TheFutureofQuantumCosmology 


S..W..Hawking
. 
DepartmentofAppliedMathematics 
andTheoreticalPhysics, 
UniversityofCambridge, 
SilverStreetCambridgeCBEW, 
UnitedKingdom. 


August9 


Abstract 


ThisisatranscriptofalecturegivenbyProfessorS..W..HawkingfortheNATOASIconference. 
ProfessorHawkingistheLucasianProfessorattheUniversityofCambridgeEngland. 


Inthislecture..IwilldescribewhatIseeastheframeworkforquantumcosmology..onthebasis 
ofMtheoryIshalladoptthenoboundaryproposalandshallarguethattheAnthropicPrinciple 
isessential..ifoneistopickoutasolutiontorepresentouruniversefromthewholezooofsolutions 
allowedbyMtheory. 


Cosmologyusedtoberegardedasapseudoscience..anareawherewildspeculationwasunconstrainedbyanyreliableobservations
Wenowhavelotsandlotsofobservationaldata..andagenerally 
agreedpictureofhowtheuniverseisevolving. 


Butcosmologyisstillnotaproperscience..inthesensethat..asusuallypracticed..ithasno 
predictivepowerOurobservationstellusthepresentstateoftheuniverse..andwecanrunthe 
equationsbackwardtocalculatewhattheuniversewaslikeatearliertimesButallthattellsusis 
thattheuniverseisasitisnowbecauseitwasasitwasthenTogofurther..andbearealscience, 
cosmologywouldhavetopredicthowtheuniverseshouldbeWecouldthentestitspredictionsagainst 
observation..likeinanyotherscience. 


Thetaskofmakingpredictionsincosmology..ismademoredicultbythesingularitytheorems 
thatRogerPenroseandIproved. 


TheUniversemusthavehadabeginningif 


EinsteinsGeneralTheoryofRelativityiscorrect 
Theenergydensityispositive 
) 
Theuniversecontainstheammountofmatterweobserve 


TheseshowedthatifGeneralRelativitywerecorrect..theuniversewouldhavebegunwithasingularity
Ofcourse..wewouldexpectclassicalGeneralRelativitytobreakdownnearasingularity, 
whenquantumgravitationale
ectshavetobetakenintoaccountSowhatthesingularitytheorems 


..
email..SWHawkingdamtpcamacuk 
1 



arereallytellingusisthattheuniversehadaquantumorigin..andthatweneedatheoryofquantum 
cosmology..ifwearetopredictthepresentstateoftheuniverse. 
Atheoryofquantumcosmology..hasthreeaspects. 


QuantumCosmology 


LocaltheoryMTheory 


BoundaryconditionsNoboundaryproposal 
) 
Anthropicprinciple 


TherstisthelocaltheorythattheeldsinspacetimeobeyThesecondistheboundaryconditions 
fortheeldsIshallarguethattheanthropicprincipleisanessentialthirdelement. 


Asfarasthelocaltheoryisconcernedthebest..andindeedtheonly..consistentwayweknow 
todescribegravitationalforcesiscurvedspacetimeThetheoryhastoincorporatesupersymmetry, 
becauseotherwisetheuncanceledvacuumenergiesofallthemodeswouldcurlspacetimeintoatiny 
ballThesetworequirementsseemedtopointtosupergravitytheories..atleastuntilBut 
thenthefashionchangedsuddenlyPeopledeclaredthatsupergravitywasonlyalowenergye
ective 
theory..becausethehigherloopsprobablydiverged..thoughnoonewasbraveorfoolhardy	enough 
tocalculateaneightloopdiagramInstead..thefundamentaltheorywasclaimedtobesuperstrings, 
whichwerethoughttobenitetoallloopsButitwasdiscoveredthatstringswerejustonemember 
ofawiderclassofextendedobjects..calledpbranesItseemsnaturaltoadopttheprincipleofpbrane 
democracy. 


