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Induced gravity with Higgs potential [Elektronische Ressource] : elementary interactions and quantum processes = Induzierte Gravitation mit Higgspotential / presented by Nils Manuel Bezares Roder

267 pages
Research Group for Cosmology and Quantum GravitationInstitute of Theoretical PhysicsFaculty of Natural SciencesInduced GravitywithHiggs PotentialElementary Interactions and Quantum ProcessesDoctoral Thesisfor obtaining the academical degree of Doctor of natural sciences(Doctor rerum naturalium)of the Faculty of Natural Sciences of the University of Ulmpresented byNils Manuel Bezares Roderfrom Mexico CityUlm, 2010RefereeProf. Dr. Frank SteinerProf. Dr. Werner BalserArbeitsgruppe für Kosmologie und QuantengravitationInstitut für Theoretische PhysikFakultät für NaturwissenschaftenInduzierte GravitationmitHiggspotentialElementare Wechselwirkungen und QuantenprozesseDissertationzur Erlangung des Doktorgrades Dr. rer. nat.(Doctor rerum naturalium)der Fakultät für Naturwissenschaften der Universität Ulmvorgelegt vonNils Manuel Bezares Roderaus Mexiko-StadtUlm, 2010Erstgutachter: Prof. Dr. Frank Steiner, Institut für Theoretische PhysikZweitgutachter: Prof. Dr. Werner Balser, Institut für Angewandte AnalysisAmtierender Dekan: Prof. Dr. Axel Groß, Institut für Theoretische ChemieTag der Promotion: 29. Juni 2010SummaryThis work is intended to first serve as introduction in fundamental subjects of physics in order to be thenable to review the mechanism of symmetry breakdown and its essential character in physics.
Voir plus Voir moins

ResearchGroupforCosmologyandQuantumGravitation
ysicsPhTheoreticalofInstituteSciencesNaturalofacultyF

InducedvityGra

with

ElementaryInteractionsHiggsPandotentialQuantumProcesses

ThesisDoctoralforobtainingtheacademicaldegreeofDoctorofnaturalsciences
naturalium)rerum(DoctoroftheFacultyofNaturalSciencesoftheUniversityofUlm

bypresentedNilsfromManuelMeBezarxicoesCityRoder
2010Ulm,

eeReferSteinerFrank.DrProf.Prof.Dr.WernerBalser

ArbeitsgruppefürKosmologieundQuantengravitation
ysikPhTheoretischefürInstitutNaturwissenschaftenfürakultätF

vitationGraInduziertemitElementareWechselwirkungenHiggspotentialundQuantenprozesse

DissertationzurErlangungdesDoktorgradesDr.rer.nat.
naturalium)rerum(DoctorderFakultätfürNaturwissenschaftenderUniversitätUlm

vorgelegtvon
NilsausManuelMexikBezaro-StadtesRoder
2010Ulm,

Erstgutachter:Prof.Dr.FrankSteiner,InstitutfürTheoretischePhysik
Zweitgutachter:Prof.Dr.WernerBalser,InstitutfürAngewandteAnalysis
AmtierenderDekan:Prof.Dr.AxelGroß,InstitutfürTheoretischeChemie
TagderPromotion:29.Juni2010

Summary

Thisworkisintendedtofirstserveasintroductioninfundamentalsubjectsofphysicsinordertobethen
abletoreviewthemechanismofsymmetrybreakdownanditsessentialcharacterinphysics.Itisdiscussed
howthismechanismisindeedfundamentalforabetterunderstandingofphysicsinallitsfields,especially
inrelationtoelementary-particleandcondensed-matterphysics,includingsuperconductivityinitsusualas
wellasdualformwhichisinvestigatedwithingluodynamics.Thisworkthenintroducestheconceptof
scalar–tensortheoriesofgravitybasedonBergmann–WagonermodelswithaHiggspotential.Themain
physicalcontextaimedistheproblemofDarkMatterandDarkEnergywhichareaddressedinthisworkin
anattempttobetterunderstandthosephenomenologicalsubjectsofastrophysicsandcosmology.
Ontheonehand,thereisgravitation.Itisespeciallyrelevantforastrophysicalphenomenaandforanalyses
ofthestructureoftheUniverseasawhole.Withinthiscontext,wehaveDarkMatterasanespecially
relevantconcept.DarkMatteristhenamegiventomostofthematterintheUniverse,anditisnecessaryto
reproducemeasuredastrophysicaldatawithinstandarddynamics.Thelatterassumes(electromagnetically
uncoupled)dynamicalmatterwhichmaystill(andshould)beproducedinexperimentsinordertoclarify
itsnature.DarkMattercomprisesca.90percentofthewholematterdensity,whereasmatterdensityonly
comprisesaboutonethirdofthetotalenergydensitythereis.Almostallotherdensity(hencearound2/3
ofallenergydensity)iscalledDarkEnergy.Thisenergyactsgravitationallyrepulsiveandleadstothe
measuredacceleratedexpansionofthecurrentUniverse.
Ontheotherhand,thereiselementary-particlephysicswheremassisexplainedasaconsequenceofrupture
ofsymmetrybymeansofaninteractionbetweenthemasslessmatterstatesandsomescalarfields.Ascalar
fieldofthiskindiscoupledheretogravitationinordertoobtainnewgravitationaldynamicsbesidestheusual
ones.Asaconsequence,antigravitationandantiscreeningofmatterwillbeexpectedasphenomenological
.MatterDarkThisworkentailsthefollowingmaincontributions:

•GeneralfeaturesofEinsteinstheoryareintroducedtogetherwithgeneralitiesofthedifferentelemen-
taryinteractionsofphysicsfromwhichtheconceptsofdarksectorsandHiggsMechanismarederived
2).and1(Chapters

•TheconceptofsymmetrybreakingandespeciallytheHiggsMechanismofmassgenerationaredis-
cussedintheirrelevanceforthemostdifferentsubjectsofphysics,especiallyinrelationtotheStan-
dardModelofelementaryparticlephysicswithelementaryHiggsfields(Chapters3).

•ThemechanismofsymmetrybreakdownwithHiggsscalarfields,asessenceforthephenomenology
ofsuperconductivityincondensedmatterphysics,isshownwithintheproblemofconfinementof
quarksinhadrons,i.e.oftheconstituentswithinnuclearparticles.ThisChaptershowstherelevance
anduniversalpropertiesofmechanismsofsymmetrybreakdowninphysics.Itwascarriedoutwithin

iii

vi

ajointworkwithDr.HemwatiNandanoftheCentreforTheoreticalPhysicsinNewDelhi.Parting
fromWick-transformedpropagatorsfordual-symmetricsystems,itcontinuesearlierworksofhis
andreenforcestheconceptofsymmetriesinnatureandtheassumptionthatHiggsfieldsmaylead
tothinflux-tubeformationforcolor-electricchargesconstrictinganalogouslytomagneticfieldsin
superconductors,whereasCooperpairsinBCSmodelsactaseffectiveHiggsbosons(Chapter4and
4.3).especially•Scalar–TensorTheoriesareintroducedhistorically(Chapter5)inordertobuildinthemtheprocess
ofHiggsMechanism.ThisisthenfulfilledwithatheoryofinducedgravitywithaHiggspotential
(Chapter6)whichseemsrenormalizableaccordingtodeWittspowercountingcriterion,andwith
mass-generatingHiggsfieldswhichonlycouplegravitationallyaswellaswithHiggsfieldswhichact
fields.cosmontoanalogously•Higgsfieldsingeneralinteractgravitationallysothattheyarecoupledheretoactwithingravitation
indeed.Further,theenergydensityofthegravitationalfieldisderivedforthespecificmodelof
inducedgravityfromananalogytoelectrodynamics(Chapters6.4and7.3).Itisshownthatanon-
vanishingvalueofpressurerelatedtothescalarfieldisnecessaryinordertoreproducestandardlinear
solar-relativisticdynamics.Withinastrophysicalconsiderationsforflatrotationcurvesofgalaxies,a
possibledark-matterbehaviorisconcludedwithinsphericalsymmetry(Chapters7.7and7.8).The
scalarfieldandthedark-matterprofileoftotalenergydensityarederived.Ananalogousrelation
betweendensityandpressureingalactodynamicstothatofsolar-relativisticbehaviorappearsforthe
dominanceofphenomenologicalDarkMatteringalaxies.
•Withinsphericalsymmetry(Chapter7),gravitationallyrepulsiveissuesofinducedgravityarecon-
cluded.ThesemayleadtoweakeningofhorizonsofBlackHoles(greystars;Chapter7.4)aswell
astoReissner–Nordström-likebehavioringalaxiesandBlackHoles(Chapter7.5).Thismayaccount
topotentiallyrelevantastrophysicalconsequencesonweak-fieldsolutionssuchasgeodesicmotion
(Chapter7.7)andsolar-relativisticeffects(Chapter7.6).
•FundamentalrelationsofcosmologywithininducedgravitywithHiggspotentialarederivedfora
Friedmann–Robertson–Walkersymmetry(Chapter8).Cosmicaccelerationanddark-matterpheno-
menologyareanalyzedinvirtueofthegeneralizedFriedmannequations,theequationsofstate,cosmic
parameters.densityanddeceleration•IndicationsofapossiblefiniteinitialstateoftheUniverseareachievedforaFriedmanncosmology
(Chapters8.7and8.8)togetherwithacceleratingbehaviorinsuchastateaswellasinthecurrentUni-
verse.Absenceofmatterleadstoanti-stiff,quintessential(antigravitational)behaviorintheUniverse
8.2).(Chapter

Übersicht

DieseArbeitbeabsichtigtalserstes,einekurzeEinleitungingrundlegendeGebietederPhysikzusein,umso
einenÜberblickdesMechanismusderSymmetriebrechungundseinerwesentlichenMerkmaleinderPhysik
wiederzugeben.Eswirdbesprochen,wiedieserMechanismustatsächlichgrundlegendfüreinbesseresVer-
ständnisderPhysikinvielenGebietenist,ganzbesondersinVerbindungmitderElementarteilchenphysik
undderPhysikkondensierterMaterie,einschließlichderSupraleitunginihrergewöhnlichen,wieauchin
ihrerdualenForm,welcheinnerhalbderGluodynamikuntersuchtwird.
DieseArbeitführtdannindasKonzeptderSkalar-TensortheorienderGravitationein,dieaufBergmann-
Wagoner-ModellenmiteinemHiggspotentialbasieren.DerwesentlichephysikalischeKontext,dendiese
Arbeitbezweckt,sinddieProblemederDunklenMaterieundderDunklenEnergie.Diesewerdenunter-
sucht,umsolchephänomenologischeFachgebietederAstrophysikundderKosmologiebesserzuverstehen.
AufdereinenSeiteliegtdieGravitationvor.Siebefasstsichimwesentlichenmitdenastrophysikalischen
PhänomenenundmitderStrukturdesUniversumsansich.IndiesemKontextistdieDunkleMaterievon
besondererRelevanz.DergrößteAnteilanMaterieimUniversumwirdalsDunkleMateriebezeichnet.
Sieistnotwendig,umdieinnerhalbderStandarddynamikgemessenenastrophysikalischenDatenwieder-
zugeben.Standarddynamikbenötigt,(nichtandenElektromagnetismusgekoppelte)dynamischeMaterie,
welcheabernochexperimentellnachgewiesenwerdenmuss.DunkleMaterieumfasstca.90Prozentder
gesamtenMateriedichte,wobeidieMateriedichtewiederumnurrundeinDritteldergesamtenEnergiedichte
darstellt.FasallerestlicheDichte,alsoknappzweiDritteldergesamtenEnergiedichte,wirdalsDunkle-
Energie-Dichtebezeichnet.DieseEnergieagiertgravitativabstoßendundwirdzurErklärungdergemesse-
nenbeschleunigtenExpansionunseresUniversumsherangezogen.
AufderanderenSeiteliegtdieElementarteilchenphysikvor,inderdieMassealsFolgegewisserSymmetrie-
brechungerklärtwirdundzwarmittelsWechselwirkungenzwischendenmasselosenMateriezuständenund
einerbestimmtenArtskalarerFelder.EinskalaresFelddieserArtwirdhierandieGravitationgekoppelt,um
somitneuegravitativeDynamikzuerlangen,diezudengewöhnlichenhinzuzufügenist.AlsFolgewerden
AntigravitationundGegenabschirmungderMateriealsphänomenologischeDunkleMaterieerwartet.
Hauptbeiträge:folgendebeinhaltetArbeitDiese

•AllgemeineBestandteiledereinsteinschenTheoriewerdenzusammenmitallgemeingültigenAspek-
tenderelementarenWechselwirkungenderPhysikeingeleitet.AusdiesenstammendieKonzepteder
dunklenSektorenunddesHiggsmechanismusab(Kapitel1und2).

•DasKonzeptderSymmetriebrechungundinsbesonderedesHiggsmechanismusderMassenerzeu-
gungwerdenimSinneihrerBedeutungfürdieunterschiedlichenGebietederPhysikbesprochen,
vornehmlichinBezugaufdasStandardmodellderElementarteilchenphysikmitelementarenHiggs-
3).(Kapitelfeldern

v

vi

•DerMechanismusderSymmetiebrechungmitHiggsfeldernwirdvorgeführ.DieseristKernder
phänomenologischenSupraleitunginnerhalbderPhysikkondensierterMaterieundwirdhierimSinne
desProblemsderEinsperrung(Confinement)derQuarksinHadronen,alsoderKonstituenteninner-
halbnuklearerTeilchen,betrachtet.DiesesKapitelzeigtdieBedeutungunduniversellenEigenschaf-
tendesMechanismusderSymmetriebrechunginderPhysik.DieswurdeunterMitwirkungvonDr.
HemwatiNandandesCentreforTheoreticalPhysicsinNeu-Delhierarbeitet.BasierendaufWick-
transformiertenPropagatorenfürdualeSystemesetztesfrühereArbeitenvonihmfortundverstärkt
dasKonzeptderSymmetrieninderNatur,zusammenmitderVermutung,dassHiggsfelderzurBil-
dungdünnerFlussröhren(fluxtubes)farbelektrischerLadungenführen,welchesichanalogzuMag-
netfelderninSupraleiternverengen,wobeiCooperpaareinBCS-ModellenalsHiggsbosonenauftreten
4.3).hauptsächlichund4(Kapitel•Skalar-Tensortheorienwerdenhistorischeingeführt(Kapitel5),umindiesedenHiggsmechanismus
einzubauen.DieswirdmiteinerTheorieinduzierterGravitationmiteinemHiggspotentialerreicht
(Kapitel6),welchegemäßdesAbzählbarkeitskriteriumsvondeWittrenormalisierbarzuseinscheint.
SowohlmassenerzeugendeHiggsfelder(dienurgravitativkoppeln)alsauchHiggsfelder(dieanalog
zuKosmonfeldernagieren)werdenalsskalareFeldergewählt.
•HiggsfelderimAllgemeinenwechselwirkengravitativ.Somitkoppelnsiehierderart,dasssiein-
nerhalbderGravitationagieren.DesweiterenwirddieEnergiedichtedesGravitationsfeldesfürdas
spezifischeModellderinduziertenGravitationauseinerAnalogiemitderElektrodynamikabgeleitet
(Kapitel6.4und7.3).Eswirdgezeigt,dasseinnichtverschwindenderWertdesmitdemSkalarfeld
verbundenesDruckesnotwendigist,umdiestandardsolarrelativistischeDynamikwiederzugeben.In-
nerhalbastrophysikalischerAbwägungenfürflacheRotationskurvenwirdmöglichesDunkle-Materie-
VerhaltenbeisphärischerSymmetrieschlussgefolgert(Kapitel7.7und7.8).DasskalareFeldunddas
ProfilDunklerMateriedergesamtenEnergiedichtewerdenabgeleitet.EineähnlicheBeziehungzwis-
chenDichteundDruckdergalaktischenDynamikzudersolarrelativistischenVerhaltenstrittbeider
DominanzphänomenologischerDunklerMaterieinGalaxienauf.
•BeizentralerSymmetrie(Kapitel7)werdenIndiziengravitativabstoßenderWirkungenschlussgefol-
gert.SolchekönnenzueinerAbschwächungdesHorizontsSchwarzerLöcher(graueSterne;Kapi-
tel7.4)führen,sowiezuReißner-NordströmVerhalteninGalaxienundSchwarzenLöchern(Kapitel
7.5).DieskannpotenziellrelevanteastrophysikalischeFolgenhaben,z.B.beiSchwachfeldlösungen
sowiebeidergeodätischenBewegung(Kapitel7.7)undsolarrelativistischenEffekten(Kapitel7.6).
•EswerdenfundamentaleBeziehungenderKosmologieinnerhalbderinduziertenGravitationmit
HiggspotentialfürdieFriedmann-Robertson-Walker-Symmetrieabgeleitet(Kapitel8).Diekosmi-
scheBeschleunigungunddiePhänomenologieDunklerMateriewerdenaufderGrundlageverall-
gemeinerterFriedmanngleichungenuntersucht,aberauchunterBetrachtderZustandsgleichung,der
kosmischenDezelerations-undderDichteparameter.
•IndizienaufeinenmöglichenendlichenAnfangszustanddesUniversumswerdenfüreineFriedmann-
Kosmologieerhalten(Kapiteln8.7und8.8).DarüberhinauswirdeinBeschleunigungsverhalten
solcherZuständeunddesjetzigenZustandesdesUniversumshergeleitet.DabeiwürdedieAbwesen-
heitvonMateriezueinemantisteifen,quintessenziellen(antigravitativen)VerhaltenimUniversum
8.2).(Kapitelführen

Contents

Summary

Contents

Abbreviations,acronymsandbasicsymbols

esFigurofList

oductionIntr

IElementaryparticlesandGravitation

iii

vii

xi

xv

xvii

1

1Onthegeometricalbasicsofgravitation3
1.1Transformationsandthemetricaltensor............................3
1.2Maxwellequationsofgravitation................................5

2ElementaryparticlesandtheStandardModel9
2.1QuantuminteractionsandtheideaofYang–Millstheories...................9
2.2WavefunctionandtheStandardModel.............................12
2.3Thetypesofmatterandthedarksectorproblem........................15
2.4Dark-energydensityanddensityparameters..........................20

3Symmetrybreakingandscalarfields25
3.1Symmetrybreakingandbreakingmodes............................25
3.2HiggsfieldsandHiggskinds..................................29
3.3SymmetryBreakingandtheSM................................30

4QCD,superconductivityandsymmetrybreaking39
4.1Dualsymmetry,monopolesanddyons.............................39
4.2Covariantformanddyons....................................42
4.3Superconductivity,dualsuperconductorsandtheHiggsfield.................46

IIInducedgravitytheorieswithscalarfields

5Alternativetheoriesofgravityandhistoricaloverview

vii

55

57

viii

CONTENTS

5.1Jordanstheory.........................................57
5.2Brans–Dicketheory.......................................59
6Scalar–tensortheorywithHiggspotential63
6.1Lagrangedensityandmodels..................................63
6.2Thefieldequations.......................................65
6.3Fieldequationsaftersymmetrybreakdown...........................67
6.4Maxwell-likeequationsandgravitationalenergydensity...................73

IIICosmologicalConsequencesofInducedGravity77
7Inducedgravitywithsphericalsymmetry79
7.1Theexactequationsforsphericalsymmetry..........................79
7.2Linearequationsandstaticweak-fieldsolutions........................80
7.3Energy-densityconstraintsonpressureandmassterms....................89
7.4Exactequationsandblack-holesolutions............................91
7.5TheReissner–Nordström-likesolution.............................97
7.6Perihelionadvance.......................................103
7.7Effectoffieldexcitationsonthegeodesicmotion.......................106
7.8Flatrotationcurves.......................................109
8Friedmann–Robertson–Walkermetric115
8.1ThegeneralizedFriedmannequationsandtheHubbleparameter...............115
8.2Equation-of-stateparametersofthescalarfield........................124
8.3DecelerationparameterandDarkEnergy............................126
8.4Thedensityparameters.....................................132
8.5Decelerationandtheequation-of-stateparameters.......................136
8.6Effectiveandbaredensityparameters.............................140
8.7BreakingofenergyconditionsandconditionsofaBounce..................146
8.8ThePlanck-lengthBounce...................................154
9Results,conclusionandoutlook159

163ppendixAIVAGeneralRelativityandGeometry165
A.1TheMetricalTensor.......................................165
A.2Lorentztransformations.....................................166
A.3ThelocalgaugeoftheLorentzgroup..............................168
A.4Einsteinequationsandmatter..................................172
BWavefunctionandelementaryparticles175
B.1QMstateandSpin-Magneticinteraction,QMpostulatesandmeasurement..........175
B.2OntheYang–Millstheory....................................178
B.3ElectroweakdoubletoftheSM.................................181

CONTENTS

ix

B.4StrongtripletoftheSM.....................................183
B.5TheSMsystem.........................................183
B.6Schematicpropertiesoffermions,bosonsandtheirinteractions................187

189CosmologyCC.1Sphericalsymmetryandtheidealliquid............................189
C.2Scalar-fieldequationwithcentralsymmetry..........................190
C.3RN-likeparameters.......................................193
C.4StandardFriedmanncosmology................................195

yBibliograph

symbolsmathematicaloflistExtended

Index

publicationsandSupport

wledgementsAckno

CurriculumScholar

199

215

229

237

239

247

x

CONTENTS

Abbreviations,acronymsandbasic
symbols

Theexactmeaningofthesymbolsmaybegottenbymeansofthecontext.Furthermore,hats,primes,
subscriptsandfurtherindicesareusedfordifferentiationwithinthetext.Sucharenotshowninthelistabove.
Forfurtherinformation,thereadermayusetheextendedlistofmathematicalsymbolsandtheindexat
theendofthiswork.Second(s),meter(m),kilogram(kg),Newton(N),Ampère(A)andsubdivisionsand
multiplesofthem(andSIprefixesingeneral)arenotshown.
AbbreviationSymbol/BasicusageAbbreviationSymbol/Basicusage

αΓ

ΔκCDMΛµξτχψΩ

e–Nordström-likReissnerchargeterm/Strength/
radiationinHe-nuclei(connection)symbolfelChristof

ratioDensity/ferenceDifEnergydensity/Levi–Civita
/parameterg–LandauurGinzbcouplingvitationalGrawithModelMatterDarkColdconstantcosmologicalMetric/parameterHiggs/Muon/componentPermeabilityxcitatione-fieldScalarDensityTauon/Eigentime/Generator
distanceariantvCo/fieldScalar/PotentialfunctionevaWsphereUnit/parameterDensity

xi

β

γ

δεΛλν

ΠσΦΨϕω

radiationwein(anti-)electrons

Diracmatrix/polytropicindex
/ratioge–couplingCharradiationinPhotonsKroneckerdelta/Deltadistri-
utionb/vityPermittiparameterGeodesictermCosmologicalLorentz/constant)(function,transformation/parameterHiggs/GaugecomponentMetricNeutrino/componentMetric

tensorPolarizationMagneticchargedensity/Cross
matrixauliP/sectionfieldScalar/PotentialPotentialScalar-fieldexcitation/Angle
couplingeJordan–Brans–Dick

xii

viationebrAb

ATLASAaal.vad.BBCSBHCCDMCMBccf.coshDDESYDMEdynEEOSEPREVed.e.g.eq.etc.weFWFLRfffGGRGSWg

usageBasic/Symbol

xindeSpinorApparatusLHCiroidalTAScaleparameter/Isospinindex
/Haloscale/Outsidesub-
scriptAdvalorem:Accordingtothe
aluevxindeSpinor/Baryonfer–SchriefBardeen–CooperHoleBlackColorMatterDarkColdCosmicMicrowaveBack-
groundLightspeedconsultcompare,:ConfercosineyperbolichDyononensyn-ElektrhesDeutsconotrhrcfectEfMeissnerDualDynamicalgyEnerstateofEquationReso-aramagneticPElectronnancealuevExpectationedition,EditorExempligratia:forinstance
equationEtcetera:andtherest,
onsoandweakelectroorceF/Field-strengthFriedmann–Lemaître–eralkRobertson–WFamilyindex(isospin);Flavor
Foliis:andfollowingpages,
pagesfromconstantcouplingvitationalGravityRelatiGeneralgeinberw–Salam–WGlashoMetric/constantCoupling

ABBREVIATIONS,ACRONYMSANDBASICSYMBOLS

AbbreviationSymbol/Basicusage

AHMUAloc.ad.

BBDBWCCCCERNCPca.cosconst.DEDMdomEiEPESRefefemal.eteVxp.eGFLAWFRffig.GLGRBGUT

ModelHiggsAbelianUnitAstronomicalAdlocum:Intheplace

amplitudefieldwtonianNeeBrans–Dickagonergmann–WBerConjectureCensorshipCosmiclapouropéenEurConseilRechercheNucléaire
arityation–PConjugabout:caCirCosineconstantgyEnerDarkMatterDarkDominancegralinteExponentialprinciplealencevEquiResonanceSpinElectrongecharElectricevfectiEfElectromagneticothersand:aliiEtoltvElectronExperimentalAnti-Length-ScaleFinitevityGraeralkFriedmann–Robertson–WFolium:andfollowingpage,
Figure/g–LandauurGinzbLinearGeneralurstbGamma-rayTheoryUnifiedGrand

AbbreviationSymbol/Basicusage

HHEh.t.ISCOi.a.ibid.JjKkLLLHCLISAllogMMEMONDmNN.B.NMRO(N)Pparpg.QQCDQM

Hubble/HiggsHilbert–EinsteinconstantsPlanckReducedHoctitulo:under/inthistitle
orbitcircularstableInnermostInteralia:amongotherthings
Ibidem:inthesameplace
JordanCurrentCurvature/Force/
parameterMass/constantBoltzmanndensityorceFsubscript"Left-handed"density)(LagrangeLagrangianColliderHadrongeLarAn-SpaceInterferometerLasertennaComptonwavelength
(lengthscale)/leptonindex
arithmLogMatterfectEfMeissnerDynamicswtonianNeModifiedxindeFermionic/masswtonNewellNote:beneNotaResonanceMagneticNuclearOrthogonalgroupofdegreeN
PerihelionparameteragePgeCharChromodynamicsQuantumMechanicsQuantum

AbbreviationSymbol/Basicusage

HDMhh.c.ii.e.JBD

KK

LLEPLIGOLNTcit.loc.lyMMOGMRIminNNFWop.cit.ppctem.proQADQEDq

MatterDarkHot/correctionMetricconstantsPlanckateconjugHermitesubscriptInsideisthat:estIdeJordan–Brans–Dick

Kaluza–Klein

/functionLagrangemomentumAngularCol-Electron–PositrongeLarlidervita-GraInterferometerLasertionalWaveObservatory
ThresholdNoLinearLococitato(l.c.):Intheplace
citedLightyearparameterMass/(Higgs)MassvityGraModifiedImagingResonanceMagneticMinimumIsobar/tensorfixingGaugequantityarro–Frenk–WhitevNaOpuscitatum:citedwork
Pressure/MomentumarsecPProtempore:Forthetime,
temporarilyAsthenodynamicsQuantumElectrodynamicsQuantum/parameterDecelerationScalar-fieldexcitation/Quark

xiii

vxi

viationebrAb

RRNrSSEPSISNeSSBGSTVSU(N)Shissins.p.TTttanhUu,VVVEVvizvs.WWIMPwXYZz

usageBasic/Symbol

Radiation/Radius/Curvature
sub-"Right-handed"/(scalar)script–NordströmReissnerOrbit/DistanceSpin/ActionprinciplealencevequiStrongdunitésInternationalSystèmeaevNoSuperSymmetrySpontaneousBreakingScalar–Tensor–VectorGravity
SpecialN-DimensionalGroupUnitarygralInteyperbolicusSinushgy–momentumEnerSineSineprole:withoutissue
otalTKinericdensitygyenerimeTtangentHyperbolicUni-/termcosmologicalBWtransformationtary/mixturePotential-termReciprocalorbit/energyden-
sity(density)PotentialVacuumexpectationvalue
Videlicet,viderelicet:namely,
wittoVersus:against
onweakgedCharboson)augeg(weakWeaklyInteractingMassive
articlePmix-potential/parameterEOSturecorrectionRNgeHyperchar/onweakNeutralfunctionartitionPRedshift

ABBREVIATIONS,ACRONYMSANDBASICSYMBOLS

AbbreviationSymbol/Basicusage

RMSWRradSBsf,SFSMSRSTTSO(N)SUSYSNesc.sinhsrTeSeVTtantotU(N)UMi

VEv.i.v.s.vWEPWMAP

xYMZw

squaremeanRooteralkRobertson–WRadiationBreakingSymmetryfieldScalarStandardModel/StiffMatter
vityRelatiSpecialTheoryensor–TScalarofGroupOrthogonalSpecialNgreedeSupersymmetryaevNoSuperScilicet:onemayknow
sineHyperbolicsteradianEnergy–stress/Kineticenergy/
emperatureT/Isospin–Scalarector–VensorTangentTotalTGroupUnitaryN-Dimensional-isMinorUrsa

xpectationeacuumVGround-statevalue/velocity
Videinfra:seebelow
Videsupra:seeabove
Weakequivalenceprinciple
WilkinsonMicrowave
ProbeyAnisotrop

Coordinate/Yukawamatrix
ang–MillsYanzigerZw

esFigurofList

4.1IllustrationofaHiggscompositeofsuperconductivitywithelectronsmediatingmassive
gaugebosonsaccordingtoaYukawamodelandfieldtheoriesofinteractionprocesses.N.B.:
TheappearanceoftheHiggsbosonleadstosuperconductivityasabroken-symmetryphe-
nomenon.............................................48
4.2BehaviorofthedifferenttypesofQEDsuperconductorsindependenceofthefieldstrength
HwithcriticalfieldsHandHandH...........................51
1c1cc

7.1Evolutionoftheparameterhfordifferentequation-of-state(eos)parameterswanddistance
coefficientsr/l=x.Cf.[23]...................................88
7.2Evolutionofthedynamical-masscoefficientMdyn/M1fordifferenteosparameterswand
distancecoefficientsr/l=x.Cf.[23]..............................89
7.3Evolutionofthemetriccomponentsinthismodel(O(r−2))fordifferent,(set)negativeval-
uesofAwithB=2andM1GN/c2=1.N.B.:TheeffectiveSchwarzschildradiusdimin-
ishesfordecayingvaluesofA..................................102
7.4EvolutionofthemetriccomponentsinthismodelforhighernegativeamplitudesAwith
B=2(v.s.).N.B.:Thereappearsaquintessentialattraction(cf.text)foreλathighervalue
ofA................................................102
7.5EvolutionofthemetriccomponentsforthepositivevaluesofA.N.B.:TheeffectiveSchwarzschild
radiusaugmentsforhighervaluesofA.............................103
7.6Evolutionofe−νandeλforw=1/5andw=1/2withM1GN/c2=1.Stiffmatter
w>1/3isrelatedtopositivesquaredchargesQ˜2>0.Forw=1/5,thedynamicalmass
forlinearappr−νoximationλreadsMdyn=13/10...............2..........104
7.7Evolutionofeandeforw=−2/5andw=−1/5withM1GN/c=1.w<0
leadstoquintessentialattraction.ThedeviationbetweenM˜GNandMdynGNishigh.For
w<2/3,whereMdynGN<0isvalid,counter-gravitativebehaviorappears,togetherwith
anakedsingularity........................................104
7.8TimelikeeffectivepotentialVeffforw=0(left)andw=0.2(right)anddifferentvalues
ofLandM1GN=1.N.B.:AnorbitforanenergyEequaltothemaximum(minimum)is
unstable(stable).Atanenergygivenbythedashedhorizontalline,forthethickcurvethere
isaboundorbitinwhichtheparticlemovesbetweentwoturningpoints............107
7.9Left:Dashedgeneralized-charge(Q˜2)plot(dashedcurve)anddynamical,effectiveM˜(dot-
dashed)andMdyn(dashed)andactual(density)massM1=1(horizontalline)inde-
pendenceofstiffnessw.h(w)Mdynasthickcontinuousline.Right:Tangentialvelocity
(dr/dt)2(withε=−1)forM1GN=1anddifferenteosparametersw............108
xv

xvi

FIGURESOFLIST

7.10Evolutionofdensitydistributionsforla=1/5(left)andla=1/35(right).N.B.:Scalar-
field(ξ)dominanceforshorterdistances,andbaryonic()dominanceforlimitsoflarge
∗scales...............................................112
7.11Densityratios:DarkMatterdominanceforl=1/36(left)andNon-newtonianbehavior
a(right)forl=1/5,l=20andl=35............................113
aaa

8.18.28.38.48.58.68.78.8

B.1

˙TotaleosparameterwT.Protem.:ξ=0,Ω=0.3,ΩΛ=0.7).N.B.:Thecontinuousline
staysforqˆ=1.Thedashedoneforqˆ=0............................136
˙TotaleosparameterwT.Protem.:ξ=0andΩ=0.3.....................137
˙MinimalvalueofqˆΩ+3(1−qˆ)ΩpforwT>0withξ=0accordingtoequation(8.5.5).
Set:ΩΛ=0.7..........................................138
ITpMinimalvalueofqˆΩ+3(1−qˆ)Ωforw>0.Left:protem.:0≤Ω≤1and
Λ−1≤ξ≤4,accordingtoequation(8.5.7).Set:Ω=0.7.Right:protem.ξ=1.......139
Exampleofapositiveeosparameterinahighlymatter-dominated,closeduniverseforΩ=
3,Ω=4,Ω=0.7fordifferentexcitationsξ.Thedashedcurvestaysforqˆ=0andthe
ΛIcontinuousoneforqˆ=1.N.B.:Lowexcitationswouldneedofhighermatter-dominance
fortotaldustorstiffnessofwtobeacquired.Thismaybetranslatedinhigherlengthscalesl.140
TTotaleosparameter.Protem.:Ω=0.03,ΩI=0.27,ΩΛ.ξ(lengthscale,cf.(8.5.3))
variable.N.B.:Fornegativeexcitationsξ,wTispositive....................141
Totaleosparameter.Protem.:Ω=0.03,ΩI=0.27,Ω+ΩI=ΩM.N.B.:Thecurveis
˙thesameasforthecaseξ=0andΩ=0.3..........................142
TotaleosparameterwT.N.B.:Ω+ΩI=ΩM,Ω:anti-screenedbaryonicparameter.Left:
ΩΛ=0.7,ΩB=0.03.Right:ΩB=0.03,ΩM=0.3.....................143

Schematicsonthepropertiesoffermions,bosonsandtheirinteractions............187

oductionIntr

Maxwellstheoryofelectrodynamicsshowsaninextricablesymmetrywhichlockstogetherelectricityand
magnetism.ThefourMaxwellsequationsdescribethefateoftheelectricandmagneticfieldsataspacetime
point,andfromtheseitispossibletoderivethateachofthefieldsobeysawaveequationrepresentingthat
lightpropagatesaselectromagneticwaves.Historically,theysignificantlycontributedtothedevelopment
ofSpecialRelativity.Furthermore,thebrokendualsymmetricstructureofelectromagnetismleadstothe
theoreticallyfascinatingaspectofthepossibilityofmonopoles.Theirexistencewouldmeanafurtherunifi-
nature.incationElectrodynamics(usualaswellascomprisingmonopoles)canbegivenonthegroundsofgaugetheoriesso
thatelectromagneticforcesmaybeinterpretedasconsequencesoflocalgaugetransformationwithgauge
potentials.Furthermore,localgaugetransformationofSpecialRelativityleadstoGeneralRelativitywith
externaltransformationsofspacetimeasconsequenceofgauge,andwithChristoffelsymbolsasgeneralized
potentialsofwhatislinearlyperceivedasthegravitationalforceatrelativelysmallvelocitiescomparedto
speed.lightIndeed,allelementaryinteractionsofnature–thefundamentalforcesbetweenelementaryparticles–canbe
givenbymeansofgaugetheories,andtheStandardModelofelementaryparticlephysicsprovidesaconcise
andaccuratedescriptionofallfundamentalinteractionsexceptgravitation.
ModernquantumtheoriesofelementaryinteractionsgroundonMaxwellstheorybymeansofYang–Mills
theorieswhichgeneralizethestructureofelectrodynamicsformorecomplex,non-abeliangaugegroupsfor
theoriesofquantumdynamics.Furthermore,theanswerofthefundamentalproblemaboutwhichmecha-
nismallowstheelementaryparticlestobecomeheavyisnowaddressedintermsoftheHiggsbosoninthe
StandardModel.TheHiggsMechanismis,therefore,apowerfultoolofmodernparticlephysicswhich
makesthemodelsmathematicallyconsistentandabletoexplainthenatureoffundamentalinteractionsin
amanifestway.ThebosonsandfermionsarebelievedtogainmassthroughaphasetransitionviaHiggs
Mechanism.Inthisway,theparticlescanbecoupledwithexperiments,andatheoreticalexplanationmay
begivenofhowthemassgenerationtakesplace.
TheHiggsparticles,belongingtotheHiggsfield,arestillnotexperimentalrealityandneedtobeobserved
tomakeanymodelcomplete.ThesearchforHiggsparticlesisaveryimportanttaskinphysics,andit
isbelievedthattheirmasswillbeachievablewiththefuturegenerationofhighenergyexperimentsasthe
LHCinGeneva,specificallyatenergieslessthan250GeVandhigherthan130GeV.Yet,thismaybemodel
dependent,whereastheexactpropertiesoftheHiggsfield,theircouplingsandtheirsourceareofspecial
relevance.Furthermore,Higgsparticlesingeneralappeareffectivelyinallbranchesofphysicsastheyare,
forinstance,basicintheunderstandingofsuperconductivitywheretheyappearascompositeswithinthe
conceptoftheCooperpairs.Thentheyleadtoaneffectivemassofphotons,whichitselfleadstotheMeiss-
nereffect.Furthermore,withinstronginteractionstheymayleadtoadualMeissnereffectandhencetoa
possibleexplanationoftheconfinementofquarksandcolorchargesinhadrons.

xvii

xviii

ODUCTIONINTR

ThenatureofHiggsfieldsisstillnotcompletelyunderstood.Universally,theyinteractinagravitativeand
Yukawaformineverymodel.However,theirexactpropertiesmaydependonthespecificmodelused.Ac-
tually,giventheirgravitativenature,iftheyarecouplednon-minimallytogravitation,unlikeintheStandard
Model,Higgsfieldsmaydecouplecompletelyfromthefermionicsectororcoupleonlyveryweaklyand
furtherevenpossessan(almost)vanishingmassinadditiontoafiniteground-statevalue.Thisisamain
issueofthiswork,whichinthiswayintendstocontributetounificationissuesofnature.
Withinastrophysics,thereexisttheproblemsofDarkMatterandDarkEnergy.Withinthestandardtheory
farmoremassisnecessarythanmassfromluminousmattercanbemeasured.Furthermore,cosmiccounter-
gravitativeinteractionsaremeasured.Thenatureoftheseissuesofcosmologyisunclearandalsoifthey
mayberelatedtostillunknownmechanismsoffurthergeneralizationsofthetheories.Thisworkrelates
themtotheconceptofscalar–tensortheoriesandtheHiggsMechanismofSpontaneousSymmetryBrea-
king.Hence,acosmon-liketheoryofinducedgravityispresentedwhichmaycontributetothephenomenon
ofthedarksectorsofcosmologyaswellastothedynamicsoftheprimevalUniverse.There,itmayaccount
tothesubjectofthecosmologicalInflation,i.e.theeraofveryhighaccelerationaftertheBigBang.Further,
itmayalsoaccounttoanunderstandingoftheBigBangitselfasnewgravitationaldynamicscomingfrom
thescalarfieldmaydominateatearlystagesoftheUniverseandactasfurthermatterwithnegativepressure
ordensity.DarkEnergyandDarkMattermaybearemanenceofsuchdynamics.
WhatcomposesourUniverse?Whicharethedynamicsofdarksectorsofenergydensity?Whichistherela-
tionbetweenscalarfieldsandastrophysicalandmicroscopicphenomena,ifany?MaytheHiggsMechanism
furthershowanevenmoreuniversalcharacterwitharelationtogeometrizedgravity?Whatconsequences
wouldthathaveinourpictureandinterpretationofastrophysicalphenomena?Whatconsequenceswould
thathavefortheearlystagesoftheUniverseaswhatwenowcanperceiveofit?Allthesearesubjectsofthis
work,whichintendstoholdontotheirphysicalcontext,especiallywithinelementary-particlephysicsand
astrophysics.Inanattempttoanswerpartofthesequestions,somereviewofnuclearandelementary-particle
physicsisnecessary,togetherwithsomegroundingoffundamentalphysicstowardsHiggsMechanismof
massgeneration,superconductivityandAbelianHiggsModelsforconfinementofquarksinbaryonsas
DualMeissnerEffect,JordanandBergmann–Wagonermodelsforinducedgravity,andcentralaswellas
Friedmann–Robertson–Walkersymmetryforgalacticdynamicsandcosmology.

Elementary

artP

I

particles

1

and

vitationGra

1Chapter

Onthegeometricalbasicsofgravitation

–GeneralfeaturesofEinsteinstheoryareintroduced.HomogeneousMaxwell-likesystemsarederivedfor
ageometricalfield-strengthtensorrelatedtotheRicciscalar.Theymaybepartlyfoundpublishedin[23].–

1.1Transformationsandthemetricaltensor

Bothquantumphysicsandgravitationalphysicscomprisealtogetherallknownelementaryinteractionsof
physics.Nuclearforces,i.e.therelevantforcesbetweennuclearconstituents(purelyofquantum-mechanical
nature),acteffectivelyonlyatshortdistances.Electrodynamics(Lorentzforcesandgeneralconsequences
ofelectromagnetismfromandonchargedparticles)andgravitation(asaconsequenceofmassandenergy),
ontheotherhand,arelong-ranged.However,electromagnetismcancelsoutbecauseofnegativeandpositive
chargessothatforlongscalesonlygravitation,theweakestofallelementaryinteractions,dominates.Fur-
therelementaryinteractions,namelyelectro-weakandstronginteractions(whicheffectivelyleadtonuclear
andelectromagneticforces)canbeunderstoodintermsofquantumphenomenawhilegravitationcannotyet
befullyunderstoodonthosegrounds.Still,allmaybeexplainedintermsofgaugetheories(cf.[56]on
QuantumCosmology)andgroundonthecovariantformalismwitha4-dimensional(lorentzian)manifold
ofspacetime.Elementary-particletheories(withquantumelectromagnetism),however,groundoninner
transformationsasaretheonesofspinandisospin,whilegravitationgroundsonexternaltransforma-
tionsasaretheonesofspacetimeitself.
Historically,innuclearphysicstheisospin,originallycalledisotopicorisotonicspin,isadefinedproperty
ofparticleswhichoriginallydifferentiatesbetweennucleonicparticlesornucleons(neutronsandprotons).
Withoutconcerningtheisospin,bothnucleonsareinterpretableasthesameparticlewithinnuclearforces
(thenucleon),giventhatforcesbetweennucleonsare(nearly)independentoftheparticlescharges[4](let
ussay,theyareisotopictoeachother).Withinnuclearforces,bothisotopicparticlesareindistinguishable
betweeneachother.Hence,sincetheisospinmakesadifferentiationbetweenprotonsandnucleons,the
isospinistheonequantumpropertywhichleadstotheexistenceofdifferenttypesofatoms(nuclides)ofthe
samechemicalelement,eachofthemhavingadifferentatomicmass(isotopes).
Thequantumstateofnucleonscanbegivenbyatwo-vectorinisospinspacewhereaseachcomponent
possessesanisospin.Nuclideswithdifferentamountofnucleonsbutthesamechemicalproperties(same
amountofprotons)areisotopesofeachother.Nuclideswiththesameamountofneutronsbutdifferent
amountofprotonsareisotones.Isotopesandisotonesdifferintheirisospin.Thereforethenameisotopic
spin.isotonicand

3

4

CHAPTER1.ONTHEGEOMETRICALBASICSOFGRAVITATION

Ifinanatomtheamountofeachnucleonisthesameandtheconfigurationofisospinsdiffers,wespeak
aboutisobars[84].Thisisthecaseinmirrornuclei(1H3–2He3,6C13–7N13etc.).NeglectingCoulomb
forces,theyshowthatprotonsandneutronshaveapproximatelythesameboundingcontributions(e.g.[91]).
Nuclearforces(asstated)dobasicallynotdifferbetweenisospinandareindependentoftheelectriccharge.
Withtheadventofelementary-particlephysics,theconceptofnucleonicisospinhasbeengeneralized.
Hence,anisospinvectororisovectorpossessNdifferentcomponentsinisospinspace[155].Theparti-
clesrelatedtoeachisospindependonthespecificgroupwithinwhichtheyareindistinguishablebetween
eachother(isotopic).Isospinisdefinedasanintrinsicpropertyofquantummechanicalstates,andtrans-
formationswithinisospinspacehaveimplicationsonparticlesthemselves.Thesetransformationscanbe
followeduptotheconsequencesonanuclearandchemicallevel.
Anotherintrinsicpropertyofparticlesisthespin.Itgivesthestatisticswhichaparticlefollows.The
spinisaquantumnumberwhichcategorizesbetweengeneraltypesofparticles.Transformationsinspin
spaceimply,amongothers,transformationssuchasbetweenfermionsandbosons1whichnotonlyfollow
differentquantummechanicalstatisticsbuthaveanantagonicalrelationtowardselementaryinteractions.
Onanelementary-particlelevel,fermionsfeelinteractionsandbosonstransmitthem[158].Suchis,
however,amatterofYang–Millstheories,especiallyinthecontextoftheStandardModelofelementary
particlephysicsbasedongaugetointroducebosonsactingonfermionicmultiplets(cf.Chapter3.3).Gra-
vitationasanelementaryinteractionofphysics,however,doesnotgroundontransformationsofmatter
itself.Transformationswithingravitationaretheonesofspacetimewhich,inanalogytospinandisospin
transformations(internal)arecalledexternalones.Thesetransformationsmaybedefinedbythelocally
gaugedhomogeneousorinhomogeneous(Poincaré)Lorentzgroup.2HomogeneousLorentztransforma-
tionsarethegeneralizationofrotationsinthe3dimensionaleuclideanspaceontotheMinkowskispace
(xµ→xµ=Λµνxν).Thegroupofproper(detΛ=+1),orthochronous(Λ00≥1)transformationsis
isomorphtothegroupSO+(3,1)(restrictedLorentzgroup)ofspecialpseudo-orthogonaltransformations
in4dimensions.Itisspannedbyusual3-dimrotationsandthespecialLorentztransformations(boost
transformations)[99].FurthercomponentsoftheLorentzgrouparegottenfromparity,timeandparity–
timetransformations(representativesofacosetclassrelatedtothefactor(orquotient)groupO(3)/SO(3)
Z2={1,−1}andtheKleinfour-groupZ2×Z2).Poincarétransformations(xµ→xµ=Λµνxν+aµ)
isthesymmetrygroupoftheMinkowskispace.Space-timetransformationsarerelatedtothepropertiesof
themetricaltensorgµν,oftensimplyknownasmetric(seeAppendixA).
Ifthemetricisspacetimedependent,thederivativealongtangentvectorsofamanifoldistointroducea
connectiononthemanifoldbymeansofadifferentialoperatorwhichentailthepropertiesofthespacetime
metricwhichwerepresentbygµν.Thisoperatoristhecovariantderivative.Itisrelatedtothelocalgauge
ofthetransformation(Lorentz)group(seeAppendixA.1).TheconnectioncomponentswithinGRarethe
ChristoffelsymbolsΓνµλ[218]asobjectswhichareanalogoustoYang–Millsgaugepotentials(fields)Aµ
(relatedtogaugebosons)ofthedifferentisospinandspincomponents.Asamatteroffact,localgauge
transformationsoftheLorentzgroupleadsto(thegeometricalpartof)gravitationinformofageometriza-
tionofgravity(seeAppendixA.2).

