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ICCUB-14-032 Pulsar Timing Arrays and the cosmological constant Domènec Espriu 4 1 0 Departamentd’EstructuraiConstituentsdelaMatèriaandInstitutdeCiènciesdelCosmos(ICCUB), 2 UniversitatdeBarcelona,MartíiFranquès,1,08028Barcelona,Spain. n a Abstract. InthistalkIreviewhowanon-zerocosmological constantL affectsthepropagationofgravitationalwavesand J theirdetection inpulsar timingarrays(PTA).IfL =0itturnsout thatwavesareanharmonic incosmological Friedmann- Robertson- Walker coordinates and although the am6 ount of anharmonicity isvery small it leads to potentially measurable 0 effects. The timing residuals induced by gravitational waves in PTA would show a peculiar angular dependence with a 3 marked enhancement around a particular value of the angle subtended by the source and the pulsars. This angle depends mainlyontheactual valueofthecosmological constant andthedistancetothesource.Preliminaryestimatesindicatethat ] O the enhancement can be rather notorious for supermassive black holemergers and infact itcould facilitatethe firstdirect detectionofgravitationalwaveswhileatthesametimerepresentinga‘local’measurementofL . C Keywords: Cosmologicalconstant,darkenergy,gravitationalwaves,pulsartimingarrays . h p - o r INTRODUCTION t s a ThereisafairamountofevidencesuggestingthatthespacetimewhereweliveisgloballydeSitterwithavaluefor [ thecosmologicalconstantestimated[1]tobearoundL ≃10−52m−2.Thishasobviouseffectsoncosmology–atvery 1 largescales.Theseeffectsareofnoconcerntoushere. v Insteadwewouldliketohelpanswerthequestion:doesL havea‘local’influence?,wherelocalmeansatmoderate 5 valuesoftheredshiftz.Inshort,canthecosmologicalconstantbemeasured‘locally’?Thisisanimportantquestion 2 becauseitmaysettletheissueastowhetherL isatrulyfundamentalpropertyofspace-time,abasicconstantofnature 9 presentatallscales,ratherthanawayofprovidingsomeeffectivedescriptionrelevantonlyatcosmologicaldistances. 7 . Alotofworkhasbeendevotedtofindingtracesoftheexistenceofthecosmologicalconstantatsub-cosmological 1 scalessuchasinclusterofgalaxies[2],sofarwithoutaclearconclusionorveryrelevantbounds.Severalworksstudy 0 theeffectofL onthegravitationalbendingofligh[3,4,5]ortheShapiroeffect[6].Someauthorshaveevenadvocated 4 1 more exoticeffectssuchas providingan explanationthe well knownPioneer anomaly[7] or the apparentincrese of : theastonomicalunitwithtime[8]. v Discussionsintheliteratureregardingtheabovepointstendtobeconfusing.Effectsrangefromsurprisinglylarge i X tozero.Notsurprisinglythesourceofmostdiscrepanciesisthemeaningofthedifferentcoordinatesystemsandtheir r physicalrealization. a Here we propose to try to find ‘local’ effects of L by studying the propagation of gravitational waves (GW) in an space endowedwitha cosmologicalconstant.Thismay seemhopelessatfirstas thereis nodirectdetectionofa GW yet,letalonepossiblemodificationsdueto L =0. Howeverbecausethe natureofthecosmologicalconstantis 6 quiteunclearitisplausiblethatitshouldbeattributedtothegravitationalinteractionitself—afundamentalproperty of space-time—and accordinglyplaced naturally on the l.h.s. of Einstein equations. Then it seems