Magnetic fields and large-scale structure in a hot

arXiv:astro-ph/9710196v117Oct1997A&Amanuscriptno.(willbeinsertedbyhandlater)Yourthesauruscodesare:02(02.13.2;02.18.8;12.12.1)ASTRONOMYANDASTROPHYSICS1.2.2008Magneticfieldsandlarge-scalestructureinahotuniverse.II.MagneticfluxtubesandfilamentarystructureE.FloridoandE.BattanerDpto.FisicaTeoricaydelCosmos.UniversidaddeGranada.Spain.February1,2008Abstract.InpaperI,weobtainedanequationfortheevolutionofdensityinhomogeneitiesinaradiationdominateduniversewhentheyareaffectedbymagneticfields.Inthissecondpaperweapplythisequationtothecaseinwhichthesubjacentmagneticconfigurationisafluxtube.Forscalesoftheorderof1Mpcorlessthedifferentialequationiselliptical.Tosolveit,wehaveusedthenumericalmethodbasedon”SimultaneousOverRelaxation”,SOR,withChebyshevaccelerationandwehavetreatedtheproblemasaboundaryvalueproblem,whichrestrictsthepredictionabilityoftheintegration.Forlarge-scalefluxtubes,muchlargerthan1Mpc,theequationcanbeanalyticallyintegratedandnoassumptionaboutthefinalshapeormagnitudeoftheinhomogeneityisrequired.Inbothcasesweobtainanevolutionwhichdoesnotdifferverymuchfromlinearintime.Theinhomogeneityinthedensitybecomesfilamentary.Largescalestructures(≥10Mpc)areprobablyunaffectedbydamping,non-linearandamplificationmechanismsafterEquality,sothatthismodelprovidesatooltointerpretthepresentobservedlargescalestructure.Filamentsareveryfrequentlyfoundinthelarge-scalestructureintheUniverse.Itissuggestedherethattheycouldarisefromprimordialmagneticfluxtubes,thusprovidinganalternativehypothesisforitsinterpretation;inparticularweconsiderthecaseofthecoma-A1367supercluster,wherethemagneticfieldisknowntobehigh.Keywords:Magnetohydrodynamics–relativity–cosmology:large-scalestructureofUniverse1.GeneralequationsInpaperI,wepresentedthebasicequationsgoverningtheevolutionofthelargescaleenergydensitystructureintheprerecombinationuniverse,whentheeffectofmagneticfieldscannotbeignored.Inthelinearregime,themagneticfieldconfigurationremainstime-independentthroughoutthisera,onlygrowingwiththeexpansion.Thedegreeandthewaythisfieldstructureaffectstheevolutionofdensityinhomogeneitieswoulddependontheparticularpatternofthemagneticfieldlines.Wewillconsiderhereasmallcellinwhichthemagneticfieldhasthesimpleststructure:amagneticfluxtube.Differentsizesofthefluxtubewillbeconsidered,forsmall,intermediateandlargescales.Wewilldealwiththeintegrationofequation(86)ofpaperI¨δ−δ−X+13e−τ∇′2δ+2e−τm=0where:δistherelativeenergydensitycontrast,definedasδǫ/ǫ,whereǫistheenergydensityandδǫthedifferencebetweenitsvaluewithintheinhomogeneityanditsmeanvalueintheUniverse.Asweareconsideringrelativisticparticles,wealsohaveδ=δp/pwherepisthehydrostaticpressureoftherelativisticparticles(eitherphotonsoranyhotdarkmatterparticles).¨δisthesecondderivativewithrespecttothetime-likevariableτ.τisdefinedasτ=ln(te/t)wheretisthetimeandteisthelasttimeconsideredinthispaper,closetoequality.MorespecificallywehaveconsideredRe=10−5,orte=3.66×109s,beforetheacousticepoch.∇′istheLaplacianoperator,whenthespatialcoordinatesarex′iinsteadofxi,theusualcomovingcoordinates.2E.Floridoetal.:Magneticfieldsandlarge-scalestructureinahotuniverse.x′iisdefinedasx′i=(K/Re)xi,whereKistheconstantintheexpansionlawR2=Kt.ThevalueofKis2.73×10−20s−1andthereforex′i=2.73×10−15xi,wherexiaremeasuredinseconds.Therelationbetweenx′iandd,thelengthmeasuredinpresent-day–Mpcisx′i=0.28(d/Mpc).XcharacterizesthemagneticenergydensityandisdefinedasX=B2024πp0B0wouldbethepresentmagneticfieldifnosourceandnolossotherthanexpansionhadtakenplaceafterthetimeperiodconsideredhere.Becausethepost-recombinationepochisprobablyverycomplicated,concerningtheevolutionofmagneticfields,Bhasinpracticenorelationwithpresentmagneticfields.B0isexactlydefinedasB0=BR2.Wemeasuremagneticfieldstrengthsins−1,withtheequivalencebeing1Gauss=8.61×10−15s−1.p0istherelativisticparticlehydrostaticpressureatpresent.Itisdefinedasp0=pR4andwehavechosenthevaluep0=8.84×10−42s−2.malsocharacterizesthemagneticfieldconfigurationm=−12∇′·23n−∇′X
wheren=−B0·∇B04πp0+3∇′XTointegrateourbasicequationitisthennecessarytospecifythefield.Inthispaperweanalyzetheinfluenceofamagnetictubeflux,i.e.B0isgivenbyB0=(0,0,A)e−r22σ2(1)whererandσaremeasuredwiththesameunitasx′i.Forinstanceσ=1isequivalenttoacomoving-lengthof0.28Mpc.Forourpurposes,thesolutionverymuchdependsonthevalueofσandwecallitlargescaleifσ≫1,intermediatescale,ifσ≈1,andsmallscale,ifσ≪1.Intheparticularcaseofafluxtube:X=ae−r2σ2wherea=A224πp0andm=2aσ2e−r2σ22−r2σ2Forintegrating,itisbettertodefineanothertime-likevariableast′=e−τ=tteUsingthisdimensionlesstimethebasicequationbecomes:δ′′t′2+δ′t′−δ−X+13t′∇′2δ+2t′m=0(2)wherenowδ′andδ′′arefirstandsecondderivativeswithrespecttot′.Fromnowonwewillwritesimplytinsteadoft′andxiinsteadofx′iwithoutanyriskofconfusion.E.Floridoetal.:Magneticfieldsandlarge-scalestructureinahotuniverse.32.TheintegrationandboundaryconditionsEquation(2)isanellipticlinearsecondorderdifferentialequationwithvariablecoefficients.Fromthephysicalpointofview,itwouldbepreferabletocarryouttheintegrationasaninitialvalueproblem,tostartwithaninitialconfigurationandcalculatetheshapeanddensityofthecloudatthefina