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1、PHYSICALREVIEWDVOLUME26,NUMBER10Bubble collisionsintheveryearlyuniverse15NOVEMBER1982S.W.Hawking,I.G.Moss,andJ.M.StewartD.A.M.T.P.SilverStreet,CambridgeCB39ER,U.I(.(Received30November1981)Itisbelievedthatfirst-orderphasetransitionsoccurredintheveryearlyuniversewhenthetemperaturedroppedbelowthegrand-
2、unificationandWeinberg-Salamenergies.Bub-blesofthenew,broken-symmetryphasewouldhaveformedsurroundedbythesymmetricphase.Theenergyreleasedinthephasetransitionwouldhavecausedthewallsofthebubblestoaccelerateoutwards.Westudywhathappenswhenthewallscollidewitheachother.Wefindthattheenergyinthewalls wouldno
3、tbethermalizedforaconsiderabletime.Intheinflationary-universescenario,inwhichthebubblenucleationrateislow,thermalizationcouldnotoccuruntillongafterthebaryonandnucleosynthesiserasandwouldnotbecomplete.Wealsoinvestigatetheformationofprimordialblackholesinbubblecollisions.TheWeinberg-Salamphasetransiti
4、onisnotlikelytoproduceblackholesbut,undercertaincircumstances,thegrand-unifiedphase transitionmightgiverisetoblackholesof103g.I.INTRODUCTIONInthecurrentlyacceptedunifiedtheories,phasetransitionsarepredictedtooccur attemperaturesoftheordersoftheenergiesatwhichthevariousinteractionsareunified.Thusinth
5、egrandunifiedtheoriessuchasSU(5)(Ref.1)orSO(10)(Refs.2and3)therewouldbephasetransitionsattempera-turesoftheorderofthegrand-unificationenergy10GeVandattheelectroweak-unificationen-ergy10GeV.Thephasetransitionsarelikelytobefirstorder(seeRef.4andreferencestherein).Thereasonisthatintheunified,symmetricp
6、haseallthevectorparticlesaremasslessandthereforehavethermalfluctuationsoftheorderofT2/12foreachdegreeoffreedom.Thesethermalfluc-tuationshelptomaintaintheexpectationvaluesoftheHiggsfieldsinthesymmetricphase.Ontheotherhand,iftheHiggsfieldsweretodevelopex-pectationvalueswhichbrokethesymmetryfromagroup6
7、toasubgroupE,thevectorparticlescor-respondingtoG/Ewoulddevelopmassesandtheirthermalfluctuationswouldbegreatlyre-duced.ThisinturnmakesiteasierfortheHiggsfieldstohavenonsymmetricexpectationvalues.Onecanthushavetwophasesatthesametem-peraturewhicharelocallystableagainstsmallchangesintheexpectationvalues
8、oftheHiggsfields.Oneofthephaseswillingeneralhavealowerenergydensitythantheotherandsoquan-tumtunnelingwilloccurgivingafirst-orderphasetransitionwiththereleaseofenergy.ThemannerinwhichsuchphasetransitionswouldoccurhasbeendescribedbyColeman.AtacertaintimetIquantumfluctuationswouldcauseabubbleofthenewph
9、asetoforminamediumwhichwas elsewhereintheoldphase.Themostprobablebubblewouldbeat=constantsectionthroughasolutionoftheclassicalscalarfieldequationinEuclideanspacewithaneffectivepo-tentialwhichincludesquantumcorrectionsandtheeffectsofthermalfluctuationsofthescalarandvectorfields.AttimesI,&tI,theexpect
10、ationvaluesofthescalarfieldswouldevolveaccordingtotheclassicalfieldequationsinLorentzianspacewiththetime-symmetricdataattIgivenbytheEuclideanbubblesolution.Thebubbleofthenewphasewouldexpandintotheold phaseuntilitcollidedwithotherexpandingbubblesofthenewphase.Itisthoughtthatsuchbubbleformationmightha
