Gravitational lensing by dark matter caustics

184 Pages·2003·7 MB·English

by  Onemli, Vakif K

GRAVITATIONALLENSINGBYDARKMATTERCAUSTICS
By
VAKIFK.ONEMLI
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
2003
ACKNOWLEDGMENTS
Thisdissertationisbasedonaresearchpaperongravitationallensingbydark
mattercaustics. CausticringsofdarkmatterwerediscoveredbyDr. PierreSikivie
whograciouslyallowedmetodrawextensivelyonhisworkasheservedasmyguide
duringthecourseoftheresearchthatyieldedthisdissertation.Iamdeeplyindebtedto
himforacceptingmeasoneofhisstudents.Ithasbeenagreatprivilegetoworkimder
hissupervisioninthisnewareaofresearchinparticleastrophysics. Thisattemptto
widentherangeofthediscussionandtomakethematerialmorepedagogicalwould
nothavebeenpossiblewithouthishelpandencouragement. Iwouldhketothank
himalsoforhispatienceandhiswillingnesstosharebothhisthoughtsandhistime
sogenerously.
Itisapleasuretoexpressmythankstotheothermembersofmysupervisory
committee:Dr.JamesPry,Dr.DavidGrossier,Dr.ZonganQiu,Dr.PierreRamond,
Dr.RichardP.Woodard,andDr.JohnYelton.Ihavebeenhonoredbytheirpresence
onmycommittee;andbenefittedfromtheircontributionsasteachers,collaborators,
andcoordinatorsthroughouttheyearsofmystayatthePhysicsDepartment.
IwouldUketothankDr.RichardP.Woodardforhisunfailinginterest,encour-
agementandsupport.Hewasalwaysavailabletoassistmewitheverykindofproblem
Ihad,fromphysicstopersonalones. Hisenthusiasmanddedicationinfluencedme
deeply.Forthe"cydonies,"Iamforevergrateful.
MythanksalsogototheothermembersoftheDepartmentofPhysicsofthe
UniversityofFloridaformakingmystudiesenjoyableaswellaseducational.Iwould
alsohketothankmyfriends,whosesupportIhavereliedon;andmycollaborators,
fortheinsightstheysharedwithme. Iamparticularlygratefultomygrandmother,
myparentsandmybrotherfortheirloveandsupport.
ii
TABLEOFCONTENTS
ACKNOWLEDGMENTS
•
ABSTRACT ^
CHAPTER
1 INTRODUCTION 1
2 COLDDARKMATTERCANDIDATES 16
2.1 WeaklyInteractingMassiveParticles(WIMPs) 16
2.2 Axions
3 COLDDARKMATTERCAUSTICS 42
3.1 CausticsinGeneral 42
3.2 CausticSurfaces 44
3.3 CDMMallinGalacticHalosandCaustics 46
3.4 OuterCausticSpheres 50
3.4.1 ExistenceofOuterCausticSpheres 50
3.4.2 DensityProfilesofOuterCausticSpheres 51
3.5 InnerCausticRings 54
3-5.1 TheExistenceofInnerCausticRings 54
3.5.2 AxiallySymmetricCausticRing 61
3.5.3 FlownearanAxiallySymmetricCausticRing 63
3.5.4 DifferentialGeometryofanAxiallySymmetricCausticRing . . 74
3.5.5 DensityProfilesofAxiallySymmetricCausticRings 86
4 GRAVITATIONALLENSINGBYDARKMATTERCAUSTICS .... 97
4.1 GeneralFormalism 100
4.2 Apphcations 106
4.2.1 LensingbyaConcaveFold 106
4.2.2 LensingbyaConvexFold 108
4.2.3 LensingbyaFoldwithZeroCurvature 112
4.2.4 LensingbyaCusp 113
5 CONCLUSIONS 119
A VACUAOFNON-ABELIANGAUGETHEORYANDINSTANTONS . . 122
A.l DescriptionofNon-AbeUanGaugeTheory 122
A.2 ClassicalInstantonSolutionsofNon-AbelianGaugeTheories 133
A.2.1 WindingNumber 141
iii
A.2.2 AnExplicitInstantonSolutionbyConstruction 145
A.3 CanonicalFormalismandQuantization 148
A.4 DiscoveringtheVacuumAngle 161
REFERENCES 170
BIOGRAPHICALSKETCH 175
iv
AbstractofDissertationPresentedtotheGraduateSchool
oftheUniversityofFloridainPartialFulfillmentofthe
RequirementsfortheDegreeofDoctorofPhilosophy
GRAVITATIONALLENSINGBYDARKMATTERCAUSTICS
By
VakifK.Onemli
December2003
Chair: PierreSikivie
MajorDepartment:Physics
Therearecompellingreasonstobehevethatthedarkmatteroftheuniverse
isconstituted,inlargepart,bynon-baryoniccollisionlessparticleswithverysmall
primordialvelocitydispersion.Suchparticlesarecalledcolddarkmatter(CDM).The
leadingcandidatesareaxionsandweaklyinteractingmassiveparticles(WIMPs).The
word "coUisionless"indicatesthattheparticlesaresoweaklyinteractingthatthey
movepurelyundertheinfluenceofgravity. GalaxiesaresurroundedbyCDMand
hence,becauseofgravity,CDMkeepsfaUingontogalaxiesfromalldirections. CDM
infallproducesadiscretenumberofflowsandcausticsinthehalosofgalaxies. There
aretwotypesofcausticsinthehalosofgalaxies:iimerandouter.Anoutercausticis
asimplefold{A2)catastrophelocatedonatopologicalsphereenvelopingthegalaxy.
