDe Laurentisand Gi- an arlo Cella for helpful advi es during my work. I have to thank the referees for their interest in my work and for pre ious advi es and omments that al- lowed to improve this paper. The European Gravitational Observatory (EGO) onsortium has also to be thanked for the using of omputing fa ilities. Referen es [1℄ http://www.ligo.org/pdf_publi / amp.pdf [2℄ http://www.ligo.org/pdf_publi /hough02.pdf [3℄ CapozzielloS-Newtonian Limit of Extended Theories of Gravity inQuan- tum Gravity Resear h Trends Ed. A. Reimer, pp. 227-276 Nova S ien e Publishers In ., NY (2005) - also in arXiv:gr-q /0412088(2004) [4℄ Capozziello S and Troisi A - Phys. Rev. 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