11、veoccurredintheearly universewhenthetem-peraturedroppedbelowthe grand-unificationtem-peratureandmaybeagainwhenthetemperaturedroppedbelowtheelectroweak-unificationtem-perature.Theenergyreleasedbythebubblefor-mationwouldbe transformedintokineticenergyofthebubblewallsandwouldcausethewallstoexpand outwa
12、rdswithuniformacceleration.Itisgenerallyassumedthatwhenthebubblewallseventuallycollidewiththoseofanotherbubble,thekineticenergywillbedissipatedandtheuniversewillbereheatedtoalmostthetemperatureithadbefore thephasetransition.Thepurposeofthis2626811982TheAmericanPhysicalSociety2682S.W.HAWKING,I.G.MOSS
13、,ANDJ.M.STEWART26paperistoexaminethecollisionofbubblesandtoseetowhatextentthisassumptionisjustified.IfthebubblenucleationrateXperunitfour-volumeislargecomparedtoHwhereH=8/8istheexpansionrateoftheuniverse,onecanneglectthespatialcurvatureoftheuniversetothefirstap-proximationandtreatthebubblesasinflats
14、pace-time.Wehaveusedanalyticalandnumericaltech-niquestostudythecollisionoftwobubblesinthissituation.Ifthetwobubblescorrespondtosignifi-cantlydifferentmembersofthefamily6/Eofbroken-symmetryphases,mostoftheenergyofthebubblewallswillbeconvertedintoapairofwavesinwhichthephaseofthescalarfieldrotatesfromi
15、tsvalueinthetwobubblestoacompromisevalue.Thesephasewaveswillpropagateatthespeedoflight.Theenergyinthemwi11eventuallybether-malizedbytheircouplingtothegaugefieldbutthismaytakealongtime.Ontheotherhand,ifthephasesinthetwobubblesarenearlyaligned,mostoftheenergyofthewallsisnotdissipatedinthecollisionandt
16、hewallspassthrougheachotherandstartacceleratingbacktowardseachother.Theregiontothefutureofthecollisionsurfaceandbetweenthewallswillagainbeintheoldsym-metricphaseandthisphasewouldcontinuetoex-istforquiteatimeasthewallsoscillatebackwardsandforwardswithincreasingfrequency.Oneachcollisiontheywilllosesom
17、efractionoftheirkinet-icenergyandtheirenergywillalsobereducedbytheexpansionofthebubbles.Eventuallytheywilldisappearleavingonlythenewbroken-symmetryphase.Ineithersituationthermalizationmaynotoccurforsometime.Thiscouldhaveabigeffectonthepossible generationofbaryonsymmetrybyCP-noninvariantinteractions.
18、Itisgenerallyassumedthatthebroken-symme-tryphasesofthedifferentbubblesareuncorrelatedbecausetheyarespacelikeseparatedwhentheyformandbecause theparticlehorizonofstandardFriedmann-Robertson-Walkercosmologicalmodelspreventcasualinfluencesfrompropagatingveryfar.However,thisassumptionwouldseemtoleadtothe
19、productionoffartoomanymagneticmono-polesofthegrand-unificationmass.Apossiblemechanismforcorrelatingthephasesindifferentbubbleswouldbethenoncausalinteractionswhichmustbepresentifspacetimehasafoamlikestruc-tureonscalesofthePlancklength.IfthebubblenucleationrateNperunitfour-volumeweresmallcomparedtoHon
20、ewouldhavetheinflationary-universescenario.Theuniversemouldgetstuckinthesymmetricphaseandtheexpansionwouldcauseittosupercool.Eventually,thethermalenergydensitywouldbereducedbelowthedifferenceinenergydensitiesofthetwophases.Becauseonedoesnotwanttohavealargecosmologicalterminthebroken-symmetryphase, o