quite natural to investigatehowthefundamentalexcitationsofgravityaremodifiedbyL .Thisissuehasbeenstudiedin[9]. HasL possibleobservationalconsequencesonGWinspiteofitscurrentlypreferredverysmallvalue?Wewillsee thattheanswertothisquestionissomewhatsurprising.Whatfollowsisanextendedversionoftheresultsfoundbyus anddescribedin[10]. Forapreviouattempttofindeffectsofanon-zerocosmologicalconstantusingGWsee[11]. LINEARIZATION OFTHE FIELD EQUATIONSIN THEPRESENCE OFL Keeping control of the different orders in L will be essential to our discussion. This will become obvious in the subsequent. LetusstartbylinearizingEinsteinequationssinceafterallGWaresolutionsofthelinearizedequations: 1 Rmn gmn R+L gmn = k Tmn (1) −2 − Inthelinearizedapproximationweassumegmn =h mn +hmn , hmn 1andthen | |≪ 1 Rmn = 2 (cid:3)hmn +h,mn −hlm ,nl −hnl ,ml . (2) (cid:16) (cid:17) Agaugechoiceismandatorytosolvetheseequations.AcommonchoiceistoselecttheLorenzgauge ¶ m hmn = 1¶ n h ¶ m h˜mn =0, (3) 2 ↔ where 1 h˜mn =hmn h mn h (4) −2 ThefieldequationinLorenzgaugeare 1 (cid:3) hmn h mn h +2L hmn = 2L h mn . (5) −2 − (cid:18) (cid:19) WhetherthetermoforderO(hL )needstobeconsideredornotdependsontherelativemagnitudeof(cid:3)handL .Ifthe L hmn termonthel.h.s.isomitted(andonlyinthiscase)thereisaresidualgaugefreedomwithintheLorenzgauge. xm xm =xm +x m (6) ′ → aslongasx m isanharmonicfunction,(cid:3)x m =0.Allthisisofcoursewellknownandstandardtexbookmaterial.Ifwe setL tozerowegetthestandardtreatmentofgravitationalwaves—inMinkowskispacetime.ItisclearthatL =0 will modify the for of the solution by terms of order L . But we also need to know what choice of coordinates6 the linearizationandtheLorenzconditionimplies,ifany.ThisisnotatotallytrivialissueindeSitter. ‘GOOD’AND‘BAD’ COORDINATESYSTEMS INDESITTER Before giving the solution to the field equation in the Lorenz gauge (the one where GW are usually treated) let us discussseveralpossiblecoordinatechoicesindeSitterspace-time.Convenientreferencesaregivenin[12].Seealso [13]. Astaticmetric:Schwarzschild-deSitter(SdS) L L 1 ds2= 1 rˆ2 dtˆ2 1 rˆ2 − rˆ2+rˆ2dW 2 (7) − 3 − − 3 (cid:20) (cid:21) (cid:20) (cid:21) Thismetrichasnotquitesphericalsymmetry. Apositionindependentmetric:Friedmann-Robertson-Walker(FRW) L ds2=dT2 exp(2 T)d~X2 (8) − 3 r This metric incorporatesthe physical principlesof cosmologicalhomogeneityand isotropy. The coordinatesXi are comovingcoordinatesanchoredin space that expandwith the universe. These are the coordinateswhere our world appearshomogeneousandisotropic. NoneofthepreviousmetricsobeytheLorenzgaugecondition.Ofcoursethereisnoreasonwhytheyshouldbecause they are notsolutions of Einstein equationslinearized aroundMinkowskispace-time, so in order to match with the discussionintheprevioussectionletusproceedtolineariarizethetwometrics.FortheSdSmetricthisisquiteeasy astheSdSmetricisexpandableinintegerpowersofL L L ds2 1 rˆ2 dtˆ2 1+ rˆ2 rˆ2+rˆ2dW 2. (9) ≃ − 3 − 3 (cid:20) (cid:21) (cid:20) (cid:21) ItshouldthereforeobeyalinearizedversionofEinsteinequations,althoughascanbeeasilyverifiednotintheLorenz gauge. OnthecontraryitisclearthattheFRWcoordinatechoicecannotbelinearizedinL becauseitcontainsoddpowers