1NamedafterEnricoFermi(1901-1954)andFermi–Diracstatistics.andafterSatyedraNathBose(1894-1974)andBose–Einstein
statistics.2AfterJulesHenriPoincaré(1854-1912)andHendrikAntoonLorentz(1853-1928).

1.2.MAXWELLEQUATIONSOFGRAVITATION

5

1.2Maxwellequationsofgravitation
4-vectorsaµinR4areclassifiedaccordingtotheirscalarproductastimelike,spacelikeorlightlike(null).
Withthesignatureusedinthiswork,thereis
(i)aµaµ>0foratimelikevectoraν.
(ii)aµaµ<0foraspacelikevectoraν.
(iii)aµaµ=0foralightlikevectoraν.
Inthecaseofaµ=dxµ,√thescalarproductisthe(squareofthe)lineelement,ds2.Inthecase(i),cdsis
calledeigentime.In(ii),−ds2iscalledeigenlength,andin(iii)theworldlinerunsthroughalightcone.
Asgenerallyknown,causallylinkedeventsliewithinalightcone(cf.[218]).
Withatimelikevectorfielduµ=dxdsµwith
uµuµ=dxµdxµ=ds2=1
2dsdsdsanduµ;σuµ=0,
thefieldequationsforagivenobservermaybewrittenas
uµ;λuλ=Kµ(1.2.1)
withKµasthenongravitationalpartofa(tomassnormalized)forcefromamasscarriedbytheobserver.
Ontheotherhand,thereisanequilibriumbetweennongravitationalforcesKµandinertialonesEµwhich
maintainthemassinageodisicaltrajectorysuchthat
Eµ+Kµ=0.(1.2.2)
TheforceEµmaybewrittenas
Eµ=−uµ;σuσ=(uσ;µ−uµ;σ)uσ=(uσ,µ−uµ,σ)uσ=F˜µσuσ.(1.2.3)
Thelatterdefinesafield-strengthtensorF˜µνofthesamestructureaswithinelectrodynamics(abelian),asa
rotationofa4-vector,
F˜µν=uν;µ−uµ;ν,(1.2.4)
withgaugevariablesuµ(cf.[64]).Forthe(gravitational)field-strength(1.2.4),Maxwell-likeequationsare
obviouslyvalid[23,64],
F˜(λµ,ν)≡F˜µν,λ+F˜λµ,ν+F˜νλ,µ=0.(1.2.5)
Inthatsense,thegravitationalorinertialforceEµwhichappearsfortheobserverhastheformoftheelectric
partoftheLorentzforce.ItisrelatedtotheRiccitensorasitcanbewrittenusingthedivergenceofF˜µν.
TheRiccitensorfromequation(A.3.24)mayberewrittenasfollows,
−Rλµuλ=(uλ;µ−uα;αδµλ);σ=Hλµ;λ.(1.2.6)

6

CHAPTER1.ONTHEGEOMETRICALBASICSOFGRAVITATION

BethisthedefinitionofatensorHµνwhichissymmetric.Thistensorgivethe(skewsymmetric)field-
strengthtensorF˜µνby
F˜µν=Hνµ−Hµν=uν;µ−uµ;ν(1.2.7)
withF˜µνasanantisymmetrictensoranduµasvectorpotentialofthegravitationalfieldstrength.Equiva-
be,lently

(1.2.10)

Hνµ+Hµν=uµ;ν+uν;µ−2uα;αgµν(1.2.8)
.Qµν≡ThisdefinesatensorQµν.Withequation(1.2.8),theRiccitensormaybewrittenintermsofthefield
ws,folloasstrength11−Rλµ=F˜µλ;λ+Qµλ;λ.(1.2.9)
22ThedivergenceofQµλisasbelow,
Qµλ;λ=uµ;λ;λ+uλ;µ;λ−2uα;α;µδλµ
=2uλ;µ;λ−2uλ;λ;µ.(1.2.10)
Consequently,thereisthefollowingequality,
Qµλ;λuµ=uµ;λ;λuµ+uλ;µ;λuµ−2λ;λ;µuµ
=uµ;λ;λuµ+uλ;µ;λuµ−2uλ;λ;µuµ(1.2.11)
=4uλ;µuµ−4uλ;µuµ;λ−2uλ;µ;λuµ−2uλ;λ;µuµ.
λ;Withgµλgλµ=1,thereisequivalently
Qµλ;λuµ=4uλ;µuµ;λ−4uµ;λuµ;λ−2uλ;µ;λuµ−2uλ;λ;µuµ,(1.2.12)
whichmaybesimplifiedforstaticfieldsinrelationtotheobserver.Thiscanfurtherbetreatedaftergoing
throughtheright-handsideoftheequationofgravitation,i.e.itsrelationtomatter,whichwillgivethesource
oftheenergy–stresstensorF˜µνrelatedtocurvature.Furthermore,therelationsderivedanddefinitionsgiven
hereareofspecialrelevanceforafurtherdefinitionoftheenergydensityofgravitationinChapter7.3,the
resultsofwhichmaybefoundunder[23].
InChapter6.4,equation(6.4.8)showstherelationofthefieldstrengthF˜µνtogravity,giventherelationof
theRiccitensorRµνtouµasfieldvariablein(1.2.6).TheEinsteintensormaybederivedthroughvariation
fromtheRicciscalarRandacosmologicalconstantintheaction.TheHilbert–Einsteinaction,entailing
bothtermsandaLagrangedensityofmatter,leadstotheequationsofgravitationas
Gµν=−κNTµν,(1.2.13)
tensorEinsteinthewith1Gµν=Rµν−2Rgµν+Λ0gµν,(1.2.14)
themetricaltensorgµνandacosmologicalconstantΛ0.Equation(1.2.13)givestheEinsteinorEinstein–
HilbertequationsofGeneralRelativity(GR).TheymaybefoundderivedintheAppendixA.4.Tµνisthe

1.2.MAXWELLEQUATIONSOFGRAVITATION

7

energy-stresstensorwhichmaybederivedfromtheLagrangedensityofmatter.κNisthecouplingconstant.
Further,themetricaltensorispresentedinAppendixA.1whileLorentztransformationsandthelocalgauge

transformationsoftheLorentzgroup,includingconnectiontermsandcurvaturetensorsarepresentedinAp-

pendicesA.2andA.3,respectively.TheseshowthegeometricalmeaningofgravitationwithinGR,which

mayberelatedto(1.2.6).Equation(6.4.8),however,givesamoregeneralapproachfromamoregeneral

iswhichactionpresented

GR

isformulationalid.v

in

Chapter

6.1.

,etY

for

the

special

case

of

anishingv

scalar

fields

(see

later),

the

8

CHAPTER

1.

ON

THE

GEOMETRICAL

ASICSB

OF

TIONAVITGRA

2Chapter

andparticlesElementaryStandardtheModel

–Generalitiesofelementaryquantuminteractionsofphysicsandquantumstatesarediscussedinviewof
YukawastheoryofmesonsandYang–MillstheorieswhichleadtotheSMofelectroweakandstrong
interactionsofphysics.Specialattentionispaidtofermionicfields,typesofmatterandcompositematter
andespeciallytothedarksectorsofmatterandenergy.Partofthisworkmaybefoundunder[24]asresult
ofthiswork.Detailsonthequantummechanicalstateandthetheoryofelementaryparticlesmaybefound
–B.Appendixin

2.1QuantuminteractionsandtheideaofYang–Millstheories

Modernparticle-physicstheorieshavetheirbeginningandinterpretationbasisintheearly20thcentury.
Backthen,H.Yukawa[246]proposedthatnuclearparticleswereheldtogetheragainstelectromagnetic
repulsionbymediationofparticlesheproposed(mesons1).Withinhismodelthereshouldexistanuclear
forcebetweennucleonswhichisgreaterthanelectromagneticrepulsion.Accordingtotheprinciplethat
forcesshouldnotactatdistance,thisforceshouldberelatedtomediationofaparticleascarrierofthe
propertiesoftheinteraction.Withinnuclearphysicsthen,Yukawaproposedin1935theparticleswenow
knowaspi-mesonsorpionsπ.Theseparticlesaremassive(withabout140MeV/c2≈2.5∙10−28kgofmass)
anddoindeedmediateshort-rangeinteractionswithinthenucleus,accordingtoHeisenbergsuncertainty
principle.Yukawastheoryultimatelystatesthatasconsequenceofmediationofmesonsbetweennucleons,stable
nucleiappear.Hence,thetypeofinteractionbetweenthemisgenerallycalledofYukawa-type.Such
interactionscanbedescribed2throughapotentialgivenbytheproductofaDiracfieldψandascalar(or
pseudoscalar)fieldφasfollows,

V∼gψ¯φψ,(2.1.1)
1Fromµσo:Inthemiddle,intermediate,andhencewithmesonsasper-definitionemintermediateparticles.
2Forπmesons,actually,φisapseudoscalar,i.e.ψ¯γ4ψφ,withtheprojectoroperatorγ4=i4!1εαβµνγαγβγµγνwithDirac
matricesγµ,theLevi–Civitatensorεαβµνandthefermionicstateψ.
9

10

CHAPTER2.ELEMENTARYPARTICLESANDTHESTANDARDMODEL

withgasacouplingconstant,andwithVwhichgivesapseudoscalarquantitywhichischaracteristicofthe
mesonsdescribedbythepotential.Further,ψ¯istheadjointconjugateoftheDiracfield,i.e.ofthequantum
mechanicalstateentailingthewholeinformationforameasurementwithinthequantummechanicalsystem
forparticleswithspin(seeAppendixB.1).
TheDiracfieldisaspinororspinvector.Itisthustobeusedforfermions.Itsadjointconjugateisdefined
byusualhermiteconjugationcoupledwithγ0,

ψ¯=ψ†γ0,(2.1.2)
sothatantimatterstatesaredescribedonlyintermsofachangedsigninrelationtomatter.γ0isoneofthe
matrices.DiracNucleonsarefermions,andthepseudoscalarmesonsdescribedbyYukawasmodelarethepions.Origi-
nally,themuonsorµparticles(whichpossessasimilarmasstotheoneofpionsandwhichareelementary,
indeed),wereassumedtobetheYukawaparticles.However,theydobasicallynotinteractwithinnucleiand
hencedonotrepresentmesonicparticles[57].Muonsareleptonsandhencemassiveisotopictoelectrons.3
Pionswerefirstdiscoveredin1947byLattesetal.[149]Forthepredictionandforthedevelopmentofex-
perimentaltechniqueswhichresultedintheirdiscovery,YukawaandPowellwereawardedwiththeNobel
prizein1949and1950,respectively.

ThetheoryofYukawamaybegeneralizedsothatotherinteractionsaredescribed.Beforetherisingof
elementary-particlephysics,itwasfurtherusedinattemptstounifynuclearforceswithgravitation,again
assumingYukawamesonsasmediatorsbutwithinahigherdimensionalspacetime(sc.KaluzasandKleins
[142].5)ChapterSeetheory.Classically,aYukawainteractionmaybewrittenintermsofaYukawapotentialwhichmaybeconstructed
startingfromCoulombpotentialsforlong-rangeinteractions,themediatorsofwhicharemassless.These
interactionsareof1/r-type.Yukawapotentialsfurtherpossessamasstermandmaybewrittenasfollows,
2V(r)∼−ge−mcr/,(2.1.3)
rwithmasthemassofthemediationparticle,i.e.ofthepionintermsofnuclearinteractions.mc/isthe
reciprocal(Compton)lengthscalerelatedtothemassmwhichgivestherangeoftheinteraction.
Pionspossessaninnerstructureanddecayinleptons[156].Theyarethusnotfundamental.Theymediate
onlyresidualinteractions.Further,thenucleus-conformingparticles(nucleons,butalsohadronsingeneral,
seeChapter2.3)possessafinitediameterofabout10−15mandalsoaninnerstructure[131].Furthermore,
theypossessmagneticmomenta[6].Inthiscontext,Gell-Mann[100]andZweig[252],independently
ofeachother,interpretedanonelementarityandintroducedconstituentparticlesofhadronsbackin1964.
Theseparticlesareknownasquarks.4Theexperimentalevidenceofthese[92],finally,wasacknowledged
withtheNobelprizein1990.
Elementaryparticlephysicsdescribesdynamicsonthebasisofquantumfieldtheoriesandhenceofquantum
mechanicalstatesaspropertycarriersformeasurements.Hence,statesmaybegivenbyDiracspinorssuch

3Itisnotirrelevanttomentionandfurthertoemphasizethathereliesanerrorinearlierliteratureandneweronecopiedfromthe
latter(seeChapter2.3,cf.[155]against[84]withµmesons)whenstatingthatµparticlesaremesons.Bestressedthatmesonsareto
transmitnuclear(effective)interactionsandmuonsdonothavesuchproperty(aselectronsdoneither).
4ThreequarksforMusterMark!Surehehasnotgotmuchofabark.Andsureanyhehasitsallbesidethemark.–JamesJoyce,
FinnegansWake

2.1.QUANTUMINTERACTIONSANDTHEIDEAOFYANG–MILLSTHEORIES11

thattheyberelatedtoconstituentparticles.Thespinorpossessesthefollowinggeneralstandardform,
ψ,A1ψψa,AL/R=.2..,A,
ψ,ANL/RIndexagivethegeneralizedisospinandAthespin.LandRbethesubscriptforleft-handedandright-
handedstates,respectively.
BeNthedimensionofthesymmetrygroupoftransformation.Nistogivetheamountofisotopicparticles
withintheinteractiongivenbythegaugegroup,i.e.itistogivetheamountofparticleswhichareindistin-
guishablewithingiveninteractions.
Isospinspacedependsonthegroupdefinedfortheinteraction.Forinstance,aneffectivenucleartheory
ofnucleonspossessesadimensionN=2whereanucleonicstatepossessestheneutronandtheproton
asisotopicelementsgivenbythestateψ.Withinatheoryofstronginteractionswhereitisdifferentiated
betweenthreedifferentcolor-quarksforeachflavororfamily,thenthereisN=3withanisospinindex
acountingeachcolor.Furthermore,electroweakinteractions(whereelectronsandneutrinosfordifferent
familiesareisotopicbetweeneachother)aregivenbyaquantummechanicalstatecomposedbytwoisospin
components.ModerntheoriesarebasedonYangsandMillsfieldtheoryof1954[245],utilizingN-dimensionalwave
functionsinisospinspace.N2further,istheamountofcomponentsofthetransformationmatrixincluding
unity.Ifeverycomponentofthestateistoberelatedtophysicalparticles,thentheamountofcomponents
ofthismatrixrelatedtoU(N)minusunityshouldgivetheamountofparticleswhichmediateinteractions.
Theseparticlesarebosonsandarecalledgaugebosons.5Theypossessanalogouspropertiestothoseof
mesonsinYukawasearliertheory.Furthermore,gaugebosonsandmesons(suchaspions)possessaninte-
gerspin.However,gaugebosonsareassumedfundamentalandtheyarerelatedtopotentials(photonsare
relatedtotheelectromagneticpotentials,forinstance).Furthermore,gaugefieldsinteractwithisomultiplets
inauniversalway[4].Hence,theyappearinthesamewayforallgaugegroupsofthedifferentinteractions.
TheYang–Millstheoryisanon-abelian(non-commutative)theorywithSU(N)transformationsandthus
withself-interactionsthatgeneralizetheMaxwellequationsof(abelianU(1)-)electrodynamicstotheso-
calledandanalogousYang–Millsequations(seeAppendixB.2,especiallyequations(B.2.1),(B.2.7)and
(B.2.9))DλFµν+DµFνλ+DνFλµ=0(2.1.4)
andDνFµν=−4πcgJµ(ψa)(2.1.5)
withanisotensorial(adjoint)field-strengthtensorandcurrentFµν(followingRicciidentities)andJµ,res-
pectively(seeAppendixB.2),andisotensorialgaugepotentials(matrices)Aµwithisospincomponentsa
andbinthecomponentnotation,
(Aµ)ab=Aµi(τi)ab,(2.1.6)
followinglocalgaugeofthetransformationgroupSU(N).BymeansofSU(N)transformations,newforces
appearforstates(andtheirrelatedparticles),withgaugefieldsAµaspotentialsofelementaryinteractions
5Realinteractionsaredescribedbyoperators,i.e.fieldoperatorsψ,Aµ,etc.;seeAppendixB.

DλFµν+DµFνλ+DνFλµ=0

(2.1.4)

12

CHAPTER2.ELEMENTARYPARTICLESANDTHESTANDARDMODEL

(cf.AppendixB.2).However,althoughYang–MillsequationsreducetoelectrodynamicsforN=1and
thenfermionicmassmaybeaddedbymeansofamasstermoftheLagrangian,themorecomplexformof
quantummechanicalinteractionsgenerallyprohibitsthesimpleadditionofmassterms.Weakinteractions,
forinstance,showparity-symmetrybreaking[244]sothatadditionofamassiveLagrangiantermleadsto
contradictionswithexperimentalfactswithinDiracequations,giventhatright-(left-)handedstatescouple
tomassthroughthesourcewithleft-(right-)handedstates.Furthermore,suchtermsassimplyaddedmasses
leadtosingularities.Aper-handmassiveYang–Millstheoryisnotrenormalizable.Toachieveaphysical
theory,itseemsnecessarytointroducescalarfieldsandtheconceptofsymmetrybreakingsothatmasses
appearinanindirectwaybymeansofnewparameters(seeChapter3,cf.[157]).

2.2WavefunctionandtheStandardModel
Theparadigmwithinelementary-particletheoriesistheunifyingGlashow–Salam–WeinbergModelofthe
QuantumAsthenodynamics(QAD)6ofelectroweakinteractions[96],Nobel-prizeawardedin1971.To-
getherwithso-calledQuantumChromodynamics(QCD)7ofthestronginteractionsofGell-Mannandoth-
ers,itleadstotheStandardModel(SM)ofelementaryparticlesunderthesymmetrygroupSU(3)C⊗
SU(2)L⊗U(1)Y.Theconstituentsarepartofamultipletorisovectorwhichisconformedbythoseparti-
cleswhichareindistinguishablewithinaspecificinteraction(i.e.theyareisotopictoeachother).Thegroup
dimensionN(andsubscript),hence,isgivenbytheparticlesrepresentedineachgroup:threedifferently
colored(C)quarksforthestronginteractions,aswellaselectronsandneutrinos(leptonsL)fortheweak
interactions,andelectronsforelectromagnetism.Ystaysforthehypercharge,whichisrelatedtotheusual
electromagneticchargeofelectronsbytheso-calledGell-Mann–Nishijimaformula.
Thisinformationisintroducedintothe(Ndimensional)Yang–Millstheory,andthelatteristhenfurther
changedempirically.Thefundamentalfirststep,however,isdefiningthepropertiesofthewavefunctionψ
group.transformationeachforTheSM,asaquantumfieldtheoryofinteractingfundamentalfields,isbasedontheso-calledgaugeprin-
cipleorgaugeinvariance,whichleadstothecovariantderivatives,paralleltransportandgaugeprinciple.
Thesemakeitpossibleforderivativestomaintaintheirtensorialcharacter,andtheycanbeintroducedin
termsofparalleltransport(andholonomy)incurvedspace(e.g.asphere).There,ausualderivativeleads
outsideofthemanifold.Anadditionaltermisneededascorrection,thatistomoveparalleltothesurfaceof
thesphereduringthederivation.Thisadditionaltermisrelatedtoso-calledconnections,suchasChristoffel
symbolsΓνµλinGRorgaugefields(orpotentials)Aµinusualgaugetheoriesofelementaryparticles.Fur-
thermore,inadditiontosimpleYang–Millstheory,theSMhasbuiltinGell-Manns[100]andZweigs[252]
ideaofquarksasfundamentalconstituentsofhadrons.8Interactionsbetweenquarks,then,areunderstood
asmediatedthroughthegaugefields,withtheso-calledgaugebosonsasthefieldquantaoftheinteractions.
Summarizing,therearetheseespeciallyrelevantinteractionswhicharegivenbytheStandardModelof
particles:elementary

6Astheno:ασθνια=weak(soft),lackofstrength.weakness.ThemodelwasproposedbySheldonLeeGlashow(1931),Abdus
Salam(1926-1996)andStevenWeinberg(1933);Nobelprize1979.
7Chromo:χρω´µα=color.
8Hadrons(˙αδρ´oς=stout,thick):Protons,neutronsandmoremassiveanalogueparticles(thehyperons),aswellasmesons,see
2.3.Chapter

2.2.WAVEFUNCTIONANDTHESTANDARDMODEL

13

•QEDwiththesymmetrygroupU(1),whichisanabelianunitarygroupwhichthenleadstoMaxwells
equations.Allthree(f=1...3)electron-likeparticles(electronse,muonsµandtauonsτ,and
antimatteranalogues)aretheoneisotopiccomponent,ef,oftheQEDisoscalarψfAwhereasall
electron-likeparticlesareelectromagneticallyindistinguishable(asidefromtheirmass).Further,Ais
thespinsothattheQEDisoscalarisavectorinspinspace.
WithinQED,theisospinindexacountsonly1.Commutatorsofgaugepotentialscomingfromcova-
riantderivatives(thecouplingtointeraction)vanish,andtheircorrespondinggaugebosons,thegauge
photons,thus,donotself-interact.QEDremainsanabeliantheory.

•Withinnuclearforcesthereexistscharge-independenceofnucleonicinteractions[84].Hence,pro-
tonsandneutronsareinterpretableasdifferentstatesofanucleonparticle.Theydifferintheisotopic
spinorisospin.Thisnucleonmaybedescribedwithinoldnuclearphysicsasa2-dimensional
isospinvector.Asalreadyclear,inelementary-particlephysics,thisconceptisgeneralizedforele-
mentaryinteractions.Yang–MillstheoriestreatNdimensionalmathematicalobjectswhichinprin-
ciplepossesssomekindofelementaryparticlesasisospincomponents.Thesecomponentsbelong
tothesameunitarygrouprelatedtoaninteraction.IfthereareNdifferentparticleswhicharein-
distinguishableandyetisotopictoeachotherforgiveninteractions,sucharegivenbyanisovector
ormultipletψafA.acounttheisospinasvectorindex,andfcountthefamilymembership.The
symmetrygroupyieldsU(N)whichentailselectrodynamicssincethegroupmaybedecomposedas
U(N)=U(1)⊗SU(N).Theamountofgaugebosonsasintermediateparticlesotherthanphotonsisgiven
byN2−1.

•ElectroweakinteractionsaregivenbyanSU(2)⊗U(1)doubletfortheweakandfortheelectromag-
neticpartsoftheinteractions,suchthatthewavefunctionisisoscalarforU(1)transformationsand
isovectorialforSU(2)transformations(forright-handedstates,though;v.i.).Further,thefactofthe
fermionicmultipletpossessingthedimensionofthegivengaugegroupentailsthatithavetwocom-
interactions.weakelectroforponentsElectroweakinteractionspersedonotdistinguishbetweenleptonsandquarks.Hence,theyforman
isospinorofelectroweakinteractions(henceanisospinindexm)withtheisodoubletψmf.Itcanbe
distinguishedbetweenleptonic(m=1=l,forelectrons,muons,tauonsandneutrinos)andquark
dynamics(m=2=qfortheelementaryconstituentparticlesofnucleonic-likematter)whereeach
componentisanisodoubletonitsown,i.e.
flψL/Rmf=ψqf,
ψL/Rwithm=1=lforleptonsandm=2=qforquarks.
Form=1,ontheonehand,left-handedelectroweakstatesofthegaugegroupSU(2)Laregivenby
ectorvisospintheψLlf=νf,
efLwhileright-handedstates,withgaugegroupU(1),aregivenbytheisoscalar
ψRl=eRf.(2.2.1)

(2.2.1)

14CHAPTER2.ELEMENTARYPARTICLESANDTHESTANDARDMODEL

Ontheotherhand,thequark-isodoubletofm=2isrepresentedbythefollowingisospinvector(see
B.3)Appendix

ψL/Rqf=uf.
dfL/R•StronginteractionsareparityconservingandaredescribedwithinQuantumChromodynamics(QCD).
Thisisatheoryofquarksdynamicsandwithinwhichallbarequarksareindistinguishableifmassis
aside.letNeutrinoscoupleonlyweaklywhileelectron-likeparticlescoupleelectroweakly,i.e.weaklyand
electromagnetically.Quarks,ontheotherhand,coupleelectroweaklybutalsostrongly.Hence,quarks
havetoappearasanisovectorwithinstrong-interactiontransformationsalso.Theisospinofthe
strong-interactionstateisgivenbyanewstrongpropertynamedcolorwhichshallbecarriedby
allquarksandtakestheplaceofasortofgeneralizedchargeofSU(3)C[101].Hencethename
chromoinQCDwiththeanalogytoQEDwith(strong)colorcharges.OtherthanwithinQED,
however,thischargeexistsinthreesortsnamedblue,redandgreen(liketheprimarycolorsfor
additivecombinationsincolortheory)andthreeanti-sorts(anti-blue,anti-redandanti-green,hence
intheanalogycyan,yellowandmagentainthesubtractivecolormixing).Sincedifferentlycolored
quarksareisotopictoeachother,thegaugegroupisofthedimensionN=3.Thegaugegroupis
calledSU(3)color=SU(3)C.Thestatethenpossessesthreeisospincomponentsforeachquark-type
whichisgivenbytheso-calledflavorf(seeAppendixB.4).Theisospincomponentsarequarksof
eachcolor-charge.Hence,thestatemaybewrittenasfollows,
frafψaf=gaf,
bawiththesubscriptacountingthecolorcharge(a=1,2,3)andtheindexfcountingtheflavor
(f=1,...,6).Thethreequarkstates,onlydifferingbytheircolorcharge,thusformatripletwithin
stronginteractions.The3×3matricesrelatedtothetransformationgroupSU(3)C,generatorsofthe
matrices.Gell-Manncalledaregroup,Colorshallhaveanalogpropertiestocharge,andtheeightgaugebosonscalledgluons9shallbeana-
loguetogaugephotons.TheyshallespeciallystaymasslesssothatSU(3)Cisanexactsymmetryand
bothQCD-andQED-interactionsarelong-ranged.However,QCDisnotabelianand,hence,gluons,
unlikephotons,self-interact.Additionally,theycarryandmediatebothacolor-andananti-color
B.4).Appendix(seegecharPhysically,gluodynamicschangethecoloroftheconstituentparticlesofhadronsinawaythat,inthe
end,thetotalcolorofhadronsisvanishing(accordingtocolortheory)[101].Thisistheprocess
ofconfinementbywhichgluonsarethoughttoself-interactinsuchawaythatcompositegluonstates
(glueballs)appear.Thesecompositestatesacquireadynamicalmasswhichthenleadstoaneffec-
tive,short-ranged,nuclearstrongforcealthoughgluonsthemselvesaremassless[8].Inconsequence,
quarksmovealmostfreelywithinhadronicranges(asymptoticfreedom)butcannotbedetectedas
freeparticlessincestrong-interaction(color)forcesshouldaugmentwithdistance.InChapter4.3,a
methodforexplainingconfinementusingsymmetrybreakdownisintroducedaspartoftheresearch
9Fromglue(orinscientifictraditionfromtheLatingl¯uten),giventhatgluonsaretobetheparticleswhichholdnucleonstogether.

2.3.THETYPESOFMATTERANDTHEDARKSECTORPROBLEM

ork.wthiswithin

2.3Thetypesofmatterandthedarksectorproblem

15

:matterExperimentalBeforescalarfieldsareintroduced,itisbettertomakeatleastsomecommentsaboutthetypesof
matterwhichareknownexperimentallytogetherwiththeirrelationtotheelementaryparticlesofthe
are:particlesTheseSM.

•Quarks:elementaryandconstituentfermionswhichappearinQCDundertheSU(3)Ctriplet
andundertheisospinindexm=qofSU(2).Thereare36kindsofthemcountingmatter,
handedness.ferentdifandantimatter•Leptons(λπτóς:thin,light10):elementaryfermionswhichappearform=lonSU(2).They
donotinteractwithinstronginteractions.Theyappearin18elementaryformscountingmatter,
antimatteranddifferenthandedness.Thefirstexperimentaldemonstrationofhigher-generation
electron-likeparticles(theµleptons)wasachievedin1937[182],whiletheexistenceofneutri-
noswasfirstdemonstratedin1956[60].Theexistenceofdifferentgenerationsofthelatterwas
demonstratedin1962[63],andnonexistenceofright-handedneutrinos(aswellasofleft-handed
antineutrinos)withintheSMfollowsfromparity-conservationandconjugation(CP)violation.
•Gaugebosons:Interactionsaregivenbyelementaryparticlesrelatedtothegaugegroup.These
arethegaugebosons.Hence,theelementarybosonicparticleswhichcarrythepropertiesof
interactionsarerelatedtogaugepotentialsandthustotransmissionofforcesonanelementary-
particlelevel.Thereare12experimentallyconfirmeddifferentkindsofgaugebosons.Elec-
troweakgaugebosonsweredemonstratedexperimentallyin1983[10].Thefirstdirectex-
perimentalevidenceofgluonswasfoundin1979(e.g.[40]alongwithotherexperimentsat
11).DESY

Quarksandgaugebosons,especiallyofstronginteractions,areelementaryconstituentparticlesof
hadrons.Theymaybecategorizedaspartons.However,whatisgenerallymeasuredismacroscopic
matterwhichisusuallycomposite.Particlescomposedespeciallybyquarksarecalledhadrons.
However,hadronsmaybefurtherunder-classifiedinbaryonsandmesons,whichgivestheirstatistics
Bose–Einstein).and(Fermi–Dirac

10Termwhichshoulddenotethesmallmassoftheseparticlesinrelationtothatofnucleons;however,tauons(ofthe3rdandmost
massivegenerationofleptons)haveca.twicethemassofaproton.
11DESY:GermanElectronSynchrotroninHamburg.

16

CHAPTER2.ELEMENTARYPARTICLESANDTHESTANDARDMODEL

ClassofcompositeConstituents(partons)Examples
HADRONS(H)QUARKSandGLUONS
H1)Baryons3quarksOR3antiquarks
H1.1)Nucleonsup-anddown-proton
neutron(antiquarks)quarksantiprotongluonsandantineutronH1.2)HyperonsStrangeness=0orΩ−
Charm=0or(3strange-quarks)
Topness=0orΛC+
Bottomness=0(up,down,charm)
H2)MesonsONEquarkandONEantiquark
H2.1)Flavorlessup-,down-quarkspionπ+
andantiquarksandgluons(up,anti-down)
H2.2)Flavoredup-,down,top-,kaonK+
anti-strange)(up,quarksstrange-bottom-,gluonsandantiquarksand

Nuclearmatterconstitutesonlyofquarksofthefirst(andlessmassive)generation(uandd).Such
combinationsaregenerallypreferredenergetically,andhyperonshence−10decayweaklyontonucleons
plusmesonsandleptonswithalifetimeoftheorderofmagnitudeof10s.
Flavoredmesons,further,decayontonormal(flavorless)mesonsplusphotonsandleptonswithana-
loguelifetimes.Hence,themaintypeofbaryonicmatterisnucleonic,andthemaintypeofmesonic
matteristheflavorlessone.
However,neitherbaryonicnorhadronicmatteringeneral,aretheonlytypes.Therearephotons
andgaugebosonsingeneralaswellasleptons.Thesemayappearasanon-baryonicclassofnon-
hadronicmatterasleptonsmayboundincompositessuchasleptonia(electron–antielectron-pairs,
forinstance).Atthesametime,gluons,forexample,boundinso-calledglueballswhicharetoac-
quiredynamicmassandmayexplainwithintheSMtheshortrangeof(effective)nuclearforces(ca.
2.5∙10−15m,incontrasttopurestronginteractions,whicharelong-ranged,sincegluonsdonotpos-
12mass).sessTheelementarynonhadronic(andthusnonbaryonic)matterislistedbelow:

12Throughtunneling,Heliumnuclei(αparticles)splitfromthenuclearparentpassingthepotentialbarriergivenbynuclearforces.

2.3.THETYPESOFMATTERANDTHEDARKSECTORPROBLEM

17

ClassofmatterConstituentsFrequentsymbolSomeproperties
BosonicmatterphotonsAMediateelectromagnetism.
orγUncharged.Nomass
gluonsGiMediatestronginteractions.
(eighttypes)Possesscolor-charge
(m=0!)ge,-anticharandweakonsW+,Mediateweakinteractions.
W−,Leadtoβ-decay.
0evMassiZLeptonicmatterelectron-e±,Massive.
(positron-)µ±,Threeleptonicgenerations
analoguesτ±withme<mµ<mτ
neutrinosνiandOnly(gravitationallyand)
(antineutrinos)ν¯iweaklyinteracting.
(i=e,µ,τ)Nonvanishingsmallmass
Thedarksectorandsupersymmetricparticles:
particles:vitygraQuantum•Withinanelementary-particlephysicstheoryofgravitation,therewouldexistanotherkindof
gaugebosonswhichistheoneofgravitonsasfieldquantaofgravitation[195].Theseare,
however,notyetexperimentallydiscovered.
Thereisbynownocompletequantummechanicaltheoryofgravitation.Yet,asgravitation
appearstobealong-rangeinteraction,analogytellsthatgravitonsaretobeassumedasmassless
gaugebosonsofgravity.
particles:Supersymmetric•However,thereareotherparticleswhichmightbebynownotofexperimentalnature.Someof
thesemightindeedbeofspecialrelevanceinastrophysicalcontextsanddoleadtoastrophysical
consequencesbymeansoflargeparticlemasses.Anespeciallyrelevantassumptionisthatthere
existsasymmetrybetweenmesonsandbaryons[164],oryetmoregenerally,betweenfermions
(particleswithoddspin)andbosons(particleswithintegerspin)[103,109,233,237].Thissym-
metry(supersymmetryorSUSY)wouldrelateeverybosontoafermionandeveryfermionto
aboson(so-calledsuperpartners).Quarkstateswouldberelatedto(new)bosonicstatescalled
squarkswhileleptonswouldberelatedto(alsonew)bosonicstatescalledsleptons.Hence,there
wouldbemoreelementarybosonswhichwould,further,notbegaugebosons.Bosons,onthe
otherhand,wouldberelatedtofermionicstatescalledbosinos(suchasgauginosforthesu-
persymmetricpartnersofgaugebosons,gravitinosforthepartnersofgravitonsetc.).
Allsupersymmetricparticles,althoughstronglyanalyzedwithinthesubjectofsupersymmetry
inelementaryparticlephysicsandsuperstringtheories,arenotyetofexperimentalnature.Their
physicalstatusisyettobeclarifiedbyexperimentsastheonesinprocessattheLHC.Yet,
massivesupersymmetricparticlesmaybeaclassofmatternecessarytocomprehenddynamics
correctly.WithintheminimalsupersymmetricextensionoftheSM,forinstance,ifthesuper-
symmetricparityispreserved,thelightestsupersymmetricparticlewillnotdecay.Thisparticle,
assumingitexists,mayaccountfortheobservedmissingmassoftheUniverse(v.i.).
•DarkMatterphenomenologyandbaryonicDM:
Actually,itwasin1933thatZwickygainedfirstevidencethataccordingtostandarddynamics,

18

CHAPTER2.ELEMENTARYPARTICLESANDTHESTANDARDMODEL

new,non-luminous(dark),typesofmatterwerenecessarytoexplainthedynamicsoftheComa
cluster[253].Missingmatterwasfurtherdeterminedintheyearsafter,firstforourLocalGroup
ofgalaxies[138]andthenforallgiantgalaxies[79,187].Furthermore,independentdetermina-
tionofrotationvelocitiesofgalaxiesatlargedistancesfromgalacticcenters[211,212]confirmed
theinterpretation:thepresenceofnonluminous(dark)matterhalosaroundgalaxies.Itsnature,
though,isunclear,althoughitmayinprinciplebesomekindofhotgas[138]orpossessastellar
origin[180].Itmightfurtherconsistofapregalacticgenerationof(verymassive)stars[43].
Moderndata,however,indicatethatstellardarkmattercannotbedominantindark-matter(DM)
phenomenology[229],andgaseoushaloscannotdominateeither[89,145,220,228].Baryonic-
gasDMcannotconsistofneutralgasandionizedgas.Further,althoughpresentasindicatedby
X-rayanalysis,itisnotsufficientingalaxiestoexplaintheirflatrotationcurves.Hence,mass-
to-luminosityratiosofgalaxiesstillindicatefarhighermassesthantheoneofvisiblematter.
Thereseemstobesomekindofmatterwhichisnonbaryonic.
DM:Leptonic•ThereisalsothepossibilitytoencounterleptonsasDM.Thepossibilityofheavystablelep-
tonicDMwasexaminedintheearlyseventies[116].SuchcandidatesforDMdominancewere,
however,rejectedintheyearthatfollowed[230].Still,anotherkindofrelevantnon-baryonic,
yetexperimentalleptonicDMtypehadstartedbeingconsideredfromtheearly70son[61]:
neutrinosasdark-mattercandidates.Asnon-baryonic,further,theywouldhelpexplainsmall
temperaturefluctuationsofthecosmicmicrowavebackgroundradiation(CMB)[55].
Neutrinosandtheirantimattercounterpartcompriseindeedarelevantcategoryofphysicalparti-
cleswhichisespeciallyrelevantinanastrophysicaldescriptionofmattertowardsDMphenome-
nology.TheypossessspecialrightsforthecategoryofDMsincethecrucialdiscussioniswhich
kindofmattermaybeperceived(almost)onlygravitationally,andneutrinosinteractonlyvery
weakly,withacross-sectionσν+n≈7.1∙10−43cm2.Neutrinosdonotcoupleelectromagneti-
callyandarethusverydifficulttodetectdirectly.Inanastrophysicalcontexttheyaretherefore
calledHotDarkMatter(HDM).Darkbecausetheylackelectromagneticcoupling(which
makesthemverydifficulttodetect–afterall,25yearspassedsincetheirpredictionbyPauliin
1930[190],whichhappensevenbeforeneutronswerediscovered,untiltheir1995Nobel-prize
awardeddiscoverybyReinesandCowanin1956[60]);hotbecauseofthehighvelocityof
neutrinosrelatedtotheiralmost,butaccordingtoneutrinooscillations[77]notvanishing,mass
ofmaximallyafeweV/c2[8].However,giventoolowmassesofneutrinos,theycannotbethe
dominantDMcontributioneither.13
Matter:DarkCold•UnderthecategoryofDarkMatter,itcanthusonlybeacknowledgedthatitmaybebaryonicor
nonbaryonic.AcategoryofnonbaryonicDMisHDM.However,withintheSMnoneofthese
typesofDMexplainstheproblemofthephenomenologyofmissingmass.Further,theremay
existothertypesofexoticDMwhicharesomekindofas-yetundiscoveredmatter.Thismatter
13Neutrinosappearespeciallyinnuclearreactors(about9MeVoftotal200MeVperfissionof235U)orproceedfromtheSun.They
appearinweakprocessesviaβdecayorelectroncapture.Hence,theycanaccompanyionizingradiationasαandβparticlesindecay
Tochains.avoidScreeningharmfuleofxposureionizingtoradiationradiation(sc.dependstheonLNTthehshieldypothesis)scrossandsecminimitionzeforexposurescatteringdoseandwhenabsorptionhandlingaswellradioactiasonveitsmaterial,thickness.a
screeningofβparticlesneedsoflightmaterialtogetherwithheavyoneforshieldingbremsstrahlung(high-energeticphotons)from
sloneutralwdownandofthehardlyionizinginteractwithparticles.otherNeutrinosparticles.cannotHence,betheydoshieldednotbutrepresenttheydonotharmfulionizeradiationanywithsubstancebiologicaleitherastheconsequences.yareelectrically

2.3.THETYPESOFMATTERANDTHEDARKSECTORPROBLEM

19

isgenerallycalledColdDarkMatter(CDM).Particularlyimportantlikelycandidatesofitare
axions14orlightsupersymmetricparticlesasneutralinosorgravitinos[30,33].
Withinsupersymmetry,gravitinosaresuperpartnersofgravitonsofaquantumtheoryofgra-
vitation,andneutralinosarequantumtheoreticalsuperpositionsofthesuperpartnersoftheZ-
bosons,ofphotons(neutralgauginos)andofneutralHiggsparticlesofsupersymmetrictheories
(higgsinos).Thelatterareassumedtomixduetotheeffectsofelectroweaksymmetrybrea-
king(whenbothelectromagneticandweakbecomeindependentinteractions,leadingtomassive
weakonscharacterizingthebrokensymmetry).Asheavy,stableparticles,neutralinos,inpar-
ticular,seemtobegoodcandidatesforColdDarkMatter(CDM)asveryweaklyinteracting
massiveparticles(WIMPs).Theyareassumedtodecayfinallyespeciallyinτ-leptons,although
decaychannelsincludingsupersymmetricparticlesasneutralhiggsinos,forinstance,arealso
expected[159].Theneutralinomassisexpectedtobeofover100GeV/c2,andevidenceof
annhihilationofsuchparticlesinregionswhichareexpectedtobehighlydark-densedishoped
willbefoundinγ-rayandneutrinotelescopes.Theexperimentalmassconstraintofneutralinos
lieatmasseshigherthan46GeV/c2formξ1˜0,mξ˜20>62.4GeV/c2,mξ˜30>99.9GeV/c2and
mξ˜40>40GeV/c2,accordingto[8].Charginoswouldhavemasseshigherthan94GeV/c2.
•Theoreticalviewpointandexoticparticles:
Fromthetheoreticalpointofview,notonlypossiblystillunobservedsupersymmetricparticles
shouldbetakenintoaccount.Therearealsocosmologicalrelicsfromsymmetry-breakingpro-
cesseswhicharepredictedbyhigh-energyphysicsthatshouldbeincludedinalistofUniverses
components[144].Alltheseparticlesandfields,asfarastheydoreallyexistinthephysical
world,shouldhaveplayedaroleinstructureformation.Theythereforeimplytheexistenceof
anexoticpartofthedarkcomponentsofthedensityoftheUniverse(thatis,ofthecomponents
suchasofdarkmatterwhichwedonotdirectlysee,orthenatureofwhichisstillunclear).
However,colddark-mattercandidatesareyettobefoundinhigh-energyexperimentsandtheir
naturehastobeclarifiedinviewofademonstrationthattheyareindeedcapableofleadingto
.phenomenologyDM•DMdominanceandmodifieddynamics:
TheconclusionwithinstandardGRdynamics,citing[79],yields:allgiantgalaxieshavemas-
sivecoronas[halos],thereforedarkmattermustbethedeterminingcomponentinthewhole
Universe(atleast90%ofallmatter).Ontheotherhand,though,althoughDMdominatesat
longranges,locally,usualtypesofmatterdominate[104,146,147,185,186]:thereisnoevidence
forthepresenceoflargeamountsofdarkmatterinthediskoftheGalaxy.Ifthereexistsdark
matternearthegalacticplane,thenitisprobablybaryonic[80].Thiscomplexityandnon-local
distributionofDMhasbeendiscussedasanindicationtodeeper,newphysics,betterdescribed
bymoregeneralmodels.WithoutknowingthenatureofCDMparticles,CDMcosmologyin
factreproducesphenomenologicaldatabutdoesnothavepredictivepowerapartfromthebare
CDMhalosthemselvesiftheeffectsofnormalmatteronCDMareneglected[163].Hence,
alternativemodelshavebeendiscussedwiththeideathatDarkMatterphenomenologyrather
reflectsdeeperphenomenawhicharenotyetrightlygivenwithinstandardtheories.Sanders
14StrongCPsymmetryshouldntinprinciplehavetobeconserved.However,abreakingwouldberelatedtoayetunobserved
neutronselectricdipolemoment.InordertoexplainstrongCPconservation,theaxions,as(pseudo-)Goldstoneparticles(cf.Chapter
v3.1)anishesfromtheundertheespontaneousxistenceofbreakingtheseoftheparticles(global)whichthenPeccei–Quinnacquiremasssymmetryby,meansareofrelatedQCDtovtheefacuumfectiefvefectsstrong(h.t.[191]).CP-violatingtermwhich

20

CHAPTER2.ELEMENTARYPARTICLESANDTHESTANDARDMODEL

model[215](FLAG),forinstance,addsaYukawapotentialtothenewtonianpotential,andre-
producesrotationcurvesofgalaxiesrangingsizesfrom5to40kpc.Furthermore,Milgroms
model(MOND,MOdifiedNewtonianDynamics)takesthephenomenologyofmissingmassas
asignalofabreakdownofnewtoniangravity[14],anditassumesamodificationofNewtons
lawbelowacriticalaccelerationa0sothat
F=mµ(a/a0)a(2.3.1)
isvalidforNewtonssecondlawofmotion,withµ(a/a0)=1forhighaccelerationsabut
withµ(a/a0)=a/a0forloweraccelerationsa<a0.Herewith,thecriticalaccelerationreads
a0=1.2∙10−10ms−2,whichisveryclosetothecosmologicalvalueprovidedbytheHubblerate
HwithaH=Hc.ItisalsoclosetotheobservedaccelerationaΛgottenfromtheexpansionrate
oftheUniverse[163].Furthermore,subsequentlytoMilgromsapproach,thereistheconstant
elocityvtangential

vt=4GNM1a0(2.3.2)
forrotationcurvesofgalaxies(withmassM1)outsideoftheirluminouscores[15].Hence,
phenomenologyofDarkMatterappearsasconsequenceofnewdynamics.
Therehavebeenapproachestofurthergeneralizemodification-approachesintocovariantfor-
malisms.Forinstance,Tensor–Vector–Scalargravity(TeVeS)reproducesMONDinthenonre-
lativisticlimitwiththepossibilitytoexplaingravitationallensing.TeVeSincorporatesvarious
dynamicalandnon-dynamicaltensor,vectorandscalarfields[15].AfurtherapproachisMof-
fatsScalar–Tensor–Vectorgravity(STVG)[165]orModifiedGravity(MOG),whichpostulates
theexistenceofavectorfieldwhileelevatingthethreeconstantsofthetheorytoscalarfields.In
theweak-fieldapproximation,thistheoryproducesaYukawa-likemodificationofthegravita-
tionalforceduetoapoint–sourcesothatfarawayfromagravitationalbody,gravitybestronger
thanaccordingtonewtonianlaw.Atshorterdistances,gravityistobecounteractedbyare-
pulsiveforcefromthevectorfield.STVGhasbeensuccessfullyusedtoreproduceflatrotation
curvesofgalaxiesamongotherphenomenawithoutthenecessityofDarkMatter[39].Itfur-
therleadstonon-singularsphericallysymmetricsolutions(greystars)[167]andtonon-singular
cosmologieswithabouncinguniversewithoutcosmologicalconstant[166,168].Furthermore,
thereareformalanalogiestofurtherapproachesas[70]whichhasalsobeenusedtoaccountto
thephenomenologyofDarkMatter[20,179].Further,following[71],Higgsparticles,which
areexpectedtobefoundintheLHCinGeneva,woulddecoupleandremainstable.Inthiscase,
negativeresultsfromhigh-energyexperimentswouldsigntosuchachangingofdynamics(see
later).and6ChapterInshort,thenatureofDMisstillunclearandamatterofdiscussion.ColdDarkMattercandi-
datesarestillnoexperimentalreality,andalternativemodelsofaltereddynamicshavebeenable
tosuccessfullyaccountforexplanationsofphenomenology.