21、nehastoadjustthenormaliza-tionofenergydensitysothatitissmallinthisphase.Onethengetsalargepostive(i.e.,repul-sive)cosmologicalterminthesymmetricphase.Thiswouldcausetheuniversetoexpandex-ponentiallyandtoapproachthedeSittermetric.Whenbubblesofthebroken-symmetryphasedidformeventually,the accelerationoft
22、heirwallsto-wardseachotherwouldbecounteractedbytheex-ponentialexpansionoftheuniverse.Itwillbeshownthattheycouldnevercollidewithmorethanaboundedcenter-of-massenergy.Despitethistherewouldbeenoughenergyinthewallstoreheattheuniversetoatemperaturenearthecriti-caltemperatureiftheenergycouldbethermalizedan
23、dspreaduniformlyoverspace.However,theprobabilitydistributionofthesizesofbubbleswouldbeveryflatwhichwouldmeanthatverylargebubbleswoulddominate.Thespeedoflightwouldpreventtheenergyreleasedinthecollisionsofthewallsoftheseverylargebubblesfrombeingspreadoverthevolumeoccupiedbythebubblesuntillongafterthec
24、onventionalbaryonandnu-cleosynthesiserasandmaybeeventhepresentday.itwouldseemthattheinflationaryuniversecouldnotaccountfortheratiosofbaryonstophotonsandheliumtohydrogenorforthe large-scalehomogeneityandisotropyoftheuniversetoday.Onemighthopethatintheintermediatesitua-tioninwhichthebubblenucleationra
25、teisofthesameorderas0,onemightgetsufficientinho-mogeneitiestoaccountforgalaxieswithouthavingtoomuchtobecompatiblewithobservation.WeshaHshowthatinsuchasituationasignificantfractionoftheuniversewouldcollapsetoformpri-mordialblackholes.Inthecaseofthegrand-unifiedphasetransitionthiswouldnotbeincon-flict
26、withobservationbecausetheblackholeswouldhavebeenoftheorderof10gandwouldhaveevaporatedveryearlyon.Ontheotherhand,anyblackholesproducedintheWeinberg-Salamphasetransitionwouldhavemassesoftheorderof10gandwouldstillbearoundatthepresenttime.Therequirementthattheirmassdensityshouldnotexceedthecriticaldensi
27、typlacesaverystringentlimitonthenumberthatcanhavebeenproduced.InturnthisimpliesthatthenucleationrateintheWeinberg-Salamphasetransitionmust26BUBBLECOLLISIONSINTHEVERYEARLYUNIVERSE2683havebeenhighcomparedtoH.Theproductionofblackholesbyfirst-orderphasetransitionshasbeenconsideredpreviouslybutonlyundert
28、heas-sumptionofsphericalsymmetrywhichwillnothold.ThereisalsoapossiblescenarioinwhichthebubblenucleationrateisnegligiblecomparedtoHuntilthetemperaturedropstosomevalueatwhichthepotentialbarriertobubbleformationbecomessmallorzero.Thissituationcouldoccur attheWeinberg-Salamphasetransitionbecauseofchiral
29、-symmetrybreaking.HoweveritseemsthatNHwouldriseonatimescalemuchshort-erthantheHubbletimeH.Thusthebubbleswouldnothaveexpandedveryfarbeforeevery-whereslidintothebroken-symmetryphase.Ontheotherhand,thechancesofcreatingprimordialblackholesatthegrand-unifiedphasetransitionseemrather betterbecauseonewould
30、notexpectNHtochangesorapidlycomparedtothemuchshorterexpansiontimescale.Theplanofthispaperisasfollows.InSec.IIwedescribetheformationofbubblesinfirst-orderphasetransitions.ThecaseofhighnucleationrateisconsideredinSec.III.Weshowthatthecol-lisionoftwobubblescanbereducedtothestudyofacomplexscalarfieldcou