P..branedemocracy 


Weholdthesetruthsasselfevident: 


	
AllPbranesarecreatedequal 


AllpbranesarecreatedequalYetforp....thequantumtheoryofpbranesdivergesforhigher 
loops. 


Ithinkweshouldinterprettheseloopdivergencesnotasabreakdownofthesupergravitytheories, 
butasabreakdownofnaiveperturbationtheoryIngaugetheories..weknowthatperturbation 
theorybreaksdownatstrongcouplingInquantumgravity..theroleofthegaugecouplingisplayed 
bytheenergyofaparticleInaquantumloop..oneintegratesoverallenergiesSoonewouldexpect 
perturbationtheorytobreakdown. 


Ingaugetheories..onecanoftenusedualitytorelateastronglycoupledtheory..whereperturbation 
theoryisbad..toaweaklycoupledone..inwhichitisgoodThesituationseemstobesimilarin 
gravity..withtherelationbetweenultravioletandinfraredcuto
s..intheAdSCFTcorrespondence. 
Ishallthereforenotworryaboutthehigherloopdivergences..anduseelevendimensionalsupergravity 
asthelocaldescriptionoftheuniverseThisalsogoesunderthenameofMtheory..forthosethat 
rubbishedsupergravityinthesanddontwanttoadmititwasbasicallycorrectInfact..asIshall 
show..itseemstheoriginoftheuniverseisinaregimeinwhichrstorderperturbationtheoryisa 
goodapproximation. 


ThesecondpillarofquantumcosmologyisboundaryconditionsforthelocaltheoryThereare 
threecandidates..theprebigbangscenario..thetunnellinghypothesis..andthenoboundaryproposal. 


2 



BoundaryconditionsforQuantumCosmology 


Prebigbangscenario 


Tunnellinghypothesis 
) 
Noboundaryproposal 


Theprebigbangscenarioclaimsthattheboundaryconditionissomevacuumstateintheinnite 
pastBut..ifthisvacuumstatedevelopsintotheuniversewehavenowitmustbeunstableAndifitis 
unstable..itwouldntbeavacuumstate..anditwouldnthavelastedaninnitetimebeforebecoming 
unstable. 


Thequantumtunnelinghypothesisisnotactuallyaboundaryconditiononthespacetimeelds, 
butontheWheelerDewittequationHowever..theWheelerDewittequationactsontheinnite 
dimensionalspaceofalleldsonahypersurfaceandisnotwelldenedAlso..the..or, 
splitisputtingapartthatwhichGod..orEinstein..hasjoinedtogetherInmyopinion..therefore, 
neithertheprebangscenario..norquantumtunnelinghypothesis..areviable. 


Todeterminewhathappensintheuniverse..weneedtospecifytheboundaryconditions..ontheeld 
congurations..thataresummedoverinthepathintegralOnenaturalchoicewouldbemetricsthat 
areasymptoticallyEuclidean..orasymptoticallyAntideSitterThesewouldbetherelevantboundary 
conditionsforscatteringcalculations..whereonesendsparticlesinfrominnityandmeasureswhat 
comesbackout. 


However..theyarenottheappropriateboundaryconditionsforcosmologyWehavenoreasonto 
believetheuniverseisasymptoticallyEuclideanorAntideSitterEvenifitwere..wearenotconcerned 
aboutmeasurementsatinnity..butinaniteregionintheinteriorForsuchmeasurements..there 
willbeacontributionfrommetricsthatarecompact..withoutboundaryTheactionofacompact 
metricisgivenbyintegratingtheLagrangian. 


Thus..itscontributiontothepathintegraliswelldenedBycontrast..theactionofanoncompact, 
orsingular..metricinvolvesasurfacetermatinnity..oratthesingularityOnecanaddanarbitrary 
quantitytothissurfacetermItthereforeseemsmorenaturaltoadoptwhatJimHartleandIcalled, 
thenoboundaryproposalThequantumstateoftheuniverseisdenedbyaEuclideanpathintegral 
overcompactmetricsInotherwords..theboundaryconditionoftheuniverse..isthatithasno 
boundary. 


