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A MODIFIED LEAST ACTION PRINCIPLE ALLOWING MASS CONCENTRATIONS FOR THE EARLY UNIVERSE

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Niveau: Supérieur, Doctorat, Bac+8
A MODIFIED LEAST ACTION PRINCIPLE ALLOWING MASS CONCENTRATIONS FOR THE EARLY UNIVERSE RECONSTRUCTION PROBLEM YANN BRENIER Abstract We address the early universe reconstruction (EUR) problem (as considered by Frisch and coauthors in [24]), and the related Zeldovich approximate model [39]. By substi- tuting the fully nonlinear Monge-Ampere equation for the linear Poisson equation to model gravitation, we introduce a modified mathematical model (”Monge-Ampere gravi- tation/MAG”), for which the Zeldovich approximation becomes exact. The MAG model enjoys a least action principle in which we can input mass concentration effects in a canonical way, based on the theory of gradient flows with convex potentials and some- what related to the concept of self-dual Lagrangians developped by Ghoussoub [27]. A fully discrete algorithm is introduced for the EUR problem in one space dimension. Introduction This paper addresses the early universe reconstruction (EUR) problem discussed by Frisch and coauthors in [24, 17]. We refer to these papers for the detailed physical back- ground of this important problem in cosmology. Here is a simplified mathematical for- mulation. We consider (for simplicity) a smooth closed bounded 3D domain D ? R3 and denote the space variable by x ? D. We are given two times t1 > t0 > 0, two probability mea- sures ?0(dx), ?1(dx) on D.

  • eulerian variables

  • monge-ampere equation

  • concave lipschitz

  • hilbert space

  • lipschitz convex

  • euler-poisson system

  • dt


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AMODIFIEDLEASTACTIONPRINCIPLEALLOWINGMASSCONCENTRATIONSFORTHEEARLYUNIVERSERECONSTRUCTIONPROBLEMYANNBRENIERAbstractWeaddresstheearlyuniversereconstruction(EUR)problem(asconsideredbyFrischandcoauthorsin[24]),andtherelatedZeldovichapproximatemodel[39].Bysubsti-tutingthefullynonlinearMonge-Ampe`reequationforthelinearPoissonequationtomodelgravitation,weintroduceamodifiedmathematicalmodel(”Monge-Ampe`regravi-tation/MAG”),forwhichtheZeldovichapproximationbecomesexact.TheMAGmodelenjoysaleastactionprincipleinwhichwecaninputmassconcentrationeffectsinacanonicalway,basedonthetheoryofgradientflowswithconvexpotentialsandsome-whatrelatedtotheconceptofself-dualLagrangiansdeveloppedbyGhoussoub[27].AfullydiscretealgorithmisintroducedfortheEURprobleminonespacedimension.IntroductionThispaperaddressestheearlyuniversereconstruction(EUR)problemdiscussedbyFrischandcoauthorsin[24,17].Werefertothesepapersforthedetailedphysicalback-groundofthisimportantproblemincosmology.Hereisasimplifiedmathematicalfor-mulation.Weconsider(forsimplicity)asmoothclosedbounded3DdomainDR3anddenotethespacevariablebyxD.Wearegiventwotimest1>t0>0,twoprobabilitymea-suresρ0(dx),ρ1(dx)onD.Welookforatime-dependentfamilyofprobabilitymeasuresρ(t,dx)onD(dependingcontinuouslyonthetimevariablet,withrespecttotheweakconvergenceofmeasures),interpolatingρ0andρ1,att=t0andt=t1respectively,andminimizingthefollowingactiont1ZZ(0.1)dtα(t)2ρ(t,dx)|v(t,x)|2+β(t)2|∇ϕ(t,x)|2dx,Dt0wherev=v(t,x)R3isavector-fieldandϕ=ϕ(t,x)ascalarfield,respectivelysubject:ot(0.2)tρ+∇(ρv)=0=1+Δϕ.HereΔ=2istheLaplaceoperator.Inthedefinitionoftheaction,αandβaretime-dependentscalingparametersrelatedtogeneralrelativity(GR).Following[24,17](caseofanEinstein-deSitteruniverse),weset:(0.3)α(t)=t3/4(t)=t1/43/2.p1
2Theformaloptimalityconditionsread2(0.4)v=v(t,x)=θ(t,x),∂tθ+|∇θ|+β2ϕ=0,α(t)22α2whichcanbealso(stillformally)written:(0.5)t(α2v)+α2(v∇)v=β2ϕ,∇×v=0,ro(0.6)t(α2ρv)+∇(α2ρvv)=β2ρϕ,∇×v=0.Theseequations,namely(0.2,0.6)are(uptothescalingfactorsα,βwhichcomefromgeneralrelativity)nothingbuttheEuler-Poissonequationsforapressure-less,curl-free,self-gravitatinggassubjecttoclassicalNewtongravitation.Theseequationscanalsobewritten,using”materialcoordinates”,(0.7)t(α(t)2tX(t,a))=β(t)2(ϕ)(t,X(t,a)),whereadenotesthematerialcoordinate,X(t,a)thepositionattimetofthemassparticlewithlabela.Inthecaseofcoefficients(0.3),wefind(0.8)2tt2tX(t,a)+tX(t,a)=(ϕ˜)(t,X(t,a)),3whereϕ˜(t,x)=ϕ(t,x)/tsatisfiesρ=1+tΔϕ˜.Remarkablyenough,atearlystaget<<1,thedensityfieldmustbeuniformlyequalto1(otherwisesolutionsareunbounded)and,evenmoresurprisingly,theaccelerationtermisdominatedbythevelocityterm,duetogeneralrelativity!(Insomesense,NewtonmodifiedbyEinsteinreturnstoAristoteles.)Asaconsequence,anamazinglysimpleapproximateformulawasproposedbyZeldovich[39]:(0.9)X(t,a)=atϕ˜0(a),whereϕ˜0isrelatedtothebehaviorofthedensityfieldρ(t,)atearlystagesρ(t,x)1ρ(0,x)=1,Δϕ˜0=lti0mt.TheZeldovichapproximateformulasuggestspossiblemassconcentrationsinfinitetime.Indeed,denotingbyΛthelargesteigenvalueoftheHessianmatrixD2ϕ˜0(a),foralla,weseethat,wheneverΛ>0,themapaX(t,a)isnolongerinvertibleatt1.Beyondtheconcentrationtime,therearemanypossibilitiesofextendingtheformulaandthisisstillacontroversialissuefromthephysicalviewpoint.Itdependsverymuchonwhetherornotwewanttopreventinterpenetrationofparticles.Ifwedoso,wearenaturallyleadtothemodelofadhesiondynamics,whereparticlesmergeaftercollisions,whichisthemostpossibledissipativebehaviorbeyondconcentrations.(See[5,23,24].)ThisissuecanbesimplyaddressedintermsofnonlinearhyperbolicPDEs[22].Indeed,givenaZeldovichsolutionXdefinedby(0.9),letusintroducethefieldu(t,x)implicitlydefined:yb(0.10)aX(t,a)u(t,X(t,a))==ϕ˜0(a),t