2.4Dark-energydensityanddensityparameters
ColdDarkMatterisusuallydefinedwithinthedarksectorofenergydensityoftheUniverse.Anothersector
isgivenbybaryonicmatterwhichcontributestoabout10%oftotalmatterdensity.Other,howeversmall
contributionstomatterdensitywouldcomefromneutrinomasses,leptonsandsoon.Auniverseinwhich

2.4.DARK-ENERGYDENSITYANDDENSITYPARAMETERS

21

matterdensitygivesthetotalenergythereis,isknownasEinstein–deSitterUniverse.There,theenergy
densityequalsexactlytheenergydensityneededfortheuniversetobeflat(c)(cf.Chapters2.4and8.4).
However,Einsteinhimselfintroducedbackin1917theconceptofthecosmologicalconstant[82]which
wouldactagainstgravitationalattractioniftheconstantwerepositive.Einsteinscosmologicalconstantacts
againstgravity,orequally,ashavinganegativepressure.Theideawastogetacloseduniversewhichwould
bestaticalso.Forthis,EinsteinreplacedGµνinhisequations(A.4.3)byGµν+Λ0gµν.Λ0,further,is
interpretableastheenergydensityofvacuum.
parametersdensitydefineweIfiiΩi=c=c,(2.4.1)
whereasi=ic2istheenergydensitytothemassdensityi,andc=3H02/(8πG)isacriticaldensity
definedintermsofG,candtheHubbleconstantH0(which,ontheotherhand,isameasureofthecosmic
expansion),thenwehave
Ωtotal=ΩBaryons+ΩCDM+ΩΛ+...,(2.4.2)
whereasΩBaryonsandΩCDMgivethemostrelevanttermsofmatterdensityΩM.ΩΛgivesthedensity
parameterofthecosmologicalconstant/energyofvacuum,withtheenergydensityΛ.Whenfinite,Λisto
representanenergyofnon-electromagneticalnature.Thus,itcanbedenotedasdark.Further,itsnatureis
notclarified.Hence,itsentitledDarkEnergy,anditistheseconddarksectorofcosmology.Furthermore,if
nonvanishing,thenitispossiblethatitbeconstant,exactlyaswithinEinsteinsapproach(Λ0),orafunction
oftimewithamorecomplexnature(Λ).
AparticularcandidateforDarkEnergyisthescalarfieldcommonlyknownasQuintessenceorcosmon
field[192,238]asatheoreticalcarrier.Thisisgenerallycoupledminimallytogravitationinmodernstan-
dardtheories,orwithascalarfieldcouplingtoRwhichstaysalmostconstant(cf.[124]).Cosmologies
containingabarotropicfluidplusascalarfieldmayleadtolate-timeattractors(cf.[58]),andacoupledsys-
temofgravityandascalarfieldmayinduceafurthertime-dependenttermintheenergy–momentumtensor
whichwouldadjustitselfdynamically[239].Hence,thereappearsacomposition-dependentgravityasa
long-rangeforce[242]mediatedbythequintessenceparticles.Quintessenceparticlescanfurtherbehave
similarlytorelativisticgases[241]andbeassociatedtoDM[240].
Quintessenceisrelatedtothecosmologicalconstant(sc.[78]).Thelatter,however,representsaspecial
caseofDarkEnergythatdoesnotchangewithtime(cf.[193])butwhichshouldalsobeexplainedwithina
vitation.graoftheoryquantumDarkEnergyisrelatedtothephenomenonofcosmicacceleration(seeChapters2.4and8),andsometheo-
riesasSupergravityleadnaturallytoantigravityindeed[217].Antigravitativeinteractionswouldleadtoa
repulsionofmatteraftertheBigBang.
Auniversewithpositivespatialcurvature(K=1)withanonvanishingcosmologicalconstantisknownas
Lemaîtresuniverse[152].Theexpansionparameterinsuchauniverseisalwaysincreasingbutthereisa
periodinwhichitremainspracticallyconstant.Thereafter,afurtherperiodofexpansionfollows.During
the1970s,thismodelinvokedtoexplaintheapparentconcentrationofquasarsataredshiftofz≈2[56].
However,giventhatsubsequentdatafalsifiedthisassumption,foralongtimeDarkEnergybecamestrongly
believedtobevanishing.Actually,Einsteinhimselfcalledthecosmologicalconstanthisbiggestblunder
(diegrößteEseleimeinesLebens).Yet,alreadyworksas[29]and[129]proposeanonvanishing,however
over-abundantcosmologicalconstantforaslightlyclosed(K=1,Ωtotal1)baryonic-matterdominated
Universe.Still,untilthelatedecadeofthe1990s,therewereonlyfewstrongempiricaldatawhichwould

22CHAPTER2.ELEMENTARYPARTICLESANDTHESTANDARDMODEL

ΩB=0.0223h−2

(2.4.4)

(2.4.5)(2.4.6)

pointtoantigravitation.Further,mostexperimentaldatauptothatpointactuallypreferredanexactlyvani-
shingΛandanEinstein–deSitter(closed)Universe(sc.[197]).
Itwasonlyinthelastdecadeofthe20thcenturythattheassumptionofavanishingΛbegantofallapart.
AnonvanishingvalueforDarkEnergywasmeasuredwithinthecontextofGRforSuperNovaeoftypeIa
(SNeIa)asextragalacticdistanceindicators[98,198,208].15Intheyearsthatfollowed,theresultswerecor-
roborated.Thus,cosmicexpansionseemstobeaccelerated,indeed.However,bynowitisunclearwhether
thevalueofthisdarkenergy(asantigravitativecomponent)staysconstantintime,asatruecosmological
constantΛ0,orwhethertodaysdark-energycomponentisaremainderofsomecosmologicalfunction.This
functionshouldcontributeasΩΛ≈0.7todaytothetotaldensityparameterΩTofthehodiernalUniverse.
Nowadaysstandardmeasuredvaluesofthemodelsare
ΩM=0.127h−2(2.4.3)
including,matterforΩB=0.0223h−2(2.4.4)
andbaryons,forΩDM=0.105h−2(2.4.5)
forDarkMatter.h=0.73givesthenormalizedmodernHubbleexpansionrate.
Forneutrinos,theconstraintliesat
Ων<0.007h−2,(2.4.6)
readsdensitycosmological-constanttheandΩΛ=0.76.(2.4.7)
Accordingtothethree-yearresultsofWMAP,thetotalenergydensityparameterliesaround[223]16
ΩT=1.003−+00..017013.(2.4.8)
Anexactvalueof1meansacurvatureK=0ofaflatuniverse,whilehighervaluesmeanacloseduniverse
withK=1,andloweronesindicateanhyperbolicuniversewithK=−1.Hence,observationalvalues
pointtoadark-energydominantUniversewithalmostonlydarksectorsandwithan(almost)flatgeometry.
Furthermore,ideasofaveryhighlyaccelerated(inflationary)phaseoftheUniversewhichexplainhorizon
andflatnessproblemsofcosmologydoaccounttothisinterpretation.
Theconceptofprimeval,cosmicInflationwasfirstproposedbyAlanGuthin1981[118],basedonideas
ofStarobinskyswork[224].ItwaslaterimprovedbyAlbrecht,Steinhardt[1]andLinde[154].Often,
anhypotheticalscalarfield,namelytheinflatonfield,isproposedinthiscontext.Further,itcanberepro-
ducedwithinducedgravitationalso[47–49].Inallways,thisphaseisinterpretableasaphaseinwhich
15SNearevariablestarswhich(simplified)resultfromaviolentexplosionofawhitedwarfstarwhichhascompleteditsnormal
thestellarstarlifeintheandsupernowherevafusionexplosion.hasFceased.ortheAftertypehaIavingespeciallyignited,thecarbonspectrumfusion,shothewsareleasedlackofenerhgyydrogenandlinessubsequentbutindicatescollapsehassingly-ionizedunbound
silicon.16ThebestfitofWMAPreadsforthefive-year+0.results:0060ΩT=1.099−+00..085100.ThebestfitofWMAPplusSuperNovae(SNe)and
baryonacousticoscillationsreadsΩT=1.0050−0.0061[128].

2.4.DARK-ENERGYDENSITYANDDENSITYPARAMETERS

23

negativepressuredominatessothatadeSitterepochappears(seealsoChapter8.7).However,itisstill
unclearwhetherthepressuretermofdark-energeticsectorsisconstantornot.IfDarkEnergycomponents

shouldchangeintime,though,thescalarfieldofQuintessencemightbeonethatactsonlocalplanetary[95]
oratgalacticscales[160].Moreover,ifcouplednonminimallytogravity,suchmassivefieldsmighteven

accounttoboththephenomenologyofDarkMatter[20,210]andDarkEnergy[21,179].Actually,thecos-
mologyofscalar–tensortheories,i.e.theorieswithcurvaturecouplednonminimallytoscalarfields,leads
naturallytocosmicacceleration[44].Thismakesscalarfieldsofsuchtheoriesthenaturalcandidatestobe
quintessential-likefields[7,32,42].

24

CHAPTER

2.

YARELEMENT

PTICLESAR

AND

THE

ARDANDST

MODEL

3Chapter

Symmetrybreakingandscalarfields

–TheconceptofsymmetrybreakinginitsdifferentmodesandespeciallytheHiggsMechanismofmass
generationarediscussedintheirrelevanceforthedifferentsubjectsofphysics,especiallyinrelationwith
theStandardModel(SM)ofelementaryparticlephysics.HiggsandGoldstonefieldsarepresentedtogether
withunitarygaugeandmasstermsoftheSM.ThisChapterisrelatedtotheworkpublishedin[22].–

3.1Symmetrybreakingandbreakingmodes
Thequestionofwhetherscalarfieldsexistatallisstillopen.However,approachesforprimevalInflationand
ofQuintessenceofDarkEnergygroundonsomekindofscalarfieldswhich,therefore,maycontributeto
somekindofdarksectorofdensity.Furthermore,theHiggsfield,aspecialkindofscalarfield,isnecessary
forsymmetrybreaking(SB)indeed,asYang–Millstheoriesforelementaryinteractionsarenon-physical
withoutsomekindofbreakdownofsymmetrywhichmayleadtotheappearanceofmassinaccordanceto
empiricaldata(vizweakCPbreaking,asin[244]).
Therearethreemainmodesofsymmetrybreaking,dependingonthepropertiesofthefieldsgroundstate.
[115]areThese(i)theWigner–Weylmode,usuallycalledonlyWignermode,
(ii)theNambu–GoldstoneorGoldstonemode,
(iii)theHiggs–KibbleorHiggsmode.
modes:symmetry-breakingtheAbout•TheWigner–Weylmode:1
Inparticular,theWigner–Weylmodeisthemostusualsymmetry-breakingmodeinquantummecha-
nics(QM),witharealinvariantvacuumwhichcanbeidentifiedwiththeclassicaloneasfollowsin
virtueoftheDiracvector|0>forvacuumandaunitarytransformation(timeevolution)Uactingon
same,the

U|0>=|0>.
1AfterEugenePaulWigner(1902-1995);HermannKlausHugoWeyl(1985-1955).

25

(3.1.1)

26

CHAPTER3.SYMMETRYBREAKINGANDSCALARFIELDS

TheWigner–Weylmodeisindeedrelatedtotheexistenceofdegeneracyamongparticlesinthemulti-
pletstructureofspectra.Theviolationofsymmetriesinvolveshereexplicitsymmetry-breakingterms
intheHamiltonianHorintheLagrangianwhichliftthemultipletdegeneracies.Suchsituationap-
pearsintheZeemaneffect:givenasphericalsymmetricsystemsuchasanatom,intheabsenceof
externalfieldsthewavefunctionsformdegenerateSO(3)multipletsasaconsequenceoftheconserva-
tionofangularmomentum.Ifwenowplaceamagneticfieldalonganaxis,therotationalsymmetryis
lostsinceapreferreddirectionhasbeenselectedinspace.Thecorrespondingnondegeneratemultiplet
structureistheZeemaneffect,andwhenitappears,SU(2)symmetryhasbeenbrokendowntoU(1)
sincethesystemisstillinvariantunderrotationsaboutasingleaxis.2
AnothercaseofaWignermodemaybegivenbytheSU(N)isovectorwhichmaybe(forinstance)the
SU(2)multipletstructureofisospin.Itistherest-groupoftheSU(3)flavormultiplet.SU(2)breaks
fromthatsymmetryduetoeffectsofhypercharges.Furthermore,thissymmetryofisospinisalso
brokentoU(1)chargesymmetrybytermsofCoulombinteractionsthatselectapreferreddirection
inisospinspace.However,theU(1)symmetryremainsunbrokenbecauseofcurrentconservation
[115].wla

•TheNambu–GoldstoneandHiggs–Kibblemodes:3
Further,intheNambu–GoldstoneandHiggs–Kibblemodes,thesymmetryisactuallynotlostbut
camouflagedandhiddeninthebackgroundofthemassgenerationbyscalarfields.Itisusuallyspo-
kenaboutspontaneousbreakingofthesymmetry.However,ondetail,itissometimesdifferentiated
betweenadynamicalandaspontaneoussymmetrybreaking(SSB)byvirtueofthenatureofthesca-
larfieldwhichleadstothebreaking.Bothkindsofthesesymmetrybreakdowns(SB)throughscalar
fieldsdifferinthefollowingway:
–DynamicalSB:TheHiggsfieldisacompositeparticlesuchasameson,forinstance,oraCooper
pairaswithinsuperconductivity.
–SpontaneousSB:TheHiggsfieldiselementary.
Bothsymmetry-breakingprocesseswhichbelongtotheNambu–GoldstoneandtotheHiggs–Kibble
modeormechanismsofsymmetrybreakingareveryimportantwithinmanyaspectsofphysics,such
ascondensed-matterphysics(wheretheyfirstappeared)andelementary-particlephysics(whereitis
spokenaboutelementaryfields).Forinstance,withinQCD,SBleadstothePeccei–Quinnmechanism
(v.s.inChapter2.3).Furthermore,theHiggsmodeofspontaneoussymmetrybreakingisofspecial
relevanceasabasisfortheSMofparticlephysicsasawhole.Further,thedifferentiationbetween
fundamentalityandcompositenessofHiggsfieldsisusuallynotdeclaredspecifically.Hence,theter-
minologyofSSBisusuallyusedforbothdynamicalandtrulyspontaneousSB.Bothmaybeexplained
analogouslytoeachother,beonthegroundsofafundamentalmechanismorofaneffectiveone.
Fortheunderstandingoftheconceptofspontaneousbreakdownofsymmetry,letusconsiderasystem
whoseLagrangianLpossessesaparticularsymmetry,whichmeansthatitsLagrangianisinvariant
underthecorrespondingsymmetrytransformations.Lmay,forinstance,besphericallysymmetric,
i.e.invariantunderspatialrotation.Twosituationsarethenpossiblewhenclassifyingenergylevels
2TheZeemaneffect(Nobelprize1902)isrelatedtoamomentum–field-strengthcoupling,ESRandNMR,cf.AppendixB.1
3AfterJ¯oichiro¯Nambu(1921),Nobelprize2008;JeffreyGoldstone(1933);PeterHiggs(1929)andThomasWalterBannermann
(1932).Kibble

3.1.SYMMETRYBREAKINGANDBREAKINGMODES

27

ofthissystem[157]:ifagivenenergylevelisnon-degenerate,thecorrespondingenergyeigenstateis
uniqueandinvariantunderthesymmetrytransformationsofL.Ontheotherhand,thelevelmaybe
degenerateandtheeigenstatesnotinvariantbutabletotransformlinearlyamongstthemselvesunder
symmetrytransformationsoftheLagrangian.Letusfurtherconsiderthelowestenergylevelofthe
system.Ifitisnotdegenerate,thestateofthelowestenergyofthesystem(thegroundstate)will
beuniqueandpossessthesymmetriesofL.Inthecaseofdegeneracy,therewillnotbeaunique
eigenstatetorepresentthegroundstate.Arbitrarilyselectingoneofthesedegeneratestatesasground
statewillleadtothegroundstatenotsharingthesymmetriesoftheLagrangian.Thesymmetrywillbe
brokenforthegroundstate.Wehavespontaneousbreakdownofsymmetry(whichmaybedynamical,
though;v.s.).Theasymmetryis,however,notduetoaddinganon-invariantasymmetrictermtoLbut
tothearbitrarychoiceofoneofthedegeneratestates.
Afurtherexampleofdynamicalorspontaneoussymmetrybreakingmaybefoundinferromag-
netism[157]:Inaferromagneticmaterial,theforceswhichcoupletheelectronicspinsandhence
theHamiltonianofthesystemarerotationallyinvariant.However,inthegroundstatethespinsare
alignedinsomedefinitedirectionresultinginafinitemagnetizationM.Theorientationofitisarbi-
trary.Thus,wehaveacaseofdegeneracy.Furthermore,excitedstatesobtainedfromthegroundstate
bysmallperturbationsalsodisplaythisasymmetry.
Inquantumfieldtheory,thestateoflowestenergyisthevacuum,andspontaneoussymmetrybreaking
isonlyrelevanttofieldtheoryifthevacuumstateisnon-unique(else,thereisaWignermode).Y.
Nambu[169–171]recognizedinthecontextofsuperconductivitythatinmodelsexhibitingsponta-
neousbreakdownofcontinuoussymmetriesnewparticleshadtoappear.Forthisdiscovery,Nambu
wasawardedtheNobelprizein2008.Furthermore,J.Goldstone[107,108]recognizedthesame
soon-afterandsystematicallygeneralizedtheconceptintoquantumfieldtheory.Itimpliesthatsome
quantityinthevacuumisnon-vanishing,notinvariantundersymmetrytransformationsofthesystem,
andcanthereforebeusedtocharacterizeaparticularvacuumstateasthegroundstate[157].Usually,
thisquantityistakenasthevacuumexpectationvalueofaquantizedfield.Thisfield,further,must
beascalarfield(φ(x))sothatthevacuumstatesareinvariantunderLorentztransformations.Further,
thevacuumexpectationvaluemustbeconstant,sothat

<0|φ(x)|0>=φ0=const=0(3.1.2)
isvalidforthemeanvaluewithaground-stateconfigurationφ0.Theappearingparticlesarespinless
bosonswhichcorrespondtothebrokeninternalsymmetrygenerators.Somearemassive(generally
calledHiggs)andtheothers,usuallycalledNambu–GoldstonebosonsorsimplyGoldstonebosons,
aremassless.Theirvanishingmassisaconsequenceofthedegeneracyofthevacuum,andsuch
bosonsfrequentlyoccurintheorieswithspontaneoussymmetrybreaking:
GoldstoneTheorem:Ifacontinuousglobalsymmetryisbrokenspontaneously,foreachgroupgene-
ratortheremustappearinthetheoryamasslessparticle[115].
However,noGoldstonebosonsareobservedinnature,anditishenceofcrucialinterestthatgauge
theorieswithspontaneous(ordynamical)symmetrybreakingdonotgeneratethem[157].Thisis
achievedviatheHiggsmode.ThetwomodesusingHiggsfields(compositeorelementary)differ
fromeachotherthroughtheirgaugesymmetrywhilebothofthemaregivenbythevacuumdefinedas
ws,follo

U|0>=|0>.

(3.1.3)

28

CHAPTER3.SYMMETRYBREAKINGANDSCALARFIELDS

TheNambu–Goldstonemode,however,worksgloballywhiletheHiggs–Kibblemodeactslocallyin
viewofgaugeinvariance.Asaconsequence,themaindifferencebetweenthemisthatintheNambu–
Goldstonemechanismbothmassive(Higgs)andmassless(Goldstone)particlesappear,whileinthe
Higgsmechanismonlythemassiveparticlesarepresentandthemassacquisitionofgaugebosonsis
atthecostoftheGoldstoneparticles,whicharetogaugeawayunitarily.Thedegreesoffreedomof
themassiveparticles,however,wontdisappearfromthephysicalspectrumofthetheory.Ingeneral
sense,thegaugefieldswillabsorbtheGoldstonebosonsandbecomemassivewhiletheGoldstone
bosonsthemselveswillbecomethethirdstateofpolarizationformassivevectorbosons[115].
TheeliminationofGoldstonebosonsfromthetheorygivingmasstothegaugequantawasindepen-
dentlyworkedoutbyP.Higgshimself[126]aswellasbyR.BroutandF.Englert[85]andbyG.
Guralnik,C.R.HagenandT.Kibble[117](hence,theHiggs–Kibblemechanismissometimescalled
Brout–Englert–Higgsmechanism).ThemassgenerationbyHiggsmechanism,however,canfurther
beidentifiedintheMeißner(or–asfollowing–Meissner)effectofconventionalsuperconductivity
(henceapplicableinnonrelativistictheories[112]informofadynamicalbreaking)[178].Goldstone
bosonscanbemadetodisappearinthepresenceoflong-rangeforces[9].Ananalogybetweenthe
HiggsmechanismandtheMeissnereffectmaybeexplainedintermsoftheYukawa–Wickinterpre-
tationoftheHiggsmechanismwherelong-rangeforcesasCoulombinteractionsaremediatedby
masslessexchangeparticles.Thelong-rangeforce,then,isshieldedbytheGoldstonefieldandbe-
comesshort-ranged.TranscribedbymeansofYukawastheory,aneffectivemassofthegaugeboson
wasgenerated.Thecondensedelectron-pairs(theCooperpairs)inthegroundstateofasuperconduc-
tormaythenbeidentifiedwithaHiggsfieldfordynamicalsymmetrybreaking.TheHiggsfieldthen
leadstothemagneticfluxexpulsionwithafiniterangegivenbythepenetrationdepth,whichfurther
givesthereciprocaleffectivemassacquiredbythephotons[115](cf.Chapter4).

Intheprocessesofsymmetrybreaking,thesymmetrygroupGbreaksdowntoarest-symmetrygroupG˜
(i.e.G→˜G)with

nG˜=G˜r,(3.1.4)
=1rwheren>1isvalidincaseofmorethanonebreakingprocess.IntheSMofparticlephysics,forinstance,
thefollowingbreakingprocessesarevalid,
SU(3)C⊗SU(2)L⊗U(1)Y→SU(3)C⊗U(1)em,(3.1.5)
whileforthegrandunifiedtheory(GUT)underSU(5)(Georgi–Glashowmodel,see[102]),togiveafurther
exampleoftheoreticalapproaches,anotherbreakingprocesstakesplaceatenergiesofabout1015GeV,

SU(5)→SU(3)C⊗SU(2)L⊗U(1)Y.(3.1.6)
WithinGUTs,athighenergies,allelementary(quantum)interactionsaretounifyintooneinteractionwhich
reliesonthespecialunitarygroupwithfiveisotopicparticles.Thebreakingprocessontorest-symmetry
groupsisrelatedtoabreakingofsymmetrywhenenergyscalesarelowenoughandtheorderedstate
becomesunstable.Thisprocessofbreakingofsymmetryischaracterizedbythescalarfieldasidenti-
fierofdisorderintermsofa(dis-)orderparameteridentifiedwiththescalarfielditself.Thisparameter

3.2.HIGGSFIELDSANDHIGGSKINDS29
istheGinzburg–Landau(alsoGinsburg–Landau)parameterwhichgivesthemostlikelystateofasystem
(cf.[105]).Itbecomesnonvanishingwhenorder,symmetry,isbroken.Furthermore,thisisidentifiedwith
theappearanceofparticleswhichareagainrelatedtothescalarfield.
TheSMisgivenbyaproductgroupentailingcolordynamicsofQCD,electroweakinteractionsandamixed
interactionofhypercharges.Forlowenergies,thisleadstoelectromagnetismwhileweakprocessesdisap-
pear.GUT,ontheotherhand,describesaunifiedinteractionwheretheleft-handisovectorentailsthefive
elementaryfermions,antielectron,neutronandthreequarksofdifferentcolorasindistinguishable,isotopic
particlesunderGUT(very-highenergy)interactions.Theright-handstate,further,isamatrix.Giventhe
stateforGUTunderSU(5),apartofgaugebosons,therehavetoexistintotal24gaugebosonswhichhave
toleadtodecayprocessesfromleptonstoquarkswhichareforbiddenundertheSM.Asaconsequence,free
protonswoulddecay.However,nosuchsignatureshavebeenfoundandthelowerlimitoftheprotonhalf-
timeliesat6.8∙1033years[183].Still,Georgi–Salamsmodelrepresentsthebestpedagogicalexampleofa
unifyingmodelwithmorethanonebreakdownofsymmetry.Furthermore,withinGUT,symmetrybreaking
isspontaneous,asitiswithintheQADinGlashow–Salam–WeinbergsmodeloftheSM.Thiscomprises
thenecessityofadditionofthetermsofanewparticleintotheLagrangianinformofascalarfieldφ(x).
ForboththeNambu–GoldstoneandtheHiggs–Kibblemechanism,anewfieldhastobepostulated.Thisis
field.Higgscalled3.2HiggsfieldsandHiggskinds
Ingeneral,thesimplestwaytogeneratethespontaneousbreakdownofsymmetryistointroduceaHiggsfield
LagrangiantermcorrespondingtoabosonicscalarparticlewithkineticenergydensityT=(1/2)φ;†µφ;µ
andself-interactiongivenbyapotentialdensityV(φ),
LH=L(φ)=21φ;†νφ;ν−V(φ).(3.2.1)
Theself-interactionpotentialdensityiscalledHiggspotentialV(φ)with
2V(φ)=µ2φ†φ+4!λφ†φ2+V˘,(3.2.2)
whereµ2<0andλ>0.Suchtheoriesarecalledφ4-theories.For
4V˘=23µλ,(3.2.3)
theminimumofthepotentialisloweredsothatenergydensityforvanishingscalarfieldsisdefinedaszero
withV(φ0†φ0)=0(3.2.4)
forthegroundstate(φ0)ofthescalarfield,andwithhermiteanconjugateφ†φ=φ∗φincaseofisoscalar
fieldsφ,andwiththetransposeTincaseofisospinors.TheadditivetermVdoesnotappearinChapter6.1
but,asitwillbeseen,thechoiceoftheminimumofthepotentialisrelatedtotheelectionofavanishing
formalcosmologicalconstantwhich,however,canbeavoidedinthetheorybyaddingaconstantterm
VΛ0=−3α˘µ2Λ0,(3.2.5)
λ4

30

CHAPTER3.SYMMETRYBREAKINGANDSCALARFIELDS

withΛ0asthecosmologicalconstantandwithatotalpotentialoftheform
VT(φ)=V(φ)+VΛ0.(3.2.6)
AcosmologicalfunctionΛ(φ)whichisdependentonthisgeneralizedHiggspotentialappears,asitmaybe
46.2.ChapterinseenTheφ4terminthepotential(3.2.2)isnotbilinear,anditiscrucialfortheapparentsymmetrybreakdown.
TheLagrangiangivenbyequation(3.2.1)isinvariantunderspatialinversion(i.e.φ→−φ)withthefeatures
ofthetachyoniccondensation(i.e.condensateforanimaginarymasswithµ2<0).Suchconditionsare
neededtostaywithintheHiggs–Kibblemode,whichotherwisebecomesaWigner–Weylmodewithclassical
vacuumwhereself-interactionslacktoproducethenecessaryHiggsmechanismattherelativelylowenergies
ofthehodiernalUniverse.Furthermore,theseconsiderationsleadtofurtherpropertieswhichareessential
general.infieldsHiggsofBeageneralHiggsfielddefinedas(cf.[93])
•afieldwithanon-trivial,i.e.nonvanishingvacuumstate.
ThiskindoffieldshavethepropertyofbreakingthesymmetryofatheoryinagroupGontherest-
symmetrytotheisotropygroupG˜ofthevacuumstatespontaneously.

•Moreover,everyHiggsfieldinafieldtheoryinteractsgravitationallywiththeparticleswithwhichit
69]).[68,sc.(couples

–AusualsymmetrygroupisGQAD=SU(2)L⊗U(1)YwhichbreaksintoU(1)eminthestandard
modelfortheelectroweakinteraction.

–Inthissense,aHiggsfieldismoregenerallydefinedasonlytheHiggsfieldwithintheSM.Fur-
thermore,ifaHiggsfieldiscouplednonminimallytothecurvaturescalarR,somecharacteristics
mayeasilydifferfromtheonesofstandardHiggsoftheSM.Otherimportantcharacteristicsare
openandhavethentobegiventhroughtheLagrangeextensionsasisthecasewithintheSM,too.

3.3SymmetryBreakingandtheSM
TheSMofelementaryparticlephysicshasbeenremarkablysuccessfulinprovidingtheastonishingsynthesis
oftheelectromagnetic,weakandstronginteractionsoffundamentalparticlesinnature[151,199].Inthe
Glashow–Salam–Weinberg(GSW)theorywithoutsymmetrybreakingneitherYang–Millsequationsnorthe
Lagrangianitselfpossessmasstermsatall.OnlyforQCDprocesses,amasstermwithmassmfmay
bedefined,givenconservationofparitysymmetryinstronginteractions.Atthispointofconsiderations,
theGSWtheorydescribesmasslessfermionsandleptons.Hence,itcannotdescribenatureasweknow
it.Suchamass,further,cannotbeachievedaddinganewmasstermtoitsLagrangian.Suchwouldbreak
withphenomenologyofelectroweakdynamics.Thesearecharacterizedbyparityviolation,andwithan
addedmassterm,left-andright-handedparticleswouldcoupleinthesamewaytovectorbosonsinorderto
4potentialThereVis.VFrom=ChapterVd3x4on,andweT=willnoTd3furtherx.difHoferwever,betweenusuallythemnoeformalxplicitly.differenceismadebetweenpotentialdensityVand

3.3.SYMMETRYBREAKINGANDTHESM

31

preservegaugeinvariance(astheydowithinQCD).Further,ifmassissimplyadded,amassivepropagator,
whichgivestheprobabilityamplitudeforaparticletotravelfromonepointtoanotherinagiventimeor
totravelwithacertainenergyandmomentum(inthiscaseformassivevirtualparticles;cf.AppendixB.1)
wouldnotloseitslongitudinalterm.Thepropagatordoesnottransformintoa(transversal)masslessonein
thelimitM1→0formassM1[139].Asaconsequence,whenaddingmasses,mostFeynmangraphswould
diverge,andthiswouldleadtothemass-containingGSWtheorynottoberenormalizable.5Forinstance,
equationProccathefor

∂νFµν−M12Aµ=−4πjµ(ψ),(3.3.1)
i.e.fortheYang–Millsequationwithmassterm,thereistheGreenfunctiongivenbytheFourier-transformed
νναδσν−pMp2σ
Gσ(p)=−pλpλ+M112.(3.3.2)
ForM1→0,thisGreenfunctiondivergesandhence,themasslessYang–Millsequationdoesnotpossessa
Greenfunction.Theonlyknownalternativeissymmetrybreakingformasstoappearasaconsequenceof
symmetrypropertiesoftheLagrangianinvacuo.Yang–Millstheoriescombinedwiththeso-calledHiggs
mechanismofsymmetrybreakdown,groundingonNambusworkasamechanismofspontaneousbroken
symmetryinsubatomicphysics,leadtotheSMofelementaryparticlephysics.Thepredictionsofthelatter,
suchastheexistenceofweakonsandgluons,havebeenverysuccessful.TheonlymissingpieceoftheSM
particles.HiggstheareAccordingtotheSM,inertialaswellaspassivegravitationalmass6areintroducedasgeneratedsimulta-
neouslywithrespecttogaugeinvariancebytheinteractionwithascalarHiggsfieldthroughtheSSB.Then,
consideringtheHiggsfieldforsmallenoughenergyscales,theHiggsfieldcouplestomatter.Bymeans
ofthisinteraction,matternolongermovesasfastasthespeedoflight.Itspontaneouslypossessesmass.
However,thelatterisgeneratedorexplainedinthetheorybyaninteractionbetweenparticles(howeveronly
withinelementary-particlephysicsandnotwithinGR).
TheHiggsmechanismofSSB[126]providesawayfortheacquisitionofmassbythegaugebosonsand
fermionsinnature,reducingmasstotheparametersoftheHiggspotential.Theseparametersandproperties
caneasilybedescribedbymeansofanisoscalarfield.FortheSM,though,anisovectorialfieldhastobe
definedfortheacquisitionofdifferentmassesforeverycomponentofthefermionicstate.
fields:HiggsIsoscalar•ForisoscalarHiggsfieldsasinChapter3.2,theEuler–Lagrangeequationswithoutextraterm(3.2.3)
give(forahermiteanfluid,thereisφ†=φ∗=φ)
3!∂ν∂ν+µ2φ+λφ3=0.(3.3.3)
Thereistheenergy–stressconservationofφ.Thecanonicalenergy–stresstensorreadsasfollows,
Tνµ=∂L(φ)φ,ν−L(φ)δνµ,(3.3.4)
φ∂,µ5TheproofofthisandhowtorenormalizeQAD,viz[132],wasNobel-awardedin1999fortHooftandVeltman.
6Inertialmassisdefinedasameasureofanobjectsresistancetothechangeofitspositionduetoanappliedforce.Passive
gravitationalmassisameasureofthestrengthofthegravitationalfieldduetoaparticularobject(see[22],especiallyinrelationwith
symmetry-breakingmodesandtheHiggsmechanism).Althoughconceptuallydifferent,Einsteinsprincipleofequivalenceassertsthat
theyareequalforagivenbody,andthishasbeenwell-groundedexperimentally.

32

CHAPTER3.SYMMETRYBREAKINGANDSCALARFIELDS

andtheenergydensityisits0–0component,
32(φ)=21(∂0φ)(∂0φ)+12(∂aφ)(∂aφ)+µ2φ2+4!λφ4,a=1,2,3.(3.3.5)
=1aWiththepossibilityoftachyoniccondensation,thegroundstateφ0becomestwicedegenerateand
φz=0hasamaximalvaluefortheenergydensity.ThegroundstatefortheHiggspotentialwithout
V˘isgivenby
40=(φ0)=−3µ≡min=−V˘,(3.3.6)
λ2andφ0(±)=±−6µ2=v.(3.3.7)
λvisthevacuumexpectationvalue.Regionswithdifferentφ0-valuesarecalledtopologicaldefects.
Thosechangingthevaluesφ=v↔−varetermedinterfacedomains.
Infact,theenergyofthesystemislowandφliesneartheminimumofenergy.Itis,therefore,possible
toexpandthescalarfieldarounditsminimalstatewithitsexcitedvaluesφˆinthefollowingform:
φ=v+φ.ˆ(3.3.8)
TheLagrangian(3.2.1)maynowbegiveninisoscalarform(onlyuptosecond-orderterms)asfollows,
2L(φˆ)=21φˆ†,νφˆ,ν−M2Hφˆ2−3!λvφˆ3−4!λφˆ4=L(−φˆ).(3.3.9)
ThefirsttermintheLagrangian(3.3.9)correspondstothekineticenergyoftheHiggsfieldwhilethe
secondonerepresentsthemasstermofthestandardHiggsfield(i.e.MH2≡−2µ2).Infact,duetothe
presenceofthetermfortheexcitedfield(i.e.φˆ3)intheLagrangian(3.3.9),thesymmetryissuddenly
brokenastheLagrangian(3.3.9)isnotspatiallyinvariantanymore.

fields:ectorialvIso•InthecaseofanisovectorialHiggsfieldwithisocomponentφa,thetreatmentofHiggsfieldsis
analogous.TheLagrangian(withoutconstantterms(3.2.3)and(3.2.5)ofthepotential)isgivenby
2L(φa)=1D†µbaφ†b(Dµacφc)−µφ†aφa+λ(φ†aφa)2.(3.3.10)
4!22TheHiggsfieldisnowcomplex(anisospinor)andbymeansofthecovariantderivative
Dµabφb≡δab∂µφb+igAµi(τi)abφb,(3.3.11)
itcoupleswiththegaugefieldAµ.
readsequationfieldHiggsTheDµabDµbcφc+µ2φa+3!λ(φ†bφb)φa=0.(3.3.12)
Analogouslytotheisoscalarform,thegroundstateφ(0)reads
2φ†(0)aφ(0)a=−6µ(3.3.13)
λ

(3.3.12)(3.3.13)

3.3.SYMMETRYBREAKINGANDTHESM

withthetheVEVasthelength,

33

2v˜=−6λµeiα≡veiα=0,(3.3.14)
andφ(0)a≡vNa(wewilltakeα=0).NaisaunitvectorwithN†aNa=1.Itisusedforgauge
fixing,i.e.tosetwhichfermionscoupletoHiggsparticles,or,whatisthesame,whichparticlesareto
mass.acquireItispossibletochooseα=0withoutmakinganyrestrictiontothesystemsincethisdoesnotdemand
anykindofphysicalchanges.However,thischoicedoesnotallowmasstogothroughthephase
transitionswithoutchangingitsvacuumvalue.Therefore,eveniftheLagrangianisinvariantunder
phasetransitions,itmustsufferthelossofinvarianceexplicitlythroughitsgroundstate,andthe
particlesthatfallinthisstateinteractwiththeHiggsbosonsandslowdown.Inparticular,inview
ofSpecialRelativity(SR),themasslessparticlestravelwiththespeedoflightc,andmassiveones
haveasspeedv<c.Sothemassgenerationoftheparticlesmaybeinterpretedinrelationtotheir
field.HiggsthewithinteractionTheisospinHiggsfieldcomponentφamaybedecomposedinaground(φ(0)a)andanexcitedstate
(φa)as
φa=φ(0)a+φa.(3.3.15)
Theminimumenergyisthengivenbythenon-vanishingHiggsground-statevalue(i.e.v=0)inthe
followingform,analogoustotheisoscalarcase:
4(φ(0)a)=−23µλ.(3.3.16)
Aftersymmetrybreaking,theLagrangiantakesthefollowingform,
L(φ)=L(φa)+const.=1(∂µδab−igAµi(τi)ab)(φ†(0)a+φ†a)(∂µδbc+igAµi(τi)abc)(φ(0)c+φc)
2−µ2(φ†(0)a+φ†a)(φ(0)a+φa)−λ(φ†(0)a+φ†a)(φ(0)a+φa)2.(3.3.17)
4!2UptothesecondorderinthefieldvariablesAµandφ,andwithouttheconstantterm(whichhas
nophysicalrelevance),thelatterequationgivesakinetictermofthescalarfield,amasstermofthe
coupledgaugebosonsandamasstermoftheparticlerelatedtothescalarfield,i.e.theHiggsfield,
L(φ)=21φ†a,µφa,µ+21g2Aµi(τi)abφ†(0)aAµi(τi)bcφ(0)c−4!λ(φ†(0)aφa+φ†aφ(0)a)2.(3.3.18)
Thesecondtermgivesthemassesofgaugebosonsinatheoryofelementaryparticlephysics.The
masstermmayberewrittensothatthemass-squarematrix,whichissymmetricandreal,bethe
parenthesis),Bach(usingwingfollo(M2)ij=g2φ†(0)τ(iτj)φ(0)=4φcg2v2N†τ(iτj)N=(M2)ji.(3.3.19)
ThecouplingconstantαYMisrelatedtothecouplingg2asfollows:
2gαYM=4πc.(3.3.20)
thediagonalelementsofthemass-squarematrixread
√M(i)=2πcαYMv(τiN)†(τi†N).(3.3.21)

34

CHAPTER3.SYMMETRYBREAKINGANDSCALARFIELDS

ThisisthemassofgaugebosonscoupledtotheHiggsfields.ForSU(2),forinstance,thegenerators
τiarerelatedtothePaulimatrices,andskew-diagonalelementsofthemass-squarematrixvanish.
Withinelectroweakinteractions,inhomogeneousYang–Millsequationsthenobtainamasstermof
4-currents.Withgauge-couplingconstantsofU(1)andSU(2)asgiandgi=2g(withi=1andi=2
forU(1)andSU(2)respectively),thegauge-bosonsquaremassofweakon-fieldsWisthensimply
(M2(2))ij=πg22cv2δij=MW2δij.(3.3.22)
istheresc.Furthermore,

(3.3.23).M(12,2)i=πg1g2cv2δi3=MW2δi3g1/g2
andM2(1)=πg12cv2=(g1/g2)2MW2
Abovemasstermscoupletothegaugefieldssuchthatelectroweakcurrents(Yang–Millsequations)
acquireaM(2)−M(1,2)term
(1+ϕ)2MW2δijWλj+gg1δi3Aλ,(3.3.24)
2withscalar-fieldexcitationϕ(cf.[74]).Further,U(1)currentsacquireanM(1)−M(1,2)term
2(1+ϕ)2MW2g1δi3Wλi+g1Aλ.(3.3.25)
gg22Bothterms(3.3.24)and(3.3.25)arenon-diagonal,whichisincontradictiontotheirinterpretationas
masssquaresofphysicalrealparticles.
Furthermore,bothmasstermsmaybetakenascomponentsofavectorX.Itdefinesatotalmass-
squarematrixofelectroweakgaugefieldssuchthat
X=M2Wλκ,(3.3.26)
withκ=1,...4withtheU(1)gaugecurrentasAλ=Wλ4.Themass-squarematrixisnon-diagonal
andpossessesavanishingdeterminant,i.e.itseigenvalueiszero.Thevanishingeigenvectorisrelated
tophotonsasnon-massiveparticles.Further,themassmatrixistobediagonalizedinordertoacquire
massofphysicalparticlesfori=3andi=4.Anorthogonaltransformationistobefulfilledsuch
thatthemasseigenstatesofthegaugefieldsyield
Zµ≡W3µcosϑW+AµsinϑW,Bµ≡−W3µsinϑW+AµcosϑW.(3.3.27)
Inthisrepresentation,physicalZbosonsarerepresentedbythefieldZµ,andphotonsarerepresented
byBµ.ϑWiscalledWeinbergangle,andthetransformationiscalledWeinbergmixture.Thereis
tanϑW=g1/g2withg1cosϑW=g2sinϑW=eforthe(positive)electricchargee.Experimentally,
theWeinberganglehasavalueofϑW≈0.50withsin2ϑW≈0.23.Hence,bymeansoftheWeinberg
mixture,neutralweakonspossessastrongercouplingtomasssuchthatMZ=MW/cosϑW>MW
alid.visfermions:evMassi•ThevectorNgivesgaugefixingandisdependentontheformofthefermionicstateψ.Within
electroweakinteractions,forinstance,ifthefirstcomponentgivestheneutrinostate,N1istobechosen

3.3.SYMMETRYBREAKINGANDTHESM

35

as0.Ifthesecondcomponentofψistheelectron,N2is1.Furthermore,giventhatφ(0)a=vNais
validforthegroundstate,N1=1thenleadstoφ(0)1=0whilethereisφ(0)2=vfortheelectron
component.Hence,electronscoupletoHiggsfieldsandbecomemassive,withamassme∼v.
Meanwhile,neutrinosremainmassless.However,withonlykineticandpotentialtermsofHiggs
particlesonlygauge-bosonmassesareactuallygenerated.Leptonicandquarkmassesarenotyet
given.ForleptonsandquarkstoacquiremassviaHiggsmechanism,afurthertermoftheLagrangian
isneeded.ThistermcouplesthefermionicstateψtotheHiggsfieldφandishencetodependonboth
fields.TherelatedtermoftheLagrangianiscalledYukawacouplinganditisoftheform
L(φ,ψ)=−kf(ψ¯Aφ†aˆxψaA+ψ¯aAxˆ†φaψA)
(3.3.28)≡mf(ψ¯AN†aψaA+ψ¯aANaψA).
WithintheGSWtheory,thesubscriptfdenotesthedifferentgenerationofquarkswhilewithinQCD
itdenotestheflavorforacolortripletofSU(3)C.kfisacouplingconstantrelatedtothefamilyandto
thefermionicmassmf∼kfvaftersymmetrybreaking.Further,xˆiscalledYukawamatrix.Itgives
themassofleptonsandquarksby
mf=kfvN†ˆx+xˆ†N.(3.3.29)
WithYukawacoupling,thepropagatorfortheexchangedboson(i.e.Higgsboson)viatheHiggs
interactionoftwofermionsturnsouttobeinthelowestorderoftheamplitudeequaltothepropaga-
torderivedfromaYukawapotential(i.e.ascreenedCoulombpotential).ThepropagatororGreen
functionofsuchKlein–Gordonequationofamassiveparticleitselfisenoughtodemonstratethatthe
HiggsinteractionisofYukawa-type.Infact,thescalarfield(φa)coupleswithfermions(ψA)through
theYukawamatrixxˆandthemassofthefermions.
SuchHiggscouplingtofermionsismodel-dependent,althoughitsformisoftenconstrainedbysome
symmetries.However,tohaveanaccuratepicture,quantummechanicalradiativecorrectionsareto
beaddedalsoinordertohaveaneffectivepotentialVeff(φ).Sincethecouplingisalsodependent
ontheeffectivemassofthefield,theλµ2φ2andλ2φ4termsfromavacuum-energycontribution
arecausedbyvacuumfluctuationsoftheφfieldandmustbeincorporatedinthesystemtohavea
correctphysicaldescription.Furthermore,thereareadditionalquantumgravitationalcontributions
andtemperaturedependencesothatVeff(φ)→Veff(φ,T)∼Veff(φ)+M2(φ)T2−T4isvalid.
Asaconsequence,symmetrymustberestoredathighenergies(ortemperatures),especiallyinthe
primordialUniverse[47],whichiscontrarytothepresentstateoftheUniverse.
•Goldstonebosonsandunitarygauge:
ThescalarmultipletintheSMbelongstoadoubletrepresentationofthegaugegroupinthefollowing
form,

+φ,=φ0φwhichisdefinedwithanon-trivialvacuumstatehavingthepropertiesofsymmetrybreakingofthe
gaugegroupGtotherest-symmetryoftheisotropygroupG˜.Thecomplexfieldφ0canbefurther
rewrittenintermsofrealfields,i.e.φ0=(σ˜+iχ)/√2.Withthespontaneousbreakdownofgauge
symmetry,theminimalvalueoftheenergydensityuistakenbytheground-statevalueφ0=v
with<σ˜>=v.Theσ˜andχfieldsmaybeidentifiedwithtwoparticles,respectively:theHiggs
andtheGoldstoneparticles.ThesymmetryoftheLagrangianisthenbrokenwhenparticlesfall

36

CHAPTER3.SYMMETRYBREAKINGANDSCALARFIELDS

fromtheirfalsevacuum(withφ=0)totherealone(φ=v).Ingeneral,forsuchSSB(assuming
elementariness),theleastenergyisthenrequiredtogenerateanewparticle(i.e.theHiggsparticle)
withtheassociatedfeaturesoftheself-interactionthanhaveitdisappear.Theseparticlesareexpected
tobefoundinhigh-energyexperimentssuchasintheLargeHadronCollider(LHC),theparticle
acceleratoroftheCERN7inGeneva,Switzerland(sc.theATLASdetector,whichshould,among
others,helptodiscovertheHiggsparticles.).Currentconstraintsarethattheyshouldbefoundat
energieslessthan250GeVandhigherthan130GeV(cf.[22]).
IntheLEP,thepredecessoroftheLHC,Higgsbosonswereexpectedtoappearatelectron–positron
ascollisions

e+e−→HZ.(3.3.30)
VerymassiveHiggsparticlessuchasthosewithintheSMareexpectedtodecayintofourjetswith
60%possibilityintheformofheavyhadrons:
¯bbH→q¯qZ→Further,Zparticlesmayalsodecayintoleptonswith6%possibility,andthereisanotherchannelin
whichHiggsparticlesdecayintoheavyhadronsandτ−τ+pairs.However,thedecaychannel(3.3.30)
hastobedistinguishedfromfarmoreprobablechannelsasthefollowing(cf.[22]),
e+e−→W+W−,e+e−→ZZ,
e+e−→W+W−γ,e+e−→γγ.
Atenergieshigherthan110GeV,though,thecrosssectionforsuchdecayas(3.3.30)isverysmallin
comparisontoallothers.Yet,intheLHCaHiggsmassofuptotwicetheZbosonmassmaybemea-
sured.Theproductionmodeisnowbasedonpartonicprocesses,andthegreatestrateshouldcome
fromgluonfusiontoformaHiggsparticle(gg→H)viaanintermediatetop-quarkloopwherethe
gluonsproduceavirtualtop-quarkpairwhichcouplestotheHiggsparticles.Furthermore,thealterna-
tivesarethechannelsofhadronicjets,witharicherkinematicstructureoftheevents.Thesechannels
arethequark–gluonscattering(q(q¯)g→qq¯H)andthequark–antiquarkannihilation(qq¯→gH).
Nevertheless,thereisstillthepossibilityofmoredecayingchannels,andthegeneralizationsofthe
SM(suchasSuSy)demandtheexistenceofmorepossibledecayswithsupersymmetricparticles.
However,experimentalevidenceisstillneeded,especiallyforsupersymmetricgeneralizations.
HiggsparticlesrepresenttheonestillunverifiedpredictionoftheSM,whichhasprovenverysuccess-
ful.Still,theSMpostulatesHiggsfieldsinordertoberenormalizable[232](i.e.especiallyavoiding
divergencesinperturbationtheory)andsotogetaphysicaldescriptionofreality.However,whereas
intheSMthereisanecessityforHiggsparticlestoappear,Goldstonebosonsarenotpredicted.Fur-
thermore,theirexistencewouldaffectastrophysicalconsiderationswithsomesortofnewmechanism
fortheenergylossinstars.
AccordingtoGoldstonestheorem,Goldstoneparticleshavetoappearwithallglobalgaugeprocesses.
However,theexcitedHiggsfielddiffersfromthegroundstatebyalocaltransformationthatcanbe
gaugedawaythroughaninverseunitarytransformationU−1.Suchunitarytransformationcontains
7CERN:EuropeanOrganizationforNuclearResearch

3.3.