Anirmercausticisaclosedtubewhosecross-sectionisanellipticumbilic (D^i)
catastrophe,withthreecusps.
Inspace,causticsaretheboundariesthatseparateregionswithdifferingnum-
bersofflows. Onesideofacausticsurfacehastwomoreflowsthantheother. The
densityofCDMparticlesbecomesverylargeasoneapproachesthecausticfromthe
sidewiththeextraflows.Darkmattercausticshavespecificdensityprofilesandthere-
forepreciselycalculablegravitationallensingproperties.Thisdissertationpresentsa
formalismthatsimplifiestherelevantcalculations,andapplyittofourspecificcases.
Inthefirstthreecases,thelineofsightistangenttoasmoothcausticsurface. The
curvatureofthesurfaceatthetangentpointispositive,negative,orzero. Inthe
fom:thcase,thelineofsightpassesnearacusp. Foreachcasewederivethemap
betweentheimageandsourceplanes. Insomecases,apointsourcehasmultiple
imagesandexperiencesinfinitemagnificationwhentheimagesmerge. Apromising
approachtorevealthelocationsofcausticsistoobservethedistortionsintheimages
ofextendedsources(suchasradiojets)passingbythecaustics.
vi
CHAPTER1
INTRODUCTION
Alargeamountofastronomicalevidenceindicatestheexistenceintheuniverse
ofmoregravitationallyinteractingmatterthanluminousmatter.Mostofthematter
intheuniverseisdark.Thestuffthatisresponsibleforholdinggalaxies(andclusters
ofgalaxies)togetherisapeculiarkindofmatterthatweneitherseenordetectby
anymeans. Thiscosmicmysterybeganpuzzhngastronomersinthe1930sandthe
casestillremainsopen.Forareviewofthesourcesanddistributionsofdarkmatter,
seeSikivie[1]andthereferencestherein.
In1932, Oort [2] studiedthemotionsofgalacticdiskstarsinthedirection
perpendiculartothedisk.Heappliedthevirialtheoremtothedistributionofvertical
starvelocities,andestimatedthedensityofthegalacticdiskaspdisk^1-2•10"^^^.
Whenthedensityduetothematterseeninstars,interstellargas,andstellarremnants
hkewhitedwarfsiscalculated,however,onefindsanestimateconsiderablylessthan
theabovevalue. ThisimpUestheexistenceofdarkmatterinthedisk. Becausethis
darkmatterisinthediskratherthanthehalo,itisexpectedtobedissipative,hence
baryonicinnature.
In1933,Zwicky[3]usedmeasurementsofthehneofsightvelocitiesofgalaxies
intheComaclustertoestimatethetotalmassoftheclusterusingthevirialtheorem:
{T)=-\{U), (1.1)
whereTandUarethekineticandpotentialenergyofthesystemrespectively.Bracket
()denotestimeaverage. Intheseestimates,however,itisimportanttoremember
thatthevirialtheoremisvalidforaclosedsysteminmechanicalequilibriiunandthat
1
—
itappliestothetimeaverageofthesystem. Whethertheobservedclusterssatisfy
theserequirementsisquestionable. Nevertheless,assumingtheapphcabilityofthe
theorem,andusingforthekineticenergyof galaxiesinthecluster:
{T)=^-N{mv'), (1.2)
thepotentialenergyofthekN{N—1)independentgalaxypairs:
(t/)=-iGAr(7V-1)M, (1.3)
andtakingiV~AT—1,onecanestimatethetotalmass:
M=N{m)~ . (1.4)
Bymeasuringthequantitiesrandv,itispossibletoestimateM.Theresultobtained
wasabout400timesthemassintheluminouspartsoftheclustergalaxies.