of√L L ds2=dT2 exp(2 T)d~X2 (10) − 3 r and therefore it is impossible that it can fulfill any linearized Einstein equation, even if t <<1/√L as it is not expandableinintegerpowersofL . Onecanworkouttheexacttransformationbetweenthetwocoordinatesystems.Weshallreservecapitallettersfor FRWcoordinatesandthehattedlowercaseonesforSdS(thereasonforthehatwillbeevidentbelow) rˆ=eT√L /3R (11) 3 √3 tˆ= log +T (12) L   r 3 L e2T√L /3R2 − q  whereT andRarethecosmologicaltimeandcomovingcoordinateswhosephysicalrealizationisclear.Thistransfor- mationisvalidinsidethecosmologicalhorizon,i.e.R< 1 . √L LINEARIZED BACKGROUNDAND LINEARIZED GWSOLUTIONS LetusnowreturntoEq.5.Weshallworkinthelinearizedapproximationbothforthebackgroundmodification(with respecttoflatMinkowskispace-time)hLmn andforgravitationalwaveperturbationshWmn .Wefollowherethediscussions in[9,13].Thetotalmetricwillbewrittenas gmn =h mn +hLmn +hWmn , hLmn ,W 1 (13) | |≪ Letusfocusonthebackground.IntheLorenzgaugeandneglectingL hLmn (cid:3)h˜mn = 2L h mn , ¶ m hñm =0, (14) − Thishasasaparticularsolution L L h˜mn = 4xm xn h mn x2 hmn = xm xn +2h mn x2 (15) −18 − ⇒ 9 (cid:0) (cid:1) (cid:0) (cid:1) Thegeneralsolutionistheformerplusanysolutionof(cid:3)h˜mn butwecallthelatter‘waves’ratherthan‘background’. IndeedtheequationforGWisparticularlysimpleinLorenzcoordinatesiftheL hWmn termisneglected. (cid:3)h˜Wmn =0 (16) i.e.itistrulyawaveequation. Now we have to answer the followingquestion:What is the physicalrealizationof the coordinatesx,t where we justsolvedEinsteinequationsintheLorenzgauge? LetustakeadvantageoftheresidualgaugeinvariancethatexistsifthetermL hmn isneglected.Thenthesolution L h˜mn = 4xm xn h mn x2 (17) −18 − or (cid:0) (cid:1) L hmn = xm xn +2h mn x2 (18) 9 canbetransformedintoastaticmetric—stillinLoren(cid:0)zgauge. (cid:1) Thefollowingchangeofcoordinates L x2 (y2+z2) x=x′+ t′2 ′ + ′ ′ x′ (19) 9 − − 2 4 (cid:18) (cid:19) L y2 (x2+z2) y=y′+ t′2 ′ + ′ ′ y′ (20) 9 − − 2 4 (cid:18) (cid:19) L z2 (x2+y2) z=z′+ t′2 ′ + ′ ′ z′ (21) 9 − − 2 4 (cid:18) (cid:19) L t=t′ (t′2+r′2)t′ −18 transformsthemetricintoastaticsolutionatorderL L L ds2= 1 r2 dt2 1 (r2+3x2) dx2. (22) − 3 ′ ′ − − 6 ′ ′i ′i (cid:20) (cid:21) (cid:20) (cid:21) ThismetrichasaZ symmetryonly.WearestillintheLorenzgauge. 3 Underthefollowingadditionalchange L L L x′=x′′+ x′′3, y′=y′′+ y′′3, z′=z′′+ z′′3, (23) 12 12 12 t =t (24) ′ ′′ themetricbecomes L L ds2= 1 r′′2 dt′′2 1 r′′2 (dr′′2+r′′2dW 2), (25) − 3 − − 6 (cid:20) (cid:21) (cid:20) (cid:21) whichisnotintheLorenzgaugeanymore.Yetanotherchange L r =rˆ+ rˆ3 t =tˆ ′′ ′′ 12 leadsto L L ds2= 1 rˆ2 dtˆ2 1+ rˆ2 drˆ2+rˆ2dW 2. (26) − 3 − 3 (cid:20) (cid:21) (cid:20) (cid:21) ThisisthelinearizedSchwarzschild-deSittermetric. NowweknowinwhichcoordinateswewerewhenwesolvedthelinearizedEinsteinequationwithacosmological constant in Lorenz gauge. A series of elementary coordinate transformations brought our solution to a linearized version of the Schwarzschild-de Sitter metric (expandedto first order in L ). Thanks to Birkhoff’stheorem[14], we knowthatthismetricisunique. TheSdScoordinatesareusefulforproblemswithsphericalsymmetry,suchasobjectsfallingontoeachother.They