SYMMETRYBREAKINGANDTHESM

theGoldstonefieldλ˜asthegeneratorofunbrokensymmetryinthefollowingform,

a˜U=eiλτa=eiχa.

37

(3.3.31)

Withsuchtransformation,Goldstonebosonsvanish.Hence,thescalarfieldaswellasfermionicfields
ψ,field-strengthtensorsFµνandgaugefieldsAµνaretobegaugedunitarily,andarepresentationof
thetheorywithoutmasslessparticlesoftheNambu–Goldstonemodeisgotten.

38

CHAPTER

3.

YSYMMETR

BREAKING

AND

SCALAR

FIELDS

4Chapter

QCD,superconductivityandsymmetry
eakingbr

–Theconceptofdualsymmetryispresentedforelectrodynamicsinviewofmagneticcharges,Dirac
stringsanddyonswhicharerelatedtoHiggsfields.Further,theconceptofabelianprojectionisusedin
viewofQCDandgluodynamics.InthesamewaythatsuperconductivityisrelatedtoAbelianHiggs
MechanismwithcompositeHiggs-fields/Cooper-pairs,dualsupeconductivityisintroducedasapossible
explanationofconfinementofquarksinhadrons.Thisworkispartlypublishedin[110,176](hereweuse
=c=1).–

4.1Dualsymmetry,monopolesanddyons
Ascommonlyknown,invacuum,Maxwellequationsingeometrical(Gauss)unitsarethefollowing:
∙B=0,∙E=0,
×B=∂∂tE,×E=−∂∂tB,(4.1.1)
withtheelectricfieldvectorEandthemagneticfield(pseudo)vectorB.ThereappearsaZ2dualsymmetry
ofsourcelessMaxwellequations,i.e.invacuumthereisadualsymmetrybetweenthebehaviorofelectric
andmagneticfields.Thisisaninvarianceundertransformationsofthetype
E→B,B→−E.(4.1.2)
However,thecompleteequationsofMaxwellinderivativeformformicroscopicsystemsarethefollowing
intheinternationalsystemofunits(SI),
FirstGausslaw:∙E=/ε0,(4.1.3)
Ampère–Maxwelllaw:×B=µ0j+µ0ε0∂E,(4.1.4)
t∂SecondGausslaw:∙B=0,(4.1.5)
Maxwell–Faradayinductionlaw:×E=−∂B,(4.1.6)
t∂withtheabsolutepermittivityε0andthepermeabilityµ0offreespaceorvacuum,andtheelectriccharge
densityandchargecurrentj.Withingeometricalunits,thereisε0=µ0=1.
39

40CHAPTER4.QCD,SUPERCONDUCTIVITYANDSYMMETRYBREAKING

Thetermε0∂∂tEinequation(4.1.4)iscalledMaxwelldisplacementcurrentjD,anditfindsitsdualanalogue
in∂∂tBofequation(4.1.6).ThisdisplacementwhichwasfoundbyMaxwellleadstoelectromagneticphe-
nomenaarebeingdescribedby(electromagnetic)waves,whichcouldbedemonstratedbyHertz.
Maxwellequations,whichhere-derivedinconjunctionwithhismolecularvertexmodelofFaradaysLines
ofForce[161],showanimpressivesymmetrybetweenelectricalandmagneticphenomenaundertheinser-
tionofMaxwellsdisplacementcurrent.However,alackofsymmetryiseasilynoticed,anditisfoundina
missingterminGausssequations(4.1.5)ofamagneticchargeandamagneticcurrent.Indeed,thisappears
tobeafundamentaldifferencebetweenelectricityandmagnetism:Itispossibletoseparatepositiveand
negativeelectricchargesbutimpossibletoseparatemagneticpoles[110].
ThebreakingofZ2dualsymmetryinelectrodynamicsintheappearanceofelectricchargesisanopenissue
ofphysicsoratleastofthephilosophyofthesame.However,magneticpolesmaybeassumedinviewof
symmetrizationofMaxwellsequations,andindeed,thisissueleadDiracin1931tointroducequantized
singularitiesofelectromagneticfields,whichdemonstratethattheexistenceofameremonopolecanex-
plainthequantizationofthewholeelectricchargeintheUniverse[75,106,202].Forthis,Diracexpanded
Maxwellsequationswithamagneticchargedensityσandamagneticcurrentdensitykbywhichdual
withedpreservissymmetry∙B=σ,×E+∂B=−k.(4.1.7)
t∂Hence,expandedMaxwellequationsshallfurtherfollowthissetoftransformations(ingeometricalunits):
(,σ)T=R(ϑ)(,σ)T,(4.1.8)
(E,B)T=R(ϑ)(E,B)T
(j,k)T=R(ϑ)(j,k)T
whereTdenotesthetransposeandR(ϑ)thesymmetryoperatorwhichisa2×2orthogonalmatrixasgiven
213][12,by

cosϑsinϑ
R(ϑ)=−sinϑcosϑ.
ϑisanarbitraryconstant.Forϑ=π/2,however,itiseasytonoticethattheMaxwellsequationsin
geometricalunitsareinvariantundertheafore-mentionedtransformationswith
E→B,B→−E,→σ
σ→−,j→k,k→−j.(4.1.9)
Further,asymmetricbehaviormayalsobeseenintheenergydensityofelectromagneticfieldswhichinSI
byenvgiis

u=12ε0E2+µ1B2.(4.1.10)
0Accordingly,ifdualsymmetryisgiven,theremustexistaparticlehavingamagneticchargewhichactsasa
sourceofmagneticfields.ThishypotheticalparticleiscalledDiracmonopole.
DualmagneticfieldsdonotsatisfytheusualrelationB=×A,demandingamodificationinthedefinition
ofthemagneticstrengthBintermsofthevectorpotentialAinpresenceofamonopole[53].Assuminga
point-likenatureofmonopolesenclosedbyavolumeτboundedbyaclosedsurfaceS,therefollows[110],
∙Bdτ=σdτ=g,(4.1.11)
τ

4.1.DUALSYMMETRY,MONOPOLESANDDYONS

41

asanintegralformofadual-extendedformofGaussslaw(4.1.7),withamagneticchargegandamagnetic
chargedensityσ.Further,followingtheGaussdivergencetheorem,
∙Bdτ=Bds,(4.1.12)
Sτforaninfinitesimalelementofareads,thereis,accordingto[53,75],
B=×A+A,(4.1.13)
withanadditionaltermA.With(4.1.12)and(4.1.13),equation(4.1.11)yieldsintermsofA,
∙Bdτ=(∙A)dτ=A∙ds=g.(4.1.14)
SττInvirtueoftheusualMaxwellsequations,magneticfieldsshouldbedefinedinsuchawaythattheybe
givenbasicallyby×A.However,Acannotbedefinedasvanishingsinceelse,theright-handsideof
equation(4.1.14)vanishes.Following,DiracpointedoutthatonemightchooseAsuchthatitwerezero
exceptatonepointonthesurfacewhereitisinfinite.Theadditionaltermwouldbeinfiniteatonepointon
eachsurfaceboundinganyvolumeτ.Hence,Awouldhavetobeinfiniteonalinejoiningthemonopoleto
infinity.ThislineofsingularityiscalledDiracstring[110].
InordertoavoidunphysicalfeaturesoftheDiracstringinquantummechanics,whichhasanimplicit
singularbehavior,aswellasarbitrarinessinitslocalizationsuchthatitmaybechosentoliealongany
directionwithasuitablechoiceofcoordinates,Diracputforwardaprinciplebywhichnochargedparticle
wastointeractwithit.Onthatground,somewaystodefineDiracsmonopolewithoutunphysicalities
arose.Furthermore,Dirachimselfwasabletoexplainthequantizationofelectricchargebasedonquantum
mechanicalprinciples.Assumingthemagneticmonopoleasapoint–particlelikeanelectron,heshowed
thatwhenanelectronmovesaroundamonopole,thereisachangeinphaseofthewavefunctionofthe
electron,whichcorrespondstothemagneticfluxandleadstoDiracsquantizationconditionwhichisgiven
ws,folloas

e∙g=n,(4.1.15)
2wherenisaninteger.Theexistenceofmonopoles,therefore,indicatesthattheelectricchargesinnatureare
theintegralmultipleoftheelectricchargeofanelectron.
Furthermore,besidesthemonopoletheremayalsoexistaparticlehavingboththeelectricandmagne-
ticcharge.Thishypotheticalparticleiscalleddyon[216,250].Suchparticlesmaybeunderstoodasa
compositeofchargeandmonopoleand,althoughbothpartsfollowBose–Einsteinstatistics,dyonsareten-
sorialbosonsorspinorialfermions[12].ThegeneralizedDiracquantizationconditiononitschargesis
duetoSchwinger,ZwanzigerandSaha[12,213,216,250],andindeed,unifyingtheoriesofelementary
interactions(GUT)doalsopredictmonopolesanddyons,asfirstpointedoutbytHooftandPolyakovin
1974[133,201].Thesemonopolesareextremelymassiveandstillofnoexperimentalreality.Further,GUTs
usuallypredictnonconservationofbaryonandleptonnumber,bywhichfreeprotondecayintoleptonsplus
otherpartslikemesonsandphotonsisexpected.Suchdecayprocesses,togetherwithmagneticmonopoles,
though,havealsonotbeenmeasured.Thebestcosmic-raysupermassivemonopolefluxlimitliesatless
than1.0∙10−15cm−2sr−1s−1[8].

42CHAPTER4.QCD,SUPERCONDUCTIVITYANDSYMMETRYBREAKING

4.2Covariantformanddyons
Withincovariantformalism,thefield-strengthtensorisdefinedthroughRicciidentitiesby
Fµνab=1[Dµac,Dνcb](4.2.1)
ig=(Aνi,µ−Aµi,ν)−gAµkAνlfkli(τi)ab,
witha4-potentialAν,andwithfkliasastructureconstantdependentonthegaugegroup.Withinelec-
trodynamicsandhencetheunitarygroupU(1),thereisfkli=0andthecovarianthomogeneousMaxwell
alid,vissystemFµν,λ+Fνλ,µ+Fλµ,ν=F(µν,λ)=0,(4.2.2)
meaningthehomogeneousMaxwellequationsfornon-appearingmagneticsources(monopoles)andelectric
chargesassourcesofelectricfieldsfollowingfromBianchiidentities.
TheinhomogeneousMaxwellequations,dependentonmatterandhenceonEuler–Lagrangeequations,are
giveningeometricalunitsby
Fµν,µ=4πjν,(4.2.3)
withthe4-currentofdensityandchargejµ=(,j).Thefield-strengthtensorisanantisymmetrictensor
with

(4.2.4)

(4.2.5)

0−BzByEx
Fµν=Bz0−BxEy.(4.2.4)
ByBx0Ez
−Ez−Ey−Ez0
ItscovariantformyieldsaftertranspositionusingtheMinkowskimetric,
0−BzBy−Ex
Fµν=Bz0−Bx−Ey.(4.2.5)
−ByBx0−Ez
−Ez−Ey−Ez0
The(Lorenz–Joule)4-forceisgivenby
Kµ=Fµνjν=(c)dpµ,(4.2.6)
dswiththecanonicalmomentumpµ.
Furthermore,thehomogeneousMaxwellsystemmayberewrittenwithhelpoftheantisymmetricLevi–
Civitatensorεµναβ.Thedualfield-strengthtensorisgivenby
Fµν∗=1εµνκλFκλ,(4.2.7)
2whereas

∗∗Fµν=−Fµν.

(4.2.8)

4.2.COVARIANTFORMANDDYONS

43

(4.2.9)

Theexplicitformofthetensorinmatrixformis
0−EzEyBx
Fµν∗Ez0−ExBy.(4.2.9)
EyEx0Bz
−Bz−By−Bz0
ThehomogeneousMaxwellsystemindualformyields
Fµν∗,µ=0.(4.2.10)
Ontheotherhand,thecontravariantinhomogeneousMaxwellsystemyields(4.2.3)
Fµν,µ=4πjν.
Hence,ruptureofdualsymmetrybetweenequations(4.2.3)and(4.2.10)fornonvanishing4-currentsiseasily
seen.Definitionofmagnetic4-currents,jgµ,though,leadstodualinhomogeneousMaxwellequationsin
form,wingfollotheFµν∗,µ=4πjνg.(4.2.11)
4-currentsjgνarerelatedtoDiracsmonopoleasmagneticcharge.Further,aparticularsolutionforthe
field-strengthtensorfortheinhomogeneousequation(4.2.11)isgivenbelow,
Fµν∗=(nµ∙∂µ)−1(n[µjgν]),(4.2.12)
whereasnisanarbitraryfixed4-vectorwithn2=0,and(nµ∂µ)−1isanintegraloperatorwithkernel
satisfyingfollowingconditionaccordingto[251],
nµ∂xµK(x−y)=δ4(x−y),(4.2.13)
,analogouslyor(nµ∂xµ)(nµ∂xµ)−1f(x)≡f(x).(4.2.14)
Thegeneralsolutionto(4.2.11)isgivenby
Fµν∗=(∂[µAν])∗+(nµ∂µ)−1(n[µjgν]).(4.2.15)
Fµν=(∂[µAν])−(nµ∂µ)−1(n[µjgν])∗
The4-potentialAµdependsonthechoiceofgauge,thechoiceonnandthedeterminationof(nµ∂µ)−1.
Similarly,thereisin[251]thegeneralsolutionto(4.2.12)whichis
Fµν∗=−(∂[µAν])∗+(nµ∂µ)−1(n[µjeν])
Fµν=(∂[µAν])+(nµ∂µ)−1(n[µjeν])∗,(4.2.16)
whereasforelectricchargesasubscriptehasbeenwritten.Further,Aµisanother4-potential.
Atthispointitisbettertopartiallyintroduceanindex-freeformalismusing(antisymmetric)wedgeoperators
oftheinnerproductinthefollowingform:
Atensorωµνisrelatedtotheformωdxµ∧dxν,withdifferentialformsd.Hence,(a∧b)givea2-form

44CHAPTER4.QCD,SUPERCONDUCTIVITYANDSYMMETRYBREAKING

(4.2.17)(4.2.18)

ω(x,a,b)=ijω(x,ei,ej)aibjwiththestandardbasise1,...,en.ωisafunctionofbothsetsaiandbj
istherethatso(a∧b)(a∧b)µν≡aµbν−aνbµ=2a[µbν].(4.2.17)
isthere,Further(a∙G)(a∙G)ν≡aµGµν
a∙(b∧c)=a∙bc−a∙cb.(4.2.18)
a∙(b∧c)∗=aµεµνκλbκcλ
Hence,forinstance,thereis(4.2.16)as
F∗=−(∂∧A)∗+(n∙∂)−1(n∧je),(4.2.19)
F=−(∂∧A)−(n∙∂)−1(n∧je)∗.
The4-potentialAµAleadstothedualfield-strengthtensordual-equivalentlytothewayAµleadsto
µν,FnµFµνn∙F=n∙(∂∧A),n∙F∗=n∙(∂∧B).(4.2.20)
Hence,thedualfield-strengthtensormaybegivenbyAµanalogouslytoelectricchargedensitiesaregiven
asthedivergenceoftheelectricfield.ThisimpliestheexistenceofmagneticmonopolesrelatedtoAµif
dualsymmetryisgiven.
Further,everyantisymmetrictensorGµνfollowsthefollowingidentitywithaµGµν=(aG)ν,
G=n12[n∧(n∙G)]−[n∧(n∙G∗)]∗.(4.2.21)
So,equation(4.2.20)leadstothefield-strengthswiththeindex-freeform
F=n12{n∧[n∙(∂∧A)]}−{n∧[n∙(∂∧A)]}∗,
F∗=n12{n∧[n∙(∂∧A)]}∗+{n∧[n∙(∂∧A)]}.(4.2.22)
Withthem,Maxwellsequationsmaybewrittenintermsofthepotentials[251]:
(1/n2)(n∙∂n∙∂Aµ−n∙∂∂µn∙A−nµn∙∂∂∙A+
+nµ∂2n∙A−n∙εµνκλnν∂κAλ)=jeµ,(4.2.23)
(1/n2)(n∙∂n∙∂Aµ−n∙∂∂µn∙A−nµn∙∂∂∙A+
+nµ∂2n∙A−n∙εµνκλnν∂κAλ)=jgµ.(4.2.24)
Foranyfield-strengthtensorFµνtoMaxwellsequationsthereexistpotentialsAµandAµsatisfyingthe
(4.2.24).equationsMaxwellFor(4.2.24),aLagrangianmaybegivenwiththeform
L=−2n12[n∙(∂∧A)]∙[n∙(∂∧A)∗]+
+12[n∙(∂∧A)]∙[n∙(∂∧A)∗]−(4.2.25)
n211−2n2[n∙(∂∧A)]2−2n2[n∙(∂∧A)]2+LI,

terminteractionthewith

LI=−jeµAµ−jgµAµ,

(4.2.25)

(4.2.26)

4.2.COVARIANTFORMANDDYONS

45

whichaddstothetotalactionforthepartitionorpropagatorfunction(seeChapterAppendixB.1).Herewith,
thereistheelectric(magnetic)chargee(g).
Theintegraloverthe4-volumeiscalledZwanzigeraction.Itgivesthedynamicsforanelectrodynamic
monopoles.DiracwithsystemApossibleapproachistotakethepartitionfunctionfollowingfromtheZwanzigeractionsincepartition
functionsencodethestaticalpropertiesofsystemsinthermodynamicalequilibriumandapartitionfunction
isnothinglessthantheWickrotation(t→it)ofFeynmanspathintegral(propagator).Thepathintegral
resemblesthepartitionfunctionofstatisticalmechanicsdefinedinacanonicalensemblewithtemperature
1/(T)(cf.AppendixB.1).
Theso-calledZwanzigerpartitionfunctionyields
ZZw[Aµ,Aµ]=DAµDAµexp[−SZw[Aµ,Aµ]],(4.2.27)
whereDdenotestheintegrationoverallpathsforthekernel.
LetthevacuumnowbenontrivialundertheincorporationofascalarfieldΦwhichleadstospontaneous
symmetrybreaking.Thepotentialofthescalarfieldbegivenby
V(Φ)=λ|Φ|2−Φ022,(4.2.28)
i.e.aHiggspotential.Thekineticenergytermoftheactionbegivenby
T(Φ)=1|DµΦ|2,(4.2.29)
2whereasthecovariantderivativebegivenhereas
Dµ=∂µ−ieAµ−igAµ.(4.2.30)
Withthisterms,apartitionfunctionfordyonswithaHiggsfieldΦcanbegivenas[178]
ZD[Aµ,Aµ,Φ]=DAµDAµDΦeSD[Aµ,Aµ,Φ],(4.2.31)

actiondyonwithSD[Aµ,Aµ,Φ]=SZw[Aµ,Aµ]+T(Φ)+V(Φ).(4.2.32)
Here,theHiggsfieldacquiresdyonicpropertieswithanelectric(e)andamagnetic(g)chargegivenbythe
covariantderivativein(4.2.30).
TheseideashavebeensuccessfulinU(1)forunderstandingsuperconductivitysothattheyhavebeenex-
tendedtonon-abelianmodelsinviewofelementaryinteractionsandespeciallycolorconfinementbysym-
metrybreaking.Thatmakesitcrucialtoformulatethetheoryintermsofitsrelevantabeliandegreesof
freedom,whicharecolor-magneticmonopoles,color-electricchargesandphotons.Innon-abeliantheories,
agaugefieldcanbeCartandecomposedintothediagonalAµandtheoff-diagonalpartaµ.Hence,aformu-
lationintermsofabeliandegreesoffreedomisachievedbyfixingtoagaugeinwhichthegaugefreedomof
themaximalabeliansubgroupremains(abelianprojections)[143].Asaconsequence,magneticmonopoles
emergewithnecessityasdegreesoffreedominabelianprojectionsforthedynamicsofgluons[114].
Diagonalgluonfieldstransformasabeliangaugefields,whereasoff-diagonalgluonstransformasadjoint
matterfields.Fromanon-abeliangaugetheory,anabelianoneisobtainedbyneglectingtheoff-diagonal
gaugefields,althoughtheycanbetakenintoaccountbyintegrationinsenseofaWilsonrenormalization

46CHAPTER4.QCD,SUPERCONDUCTIVITYANDSYMMETRYBREAKING

group,reducingtothenormalizationoftheeffectiveabeliangaugetheory.Theseareabelianprojections
andinthoseofgluodynamicsforQCD,magneticmonopolesnecessarilyemergeasdegreesoffreedom.An
abelian-projectedeffectivegaugetheoryisthenconsideredasthelow-energyeffectivegaugetheoryofthe
originalnon-abeliangaugetheory,e.g.QCD.Thelatterespeciallybecausetheoff-diagonalgluonsbecome
massiveafterthemaximalabeliangauge.
Further,ifthevacuumisnotassumedtobetrivialandspontaneoussymmetrybreakingisincorporated
intoZwanzigersformalism,thenunphysicalsingularitiesarise.Theyhowevervanishfollowingsomefor-
mulationoftheAbelianHiggsModel(AHM),i.e.theHiggsmodelwithinelectrodynamics(abelian).In
theabelianprojection(i.e.inprincipletakingonlyabeliancontributions)andtakingelementary-particle
processesinscope,quarksareelectricallychargedparticles,andifmonopolesarecondensed,thedual
Abrikosovstringcarryingtheelectricfluxisformedbetweenquarksandantiquarks.Duetoanon-zero
stringtensionthequarksareconfinedbythelinearpotential[5].
AccordingtoAkhmedov,forthe(anti-)self-dualfieldstheabelianmonopolesbecomeabeliandyons.Fur-
ther,theinfraredpropertiesofQCDintheabelianprojectioncanbedescribedbytheAHMinwhichdyons
[5].condensedareLetusconsideralineartransformationofthegaugefieldsas(A˜µ,A˜µ)T=R(ϑ)(Aµ,Aµ)T,whereTdenote
withtransposetheϑsinϑcos−R(ϑ)=sinϑcosϑ.
suchthatϑ=g/e.Theintegrationofthedyonpartitionfunction(4.2.32)overthetransformeddual-electric
gaugepotentialA˜µthenleadstothepartitionfunctionoftheAHMofQCD,
ZAHM=DA˜µΦe−SAHM[A˜µ,Φ].(4.2.33)
TheAHMactionwithtransformedmagneticgaugefieldA˜µisgivenby
SAHM[A˜µ,Φ]=d4x−1C˜µνC˜µν+|∂µ−ie2+g2A˜µΦ|2+λ(|Φ|2−Φ02)2,(4.2.34)

4withageneralizedmagneto-chargeQ=e2+g2andthedualfieldstrengthasfollows,
C˜µν=∂µA˜ν−∂νA˜µ=A˜ν,µ−A˜µ,ν.(4.2.35)
∂µ−iQA˜µgivesthecovariantderivativeshowingthatΦinaction(4.2.34)isdyonicinnature.ForΦ,there
isDµDµΦ−4λ|Φ|2−Φ02Φ=0.(4.2.36)
ThetensorCµν,further,isdualtotheusualfield-strengthtensorFµνandofthesamestructurewithgauge
potentialA˜µ.Itsfieldcontentsarecolor-electricfieldsE˜andcolor-magneticfieldsB˜[174].Hence,the
actiongivenby(4.2.34)coincideswiththeGinzburg–Landauactionofsuperconductivity,howeverindual
gluodynamics.forform

4.3Superconductivity,dualsuperconductorsandtheHiggsfield
•SuperconductivityandHiggsfields:InwordsofStephenWeinberg,asuperconductorismoreorless
amaterialinwhichaparticularsymmetryofthelawsofnature,electromagneticgaugeinvariance,is

4.3.SUPERCONDUCTIVITY,DUALSUPERCONDUCTORSANDTHEHIGGSFIELD47

spontaneouslybroken.Thesymmetrygrouphereisthegroupoftwo-dimensionalrotations.These
rotationsactonatwo-dimensionalvectorwhosetwocomponentsaretherealandimaginarypartsof
theelectronfield,thequantummechanicaloperatorthatinquantumfieldtheoriesofmatterdestroys
[236].electronscreatesorThesymmetrybreakinginasuperconductorleavesunbrokenarotationby180◦,whichchangesthe
signoftheelectronfield.Inconsequence,productsofanyevennumberofelectronfieldshave
non-vanishingexpectationvaluesinasuperconductor.Asingleelectronfield,however,doesnot
[236].Phenomenologically,electronsaresaidtobe1boundintoacompositewhichisknownasBCS
(Bardeen–Cooper–Schrieffer)orsimplyCooperpair.Consequently,allexperimentalphenomena
suchastheMeißner–Ochsenfeld(orsimplyMeissner)effect(ME),zeroelectricalresistance,the
expellingofmagneticfieldsandsoonappearfollowingtheassumptionthatelectromagneticgauge
invarianceisbroken.
Superconductivityistracedbacktoanorderparameterwhichisthenonvanishingvalueofthepro-
ductoftwoelectronfields.Thisorderparameter,further,isrelatedtotheHiggsfieldofspontaneous
symmetrybreaking.Thescalarfieldasanorderparametergivestheorderofthesysteminterms
ofthebrokensymmetryandtheunbrokensubgroupsinceanonvanishingexpectationvalueofthe
Higgsfieldaccompaniesabrokenmodeofsymmetry.TheappearingHiggsbosonsofthefield,fur-
ther,arerelatedincondensed-matterphysicstotheappearingbosonicstateofelectron(BCS)pairs.
Field-theoretically,then,electronsareboundtogetherbymediationofvirtualphotonswhichacquire
aneffectivemassfollowingsymmetrybreaking,inanalogytoHiggsmass,giventhefactthatCooper
pairsarephenomenologicalanaloguesofHiggsbosonsforcondensed-matterphysics.Theypossess
aneffectivemasswhichisthenrelatedtothepenetrationdepthofmagneticfieldsinsuperconductors
fect).ef(Meissner•Ginzburg–LandauandMeissnereffect:Aphenomenologicalapproachtosuperconductivityis
givenbytheGinzburg–Landaumodel.Further,theGinzburg–Landauactionmaybestatedbythe
Lagrangian,wingfolloL=1FµνFµν+(Dµφ)†(Dµφ)+µ2φφ∗−λ(φφ∗)2,(4.3.1)
424withacovariantderivativeasfollows,

andafield-strengthtensorgivenby

Dµ=∂µ−ieAµ,

(4.3.1)

(4.3.2)

Fµν=∂µAν−∂νAµ.(4.3.3)
Thus,φisaHiggsfieldwhichisherecoupledtoQED(AHM).Forthestaticcase,forwhichthereis
∂0φ=∂0A=0andA0=0,thefieldequationforthepotentialAcanbegivenasbelow,
×H=j=ie21φ†(−ieA)φ−(+ieA)φ†φ,(4.3.4)
1Followingthemany-bodyBCStheory[13],Nobelprize1972.ThesolutionsofBCStheoryinahomogeneoussystemarefound
usingalinearcanonicaltransformationcalledBogoliubovtransformation[31],whichisoftenusedtodiagonalizeHamiltonians,i.e.to
makethemequivalenttoasetofnon-interactingharmonicoscillators(cf.[219]).

48

CHAPTER4.QCD,SUPERCONDUCTIVITYANDSYMMETRYBREAKING

Figure4.1:IllustrationofaHiggscompositeofsuperconductivitywithelectronsmediatingmassivegauge
bosonsaccordingtoaYukawamodelandfieldtheoriesofinteractionprocesses.N.B.:Theappearanceof
theHiggsbosonleadstosuperconductivityasabroken-symmetryphenomenon.

whereHisthemagneticfieldforamacroscopicsystem.Inthespontaneouslybrokenphaseofsym-
metry,thecurrentsatisfiesthefollowinglocalrelationwhichisknownastheLondonequation,
j=e2v2A,(4.3.5)
wherev=µ2/λ.Equation(4.3.5),further,leadsto
2H=e2v2H,(4.3.6)
wherethereisGausssequation∙H=0.Finally,(4.3.6)issolvedforx≥0by
H(x)=H(0)e−x/lA,(4.3.7)
wherelA=cmA−1=(ev)−1isthepenetrationdepthwhichistheinverseofthevectorgaugefield
mass.Further,equation(4.3.7)impliestheMeissnereffectindicatingthatthemagneticfielddecaysin
adistancelA.Phenomenologically,theGinzburg–LandaumodelgivesanexplanationoftheMeissner
effectbymeansofphotonsacquiringeffectivemassviaHiggsfields.Symmetrybreakingleadstoef-
fectivemassesrelatedtoshortrangesofinteractionoftheparticlescoupledtothescalarfield.Hence,
Ginzburg–Landauphotonsdonotentersuperconductorsmorethanadistancegivenbythepenetration
depthlA.Magneticfieldsareavoided.

•DualMeissnereffect:Asalreadyshown,thephenomenologyofsuperconductivitymaybeunder-
stoodintermsoffieldtheory,anditindeedfindsitsnatureintheconceptofsymmetrybreakingand

4.3.SUPERCONDUCTIVITY,DUALSUPERCONDUCTORSANDTHEHIGGSFIELD49

henceintheappearanceofsomekindofHiggsfieldcoupledtoelectrodynamics.Hence,superconduc-
tivityisacondensed-matterphenomenonwhichisactuallyusualwithinallrangesofphysicsfinding
itsrootsinelementary-particlephysics.Furthermore,asdualsymmetrytotheGinzburg–Landaumo-
delshows,theZwanzigerformulationmaybeusedtounderstandissuesfromnuclearandelementary
particlephysicsifitisinterpretedintermsofelementaryfields.Actually,colorconfinementcanbe
understoodintermsofacolor-magneticsuperconductorinwhichcolorchargesareconfined(cf.[54]).
Thispictureisdualtoordinarysuperconductors[143]inwhichelectricchargescondenseandmag-
neticmonopolesareconfinedthroughtheMeissnereffect.Someconceptsabouttheideasofdual
QuantumChromodynamicsmaybefoundunder[177]whilethereadermayfurtherfindathorough
reviewonColorConfinementin[209].
Zwanzigersformalismallowstoconsistentlydescribeaphotoninteractingwithmagneticandelectric
charges[114].Inthedualdescription,quarksaretheelectricallychargedparticleswhichareconfined
withinhadrons.Gluonsasmediatedgaugefieldsacquireeffectivemass,whichisusuallyunderstood
undertheappearanceofglueballsascompositestateofgluonswhichself-interactduetonon-abelian
propertiesofthegaugegroup.Hence,adualdescriptionofsuperconductivitymayhelpunderstanding
dynamicsofgluonswhenQCDisabelianprojected.Theactiongivenby(4.2.34)canbeapproximated
ws,folloas

1H0=Kg+2Q2Φ02A˜i2,(4.3.8)
whereKg=E˜2/2isthegluon-fieldenergyandQthemagneto–electricchargeofdyons.Φbea
HiggsfieldforsymmetrybreakingwhileA˜µbeatransformeddualgaugefieldofAµsotomaintain
.symmetrydual

InthedualformoftheGinzburg–Landauactionwetakedualfield-strengthtensorsrelatedtothe
furtherpotentialAµwhichleadtomagneticmonopoles.ThescalarfieldΦrepresentsthemonopole
(ordyon)field,andithasanon-zeromagneticchargeg(orbotheandg).ThepotentialV(Φ)
istheeffectivepotentialwhichgeneratesthemassofthedualgaugefieldinthebrokenphaseof
symmetryandconsequentlythefeaturesofmagneticsuperconductivityinthecondensedmodeof
QCDvacuumwhenthemodelisusedforelementaryinteractions.Infact,asusualwithinaHiggs
mode,theHiggspotentialensuresthattheaveragevalueofthescalarfieldisnonvanishing(<Φ>=
0)invacuumandthatthemonopolefieldplaystheroleoftheGinzburg–Landau(GL)orderparameter
inthewaythescalarfieldtakesthephenomenologicalroleofmacroscopicCooper-pairwavefunctions
inconventional(electric)superconductivity.
InordertoanalyzescreeningcurrentsandtheirimplicationsonthenatureofQCDvacuum,thefield
equationscorrespondingto(4.2.35)arederivedintheformgivenbelow[178]with=c=1,
∂νC˜µν−i2Q(Φ∗∂µΦ−Φ∂µΦ∗)−Q2(ΦΦ∗)A˜µ=0,(4.3.9)
DµDµΦ−4λ|Φ|2−Φ02Φ=0.(4.3.10)
TheseequationsgovernthedynamicsofQCDvacuuminthebrokenphaseofsymmetry.Furthermore,
equations(4.3.9)and(4.3.10)areidenticaltotheGL-typefieldequationsinconventionalsupercon-
ductivitywhenC˜µν→FµνandA˜µ→Aµ.
Sincethemacroscopicdescriptionoftheformulationinvolvesanumberofdyons,itisbettertospe-
cifythemassmodesandothercrucialparametersintermsofthedensityofthecondenseddyonsor

50

CHAPTER4.QCD,SUPERCONDUCTIVITYANDSYMMETRYBREAKING

monopoles.ThescalarfieldΦwouldbesuchthatitremaineffectivelyunperturbedbythecolor-
electricfield,andthedensityofsuperconductingdyonsormonopolesmustbedefinedbyitsconstant
modulusgivenintermsofΦ0.InthedualQCDvacuum,theparametersspecifyingtheconfiningme-
chanismofvacuumare,indeed,closelyrelatedtosuchdensityprofileofdyon/monopolepairs.The
vacuum,asacoherentcondensateofallsuchpairs[4],maythenbenormalizedto
ns(Φ)=|Φ|2=Φ02.(4.3.11)
Thedensityofcondenseddyonsgivenby(4.3.11)cannotbedefinedinthiswayinthelargepertur-
bativesectorofQCDastheVEVofdyonfieldswoulddisappearcompletelyintheultravioletregion.
Thedensityprofilealongwithotherconfinementparametersinthenon-perturbativeinfraredsector
canbethereforeusedforthecorrectphysicalexplanationoftheconfiningbehaviorofQCDvacuum.
Letusconsiderthevariationsinthedyonfieldsuchthat∂µΦ=0=∂µΦ∗sinceithasafinitevalue
ateachspacetimepoint.Equation(4.2.35)thenleadsto[176]
(+mV2)A˜µ−∂µ(∂νA˜ν)=0,(4.3.12)
wheremV=QΦ0isthemassofdualgaugefields.
Equation(4.3.12)isofmassivevectortypeandmaybeidentifiedwiththatofthecondensedmodeof
QCDvacuum.Forthisformulation,twomassmodesmaybegiven,i.e.ofavectorandascalarmass,
mV=Qns(Φ),andmΦ=2λns(Φ).(4.3.13)
ThesemassmodesappearasinanystandardHiggsmechanism,andthemassivevectorequation
(4.3.12)showsthatQCDvacuum,asaresultofsymmetrybreaking,acquirespropertiessimilarto
thoseofarelativisticsuperconductorwherequantumfieldsgenerateanon-zeroVEV.Theinteraction
betweenthemacroscopicfieldΦandA˜µleadstoacolor-fluxscreeningarisingbecauseofascreening
currentduetothestrongcorrelationamongthedyonicorthepuremagneticcharges.Further,letus
makesomecommentsabouttherelationbetweenthemassmodesandthesuperconductingphase:In
usualsemiconductorsthereexiststheGLparameterκwhichdescribesthetypeofsuperconductorone
has.Thisparameterisgivenbytheratioofthepenetrationdepthandthecoherentlengthξwhichisa
naturallengthscaleforspatialvariationsoftheorderparameter.Fordualsuperconductors,ξmaybe
relatedtothecoherentlengthofmonopolecondensatesandthustothereverseofthescalar-fieldmass
mΦ.Hence,thedualGLparametermaybedefinedas
κ=mΦ.(4.3.14)
mVQCDvacuumthusbehavesasatype-IIsuperconductorformΦ>mVwhileitbehaves√asatype-I
superconductorformΦ<mV.Further,bothmassespossessanequalvalueforQ=2λ.Inthat
case,theQCDvacuumundergoesatransitionfromatype-IItotype-Isuperconductingstate[4].
InQED,type-Isuperconductorsarethosewhichcannotbepenetratedbymagneticfluxlines,ac-
cordingtotheMeissner–Ochsenfeldeffect.Theyhaveonlyasinglecriticaltemperatureatwhich
thematerialceasestosuperconduct.Elementarysuperconductorsareofthistype,whichgenerallyis
exhibitedbymaterialswitharegularlystructuredlattice.Thisallowselectronstobecoupledovera
relativelylargedistanceontoCooperpairs.Ontheotherhand,type-IIsuperconductorsofQEDare
characterizedbyagradualtransitionfromthesuperconductingtothenormalstatewithinanincreasing
magneticfield.Typically,theysuperconductathighertemperaturesandmagneticfieldsthantype-I

4.3.SUPERCONDUCTIVITY,DUALSUPERCONDUCTORSANDTHEHIGGSFIELD51

superconductors.Inthedualpicture,then,onlytype-Idualsuperconductorsleadtoastrictconfine-
mentofcolor-electricfields.ThisisthecaseforsmallerpenetrationdepthslAinrelationtothedyon
.Qgechar

Figure4.2:BehaviorofthedifferenttypesofQEDsuperconductorsindependenceofthefieldstrengthH
withcriticalfieldsHcandHc1andHc1.

Thedivergenceofequation(4.3.12)leadsto∂µA˜µ=0formV=0[176].Themasslessdualgauge
quantumwhichpropagatesinthedyonicallycondensedQCDvacuumthensatisfies
A˜=jsµ,(4.3.15)
wherejsµisthescreeningcurrentthatresidesinvacuum.Comparing(4.3.12)and(4.3.15)usingthe
istherecondition,Lorentzjsµ=−mV2A˜µ,(mV=Qns(Φ)),(4.3.16)
whichreducesinthestaticcasetotheLondonequationwhichinQEDgivesj∝nv2A(viz4.3.5).
Thesimplestsolutionof(4.3.12)maybederivedinthehalf-spaceofallspace(x≥0,y=z=0).
ThedualgaugefieldhasthenonlyadependenceonxandA˜µasfollows,
∂x2−mV2A˜µ=0,(4.3.17)
inresultsthenwhichA˜µ=A˜0µe−mVx,(4.3.18)

inresultsthenwhich

52

CHAPTER4.QCD,SUPERCONDUCTIVITYANDSYMMETRYBREAKING

whereA˜0µisaconstantvector.
InanalogytoQED,applyingAmpèreslaw,forcolor-magnetic(B˜)andcolor-electric(E˜)fields,E˜
satisfies×E˜=jswithE˜=×A˜.Undersuchconsiderations,onecanobtain(inanalogyto
(4.3.6))2E˜−(∙E˜)−mV2E˜=0.(4.3.19)
IfonetakesavectorfieldE˜≡(0,0,E˜z(x)),thecondition∙E˜=0issatisfiedsothatadualform
of(4.3.5)isachieved.OnecancontinueanalogouslytoQED.Equation(4.3.19)reducesto
∂x2E˜z(x)−mV2E˜z(x)=0,(4.3.20)
whichisadualHelmholtzequationofQCD.Itpossessesthegeneralsolution(viz(4.3.7))
E˜x(x)=D1e−mVx+D2emVx,(4.3.21)
whereD1andD2areintegrationconstants.TheinitialconditionsareE˜z(0)=E0atx=0whileEz
cannotincreasetoinfinityfarfromx.Hence,thereisD1=E0andD2=0.Thecolor-electricfield
thuspenetratesthevacuumuptoafinitedepthgivenbymV−1,equivalentlytoQED[176].Equation
(4.3.21)indicatesthattheelectricfieldisscreenedoutadistancel=∼mV−1whichisthepenetration
depthwheremVisthedualgaugefieldmass.ThisequationguaranteesadualMeissnereffect(DME).
Withincreasingdensityofthecondenseddyons,theelectricfielddiesoffmorerapidly.
Inthecasee=0,thedyonicvectormassmodegoesthroughthepuremagneticdualcounterpartof
electriccharges,i.e.tomagneticmonopoles.Therefore,thedyonicmassmodeisalwaysgreaterthan
counterpart.magneticpureitsLetustakethedimensionlessquantity

Q(4.3.22)=γgIthasthevalue1fore=0,andγ>1fore=0.Thesecasescorrespondtomonopoleanddyon
condensation,respectively.Incaseofdyoncondensation,thedecayofcolorfluxisalwaysfaster
thanthatofmonopolecondensation(cf.(4.3.21)).Thecolor-electricfluxthusconstrictsitselfmore
rapidlyinasmallerregion.Wecanconsidertheradiusofsuchfluxtubeastheinverseofthevector
mass[54,172,173,189,227].Forit,thereis
r1=mg−1>r2=mV−1.(4.3.23)
Inordertohaveacomparisonoftheroleofpuremagneticanddyoniccondensationontheconfining
mechanism,thestringtensionofthefluxtubemaybeanotherguidingparameter.Hence,letuscon-
siderthespin(J)andmass(MJ)relationshipofafluxtubeasJ=α0+αMj2whereα=(2πσ)−1
istheReggeslopeparameter,andσisthestringtensionofthefluxtube.SincethedualGLfreeenergy
givenby(4.3.8)isalwaysgreaterforthedyoniccasethanforthemonopolecase,thestringtension
forthelatterwillbenaturallylessthanthepreviousone.Thedyoniccasemay,therefore,leadtothe
lowestlyingstatesoftheReggetrajectoriesforhadrons(formoredetails,see[174,175]).

