garcia-bellido j. cosmology for string theorists, a crash course (2002-05)(93s)

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1、ARNOWITT-DESER-MISNERFORMALISMIntheArnowitt{Deser{Misnerformalismthefourdimensionalmetricgisparametrizedbythethree-metrichijandthelapseandshiftfunc-tionsNandNi,whichdescribetheevolutionoftime-likehypersurfaces,2ijg00=N+hNiNj;g0i=gi0=Ni;gij=hij:(1)Theactionforthein

2、 atonscalareldwithpotentialV()intheADMformalismhastheform23Zp11S=d4xg4R(@)2V()522223Zp11=d4xNh4(3)R+KKijK2+()2jiV()5;22ij2ji(2)where2=8G=8=M2isthegravitationalcouplingdeningthePPlanckmassandisthescalar-eld'sconjugatemomentum1i=(_Nji):

3、(3)NVerticalbarsdenotethree-space-covariantderivativeswithconnectionsderivedfromh;(3)Risthethree-spacecurvatureassociatedwiththeijmetrichij,andKijistheextrinsiccurvaturethree-tensor1Kij=(Nijj+Njjih_ij);(4)2Nwhereadotdenotesdierentiationwithrespecttothetimecoordinat

4、e.Thetracelesspartofatensorisdenotedbyanoverbar.Inparticular,11pK=KKh;K=Ki=N@lnh:(5)ijijijiijit3NThetraceKisageneralizationoftheHubbleparameter,aswillbeshownbelow.VariationoftheactionwithrespecttoNandNiyieldstheenergyandmomentumconstraintequationsrespectively2(

5、3)R+KKijK2+22=0;ij3(6)Kj2K+2=0:ijj3jijiijequations,whichcanbeseparatedintothetraceandtracelessparts011312K_NiK=Nji+NB@(3)R+KKij+K2+TCA;(7)jijiij4422K_iNkKi+NiKkNkKi=Nji+1Njkijjjkjkjjjkjj3jkj(8)+N(3)Ri+KKi2Ti:jjjVariationwithrespecttogi

6、vesthescalar-eld'sequationofmotion1i1jiji@V(_Nji)KNjiji+=0:(9)NN@Theenergydensityonaconstant-timehypersurfaceis12ji=()+ji+V();(10)2andthestressthree-tensor231T=+h4()2jkV()5:(11)ijjijjijjk2Itisextremelydiculttosolvethesehighlynonlinea

7、rcoupledequa-tionsinacosmologicalscenariowithoutmakingsomeapproximations.Theusualapproachistoassumehomogeneityoftheeldstogiveabackgroundsolutionandthenlinearizetheequationstostudydeviationsfromspatialuniformity.Thesmallnessofcosmicmicrowavebackgroundanisotropiesgive

8、ssomejusticationforthisperturbativeapproachatleastinourlocalpartoftheUniverse.However,thereisnoreasontobelieveitwillbevalidonmuchl