donotadmitaNewtonianlimitifL =0,i.e. 6 ds2=(1+2F )dt2 (1 2F )dx2 (27) 6 − − becauseinadditiontothescalarpotentialF (whichitselfhasacorrectionofO(L ))thereisatensorpotentialt L . ij ∼ HoweveritisstilltruethatifamassivesourceofmassMisintroducedvia T =Md (rˆ) (28) 00 theequationofmotionforanon-relativisticbodyaregovernedbyF alone GM r¨ˆ= +O(L ) (29) − rˆ Solutions are periodic in coordinates rˆ,tˆ (up to O(L )) and the wave equation will lead to harmonic GW in these coordinatesuptocorrectionsofO(L ). However,thecoordinateswherethemetricisSchwarzschild-deSittercenteredinaremoteblackholearenot‘useful’ forcosmologysimplybecausewedonotperformmeasurementshereusingthese.Butweknowhowtogofromthese SdS coordinated to the ones (FRW) where cosmological measuments are made. Of course none of these subtleties occurifL =0. GRAVITATIONALWAVESIN COSMOLOGICALCOORDINATES LetusgobacktoLorenzgauge.Recallthatwewanthmn =hLmn +hWmn andthathWmn shouldfulfill,ifthetermL hW is neglected, h˜Wm m =0, ¶ m h˜Wn m =0, (cid:3)h˜Wmn =0 (30) whilehLmn shouldfulfillifL hL isneglected h˜Lmm =0, ¶ m h˜Ln m =0, (cid:3)h˜Lmn = 2L h mn . (31) − Thegeneralsolutionatlowestorder(writtenforhmn )willbe L hmn =hLmn +hWmn = xm xn +2h mn x2 +EmWn coskx+DWmn sinkx (32) 9 (cid:0) (cid:1) where Emn and Dmn are polarization tensors having vanishing traces EW = DW = 0 and obeying the condition km Enm W =km Dnm W =0withk2=0. ItispossibletoderivethefullsolutionincludingL hmn termsbutweshallnotconsiderithereasthemodifications areunrealisticallysmalltobeseen,buttheyhavesomeinterestingaspectsnevertheless.Theinterestedreadercansee [9]fordetails. ThiscoordinatesystemiseasilyrelatedtoSdScoordinates.Thesecoordinatesarewellsuitedtodescribeproblems withsphericalsymmetry(rˆ=0isa‘special’point)suchasthesolarsystemorcollapseontoablackholebutthese coordinatesarenottheoneswhereweobservethe(expanding)universe.Whenthesphericalsymmetryislost(away fromthesource),wehavetomatchtosuitablecoordinates,physicaltotheobserver. We have to transformnowthe solutionsfoundin SdS-likecoordinatesto FRW coordinates.Letus showhere for simplicityjusthowthelowestordersolution(i.e.theoneobtainedneglectingL hWmn terms)looksoncetransformed.A planewavepropagatinginthezˆdirectiontransformsinto 0 0 0 0 L L 0 E11 1+2 3T E12 1+2 3T 0 hWmn FRW = (cid:18) q (cid:19) (cid:18) q (cid:19)  L L × 0 E12 1+2 3T −E11 1+2 3T 0 0 (cid:18) 0 q (cid:19) (cid:18) 0 q (cid:19) 0    L Z2  cos w(T Z)+w TZ +O(L ) +O(L ) − r3 (cid:18) 2 − (cid:19) ! (33) 0 0 0 0 L L 0 D11 1+2 3T D12 1+2 3T 0 + (cid:18) q (cid:19) (cid:18) q (cid:19)  L L × 0 D12 1+2 3T −D11 1+2 3T 0 0 (cid:18) 0 q (cid:19) (cid:18) 0 q (cid:19) 0    L Z2  sin w(T Z)+w TZ +O(L ) +O(L ) − r3 (cid:18) 2 − (cid:19) ! Themaximaofthewavewillbereachedwhen L Z2 w(T Z)+w TZ =np (34) − 3 2 − r (cid:18) (cid:19) np T2 L n2p 2 L Z (n,T) T + (35) max ≃ − w − 2 3 2w2 3 r r Thephasevelocityofthewaveis dZ L v (T) max =1 T +O(L ) (36) p ≡ dT − 3 r Incomovingcoordinatesthephasevelocityissmallerthan1.Thisdoesnotmeanthatthewavesslowdown.Wecan calculatethevelocityin‘ruler’distance. L dl d L dl2= 1+2T dZ2, = 1+T dZ 1 (37) max − − r3! dT dT " r3! #≃ Notice that the