Furthermore,followingnotionsofmagneto-staticsforadualrepresentation,thereisamagneticcur-
rentjs=×E˜forthedualelectricfieldE˜andassumingelectricvacuumsothat∙E˜=0isvalid,

4.3.SUPERCONDUCTIVITY,DUALSUPERCONDUCTORSANDTHEHIGGSFIELD53

togetherwithaconstantscalar-pressureanaloguePwhichisidentifiedtoacolor-forcedensityE˜×js
tube,fluxtheof∙P=js×E˜,(4.3.24)
[176]showsthattheminimizationofkineticenergyleadstoaquantizationofcolor-electriccharge
thatsuch

ΨE=×A˜dS=E˜∙dSnΨ0(4.3.25)
isvalidinordertomaintainanequilibriumbetweenthecondensateandcolorforce.HereΨ0=2π/Q.
Thisshowsthattheelectricfluxisquantizedintermsofthedyoniccharge.

Further,asshownin[176],thepresenceofmagneticdyonicchargesinQCDimpartsadielectric
naturetoitduetotheirvacuumpolarization.QCDvacuumbehavesasaperfectdielectricmedium
independentlyofthetypeofcondensateinthevacuum.Additionally,thereexistsaphenomenologi-
calrelationbetweentheflux-tubestructuresforthemonopoleanddyoncondensationcasefromthe
viewpointoftheDMEasanonsetofscreeningcurrentsandthedielectricparameters[176],whereas
µisconcludedtobedependentonthesquareofthemomentumpandthereciprocalsquaredvalueof
,mV

µ(p2,Φ0)=1+˜Π(p)=1−p−2mV2,(4.3.26)
whileεisthereciprocalofµaccordingtoεµ=c−2.Thisisachievedbymeansofadualmag-
neticpolarizationtensorwhichisgivenas˜Π(p2,Φ0)=−mV2/p2,following[134].Higherdyonic
chargesQleadtosmallerdielectricpermittivityandalargerpermeabilityµ.Thedualmagnetic
fieldH˜=B˜/(µ0µ)anddisplacementfieldD˜=ε0εE˜(givenisotropyandnondispersivebehavior)
arescreened.Atthesametime,thisisrelatedtodenserflux-tubestructuresbetweenthecharges,
relatedtohigherdualpolarizations(∼−mV2/p2)andsmallerflux-tuberadii.
Hence,somepropertiesneededforconfinementofquarksinhadronsmaybegivenusingadualap-
proachwithsymmetrybreakdownusingHiggsfields.Nevertheless,themechanismwhichisactually
responsibleforthevanishingofcolor-dielectricfunctionofacolor-confiningmediumisstillunclear
discussion.ofsubjectaand

54

CHAPTER

4.

QCD,

SUPERCONDUCTIVITY

AND

YSYMMETR

BREAKING

Induced

artP

vitygra

II

theories

fields

55

with

scalar

Chapter5

Alternativetheoriesofgravityand
historicalviewervo

–Scalar–tensortheoriesareintroducedhistoricallyinviewoftheJordan–Brans–Dicketheoryandthe
Bergmann–WagonerclasstogetherwithHiggsgravitationandbroken-symmetrictheoriesofgravitation.
Thismaybepartlyfoundin[24](hereweset=c=1).–

5.1theorysordanJ

Inmodernquantumtheories,interactionsbetweenequallychargedparticlesmediatedbybosonswithodd
spinarerepulsive,andinteractionsbetweendifferentlychargedparticleswiththemediationofodd-spined
e:vattractiarebosons

•InQED,photonspossessspin1andequallychargedparticlesrepulseeachother.
•QCDconfinementderivesfromanattractiveforcewhichactsbetweendifferentlycolor-chargedquarks
hadrons.in

Interactionswhicharemediatedbyeven-spinbosonsareattractive:

•Higgsparticlespossessspin0andthusmediateattractiveforcesbetweenparticleswhichcoupleto
them.

•Pions,asspin-0-bosons,mediateanattractiveeffectivenuclearforcebetweenisotopicparticles.
Classically,todescribegravitationalinteraction,thegravitationalLagrangianofthetheory(whichobeys
theEuler–Lagrangeequationsforafield)describesthepropagationandself-interactionofthegravitational
fieldonlythroughtheRicciscalarR(see(A.4.6)).FromEinsteinsGR(inanalogytoquantumtheories)it
followsthatthegravitationalinteractionis,initsquantum-mechanicalnature,mediatedbymasslessspin-
2-excitationsonly[88].Thisisexpectedtoberelatedtothestill-hypotheticalgravitonsasintermediate
particlesofaquantumtheoryofgravity.Scalar–tensortheories(STTs),ontheotherhand,postulatein
thiscontexttheexistenceofmorecomplexdynamicsfromfurthermediatingparticles,namedinthiscase
graviscalarswithinthecontextofquantumtheories.ThismeansthatSTTsmodifyclassicalGRbythe

57

58CHAPTER5.ALTERNATIVETHEORIESOFGRAVITYANDHISTORICALOVERVIEW
additionofscalarfieldstothetensorfieldofGR.Theyfurtherdemandthatthephysicalmetricgµν
(coupledtoordinarymatter)beacompositeobjectoftheform
gµν=A2(ˆφ)g∗µν,(5.1.1)
withacouplingfunctionA(φˆ)ofthescalarfieldφˆ[62].
Thefirstattemptsofascalar–tensortheorywerestartedindependentlybyM.Fierzin1956[90]andby
PascualJordanin1949[136].Thelatternoticedthroughhisisomorphytheoremthatprojectivespacesas
Kaluza–Kleins(five-dimensional)canbereducedtousualRiemannian4-dimensionalspacesandthatasca-
larfieldasfifthcomponentofsuchaprojectivemetriccanplaytheroleofavariableeffectivegravitational
constantG˜,whichistypicalforSTTsandbywhichitispossibletovarythestrengthofgravitation[87]
(thus,obviouslyviolatinginsomeaccountthestrongequivalenceprinciple(SEP)).Thesamegravitational
interactionsmightnotholdonallphysicalsystems.Furthermore,thiskindofgeneral-relativisticmodelwith
ascalarfieldisconformequivalenttomulti-dimensionalgeneral-relativisticmodels[59].Manytheoriesin-
volvethisphysics(e.g.stringtheoriesorbranetheories),butscalar–tensortheoriesaretypicallyfoundto
representclassicaldescriptionsofthem[113].
Inhistheory,Jordanintroducedtwocouplingparametersofthescalarfield.Oneparameterproduceda
variationofthegravitationalconstant.Theotheronewouldbreaktheenergyconservationthroughanon-
vanishingdivergenceoftheenergy–momentumtensortoincreasethemassintime,inaccordancewiththe
ideasofJordanandDirac[137].However,the1cosmicmicrowavebackgroundradiation(CMB)asareal
black-bodyradiationdiscoveredin1965[196]forcestoacceptgeneralenergyconservationasexperimen-
[130]).(seeactftalInJordanstheory,thereappearsag55componentofthemetricaltensorwhichisingeneraldependenton
ascalarquantity.Inamoreusualformulation,however,thelatterisequivalenttohavingκ(orthegravi-
tationalcouplingG)asafieldquantity.Hence,inthissensehespokeaboutanaugmentedorgeneralized
gravitationaltheory(erweiterteGravitationstheorie)whichhealsoderivedfollowingthevariationprinciple.
Therefore,heexplicitlyassumedκ=const.andallowedderivatives
κ∂κ,µ=∂xµ=0(5.1.2)
ofthesame.Heassumed,inabsenceofmatter,anaction2
SJ=κηR−ζκ,µκ2,νgµν√−gdt.(5.1.3)
κHere,wehavethedeterminantgofthemetricaltensor,the(Ricci)curvaturescalarR,thegravitational
couplingκandηandζasempiricalvalues.
Takingmatterintoaccount,JordanderivedgeneralizedEinsteinequationsasfollows,
κκηR+ζ(η−2)κ,µ2κ,µ+2κ,µ2κ,µ=0,
gνλ21R+(κηκ)η,µ;µ−2ζκ,µκ2κ,µ−Rνλ−(κηκ)η,ν;λ+ζκ,νκ2κ,µ=κTνλ.(5.1.4)
Forcentralsymmetry,thereappearsacorrectionforeλ=e−νaswellasanonstaticalcoupling
κ=κ0τβ0/B(5.1.5)
1Wcorroborationorkwhichoftheresultedqualitiesinof1984CMB,inthetogetherNobelwithprizetheforphsmallysicsanisotropforA.A.ypresentPenziasinit,andledR.Wto.theWilson.NobelprizeFurthera,wtheardetoxactJohnC.analysisMatherand
andGeorgeF.Smootin2006.
2JafterErnstPascualJordan(1902-1980).

BRANS–DICKE5.2.YTHEOR

59

forvacuum,withtheeigentimeτ,aconstantβ0andB=1+2ηβ0.Time-dependenceofκ,heconcluded,
however,wouldbeespeciallyweak,with
0=β0≈0(5.1.6)
accordingtosolar-relativisticeffects.Furthermore,heconcludedη=1and|ζ|1accordingtoempirical
datawithinsomeapproximationofthetheory.

theoryeBrans–Dick5.2JordanstheorywasworkedoutindependentlybyBransandDickein1961[41]withoutbreakingenergy
conservation,butagainintroducingascalarfieldwithaninfinitelengthscalewhichnowexplicitlyplayed
theroleofavariablegravitationalcoupling.ThegeneralizationtoGRsaction(A.4.6)wasthenproposedas
3ws,follo

SJBD=ˆφR+(16π/c4)LM−ωˆ∂φˆ∂φˆµ√−gd4x.(5.2.1)
φ∂xµ∂x
Here,wehavethedeterminantgofthemetricaltensor,the(Ricci)curvaturescalarR,thematterLagran-
gianLMandascalarfieldφˆwhichplaystheroleofthereciprocalnewtonianconstantG−1.Thefirstterm
of(5.2.1)couplesthescalarfieldandgravitationgivenbyR,whilethethirdtermrepresentsthekinetic
energyofφˆ,sincetheLagrangedensityL(conceptuallyderivedfromtheLagrangefunctionofmechanics)
isusuallydefinedintermsofthesubtractionofthepotentialfromthekineticenergyoftheanalyzedsystem.
OtherthanintheoriginaltheoryofJordan,BransandDickestheoryinequation(5.2.1)doesnotcontaina
mass-creationprinciple.Thewaveequationofφˆcanbetransformedsoastomakethesourcetermappear
asthecontractedenergy–momentumtensorofmatteralone.Inotherwords,theinhomogeneouspartof
thewaveequationisonlydependentonthetraceTofthetensorTµν,andthisisinaccordancewiththe
requirementsofMachsprinciple:φˆisgivenbythematterdistributioninspace.
In1968,P.Bergmann[18],andin1970R.Wagoner[234],discussedamoregeneralscalar–tensortheory
whichpossessesanadditivecosmologicalfunctiontermΛ(φˆ)intheLagrangian.Furthermore,thelatter
maynowpossessafunctionalparameterω=ω(φˆ)forascalarfieldφˆ.Thisgeneralkindoftheories,now
oftencalledBergmann–Wagoner(BW)classofSTTs,possessestheJordan–Brans–Dicke(JBD)classasa
specialcaseforω=const.andΛ(φˆ)=0.
Inphysicsatheoryissaidtobeinacanonicalformifitiswritteninaparadigmaticformtakenfromclassical
mechanics(asidealwhichisinprinciple,however,freelyeligibleandamatterofdefinition).4Theequation
(5.2.1),calledtobeinJordanframe,isnotinthisform.TheBergmann–Wagoner-formedmodelsarenot
canonical.However,STTscanbetransformedconformallyintoacanonicalform(Einsteinframe)inwhich
acosmologicalfunctionstillappears,butφˆisminimallycoupled.
CanonicalformisachievedbychangingfromtheJordanframe(withmixeddegreesoffreedomofmetric
andscalarfield)totheeinsteinianone(withunmixeddegreesoffreedom).Inthefour-dimensionalcase,
suchisfulfilledthroughgµν→φˆ−1gµν(cf.(5.1.1))andaredefinitionofthescalarfieldandcosmological
3JBDafterJordanandCarlHenryBrans(1935)andRobertHenryDicke(1916-1997).
further4Cf.anrelatestointerestingHalbwachsanalysismémoireaboutvtheolontairehistoricalofaoriginsocietyand.meaningoftheconceptofcanonbyJ.Assmannin[11],whichhe

60CHAPTER5.ALTERNATIVETHEORIESOFGRAVITYANDHISTORICALOVERVIEW

function.However,itisstillsubjectofdiscussionwhichframeisbest.TheJordanone,though,isusually
calledthephysicalframe[45].
ThescalarfieldintheJordan–Brans–Dickestheoryismassless.However,agenerallycovarianttheoryof
gravitationcanaccommodateamassivescalarfieldinadditiontothemasslesstensorfield[2,120].Thus,a
versionoftheJBDorBWtheorywithmassivescalarfieldsmaybepostulated[94];indeed,A.Zeeincorpo-
ratedasfirsttheconceptofSSBtogravitywithinaSTT[247],suggestingthatthesamesymmetry-breaking
mechanismwasresponsibleforbreakingaunifiedgaugetheoryintostrong,weakandelectromagneticin-
teractions(mediatedbytheircorrespondinggaugebosons).Spontaneoussymmetrybreaking(SSB)causes
somescalarfieldtohaveavacuumexpectationvaluev,thusgeneratingthemassoftheintermediatebosons
andoffermions,relatingthemtothegroundstateofthescalarfieldafterthebreakdownofsymmetry.Zee
attributedthesmallnessofNewtonsgravitationalconstantGN(oftheorderofmagnitudeofabout10−11
m3kg−1s−2)tothemassivenessofsomeparticle(thismaybecomparedwiththeresultof[68])withNew-
tonscouplingconstantGN∼(1019mN)−2as
GN∼1/v2,(5.2.2)
where

(5.2.2)

√v2=(8πG2)≈6.07∙104(GeV)2.(5.2.3)
FThus,SSBgeneratesthemassoftheintermediatebosonsuchthatforFermiscouplingconstant,
1GF≈2π(294GeV)2∼(300mN)−2,(5.2.4)
withweakonmassMW(∼80GeV/c2)andelementarychargee(∼10−19C))thereis
GF∼e2/MW2∼1/v2,(5.2.5)
whichmaybecomparedwith(5.2.2).SinceSSBhasprovenextraordinarilyfaithfulinmanyareasofphysics,
Zeeconsidereditworthwiletoincorporatethismechanismintogravitation[247]andexplainthesmallness
ofNewtonssconstantthroughthemassofHiggsparticles.5ThroughtheincorporationofSSB,thescalar
fieldisanchoredinadeeppotentialwellV(φ).Thephysicalconsequencesarethenindistinguishability
betweenEinsteinsmodelandZeesSTTexceptunderextremeconditionsofspacetimecurvature[247];in
sharpcontrasttoearlierworkofBransandDicke.
Zeesmechanismincludesaself-interactionofthescalarfieldand,thus,apotentialVaspartofthecosmo-
logicalfunctionoftheBWclassofSTTswhich,however,lacksinusualJBDtheories.Asaresultofthe
missingpotential,Brans–Dickestheoryisinconsistentwithobservationunlessacertainparameterisvery
large[235].Infact,frommeasurementsofradio-signalcurrenttimedelaywithVikingprobesformMars,
thecouplingparameterωoftheusualJBDtheoryin(5.2.1)(ameasureofthestrengthofthescalarfield
couplingtomatter)isrequiredtobeinggreaterthanabout500[205].
Inanysensitivetheory,BransandDickeprovedintheiroriginalwork[41],thedimensionlessconstantω
mustbeofthegeneralorderofunity.Forω→∞,however,GRisobtained,whichentailsthattheJBD
theoryleadsnearlytothesameresultsasGR.Incontrast,however,inaSTTwiththescalarfieldanchored
bytheSSBpotential,thisstrengthofthescalar-fieldcouplingmaynaturallybesmaller.Thus,newphysics
strong5ThecouplingsmallnessoftheofGscalarcanfieldalso,toasgrawillbevitation,seen,beeanalogouslyxplainedto[69].throughahighexpectationvalueofthescalarfieldaswellasthrougha

5.2.YTHEORBRANS–DICKE

61

ofhigherorderispossible.GiventhegravitationalpropertiesofHiggs-likefields(seeChapter3),forins-
tance,itseemsnaturaltocouplethemtogravitationandanalyzenewpropertiesofthemodel.Thiskindof
Higgsfields(becauseoftheircouplingtoSSBandthepossessionofanontrivialvacuumstate)maythenbe
relevantinviewofagravitationaltheorywhichmightentaillong-rangechangesinthedynamicstoexplain
darkcomponents,anchoredornotwithelementaryparticlephysics.
Indeed,thesimplestHiggs-fieldmodelbeyondthestandardmodelconsistsintheadditionofasingletpar-
ticlethatonlyinteractwiththeHiggssectoroftheSM,inwhichthesectordoesnotcoupledirectlytovector
bosons.Withafundamentalgauge-invariantconstructionblockφ†φ,thesimplestcouplingofaparticletoa
HiggsorHiggs-likefieldis[127]
LagrangiantermofHiggssector=˜λXφ†φ,(5.2.6)
whereXisascalarfieldand†representsthehermiteanconjugation,thetranspositionofatensorforreal-
valuedcomponents,andcomplexconjugationforpurelyscalarquantities.
TheHiggsfielddevelopsavacuumexpectationvalueand,aftershiftingit,thevertex(5.2.6)leadstoa
mixingbetweenthescalarfieldandtheHiggsfield.Thus,itmaygiverisetoneweffectsthatdonotinvolve
thescalarexplicitly[127].Furthermore,theX-fieldmaybeconsideredasnotfundamental,butaneffective
descriptionofanunderlyingdynamicalmechanism,andarelationbetweengravityandthegeneralized
Higgssectormaybeassumed.BothgravityandaHiggsparticlepossesssomeuniversalcharacteristics:
•Gravitycouplesuniversallytotheenergy–momentumtensorandtheHiggsparticletomass,which
correspondstothetraceoftheenergy–momentumtensor.Thissuggestsarelationbetweenthegene-
ralizedHiggssectorandgravity,whichisindeedgivenbyHiggsgravityin[69].

•Furthermore,thereisasimilaritybetweenXandthehypotheticalgravitonsincebotharesinglets
underthegaugegroup(see[26]).
Becausetheyhavenocouplingtoordinarymatter,singletfieldsarenotwellconstrainedbyexperiments.
Typically,onecanarguethattheyareabsentfromthetheorybecausetheycanhaveabaremasstermwhich
canbemadetobeoftheorderofthePlanckmassMP,makingthesefieldsinvisible.However,onecan
taketheattitudethatthePlancklengthbenotafundamentalconstantbutratherapropertyoftodaysstate
oftheworld,whichevolveintimeandbetypicallygivenbyavacuumexpectationvalueofsomescalar
field[238].WithaHiggscouplingtogravity,then,allmasses,includingthePlanckmass,shouldbegiven
bySSB.InthiscasethereisahierarchyofmassscalesMPv.Giventhesesimilarities,Xcanbe
consideredtobeessentiallythegravitonandbeidentifiedasconstant∙R,withthecurvaturescalarR(as
doneby[26]).Moreover,thispossibilitymaybeusedtoexplainthenaturalnessproblem,especiallysince
othercandidatessuchastop-quarkcondensationortechnicolor(inwhichquarksarenolongerprimordial)
havenotfunctionedsofarandsupersymmetrydoublesthespectrumofelementaryparticles,replacingBose
(Fermi)degreesoffreedomwithFermi(Bose)degreesoffreedom,whereasallsupersymmetricparticlesare
bynowbeyondphysicalreality(cf.Chapter2.3).
Makingalow-energyexpansion[26]andignoringhigherderivativeterms,aspontaneoussymmetrybreaking
theoryofgravitywithaHiggsfieldastheoriginofthePlanckmassmaybederived[26,27].Moreover,this
isthetheorywhichwasfirstderivedin[70]and[71].Theremnantoforiginallyverystronginteractions
istheparameterα˘,whichinChapter6.1willbeintroducedasthecouplingstrengthoftheHiggsfieldto
gravitation.ItwillessentiallygiveNewtonsgravitationalconstant,anditshighvaluewillenablethemodel
tobedistinguishabletogravityatlowenergyscales,otherthanthecasewithinusualJBDtheories.

62CHAPTER5.ALTERNATIVETHEORIESOFGRAVITYANDHISTORICALOVERVIEW

TheclassofSTTswithmassivescalarfieldsisgivenwithintheBergmann–Wagoner(BW)classwiththe
6Lagrangian,wingfollo

ˆˆLBW=16φπR+ωˆ(2φ)φˆ,λφˆ,λ−2ˆφU(φˆ)+LM√−g,(5.2.7)
φFurther,ˆφU(φˆ)=˜Λ(φˆ)givesacosmologicalfunctionand,λthederivativeinrespecttotheλ-coordinate.
WithintheBergmann–Wagonerclass,thereisawideaccountofanalyses,althoughmostofthemfocuson
U(φˆ)=0asspecialcase.However,analyseswithinthegeneralBWclasssuchasontheexistenceof
blackholesaswellasglobalpropertiesofstatic,sphericallysymmetricconfigurationscanbefound,for
instance,in[36–38],andondeSitterandwarminflationmodelsintheframeworkofSTTsin[19],andwith
Higgspotentialin[48,49].Friedmann–Lemaître–Robertson–Walker(FLRWorsimplyRW)modelsfor
Friedmann–LemaîtreUniversesforcosmology,further,areanalyzedin[200],obtainingaclassofseparable
Wheeler–deWittequationsafteraquantizationofthemodels.Thatis,weobtainequationswhichawave
functionoftheUniverseshouldsatisfyinatheoryofquantumgravity.

6Forpurposesofcompleteness,theBWclasscanbegiveninanevenmoregeneralformforDdimensionsandwithanon-minimal
couplingf(φ)R(see[38]).

6Chapter

Scalarpotential–tensortheorywithHiggs

–AtheoryofinducedgravitywithHiggspotentialisintroducedpartingfromtheBergmann–Wagoner
classofscalar–tensortheoriesandSpontaneousSymmetryBreaking.Thefieldequationsofgravitybefore
andaftersymmetrybreakdownarepresentedtogetherwithMaxwell-likeequationsforgravitywitha
gravitationalenergydensity.Partsofthisworkmayfoundpublishedespeciallyin[20,23,24].–

modelsanddensityLagrange6.1LetustakeacloserlookataBergmann–Wagoner(BW)modelwithaningeneralnonvanishingcosmological
function.LetthenthescalarfieldbedefinedthroughaU(N)isovectorwhichisascalarfieldalso,with
φˆ=α˘φ†φandthedefinitionω=2α˘π=const.,(6.1.1)
withthegravitationalstrengthα˘(asremnantofstronginteractions,cf.[26]),andthecosmologicalfunction
oftheBWclassgivenby
U(φˆ)=U(α˘φ†φ)=18πV∗(φ†φ)=8πV∗φˆ,(6.1.2)
α˘φ†φφˆα˘
whereasV∗(φ)≡V∗(φ†φ)bethepotential(density)ofthescalarfield.1
Themodelpartingfromequations(6.1.3)and(6.1.6)doesnotpossesssolelygravitativeverticesasan
einsteinianquantumtheorywould.Thislackingofonlygravitativeverticesshouldfurtherexcludetheap-
pearanceofoutergravitationallines(aslongasnoprimordialgravitationalconstantisassumed2)[93].If
aprimordialgravitationalconstantappeared,gravitationalsourceterms(vertices)wouldfollow,andthen
therenormalizationargumentswouldnotapplyanymore.Ascanbeeasilyseen,suchamodeldoesnot
possessadimension-loadedcouplingconstantasG,whichisthemainproblemforrenormalizingEinsteins
1ForarelationbetweenZees[247]andDehnensmodels,onemaylatertake˘α/=8πandisoscalarfieldsφ=ϕwithα˘and
=const.Further,thereisλZee=(1/3)λ.
2Thiswouldbethecaseinageneralizationwithmanyscalarfields.Then,R2-termswouldbenecessaryforrenormalization,asis
thecaseintheworkofK.Stelle[225]

63

64

CHAPTER6.SCALAR–TENSORTHEORYWITHHIGGSPOTENTIAL

theory.Through(6.2.4),Gwillbereplacedwiththereciprocaldimensionlessconstantα˘multipliedwith
φ†φ.Thus,thedimensionproblemforrenormalizationdisappears.DeWittspowercountingcriterion[243]
fornormalizabilitymaybeused[93]andthetheoryshouldberenormalizable[70].

ThescalarfieldshallcouplenonminimallywiththeRiccicurvaturescalarRwiththegravitationalstrength
α˘.Inthisway,wecangivetheLagrangianofascalar–tensortheoryinJordanframeoftheform
L=1α˘φ†φR+1φ;†µφ;µ−V∗(φ)+LM√−g,(6.1.3)
2π16whereas=1andc=1areset,and;µmeanthecovariantderivativewithrespecttoallgaugegroups.The
subscript,µrepresenttheusualderivative(seediscussioninrelationwiththeLagrangian(5.2.7)).The
Lagrangian(6.1.3)postulatespossiblegravitationalinteractionsnotonlymediatedbymasslessspin-2-
excitationsasispostulatedontheonehandinusualGR,buttakesintoaccountgravitationalinteractions
ofmassivescalarfields.Further,letthepotentialV∗(φ)ofthescalarfieldbeoftheformoftheoneofthe
Higgsfieldofelementaryparticlephysics,thatisaφ4-potentialwith
2224
24λ2242λ
V(φ)=λφ†φ+6µ=µφ†φ+λφ†φ2+3µ.(6.1.4)
Thepotentialin(6.1.4)possessestheadditivefactor3µ4/(2λ)ofequation(3.2.3).Theadditivetermisthus
relatedtotheelectionofavanishingformalcosmologicalconstantwhich,however,canbeinsertedinthe
theorybyaddingtheconstantterm(3.2.5),
2µα˘3V0=−4πλΛ0
withΛ0asatruecosmologicalconstantsothatthetotalHiggspotentialisgivenbyequation(3.2.6),
V∗(φ)=V(φ)+V0(6.1.5)
withacosmologicalfunctionΛ(φ)dependentonthisgeneralizedHiggspotential,aswillbeseeninequa-
tions(6.2.5)and(6.3.25).ThecosmologicalconstantΛ0isoftenexpectedtobevanishingforphysical
economy.However,togetherwithQuintessenceingeneral,itisrelatedtoourunderstandingofthenatureof
gravity.Itmightindeedbealow-energyappearancecomingfromprimarygravitationintheearlyUniverse,
asproposedin[188],butrelatedtodynamicalquintessentialfields.Nevertheless,theconstantpartofthe
cosmologicalfunctioncomingfromtheHiggspotential(6.1.5)(i.e.thetruecosmologicalconstant)will
furtherbetakenasvanishingand,ifwritten,thenonlyforpurposesofcompleteness.
Inprinciple,equation(6.1.3)representsamodelbetweentheJBDandtheBWclassofSTTs,withaconstant
couplingω=const.andΛ(φ)=0(unlessforSTT→GR).
In(6.1.3),LMistheLagrangedensityofthefermionicandmasslessbosonicfields,
1iLM=2ψ¯γµL,Rψ;µ+h.c.−16πFµνFµν−(1−qˆ)kψ¯Rφ˘†ˆxψL+h.c.,(6.1.6)
whileψin(6.1.6)arethefermionicfields,and
Fµν=ig1[Dµ,Dν]=Aν,µ−Aµ,ν+ig[Aµ,Aνa]
=A[ν,µ]+igAµa,Aµ


6.2.THEFIELDEQUATIONS

65

isthematrixrepresentationofthefield-strengthtensorforthegaugepotentialsAµ(see(4.2.1),(B.5.1)and
(B.5.11)).ItisdefinedbythecommutatorofthecovariantderivativeDµ(Ricciidentities),analogouslyto
withinelectrodynamicsfortheelectricandmagneticstrengthsEandB.Theexactformofcovariantderiva-
tives,thatisofthepotentials,however,dependsonthechiralityandformoftheactualfermionicfield.For
theelectroweakinteractions,left-handedwavefunctionsarethusdescribedby(iso-)doublets,whileright-
handedonesaredescribedby(iso-)singlets(cf.AppendixB.3).
Withinelectrodynamics,thehomogeneousMaxwellequationsarederivedusingJacobiidentitieswithco-
variantderivatives(Bianchiidentities).TheinhomogeneousonesdependontheLagrangianandthuson
theexactsystem(andthusontheenvironment,asreflectedintheappearanceofmagnetizationMand
polarizationPinthefieldequations).ThemoregeneralequationsofYang–Millsstheories,forthedyna-
micsforFµνandisovectorialψ,arederivedanalogously.However,unlikewithinQED,thecommutator
[Aµ,Aν]≡AµAν−AνAµisnotvanishing.Itpresentsself-interactionsofthegaugepotentials.Through
them,inQCD,forinstance,gluonsdointeractwitheachother,whilesuchinteractionsvanishwithinQED
giventheabelian(commutative)characterofthesymmetrygroupU(1).Photons,asgaugebosonsinQED,
self-interact.notdoInequation(6.1.6),xˆgivetheYukawacouplingoperator,k(orkfwhentakingthedifferentfamiliesorfla-
vors)beaconstantfactor,andthesubscriptsRandLrefertotheright-andleft-handedfermionicstatesof
ψ.Theindexabetheiso-spinindex,whichcountstheNisotopicelementsofthemultipletψ.Formatters
ofcomplementarity,inadditiontoφˆandφ,wehavetakenascalarfieldφ˘in(6.1.6).However,letusfurther
takeφ˘=φinthefollowing;thismeansthesamescalarfieldcoupledwiththeRicciscalarRandmatterfor
thecaseqˆ=1.
Equation(6.1.6)togetherwith(6.1.3)leadstothefieldequationsasderivedfirstin[26,27,70,71].The
parameterqˆisdefinedtogivethefermioniccouplingwiththescalarfield.ItwillbeessentialfortheKlein–
GordonequationoftheHiggsfieldofthemodelaswellasfortheDiracequation(v.i.inChapter6.2).

Concluding,following(6.1.3),(6.1.4),(6.1.5)and(6.1.6),wehavethefollowingLagrangian,
2L=1α˘φ†φR+1φ;†µφ;µ−λφ†φ+6µ2+3α˘µ2Λ0+iψ¯γµL,Rψ;µ+h.c.−(6.1.7)
116π224λ4πλ2
√−16πFµνFµν−(1−qˆ)kψ¯Rφ˘†ˆxψL+h.c.−g.
α˘isthefieldstrength,RistheRiccicurvaturescalar,Λ0isacosmologicalconstant,Fµνisthefield-strength
tensorinmatrixnotationforisocomponentsawithfieldvariablesAµ,ψisthefermionicwavefunctionof
thematterLagrangianLM,kistheYukawacoupling,xˆistheYukawamatrixandgisthedeterminantof
themetricgµν.SubscriptsRandLdenotetheright-andleft-handedwavefunction.φisafieldwithHiggs
potentialofparametersµ2<0andλ>0.Ingeneralterms,φ˘maybeafurtherHiggsfieldaddedtoφ.It
leadsinthecaseofqˆ=0tomassofelementaryparticles.α˘isrelatedtothegravitationalcouplingG˜which
willbeinducedbysymmetrybreaking.

equationsfieldThe6.2UsingtheHamiltonPrincipleofLeastActionandtheEuler–Lagrangeequationsforrelativisticfields,one
acquiresgeneralizedEinsteinfieldequationsandaHiggsfieldequationwithacouplingofthescalarfieldφ

66

CHAPTER6.SCALAR–TENSORTHEORYWITHHIGGSPOTENTIAL

withthecurvaturescalarRandthesymmetricmetricalenergy–momentumtensorTµν:3
18π8π1
Rµν−2Rgµν+Λ∗(φ)gµν=−α˘φ†φTµν−α˘φ†φφ(;†µφ;ν)−2φ;†λφ;λgµν−
−†1(φ†φ),µ;ν−φ†φ,β;βgµν,(6.2.1)
φφ∗φ;µ;µ−α˘φR+2δV(φ)=−2δLM,with2δV(φ)=µ2φ+λ(φ†φ)φ,(6.2.2)
8πδφ†δφ†δφ†6
whereTµνwillbegiveninequation(6.3.16),andwithΛ∗(φ)givenin(6.2.5).
Thetermontheright-handsideofscalar-fieldequation(6.2.2)isthesourceofthescalarfieldwith
†2δL†M=2δLM=−2k(1−qˆ)ψ¯RˆxψL.(6.2.3)
φδφδEquation(6.2.3)dependsonthefermionicLagrangianandthusontheparameterqˆ,consequencesofwhich
willbediscussedinChapter6.3.
InanalogytoGR(seeequation(A.4.3)),wemaydefinein(6.2.1)agravitationalcouplingtermasfollows,
1G(φ)=α˘φ†φ,(6.2.4)
whereasG(φ)hereisafieldquantityandthuslocal.Itisdependentonthescalarfieldφandthegravitational
strengthα˘.Analogously,ageneralcosmologicalfunctionwasdefinedin(6.2.1)as
8π6µ2Λ0
Λ∗(φ):=α˘φ†φV∗(φ)=8πG(φ)V(φ)−λφ†φ,(6.2.5)
mainlygivenbythepotentialofthescalarfieldanditsexcitations(viz(6.3.2)and(6.3.1)),andrelatedto
thecosmologicalfunctiontermα˘φ†φU(φ)oftheBWclassofSTTs.Thefieldequationsforthefermionic
fieldsandthebosonicYang–Millsfieldsareneglected.
BoththeRiccicurvaturescalarRinthefieldequationsofgravityandofthescalar-fieldequationarecoupled
tothescalarfielditself.R=gµνRµν≡3µ,ν=0gµνRµνcanbederivedfromequation(6.2.1),withthe
formR=8†πT+4V∗(φ)−φ;†βφ;β−†3(φ†φ);β;β,(6.2.6)
α˘φφφφ
whereasV∗(φ)=V(φ)+V0isvalidfrom(6.1.5)andTisthetraceofthetensorTµν.

Thefieldequationsofthetheoryofelementaryparticlesarevalid.Forinstance,theDiracequationsread4
µˆxφψR
iγ(RL)ψ;µ−kφ†ˆxψL=0,h.c.,(6.2.7)
withtheYukawacouplingoperatorxˆandtheDiracmatricesγµ,whicharegivenbytheCliffordalgebra
γµγν+γνγµ=2gµν1.(6.2.8)
TheinhomogeneousYang–Millsequationsforthegauge-fieldstrengthread
Faµν;ν=4πjaµ,(6.2.9)
3(...)aretheantisymmetricBachparenthesisgivenbyA(iBk)=21(AiBk+AkBi).
4Asusual,thesecondterminequation(6.2.7)belongstothemassofthe(other-handed)particles.

6.3.FIELDEQUATIONSAFTERSYMMETRYBREAKDOWN67
withthegaugecurrentsjaµgivenby
jaµ=jaµ(φ)+jaµ(ψ)
=gψ¯γµL,Rτaψ+2igφ†τaφ;µ+h.c.,(6.2.10)
withafermionicpartjaµ(ψ)andjaµ(φ)oftheHiggsfields.Further,theHiggscurrentsjaµ(φ)aregotten
throughequation(6.1.3)asgaugefieldsoftheinnersymmetrygroup,withthegaugepotentialAµa:
jaµ(φ)=δδLAµaM=2igφ†τaφ;µ+h.c.(6.2.11)
Further,theenergy–stresstensorisdefinedinequation(A.4.9).Itisofthefollowingform,
Tµν=2iψ(µL,Rψ;ν)+h.c.−41πFµλaFaνλ−41FaαβFaαβgµν.(6.2.12)
6.3Fieldequationsaftersymmetrybreakdown
Inthespontaneouslybrokenphaseofsymmetry,developingthescalarfieldφarounditsground-statevalue
,vφa=vNa+φa,(6.3.1)
theground-statevalueofthescalarfieldisgivenby
φa†0φa0=v2=−6µ2,(6.3.2)
λwithvreal-valued,µ2<0andλ>0.Thiscanfurtherberesolvedingeneralasφ0≡φa0=vNawith
Na=const.,withthetensorNsatisfyingN†aNa=1,with
ρφ=ρUN=vUφ0.(6.3.3)
Here,Uisaunitarytransformationandρisareal-valuedfunctionwhichtakesintoaccountthatnoconser-
v2ation2ruleisvalid2forHiggsfieldsalone.GiventhepropertiesofthetransformationUandofN,thereis
φ=ρsothatρgivesthesquaredvalueofthescalarfield.
Forthegroundstateφ0,thepotentialvanishesfortheelectionofnofurtheradditivefactorΛ0ofthecosmo-
logicalfunction,following(6.1.4)and(6.1.5):
1V(φ0)=u0≡8πG(φ0)Λ0.(6.3.4)
Thisistheenergydensityofthegroundstateofthescalarfield,whichisV˘∗(φ0)=−(3/2)(µ4/λ)+
(1/(8πG(φ0))Λ0ifthelastfactorofequation(6.1.4)isnottaken.Suchwouldleadtoaformalcosmological
constantaddedtothecosmologicalfunctionitself,whichwewanttoavoid.Hencethechosenformofthe
(6.1.4).inpotentialAccordingtotheusualmechanism,aftersymmetrybreaking,twoparticlesappear:amasslessparticle,
calledGoldstone,andamassiveparticleusuallycalledHiggs(cf.[22]).Thefirstoftheseparticlescanbe
gaugedawaythroughtheso-calledunitarygauge[20,69].i.e.
φ→U−1φ=ρU−1UN=ρN
Aaµτa→U−1AaµUτaU−1+igU,µ−1U.(6.3.5)

68

CHAPTER6.SCALAR–TENSORTHEORYWITHHIGGSPOTENTIAL

(6.3.7)

Afterunitarygauge,thereisstillφ†φ=ρ2.Further,thescalarfieldφcanbewrittenintermsofthe
real-valuedexcitedHiggsscalarfieldξ(areal-valuedscalarvariable)inthefollowingform:
†φ=ρN≡φ01+ξ=v1+ξNwithξ=φv2φ−1,(6.3.6)
withthedimensionlessparameterα˘from(6.1.3)andtheground-statevaluevwhich,following(6.3.2),is
relatedtotheHiggspotentialandtheHiggsparametersasfollows,
V∗(ξ)=3µ4ξ2−8πλ−1Λ0
2λ6µ2−1
=−41µ2v2ξ2+α˘8vπ2Λ0(6.3.7)
1−=λv4ξ2+8πΛ0.
2vα˘24Thegravitationalstrengthα˘mayfurtherbedefinedintermsoftheratio
α˘(MP/MB)21,(6.3.8)
whereMPandMBarethePlanckmassandthemassofthegaugeboson,respectively.Themassofthe
gaugebosonisgivenby√
MBπgv,(6.3.9)
wheregisthecouplingconstantofthecorrespondinggaugegroup.
Inrelationtothefermionicmass,thusinthecaseofacouplingofφtothefermionicLagrangian(6.1.6),the
couplingconstantsgandtheground-state(vacuumexpectation)valueareindirectlyknownfromhigh-energy
experiments.Fromacomparisonbetweencurrent–currentcouplingwithinFermistheory,low-energetic
limitsofW+-couplingsandtheweakonmassMW,theground-statevaluevcanbewrittendependenton
FermisconstantGFandbeexperimentallydeterminedasv2≈6∙104(GeV)2.However,therelation
betweenthevacuumexpectationvalue(v)andthemassMoftheparticlerelatedtoHiggsmechanismis
nowdifferenttowithintheSMsothatconstraintsonvaffectMinadifferentwaywithinthisscalar–tensor
theory(v.i.massanddiscussion).
Letusdiscussthefieldequationsofthismodelafterbreakdownofsymmetry:
equation:Dirac(i)TheDiracequations(6.2.7)(withDiracmatricesγµ)acquirethefollowingformaftersymmetrybrea-
king:iγ(µRL)ψ;µ−(1−qˆ)1+ξˆmψ(LR)=0.(6.3.10)
Theparameterqˆisdefinedsuchastoshowthefermioniccouplingofthescalarfield.Hence,thereis:
•(i)qˆ=0inthecasethatthisHiggsfieldcouplestothefermionicfield(ψ)intheLagrangian
and(6.1.6),•(ii)qˆ=1whenitcouplesonlywithcurvatureR.
Inthecase(ii),thescalarfieldmaybecosmon-likeorelsebeisovectorialandcoupleanalogouslyto
HiggsfieldsinGUTs(cf.[48]).Inthecase(i),thescalarfieldmayleadtomassgenerationanalogously

6.3.FIELDEQUATIONSAFTERSYMMETRYBREAKDOWN

69

toSM(cf.[49]).Consequently,mˆisthefermionicmassmatrixwhichisrelatedtotheYukawa
couplingoperatorxˆandtotheground-statevalueasfollow(cf.Chapter3.3),
2mˆ=1kvN†xˆ+xˆ†N.(6.3.11)
Here,formattersofsimplicitywehaveletthefamilysubscriptfaside.
Ifthescalarfieldiscoupledtothefermionicfield(ψ)in(6.1.6)(i.e.inthecaseqˆ=0),wehavethe
samestructureaswithintheSM.Thediagonalelements(eigenvalues)ofthemass-squarematrixread
(withc==1)
M¯(i)≡2πgˇv(τiN)†(τiN),(6.3.12)
withgauge-couplingconstantgandthegeneratorsτiofthesymmetrygroup(seeChapter3.3),for
whichthefollowingalgebrarelationisvalid,
[τi,τj]=ifijkτk
{τi,τj}=cij1+dijkτk.(6.3.13)
(ii)Scalar-fieldequationandmassparameter:
ThedynamicsoftheHiggsparticlesisgivenbyequation(6.2.2).Insertionof(6.2.6)inthesameleads
aftersymmetrybreakingtotheHiggsfieldequationwhichnowreadsasfollows,
,µ94α˘πλv218π44π−1
ξ;µ+1+34α˘πξ=1+34α˘π∙3α˘v2Tˆ−ξ+1ψ¯ˆmψ+3Λ01+3α˘,(6.3.14)
withtheenergy–momentumtensorTµν(analogouslytotheSM)withthetrace(sc.[71])
T=iψ¯γµL,Rψ;µ+h.c.=1+ξψ¯ˆmψ(6.3.15)
2andfermionicmassmatricesmˆasdefinedaccordingtoequation(6.3.11).Furthertermsfromgauge
bosonswhichwouldappearwithintheenergy–momentumtensoraftersymmetrybreakinghavebeen
glected.neTheenergy-stresstensorsatisfiesthefollowingequationlaw(seediscussionbelow),
Tˆµν;ν=(1−qˆ)21ξ,µ(1+ξ)−1Tˆ.(6.3.16)
Inequation(6.3.14),whichisaYukawaequation,agravitationalcouplingconstant
λ11G0=α˘v2=−α˘6µ2(6.3.17)
maybedefined(v.i.equation(6.3.26)).Further,the(Compton-)lengthscaleofthescalarfieldunder
validityof(6.3.11)maybedefinedusingequation(6.3.17).With=c=1,thelengthscalereads
l=16π1G+(3µα˘4/λ)=M−1∙c(6.3.18)
4π1/2
0anditis(geometrically)thereciprocalofthescalar-fieldmassM,whichintheSMisonlygivenby
|2µ|2(v.s.).ForthemassoftheHiggsparticles,hence,wehave
M2=−83πµα˘21+34α˘π−1c2
=94απ˜λv21+34α˘π−1c2.(6.3.19)

70

CHAPTER6.SCALAR–TENSORTHEORYWITHHIGGSPOTENTIAL

Thescalar-fieldmassisdependentonthereciprocalgravitationalcouplingstrengthα˘−1.Hence,in
contrapositiontotheSM,regardlessofhighvaluesofµ,Mmaybesmall-valuedindeed.
Afterinsertionofthelengthscale(6.3.18),theHiggspotential(6.3.7)reads
3ξ24π
V∗(ξ)=32l2πG01+3α˘+(8πG0)−1Λ0.(6.3.20)
Further,afterinsertionof(6.3.18)and(6.3.15)in(6.3.14),theHiggsfieldequationreads
ξ,µ;µ+ξ2=14π∙8πG0qˆTˆ+4Λ0.(6.3.21)
l1+3α˘33
Now,with(6.3.21)and(6.3.16),thefollowingisclear(cf.[23]):
•(I)Inthecaseqˆ=0,(6.3.21)willnotpossessasource,andfortheSM,thelattermeansthe
productionoffermionicmassthroughthisHiggsfield.Thisfactleadstoabreakingofthe
conservationlaw(6.3.16)throughanewHiggsforce.Ifthescalarfield(φ)couplestoψ
indeed,thenforΛ0=0thesourceofequation(6.3.21)vanishesexactly(sc.[71]).
•(II)fornosuchcoupling(qˆ=1)thesourceisweak(cf.[20])(thismeansproportionaltoG0),
andthereisnoentropyprocessfromtheconservationequation(6.3.16);sc.[70].
ForthephysicalpropertiesoftheparticlesrelatedtothisHiggsfield,thecase(I)meansthatthe
particles,whichareresponsibleformassofelementaryparticles,decoupleandinteractonlygravi-
tationally.Hence,theycannotbegeneratedthroughhigh-energycollisionexperimentsasexpected
intheforthcomingLHCexperiments.Ontheotherhand,thecase(II)meansnewparticleswhich
interactwithotherparticlesindeed,howeverweakly.Theseparticlesarerelatedtoadarksector(cf.
2.3).ChapterFurthermore,accordingto[27](withξBij=α˘/(16π)),thePlanckscalearisesafterSSB,thusre-
solvingthediscrepancyproblembetweenthePlanckandtheelectroweakscale.Additionally,wave-
functionrenormalizationofthescalarfieldresultsintheeffectivecouplingofthisHiggsfieldtomatter
becomingofgravitationalstrengthO(M/MP)(loc.cit.).5Forthisreason,theHiggsbecomesessen-
tiallyastableparticle,whichmayhavesomecosmologicalconsequences.WehavebasicallytheSM
withoutHiggsparticles(loc.cit.),especiallyforqˆ=0butalsoforqˆ=1,forwhich,however,a
furtherscalarfieldistobeaddedfornotonlyastrophysicalconsiderations.