NoBoundaryProposal 


Theboundaryconditionoftheuniverseis 


	
thatithasnoboundary 


TherearecompactReechiatmetricsofanydimension..manywithhighdimensionalmoduli 
spacesThuselevendimensionalsupergravity..orMtheory..admitsaverylargenumberofsolutions 
andcompacticationsTheremaybesomeprinciple..thatwehaventyetthoughtof..thatrestrictsthe 
possiblemodelstoasmallsubclass..butitseemsunlikelyThusIbelievethatwehavetoinvokethe 
AnthropicPrincipleManyphysicistsdisliketheAnthropicPrincipleTheyfeelitismessyandvague, 
thatitcanbeusedtoexplainalmostanything..andthatithaslittlepredictivepowerIsympathize 
withthesefeelings..buttheAnthropicPrincipleseemsessentialinquantumcosmologyOtherwise, 
whyshouldweliveinafourdimensionalworldandnoteleven..orsomeothernumberofdimensions. 
Theanthropicansweristhattwospatialdimensionsarenotenoughforcomplicatedstructures..like 
intelligentbeings. 


3 



Ontheotherhand..four..ormore..spatialdimensionswouldmeanthatgravitationalandelectric 
forceswouldfallo
fasterthantheinversesquarelawInthissituation..planetswouldnothavestable 
orbitsaroundtheirstar..norwouldelectronshavestableorbitsaroundthenucleusofanatomThus 
intelligentlife..atleastasweknowit..couldexistonlyinfourdimensionsIverymuchdoubtwewill 


ndanonanthropicexplanation. 
TheAnthropicPrinciple..isusuallysaidtohaveweakandstrongversionsAccordingtothestrong 
AnthropicPrinciple..therearemillionsofdi
erentuniverses..eachwithdi
erentvaluesofthephysical 
constantsOnlythoseuniverseswithsuitablephysicalconstantswillcontainintelligentlifeWiththe 
weakAnthropicPrinciple..thereisonlyasingleuniverseButthee
ectivecouplingsaresupposedto 
varywithposition..andintelligentlifeoccursonlyinthoseregionsinwhichthecouplingshavetheright 
valuesEventhosewhorejecttheStrongAnthropicPrinciple..wouldacceptsomeWeakAnthropic 
argumentsForinstance..thereasonstarsareroughlyhalfwaythroughtheirevolution..isthatlife 
couldnothavedevelopedbeforestars..orhavecontinuedwhentheyburntout. 


Whenonegoestoquantumcosmologyhowever..andusesthenoboundaryproposal..thedistinction 
betweentheWeakandStrongAnthropicPrinciplesdisappearsThedi
erentphysicalconstantsare 
justdi
erentmodulioftheinternalspace..inthecompacticationofMtheory..orelevendimensional 
supergravityAllpossiblemoduliwilloccurinthepathintegralovercompactmetricsBycontrast, 
ifthepathintegralwasovernoncompactmetrics..onewouldhavetospecifythevaluesofthemoduli 
atinnityEachsetofmoduliatinnitywoulddeneadi
erentsuperselectionsectorofthetheory, 
andtherewouldbenosummationoversectorsItwouldthenbejustanaccidentthatthemoduliat 
innityhavethoseparticularvalues..likefouruncompactieddimensions..thatallowintelligentlife. 
ThusitseemsthattheAnthropicPrinciplereallyrequiresthenoboundaryproposal..andviceversa. 


OnecanmaketheAnthropicPrincipleprecise..byusingBayesstatistics. 


BayesianStatistics 


P
matter

..
jGalaxy	/ 


	
PGalaxyj
matter

..
	P
matter

..
) 


Onetakestheaprioriprobabilityofaclassofhistories..tobetheetotheminustheEuclidean 
action..givenbythenoboundaryproposalOnethenweightsthisaprioriprobability..withthe 
probabilitythattheclassofhistoriescontainintelligentlifeAsphysicists..wedontwanttobe 
drawnintotothenedetailsofchemistryandbiology..butwecanreckoncertainfeaturesasessential 
prerequisitesoflifeasweknowitAmongthesearetheexistenceofgalaxiesandstars..andphysical 
constantsnearwhatweobserveTheremaybesomeotherregionofmodulispacethatallowssome 
di
erentformofintelligentlife..butitislikelytobeanisolatedislandIshallthereforeignorethis 
possibility..andjustweighttheaprioriprobabilitywiththeprobabilitytocontaingalaxies. 