Duringtheseventies,therotationcurves(velocityasafunctionofradialdistance
fromthecenter)ofspiralgalaxiesbegantobemeasuredovermuchlargerdistances
thanbefore. Spiralgalaxiesarelargescalestructurescontainingbillionsofstars,
arrangedintheformofarotatingdiskwithacentralbulge.Assumingthatthestars
haveacircularorbitaroundthegalacticcenter,therotationvelocityofastarcanbe
calculatedbybalancingthecentrifugalandthegravitationalforces:
—GM{r)m v^.m
;:2 ^~r^ (1-5)
whereM(r)isthemassinteriortotheorbitradiusr,misthemassofthestarand
Urot('")istherotationvelocityatr. Here,weusedthefactthatforcyhndricallyor
sphericallysymmetricdistributions,theforcesduetothemasslyingoutsidetheorbit
3
compensateexactly.Wehave,therefore
Ifweassumethatthebulgeissphericallysymmetricwithaconstantdensityofp,
thenM(r) = ^irr^p. Hence,fortheinnermostpartofagalaxy,arotationcurve
wherev^ot~risexpected.Fortheoutsideofthegalaxy,wherer>discradius,M(r)
isequaltothetotalmassofthegalaxyM(r)=Mgaiaxy,whichisaconstant.Inthis
case yC^.-L.
= (1.7)
Thedatashow,however,thati>rot~constantforlarger.ThismeansthatM(r)~r,
whichimpliestheexistenceofahugeunseenmassextendingfarbeyondthevisible
region. Thedistributionofglobularstarclusterssuggestsasphericaldistribution.
Theoreticalmodelcalculationsshowthatpurediskgalaxieshaveatendencytobecome
bars(i.e.,withinthecentralnucleus,abar-hkestructureforms)[4]. Barredgalaxies
doexist,buttheyarerelativelyrare. Asphericalhaloofdarkmatterincreasesthe
stabilityofthepurediskstructure,producingaratioofbarredspiralsconsistent
withobservations[5].ConsiderationsofthestellardynamicsineUipticalgalaxiesalso
implythattheycontainasignificantfractionofdarkmatter[6].
Theseobservationsleadtothehypothesisofahaloofdarkmatterwhosedensity
isddm{r)~^atlarger.Thehalodistributionisusuallymodelledbythefunction
—
pdm{,r)^= Pdllrl(TO)2' y^-^ON)
1+
whereroiscalledthecoreradius.Forourowngalaxy,Urot—220 Tqisafewkpc,
andpdm(O) 10-23-^3.
4
AccordingtoWilkinsonMicrowaveAnisotropyProbe(WMAP)results [7-9],
thedensityparameter
oftheuniversehasthenumericalvalue =1.02±0.02.InEq. 1.9Pcisthecritical
densityforclosingtheuniverseandHqisthepresentHubbleexpansionrate.Mostof
theenergydensityoftheuniverseisintheformofvacuumenergy0,^=0.73±0.04.
Thedensityduetomatteris =0.27±0.04and0.17±0.01of isbaryonic:
Q,b=0.044±0.004.SincethedensityofluminousmatterisQium.<0.006,weconclude
thatsomebaryonsaredark.Recallthatthereisdarkmatterassociatedwiththedisk
ofagalaxy.Becauseitisinthediskratherthaninahalo,thisdarkmattermustbe
dissipativewhichpresumablymeansthatitisbaryonic(diskdarkmattermusthave
sufficientlystronginteractionstobeconcentratedinadiskbydissipatingitsenergy
whileconservingitsangularmomentmn).Likelyhidingplacesforthesedarkbaryons
maybeblackholes,coldwhitedwarfs,andbrowndwarfs(i.e.,starstoolowinmassto
burnbynuclearfusion).Objectsofthiskind,genericallycalledMACHOs(formassive
compacthaloobjects),havebeensearchedforbylookingforthegravitationallensing
ofbackgroundstarsbyMACHOsthathappentopassclosetothehneofsight. Many
ofthemhavebeendiscovered[10-13]. Westilldonotknowwhatmakesupmostof
themissingmassintheuniverse.
ThesuccessofnucleosynthesisinproducingtheprimordialabundancesofUght
elements[14],alsorequiresthat0.011<Qb<0.12. Studiesoflargescalestructure
formationsupporttheviewthatmostofthedarkmatterconsistsofnon-baryonic,
collisionlessbutgravitationallyinteractingparticlessuchasaxionsand/orWeakly
InteractingMassiveParticles(WIMPs) (likeneutralinos,photinos, higgsinos, etc.)
[14]. Theparticlesmusthavesmallprimordialvelocitydispersion;forthisreason,