modifications due to the cosmological constant are not of order L as a naive consideration of the linearizedfieldequation(obeyedbytheGW)wouldleadustobelieve.Aswediscussedinmuchdetail,thislinearized equationintheLorenzgaugeisinanessentialwayrelatedtoSdScoordinates.Theeffectofthecoordinatechangeto thecosmologicalFRWcoordinatesisactuallyaneffectoforder√L andthereforethemagnitudeofthechangecanbe verydifferent. Severalphysicalmagnitudesappearinthemodifiedexpressionforthewaves;inparticularL .Itseffectsareshown intheaccompanyingfigureandthemostrelevantquestionis:arethesechangesdetectable? h++ 1.0 0.5 4´1024 6´1024 8´1024 1´1025 -0.5 -1.0 -1.5 FIGURE1. Dependenceoftheamplitudeandwave-lengthonZ(expressedinmeters)foraconstantvalueofT andfordifferent valuesofL .Dashedline:L =0,dottedline:L =10 52m 2,solidline:L =10 51m 2.Waveswith103Hz<w<10 10Hzcannot − − − − − bepracticallyplottedintherelevantZ-range.Herew=4 10 16Hz − · TIMINGRESIDUALS IN PULSAR ARRAYS Pulsarsareverystableclockswithperiodsrangingfrommillisecondsuptoabout10seconds.About600pulsarsare known.Theaverageperiodis0.65s.Fastrotatingpulsarscanbemoreregularthanatomicclocksanddisruptionsof theorderof1m scanbemeasuredonEarth. The passage of a gravitational wave disrupts this array of clocks and indeed PTA’s may provide the first direct evidence of gravitational waves in <10 years. The idea behind the PTA collaborations is to detect the correlated disruption of the periods measured for a significant number of pulsars due to the passing of a gravitational wave throughthesystem [15, 16,17, 18].PTA aresuitabledetectorsforlowfrequencyGW, i.e.fortherange10 9 Hz to − w 10 7Hz[15]andthesignalisexpectedtofollowapowerlaw[16,19].Akeyprobleminmakingpredictionsis − ≤ ≤ modelinginarealisticwaythewavefunctionsproducedinthedifferentsources,inparticularthevalueoftheamplitude ofthemetricperturbationhisafreeparameterinprinciple.Someboundsintherangeof10 17 h 10 15havebeen − − ≤ ≤ setalready[19]. RoughestimatesfromtheexpressionsintheprevioussectionindicatedthatcorrectionsofO(√L )canberelevant forpulsar timingarrays(typicallysituatedatL 1 kpc)beingdisturbedby extragalacticbinaryblackholesystems ∼ (typicallyatl 100Mpcormore). ∼ Iff (t)isthefieldphaseofthepulsar,thepulsedemissionmeasuredwithanEarth-basedradiotelescopewillbe 0 L f (t)=f (t t (t) t (t)) (38) 0 E+S GW − c − − wheret arelocalcorrectionsduetothemovementoftheEarthandsolarsystemandt (t)istheshiftduetothe E+S GW passageofaGW. Letusconsiderthe observationalset-upshowninthe figureThe fieldphaseshiftdueto theGWwill beapproxi- Source P ZE Pulsar a L Earth matelygivenby[20] 1 t (t)= nînˆjH (t), (39) GW ij −2 where 0 H (t)=L dxh (t+Lx,~P+L(1+x)nˆ) (40) ij ij 1 Z− and~Pisthepulsarlocation,nˆ=( sina ,0,cosa )andZ =cz 3 (zistheredshift).Hereandinwhatfollowswedo E L − nottakematterintoaccount,thatisW M=0.Weshallreturntoqthispointlater. Itisinterestingtonotethatthisexpression(whichincludestheleadingcorrectionanyway)isvalidonlyifthetime componentsoftheperturbedmetricareallzero.Thisissoatorder√L ,whichbyfardominates,butceasestobetrue whenoneconsiderstheO(L )termsindicatedin(33).See[9]forthecompleteexpressions. Tokeepthingsassimpleaspossibleletusassumethataremotemergingofgalaxieseventuallyleadstothemerging