Gravitational Lensing and Structural Stability of

arXiv:astro-ph/0510414v228Oct2005astro-ph/0510414LPT-ORSAY-05-59GravitationalLensingandStructuralStabilityofDarkMatterCausticRingsV.K.Onemli121DepartmentofPhysics,UniversityofCrete,GR-71003Heraklion,Greece2LaboratoiredePhysiqueTh´eorique,Universit´eParisXI,Bˆat.210,91405Orsay,FranceABSTRACTGravitationallensingbythedualcusp(A−3)catastrophesofthecolddarkmatter(CDM)causticringsatcosmologicaldistancesmayprovidethetantalizingopportunitytodetectCDMindirectly,anddiscriminatebetweenaxionsandweaklyinteractingmassiveparticles(WIMPs).TheinfallofCDMontoisolatedgalaxiessuchasourownproducesdiscretenumberofflowsandcausticsinthehaloCDMdistribution.CausticsareplaceswheretheCDMparticlesarenaturallyfocussed.IntheCDMcosmology,causticsinthedistributionofdarkmatterareplentifuloncedensityperturbationsenterthenonlinearregime.Ourfocusisuponthecausticringswhichareclosedtubeswhosecross-sectionisanellipticumbilic(D−4)catastrophewiththreedualcusps.Wecallthiscross-sectiona“tricusp.”Causticringsarelocatednearwheretheparticleswiththemostangularmomentuminagiveninflowreachtheirdistanceofclosestapproachtothegalacticcenterbeforegoingbackout.WereparameterizetheCDMflownearacausticring,andobtaintheringequationsinspace,assinglevaluedfunctionsofanangularvariableparameterizingthetricusp.Acausticringhasaspecificdensityprofile,aspecificgeometryand,therefore,preciselycalculablegravitationallensingsignaturesthatvaryonthecausticsurface,dependingonthelocationofthelineofsight.Wecalculatetheimagemagnificationofpoint-likesourcesasafunctionoftheangularparameterofthetricuspforthelineofsightsthatareparalleltothegalacticplaneofthecausticringandneartangenttothesurface.Themagnificationmonotonicallyincreases1asthelineofsightapproachestothecuspswhereitdivergesinthelimitofzerovelocitydispersion.Ifthevelocitydispersionissmallbutnonzero,thedivergenceiscutoff,becausethecausticsurfacegetssmearedoversomedistanceinspaceandthecuspsaresmoothedout.Thelensingeffectsarenolongerinfiniteatthecusps,butmerelylarge.Toestimatethemagnificationnumerically,fortheringsatcosmologicaldistancesandfortheringclosesttous(thefifthcausticringoftheMilkyWay),wechoosesamplepointsnearthecusps.Wefindthatthelensingeffectsarelargestattheoutercuspwhichliesin(or,near)thegalacticplane.Aroundtheoutercusp,thesurfaceiscurvedtowardthesidewithtwoextraflows.Wecallsuchasurfaceconcave.Inthelimitofzerovelocitydispersion,atasamplepointneartheoutercusp,wefind37%magnificationfortheCDMcausticringsatcosmologicaldistances.Thisisabout250timesstrongerthanthelensingeffectwepredictedfortheconcavefolds(consideringtheA2catastrophesenvelopingthegalaxy),and37timesstrongerthanthelargestlensingeffectwepredictedfortheCDMcaustics(consideringtheconvexfoldsoftherings)before.Inthepresenceoffinitevelocitydispersion,thelowerandupperboundsoftheeffectivevelocitydispersionsoftheaxionandWIMPflowsingalactichalosmaybeusedtoconstrainthelensingeffectsatthesmoothedcusps.Foracosmologicalaxioncausticring,wefindthatthemagnificationmayrangebetween3%and2800%attheoutercusp,andbetween2%and46%atthenon-planercusps.ForacosmologicalWIMPcausticring,ontheotherhand,weconstrainthemagnificationbetween3%and28%attheoutercusp,andbetween2%and5%atthenon-planercusps.BecausetheupperboundsforthemagnificationatthecuspsareobtainedconsideringtheminimumprimordialvalueofthevelocitydispersionsfortheCDMcandidatesinspacewherenosmallscalestructurehasformed,andalsobecausetheobserver’slineofsightmaynotexactlybeparalleltothegalacticplaneofthecausticringingeneral,theyshouldberegardedwithprecaution.Theimagesofextendedsourcesmayalsoshowdistortionsthatcanbeunambiguouslyattributedtolensingbydarkmattercaustics.Finally,wederivetheCatastropheFunctionofthetriaxialcausticrings.WeobtaintheflowequationsastheequilibriumpointsofthisCatastropheFunction.TheanalysisoftheStability(Hessian)Matrixshowthatthecausticringsarestructurallystable.PACSnumbers:98.80.Cq,4.62.+v2I.INTRODUCTIONAlargeamountofastronomicalevidenceindicatestheexistenceintheuniverseofmoregravitationallyinteractingmatterthanluminous(baryonic)matter.Mostofthematterintheuniverseisdark(23%ofthecomposition,accordingtothefirstyeardata[1]oftheWilkinsonMicrowaveAnisotropyProbe,isdarkmatter,whereasonly4.4%isordinarybaryonicmatter).Thestuffthatisresponsibleforholdinggalaxies,andclustersofgalaxiestogether,isapeculiarkindofmatterthatweneitherseenordetectbyanymeans.Thedetectiondilemmaofdarkmatterparticleshasbeenoccupyingphysicistsformanydecadesbynow,andthecasestillremainsopen.Thereare,however,significantreasonstobelievethatthedarkmatteroftheuniverseisconstituted,inlargefraction,bycollisionlessparticleswithverysmallprimordialvelocitydispersion.Suchparticlesarecalledcolddarkmatter(CDM).Theattribute“collisionless”indicatesthattheparticlesinteractsoweaklythat,practically,theymovepurelyundertheinfluenceofgravity.Hence,unlikeordinarybaryonicmatter,theyareundissipative.ThesmallprimordialvelocitydispersionallowsCDMtoclumpingalacticscales.ItisbelievedthatgalaxiesaresurroundedbyCDMwhichkeepsfallingintothegravitationalpotentialwellsofgalaxiesfromalldirections,andformshalosaroundtheirbary