4 



EuclideanFourSphere 




ds

d


Hsin

Hd

sind

) 



North 
Pole 
South 
Pole 
) 
Thesimplestcompactmetric..thatcouldrepresentafourdimensionaluniverse..wouldbethe 
productofafoursphere..withacompactinternalspaceBut..theworldweliveinhasametricwith 
Lorentziansignature..ratherthanapositivedeniteEuclideanoneSoonehastoanalyticallycontinue 
thefourspheremetric..tocomplexvaluesofthecoordinates. 


Thereareseveralwaysofdoingthis. 


AnalyticalContinuationtoaClosedUniverse 


Analyticallycontinue
equator
it 



Equator 

s = 0 

ds

dt




Hcosh

Htd

sind

) 


) 
Onecananalyticallycontinuethecoordinate....as
equator
itOneobtainsaLorentzianmetric, 
whichisaclosedFriedmannsolution..withascalefactorthatgoeslikecoshHt	Sothisisaclosed 
universe..thatstartsattheEuclideaninstanton..andexpandsexponentially. 


5 




Analyticalcontinationofthe 
fourspheretoanopenuniverse 


Anaylticallycontinueit..i. 
) 


ds

dt


1 


HsinhHt	

d

sinh

d

) 



However..onecananalyticallycontinuethefoursphereinanotherwayDeneti..andi. 
ThisgivesanopenFriedmannuniverse..withascalefactorlikesinhHt	. 


Penrosediagramofanopenanalyticalcontinuation 



) 
Thusonecangetanapparentlyspatiallyinniteuniverse..fromthenoboundaryproposalThe 
reasonisthat..oneisusingasatimecoordinatethehyperboloidsofconstantdistance..insidethelight 
coneofapointindeSitterspaceThepointitself..anditslightcone..arethebigbangoftheFriedmann 
model..wherethescalefactorgoestozeroButtheyarenotsingularInstead..thespacetimecontinues 
throughthelightconetoaregionbeyondItisthisregionthatdeservesthenamethePreBigBang 
Scenario..ratherthanthemisguidedmodelthatcommonlybearsthattitle. 


IftheEuclideanfourspherewereperfectlyround..boththeclosedandopenanalyticalcontinuations 
wouldinateforeverThiswouldmeantheywouldneverformgalaxiesAperfectlyroundfoursphere 
hasaloweraction..andhenceahigheraprioriprobabilitythananyotherfourmetricofthesame 
volumeHowever..onehastoweightthisprobabilitywiththeprobabilityofintelligentlife..whichis 
zeroThuswecanforgetaboutroundspheres. 


Ontheotherhand..ifthefoursphereisnotperfectlyround..theanalyticalcontinuationwillstart 
outexpandingexponentially..butitcanchangeoverlatertoradiationormatterdominated..andcan 
becomeverylargeandat. 


ThismeansthereareequalopportunitiesfordimensionsAlldimensions..inthecompactEuclidean 
geometry..startoutwithcurvaturesofthesameorderButintheLorentziananalyticalcontinuation, 
somedimensionscanremainsmall..whileothersinateandbecomelargeHowever..equalopportunities 
fordimensionsmightallowmorethanfourtoinateSo..wewillstillneedtheAnthropicPrinciple..to 
explainwhytheworldisfourdimensional. 


