Quantum creation and inflationary universes a crit

arXiv:gr-qc/0007037v117Jul2000Quantumcreationandinflationaryuniverses:acriticalappraisalD.H.CouleSchoolofMathematicsUniversityofPortsmouth,MercantileHouseHampshireTerrace,PortsmouthPO12EGAbstractWecontrastthepossibilityofinflationstartinga)fromtheuni-verse’sinceptionorb)fromanearliernon-inflationarystate.Nei-thercaseisidealsincea)assumesquantummechanicalreasoningisstraightforwardlyapplicabletotheearlyuniverse;whilecaseb)re-quiresthatasingularitystillbepresent.Further,inagreementwithVachaspatiandTrodden[1]caseb)canonlysolvethehorizonprob-lemifthenon-inflationaryphasehasequationofstateγ4/3,soexcludingradiationormasslessscalarfielddominatedcases.Otheralternativemodels,suchasthesmoothbranchchangeinthepre-bigbangmodel,haverelatedproblemsofrequiringhomogeneityovernon-causallyconnectedlargescales.PACSnumbers:04.20,98.8011IntroductionThestandardbigbangmodel(SBB)modelhasanumberofpuzzles,particularytheso-calledhorizonandflatnessproblems,thatarebelievedtorequireexplanationandnotjustbeacceptedbyfiat,seeeg.[2,3].However,formatterthatobeysthestrongenergycondition,soincludingradiationordustsources,thereisbynecessityaninitialsingularity,seeforexample[4].Onsuchasingularsurfaceonecanimposeanysuitabledataonewantstogiverisetoouruniverseafteritevolvesintothefuture[5,6].Sostrictlyspeakingthepresenceofasingularityobviatesanytalkofbeingunnaturalsincewhatisnaturalatasingularsurface.Asomewhatrelatedargumenthasbeenmadethatatthesingularsurfaceeverythingcannaturallybecomessmoothbyentropyarguments[7].ThemoregenerallyacceptedsolutiontosolvingtheproblemsoftheSBBistoallowamattersourcethatviolatesthestrongenergyconditionandwhichcangiverisetoarapidlyexpandingregime,socalledinflatione.g.[2,3].Oneimaginesthatasmallhomogeneousinitialpatchcanbetakenandmassivelyexpandedtogiveourpresentuniverse.However,alowerboundonthesizeofthispatchhasbeenmadebyVachaspatiandTrodden(VT)andwhichcouldactuallypreventthehorizonproblembeingsolvedbyinflation[1].Wewillconsidertheseclaimsindetail.Throughoutthisarticlewewilltakeso-calledchaoticinflationasthepro-totypeforinflation[2,3].ItcanbecausedbyascalarfieldwithpotentialV(φ)sufficientlydisplacedfromitsminimum.However,onestillneedstoknowthedistributionfortheinitialconditionsofthisscalarfield.Onenat-uralmeasureforaclassicalsystemisthecanonicalone[8].Butusingthismeasure,andevenafterassuminganhomogeneousscalarfield,theproba-bilityforinflationisfoundtobeambiguous:aninfinityofbothinflationaryandnon-inflationarysolutions[9].Further,thepredictedflatnessbecomesarbitraryifthereisanupperboundinthepotentialV(φ),comingfromper-hapsaconformalanomalycorrectionorfromahigherderivativeR2termintheaction[10].Oneobviouslimitationtotheseargumentsisthattheclassicalequationswillfailasthequantumgravityepochisencounteredneartheinitialsingu-larity.Theinitialconditionsshouldthenbesuppliedbyasuitablequantummeasure.OneapproachusingtheWheeler-DeWitt(WDW)equationaimstoachievethis,seeeg.[2,3].Althoughdifferentresultsareobtaineddependinguponwhichboundaryconditionsarebelievedcorrect.Thetwomostcommon2approachesarethe“noboundary”ofHartle-Hawking[11]andTunnellingones[12].ThereissomedisagreementastowhetherinflationisactuallypredictedwiththeHartle-Hawkingboundarycondition[13,14],buttheTunnelingoneseemsmorecertain.Wejustnotethatthequantummeasurecangiveamorecertainpredictioncomparedtoapurelyclassicalapproachbecausetheinitialsizeoftheuniverseisnecessarily“small”whenquantummechanicsisrele-vant.Aninitial“quantumofenergy”canbedistributedamongstthekineticandpotentialenergyofthescalarfield.Iftheboundaryconditionsallowthepotentialtodominatethenarapidinflationaryexpansionensues.Roughlyspeaking,largepositiveenergyisproducedbyproducinglargevolumeswithnegativegravitationalenergy,whichcansubsequentlycreatelargeamountsofmatter.Thepossibilityofinflationisofcoursedependentontherebeingpresentasuitablescalarpotential.Thisisputinbyhandandnotapredic-tionofthequantumcosmologicaltheory.Theactualsourceofthispotentialormoregenerallyareasonfortheinitialviolationofthestrongenergycon-ditionneedstobeunderstoodfromtheactualquantumgravitationaltheoryonceknown,perhapseventuallyfromstringtheory.Untilthistimeinflationcannotbethoughtofasatrueexplanationoftheuniverse’sexistencebutratherapossiblemechanismforamplifyingaquantumcreateduniversethatisunnaturallysmall.Sincethecreationmechanismisnotknownitcouldjustbethecasethatthismechanismitselfcouldgiveareasonforexplainingthevariouspuzzlesandallowingalargerinitialuniverse.RecallthepresentvisibleuniverseextrapolatedbacktothePlancktimeisasize∼10−3cm,afactor∼1030largerthanthePlancklength∼10−33cm.Inotherwordsthecreationmechanismisassumedtobedrivenbyquantumeffectsthataremoreprobableonlyoversmallscales,butsincenotionsoftimeandenergyareonlypropertieswithinouruniverseitisn’tclearwhetherquantummechanicsconstrainstheinitialcreationinsuchaway.2InflationstartingfrominceptionGiven,thatsomequantumprocessdoeslaydowntheinitialconditionsonecantrytojustifythelikelihoodofinflationbeingpresent.ThisisusuallydoneatthePlanckepochwherethesemiclassicalequationsarefirstvalidalthoughasmentionedthisshouldon