(iii)Higgsmass:Cosmologicalconsequenceswoulddependonthelengthscaleofthescalarfield[25].
Especially,thescalar-fieldparticlesshouldeffectivelydecoupleforasmallmassM.Meanwhile,α˘,
asremnantofanoriginalstronginteraction(cf.[26]),wouldbetheessentialcauseforthegravitatio-
nalcouplingG0beingsosmall.Particularlyinthecaseqˆ=0,thescalarfieldpossessesqualitiesas
in[17]asacandidateofself-interactingDM,andmightbeinthiswayrelatedtoworkslike[73].
ThewaytheHiggsisremovedfromthetheoryherebymakingitscouplingtomattersmallistobe
contrastedwiththeusualwaywherethemassoftheHiggshastobetakentobelarge.
Further,accordingto[25],forλ=O(1),themassMofthe√Higgsbecomesverysmallandthisresults
inacontributiontothegravitationalforcewitharange∼1/λ.
Additionally(loc.cit.),theHiggsparticlethenbehavesasthecosmonofQuintessence.Withhigh
5Hence,themodelisrenormalizableindeed,asalreadystatedin[93].

6.3.FIELDEQUATIONSAFTERSYMMETRYBREAKDOWN

71

λ-values,assumedofO(α˘),MbecameoftheorderoftheelectroweakscalebecausetheHiggscou-
plingwouldhavebeenreducedtogravitationalstrength.
Above-resultspointtoasmallmassoftheparticlesrelatedtothescalarfieldwithininducedgravity
ofascalar–tensortheorywithHiggspotential,andindeed,iftheparticlesrelatedtothescalarfield
areasmassiveasindicatedin[20,23,24,50],thentheyhardlydecayinlessmassiveparticles.Fur-
thermore,withthoselowmasses,theystillliebelowtheaccuracyrangeof5th-forceexperimentsas
discussedin[3].Suchmasseswouldto-datefulfillthestrongequivalenceprinciple(SEP).Ifaneffec-
tivecouplingisfurtheralsodependentonstiffness(sayinternalpropertiesofmatter),thentheweak
equivalenceprinciple(WEP)willalsobebroken.Experimentsinthatmatter,whichtrytomeasure
thecorrelationbetweeninertialandgravitationalmass,areknownundertheconceptofEötvösexpe-
riment(viz[86],orAdelbergersmolybdenumEöt-Washexperiment[140]fromwhichthegreatest
compactifieddimensioninstringtheoryhastobesmallerthan44µm).
Thesquaredmass(6.3.19)ofthescalarfielddependsessentiallyonthegravitationalcouplingstrength
G0,whichisveryweak.Thus,theComptonlengthgivenbyl=M−1mayatthispointbeexpected
tobehigh-valued.WithintheSM,thiswouldmeanaverysmallvalueof|µ2|.Theconstraintsof
aHiggsfieldmass,though,maychangehereinrelationtothoseinthestandardtheory.Especially,
nonvanishingvaluesofvarepossibleforsmallmassesMwhichheremaybesmall-valuedwithout
thenecessityofasmall|µ2|parameter.Thisisthecaseforvanishingvaluesofbothµandλleading
tov=∼µ2/λ=0(seediscussionaboutqˆandlandMin[20,21,23,179]also).
Inviewofthestructureofl,relativelylargevaluesofthelengthscalearepossible,indeed.Thus,the
solutionofthefieldequationsfortheextremecasel→∞isworthanalyzing(seeChapter7.4).Only
relativelysmallvalues(whichshould,however,befinite,seeChapter7.2)seemtobeabletohelp
explainingproblemsastheoneofDarkMatter,andearlyanalyses(viz[20])doleadtohighvaluesof
ltoexplaincosmologicalproblemsthroughlong-rangechangesofdynamics.Theextremebehavior
ofthelimitingcasel→∞canhelpinthecharacterizationoftheusualone.
Forvalueslikein[20],l12ξ-termsarenegligibleindeed,andthestrongequivalenceprincipleisthen
validevenforsupra-solaraswellasmicroscopicdistances.Inthework[20],further,linearizationinν
andλandnotinξ(whichisvalid),leadsforlengthscalesloftheorderofmagnitudeofsomegalaxy
radii(somekiloparsecwith1kpc∼1013km)toflattenedrotationcurvesinamodelofgalaxieswith
polytropicdensitydistributionwithpolytropicindexγ=2,withorwithoutassumingaverymassive
core[20].Further,forthestrongestbarsinisolatedgalaxies,asimilarvalueofthelengthscale,of
about10kpc,isgottenin[50]withinthegeneralBWclassandwithanarbitrarypotential(butana-
logousfieldequations,withp=0).Thisvalueisbeyondtheaccuracyoftheexperimentspresented
in[3],andrepresentsamassM=lc∼10−26eV/c2.

equations:Einsteinv)(iThegeneralizedEinsteinfieldequations(6.2.1)readnowasgivenbelow,6
11πRµν−2Rgµν+Λ∗(ξ)gµν=−8πG˜Tˆµν−α˘(1+ξ)2[2ξ,µξ,ν−
1−ξ,λξ,λgµν−1+ξξ,µ;ν−ξ,λ;λgµν.(6.3.22)
6Thesemaybecomparedwiththefieldequationsin[50,210]withintheBWclasswitharescaledpotential.Thenewtonian
approximationofitleadstoessentiallythesameequationsashere.

72

CHAPTER6.SCALAR–TENSORTHEORYWITHHIGGSPOTENTIAL

Throughsimilitudewiththestandardtheory,aneffectivegravitationalcoupling(asscreenedgravita-
tionalstrength)wasdefinedas7
G˜≡G(ξ)=(1+ξ)−1G0.(6.3.23)
ThelatterisrelatedtothelocalcouplingG(φ)in(6.2.4)anditreducesto(6.3.17)intheabsenceof
aHiggs-likescalar-fieldexcitationξ(thatisforξ=0withthechosenformofHiggsexcitations,see
isthereHence,(6.3.6)).G(vN)≡G(φ0)=G0.(6.3.24)
Further,G(ξ)becomessingularforavanishingHiggs-likescalarfieldwithξ=−1.
Furtherinequation(6.3.22)acosmologicalfunctionisgivenby
∗8πG0Λ012πµ4ξ2Λ0
Λ(ξ)=1+ξV(ξ)+1+ξ=α˘v2λ1+ξ+1+ξ.(6.3.25)
Itisclearthat,forthespecialcaseofvanishingscalar-fieldexcitationsξ,equation(6.3.22)goes
throughtotheusualEinsteinfieldequations
Gµν=Rµν−1Rgµν+Λ0gµν=−κTµν,(6.3.26)
2withκ=κ0=8πG0/c4,giventhatG(ξ=0)=G0.
Asaresultofequation(6.3.17),thegravitationalcouplingstrengthgivenbyα˘isveryhigh,sothatthe
secondtermontheright-handsideofequation(6.3.22),π/(3α˘)-proportional,canbeneglecteddueto
thesmallnessoftheterm4π/(3α˘).
Equation(6.3.22)canberewrittenforα˘1.ThisandinsertionoftheHiggs-likefieldequation
(6.3.14)intheEinsteinfieldequationsleadtothefollowing,
Rµν−1Rgµν+12(1+ξ)−1ξ1+3ξgµν−1(1+ξ)−1Λ0gµν
2l43
=−8πG˜Tˆµν−3qˆTˆgµν−(1+ξ)−1ξ,µ;ν,(α˘1).(6.3.27)
ThecosmologicalfunctionΛ(φ)aftersymmetrybreaking(6.3.25)isessentiallyquadraticinξ.For
α˘1,ityields
∗312Λ0
Λ(ξ)=4l21+ξξ+1+ξ.(6.3.28)
withHence,

ξ=G(v)−G˜
˜Gitcanbewrittenintheformasbelow,
∗3G(v)2+G˜2G˜
Λ(ξ)=4l2G(v)G˜−2+G(v)Λ0.
7WewillusethetildeofG˜onlyforthecaseswhenwedonotwritetheξdependencyexplicitly.

(6.3.29)(6.3.30)

6.4.MAXWELL-LIKEEQUATIONSANDGRAVITATIONALENERGYDENSITY73

Inequation(6.3.30),apartfromtheconstantΛ0term,thecosmologicalfunctionisclearlyaconse-
quenceofthelocalityofthegravitationalfunctionG˜.
Further,thetraceofequation(6.3.27)leadstotheRicciscalar,
R=32ξ+8πG˜(1−qˆ)Tˆ=32G(v)−1+8πG˜(1−qˆ)Tˆ.(6.3.31)
˜llGRisindependentonΛ0,sinceitappearsintheHiggs-likefieldequation(6.3.21)andintheEinstein
fieldequations(6.3.22).ThetraceoverEinsteinsfieldequations,usingtheHiggs-likefieldequation,
leadstoacancelationoftheΛ0-term.
Usingequation(6.3.31),(6.3.27)canberewrittenintotheform
3Rµν−2l1211++2ξξξgµν−31(1+ξ)−1Λ0gµν
=−8πG˜Tˆµν−211−31qˆTˆgµν−(1+ξ)−1ξ,µ;ν,(α˘1),(6.3.32)


with

1+23ξ=13−G˜.
1+ξ2G0
Obviously,forvacuumandforqˆ=0ingeneral,theRicciscalarisgivenonlybythescalarfieldif
1/α˘termsareneglected.Themattertermofequation(6.3.31)leads,however,toadifferentright-hand
sideinthesquarebracketoftheeinsteinianfieldequationswheretheRiccicurvaturehasbeeninserted.

6.4Maxwell-likeequationsandgravitationalenergydensity
Letustakeageneralα˘togetherwithc==1.Accordingto(6.3.22),thegeneralizedEinsteinequations
aftersymmetrybreakingaregivenby
Rµν−2Rgµν+4l21+ξgµν+1+ξgµν=−c4Tµν−1+ξξ,µ;ν−ξ;λgµν−(6.4.1)
13ξ2Λ08πG˜1,λ
1π−α˘(1+ξ)22ξ,µξ,ν−ξ,λξ,λgµν.
MultiplyingwithgµνleadstotheRicciscalar,
3ξ2Λ0321
R=l21+ξ+41+ξ+8π˜GT−(1+ξ)2ξ,λ;λ−α˘(1+ξ)2ξ,λξ,λ.(6.4.2)
Thelattertakesfollowingformafterintroducingthescalar-fieldequationforξ,λ,λ(cf.(6.3.31)):
3˜qˆ2ξ,λξ,λΛ01
R=l2ξ+8πGT1−1+34α˘π−α˘(1+ξ)2+41+ξ1−1+34α˘π.(6.4.3)
Now,itisclearthatqˆ=1doesnotleadtoanexactlyvanishingcouplingbetweentheRicciscalarand
energy–stress.Itis,however,weak.
Letusnowtransformequation(6.4.1)further.Makinguseofsymmetrygµν=gνµandorthonormality

74CHAPTER6.SCALAR–TENSORTHEORYWITHHIGGSPOTENTIAL
gµνgµλ=δνλofthemetric,multiplyingequation(6.4.1)bygµσuσleadstotheform
Rνuσ−2Ruν+4l21+ξuν+1+ξuν=−8π˜GTνuσ−1+ξξ;νuσ−ξ;λuν−(6.4.4)
σ13ξ2Λ0σ1,σ,λ
−π(1+ξ)−22ξ,σξ,νuσ−ξ,λξ,λuν.
α˘Equation(6.4.4)mayfurtherbetransformedusingthescalar-fieldequationonto
σ11−13Λ044π−1
Rνuσ−2Ruν+l2ξ(1+ξ)1+4ξuν+1+ξ1−31+3α˘uν=−˜κjν+(6.4.5)
+1+3α˘qˆ˜κ3uν−(1+ξ)2ξ;νuσ−α˘1+ξ2ξξ,νuσ−ξλξuν.
4π−1T1,σπ1,σ,λ
Weuseκ˜=κ0/(1+ξ)andκ0=8πG0/c4.Furthermore,wehavedefinedthecurrentbymeansofthe
energy–momentumdensityofmattermeasuredbytheobserverinthefollowingform,
jµ≡Tσµuσ.(6.4.6)
ItmayberelatedtoanequationanaloguetoMaxwellsonesofelectrodynamicswhichweknowfrom
Chapter2.1.WeknowthatasetofhomogeneousMaxwell-likeequationarevalidforgravity.Here,we
intendtorewritethegeneralizedEinsteinfieldequationontoasetofinhomogeneousMaxwell-likeequations
ofgravity(vizas[64,65],cf.[23]).Inordertofulfillthis,theleft-handsideofequation(6.4.5)istobewritten
asadivergenceofthefield-strengthtensorF˜µν.Usingequation(6.4.6)and(1.2.9),thereis
F˜µσ;σ=−2Rσνuσ−Qµσ;σ(6.4.7)
11qˆ4πˆ11+23ξ24πΛ0
=2κ˜jν−2κ˜1+4π1−3+3α˘T−l21+ξξ−31+3α˘1+ξuν+
α˘3+1ξ,σ;νuσ−1Qν,σ;σ+1π(1+ξ)−12ξ,σξ,ν−ξ,λξ,λδσµuσ,(6.4.8)
κ02˜κκ0α˘
Thelatterdefinesatensorsµwith
ˆ1qˆ4π11+23ξ2πΛ0σ
sν=−2κ˜κ˜1−3+3α˘T−l21+ξξ−31+4α˘1+ξuν+Qν;σ+
+1ξ,σ;ν+π(1+ξ)−12ξ,σξ,ν−ξ,λξ,λδσνuσ(6.4.9)
α˘κ0sothatfollowingMaxwell-likeequationsarevalid:
F˜µσ;σ=2κ˜(jµ+sµ).(6.4.10)
Thefield-strengthF˜µνhastwosources:4-currentsjνasenergy–momentumdensityofmatterandsνas
energy–momentumdensityofthegravitationalfield(cf.[64]).Consequently,momentumconservationis
withalidv(jµ+sµ);µ=0.(6.4.11)
Theenergy–momentumdensitysµasdefinedheredependsontheaccelerationstateoftheobserver.For
α˘1,itsimplifiesto
1qˆ11+23ξ2Λ0σ1,σ
sµ=−2κ˜κ˜1−3Tˆ−l21+ξξ−31+ξ+uν+Qµ;σ+κ0ξ;µuσ.(6.4.12)

6.4.MAXWELL-LIKEEQUATIONSANDGRAVITATIONALENERGYDENSITY75
Theenergydensitymeasuredbyanobserveris
s=sµuµ.(6.4.13)
3Hence,forα˘1,thelatterleadsto
s=21κ˜κ˜1−3qˆT−l1211++2ξξξ−321Λ+0ξδνµ+Qµσ;σuµ+κ1ξ,σ;µδσµ.(6.4.14)
0Further,thereisQµσ;σuµasinequation(6.4.7).Inastaticalcase,itsimplifiesto
Qµσ;σuµ=−4uµ;σuµ;σ−2uσ;µ;σuµ.(6.4.15)
Afterrewritingwithξ,σ;µδσµ=ξ,µ;µ,thegravitationalenergy–momentumdensityasmeasuredbythe
observerthenreadsforα˘1asfollows,3
s=+κ˜2uµ;σuµ;σ+κ10ξ,µ;µ+κ˜1uσ;µ;σuµ−2κ˜1−3qˆT+2l1211++2ξξξ+311Λ+0ξ.
(6.4.16)Forweakdynamicalbehavior,themetricisnearlyconstantandforstaticalfieldswith
√uν;ν=(−√guν),λ=0,(6.4.17)
g−toleadidentitiesRiccitheRµνuµuν=−uσ;µ;σuµ,(6.4.18)
whichmaybefoundinequation(6.4.16).Further(6.4.18)readsexplicitlyasfollows,
Rµνuµuν=−uσ;µ;σuµ(6.4.19)
3=−2l1211++2ξξξ−311Λ+0ξ+˜κTµνuµuν−2κ˜1−3qˆT+1+1ξξ,µ;µ.
Ifweinsertthescalar-fieldequationin(6.4.16)fortimelike4-vectors(withgµνuµuν=1),thereis
s=2uµ;σuµ;σ+1ξ,µ;µ+1ξ,µ;νuµuνδνµ+Tµνuµuν−1−qˆT.(6.4.20)
κ˜κ0κ03
Multiplyingwiththeunitygµνgµν=1,thereisdirectly
ξ,µ;νuµuν=ξ,ν;ν(6.4.21)
sothattheenergydensitytakesthesimplifiedformasfollows,
Λ822s=+κ˜uµ;σuµ;σ+Tµνuµuν−(1−qˆ)T−κ0l2ξ+3κ00.(6.4.22)
Theenergydensityofthegravitationalfielddependsonthegeometry,ascalarfieldΛ0andtheenergy-stress
tensorTµνofmatter.
Forl→∞orlowscalar-fieldexcitations,togetherwithqˆ=0andΛ0=0,equation(6.4.22)givesthe
usualenergydensity.Forqˆ=1,however,thereisnotermproportionaltoT.Furthermore,undersuch
circumstancesofl→∞inqˆ=1,weknowfromearlierworks(e.g.[20,179])thatκ0istoberescaledto
κN.Thedemonstrationofthisfactwillfollowforcentralsymmetryinlinearapproximationofthetheory.
Further,inChapter7wewillusecentralsymmetrytoanalyzethismodel.InChapterChapter7.3,wewill
useitinrelationtotheenergydensityofequation(6.4.22)sotocomparetheresultswiththoseofstandard
dynamicsandtogainconstraintstothemodelpresentedhere(cf.[23]).

76

CHAPTER

6.

SCALAR–TENSOR

YTHEOR

WITH

HIGGS

TENTIALPO

Cosmological

artP

III

Consequences

vityGra

77

of

Induced

7Chapter

Inducedsphericalwithvitygrasymmetry

–ThemodelofinducedgravitywithHiggspotentialisanalyzedforcentralsymmetryinvirtueofits
phenomenologicalconsequencesforweakfieldsandBlackHoles.ThisisrelatedtothephenomenaofDark
MatterandDarkEnergy.Thegroundingandresultspresentedheremaybefoundpartlyin[21,23,24,179]
asconsequenceofthiswork.–

7.1Theexactequationsforsphericalsymmetry

LetusmaketheassumptionofavanishingtruecosmologicalconstantΛ0andagravitationalstrengthα˘1.
Further,letusassumethatsphericalsymmetry(xµ={x0=ct,x1=r,x2=ϑ,x3=ϕ})as
ds2=eν(cdt)2−eλdr2−r2dΩ2(7.1.1)
isgiven,withνandλasfunctionsoftherandtcoordinatesonlyanddΩ2=dϑ2+sin2ϑdϕ2asthe
metricofa2-dimunitsphere.Furthermore,letustakenowc==1intheequations.
Fortheenergy–momentumtensorTµν,letusassumephenomenologicallyanidealliquidwiththeenergy–
momentumtensoringeometricalformas

Tµν=(+p)uµuν−pgµν,uµuµ=1,(7.1.2)
vwithpressurepandenergy-densitydistribution,andwiththe4-velocityuµ=(u0,u1,0,0)andu1:=u0c1
(velocityv1)andthus

u02=e−ν−v12e−λ.(7.1.3)
−1
cThenonvanishingcomponentsofTµνaswellasthenonvanishingtermsoftheRiccitensorandscalarmay
befoundinAppendixC.1.Forthelineelement(7.1.1),withequations(C.1.8)through(C.1.12)andmetric

79

80

CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

componentsνandλ,theexactfieldequations(6.3.27)readnow
eν−λν+ν−νλ+ν−1λ−1λ+1λν˙+1(1+ξ)−1ξ1+3ξeν=
2¨˙2˙
244rc22c24c242l22
=8πG0e−ν−v1e−λ+v1peν−λ−11−1qˆ(−3p)eν+(7.1.4)
2−12
(1+ξ)c4c2c223
+(1+ξ)−1ξ¨−ν˙ξ˙−νeν−λξ,
c22c22
eλ−ν1λ¨+λ˙2−λ˙ν˙−ν−ν2+νλ+λ−1(1+ξ)−1ξ1+3ξeλ=
c2244244r2l22
=8πG40e−ν−v21e−λv21+peλ−ν+11−1qˆ(−3p)eλ+(7.1.5)
2−12
(1+ξ)ccc23
˙+(1+ξ)−1ξ−λeλ−νξ˙−λξ,
22c21λ˙=−8πG0e−ν−v1e−λ(+p)v1−(1+ξ)−11ξ˙−νξ˙−λ˙ξ(7.1.6)
2−1
cr(1+ξ)c4c2cc22

and

2e−λ1+2r(ν−λ)−1+2rl2(1+ξ)−1ξ1+23ξ=
=−8πG40pr2+11−1qˆ(−3p)r2−(1+ξ)−1re−λξ.(7.1.7)
(1+ξ)c23
Theunderlinedtermsshowadifferencebetweenboththemodelsqˆ=0andqˆ=1inthepresenceofmass
givenbyenergydensityandpressurep.Further,forξ=0,theoriginalEinsteinfieldequationsforcentral
symmetryarerestored.Forthem,theBirkhofftheoremisvalid.Thus,forvacuum(=0)allfieldsare
staticandν=ν(r)andλ=λ(r).Fornonvanishingexcitationsξ,however,thiscannotbestateddirectly.
TheHiggs-likeequationfortheexcitedHiggsfieldξyields(cf.AppendixC.2)
12ξ¨e−ν−ξe−λ−12ν˙−λ˙e−νξ˙−ν−λe−λξ−2e−λξ+12ξ=+qˆ8πG40(−3p),(7.1.8)
cc22rl3c
Thecouplingofthescalarfieldtomatterisgivenbyqˆ,whichhasspecialrelevanceinconnectiontothe
sourceandstabilityoftheHiggs(orHiggs-like,inthesenseoftheSM)particlesofthemodel.

7.2Linearequationsandstaticweak-fieldsolutions
Alinearizationinthepotentialsνandλisexpectedasagoodapproximationformanyphysicalcircum-
stances.Forverymassivecenters(r=0),however,itmaybenecessarynottolinearizeinthescalarfieldξ.
ThisispossiblebecausetheeffectivegravitationalcouplingG˜∼(1+ξ)−1runstozeroforr→0(withthe
scalar-fieldexcitationξasaYukawasolution,giventheformoftheHiggsfield)withoutthegravitational
potentialgettingsingular(themassesdecouplefromgravitationsincetheircouplingvanishes).Sincethe
gravitationalcouplingdisappearsatr=0,itispossibletolinearizeinνandλ,withoutthenecessityoflin-
earizinginξindeed(cf.[20]).Undersuchcircumstances,thefieldequations(7.1.4)through(7.1.8)reduce

7.2.LINEAREQUATIONSANDSTATICWEAK-FIELDSOLUTIONS81
3tothefollowingones,
1Δν+121+2ξξ−12λ¨+(1+ξ)−1ξ4¨=8π2G0−11−1qˆ(−3p)(1+ξ)−1,
22l1+ξc2cc23
(7.2.1)−ν2+λr−2l1211++2ξξξ+c122λ=8πc4G0p+211−31qˆ(−3p)(1+ξ)−1+
3¨
+ξ(1+ξ)−1,(7.2.2)
3−λ+r(ν−λ)+r221+2ξξ=−8π4G0p+11−1qˆ(−3p)r2(1+ξ)−1−
22l1+ξc23
−rξ(1+ξ)−1,(7.2.3)
andforthescalarfield,equation(7.1.8)leadstothefollowing:
ξ+l12ξ=qˆ83πcG40(−3p),(7.2.4)
1operatordAlembertthewith1∂∂22∂
=c2∂t−ΔwithΔ=∂r2+r∂r.(7.2.5)
Herewith,=c2istheenergydensityofthemassiveobject,andpthepressure.Itwouldbeexpectedto
actaspartofthemeasuredmassofwhatwewillcalltheeffectivemasslaterinthisChapter.
Forfurtherlinearizationinthepotentialsν,λandinthescalar-fieldexcitationξaswell,especiallyfor
r→∞,thereis[20]
3ξˆ=11++2ξξξ≈ξ.(7.2.6)
Thefieldequationsnowreadasfollows,
21ν+r1ν+2l12ξ−c122λ=8πc4G0−211−31qˆ(−3p)+cξ2,(7.2.7)
¨¨
−ν+λ−12ξ=8π4G0p+11−1qˆ(−3p)−λ¨2+ξ,(7.2.8)
2r2lc23c
−λ+r(ν−λ)+r2ξ=−8πG0p+11−1qˆ(−3p)r2−rξ,(7.2.9)
22l2c423
ξ+r2ξ−l12ξ−c12ξ¨=−qˆ83πcG40(−3p).(7.2.10)
Forξ=0,theoriginalEinsteinequationsforcentralsymmetryarerestored.
Forweakfields,i.e.forthenewtonianapproximation,themetriciswell-giventhroughsmall-valueddevia-
withhtionsµνgµν=ηµν+hµν,|hµν|1(7.2.11)
oftheMinkowskimetricηµν=diag(1,−1,−1,−1).Anewvariableψµν=hµν−21hηµνleadstoan
inhomogeneouswaveequation
ψµν,ν=−2κTµν.(7.2.12)
1ThedAlembertoperatorisdefinedaccordingtothesignature(+,-,-,-)ofthemetric.

82

CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

Itleadstotheexistenceofgravitationalwavescorrespondingtohµνwhichpropagatewithlightspeed.2
Further,thenewtonianpotentialΦisgiventhrough
Φh00=2c2,(7.2.13)
withh00=ν.Equation(7.2.7)canbewrittenas
c211p1λ¨ξ¨
ΔΦ+2l2ξ=8πG0−21−3qˆ−3c2+c22+c2
2p1λ¨ξ¨
=8πG03+c2+c22+c2,(qˆ=1),(7.2.14)
p1λ¨ξ¨
=4πG0+3c2+c22+c2,(qˆ=0),
togetherwiththescalar-fieldequationasfollows,
Δξ−c2ξ¨−l2ξ=−qˆ3c2−3c2.(7.2.15)
118πG0p
Equation(7.2.4),whichisequalto(7.2.15),isaKlein–Gordonequation.Calculationsforthecaseqˆ=0
maybefoundinChapterC.1.Furthermore,inthestaticcase,thescalar-fieldequation(7.2.4)forqˆ=1
equalsthemodifiedHelmholtzequationderivedinotherworks(viz[210]andrelated)foranormalized
gravitationalconstantG/c4=1andαRR=(−3p)/3.
Here,letustakethetimeindependentcaseoftheYukawaequationforthevacuumsolution.Equation
(7.2.15)maybesolvedthroughthefollowingansatz(weusethesubscriptaforr≥R),
ξa(r)=ae−r/l(7.2.16)
rwiththeComptonwavelengthl=Mcofthescalarfieldandneglectingananti-Yukawasolutionsothatthe
scalarfieldvanishesforspatialinfinity(boundarycondition).Thevalueofadependsonwhetherqˆ=0or
qˆ=1isset.TheconstantaisfurtherdependentonthemassM1.
Thepressurep,whichderivesfromtheequationofstateinT,isingeneralafunctionofthecoordinater,and
itoftendependsonthedensity(r).Itisoftengiventhroughthebarotropicparameterwwithp=wc2,
assumingalinearproportionalitybetweendensityandpressure.Specifically,itisspokenaboutdustmatter
forw=0andradiationforw=31.Mattermodeledwithw=1isfurthercalledultrastiff,andwith
w=−1itiscalledanti-stiffmatter.
Further,forthestaticalcasethelinearsolutionsofequations(7.2.1)through(7.2.4)forvacuumareknown.
Theyaregivenbyequation(7.2.16)andthefollowing,
νa=−ae−r/l−b,λa=−ν−2ξa(1+r)=−a1+2re−r/l+b,(7.2.17)
rrlrlr
with

(λa+νa)=rξa(7.2.18)
2Pulsarsarehighlymagnetized,rotatingneutronstarsthatemitabeamofelectromagneticradiation.Insomecases,theregularity
oftheirpulsationisaspreciseasanatomicclock.Adoublepulsarsystemshallproducestronggravitationalradiation,causingtheorbit
tocontinuallycontractasitlosesorbitalenergy.Indeed,indirectproofofgravitationalwavescomesfrommeasurementsonthedouble
pulsarPSR1913+16[135](Nobelprize1993)aswellasontheQuasarOJ287[231].Laser-interferometryonEarthbyGEO600,Virgo
orLIGOmightprovetheirexistencedirectly.ThebestupperlimitforthewaveamplitudebyLIGOliesabout2∙10−26[153].More
likelyisadetectionbymeansoftheLISAinspace,though.

7.2.LINEAREQUATIONSANDSTATICWEAK-FIELDSOLUTIONS

83

andtheintegrationconstantsaandb.Boundaryconditionsareusedtoreestablishtheminkowskianmetric
.infinityspatialatTheexactphysicalmeaningoftheintegrationconstantscanbedeterminedbysolvingtheinhomogeneous
equations,i.e.thelinearizedequationsinthepresenceofthesource.Examplesare–apolytropicdensity
distributionasshownin[20],aFreeman-discprofileasin[52]orahomogeneousmassdistributionwhich
thengivesthesolutionforapoint–particlewhentheradiusR0ofthegravitatingbodyistakenasR0→0.
e:takusLetdensity:opicolytrP(i)Apolytropicequationofstateisgivenbyp=wPγwithapolytropicamplitudewPandapolytropic
indexγ.Densityisgivenby[66]
1=1γ−11γ−1(νs−ν)γ−11,(7.2.19)
γ2wPwithνs=ν(R0).
Forγ=2,theEinsteinequationsbecomelinearinνconcerning,inaccordancewiththelinearized
Einsteintheory.Then,thepolytropicequationbecomes
1p=wP2,=4wP(νs−ν).(7.2.20)
Therefore,pressurepcanbeneglected(cf.[20]).
Asaresultofthisassumption,theinnerfieldsνi,λiandξiforr≤R0leadtoboundaryconditions
whichfurtherleadtoflattenedtangential-velocitycurves.Hence,forlargedistances,DarkMatteris
partlygiven,assumingornotaverymassivecenterofthegalaxies(sc.[20]).Inthiswork,however,
wewilluseadifferentapproach.
(ii)Constantdensitiesandnegligiblepressure:
AsimplemodelfortheinnerfieldsmaybegiventhroughahomogeneoussphereofradiusR0,withan
inside-constantdensitydistribution=const.=3M1/(4πR03)=/c2andanegligiblepressure
p.Forthisinnersolution,wetakeqˆ=1.Accordingly,wewillwritea=A1,b=B1andsoonfor
constants.grationintetheTheinnerscalarfield(r≤R0)isthengivenby(weusethesubscriptiforr≤R0)
rll3c4
ξi=1C1sinhr+C2coshr+8πG0l2,(qˆ=1).(7.2.21)
Usingcontinuityconditionsatr=R0withthevacuumfield,onegetsfortheintegrationconstants
condition:wingfollotheC1=−C2−8π4G0l2(R0+l)eR0/l,(7.2.22)
c3whereasC2isdependentontheexistenceofapoint–massinthecenterofthesphere.ForC2=0(i.e.
nosuchmass),theintegrationconstantA1reads
2A1=2M21G0l2coshR0−lsinhR0,(7.2.23)
cR0lR0l
withthemassM1ofthegravitationalobjectwithradiusR0.
Forlargelengthscalesinrelationtotheradiusofthegravitationalsource,thereis
l2R0lR01
R0coshl−R0sinhl−→3forlR0,(7.2.24)

84

CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

Further,thescalarfieldisgivenby

ξa=2M1G0e−r/l.(7.2.25)
2rc3ThemassM1isgivenbytheintegraloverdensityinthegravitationalobjectwithM1=4πR03/3.
TheamplitudeoftheYukawatermisoftheorderofmagnitudeofunityforrl.Thecoupling
constantG0,however,wasdefinedhereinanalogytothefieldequationsofGRandtonewtonian
theory.Itsvalueasacouplingconstant,though,hastobemeasuredinthelaboratory,anditsrelation
totheconstantofnewtoniandynamicsisdependentontheactualtheoryused.Furthermore,itsvalue
ishereconstrainedbythelengthscalel.Thecouplingconstantisimportantforallcalculationsof
astrophysicalmasses,anditcannotbegiventhroughastrophysicalconsiderations.
Theactualvalueofthegravitationalconstantwithinamodelcanbegainedonlythroughexperiments.
Thismeasurementwithintheinducedgravity,aswillbeseenbelow,shouldleadtoarescaling(cf.
(7.2.30))forl→∞inequation(7.2.29).Further,theexactvalueofthecouplingshouldbematerial-
independent,anditshouldbegiveningoodapproximationbythepotentialforpoint-likeparticles,for
whichtheinnerstructureisnegligible(seeequation(7.2.29)).Theeffectivevalueofthecoupling(G˜),
however,ismaterial-dependent,anditchangeswithdistance(itisnotoffirstorderinξ,though).The
couplingisgivenwithG0/(1+ξ).Thevariabilityofthegravitationalcouplingdecreasesinviewof
equation(7.2.25)withdecreasingdistancefromamass(sinceξ∼1/rinvacuum).Thecosmological
functionΛ,further,increasesitsabsolutevaluewithdecreasingdistancefromamass.However,itis
ofsecondorderinξandthereforenotyetcontainedinthelinearapproximationgiventhroughequa-
(7.2.29).tion

AnalogouslytoA1,theintegrationconstantB1isgivenby
B1=2M1G01+l1+l1−e−R0/lcoshR0−lsinhR0.(7.2.26)

c2R0R0lR0l
Itiseasilyseenthatforavanishinglengthscale(i.e.veryhighmassesofthescalar-fieldparticle),
A1disappearsandB1readsB1=2Mc12G0=2Mc12GNandgivesboththepotentialsλandνexactly
aswithinGR.Forlargelengthscalesl→∞(i.e.vanishingmassesofthescalarfieldparticle),from
(7.2.24)itisseenthatA1=23Mc12G0isvalid,andB1=2Mc12G0.

TheinnergravitationalpotentialΦireadsinlinearform
Φ=−M1G0l21+R0e−R0/lsinhr+1r2+const.(7.2.27)
iR0R0lrl2R0
2withaconstanttermconst.=−MR10G023+Rl0.Theoutersolution(r≥R0)is
Φa=−M1G01+l2coshR0−lsinhR0e−r/l+
rR02lR0l
+l1+l1−e−R0/lcoshR0−lsinhR0.(7.2.28)
R0R0lR0l
ForlR0andforlR0(see(7.2.24)),equation(7.2.28)goesthroughtoa1/rpotential.Forl
R0,thenewtonianpotential(withasmallcorrectionfromlthatshouldbedetectableinalaboratory)

7.2.LINEAREQUATIONSANDSTATICWEAK-FIELDSOLUTIONS

85

withG0=GNisgiven.Ontheotherhand,forlR0(andrl),thereis
Φaextreme=−M1G01+1e−r/l≈−M1G04−1r+...,(lR0).(7.2.29)
r3r33l
Ifequation(7.2.29)istogiveNewtonsgravitationalpotentialatsmallscales,then3
3G0=4GN(7.2.30)
shouldbevaliduptolrcorrectionswhichpullthisvaluedown.Correctionswouldbeexpectedfor
largescalesonly.Forthem,thegravitationalpotentialmayberewrittenas
Φa=−M1∗GN,(7.2.31)
rbeingM1∗aneffectivemasswith
M1∗≡M1(1−f1),f1=411−e−r/l.(7.2.32)
Itisvalidforvacuasuchthatgravitationalsourcesmaybetakenaspoint-like.Theeffectivemass
givestheradialdependenceforthecasep=0.Thefunctionf1isacorrectionwhichforhighlength
scalesinrelationtothedistancehasthefollowingform,
rrf1≈4l1+2l,rl.(7.2.33)
Consequently,forlargerdistances,theeffectivemasswouldbemeasuredassmallerthantheactual
one.Thecorrectionterm,however,isverysmallaslongasrl.Forlargedistances,ontheother
hand,f1tendstoavalueof1/4.

ThegravitationalpotentialΦisgivenbyν.Thereisstill,however,thesecondpotentialtoanalyze
inequation(7.2.17).Ityieldsinthecaser≥R(forqˆ=1forahomogeneoussphereofnegligible
pressure),λa=2M1G01−l2coshR0−lsinhR01+2re−r/l+
2
rR0lR0ll
+l1+l1−e−R0/lcoshR0−lsinhR0.(7.2.34)
R0R0lR0l
ForsmallvaluesofthelengthscalelR0,equation(7.2.34)leadsto
λa=2M1G0=2M1GN,(lR0),(7.2.35)
rrwhile,forlR0,thereis
λa=2M1G01−11+2re−r/l−→4M1G0=M1GN,(rl).(7.2.36)

r3l3rr
Furthermore,forhighlengthscales,λmayberewrittenforGNandforaSchwarzschildradiusrS=
2Mc12GNtotheform

λa=rS1−11+1+2re−r/l.

l4r3Actually,thesamere-scalingcanbefoundin[50].

(7.2.37)

86CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY
Letusdefinethefollowingfunction;
f2=11+1+re−r/l.(7.2.38)
l4Forrl,ityields
f2≈11−r−5r22forrl.(7.2.39)
l4l2Usingequations(7.2.33)and(7.2.38)aswellastheeffectivemassM1∗fromequation(7.2.32),λ
acuum,vforthenyieldsλa=2M1∗GN1−f2.(7.2.40)
rc21−f1
Hence,thereistheSchwarzschildradiusfortheeffectivemassM1,
∗rS∗=2Mc12GN.(7.2.41)
Accordingtoequation(7.2.40),λa=−νaisnotpossibleaslongastherearenopressureterms.
•Inshort:G0isnotapriorideterminedtobethenewtonianconstantofnewtoniandynamics.For
GNasthecouplingconstantasdeterminedbyatorsion-balanceexperimentinthelaboratory,if
dynamicsaretobenewtonian,wehave
(a)G0=GNforadistancerwhichislargeinrelationtothelengthscalelinthetorsion-
xperiment.ebalance(b)G0=43GNfordistancesrwhichareshortinrelationtolengthscalel.
(c)Radialdependenceappearsforaneffectivemasswhichdiffersfrombare,luminousmass.
Thismassisingenerallowerthanluminousmass.
(d)Fordust,withoutpressureterms,thereisν=−λ.
(iii)Nonvanishingpressures:
Letusnowtakepressuretermspwhichmayberelevantforastronomicalconsiderations.Now,with
κ0=8πc4G0andafterinsertionofequation(7.2.4)ofthescalar-fieldexcitationinequation(7.2.1)
forthegravitationalpotentialΦ,wehavethePoissonequationlinearizedinνandλofthefollowing
form,2ΔΦ(1+ξ)+c21+23ξΔξ=κ303−23(−3p)−43ξqˆ(−3p).(7.2.42)
Further,letusdefineageneralizedpotential
Ψ=Φ+c2ξ(7.2.43)
2forthejointactionofthenewtonian-likeΦpotentialandthescalarfieldξ.Nowequation(7.2.42)
reads2ΔΨ(1+ξ)=κ20+3p−21ξqˆ(−3p)−c4ξΔξ.(7.2.44)


7.2.LINEAREQUATIONSANDSTATICWEAK-FIELDSOLUTIONS

87

Inlinearapproximation,wehavetheusualPoissonequationforΨ:
ΔΨ=2Ψ=κ0(+3p).(7.2.45)
2Itisindependentonqˆ.Further,onemayeasilynoticethatforavanishingscalar-fieldexcitation,equa-
tion(7.2.45)reducestothatofusualGR.Consequently,thescalarfieldactsasafurthergravitational
interactionwhichatlowscalesisofnewtonianform.Further,itleadstoaneffective(measured)mass
whichpossessesscalarfieldcontributionsviapressurep(cf.[23,179]).
LetusnowsolvethePoissonequationforΨ,whereasΨrepresentstheclassicalfieldwhichfulfils
theclassicalPoissonequation,whileΦrepresentstheactualgravitationalpotential,dependentona
Yukawatermofthescalarfieldexcitationξ.Itssolutioncanbewrittenasatermequaltothestandard
newtonianoneplusatermofthepressurep.WehaveforscalesrR0andgeneraldensities,
Ψ=−G201+1e|r−rs|/l(rs)drs−G203−e|r−rs|/lp(rs)drs.(7.2.46)

c3|r−rs|c|r−rs|
Forapoint–particlewithbarotropicequation-of-stateparameterw=/p,wehave
Ψ=−MrG01+31(1−3w)e−r/l,(7.2.47)
whichpossessesnewtoniancharacterwhenthenewtoniangravitationalconstantGN=4G0/3is
takenforR0M1(whichisvalidintorsionexperimentsofGN).Obviously,theresultsaboveare
stillvalidforp=0.
Theequations(7.2.16)and(7.2.17)arestillvalid.Hence,letusgivetheintegrationconstantsvalid
forp=0.Insteadof(7.2.23),nowwehave
C1=−C2−8πG20l2(R0+l)(1−3w)eR0/l.(7.2.48)
c3Furthermore,thereisinsteadof(7.2.23),
A1=2M1G0(1−3w)l2coshR0−lsinhR0,(7.2.49)
c2R02lR0l
reads(7.2.23)toanalogouslywhich,A1=23Mc12G0(1−3w)forR0l.(7.2.50)
Further,thereisinsteadof(7.2.26),
B1=2M21G0(1+3w)1+l1+l1−eR0/lcoshR0−lsinhR0.

cR0R0lR0l
(7.2.51)Forapoint–particleofmassM1withbarotropicequationofstateatrestintheorigin(equivalentlyto
R0r),thecorrectionterms(7.2.33)and(7.2.38)ofthemetricarestillvalid:
f1=411−e−r/l,(7.2.52)
f2=11+1+2re−r/l,(7.2.53)
l4

88CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

Wemaynowdefinefollowingcorrectionparameter,
h=1−f2+32w(1+12f2),(7.2.54)
1−f1+2w1−2f1
togetherwithadynamicalmasswhichnowyields
Mdyn=M11−f1+23w1−21f1.(7.2.55)
Forr/l1,(7.2.55)reducesto
Mdyn=(1+3w)M1.(7.2.56)
Forr/l1,ontheotherhand,itreducesto
Mdyn=1+3wM1forlr.(7.2.57)
2Clearly,fornonvanishingpressures,theequation-of-stateparametermayleadtohigherdynamical
masstermsthanthoseofluminous(bare)massonlyfromdensities.Adiscussionabouteffective
massesinrelationtoDMmaybefoundin[23,179].Further,ther/lcorrectionsshallleadtodeviations
fromthestandardnewtonianpotential.Inthissense,workastheonein[3]isimportant.
Now,wedefineadynamical(effective)SchwarzschildradiusforthedynamicalmassMdyn:

Figure7.1:Evolutionoftheparameterhfordifferentequation-of-state(eos)parameterswanddistance
coefficientsr/l=x.Cf.[23].

r˜S=2Mdy2nGN.(7.2.58)
cForthevacuumsolutionsforrR0withradiiR0ofthegravitationalobjects,thepotentialsinthe
metric(C.1.1)maynowbegivenasfollows,
ν=−rdyn,λ=hrdyn.(7.2.59)
rrClearly,forweak-fieldapproximation,consistencywithaPPNframeworkisgivenforh=1sothat
ν=−λisvalid.
Forr/l1,hsimplifiessothatonegets
h=1+8wforl→∞.(7.2.60)
w3+2

7.3.ENERGY-DENSITYCONSTRAINTSONPRESSUREANDMASSTERMS

89

Figure7.2:Evolutionofthedynamical-masscoefficientMdyn/M1fordifferenteosparameterswanddis-
tancecoefficientsr/l=x.Cf.[23].

Termsofh(w)whichareactuallydependentondistance–length-scalerelationarenegligiblefor
rl,andderivativesofh(w,r)areofthemagnitudel2(andhenceofM2parametrizedbyc
).andEvidently,forw=0,thelinearapproximationshowsν=−2λasgottenfromequations(7.2.40)and
(7.2.31)inthecaseofvanishingpressureterms.Afinitevalueoftheparameterwseemsnecessary
withinaPPNframeworkifν=−λistobevalid.Suchvalue,itcanbesaidhere,isofw=1/5
withinalinearsolution(cf.[23,179]).Thismaybeseenasarelevantempiricalconstraintwhichis
furtherrelatedtotheenergydensityinChapter7.3andwhichwillbealsoimportantinthecontextof
solar-relativisticeffects(cf.Chapter7.6)aswellasflatrotationcurves(cf.Chapter7.8),forinstance.

7.3Energy-densityconstraintsonpressureandmassterms
Weknowfrom(7.2.59)thatnonvanishingpressuretermsarenecessaryforconsistencywithinaPPNframe-
work.Suchpressuretermsappearindynamical(7.2.55)andeffectivemasses(7.5.14)(v.i.Chapter7.5)and
shouldhavemeasurableeffectsonmassunlikeluminousmassfromdensityalone.
Thedifferentiationbetweenbareandeffectivemassesisrelatedtothefinitenessofscalarfields(v.i.Chapter
7.4,especiallytheamplitude(7.4.10)).Furthermore,pressuretermsarevisibleintheenergydensityof
gravity(6.4.22)withintheenergy–stresstensor.
Now,wewillanalyzetheconsequencesofthescalarfieldontheenergydensityforcentralsymmetryby
meansofcomparisonwiththeusualvaluefromGR.Bothhavetobeequalinordernottocontradictem-
piricalfactsfromnewtoniandynamics.Hence,itispossibletoconstraintthevaluesoftheequation-of-state
parametersforsolar-relativisticregimes.