31、pledtoaU(1)gaugefieldinonespaceandonetimedimensionandweuseanalyticapproximationstostudythedetailsofthecollision.Ourresultsarecomparedwithnu-mericalcalculationsinSec.IV.Theinflationary-universecaseoflownucleationrateisstudiedinSec.VandtheproductionofprimordialblackholesisconsideredinSec.VI.II.PHASETR
32、ANSITIONSWeshallconsideraHiggsscalarfield4whichisinthefundamentalrepresentationofaYang-Millsgroup6withapotentialA.TheeffectiveLagrangianforthescalarfieldwillhavetheformL(D4).(D4)V(4),whereDisthegauge-covariantderivativeandV(4)istheeffectivepotentialincludingquantumandthermalcorrections.Thedetailedfo
33、rmofV(4)willdependontheparticularmodelinques-tionbutinordertohaveageneraldiscussionweshallassumethatV(4)hastwolocalminima,anisolatedoneat4=-4&whichisinvariantunderthefullgroupandadegenerateoneat4=42whichisinvariantonlyunderasubgroupE.Infacttherewillbeawholedegeneratefamily6/Eofsuchlo-calminima.Inord
34、ertoobtainafirst-orderphasetransitionweshallassumeV(i)V(i)=&,whereehasthedimensionofenergyinunitsinwhichc=h=1.Weshallassumethattheeffec-tiveheightofthepotentialbarrierbetweenthetwominimaisgwhereg&ye .Thequantumtunnelingfromthesymmetricphase4tothebroken-symmetryphase42occursthroughabubble,asolutionin
35、Euclideanspaceoftheclassicalfieldequationsderivedfrom(1)whichhasasphericalregionofthenewphase42surroundedbytheoldphase4.Theradiusofthesphericalregionwillbeofordergvewherev=4i42.Thethicknessofthetransitionre-gionorbubblewallseparatingtheregions4and42isoforder,vg,andtheactionperunitareaofthewall03gv.T
36、herateofbubbleforma-tionperunitfour-volumeisN-ee4Bwhere8gveisthe differencebetweentheactionofthebubblesolutionandthatofasolutioninwhich4=4everywhere.Ifonelooksatthe situationinLorentzian,asopposedtoEuclidean,space,theideaisthat thequantumfluctuationsleadtoascalarfieldconfigu-rationonasurface t=t&whi
37、chis thesameasthatonanequatorialslicethroughthe Euclideanbubblesolution.Attimest&t&,thescalarfield,orratheritsexpectationvalue,evolvesaccordingtotheclas-sicalfieldequationswiththeinitialdatagivenatt=t&.Thebubblewallacceleratesuniformlyout-wardsintotheold4=4&phase.Thephase-transitionenergyreleasedbyt
38、heexpansionofthebubbleis convertedintokineticenergyofthewall.ThesolutionisinvariantunderLorentztransfor-mationsaboutthepointt=tiatthecenterofthebubble.ThescalarfieldinLorentzianspaceisinfacttheanalyticcontinuationofthebubblesolu-tioninEuclideanspace.TheexpectationvalueofthegaugefieldA&willbezeroduri
39、ngtheformationandexpansionofthebubble.ThegaugefieldwillaffectthebubbleonlythroughthequantumcorrectionsitproducesintheeffectivepotentialV(4).Eventually,however,thebubblewallwillcollidewiththatofanotherbub-ble.Ifthevalueof4insidethesecondbubblecorrespondstoadifferentmemberofthefamily6/Eofdegeneratemin
40、imaofV(4),thecollision2684S.W.HAWKING,I.G.MOSS,ANDJ.M.STEWART26ofthebubbleswillgenerateanonzeroexpectationvalueofA&whichinturnwillaffecttheevolutionof4.WeshallconsiderthisinmoredetailinSec.III.Weshallassumethattheuniversewasspatiallyhomogeneousandisotropicbefore thephasetransi-tionandcontainedpredom