Inthesemiclassicalapproximation..whichturnsouttobeverygood..thedominantcontribution 
comesfrommetricsnearinstantonsThesearesolutionsoftheEuclideaneldequationsSoweneed 
tostudydeformedfourspheresinthee
ectivetheoryobtainedbydimensionalreductionofeleven 


6 



dimensionalsupergravity..tofourdimensionsTheseKaluzaKleintheoriescontainvariousscalar 
elds..thatcomefromthethreeindexeld..andthemodulioftheinternalspaceForsimplicity..Iwill 
describeonlythesinglescalareldcase. 


EnergyMomentumTensor 




T
..

..




g
..




V	] 
) 


Thescalareld....willhaveapotential..V	Inregionswherethegradientsofaresmall, 
theenergymomentumtensorwillactlikeacosmologicalconstant..GV..whereGisNewtons 
constantinfourdimensionsThusitwillcurvetheEuclideanmetric..likeafoursphere. 


However..iftheeldisnotatastationarypointofV..itcannothavezerogradienteverywhere. 
ThismeansthatthesolutioncannothaveO	symmetry..liketheroundfoursphereThemostit 
canhaveisO	symmetryInotherwords..thesolutionisadeformedfoursphere. 


OInstantons 


ds

d

b

	d

sin

d

) 


b f 
s = 0 ssmax
max 

) 
OnecanwritethemetricofanO	instanton..intermsofafunction..b	Herebistheradius 
ofathreesphereofconstantdistance....fromthenorthpoleoftheinstantonIftheinstantonwere 
aperfectlyroundfoursphere..bwouldbeasinefunctionofItwouldhaveonezeroatthenorth 
pole..andasecondatthesouthpole..whichwouldalsobearegularpointofthegeometryHowever..if 
thescalareldatthenorthpoleisnotatastationarypointofthepotential..itwillvaryoverthefour 
sphereIfthepotentialiscarefullyadjusted..andhasafalsevacuumlocalminimum..itispossibleto 
obtainasolutionthatisnonsingularoverthewholefoursphereThisisknownastheColemanDe 
Luciainstanton. 


However..forgeneralpotentialswithoutafalsevacuum..thebehaviorisdi
erentThescalareld 
willbealmostconstantovermostofthefoursphere..butwilldivergenearthesouthpoleThisbehavior 
isindependentofthepreciseshapeofthepotential..andholdsforanypolynomialpotential..andfor 
anyexponentialpotential..withanexponent..a..lessthenThescalefactor..b..willgotozeroatthe 
southpole..likedistancetothethirdThismeansthesouthpoleisactuallyasingularityofthefour 
dimensionalgeometryHowever..itisaverymildsingularity..withanitevalueofthetraceKsurface 
term..onaboundaryaroundthesingularityatthesouthpoleThismeanstheactionsofperturbations 


7 



ofthefourdimensionalgeometryarewelldened..despitethesingularityOnecanthereforecalculate 
theuctuationsinthemicrowavebackground..asIshalldescribelater. 


ThedeepreasonbehindthisgoodbehaviorofthesingularitywasrstseenbyGarrigaHedimensionallyreduced
vedimensionalEuclideanSchwarzschild..alongthedirection..togetafour 
dimensionalgeometry..andascalareld. 



) 
Theseweresingularatthehorizon..inthesamemannerasatthesouthpoleoftheinstantonIn 
otherwords..thesingularityatthesouthpole..canbejustanartefactofdimensionalreduction..and 
thehigherdimensionalspace..canbenonsingularThisistruequitegenerallyThescalefactor..b, 
willgolikedistancetothethird..whentheinternalspace..collapsestozerosizeinonedirection. 


WhenoneanalyticallycontinuesthedeformedspheretoaLorentzianmetric..oneobtainsanopen 
universe..whichisinatinginitially. 