Theenergy-densityofgravitationfromequation(6.4.22)ofChapter6.4and1.2reads
Λ822s=+κ˜uµ;σuµ;σ+Tµνuµuν−(1−qˆ)T−κ0l2ξ+3κ00.
Forl→∞orlowscalar-fieldexcitations,togetherwithqˆ=0andΛ0=0,equation(6.4.22)givestheusual
energydensity,assumingκ0=κN.Forqˆ=1,however,thereisatermmissingfromTanditisclearfrom
earlierworks(viz[51,179])thatκistoberescaledtoκN.

90

CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

Ifweconsidertheenergy–momentumofanidealfluidthenthereis
T=Tµνuµuν−3p
withalong

(7.3.1)

T=−3p,(7.3.2)
withenergydensities=c2andpressuresp.Now,energydensityofgravitationasin(6.4.22)leadstothe
,wbeloform822s=+κ˜uµ;σuµ;σ+qˆ+3(1−qˆ)p−κl2ξ+3κΛ0.(7.3.3)
00Letustakecartesiancoordinatesforcentralsymmetryasfollows,
ds2=(1+eν)dt2−1+eλ(dx12+dx22+dx32).(7.3.4)
Forthescalarfield,thereis
1ξ¨e−ν−ξe−λ−1ν˙−λ˙e−νξ˙−ν−λe−λξ−2e−λξ+1ξ=+qˆκ0(−3p).(7.3.5)
c2c422rl23
Foranobserverwhichisstatictomatter,thereisa4-velocityinlinearapproximationasfollows,
1uµ=1−2ν,0,0,0.(7.3.6)
Withνc2/2=Φ(thegravitationalpotential),wehave
u;µσuµ;σc4=−(u0,1)2c4=−(gradΦ)2.(7.3.7)
Therefore,equation(7.3.3)readsinthestaticcaseasfollows,
2/c4228
s=−κ˜(gradΦ)+qˆ+3(1−qˆ)p−κ0l2ξ+3κ0Λ0.(7.3.8)
Forcoherentmatterandusingthepressure-comprisingPoissonequation(vizequation(7.2.42)inChapter
7.2),wemaywritethefollowingforweakfieldsandnondominantξexcitations,
3ΔΦ2gradp=−κ0gradΦc4−2pc2.(7.3.9)
ThisproblemisanalogtotheoneofGRplusapressureterm.Now,undertheassumptionl→∞,usingthe
Gausstheoremseveraltimesandtakingintoaccountthatthepressurepissupposedtovanishatthesurface
ofmatterdistribution,equation(7.3.9)leadstoarelationbetweenthegravitationalpotentialandpressure
termspasbelow,
3c4pdV=1(1+3w)−1(gradΦ)2dV,(7.3.10)
κ0usingp=wwithanequation-of-stateparameterw,analogouslyto[65].Thisisarelationshipbetween
thenewtoniangravitationalpressureinmatterandthegravitationalfieldstrength.Theequation(7.3.10)is
relatedtothegravitationalenergy–momentumdensitybyequation(7.3.8).Actually,forthefieldenergy,
thereisforweakξfieldswithl→∞andavanishingcosmologicalconstantΛ0,
2/c4
sdV=−κ(gradΦ)2dV+qˆdV+3(1−qˆ)pdV,(7.3.11)
0

7.4.EXACTEQUATIONSANDBLACK-HOLESOLUTIONS

91

withanenergytermasdefinedbelow,
11c4dV=3κ0w(1+3w)(gradΦ)2dV.(7.3.12)
Insertingequation(7.3.12)in(7.3.11),thefieldenergyyields
1qˆ(1−qˆ)2
sdV=−κ0c42−3(1+3w)−(1+3w)(gradΦ)dV.(7.3.13)
ThesolutionwithinGRyieldssdVc4=−(1/κN)(gradΦ)2dV.Itgivesthesamevalueasthepotential
energyofabodywithinNewtonsgravitationaltheory,andsuchisnecessarytoavoidconflictswithelemen-
65].[64,mechanicstaryTheequation(7.3.8)givesGRssolutionforqˆ=0andw=0withκ0=κNindeed.Forqˆ=1,ontheother
hand,afterrescalingwithκN=4κ0/3,equalitybetweentheusualgravitationalenergyofGRandinduced-
gravitysgravitationalenergy(asofinterpretationgivenhere)leadstoaconstraintofw≈0.17≈1/6.This
pressurevalue,necessaryforconsistencywithphenomenologyandfornonvanishingscalarfields,should
appearinsomespecificcontexts,andvariationsfromitshouldleadtomeasurableconsequencesespecially
forlarge-scaledynamics(cf.[23]).
Indeed,theconstraintvalueofpressureisneartothevaluegiveninChapter7.2forthepressure.Hence,if
pressure(whichappearsbymeansofthescalarfieldinrelationtoA1)isgivensuchthatthetheoryiscon-
sistentwithnewtoniandynamics.Further,newtoniandynamicsisdirectlygivenforsolar-relativisticeffects
(furthergeneral-relativisticeffectsareusuallyobtainedasfurthercorrections).However,thishasnottobe
thecaseforlargedistances,aspressureactsantiscreeningformassesleadingtoeffectivemassesandthus
tophenomenologicalDarkMatterifunderstoodasthedeviationbetweenmeasuredandluminousmatter
withinGR.Newtoniandynamicsarenofurtherboundaryconditionforthegravitationalpotentialsatlong
ranges.Further,thismayrespondtonatureofQuintessencefieldswhosefluctuationsmaybehavesimilarto
arelativisticgas(sc.[241]).
GiventhatmodelsofQuintessenceusuallypredictcomposition-dependentgravitylikelong-rangeforces
mediatedbythefields[242](i.e.breakingoftheWEP),measurableconsequencesshouldappearforlarge
distancesindeed.There,scalarfieldsmayactsimilarlytoacosmologicalconstant(cf.[78])orasrelated
tothehalomassofgalaxiesandhencetoDarkMatterphenomenology[240].Toanalyzesuchdynamics,
wewilldiscussexactsolutionsofthemodelbeforecomparingthemintheirweak-fieldlimitwiththelinear
results.

7.4Exactequationsandblack-holesolutions
WithinGR,theexactvacuumsolutionforcentralsymmetrywithavanishingcosmologicalconstantΛ0is
metricarzschildSchwtheds2=1−rSc2dt2−dr2rS−r2dϑ2−r2sin2ϑdϕ2,(7.4.1)
1rr−whereastheconstantrS=2Mc12G0=Bistheso-calledSchwarzschildradius,validforaconstantgravita-
tionalcouplingG0whichinGRisG0=GN.rSrepresentstheradiuswhichabodyofmassM1musthave
sothatitsrest-massM1c2beequaltoitsinternalgravitationalpotentialenergyVNG0M12/rS(cf.[56]).
WithinGR,anyparticle,notevenaphoton,cannotescapefromaregionofradiusrSaroundabodyofmass
M1.Hence,theSchwarzschildradiusdefinesthehorizonofablackhole,sothatforr=rS,thereappears
anhorizonsingularity.Then,eλdiverges.

92

CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

Here,ananalysisofblack-holesolutionswithinthismodelisfulfilledforvanishingscalar-fieldmasses.In
viewofthediscussioninChapter6.3,thisisofspecialrelevanceindeed.
ThelimitingcaseofavanishingHiggsfieldmass(6.3.19)ofthenonminimallycoupledHiggsfieldas
scalarfield(l→∞)canbeunderstoodasadoublelimitµ2→0andλ→0,sothatµ4/λ=0and
v2=µ2/λ=finiteremainvalidthroughout.Thus,theground-statevaluekeepsthedegeneracy(remains
theoneofaHiggsmodeanddoesnotgothroughtooneofaWignerone)evenforthemasslesscaseofthese
particles(whichisnotthecaseinthestandardmodel),andthesymmetrystaysbrokenatlowenergies.The
scalarfieldstillchangestheusualdynamicsaftersymmetrybreakdownandtheexcitationsareingeneral
nonvanishing.Thus,thefieldequationsdonotreducetotheusualonesofGRaslongastheexcitationsξ
donotvanish,andnewchangesinthedynamicscanbeacquaintedtothescalarfieldanditsgravitational
interaction.awukaY

Inthestaticcaseandundertheassumptionofapoint–massatr=0(oronadistancer>R0foraradius
R0ofthemassiveobject),theHiggsfieldequationtakestheform(withl→∞)
ξa−1(λa−νa)ξ+2ξa=0,(7.4.2)
r2wheretheprimedenotesthedifferentiationwithrespecttotheradialcoordinaterasbefore.Thefirst
derivativeoftheexcitedscalarfieldξafromequation(7.4.2)inthecaseofapoint–mass(withinternal
structure(pressure))atr=0thenreads
ξa=A2ew/2=A2e(λa−ν)/2(l→∞),(7.4.3)
rrwhere(allsubscriptsawillbeletasideafterwards)
u:=λa+νaandw:=λa−νa(7.4.4)
aredefined.Here,thesubscriptameansforvacuumr>R0witharadiusR0ofthemassiveobject.The
integrationconstantA,whichappearsinequation(7.4.3),isderivedfromequation(6.3.21)inthelimitof
r→∞andl→∞through
ξ,µ;µ=√1(√−gξ,µ),µ=8πG20T(7.4.5)
c3g−withg=detgµνforthemetricgµν.Withg=g00k,afterdefiningk=detgik,equation(7.4.5)canbe
aswrittenre√1√(√g00√−kξ,µ),µ=8πG0T,(7.4.6)
g00−b3c2
whichisvalidforqˆ=1andwhich,aftermultipliedwith√g00√−bd3x,leadstothefollowingequation:
√√(√g00−kξ,µ),µd3x=8πG20T−kd3x√g00.(7.4.7)
c3Forthelimitr→∞,thereis
limr→∞√g00ξ,µ4πr2=8πG20√−gTd3x.(7.4.8)
c3Forequation(7.4.8),limr→∞g00=1isvalidaccordingtotheboundaryconditionsofminkowskianspace-
time.Forthesignature(+,-,-,-),equation(7.4.8)leadsthento
√−ξ,µr2=2G20−gTd3x(r→∞).(7.4.9)
c3

7.4.EXACTEQUATIONSANDBLACK-HOLESOLUTIONS93
Accordingtoequation(7.4.3)for|ew/2|≈1approximatingtotheminkoswkianlimit,theintegrationcons-
tantAisthengivenaccordingtoequation(6.3.21)and(7.4.9)by
A=−2G20T√−gd3x,(l→∞).(7.4.10)
c3ThevalueofAisrelatedtotheconstantA1oflinearapproximationofChapter7.2forsmallr/lvaluesfor
thehomogeneoussphereorpoint–masses.Itisrelatedtoinnersolutionsthroughcontinuityconditionsat
thesurfacer=R0and,aseasilyseen,dependentonp.Hence,itmaybeexpectedthatpdonotvanish
forvacuumsolutions.Otherwise,thescalar-fieldexcitationwouldvanish.Consequently,thescalarfield
shallconstraintherelationbetweenenergydensityandp,asalreadyseenforlinearsolutionsandaswill
beseenforsolar-relativisticeffectsandforthedynamicsofflatteningofrotationcurves.Suchshouldhave
implicationsformeasurablemasswhichisnottheintegraloverdensityalonebutoverpressuretermstoo.
SuchisseenunderthedefinitionofadynamicaleffectivemassinChapter7.2underequation(7.2.55),and
willlaterbeseeninChapter7.5.Further,thisconstraintsinrelationtogravitationalenergymaybefoundin
7.3.Chapterdefinewe,wNoq:=ln(1+ξ)(7.4.11)
fortheexcitationofthescalarfield.Insertingitinthenon-trivialfieldequationsassociatedtotheLagrangian
(6.1.3)forthemetric(7.1.1)(see[23]),andmakinguseofequation(7.4.3),leadsinthecaseofapoint–mass
invacuo(orequivalentlyfortheouterregionofamassiveobject)tothefollowingequationsforl→∞(the
subscriptaissuppressed)andforthestaticcase[21]:
ν+ν2−νλ−λ=−1ξ−λξ,(7.4.12)
244r1+ξ2
1−r2λ+r2νe−λ−1=−1+rξe−λξ=−re−λq,(7.4.13)
1ν−1λν+ν+ν2eν−λ=−νqeν−λ(7.4.14)
24r42Subtractionofequation(7.4.14)fromequation(7.4.12),andinsertionofwandu,definedinequation(7.4.4),
toleads21rw=1−e(u+w)/2+rq,(7.4.15)
u(1+2rq)=r2q(w−r4),(7.4.16)
21(u−w)=rB2ew/2−q=ABq,(7.4.17)
whereasequation(7.4.16)isthesubstractionoffieldequations,andequation(7.4.17)isthetotalintegralfor
νwithBasanintegrationconstant(theseresultsweremainlypublishedinreference[21]).Especially,it
thatseenbecanν=−λ−q(rν+2),u=−rABq2−2q.(7.4.18)
Usingthevalueofugiveninequation(7.4.16),equation(7.4.17)leadstothefollowingdecoupledequation:
w=−2(A+2B)ew/2−q−AB3ew−2q.(7.4.19)
rr

94CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY
Now,usingtheequations(7.4.15)and(7.4.19)oneimmediatelydeduces
eu/2+q=e−w/2+q+(2Ar+B)+2rAB2ew/2−q,(7.4.20)
and,therefore,onlythedifferentialequation(7.4.19)remainstobesolved.Theseconsiderationsfurtherlead
tothesolutionoftheexcitedHiggsfieldgivenbyequation(7.4.3)inthefollowingformforB=0:
ξ=eq−1=−1+e2AB(u−w).(7.4.21)
Equation(7.4.21)clearlyshowsthatsuchexcitationsofthescalarfieldareonlypossibleforanonvanishing
valueoftheintegrationconstantAgivenbyequation(7.4.10).
AsboundaryconditionwepostulatetheMinkowskimetricatspatialinfinity.Inordertodeterminethe
meaningoftheintegrationconstantBweconsideratfirsttheasymptoticcaser→∞ofthepotentials
again,i.e.|w|1,|u|1.Then,wegetfromequation(7.4.19):
ABAu=2r+2r2,(7.4.22)
w=2(A+B)+AB2.(7.4.23)
r2rinresultsThisν=(u−w)=−B(7.4.24)
r2andλ=(u+w)=AB2+(2A+B),(7.4.25)
r22r,consequentlyandB=2McS2GN=rS(7.4.26)
isvalidinviewoftheequationofmotionofthelineelement(7.1.1),whereMSistheasymptotic(r→∞)
visiblemassoftheparticle(andrepresentstheSchwarzschildmass),GNisthenewtoniangravitational
couplingconstant(sincethenewtoniancharacterisexpectedatspatialinfinity)andrSthedistancebelonging
toMS,belongingtotheSchwarzschildradius.Further,thedifferentialequation(7.4.19)isanabelianone
andcanbesolvedexactly.Withthesubstitution
ew/2−q=:rg˜(r)=:rg˜,(7.4.27)
equation(7.4.19)acquiresamuchsimplerformasgivenbelow,
rg˜=αg˜3−Kg˜2−g˜,(7.4.28)
whereasGK:=2A+B=2A+c2NMSand(7.4.29)
α:=−2AB=−cA2MSGN.(7.4.30)
Equation(7.4.28)canbeintegratedbyusingthemethodofseparationofvariables,whichforα=0reduces
totheformgivenas
g˜2√K2+4α+K−2αg˜√K2K+4αC
1+Kg˜−αg˜2√K2+4α−K+2αg˜=r2.(7.4.31)

7.4.EXACTEQUATIONSANDBLACK-HOLESOLUTIONS

95

TheintegrationconstantCinequation(7.4.31)canbecalculatedintheminkowskianlimit[121]as
K√√C=√K2+4α+KK2+4α.(7.4.32)
K2+4α−K
Thus,thenonminimallycoupledmasslessHiggsfieldwithininducedgravitywithHiggspotentialactsin
ananalogouswaytoamasslessscalarfieldwithinEinsteinstheoryofgravityin[121].Theintegration
constants,however,areofdifferentnaturetothosein[121]sinceKandthechargeparameterαaregiven
byboththeparametersofthefields.
Inviewoftheequations(7.4.20),(7.4.21)and(7.4.27),themetriccomponentsgivenbyequation(7.1.1)
andthescalarfieldbytheequation(7.4.10)forthecaseB=0maythenbeexpressedintermsofg˜inthe
followingform:B
eν=r21g˜2(1+Kg˜−αg˜2)K,(7.4.33)
eλ=1+Kg˜−αg˜2,(7.4.34)

ξ=−1+eKAlnr21g˜2(1+Kg˜−αg˜2)
AK=−1+r21g˜2(1+Kg˜−αg˜2).(7.4.35)
Theonlyeffectivephysicalparametersremaininginthetheoryofthepresentmodelaretheintegration
constantsAandBdefinedbytheequations(7.4.10)and(7.4.26),respectively.Unfortunately,itisquite
difficulttosolveequation(7.4.31)forg˜explicitly.However,atransparentdiscussionofthepropertiesof
thesolutionisfeasibleinconnectionto[121].ForthelimitingcaseA=0,i.e.fortheequationofstate
p=31c2(seeequation(7.4.10))andB=0(i.e.α=0andK=B),usingequations(7.4.20)and(7.4.27),
equation(7.4.28)canbeexactlysolvedforg˜inthefollowingform:[21]
g˜=11−B−1,(7.4.36)
rrandthusforthepotentials,usingequations(7.4.33)and(7.4.34):
eλ=1−B−1,
reν=1−B.(7.4.37)
rTheequation(7.4.37)indicatesthatthemetriccomponentsoflineelementgivenbyequation(7.1.1)corres-
pondtotheusualSchwarzschildmetric(withassociatedfeatures)whichappearsinthisformonlyforthe
limitingcaseofthevanishingscalar-fieldexcitations(i.e.ξ=0)[21].However,forthegeneralvaluesof
A<0,thequalitativeresultsshownintheworkofHardellandDehnen[121]arevalid.Itisworthmen-
tioningthatthehighervaluesof|A|,equation(7.4.33)leadtoadecreaseinνthroughtheexponentB/K.
Infact,themetricandscalarfieldareregulareverywherewithexceptionofr=0asnakedsingularity,and
thereexistsnoSchwarzschildhorizonexceptforthecaseofvanishingscalar-fieldexcitations.Therefore,
BlackHoles(intheusualsense)donotappearforthecaseA=0.TheymaythereforebecalledGreyStars.
NakedsingularitiesbreaktheCosmicCensorshipConjecture(CCC)andmayberelatedtobrighter,weaker

96CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

novaeandmightbeinterpretedintermsofsomegamma-raybursts(GRBs)[122,179,221].
Herewith,thescalarfieldleadstoascreeningoftheusualgravitationalinteraction(vizQuintessenceand
DarkEnergy).Forhigherscalar-fieldmasses,thisfeatureshouldbealsovalid,althoughinaweakenedform.
Forthederivativesorthepotentials,thereis
B22+Kg˜g˜
ν=Kr+1+Kg˜−αg˜2g˜,(7.4.38)
K−2αg˜
λ=1+Kg˜−αg˜2g˜,(7.4.39)
A22+Kg˜A
q=−Kr+1+Kg˜−αg˜2g˜=Bν,(7.4.40)
whereg˜isgivenby
g˜=1(αg˜3−Kg˜2−g˜).(7.4.41)
r

whereg˜isgivenby
isthereThen,

qrν=−ν−λ−2q
ABα=−2g˜2=g˜2.
rr

qrν=−ν−λ−2q(7.4.42)
=−2αg˜2=ABg˜2.
rr(7.4.43)Thistermvanishesinalinearapproximation.Further,withequation(7.4.28),itispossibletowritethe
derivativesofthefieldcomponentsintermsofg˜andtheconstants,
ν=Bg˜,λ=−Kg˜+2αg˜2,(7.4.44)
rrrq=rAg˜.(7.4.45)
Withequation(7.4.40),equation(7.2.36)canfurtherbediscussed.Aninsertionofqindependenceofν
toleadsν=−λ−Aν(rν+2)
B=−λ−q(rν+2).(7.4.46)
Insertingequation(7.4.42)inequation(7.4.46)leadsimmediatelytotheλequationin(7.4.44),sinceK=
2A+Bisvalid.Fromthispoint,forweakfields,itfollows
ν1+2AB=−λ,(forweakfields).(7.4.47)
Fromthisequation,itcanbeseenthatA=0leadstoν=−λofusualGR,asalreadydiscussedparting
results.linearthefrom

SOLUTIONÖM-LIKEREISSNER–NORDSTRTHE7.5.

97

7.5TheReissner–Nordström-likesolution
Theexactsolutionofequation(7.4.28)asanalyzedearlier(viz[21])showsqualitatively,togetherwith[121],
thatSchwarzschildhorizonsvanishfornonvanishingexcitationsξ=0.However,ananalyticalapproach
ofexactsolutionsforsuchfieldsisverydifficultandwehavethereforelookedforsomeapproximated
solutionsbyusingaseries-expansionmethod.Thisshowsbasicpropertiesofthesolutionandofthescalar-
fieldinteractionforsmallbutnon-vanishingexcitationsofthescalarfield,whichisespeciallyvalidfor
longdistancestothegravitativesourceaswellasforrelativelylowlymassivebodies.Anexactsolutionof
(7.4.28)forthecaseξ=0forg˜isdiscussedin[21],togetherwiththeformalsolutionofthemetric.
Forthepurposeofbehavioralanalysis,letusconsidertheseries-expansionmethodasansatzforg˜tofurther
(7.2.1)–(7.2.4),equationsthesimplify∞g˜=Cnn=1C1+C2+C23+C34+...,(7.5.1)
n=1rrrrr
Usingequation(7.5.1)in(7.4.28)andthencomparingtheleft-andright-handsidesofthisequation,the
coefficientsofr−1,r−2andsoon(uptothefifthorderin1/r)canbeobtainedwiththefollowingsimple
recursionrelationswithstraightforwardcalculationsasfollows(see[179]),
C1=1
C2=2A+B
C3=(2A+B)2+4AB.(7.5.2)
C4=(2A+B)3+23AB(2A+B)
C5=(2A+B)4+2924AB(2A+B)2+3(32AB)2
Clearly,theconstantsCiappearasadditiveandmultiplicativetermsofAandB,andthesearetheonlytwo
parametersofphysicalinterestofthepresentmodel.Consequently,werestructure(7.5.1)asfollows,
1−g˜=r11−(2Ar+B)+2ABrX(A,B;r−n);(n≥2),(7.5.3)
whereX(A,B;r−n)isfunctionofr,AandBonly,withvaluesuptothefifthorderin1/ras
X(A,B;r−n)=1+4(2A+B)+29(2A+B)2+3AB+...;(n≥2).(7.5.4)
2r23r36r48r4
Thereisn≥2.Accordingly,X(A,B;r−n)isnegligibleforextremelylargedistances(whichisinmostof
thecasestheregionofinterest).Lowpotencytermsappearassmallcorrectionsforsmallerdistancestothe
gravitationalsource.Suchasituationcanphysicallybeunderstoodintermsoftheweakeningofgravity
onceonemovesawayfromthecenterofagravitatingmass.Andintermsofthescalar-fieldmassMas
discussedin[23]andChapter6.3,furthercorrectionsappearatlargescales.
Fromthesubstractionof(7.2.1)from(7.2.3)andusing
ν=B21e(λ−ν)/2,(7.5.5)
ξ+1r

weget(cf.[21,179])

e(λ+ν)/2=(1+ξ)1+2A+B+ABg˜.
rg˜r2r

(7.5.6)

98

CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

Using(7.5.3)andequation(7.5.6)furtherleadsto
λ2A+B−1AB(2A+B)−2
e=1−r−2r21−r+(7.5.7)
2+2ABr(2A+B)X(A,B;r−n)−2rAB2X1(A,B;r−n),
whereX1(A,B;r−n)isafunctionofX(A,B;r−n)itself.
X1(A,B;r−n)=X(A,B;r−n)21−2A+B+ABX(A,B;r−n)(7.5.8)
−1
2rEquation(7.5.7)mayberewrittenontofollowingform,
eλ=1−(2A+B)−AB1−(2A+B)−2+(7.5.9)
2r2r2r
+AB(2A+B)X(A,B;r−n)−AB2X1(A,B;r−1),
r2r2whichaftersomecalculationsleadsto
1−eλ=1−(2A+B)+AB+AB(2A+B)X(A,B;r−n)−(AB)2X2(A,B;r−n),(7.5.10)
r2r22r4r2
whereX2(A,B;r−n)isafunctionofX1(A,B;r−n)andfurthertermsrelatedtothefirsttermof(7.5.10).
isThere(2A+B)AB2(2A+B)AB−1
X2a(A,B;r−n)=121−r−2r221+(AB4)1−(2Ar+B)−2ABr2(7.5.11)
r1−(2Ar+B)+2rAB24r1−r+2r2
andX2(A,B;r−n)=X1(A,B;r−n)+X2a(A,B;r−n).(7.5.12)
Forthepotencyn,thereisagainn≥2.Finally,uptothesecondorderin1/r,theequation(7.5.10)yields
1−eλ=1−2M˜GN+Q˜2,(7.5.13)
22rcrforwhichwehavedefinedaneffectivemassterm
M˜GN=A+B(7.5.14)
22candaReißner–Nordström-like(orReissner–Nordström(RN)-like)chargeparameter
Q˜2=2AB.(7.5.15)
TheseparametersarerelatedtotheparametersasdefinedinChapter7.4byreasonsof
˜K=2A+B=2M2GN,andα=−AB=−Q˜2.(7.5.16)
2cTheassumptionofKtakingtheplaceofageneralizedmassparameterandofαactingasrelatedtoagene-
ralizedchargeparameterarecorrectinthesenseofaReissner–Nordströmsolutionasisachievedbymeans

THE7.5.SOLUTIONÖM-LIKEREISSNER–NORDSTR

99

ofseriesexpansion.Hence,themetriccomponentλactsasthemetriccomponentforanelectricallycharged
particleinagravitationalfield(sc.[206])withamassM˜andachargeQ˜whichis,however,imaginary.
GNisthenewtoniangravitationalcouplingconstantwhichfortheoreticalcalculationsmayusuallybetaken
as1.Theeffectivemass,asageneraldynamicalmassandincontrapositiontotheactualmass,isde-
pendentonthemasswhichcomesfromenergydensity()itselfaswellasonpressurepwhichentersa
measuredmasstermthroughtheintegralofthetraceoftheenergy–momentumtensorT.Actually,bothA
andBshouldcontainapressurefactorindeed.Further,thegeneralizedchargeparameterQ˜,whichappears
asaconsequenceofusualgravitationaltermshiddeninBcoupledtogravitationalscalar-fieldtermsinA,
mayactagainstusualgravitationofGRinthesamewaytheRNchargedoesforachargedpoint–particlein
agravitationalfield[206].Thisquintessentialbehaviorgrowsforhighercharges|Q2|(i.e.forhigher
massesorfieldexcitations)andsmallerdistancestothegravitatingbody.Physicalconsequencesoftheso-
lutionsarevisible,althoughtheexactvanishingofhorizonsisduetoat-low-distancedominanttermswhich
appearforhigh-ordercorrections.Yet,itispossibletoanalyticallyglanceatthenaturewhichultimately
leadstothevanishingofusualSchwarzschildhorizonsontoGreyStarswhichappearsinexactsolutions
(seeChapter7.4and[21]).Yetmoreimportant,itispossibletointerpretthetermswhichactindynamics
withhelpofweak-fieldbehavior.Further,thesecondcomponentofthemetric(cf.Chapter7.4and[21])
asresults21−2M˜GN2eλ2˜BMcGN
eν=rc22;(n≥2).(7.5.17)
1−1−2rM˜c2GNQ˜2X(A,B;r−n)
Withsomestraightforwardcalculations,theequation(7.5.17)mayberewrittenontothefollowingform,
Bc2
ν1−2rM˜c2GN−rQ˜22+1−2rM˜c2GN2Q˜r22M˜c2GNX(A,B;r−n)−rQ˜24X1(A,B;r−n)2M˜GN
,=ecr1−1−2M˜2GNQ˜2X(A,B;r−n)2
(7.5.18)whichmaythenfurtherberestructuredforn≥2asfollows,
2
eν=1−2M˜2GN−Q˜22+Q˜21−2M˜2GN2M˜2GNX(A,B;r−n)−
rcrrrcc
2Bc2
Q˜21−2rM˜c2GNQ˜4X(A,B;r−n)22M˜GN
2cr−rX1(A,B;r−n)1+1−1−2M˜GNQ˜2X(A,B;r−n).(7.5.19)
Unlikeintheequation(7.5.13),uptosecondorder,thegeneralizedchargeparameterQ˜2cancelsoutin
equation(7.5.19),andforthemetriccomponentνthusevolvesasgivenbelow,
˜Bc2
eν=1−2M˜GN2MGN.(7.5.20)
2crThepotencytermof(7.5.20)maybewrittenasaneffective-massratioasfollows,
rdynBc2
r˜S=2M˜GN.(7.5.21)

100

CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

TheeffectiveSchwarzschildradiusr˜Sandthedynamicalradiusrdynofthelinearapproachmaybefurther
wing,follothethroughrelatedr˜S=2A+rdyn=2M12GN1+3w≈h(w)rdyn.(7.5.22)
2cActually,gravitationalenergy-densityanalyses(vizChapter7.2,[23])constraintwtoca.1/5to1/6,and
indeed,forsuchvaluesboththeeffective(r˜S)andthedynamical(rdyn)massesarenearlythesame,andfor
w=1/5thereish(w)=1,andthemetriccomponentsvalidinlinearapproximationhavethefollowing
form,eν=1−2Mdyn2GN,(7.5.23)
creλ=1+2Mrdyc2nGN.(7.5.24)
Furthermore,solar-relativisticeffectscanthenbeexpectedtobegivenastheyaremeasuredforalllow-
energysystemsandwithadvancesofperiheliondependentonthesystemsinternalstructure(p)(cf.Chapter
7.6,[23]).Forlowgravitatingsystems,effectivemassesM˜andMdynareapproximatelythesameandthe
dynamicalmassMdyntakestheplaceoftheactualmassM1.
Inlinearapproximation,from(7.5.20)wehave
eν=1−rB,(7.5.25)
withBforthedynamicalmassparameter(seeChapter7.2).Inviewofequation(7.2.59),Bcanbewritten
directlyintheformgivenbelow(cf.[179]),
B=2M1GN1+3w.(7.5.26)
22cHence,thelinearapproachisconsistentwiththeseries-expansionmethodasusedabove.4
InlinearapproximationforM→0,accordingtoequation(7.4.10)and(7.2.23),Aisgivenasfollows,
A=−1M1GN(1−3w).(7.5.27)
2c2Hence,forweak-fieldregimesitisthenequalto−A1aslongasM→0.Further,forpoint–particleswith
Mr1,linearapproachleadsbackto
λ=h(w)2Mdy2nGN,(7.5.28)
crwitha(lowlyMr-dependent)parameterh(w)well-givenby
h(w)=1+8w,(7.5.29)
w3+2forM→0andfornon-dominantQ˜-charges.
ForA=0,thepowercoefficientBc2/(2M˜GN)maybewrittenas(1+2A/B)−1,thusshowingthe
deviationfromausualSchwarzschildvalueofeν(withB=rSasSchwarzschildradius),andpointingto
differentconceptsofeffectivemassfordifferentranges.Evenforweakgravitationalfields,anonvanishing
scalarfieldappearsrelatedtothedensityandpressuretermsasadynamicalcorrectiontothebaremassM1
4Further,higher-ordercorrectionsarerelevantforconsiderationsneartoaswellasbeyondtheSchwarzschildandchargeradius.

(7.5.29)

ÖM-LIKEREISSNER–NORDSTRTHE7.5.SOLUTION

101

(cf.(7.4.10)andChapter7.2).ForA/B1,forrelativelyweakfieldregimesthereisclearlyaRN-like
solutionforλin(7.5.13)withageneralizedchargeparameterQ˜.Forνin(7.5.20),aquadraticterminr
mayonlyappearasconsequenceofthepotencyterm,i.e.fromtherelationbetweentheamplitudeof(7.5.5)
andtheeffectivemassparameter(7.5.14).AneffectivemassappearsfromananalogytotheSchwarzschild
solution.Thiseffectivemass,though,dependsonscalar-fieldcontributionsrelatedtothepressurep.
TheexcitationofthescalarfieldforsmallmassM˜andchargeQ˜inrelationtothedistance(i.e.beyond
high-fieldregimes)nowyieldsasfollows,
2M˜AcG2
ξ=1−2M˜2GNN−1.(7.5.30)
crItisexactlyvanishingfortheSchwarzschildmetric(A≡0).However,foraRN-likesolution(A=0),
thereisξ0forthetypicalvalueM˜>0withA<0.
Clearly,foravanishingexcitationparameterA,theSchwarzschildmetricisvalid.NegativevaluesofA,on
theotherhand,leadtoapositivefieldwithasingularvalueatr=0andthetendencyξ→0forspatial
.infinityThemetriccomponent(7.5.13)showsaRN-likeform.Hence,uptosecondordereλvanishesfor
2M˜GN±(2M˜GN)2−(2˜Qc)2
r±=2c2.(7.5.31)
GiventhevanishingofhorizonforA<0accordingto[21],thisgivesaregimewherethevalidityofappro-
ximationclearlybreaks.However,itshowsachangedbehaviorfromusualquasi-Schwarzschildcharacter
foranalmostflatmetrictowardsthevanishingofthesingularityfortheexactsolution.Second-orderappro-
ximationhasaReissner–Nordströmcharacterandthuspretendsfollowingcasesofbehavior:(i)-extremal
BHwhen(M˜GN)2=Q˜2(forwhichtheconcentriceventhorizonbecomesdegenerate),(ii)-anakedsingu-
laritywith(M˜GN)2<Q˜2,and(iii)-aSchwarzschildcasefor(M˜GN)2>Q˜2.Thecase(iii)alsoappears
whenthefieldexcitationsξvanishcompletely(forwhichQ˜iszeroexactly),whichisclearfromequations
(7.5.30).and(7.5.20)(7.5.13),Clearly,following[21,121]suchdegeneracyofthehorizonasgivenin(i)isnotgivenexactly.Nevertheless,
withinRNapproximationforA<0andB>0,onlythecase(iii)ispossibleindeed(Q˜2<0),leading
toaquasi-Schwarzschildbehaviorforlow-fieldregimes.Nevertheless,theanalogytoRNsolutionsisan
interestingsubjectwhichremindsthatforamassiveobjectwhosechargeisnotneutralizedbyfurthereffects,
theSchwarzschildradiusitselflosesitsmeaningofdominantpropertyofthesystem.Here,thegeneralized
chargeisanintrinsicqualitywhichaffectstheSchwarzschildradiusitself,andtheweakeningofthelatter
appearsindeedasconsequenceofthecorrectiontermswhichalreadyweakengravitationalfieldsforweak-
fieldregimes(cf.figures).Takingthisfactintoaccount,itmaybepossibletoestablishmeasurablyrelevant
distinctionsofthisinduced-gravitymodeltousualdynamicsevenatlong-scaleregimessuchasthoseof
galacticbulgesaswellasrelevantindicationsforintermediateregimestowardsstronggravitationalfields.
Itmaybeestablishedthatinallorders,theevolutionofgravitationalpotentials(i.e.themetriccomponents)
stronglydependsonthepossiblerelationsbetweenAandB.Suchrelationsarehelpfultounderstandhow
newphysicalcorrectiontermsactwithinlowgravitationalregimesinordertofinallybreakthegravitational
collapseontoaGreyStar.
LetustakeB>0throughoutforthepurposeofouranalysis.Further,letusnowconsidernegativevalues
(amplitudes)ofA.Accordingly,thechargeQ˜isimaginaryanditsnormfallingforpositivescalar-field
excitations.WetakeB=2foraSchwarzschildradiusrSgivenbyB=2M1GNforthecaseA=0(with

102

CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

Figure7.3:Evolutionofthemetriccomponentsinthismodel(O(r−2))fordifferent,(set)negativevalues
ofAwithB=2andM1GN/c2=1.N.B.:TheeffectiveSchwarzschildradiusdiminishesfordecaying
.Aofvalues

Figure7.4:EvolutionofthemetriccomponentsinthismodelforhighernegativeamplitudesAwithB=2
(v.s.).N.B.:Thereappearsaquintessentialattraction(cf.text)foreλathighervalueofA.

M1GN=1).InFig.7.3,itcaneasilybenoticedthatforverysmallexcitationamplitudesA,Schwarzschild
behaviorappearsalmostexactlywhileforgrowingvaluesof−A,i.e.of|Q˜2|,theSchwarzschildradius
diminishes(i.e.thesingularitydistancefromthegravitatingbodycenter)decreases.Aweakeningofthe
horizonistractableinviewof[121]foralowratioofthescalar-fieldparametertotheusualSchwarzschild
radius(A/B<1).Furthermore,theSchwarzschildradiusisnowgivenbytheeffectivemassM˜whichhas
decreased(hence,weconsideraneffectiveradius).Therefore,itisclearthatforlowgravitationalregimes,
itisaneffective,dynamicalmasswhichistotaketheroleofbare,luminousmass.Further,forevenhigher-
ordertermsof−A(Fig.7.4),thecurveofeλbecomesflatterastheeffectivemasstendstozero.Therise
ofquintessentialtermsforthedominanceofdynamicalbehaviorisclearastheSchwarzschildmassisnot
adominanttermanymore.Actually,formid-fieldregimesofhigh|A|withdecayingλfieldtowardsthe
origin,quintessentialattraction(v.i.)showsabehavioranaloguetotheexactonein[121].Suchbehavior
ishenceinterpretablebymeansofQ˜2.ForthevalueA=−1andhenceM˜GN=0,atlargescales,we
obtainaflatcurve(howevernon-minkowskianneartorQ≡|˜Q2|wheretheapproximationisbroken.
Yet,itisrQandnotrSitselfwhichmarksthesingularitywhichpretendstoappearatthisorder,analogously
toRNsolutions).Antigravitationalpropertiesappearforeλ,exactlyashappensforaRNcasewithmass
andchargeasgivenbytheeffectiveparametersasdefinedinequations(7.5.14)and(7.5.15)(inFig.7.4for

ANCEVADPERIHELION7.6.

103

M˜GN/c2=1andQ˜2=−2).Thisisstillofrelevanceforlowgravitationalregimes.
AlbeitshowingashiftofSchwarzschildradius,eνdoesnotshowgravitationallyrepulsivebehaviorfor
r>2M˜GNnorsingularpretensionatrQ(whichisnotshownin[121]either).However,aweakening
ofgravitationalcollapseappearanceisalsovisible,andthepatternsoftheexactqualitativebehaviorofeν
accordingto[121]areherealreadynoticeable,especiallyforlargeexcitationparametersA.
Inbrief,forA<0thesystemappearsaslessmassivethantherelatedSchwarzschildsystem(wheredy-
namicsisgivenstrictlybyM1).ThecaseA≤−1,however,isespeciallyinterestinginmanyaspects:It
showsbestthebehaviorformid-strongregimesaccordingto[21,121].Further,dynamicallyspeaking,let
uscallsuchsystemswithM1>0andM˜<0asquintessentiallyattracted.Quintessentialbecauseof
theantigravitationalbehaviorofeλfollowingthenegativeeffective(yetpositiveactual)mass.Attractive
becauseeνstillshowsattractionofthegravitatingbodylyingatr=0.ItsSchwarzschildradius,however,
isvanishing.Inthisanalysis,theroleofthechargeradiusrQisimportant.Athoroughdiscussion,though,
needsofthevaluesofAandBintermsofmassandpressure(v.i).

Figure7.5:EvolutionofthemetriccomponentsforthepositivevaluesofA.N.B.:Theeffective
SchwarzschildradiusaugmentsforhighervaluesofA.

LetusconsiderpositiveamplitudesA>0withB>0.Now,thegravitationalfieldisstrengthenedandthe
effectiveSchwarzschildradiusmovestor>2M1GN/c2,andgravitationalattractionbecomesgreateras
relatedtoarelativelyhigherdynamicalmassM˜>M1.Here,thescalar-fieldexcitationleadstoastrength-
eningofthegravitationalcoupling(cf.Fig.7.5)andmaythusbeofspecialrelevanceintermsofDarkMatter.
Using(7.5.26)and(7.5.27),acloserlookateλandeνindependenceofwmaybetakenintoaccount.In
secondapproximation,forpositivepressuresp(Fig.7.6),theeffectiveSchwarzschildradiusdecreasesinres-
pecttotheonegivenbyM1,correspondingtothecaseA<0asdiscussedearlier.Forstiffmatterw>1/3
(A>0),e−νhaslowervaluesthaneλ.Fornegativepressures,ontheotherhand(seeFig.7.7),w<−1/6
leadstoquintessentialattractionforeλ.Forw<−2/3thereisalsoagravitationalrepulsion.Q˜2isalways
smallerthanM˜unlessforw−0.7,forwhicheνisnearlyflat.

anceadverihelionP7.6Solar-relativisticeffectsneedofhigher-ordercorrectionsofthetime-coordinaterelatedtothemetriccom-
ponent.Hence,wewillconsiderthesolutionalreadyderivedinChapter7.5(cf.[179])forfurtheranalysis.

104

CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

Figure7.6:Evolutionofe−νandeλforw=1/5andw=1/2withM1GN/c2=1.Stiffmatterw>1/3
isrelatedtopositivesquaredchargesQ˜2>0.Forw=1/5,thedynamicalmassforlinearapproximation
readsMdyn=13/10.

