Thisbookisaneditedversionofthelecturesdeliveredduringthe1stAegean SummerSchoolonCosmology,heldonSamosisland,Greece,inSeptember 21-29,2001,andorganizedjointlybytheDepartmentofMathematics,U- versity of the Aegean and the Department of Physics, National Technical UniversityofAthens. Cosmology,thescienceoftheuniverse,standsatthecrossroadsofmany ?eldsofphysicsandmathematicsandpresentsuswithchallengingproblems of many forms. Although there are by now many textbooks discussing the subjectatmanylevels,itistruethatnosinglebookhasthecharacteristics wehadinmindwheneditingthisvolume. Wehavetriednottoproducea proceedingsvolumebutmoreamultiauthoredtextbookwhichcouldserveas areferencesourceofcurrentideasincosmology. Webelievethisbookcovers atanintroductorylevelmostoftheissueswhichareconsideredimportant inmoderncosmologicalresearchandcanbereadbyagraduatestudentor researcherwhowishestoacquireareasonableknowledgeofcosmologythat will,wehope,continuetobeofvalueforyearstocome. The 1st Aegean School on Cosmology, and consequently this book, - camepossiblewiththekindsupportofmanypeopleandorganizations. We received ?nancial support from the following sources and this is gratefully acknowledged: the Municipality of Karlovassi, the North Aegean Regional Secretariat, the Prefecture of Samos, the Ministry of the Aegean, and the NationalBankofGreece. TheadministrativesupportoftheSchoolwastakenupwithgreatcare byMrs. EvelynPappaandMantoKatsianiandwewouldliketothankthem bothfortheirkinde?ortstoresolvemanyissueswhicharosebefore,during andaftertheSchool. WeacknowledgethehelpofMr. NectariosBenekoswho designedandmaintainedthewebsiteoftheSchool. Last,butnotleast,wearegratefultothesta?ofSpringer-Verlag,resp- siblefortheLectureNotesinPhysics,whoseabilitiesandhelpcontributed greatlytothe?neappearanceofthisbook. Karlovassi,Samos, SpirosCotsakis March2002 EleftheriosPapantonopoulos TableofContents PartI HistoryandOverview 1 IsNatureGeneric? SpirosCotsakis,PeterG. L. Leach...3 1. 1 Introduction...3 1. 2 PrinciplesofCosmologicalModelling...4 1. 2. 1 Spacetimes...4 1. 2. 2 TheoriesofGravity...5 1. 2. 3 MatterFields...6 1. 3 Cosmologies...6 1. 4 CosmologicalProblems...8 1. 4. 1 TheSingularityProblem...8 1. 4. 2 TheProblemofCosmicTopology...9 1. 4. 3 TheProblemofAsymptoticStates...9 1. 4. 4 GravityTheoriesandtheEarlyUniverse...11 1. 5Outlook ...12 References...14 2 EvolutionofIdeasinModernCosmology AndreasParaskevopoulos...16 2. 1 Introduction...16 2. 2 TheBeginningsofModernCosmology(1917-1950)...17 2. 3 Cosmology1950-1970:HotBigBang, SingularitiesandQuantumApproach...20 2. 4 Cosmology1970-Thisbookisaneditedversionofthelecturesdeliveredduringthe1stAegean SummerSchoolonCosmology,heldonSamosisland,Greece,inSeptember 21-29,2001,andorganizedjointlybytheDepartmentofMathematics,U- versity of the Aegean and the Department of Physics, National Technical UniversityofAthens. Cosmology,thescienceoftheuniverse,standsatthecrossroadsofmany ?eldsofphysicsandmathematicsandpresentsuswithchallengingproblems of many forms. Although there are by now many textbooks discussing the subjectatmanylevels,itistruethatnosinglebookhasthecharacteristics wehadinmindwheneditingthisvolume. Wehavetriednottoproducea proceedingsvolumebutmoreamultiauthoredtextbookwhichcouldserveas areferencesourceofcurrentideasincosmology. Webelievethisbookcovers atanintroductorylevelmostoftheissueswhichareconsideredimportant inmoderncosmologicalresearchandcanbereadbyagraduatestudentor researcherwhowishestoacquireareasonableknowledgeofcosmologythat will,wehope,continuetobeofvalueforyearstocome. The 1st Aegean School on Cosmology, and consequently this book, - camepossiblewiththekindsupportofmanypeopleandorganizations. We received ?nancial support from the following sources and this is gratefully acknowledged: the Municipality of Karlovassi, the North Aegean Regional Secretariat, the Prefecture of Samos, the Ministry of the Aegean, and the NationalBankofGreece. TheadministrativesupportoftheSchoolwastakenupwithgreatcare byMrs. EvelynPappaandMantoKatsianiandwewouldliketothankthem bothfortheirkinde?ortstoresolvemanyissueswhicharosebefore,during andaftertheSchool. WeacknowledgethehelpofMr. NectariosBenekoswho designedandmaintainedthewebsiteoftheSchool. Last,butnotleast,wearegratefultothesta?ofSpringer-Verlag,resp- siblefortheLectureNotesinPhysics,whoseabilitiesandhelpcontributed greatlytothe?neappearanceofthisbook. Karlovassi,Samos, SpirosCotsakis March2002 EleftheriosPapantonopoulos TableofContents PartI HistoryandOverview 1 IsNatureGeneric? SpirosCotsakis,PeterG. L. Leach...3 1. 1 Introduction...3 1. 2 PrinciplesofCosmologicalModelling...4 1. 2. 1 Spacetimes...4 1. 2. 2 TheoriesofGravity...5 1. 2. 3 MatterFields...6 1. 3 Cosmologies...6 1. 4 CosmologicalProblems...8 1. 4. 1 TheSingularityProblem...8 1. 4. 2 TheProblemofCosmicTopology...9 1. 4. 3 TheProblemofAsymptoticStates...9 1. 4. 4 GravityTheoriesandtheEarlyUniverse...11 1. 5Outlook ...12 References...14 2 EvolutionofIdeasinModernCosmology AndreasParaskevopoulos...16 2. 1 Introduction...16 2. 2 TheBeginningsofModernCosmology(1917-1950)...17 2. 3 Cosmology1950-1970:HotBigBang, SingularitiesandQuantumApproach...20 2. 4 Cosmology1970-1990:Chaotic,In?ationary, QuantumandAlternative...22 2. 5ConclusionsandOutlook ...25 References...26 VIII TableofContents PartII MathematicalCosmology 3ConstraintsandEvolutioninCosmology YvonneChoquet-Bruhat,JamesW. York...29 3. 1 Introduction...29 3. 2 MovingFrameFormulas...30 3. 2. 1 FrameandCoframe...30 3. 2. 2 Metric...31 3. 2. 3 Connection...31 3. 2. 4 Curvature ...32 3. 3 (n+1)-SplittingAdaptedtoSpaceSlices ...33 3. 3. 1 De?nitions...33 3. 3. 2 StructureCoe?cients...34 3. 3. 3 SplittingoftheConnection ...

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