Hawking..TurokInstanton 


Singularity 
t 
Instanton 
Region II 
Region I: Open Universe 
Surfaces of homogeneity 
Null surface 
) 
Onecanthinkofthisasabubbleinaclosed..deSitterlikeuniverseInthisway..itissimilartothe 
singlebubbleinationaryuniverses..thatoneobtainsfromColemanDeLuciainstantonsThedi
erence 
is..theColemanDeLuciainstantons..requiredcarefullyadjustedpotentials..withfalsevacuumlocal 
minimaButthesingularHawkingTurokinstantonwillworkforanyreasonablepotentialTheprice 


8 



onepaysforageneralpotential..isasingularityatthesouthpoleIntheanalyticallycontinued 
Lorentzianspacetime..thissingularitywouldbetimelike..andnakedOnemightthinkthatanything 
couldcomeoutofthisnakedsingularity..andpropagatethroughthebigbanglightcone..intotheopen 
inatingregionThusonewouldnotbeabletopredictwhatwouldhappenHowever..asIalready 
said..thesingularity..atthesouthpoleofthefoursphere..issomildthattheactionsoftheinstanton, 
andofperturbationsaroundit..arewelldened. 


Thisbehaviorofthesingularity..meansonecandeterminetherelativeprobabilitiesoftheinstanton
..andofperturbationsarounditTheactionoftheinstantonitselfisnegative..butthee
ectof 
perturbationsaroundtheinstantonistoincreasetheactionThatis..tomaketheactionlessnegative. 
Accordingtothenoboundaryproposal..theprobabilityofaeldcongurationisetominusitsaction. 
Thusperturbationsaroundtheinstanton..havealowerprobability..thantheunperturbedbackground. 
Thismeansthatthemorequantumuctuationsaresuppressed..thebiggertheuctuation..asone 
wouldhopeThisisnotthecasewithsomeversionsofthetunnelingboundarycondition. 


HowwelldothesesingularinstantonsaccountfortheuniverseweliveinThehotbigbangmodel 
seemstodescribetheuniverseverywell..butitleavesunexplainedanumberoffeatures. 


ProblemsofaHotBigBang 


Isotropy 


Amplitudeofuctuations 


	
Densityofmatter 


Vacuumenergy 


Thereistheoverallisotropyoftheuniverse..andtheoriginandspectrumofsmalldeparturesfrom 
isotropyThentheresthefactthatthedensitywassucientlylowtolettheuniverseexpandtoits 
presentsize..butnotsolowthattheuniverseisemptynowAndthefactthatdespitesymmetry 
breaking..theenergyofthevacuumiseitherexactlyzero..oratleast..verysmall. 


InationwassupposedtosolvetheproblemsofthehotbigbangmodelItdoesagoodjobwith 
therstproblem..theisotropyoftheuniverseIftheinationcontinuesforlongenough..theuniverse 
wouldnowbespatiallyat..whichwouldimplythatthesumofthematterandvacuumenergieshad 
thecriticalvalue. 


Butination..byitself..placesnolimitsontheotherlinearcombinationofmatterandvacuum 
energies..anddoesnotgiveananswertoproblemtwo..theamplitudeoftheuctuationsThesehave 
tobefedin..asnetuningsofthescalarpotential..VAlso..withoutatheoryofinitialconditions..it 
isnotclearwhytheuniverseshouldstartoutinatingintherstplace. 


TheinstantonsIhavedescribedpredictthattheuniversestartsoutinaninating..deSitter 
likestateThustheysolvetherstproblem..thefactthattheuniverseisisotropicHowever..there 
aredicultieswiththeotherthreeproblemsAccordingtothenoboundaryproposal..theapriori 
probabilityofaninstanton..isetotheminustheEuclideanactionButiftheReechiscalarispositive, 
asislikelyforacompactinstantonwithanisometrygroup..theEuclideanactionwillbenegative. 


Thelargertheinstanton..themorenegativewillbetheaction..andsothehighertheapriori 
probabilityThusthenoboundaryproposal..favourslargeinstantonsInaway..thisisagood 
thing..becauseitmeansthattheinstantonsarelikelytobeintheregimewherethesemiclassical 
approximationisgoodHowever..alargerinstantonmeansstartingatthenorthpolewithalower 
valueofthescalarpotential..VIftheformofVisgiven..thisinturnmeansashorterperiodof 
inationThustheuniversemaynotachievethenumberofefoldings..neededtoensure
matter

. 
isneartoonenow. 


