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arXiv:astro-ph/9803172v116Mar1998astro-ph/9803172UFIFT-HEP-98-2CRETE-98-11CosmologicalDensityPerturbationsFromAQuantumGravitationalModelOfInflationL.R.Abramo†andR.P.Woodard‡DepartmentofPhysicsUniversityofFloridaGainesville,FL32611USAN.C.Tsamis∗DepartmentofPhysicsUniversityofCreteHeraklion,GR-71003GREECEABSTRACTWederivetheimplicationsforanisotropiesinthecosmicmicrowavebackgroundfollowingfromamodelofinflationinwhichabarecosmo-logicalconstantisgraduallyscreenedbyaninfraredprocessinquantumgravity.ThemodelpredictsthattheamplitudeofscalarperturbationsisAS=(2.0±.2)×10−5,thatthetensor-to-scalarratioisr≈1.7×10−3,andthatthescalarandtensorspectralindicesaren≈.97andnT≈−2.8×10−4,respectively.Bycomparingthemodel’spowerspectrumwiththeCOBE4-yearRMSquadrupole,themassscaleofinflationisdeterminedtobeM=(.72±.03)×1016GeV.Atthisscalethemodelproducesabout108e-foldingsofinflation,soanotherpredictionisΩ=1.PACSnumbers:04.60.-m,98.80.Cq†e-mail:abramo@phys.ufl.edu∗e-mail:tsamis@physics.uch.gr‡e-mail:woodard@phys.ufl.edu1IntroductionTheviewthattheveryearlyuniverseunderwentaperiodofinflationatsomelargemassscaleMisstronglysupportedbythehomogeneityandisotropyofthecosmicmicrowavebackground,andbytheabsenceofrelicssuchasmagneticmonopoles[1].Anenormousvarietyofmodelshavebeenproposedtoimplementinflation[2,3,4,5,6,7,8,9],allofwhichinvolveadynamicalscalardegreeoffreedominsomeform.Anothercommonfeatureofthesemodelsisthatthecosmologicalconstantmustbefinetunedsothatinflationcanend.Manymodelsrequireadditionalfinetuninginordertomakeinfla-tionlastlongenoughandinorderthatquantumfluctuationsneartheendofinflationcangenerateaplausiblespectrumofprimordialdensityfluctuations.Recentlyamodelhasbeenproposedinwhichfundamentalscalarsplaynoroleandforwhichthecosmologicalconstantisnotfinetunedtozero[10].Indeed,inflationbeginsinthismodelfornootherreasonthanthatthecosmologicalconstantisnotunreasonablysmall.Itendsduetothesecu-laraccumulationofgravitationalbindingenergybetweenvirtualgravitonswhichhavebecometrappedinthesuperluminalexpansionofspacetimeandarethereforeunabletorecombine.Thiseffectisuniquetoparticlesthatareeffectivelymasslessandyetnotconformallyinvariant,theonlydefinitivelyknownexampleofwhichisthegraviton[11].Theprocessisslowbecausegravityisaweakinteraction,evenatGUTscales.However,itmusteven-tuallynullthebarecosmologicalconstantsincetheeffectiscoherentandpersistsforaslongasinflationdoes.Becausethemechanismoperatesinthefarinfrared,itcanbestudiedperturbativelyusingquantumgeneralrelativity:L=116πG(R−2Λ)√−g+counterterms,(1)withoutregardtoultravioletdivergencesormodificationsatthePlanckscale.WedidthisonthemanifoldT3×ℜ,inthepresenceofahomogeneousandisotropicstateforwhichtheexpectationvalueofthemetrichastheform:h0|gμν(t,~x)dxμdxν|0i=−dt2+e2b(t)d~x·d~x,(2)withinitialconditionsb(0)=0and˙b(0)=H≡qΛ3.Theresultis[12]:b(t)=Ht(1+...)+12ln1−1729ǫ2(Ht)3+....(3)1Thesmallparameterisǫ≡GΛ3πandtheneglectedtermsturnouttobeirrelevantuptoandincludingthebreakdownofperturbationtheory.Theeffectistwo-loopbecauseitrequiresonelooptoproduce0-pointenergythroughsuperadiabaticamplification[13]andanotherloopforittoself-interact.Perturbationtheorybreaksdownwhentheargumentofthelogarithmin(3)approacheszero,atwhichtimehigherloopeffectsarestillnegligible[12].Weaccordinglyestimatethenumberofe-foldingsofinflationas:Npert=917213ǫ−23.(4)Onecanalsousetheperturbativeresulttoshowthatinflationendssuddenlyoverthecourseofaboutfivee-foldings[12].Ofcourseperturbationtheorycannotbetrustedpastthetimewhenloopeffectsbecomecomparabletotheclassicalresult.Onewaytoevolvebeyondthispointisbyusingeffectivefieldequationsfortheexpectationvalueofthemetricgμν.Thesecanalwaysbewrittenastheclassicalfieldequationsplusaquantum-inducedstresstensorTμν[g]:Rμν−12gμνR+gμνΛ=8πGTμν[g].(5)ComputingTμν[g]foranarbitrarymetricisasdifficultassolvingquantumgravity.However,forthepurposesofcosmologyonelosesnothingbyrestrict-ingtothestresstensorofaneffectivescalarφ[g]whichisitselfanon-localfunctionalofthemetric:Tμν[g]=∂μφ[g]∂νφ[g]−gμν12gρσ∂ρφ[g]∂σφ[g]+P(φ[g]).(6)Whenspecializedtoahomogeneousandisotropicmetric(2),theevolutionequationisindependentofthepotential:¨b=−4πGdφdt!2.(7)Theinducedstresstensoristhereforecompletelyspecifiedbygivingtheeffec-tivescalarasafunctionalofthemetric.Thepotentialcanbereconstructedasafunctionoftimefromthesolutionb(t)[14]:P=18πG¨b(t)+3˙b2(t)−3H2,(8)2andthenexpressedasafunctionofthescalar.Carefulconsiderationofthephysicalmechanism,plusgeneralprinciplessuchascoordinateinvarianceandcausality,alongwiththerequirementofreproducingtheknownperturbativeresult(3),haveledustothefollowingans¨atzfortheeffectivescalar[14]:φ=1√8πGln1−4348ǫ212R12R2!#.(9)Here2−1istheretardedGreen’sfunctionassociatedwiththescalarcovariantd’Alembertian:2≡1√−g∂μgμν√−g∂ν.(10)Ourans¨atzfortheinducedscalarφ[g]isnotunique.However,itcanbeshownthatthebehaviorbeforethebreakdownofperturbationtheoryisuniversalandthatthepost-inflationaryevolutiondependsonlyuponhowmanyfactorsofRstandimmediatelytotherightoftheoutermost2−1[14].ForonesuchfactorofR,theasymptoticlatetimebehavioroftheeffectiveHubblecon