of their centralblack holes. Characteristically,the two black holeshave very differentmasses and thereforewe can thinkofoneobjectorbitingaroundthemostmassiveone,andeventuallycollapsinontoit.Theproblemistherefore a keplerian one in essence, with approximatespherical symmetry.The two spiraling black holes produceGW with a characteristictimeofemissionthatisoftheorderofonetoseveralyearsandtheperiodofthesignalrangesfrom days to months. The coordinates where the emission is just a collection of a few harmonics will of course be SdS coordinates.Considerthesimplestpossiblecase(justoneharmonic): 1 hSmndS= Emn cos[w(t r)]+Dmn sin[w(t r)] +O(L ) (41) r − − (cid:0) (cid:1) When we say days or months we have to be definite about which clock we are talking about; namely in which coordinatesystemtandraremeasuredinthepreviousequation.Itshouldbeobviousthattheyrefertothecoordinates associatedtothesupermassiveblackhole(locatedatr=0).ItisonlyinthosecoordinatesthattheemissionofGWis periodic.(Ofcourseitisnotexactlyperiodicasthesmallestblackholelosesenergyandeventuallycollapsesandthe emissionoftheGWcannotbereallyattributedtothepointr=0,butthisdoesnotchangetheessenceoftheargument becausetheuncertaintyinthepointofemissionismuchsmallerthanothermagnitudesrelevantforthediscussion.)In conclusion,thepreviousequationisthedistortiontothemetricproducedbyGWinSdScoordinates. If the metric would be exactly Schwarzchild, that is without any cosmological constant at all, the metric would asymptotically become Minkowski and these coordenates are roughly speaking also the ones of a remote observer (rememberthatwehaveneglectedthepresenceofmattercompletely).However,thisiscertainlynotthecaseifL =0 6 becausethenthemetricisnotasymptoticallyflat.Thenitisclearthataterrestrialobserverisnotusingthecoordinates r and t but rather describes their observations of the universe (in particular the distant place where the black hole mergingtookplace)usingFRWcoordinatesRandT. AswelearnedintheprevioussectionsthentheGWbecomesinFRWcoordinates Emn L L R2 hFmn RW = 1+ T cos w(T R)+w TR R r3 ! " − r3 (cid:18) 2 − (cid:19)# (42) Dmn L L R2 + 1+ T sin w(T R)+w TR +O(L ). R r3 ! " − r3 (cid:18) 2 − (cid:19)# AswediscussedindetailintheprevioussectiontherearecorrectionsoforderL totheexpressionforthewavesinSdS coordinates,butafterchangingtoFRW coordinates,thesecorrectionswillstill beoforderL .Itisreallythechange fromSdStoFRWcoordinatesthatmattersandintroducescorrectionsoforder√L . From the pulsar to the Earth the electromagnetic signal follows the trajectory given by the line of sight ~R(x)= ~P+L(1+x)nˆ.Listhecomovingdistance(replacingitbytherulerdistancemakesnosignificantdifferences). R(x)= Z2+2xLZ cosa +x2L2 Z +xLcosa , (43) E E ≃ E Intheusualtreatment,thecosmologicalcqonstantisneglectedandtheeffectofL wouldtakenintoaccountonlythrough theredshiftw w =w/(1+z).TheimportantquestionisofcoursewhetherL isreallyrelevantafterall. eff → OBSERVING COSMOLOGICALCONSTANT EFFECTS IN GRAVITATIONALWAVES Letusdefine L Z L Q (x,T ,L,a ,b ,Z ,w,L ) w(T + x E x cosa ) E E E ≡ c − c − c L (ZE +xLcosa )2 L Z L (44) +w c c T + x E +x cosa . E r3 2 −(cid:18) c (cid:19)(cid:18) c c (cid:19)! Then 1Le H t (T )= sin2a cos2b +2sina sinb cos2b sin2a sin2b GW E ≡ −2 c − (45) 0 (cid:0) 1 L L (cid:1) dx 1+ (T + x) (cosQ +sinQ ), ×Z−1 (ZE+xLcosa ) r3 E c ! wheree isacharacteristicGWamplitude.Allthevariableshavealreadybeendefinedexceptb thatcorrespondstothe azimuthalangleofthepulsarreferredtotheplaneperpendiculartothelineEarth-source. The first indicationthat the cosmologicalconstantmatterscomesfromconsideringthe theoreticaldependenceof thesignalontheanglea subtendedbythesourceandthepulsar,theEarthbeingthevertex.Thisisshowninthefigure whereanenhancementisfoundatlowangles(atleastforthevaluesselectedfortheastrophysicalparameters) 2.´10-7H 2.´10-7H 1.´10-7 1.´10-7 Α Α 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 -1.´10-7 -1.´10-7 -2.´10-7 -2.´10-7 -3.´10-7 -3.´10-7 FIGURE2. OnthelefttherawtimingresidualforL =10 35s 2 asafunctionoftheanglea subtendedbythesourceandthe − − measuredpulsarasseenfromtheobserver.OntherightthesametimingresidualforL =0.Inbothcaseswetakee =1.2 109m, L=1019mandTE = ZcEsforZE =3×1024m;withthesevalues|h|∼ Re ∼10−15 whichiswithintheexpectedaccuracy×ofPTA. SimilarresultsareobtainedforotherclosevaluesofTE Letusnowdefinethefollowingstatisticalsignificance s = 1 (cid:229)Np (cid:229)Nt H(TEi,j,Li,a i,b i,ZE,w,e ,L ) 2 (46) vN N s u p t i=1j=1 t ! u t wheres istheaccuracywithwhichweareabletomeasurethepulsarsignalperiod.Wetakes 10 6s(thisisthe t t − ≃ averageofthebestmeasuredpulsarsincludedintheInternationalPTAProject[17]). Weassumeanobservationtimeof3years,startingatthetimethesignalis1016sold(timeofarrivalatourGalaxy) andobservationsevery11days(N =101);1016s T 1.00000001 1016s.Thecoalescencetimesofsupermassive t E black holes is taken to be O(107) s; that is abou≤t one≤year (much sh×orter time scale than the time of arrival of the perturbationtothelocalsystem). The galactic latitude and longitude of each pulsar are transformed to (a ,b ), where a is the angular separation betweenthelineEarth-GWsourceandthelineEarth-pulsar. Weplots (a )usingasetof5fixedpulsarssupposedtobeexactlyatthesameangularseparationfromasourcethe positionofwhichwevary.Thiscouldbedoneforanysetoffivepulsars.Thepositionofthepeakdoesnotdependon L andb .WeusethepulsarswhichareallclosetoeachotheratadistanceL 1020m: i i ∼ TABLE1. PulsarsfromtheATNFCatalogue J0024-7204E J0024-7204D J0024-7204M J0024-7204G J0024-7204I Theresultsaresummarizedinthefollowingfigures: Σ Σ Σ 0.14 0.14 4 0.12 0.12 3 0.10 0.10 0.08 0.08 2 0.06 0.06 0.04 0.04 1 0.02 0.02 Α Α Α 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 FIGURE 3. s (a ) for L =10 35s 2 (left). Zoom on the lower values for L =10 35s 2 (middle) with the prominent spike − − − − removed(noticetheverydifferentverticalscale,andcomparisontotheL =0case(right). Asisobviousfromthefigure,thesignalshowsaremarkablesimilaritybetweenthecasesL =0andL =0except fortheveryprominentspikethatthecaseshownforL =10 35s 2,thecurrentlypreferredvalueforL ,ata6 relatively − − lowangle. Nowtakealistofobservedpulsarswelldistributedinthegalaxy.Theangles(a ,b )arecalculatedforallofthem consideringtwohypotheticalsourcesofGW.Onelocatedatgalacticcoordinatesq =300 ,f = 35 andanother S1 ◦ S1 ◦ atq =4 ,f =10 . − S2 ◦ S2 ◦ We orderthem from the lowest a to the largest. We groupthem in sets of five pulsars. We consider 27 sets of 5 pulsars;thatisalistof135pulsars.Foreachsetwecalculatethesignificance s = 1 (cid:229)5k 1(cid:229) 01 H(TEi,j,Li,a i,b i,1024,10−8,1.2×109,10−35) 2 (47) k v5 101 10 7 uu · i=1j=1 − ! t andplotitasafunctionoftheaverageangleoftheset,a¯ =(cid:229) 5k a i with1 k 27.Wethinkthatthefiguresspeak k i=1 5 ≤ ≤ by themselves.It is clearthat PTA observationsthataim at observingthe ‘normal’GW spectrumshouldabsolutely seethe‘abnormal’enhancementatagivenvalueoftheangle. WHYTHISENHANCEMENT? In order to understand why this effect comes about let us examine the behaviour of the differentialtiming residual aswemovealongthelineofsight.Thewindowintheangularvariablecorrespondstoa‘valley’wherethephaseis (nearly)stationary.Atthisvalueoftheangularvariablea iswheretheenhancementtakesplace. Σ Σ Σ á 3.5 0.15 0.15 3.0 çççç 2.5 2.0 0.10 á 0.10 çççç 0011....0505 ááááááááç0.5áççááçáááçáçá1.0áçççáççááçááááççç1.á5çççáçççç2á.0ç Α00..00050.0á ááç0.5áççááçáááçáçá1.0áçççáççááçááááççç1.á5çççáçççç2á.0ç Α00..00050.0ááááááááç0.5áççááçáááçáçá1.0áçççáççááçááááççç1.á5çççáçççç2á.0ç Α FIGURE 4. Plot of s k(a¯k), k=1,27. L =10−35s−2. Circles correspond to Source 1 and squares to Source 2. Full range is showedontheleft,zoomonthelowervaluesforL =10 35s 2isshowninthemiddle(againthespikeisremoved-verticalscale − − isdifferent)andcomparisontoL =0isshowontherightfigure,respectively. 0.0 -x0.5 0.0 x -1.00.2 -0.5 -1.0 -1.0-x0.50.0 0.05 0.0dH 00..0005dH 0.00 dH -0.05 -0.2 -0.05 0.0 0.1 Α 0.2 0.3 0.0 0.1 Α 0.2 0.3 0.0 0.1 Α 0.2 0.3 FIGURE5. EvolutionofthephaseoftheGWalongthelineofsighttothepulsarfordifferentangles.LeftfigureL =0.Middle: L =10 35s 2.Right:sameasthepreviousonebutrotatedsoastoshowthe‘valley’ofstationaryphase,whichisabsentwithout − − thecosmologicalconstant. Amongallthedependencies,andwhenthedistancetothesourceiswellknown,themostrelevantappearstobethe onerelatedtothevalueL .ThepositionofthepeakdependsstronglyonthevalueofL .Itmovestowardsthecentral valuesoftheangleforlargervaluesofthecosmologicalconstant. Thepositionofthe‘valley’(andthereforeL )canbefoundanalyticallyintwoways 1. Lookingforthestationaryphasecondition 2. ExaminingthebehaviouroftheFresnelfunctionsandprefactorsobtainedafterintegration The prefactor becomesquite large for a specific value of the parameters involved.This particular value rendersthe Fresnelfunctionclosetozeroandtheproductisanumbercloseto2.Awayfromthispointthenetresultissmall. Using the series expansionof the Fresnel functionsat first order we are able to obtain an approximateanalytical expressionfortherelationL (a ) 12c2sin4 a 12c2sin4 a L (a )= 2 2 , (48) ((cT Z )cosa +Z )2 ≃ Z2 E E (cid:0) (cid:1) E E (cid:0) (cid:1) − L 1.8´10-35 1.6´10-35 1.4´10-35 1.2´10-35 1.´10-35 8.´10-36 Α 0.17 0.18 0.19 0.20 0.21 0.22 FIGURE6. L (a )obtainednumericallyfromthepositionsofthepeaksinthes (a )plotsfordifferentvaluesofthecosmological constant(dots)andobtainedanalyticallyfromanapproximationoftheFresnelfunctionsinvolvedinthetimingresidual(line).