9 



InthecaseoftheopenLorentziananalyticalcontinuationconsideredhere..thenoboundaryaprioriprobabilitieswouldbeheavilyweightedtowards

matter


Obviously..insuchanempty 
universe..galaxieswouldnotform..andintelligentlifewouldnotdevelopSoonehastoinvokethe 
anthropicprinciple. 


Ifoneisgoingtohavetoappealtotheanthropicprinciple..onemayaswelluseitalsoforthe 
othernetuningproblemsofthehotbigbangThesearetheamplitudeoftheuctuationsandthe 
factthatthevacuumenergynowisincrediblynearzeroTheamplitudeofthescalarperturbations 
dependsonboththepotentialanditsderivativeBut..inmostpotentialsthescalarperturbationsare 
ofthesameformasthetensorperturbations..butarelargerbyafactorofabouttenForsimplicity, 
IshallconsiderjustthetensorperturbationsTheyarisefromquantumuctuationsofthemetric, 
whichfreezeinamplitudewhentheircomovingwavelengthleavesthehorizonduringination. 


Thus..thespectrumofthetensorperturbationwillberoughlyoneoverthehorizonsize..inPlanck 
unitsLongercomovingwavelengths..willleavethehorizonearlierduringinationThusthespectrum 
ofthetensorperturbations..atthetimetheyreenterthehorizon..willslowlyincreasewithwavelength, 
uptoamaximumofoneoverthesizeoftheinstanton. 


Amplitudeofperturbationswhenthey 
comeintothevisibleuniverse 


litud

e

Amp

1 
size of instanton 


) 
Thetimeatwhichthemaximumamplitudereentersthehorizon..isalsothetimeatwhich. 
beginstodropbelowoneTherearetwocompetinge
ectsOneistheaprioriprobabilityfromthe 
noboundaryproposal..whichwantstomaketheinstantonslargeTheotheristheprobabilityofthe 
formationofgalaxiesThisrequiressucientinationtokeepomeganeartoone..andasucient 
amplitudeoftheuctuationsBoththesefavoursmallinstantonsizesWherethebalanceoccurs 
dependsonwhetherweweightwiththedensityofgalaxiesperunitpropervolume..orbythetotal 
numberofgalaxiesIfweweightwiththepresentproperdensityofgalaxies..theprobabilitydistribution 
for..wouldbesharplypeakedatabout

3 
. 


Predictionsfor. 


Weightingwithproperdensityofgalaxies..	1 


	
Weightingwithtotalnumberofgalaxies..

Thisistheminimumvalue..thatwouldgiveonegalaxyintheobservableuniverse..andclearlydoes 


0 


Time 
Time when O < 1 


notagreewithobservationOntheotherhand..onemightthinkthatoneshouldweightwithafactor 
proportionaltothetotalnumberofgalaxiesintheuniverseInthiscase..onewouldmultiplythe 
probabilitybyafactore 

n 
..wherenisthenumberofefoldingsduringinationThiswouldleadto 
thepredictionthat..whichseemstobeconsistentwithobservation..asIshalldiscuss. 


SofarIhaventtakenintoaccounttheanthropicrequirement..thatthecosmologicalconstantisvery 
smallnowElevendimensionalsupergravitycontainsathreeformgaugeeld..withafourformeld 
strengthWhenreducedtofourdimensions..thisactsasacosmologicalconstantForrealcomponents 
intheLorentzianfourdimensionalspace..thiscosmologicalconstantisnegativeThusitcancancelthe 
positivecosmologicalconstant..thatarisesfromsupersymmetrybreakingSupersymmetrybreaking 
isananthropicrequirementOnecouldnotbuildintelligentbeingsfrommasslessparticlesThey 
wouldyapart. 


Unlessthepositivecontributionfromsymmetrybreakingcancelsalmostexactlywiththenegative 
fourform..galaxieswouldntform..andagain..intelligentlifewouldntdevelopIverymuchdoubtwe 
willndanonanthropicexplanationforthecosmologicalconstant. 