41、inantlymasslessparti-cleswhoseinteractionscouldbeneglectedandwhichwereinthermalequilibrium.Thetotalen-ergydensityisthenp=g(&)&+e,30(4)whereg(T)isanumericalfactorwhichdependsonthenumberofbosonandfermiondegreesoffree-dom.Itisreasonabletoassumethatthefirsttermin(4)representingthethermalenergy,dominates
42、thesecondterm,representingthelatentheatatthecriticaltemperatureT,.BytheEinsteinequation,H=2(p+8),3mpwhereH=R/Ristheexpansionrate,mpisthePlanckmass,and8istheexpansionenergyperunitvolume.Astheuniverseexpands,TwillbeproportionaltoRand8willbeproportionaltoR.Frommeasurementsoftheexpansionrateandthedecele
43、rationparameterthepresentvalueg0/poislessthan10wherepois thepresentdensityofthemicrowavebackgroundradiation.Thustheexpansionenergycanbeneglectedatear-lytimessuchthatR/Ro&10III.HIGHNUCLEATIONRATEInthecaseofahighnucleationrate,theaverageseparationbetweenbubbles-Eismuchlessthantheexpansiontimeorthehori
44、zonsize-H.Inthissituationonecanignoretheexpan-sionandcurvatureoftheuniverseandtreatthebubblesasifinflatspacetime.AsinglebubbleisinvariantunderaLorentzgroupO(3,1)aboutitscenter.Whenoneconsiderstwobubbles,thelineadjoiningtheircentersisapreferredaxis(saythexaxis).Thesituationisthereforeinvariantunderan
45、O(2,1)groupconsistingofLorentzboostsintheyandzdirectionsandspatialrotationsintheyzplane.Itisconvenienttodefinenewcoordinatess,g,.8byt=scosh&/i,y=ssinhgsin8,z=ssinhgcos8,4geg4g,wheregisthecouplingconstantoftheYang-Millsgroupandwisthegeneratorof8normalizedsothate=1.Thesubgroup8isdefineduptoconjugacyin
46、EprovidedthategisnotinthecenterofG.TheexpectationvalueoftheYang-MillsfieldA&isinitiallyzeroandwillremainnearlysoas thebubblewallsacceleratetowardseachotheralongthexaxis. AftertheycollidetheYang-Millspo-tentialsAandA,willdevelopnonzeroexpectationvalueswhichwilllieintheU(1)subgroupgenerat-edbyw.Therei
47、sthusnolossofgeneralityinthetwo-bubblecollisioninrestrictingattentiontotheAbelianHiggsmodelconsistingofasinglecorn-plexscalarfieldPcoupledtoanelectromagneticfield.ThesymmetricphasewillcorrespondtoalocalminimumofV(P)at/=0andthebroken-symmetryphasewillcorrespondtoacircleofde-generateminimaatP=v.Ifonet
48、akestheoriginofthecoordinatetoliemidwaybetweenthebub-bles,theinitialdatawillsatisfytheconditionsP(x)=eP(x),A(x)=A(x),(9)Ag(x)=A,(x).Theconditions(9)willremaintrueatsubsequentsothats=tyz.InthesecoordinatestheAat-spacemetrictakestheformds+s(dg+sinhPd8)+dxThesolutionrepresentingthecollidingbubbleswillb
49、eindependentofthecoordinatesPand8.Inor-dertospecifytheinitialdataonthesurfaces=0forthetwo-bubblecollisiononemustfirstpickasuitablegaugefortheYang-MillsfieldsuchasA,=0orB&A&0.TheinitialdatafortheYang-MillsfieldarethenA&0,andBAz/Os=0.Theinitialdataforthescalarfieldconsistofthesuit-ablysmootheddatafort
50、wosinglebubblesseparat-edbyadistance2balongthexaxis.IngeneralthevaluesC&Land4zintheleft-andright-handbubbleswillcorrespondtodifferentmembersofthefamilyG/KofdegenerateminimaofV(4).ThenonecanfindaU(1)subgroupWofGsuchthat26BUBBLECOLLISIONSINTHEVERYEARLYUNIVERSE2685valuesofsbecausethefieldequationsarein