Intheelevendimensionalgeometry..theintegralofthefourformoveranyfourcycle..oritsdual 
overanysevencycle..havetobeintegers. 


Thismeansthatthefourformisquantized..andcannotbeadjustedtocancelthesymmetry 
breakingexactlyInfact..forreasonablesizesoftheinternaldimensions..thequantumstepsinthe 
cosmologicalconstantwouldbemuchlargerthantheobservationallimitsAtrst..Ithoughtthiswas 
asetbackfortheideatherewasananthropicallycontrolledcancellationofthecosmologicalconstant. 
Butthen..IrealizedthatitwaspositivelyinfavourThefactthatweexist..showsthattheremustbe 
asolutiontotheanthropicconstraints. 


Butthefactthatthequantumstepsinthecosmologicalconstant..aresolarge..meansthatthis 
solution..isprobablyuniqueThishelpswiththeproblemsoflow..orexactlyone..Idescribed 
earlierIftherewereacontinuousfamilyofsolutions..thestrongdependenceoftheEuclideanaction, 
andtheamountofination..onthesizeoftheinstanton..wouldbiastheprobability..eithertothelowest 
..orThiswouldgiveeitherasinglegalaxyinanotherwiseemptyuniverse..orauniversewith 


exactlyone. 
Butifthereisonlyoneinstantonintheanthropicallyallowedrange..thebiasingtowardslarge 
instantonshasnoe
ectThus
matterand

couldbesomewhereintheanthropicallyallowedregion, 
thoughitwouldbebelowthe
matter


line..iftheuniverseisoneoftheseopenanalytical 
continuationsThisisconsistentwiththeobservations. 


TheredelipticregionisthethreesigmalimitsofthesupernovaobservationsTheblueregionis 
fromclusteringobservations..andthepurpleisfromtheDopplerpeakinthemicrowaveTheyseem 
tohaveacommonintersection..onorbelowthe
total
line. 


1 



ComparisonofSupernovaMicrowaveBackgroundandClusteringregions 


this Supernova 
Microwave background 
Galaxiescannot 
form inregion 
Matter density 
Vacuum energy 
Anthropic 
line 

Clustering 

) 
Assumingthatonecanndamodelthatpredictsareasonable..howcanwetestitbyobservation. 
ThebestwayisbyobservingthespectrumofuctuationsinthemicrowavebackgroundThisisa 
verycleanmeasurementofthequantumuctuations..abouttheinitialinstantonHowever..there 
isanimportantdi
erencebetweenthenonsingularColemanDeLuciainstantons..andthesingular 
instantonsIhavedescribed. 


AsIsaid..quantumuctuationsaroundtheinstantonarewelldened..despitethesingularity. 
PerturbationsoftheEuclideaninstantonhaveniteaction..ifandonlytheyobeyaDiricheletboundary 
conditionatthesingularityPerturbationmodesthatdontobeythisboundarycondition..willhave 
inniteaction..andwillbesuppressedTheDiricheletboundaryconditionalsoarises..ifthesingularity 
isresolvedinhigherdimensions. 


WhenoneanalyticallycontinuestoLorentzianspacetime..theDiricheletboundaryconditionimplies 
thatperturbationsreectatthetimelikesingularity. 


Thishasane
ectonthetwopointcorrelationfunctionoftheperturbationsItisverysmallfor 
thedensityperturbations..butcalculationsbyHertogandTurok..indicateasignicantdi
erencefor 
gravitationalwaves..ifislessthanone. 



) 
2 



Thepresentobservationsofthemicrowaveuctuations..arecertainlynotsensitiveenoughtodetect 
thise
ectButitmaybepossiblewiththenewobservationsthatwillbecominginfromthemap 
satellitein..andthePlancksatelliteinThusthenoboundaryproposal..andthesingular 
instanton..arerealscienceTheycanbefalsiedbyobservation. 


Iwillnishonthatnote. 


3