Figure7.7:Evolutionofe−νandeλforw=−2/5andw=−1/5withM1GN/c2=1.w<0leads
toquintessentialattraction.ThedeviationbetweenM˜GNandMdynGNishigh.Forw<2/3,where
MdynGN<0isvalid,counter-gravitativebehaviorappears,togetherwithanakedsingularity.
ItmayhoweverbecomparedtoChapter7.2inthelightoftherelationofmassparametersinthecontextof
themasscoefficientrdyn/r˜SwhichisunlikeoneforA=0.
PartingfromthecaseofsmallSchwarzschildradiiincomparisontodistance,letustaketheresultfrom
Chapter7.5(cf.[179])whichshowsaRN-likesolutionasgivenbelow(cf.[23]),
rdyn/r˜S2−1
eν=1−r˜S;eλ=1−r˜S+r2Q,(7.6.1)
rrrwiththedynamicalradiusrdynofthelinearapproach,theeffectiveSchwarzschildradius(withcexplicitly)
r˜S=2A+rdyn=2M12GN1+3w≈h(w)rdyn,(7.6.2)
2cforthebare(luminous)massM1andthesquaredgeneralizedcharge-parameterradius
rQ2=|Q˜2|=|Ar˜S|.(7.6.3)
2Geodesicsaretheapplicabletrajectoriesforthetheory(cf.(6.3.16)).Forawell-chosensysteminorderto
getcurvesalongaplane,forr˜S/r1,equation(7.6.1)leadstoaLagrangefunctionofgeodesicmotionof

7.6.PERIHELIONADVANCE105
thefollowingformforqˆ=1,
23mr˜Srdyn/r˜Sdx02r˜Sdr2dϕ2
Ldx=21−rdτ−1+rdτ−rdτ,(7.6.4)
withtheeigentimeτandacycliccoordinateϕandthusaconstantconjugatemomentum,
L=mr2dϕ=mCb2=constant,(7.6.5)
dτaswellasacycliccoordinatex0=ctsothat
−m1−r˜Sdct=mCa=constant(7.6.6)
rdyn/r˜S
dτrisvalidforaparametrizedenergyterm.Consequently,equation(7.6.4)leadstofollowingrelation,
r˜Sdr22dϕ2r˜Srdyn/r˜Sdct22
1+rdτ+rdτ−1−rdτ=−c.(7.6.7)
Usingthedefinitionu=r−1and=d/dϕ,withtheinsertionof(7.6.5)and(7.6.6),theequation(7.6.7)
ws,folloasreads22222Ca2
−c=(1+r˜Su)Cbu+Cbu−(1−r˜Su)rdyn/r˜S.(7.6.8)
Therelationbetweentheeffectiveandthedynamicalradiireads
r˜S−rdyn=2A.(7.6.9)
Br˜SHence,equation(7.6.8)readsforsmallSchwarzschildradii,
Cb2u2+Cb2u2(1−r˜Su)−Ca2(1−r˜Su)2A/B=−c2(1−r˜Su).(7.6.10)
Afterafurtherderivativeinϕ,andtakingsmalleffectiveSchwarzschildradii,equation(7.6.10)leadsto
22u+u1−C2aAr˜S=3r˜Su2+r˜S2X¯c2,(7.6.11)
Cbrdyn22Cb
withtheparameterXdependentonCaasfollows,
2X¯=1−2ACa.(7.6.12)
2crndyClearly,forthelinear(quasi-newtonian)approximation,(7.6.11)alreadyleadstoatrajectorywhichshows
aperihelionshiftdependentonthescalarfieldviaCa2Ar˜S2/(Cb2rdyn).Forlow-energeticsystems,however,
thenewtonianKeplerorbitappearsasfirst-ordersolution,
r˜u0=S2c2(1+εcosϕ).(7.6.13)
C2bInthenext-orderapproximationandonlyforlineartermsinεϕ,thereis
2u1=r˜Sc21+εcos1−3r˜Scϕ.(7.6.14)
2Cb24Cb2

106CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY
Equations(7.6.13)and(7.6.14)givetheusualvalueonlyforw=1/6.Theperihelionadvanceforlow-
energeticsystemsisthenobviouslygivenby
˜ΔϕP=6MC2GNπ,(7.6.15)
bwhichisformallytheusualvalue.Itreadsasusualforw=1/6sothatM˜=M1.Ataboutsuchpressures,
thereisinhrdyn≈r˜S≈rS.Forhigherpressures,effectiveanddynamicalmassesarehigherthanthe
mass.luminous7.7Effectoffieldexcitationsonthegeodesicmotion
Wenowtrytoanalyzethesingularitiesinviewofthecompletenessofgeodesicsonthegroundsofequation
(7.1.1)withthemetriccomponentsasmentionedinChapter7.5(whereastheRN-likechargeparameter
basicallyarisesbecauseofthenonvanishingfieldexcitations).Herewehavegµνgµν=−εwhereε=0and
ε=−1representtheconstraintsforthenullandtimelikegeodesicsrespectively.Letustakec=1.The
geodesicequationscorrespondingtothemetric(7.1.1)aregivenasfollows,
t¨=−t˙r˙ν,(7.7.1)
r¨=−21e−λ−2r(ϑ˙2+sin2ϑϕ˙2)+eλr˙2λ+eνt˙2ν,(7.7.2)
ϑ¨=−2r˙ϑ˙+cosϑsinϑϕ˙2,(7.7.3)
rϕ¨=−r2r˙ϕ˙−cotϑϑ˙ϕ,˙(7.7.4)
wherethedotsandprimesrepresentthedifferentiationswithrespecttotheaffineparameterτandrres-
pectively.Equation(7.7.1)hasthesolutiont˙=Ee−ν,andusingϑ=π/2(equatorialplane)itleads
toϕ˙=L/r2whereEandLareintegrationconstants.Now,usingtheconstraintfortimelikeandnull
geodesics,weobtain22νν
v2=dtdr=e−(ν+λ)E21−EL2re2+εEe2.(7.7.5)
However,forthetangentialvelocity,wegetfromthegeodesicsequations,
Ω=dtdϕ=r12eνEL.(7.7.6)
Using(7.7.5)and(7.7.6),wecanwritetheangularvelocity,
dϕ=Le(ν+2λ)1−L2eν+εeν−21,(7.7.7)
drEr2E2r2E2
whichcountstheradialorbitchanges.
Using(7.7.5)andfirstconsideringtheRN-likechargeparametersuchthat|Q˜2|r2,aneffectivepotential
maybedefinedinthefollowingway,
Veff=εM˜GN+L22−M˜G3NL2.(7.7.8)
rr2r

7.7.EFFECTOFFIELDEXCITATIONSONTHEGEODESICMOTION

107

Theequations(7.7.5)and(7.7.8)satisfythefollowingenergylaw,
2dr211−2M˜rGN
1E=dt+Veff=22M˜GN2M˜BGNE+ε.(7.7.9)
r−Inequation(7.7.9),forABthereis,
2N1−2M˜rGN2M˜G2BA−4BA2
2M˜GN2M˜BGN=1−r.(7.7.10)
1r−Theeffectivepotential(7.7.8)hasnewtonianformforr→∞,anditpossessesanextremalvaluefor
2˜2
2εM˜GNL
r=−L11+3(2MG2N)ε.(7.7.11)
√Theinnermoststablecircularorbit(ISCO)isthengivenbyr=6M˜GN,whichisrelatedtoL/(M˜GN)=
12.Fortimelikegeodesicsthemaximalmomentum(Lx)–massrelationfortheextremumisthengivenby
2Lx(M1GN)2=3(1+6w).(7.7.12)

Figure7.8:TimelikeeffectivepotentialVeffforw=0(left)andw=0.2(right)anddifferentvaluesofL
andM1GN=1.N.B.:AnorbitforanenergyEequaltothemaximum(minimum)isunstable(stable).At
anenergygivenbythedashedhorizontalline,forthethickcurvethereisaboundorbitinwhichtheparticle
movesbetweentwoturningpoints.
Boundstatesappearforhighenoughstiffness(givenbytheequation-of-stateparameterw)andmomentum
givenbyL.For(parameterized)energiesEbelowthemaximumthereappearstableboundstates.ForE<0
thereareorbitswhichoscillatebetweentwoturningpoints,theperihelionandtheaphelion(cf.Fig.7.8)as
giveninusualGR.ThedifferencetousualGRisadependenceonw.Suchdependenceisinfactrelatedto
thedifferencebetweenluminousanddynamicalmass,tangentialvelocityandangularvelocity.Infact,the
presenceofascalarfieldforQuintessencegenerallychangesthesingularityofBlackHolesolutions[240],
andfurther,modelsofQuintessenceusuallypredictlong-rangeforcesmediatedbythefields[242]indeed,
leadingtodifferentconceptsofeffectivemass,especiallyasfluctuationsofthescalarfieldwhichmaybehave

108CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

similarlytorelativisticgas[241]and/orbeassociatedtohalomassofgalaxies[240](hencerelatedtoDM).
Tangentialandangularvelocitiesareachievedbyusingequations(7.7.5)and(7.7.7)alongwith(7.5.20)and
(7.5.13)suchthat(c=1)
B2M˜G2M˜BG
dr22M˜GN2M˜GN2M˜GNQ˜21−rNNL2−r2ε
dt=1−r1−r+r21−Er2,
(7.7.13)rrB−1
rdϕ2L21−2M˜GN2M˜GN+1−2M˜GN+Q˜22
dr1−2M˜GN2M˜GN
=B.(7.7.14)
E2r41−rE2r2(L2−r2ε)

Figure7.9:Left:Dashedgeneralized-charge(Q˜2)plot(dashedcurve)anddynamical,effectiveM˜(dot-
dashed)andMdyn(dashed)andactual(density)massM1=1(horizontalline)independenceofstiffness
w.h(w)Mdynasthickcontinuousline.Right:Tangentialvelocity(dr/dt)2(withε=−1)forM1GN=1
anddifferenteosparametersw.
FornonvanishingvaluesofAthereisadeviationbetweenMdynandM˜,whichisvisibleinB/(2M˜GN)
(vizfurther[23]).Aseasilyseenfromthelinearanalysisanditsrelationwiththegeneralone,forw=0,M˜
givesonlyhalfofthedynamicalmass.ThiscanbeseeninFig.7.9.M˜≈Mdynasmeasuredmassisvalid
forw≈1/3(i.e.A≈0).Thedeviationbetweenthemthengrowswithstiffness(w)asM˜growsmore
rapidlythanMdyn.Effective(M˜andMdyn)massesarethenhigherthantheactual(density,luminous)mass
becauseofpressuretermsthemselves.Forhigherstiffnessofmatter(pressure,innerstructure),effective
massM˜ishigherthanMdyn.Withinlineardynamics,itisthedynamicalmassMdynwhichdominates,
howeverwithnonvanishingvaluesofwaccordingtoaPPNframework.Theactualeffectivemassisgiven
forsmallpressures(w)approximatelyby
M˜=h(w)Mdyn,(7.7.15)
withM˜asactualeffectivemassfordynamicswithinanexactsolution.Atshortdistancetogravitational
sourcesandforastrophysicalconsiderations,itgivesameasuredmasswhichisunlikethebare,luminous
massfromdensity(M1).Forw=1,Mdynisover3timeshigherthandensitymassM1anditgivesonly
athirdoftheactualdynamicalmasswhichisforw=1about10timeshigherthanluminous(density)
mass!Hence,itisreasonabletospeculateaboutarelationtodark-matterphenomenologywithinthismodel

7.8.FLATROTATIONCURVES

109

ofgravitywithHiggspotential.
Further,inFig.7.9thetangentialvelocitydr/dtshowsaflatteningbehaviorwhichisgreaterasgreaterthe
stiffness(partingfromscalarfieldsgeneratedpurelybyhadronic–mainlybaryonic–matterandnotbythem-
selves,energy-densitydistributionandscalarfieldforexactlyflatcurvesisgivenin[24]whichtooshowsthe
relevanceofinnerstructureofgalaxiesinrelationtoscalar-fieldexcitationsandthephenomenologyofDark
Matter).Hence,assumingR0M1withR0asagalaxyradius,flattenedrotationcurvescanbeobtained
from(7.5.10)and(7.5.20)evenwithsimpledensityprofiles(vizFig.7.9).Dynamicsofgalaxiesaresuch
asifdynamicalmasswerehigherthantheluminousmass.ThisisinprinciplethephenomenonofDark
Matter.ThepotencytermB/(2M˜GN)in(7.5.20),togetherwiththegeneralizedRNchargeQ˜2maylead
phenomenologicallytoahaloofnon-luminous(effective)mattersurroundingagalaxycore.Fourth-order
corrections,further,donotchangetheseresultsmuchfortherelevantvaluesr>r˜S.

7.8Flatrotationcurves

In[20],theauthorderivedagalacticmodelforcentralsymmetryofthescalar–tensortheorywithHiggs
potentialinwhichflatrotationcurvesappearforapolytropicdensityofpolytropicindex2inwhich,parting
fromthegalacticcenter,spacefartherthanthegalacticluminousdiscisassumedasvacuum(i.e.witha
negligibleamountofmatterinthegalacticbarssothateachgravitationalbodymaybetakenassurrounded
byvacuum).DarkMatterphenomenologycouldbepartlyreproduced,howeverwithapeakatr=R0(for
abulgeradiusR0)whichisnotalwaysobservationallyverified.Yet,apolytropicdensitydistributionfor
galaxiesisusefultoachieveasatisfactoryagreementbetweentheoreticalandempiricaldata,postulatingor
notpostulatingacentralmassivecoreforgalaxies.
Further,inthelatterChapter(7.7),itwasshownthatundersomecircumstances,flatrotationcurvesare
obtainedfrominducedgravitydirectly,whereastheinternalpropertiesofthebulgearerelevantinform
oftheequation-of-stateparameterwasafactorofflattening.Actually,inChapter7.5,thisisshownasa
consequenceofpressuretermswhichactaspartofeffectivemasstermschanginggeodesicmotion.For
thesedynamics,further,theconstraintw≈1/5fortheusualequation-of-stateparameterwdoesnothave
tohold,sinceGRdynamicsareonlytobevalidatsolar-relativisticranges.
FlatrotationcurvesareusuallyrelatedtothephenomenologyofDarkMatter,asmentionedinChapter
2.3.Thecitedwork[20]leadsinthisdirection.Thecomparisonofthetheoreticalrotationcurveswiththe
rotationcurvesforseveralgalaxiesthereindicatesthatthescalar–tensortheorywiththeHiggsMechanism
isabletoexplainandcontributetotheflatrotationcurvesindeed.
ComparableapproachesareaFreeman-diskprofileasin[52]orahomogeneousmassdistributionwhich
thengivesthesolutionforapoint–particlewhentheradiusR0ofthegravitatingbodyistakenasR0→0.
Here,wewillassumelargedistancesrinrelationtotheradiusR0sothatthesolutionforapoint–particle
withinnerstructure(i.e.pressurewhichisrelatedtothescalar-fieldexcitationamplitudeandwhichshould,
thus,notbeneglected)willbegiven.Thedark-matterprofileforexactlyflatrotationcurves,forinstance,
wasanalyzedin[24],whereasDarkMatterdensitymayberelatedtoapressuretermwhichisrelatedtothe
scalarfield,theonlysourceofwhichisusualhadronicmatter.Hence,theremaybearelationbetweenpand
DarkMatterviaHiggsdynamics.Wewillgotosomedetailsofthisinthenextpages.
Densityprofilesforoutsidegalaxysbulgesareusuallytakensuchthattheyleadtoflatteningofrotation
curves.Thus,theyarecalledDMprofiles.UsualprofilesaretheNavarro–Frenk–White(NFW)[181]and

110

CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

the(moregeneral)Dehnenorγprofile[72]:
1+1rNFW:ˆ(r)∝r2,(7.8.1)
rsrDehnen:ˆ(r)∝rγ(1+srs)4−γ,(7.8.2)
rsistheprofileradiuswhichisalengthscaleofthesphericalsystem.Clearly,Dehnensmodelwithγ=2
isrelatedtotheNFWprofile.Thesemodelsseekforuniversalhalodensitiesinthecontextofflatrotation
curves.Hence,suchareoftencalleduniversalhalodensitylaws.
Usedwithinascalar–tensortheory,suchdensitiesmaybeusednotasanalternativetoDarkMatterinorder
tosolvethemissingmassproblembuttocomputetheinfluenceofscalarfieldsinrotationcurvesandve-
locitydispersionswithgalaxieswhichpossessNFWorDehnenprofiles,forinstance(cf.[210]foralinear
analysiswithγdensity).Then,universaldensitylawsleadtohigherrotationcurvesthanwithinGR,given
masspropertiesofthenonminimallycoupledscalarfield.Hence,inthatspirit,thepresentapproachdoes
notintendtoreproducerotationanddispersioncurvesforwell-knowndensitiesbuttoderivethenecessary
densityprofileforflatrotationcurvesasanalternatemodeltoCDMindeedbutwiththescalarfieldasso-
ciatedwithDarkMatterphenomenologicallyandhenceactingasadensitycontributionthenon-newtonian
dynamicsofwhicharedominantatgalacticranges.
Letusconsidertheweakfieldsforgalacticrangesandthetangentialvelocityofgalaxiesasgivenbelow,
vt=rdΦ.(7.8.3)
drNow,thePoissonequationmaybewrittenasfollows,
ΔΦ+c2ξ=3πGN(+3p).(7.8.4)
2c2readsequation-fieldscalarTheΔξ−12ξ=−2π4GN(−3p).(7.8.5)
clPhenomenologically,rotationvelocityofespeciallyspiralgalaxiesisnearlyconstant(problemofflatrotation
curves)outsidetheluminouscoreasifasphericalhaloofnonluminousmatterwithanextensionmuch
greaterthanthegalaxysvisiblediscsurroundedthem(cf.[187]).Hence,assumenowthattherotation
velocityisconstantsotoanalyzethenecessaryconditionsforsuchcase.Then,thegravitationalpotentialto
giveflatrotationcurvesisofthefollowingform,
Φ=vt2ln(r).(7.8.6)
ThePoissonequationofthemodel(7.8.4)togetherwiththescalar-fieldequation(7.8.5)leadsto
22vr2t+2cl2ξ=4πc2GN.ˆ(7.8.7)
Equation(7.8.7)definesadensityprofilewhichisthefollowing,
3(vtc)2ξc4
ˆ=+2p=4πGNr2+8πGNl2.(7.8.8)

(7.8.4)

7.8.FLATROTATIONCURVES111
Itpossesses,ontheonehand,acontributionofmatterdensityingeneralandacontributionpofpressure
(comingfromtheinnerstructureofmatter).Ontheotherhand,itpossessesanewtonian-typeenergydensity
andascalar-fieldcontributiontodensitydistribution.Hence,wedefinetwoenergy-densitycomponentsas
ws,follo2∗=4(πvtGcN)r2,(7.8.9)
4cξξ=8πGNl2.(7.8.10)
Both∗andξtogethergivethedensityprofileusuallycalleddark-matterprofileDM(cf.[51],[24]).In
theseterms,thescalar-fieldcontribution(ξ)shallactasdark-matterdensitycontributiontothetotalenergy
density.Theothercontribution(∗)ispurelynewtonianandrepresentsenergydensityespeciallyofbaryons.
Furthermore,thescalarfieldcannotbeitsownsource,whichmeansthatitwouldhaveonlyusual-matter
density(∗)assource.Hence,forequation(7.8.5)theremustbe
Δξ−12ξ=−2π4GN∗.(7.8.11)
clAccordingly,pressureisgivenby
p=92ξ.(7.8.12)
Thepressureislinearlydependentonthescalar-fielddensityandonthescalarfielditself.Thescalar-field
excitationishencegiveninsuchcasesasfollows,
c2ξ=36πGNl2p.(7.8.13)
Phenomenologically,thereisarelationofabouttentoonebetweenhadronicmatter(∗)andDarkMatter.
Accordingtotheequations(7.8.9)and(7.8.10),DarkMatterisgivenbythescalar-fieldcontributionof
density.Hence,arelationastheonefollowingshouldbevalid:
≈10∙∗,(7.8.14)
whereasthetotalenergydensityˆfordark-matterdensityprofileisgivenaccordingtoequation(7.8.8).
Interestingly,followingtheequation(7.8.12),therelationbetweentotalenergydensityˆandpressuregives
anequation-of-stateparameterasfollows,
1pwˆ=ˆ≈5.(7.8.15)
Forlarge,galacticscales,hence,wˆisgivenbytheDarkMattercontributionwhichcomesfromthe∗scalar
field.Furthermore,forvanishingcontributionsofthescalarfield,p/ˆξtendstozero,andfor=0,
i.e.foracompletedominanceofthescalar-fieldexcitation,thetotalequation-of-stateparameterreads
exactly1/4.5.Abaryonicdensityof1/9ofthescalar-fielddensityleadstowˆ=1/5.Astonishingly,these
valueswhicharenecessarywithindark-matterphenomenologyofflatrotationcurvesarecomparabletothe
equation-of-stateparameterwwithinthecontextofsolar-relativisticeffects,andespeciallywithinthelinear
approachaswellaswheredensityismainlygivenbyusualmatter∗andnewtoniandynamics.Apparently,
anequation-of-stateparameterofabout1/5isaweak-fieldconstraintnotonlyforsolar-relativisticeffects

112CHAPTER7.INDUCEDGRAVITYWITHSPHERICALSYMMETRY

butalsowithindark-matterphenomenologyderivedfromthepresentinduced-gravitymodelwithaHiggs
potential.Thebehaviorofthecontributionsofpressure,however,differforbothcases.Wewillnow
investigatethebehaviorofdensitycomponentsforgalacticdynamics.
Afterparametrizingdistancebyalengthscaleaofthesphericalsystem(alengthrelatedtothedistanceat
whichgalaxiespossessflatrotationcurves),intheintervalbetweenr=0andr=rhwithrhashaloradius
withrh>landrh>a(see[24],cf.[51]),thesolutionofthescalarfieldreads
ξ=1v2te−lraaShira−sinhraEi−ra,(7.8.16)
2
2raclalala
whereasr/a=raandl/a=la.Shi(x)isthehyperbolicsineintegralfunction(i.e.SinushIntegral(x))and
Ei(x)istheexponentialintegralfunction.Forthedark-matterprofile(totaldensitydistribution),thereis
(vtc)211−rararara
ˆ=4πGNa2ra2+4la2raelaShila−sinhlaEi−la.(7.8.17)
ItgivesthehalostructureinawayanaloguetoNFWorDehnenprofileswithscaleradiirs=a[72,181].
ThescaleradiusaisoftheorderofmagnitudeofagalacticcoreR0(i.e.theluminous-discradiusof

Figure7.10:Evolutionofdensitydistributionsforla=1/5(left)andla=1/35(right).N.B.:Scalar-field
(ξ)dominanceforshorterdistances,andbaryonic(∗)dominanceforlimitsoflargescales.
galaxies),andthescalarfieldisnegligiblefortoohighavalueofloftheorderofmagnitudeofa.Ifthe
lengthscalelislowerthana,though,fromshortdistancesuptosometimesthescalea,thenthereisa
dominantcontributionofthescalarfield,asmaybeseeninFig.7.10.Forlongerdistances(vizFig.7.10left
panel),usualmatter(∗)dominatesthedynamicswithinthetotalenergydensityˆ.Thus,thereisdust-matter
dominanceoftheUniverse.
Letusnowdefinetheratioofdensityparameter,
Δ≡ˆ/∗=1+r2ae−ra/laShi(ra/la)−sinh(ra/la)Ei(−ra/la).(7.8.18)
laThedensityratiogivesnon-baryonicbehavior(Δ−1=0),anditshowsthreespecialbehaviorcases.At
lowerscales(asshownintherightpanelofFig.7.10),alinearlygrowingfunctionwithrelativelyhighslope,
athighscalesaconstantvalue,andanintermediatephasewithamaximum(adloc.figure7.11right).For
alllengthscalesla,thenonbaryonicbehaviorΔ−1isnegligibleatshorterranges,eventhoughscalar-field
densitiesdodominate.Hence,thedominantscalar-fieldcontributionofdensityactsasabaryoniccontribu-
tionforshorterdistances(evenra>1).Newtonianbehaviordominatesatshortranges.
Forl/a≈35(Fig.7.11leftpanel),thereisΔ≈10(i.e.long-rangedynamicsareasiftherewere10

7.8.FLATROTATIONCURVES

113

Figure7.11:Densityratios:DarkMatterdominanceforla=1/36(left)andNon-newtonianbehavior
(right)forla=1/5,la=20andla=35.

timesthebaryonicdensity).Thereisscalar-fielddensity(ξ)dominanceatdistancesofgalacticbars,and
therelationwˆ=1/5isthusvalidandnon-newtonianbehaviorofthescalarfieldisdominantforflattening
dynamicsofgalaxies.DarkMatterbehaviorappearsatlongranges.

114

CHAPTER

7.

INDUCED

VITYGRA

WITH

SPHERICAL

YSYMMETR

8Chapter

alkFriedmann–Robertson–Wmetricer

–ThemodelofinducedgravitywithHiggspotentialisanalyzedfortheFriedmann–Lemaîtrecosmology
withRobertson–WalkersymmetryinvirtueofDarkMatterandQuintessencefromgeneralizedFriedmann
equations.SignaturesfortheprimevalUniverseandInflationarealsodiscussed.Introductoryaspectsmay
alreadybefoundpublishedin[24].–

8.1ThegeneralizedFriedmannequationsandtheHubbleparameter

LetusnowtakealookattheFriedmann–Lemaître–Robertson–Walker(RW)metric,usedforgeneralcos-
mologyandcosmicevolution:
ds2=(cdt)2−a(t)2dχ2+f(χ)2dϑ2+sin2ϑdϕ2.(8.1.1)
Here,χisthecovariantdistance,a(t)isthescaleparameter(else,manytimesfoundespeciallyasR),
K∈{1,0,−1}isthecurvatureconstantandf∈{sinχ,χ,sinhχ}isaparameterthatdependsonK.This
lastsymmetryisbasedonthelong-rangewell-realizedassumptionthatthecosmosishomogeneousand
Principle).Cosmological(theisotropicInthefollowing,letuswritedowntheEinsteinandscalar-fieldequationsforthismetric.Thesewillbe
aHiggs-likeequationandgeneralizedformsoftheFriedmann–Lemaître(orsimplyFriedmann)equations
whichhaveanewtermofthescalar-fieldexcitationsξandderivativesofthesame.Further,sincetheseexci-
tationsleadtotheeffectivecouplingG˜,thewholesetofequationscanalsobewrittenintermsofchangings
ofthegravitationalcouplinginsteadofthescalarfield.Inthisway,effectsongravityofthenonminimal
couplingwithξmaybeclearer.

•Equationsindependenceofξ:
OnthegroundingoftheRWmetric,thecontinuitycondition(6.3.16)fortheenergydensity=c2
takesthefollowingform,

˙ξ1a˙˙+3a(+p)=(1−qˆ)21+ξ(−3p).(8.1.2)
Thetotalenergyisconservedandthescalarfieldproducesnoentropyprocessforqˆ=1,other
thanwithqˆ=0.Inthelattercase,however,suchprocessesbecomeminimalwhentheeffective

115

116

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

gravitationalcouplingtendstoaconstantbehavior,i.e.forscalarfieldswithtendencytoaconstant
term.Forabarotropicpressurep=w,equation(8.1.2)mayfurtherbewrittenas
˙ξ1˙=−3H(1+w)+(1−qˆ)21+ξ(1−3w),(8.1.3)
withtheHubbleparameterH=˙a/a.
TheHiggs-likefieldequationfor(8.1.1)reads
ξ¨+3a˙ξ˙+c2ξ=8πG0qˆ(−3p).(8.1.4)
al23c2(1+34α˘π)
Asexpected,itssourcevanishesforqˆ=0,andinsuchcaseHiggsparticlesonlyinteractgravitatio-
.nallyWith(8.1.4)and(6.3.22),equation(8.1.1)leadstogeneralizedFriedmannequationsinformsinde-
pendentonthesourceparameterqˆ.Explicitly,theyread
a˙2+Kc2=18πG0(+V(ξ))−a˙ξ˙+πξ˙2(8.1.5)
a21+ξ3c2a3α˘1+ξ
−18πG0c224πa˙πξ˙2
=(1+ξ)c23+4l2ξ1+3α˘−aξ˙+3α˘1+ξ,
and2a¨+a˙2+Kc2=−18πG0(p−V(ξ))+ξ¨+2a˙ξ˙+πξ˙2
aa21+ξc2aα˘1+ξ
=−8πG0p−(1+ξ)−1ξ¨−3c2ξ21+4π−(8.1.6)
1+ξc24l23α˘
a˙ξ˙πξ˙2
−2a1+ξ+α˘1+ξ,
withdensitydistribution,energy-densitydistribution=c2andpressurepandthecosmological
functionΛ(ξ).Further,theymayberewrittenontothefollowingformwhichishelpfulforsome
(8.2.7)):equation(seeanalysesa˙2+Kc2=8πG˜+f1+Λc2(8.1.7)
a2a23c23
(8.2.8))(seeand2a¨+a˙2+Kc2=−8πG˜p+Λc2+f2.(8.1.8)
22caaHerewefindthecosmologicalfunctionΛandcorrectiontermsf1andf2totheusualFriedmann
equationsofusualGR.Thesecorrectiontermsreadasfollows,
a˙ξ˙πξ˙2
f1(t)≡f1=−a1+ξ+3α˘(1+ξ)2(8.1.9)
and

ξ¨˙aξ˙πξ˙2
f2(t)≡f2=−1+ξ−2a1+ξ−α˘(1+ξ)2.

(8.1.10)

8.1.THEGENERALIZEDFRIEDMANNEQUATIONSANDTHEHUBBLEPARAMETER117

ThecosmologicalfunctionΛ≡Λ(ξ)reads
G˜3ξ24π
Λ=8πc4V=4l21+ξ1+3α˘,(8.1.11)
potential-fieldscalarthewith3c4ξ24π
V(ξ)≡V=32πG0l21+3α˘(8.1.12)
andtheeffectivegravitationalcoupling
G˜≡G(ξ)=1G+0ξ.(8.1.13)
Inform,ΛtakestheplaceofthecosmologicalconstantΛ0ofusualGR.Whetheritactsasoneornot
willdependonitsdependenceonthescaleparameter,i.e.ontime.Incasewemeanthecosmological
constant,Λwill,asthroughout,besubscriptedasΛ0.Furthermore,undertheassumptionofavani-
shingscalarfieldanditsderivatives,ofcourse,allcorrectionsandcosmologicalfunctionvanish.The
cosmologicalfunctionappearsascorrectionfornonvanishingexcitationsofthescalarfield.However,
furthercorrectionsappearasconsequenceofitstimedependenceandhenceoftheonesofthegravi-
tationalcouplingitself.ThepropertiesofsuchcontributionsareanalyzedfurtheroninthisChapter.
Letusfirsttakethecosmologicalfunction.Itisrelatedtothelengthscaleofthescalarfieldandtothe
23ξ24π
valueofthefieldexcitationitself.Thus,thelengthscalecanbegivenasfollows,
l=4Λ1+ξ1+3α˘.(8.1.14)
Forα˘1,highexcitationsξ1aregivenby
ξ=4l2Λ.(8.1.15)
3ForΛ10−50cm−2,inacaseas(8.1.15),however,l2Λhastobegiven.Therewouldbe
l21050cm2(forξ1).(8.1.16)
declare:mayweHence,–IfΛgivesthedominanttermofthemeasuredcosmologicalconstant(cf.[198,208]),this
givesaconstraintonthelengthscale(seedensityparametersinChapter8.4).
However,thisconstraintisstronglydependentonthevalueofξ,sincelowexcitationsconstrain
ltolowervaluesinthecaseofΛbeingdominant.ForlowexcitationsandΛ=10−50,thereis
l<1025cm(i.e.lessthan10kpc).
Animportantcosmologicalparameteristheso-calleddecelerationparameterq.Itisdefinedby
q=−a¨2.(8.1.17)
aHItcanbefoundintheFriedmanequations,relatedtoa¨.Bothequations(8.1.5)and(8.1.6)give,with
V(ξ)≡V,
˜a¨=−42πG(+3p−2V)+f
3ca=−4πG˜2(+3p)+1Λc2+f(8.1.18)
3c3=−4πG20(∗+3p∗)+1Λc2+f,
3c3

(8.1.18)

118

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

witheffectivedensityandpressure
∗=/(1+ξ)andp∗=p/(1+ξ)(8.1.19)
and2˙f≡f(t)=21(f2−f1)=−21(1+ξ)−1ξ¨+aa˙ξ˙+34α˘π1ξ+ξ.(8.1.20)
ApartfromthecosmologicalfunctiontermΛandthescreeningofdensitiesandpressuresthroughthe
effectivegravitationalcoupling,thetermfgivesthecorrectionsofthetheory,andespeciallythose
causedbythetime-dependenceofthescalarfield(andhenceofΛ)itself.
Further,usingthescalar-fieldequation,(8.1.20)yields
222al3c1+3α˘3α˘1+ξ
f(t)=1(1+ξ)−12a˙ξ˙+c2ξ−8πG20qˆ−43πp+4πξ˙.(8.1.21)
Furthermore,inthecaseqˆ=1forhighvaluedscalar-fieldlengthscales,itisknownthatthenewtonian
gravitationalconstantisgivenbyaparameterizedcouplingconstantG0.Hence,forhighlengthscales
asareexpectedhere,introductionofNewtonsgravitationalconstanttogetherwithamaintenanceof
acanonicalformoftheequationsmayleadtotheintroductionof
43=˜and43p=p,˜(l→∞,andqˆ=1)(8.1.22)
sothat(cf.equation(8.2.8))
a¨=−4πG2N(˜∗+3p˜∗)+1Λc2+f(8.1.23)
3c3amaybewritten,incanonicalformwithcosmologicaltermΛandcorrectiontermf.However,andp
aretheactualdensityandpressure.
Inthecaseofaneglectionofcosmologicalcorrections,measuredterms˜andp˜wouldbeeffective
valuesforwhichdensity-likeandpressure-liketermscomingfromΛandfaretakenaspartofdensity
andpressureandnotasscalar-fieldterms.Insuchacase,ξispartlytobetakenasacontributionof
matterdensity.Hence,quintessentialcosmologicalpropertiescomingfromit,forinstance,would
appearasmatter-caused(seeChapters8.3and8.6).
ApartfromthefactofGandΛbeingfunctional,itisfwhichmakes(8.1.18)formallydifferentto
theusualequationinGR,wherethereisacceleration(q<0)merelyforΩΛ>Ω/2(see(8.4)).This
newtermgivesthechangesofdynamicscausedbythetimedependenceoftheeffectivecoupling
constantand(togetherwiththecorrectiontothefirstFriedmannequation)itcanbecomparedtoan
analogfunctionderivedwithinModifiedGravity(MOG)byMoffat[168],butherewithafunctional
cosmologicaltermΛanddefiningascalarfieldwhichwemightwriteasx=1+ξforadirect
1[168].withcomparisonWiththeassumptionofanequationofstatewithbarotropicequation-of-stateparameter,
w=c2=,(8.1.24)
pp¨1In˙2[166],thereis,forinstance,aa¨=−4π3G[(1+3w)+K(1+3w)+V(1+wV)+Λ(1+3wΛ)+G(1+3wG)]+
G−G,withusualmatterdensity,densityrelatedtothescalar-fieldpotential,V,densityfromthecosmologicalfunction,Λ,and
2GdensityG2relatedtoHG˙terms,G(cf.equation8.1.57with8.1.58).Similarly,thereisH2+aK2=8π3G[+K+V+Λ+G].

8.1.THEGENERALIZEDFRIEDMANNEQUATIONSANDTHEHUBBLEPARAMETER119
thesolutionforthedensityisgivenby(see[47])
2w=aM3(1+wcw)(1+ξ)21(1−3w)(1−qˆ).(8.1.25)
Herewith,Mwisanintegrationconstantwhichweparameterizeexplicitlybyasubscriptw.Inthis
way,itispossibletoanalyzedifferentmatter-typebehaviorandmatterdominance.Consequently,we
writeasubscriptwintoshowitastheenergydensityrelatedtoaspecifickindofmatterormatter
dominancewithintegrationconstantMw.
MatterdominanceasgiveninthehodiernalUniverseisgottenwiththeparameterw=0(dust-
matter).2Thenthereisadependenceasfollows,
∼a−3,(matterdominance),(8.1.26)
whileradiation-dominatedepochsaregivenbyw=31suchthat
∼a−4,(radiationdominance).(8.1.27)
Anti-stiffmatterwithw=−1representsdark-energy-interactingmatter,anditisrelatedtoacosmo-
constant,logical∼a0,(darkenergy).(8.1.28)
w=−1isthusrelatedtoadensitywhichevolvesindependentlyofthescalefactora.Thisdensityis
constantintime,relatedtoΛ0=const.
Forqˆ=1,theusualrelationofthestandardFriedmannmodelsisvalid,
wa3(1+w)=const.=0wa03(1+w).(8.1.29)
Forqˆ=0,ananalog,however,changedrelationisvalid(cf.equation(8.1.60)):
w(1+ξ)21(1−3w)a3(1+w)=const.=0w(1+ξ0)21(1−3w)a03(1+w),(8.1.30)
Timet=t0usuallymeanstobetheoneofthecurrentUniverse(oftheobserver).a0istheobservers
currentscalefactora(t0).0wistheobserversenergydensityw(t0)forw-typedmatter.3
Consequently,theintegrationconstantMwisrelatedtoa0infollowingwayforqˆ=1(cf.equation
(8.1.61)):Mα=a03(1+w)0w,(8.1.31)
andforqˆ=0andw=1/3,
)w3(1+a0Mw=(1+ξ0)21(1−3w)0w.(8.1.32)
Ascommonlyknown,signsmeasuredbyanobserveratatimet0fromastronomicalobjectssuchas
distantgalaxiesatagenerictimetareredshiftedinrelationtotheoriginallysentones.Thisredshift
z=λ0−λemission(8.1.33)
λemission2Ingeneral,thehodiernalUniversewillbetakenasdusty.Quintessential(anti-stiff)propertieswillbetakenascomingnotbasically
fromusualmatteritself,i.e.notfrommatterdensity,butfromthescalarfieldleadingtoatotaldark-energydominance.
3Thesubscriptwmayusuallybeletasideandtheparametrizationbeimplicitlygiven.

120

isrelatedtothescalefactorby

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

1+z=a0.(8.1.34)
aTherefore,therelationbetweenadensityatgenerictimeandtheoneatthepointofobservation
(usually,thepresenttime)maybeexpressedbytheredshiftwith
˜21(1−3w)(1−qˆ)
=0G˜0(1+z)3(1+w).(8.1.35)
GTheredshiftparameterzistheredshiftasmeasuredatthepresenttimet=t0.
Fort<t0,thereisa<a0andacosmicexpansionleadstoaredshiftofthesentwaves.
From(8.1.2)andafterdefinitionoftheHubbleparameterasameasureofcosmicexpansion,
H(t)≡H:=aa˙,(8.1.36)
theHubbleparametermaybedirectlygivenby
c2H=−1˙(+p)−1+ξ˙c2(1−qˆ)(1+ξ)−1−3p,(8.1.37)
36(+p)
Itnotonlydependsonthetimederivativeofdensitybutmayalsodependontheoneofthescalar-
fieldexcitation,togetherwithatermfromdensityandpressurethemselves.Forthelasttermofthe
right-handsideofequation(8.1.37),forp,thereis
p2+c2
+3pc2≈1+2p−2p2+...(8.1.38)
Equation-of-statetermsmaythenbeneglectedforthelasttermoftheright-handsideofequation
(8.1.37)forp.
Insertionofequation(8.1.24)into(8.1.37)leadstothelatterinthefollowingform(withwexplicitly),
c2Hw=−1˙(1+w)−1+(1−qˆ)ξ˙(1−3w).(8.1.39)
36(1+w)(1+ξ)
Itiseasilynoticedthatanincrease(decrease)inthedensityisrelatedtoacontraction(˙a<0)(expan-
sion(a˙>0))ofthecosmosandthatforqˆ=0thetimevariationofthescalar-fieldexcitationplaysa
roleincosmicexpansion,too:higherderivativesreducethevalueofH.
Explicitly,forthehodiernal,matter-dominatedUniverse(w=0),equation(8.1.39)givesdirectly
c2H0=−1a03˙0(1+ξ0)−21(1−qˆ)+(1−qˆ)1ξ˙0c2(1+ξ0)−1(8.1.40)
3M0c26
withtheintegrationconstantM0≡Mw=0andξ0≡ξw=0and0≡w=0.Forradiation(rad,
w=1/3),thereis
21ar4ad
cH1/3=−4M1/3c2˙rad,(8.1.41)

andforstiffmatter(SM),
26c2H1=−1aSM˙SM(1+ξSM)(1−qˆ)−(1−qˆ)cξ˙SM(1+ξSM)−1.(8.1.42)
6M1c26

8.1.THEGENERALIZEDFRIEDMANNEQUATIONSANDTHEHUBBLEPARAMETER121

Forw>1/3,andespeciallyforstiffmatter,anegativechangingofthescalar-fieldexcitationleadsto
ahigherHubbleparameterifqˆ=0.
ThepresentUniverseshouldbegivenbyw=0.SincetheHubblefunctionatthepresenttimeis
quasiconstant,thentime-changesofdensityshallnotbetoohigh.

(8.1.43)

•EquationsindependenceofG˜:
Sincetheeffectivegravitationalcouplingisrelatedtothescalarfieldvia
G(ξ)≡G˜=1G+0ξ,(8.1.43)
itpossessesatime-dependencecoefficientasfollows,
G˙eff=−ξ˙.(8.1.44)
Geff1+ξ
Thegravitationalcouplingisdirectlydependentonthescalarfieldanditsbehaviorintime.Withit,
equation(8.1.2)mayaswellbewrittenasfollows,
˙+3a˙(+p)=−(1−qˆ)1G˙(ξ)(−3p),(8.1.45)
a2G(ξ)
sothatthedependenceoncouplingdeviationsisexplicitlygiven.Smalltimedeviationsoftheeffec-
tivecoupling,thus,meansforqˆ=0asmall-valuedsourceforcontinuitycondition.Thereappear
entropy-productionprocesseswhich,however,becomeminimalwhentheeffectivegravitationalcou-
plingtendstoconstantbehavior.Forscalarfieldswithtendencytoaconstantterm,entropyproduction
anishes.vFurther,withabarotropicequation-of-stateparameter,equation(8.1.45)maybewrittenasfollows,
˙=−3H1+w−sG˙(ξ),(8.1.46)
6HG(ξ)

whereaswehavedefined

s≡(1−qˆ)(3w−1).(8.1.47)
Hence,forqˆ=0,aneffectiveequation-of-stateparameter
s˙G(ξ)
weff=w−6HG(ξ)(8.1.48)
maybedefined.Itisrelatedtothechangeddensitydistributioninequation(8.1.25)inthecaseof
qˆ=0.Asacosmologicalparameter,weffmustdependonthedynamicalscaleH−1,asdoesalso
G(ξ).ThevalueofweffmaythendifferfromwandleadtoQuintessencewithinamodelofonlygra-
vitationallycoupledHiggsbosons(qˆ=0)forw=0ands=−1.Thismayberelatedto[248,249]
inwhichΛCDMandChaplygingasprofilesarederivedwithoutacosmologicalconstant.

Further,thesecondderivativeofthescalar-fieldexcitationmayalsoberewrittensothatfollowingis
alid,v

¨1ξ1+ξ=G(ξ)22G˙(ξ)2−G¨(ξ)G(ξ).


(8.1.49)

122

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

Thismeansthatthesecondderivativeofthegravitationalcouplingisrelatedtotheonesofscalar-field
asxcitationseG¨(ξ)ξ¨ξ˙2
G(ξ)=−1+ξ+21+ξ.(8.1.50)
Withequation(8.1.49),thescalar-fieldequation(8.1.4)mayberewrittenontothefollowingform,
2G21G¨(ξ)G(ξ)−2G˙(ξ)2+3aa˙GG˙((ξξ))+lc2GG((vξ))−1=−c82πG3(ξ)qˆ(−43πp).(8.1.51)
eff(1+3α˘)
Thedynamicsofthecosmologicalscalar-fieldexcitationaregivenbythetimedependenceoftheef-
fectivegravitationalcouplingG˜.Forvanishingdeviationsoftheeffectivegravitationalcouplingfrom
theground-statecouplingconstant,itisdirectlyseenin(8.1.51)thatthelength-scaletermvanishes.
Furthermore,theFriedmannequations(8.1.5)and(8.1.6)intheforms(8.1.7)and(8.1.8)maybe
written,too,intermsofcorrectionscomingfromtimederivativesofthegravitationalcoupling(cf.
(8.1.9)):equation

˙˙f1(G)≡f1=GG((ξξ))aa˙+3πα˘GG((ξξ)),

and(cf.equation(8.1.10))
aG(ξ)G(ξ)2α˘G(ξ)
f2(G)≡f2=2a˙G˙(ξ)−G˙(ξ)22+π+G¨(ξ).
Explicitly,theFriedmannequations(8.1.5)and(8.1.6)readnow
a˙2+Kc2=8πGeff+1Λc2+a˙G˙(ξ)+πG˙(ξ)2
a23c23aG(ξ)3α˘G(ξ)2
and2222a¨+a˙+2Kc=−8πGeffp2+Λc2−(1+ξ)−1ξ¨+2a˙G˙(ξ)+πG˙(ξ)2.
aacaG)(ξ)α˘G(ξ)
Thecosmologicalfunction(8.1.11)nowreadsasfollows,
3G02+G˜24π
Λ≡Λ(G˜)=4l22−G0G˜1+3α˘.
Again,(8.1.7)and(8.1.8)givetogetherequation(8.1.18),
a¨=−42πG˜(+3p−2V)+f
3ca=−4πG˜2(+3p)+1Λc2+f
3c3=−4πG20(∗+3p∗)+1Λc2+f,
3c3againwitheffectivedensityandpressure
∗=/(1+ξ)andp∗=p/(1+ξ).

(8.1.52)

(8.1.53)(8.1.54)

(8.1.55)(8.1.56)

(8.1.57)

8.1.THEGENERALIZEDFRIEDMANNEQUATIONSANDTHEHUBBLEPARAMETER123

Forexplicitgravitational-couplingderivatives,thereisnowequation(8.1.20)intheform
111a˙G˙(ξ)22π
f≡f(G(ξ))=2(f2−f1)=2G(ξ)G¨(ξ)+aG˙(ξ)−2G(ξ)21+3α˘.(8.1.58)
Inthesameway,energydensitymaybewrittenintermsofthegravitationalcoupling,
=Mwc2G(v)21(1−3w)(1−qˆ).(8.1.59)
wa3(1+w)G(ξ)
Forqˆ=0andw=1/3,higheffectivegravitationalcouplings(i.e.ξ<0)inrelationtotheground-
statecouplingG(v)=G0leadtosmaller(asmeasured)valuesoftheenergydensity,with
˜G>G0.
Analogously,smalleffectivecouplings(ξ>0)inrelationtoG0leadtohigherdensitiesthanwithin
thestandardFriedmannmodels,assumingw<1/3.Theevolutionofξ,thus,isofspecialrelevance.
Furthermore,thereis(cf.[24]and(8.1.30))
G021(1−3w)(1−qˆ)a3(1+w)=const.=G021(1−3w)(1−qˆ)a3(1+w),(8.1.60)
wG˜0wG˜00
withG˜0astheeffectivecouplingforthetimet=t0,usuallymeanttobetheoneofthecurrent
Universe(oftheobserver).a0istheobserverscurrentscalefactora(t0).0wistheobserversenergy
densityw(t0)forw-typedmatter.4
Consequently,theintegrationconstantMwisrelatedtoa0infollowingwayforqˆ=1:
Mw=G˜0a3(1+w)0w.(8.1.61)
21(1−3w)(1−qˆ)
G0

(8.1.62)

HubbleThereadsparameter˙c2Hw=−31˙(1+w)−1−6(1(1−+qˆw))GG((ξξ))(1−3w).(8.1.62)
Further,equation(8.1.25)leadsto(8.1.39)as
c2Hw=−1˙a3(1+2w)G(v)2−(1−qˆ)G˙(ξ)c2(1−3α).(8.1.63)
−1(1−3w)(1−qˆ)
3MwcG(ξ)6(1+w)G(ξ)
Thevariationofthecouplingconstantleadstoascreeningofthedensityparameterin(8.1.59)in
relationtothecasewheretheG-coefficientisnegligible.Thevalueofthedensityforqˆ=1issmaller
ifξ<0,i.e.G(ξ)>G(v)(anti-screeningofthegravitationalconstant).
Forw>1/3,andespeciallyforstiffmatter,apositivederivativeofthecouplingconstantleadstoa
higherHubbleparameterifqˆ=0.

4Thesubscriptwmayusuallybeletasideandtheparametrizationbeimplicitlygiven.

124CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC
8.2Equation-of-stateparametersofthescalarfield
Letusdefineascalar-fieldcomponentofthetotaldensityintermsofthescalar-fieldexcitations,
2˙22Λ=V−8π3cG0a˙aξ˙+v81ξ+ξ.(8.2.1)
Thisdensityisdependentonthescalar-fieldpotential,onatermofthescalar-fieldderivativeandon
.parameterHubbletheIntermsofgravitational-couplingchangings,equation(8.2.1)yields(relatedtotheparameterΩΛ∗in
(8.4.3)),with(8.4.2)3c2G˙(ξ)1G˙(ξ)2
Λ=V+8πG(ξ)HG(ξ)+3α˘G(ξ)2.(8.2.2)
Further,thetotaldensityread
T=+Λ.(8.2.3)
densityscreenedthe,Analogously˜GT∗=1+Tξ=G0T(8.2.4)
defined.bemayFurthermore,wedefineascalar-fieldtermofthepressureforthesecondFriedmannequation(8.1.6),
c2a˙πξ˙2
pΛ=−V+8πG0ξ¨+2aξ˙−3˘α1+ξ,(8.2.5)
orequivalentlyindependenceofthegravitationalcoupling,
c2G¨(ξ)G˙(ξ)G˙(ξ)2π
pΛ=−V−8πG(ξ)G(ξ)+2HG(ξ)−2G(ξ)21+6α˘.(8.2.6)
So,equations(8.1.5)and(8.1.18)yield
a˙2+2Kc2=82πG˜T(8.2.7)
3ca=82πG0T∗,
3ca¨=−42πG˜(T+3pT)(8.2.8)
3ca=−c42πG30(T∗+3pT∗).
Allchangesondynamicsarenowwrittenintermsofscreeneddensitiesandpressuresoreffective
densitiesandpressureswithscreenedgravitationalcoupling.Furthermore,thereisanequationof
stateforscalar-fielddominanceinthefollowingway:
Λ+3pΛ=−2V−8π3Gc(ξ)HGG((ξξ))+GG((ξξ))−2GG((ξξ))21+32α˘π.(8.2.9)
2˙¨˙2

8.2.EQUATION-OF-STATEPARAMETERSOFTHESCALARFIELD

125

Introducingthescalar-fieldequation,itisseenthatthelatterisequalto
(−3p)3c2c2
Λ+3pΛ=−2V+qˆ1+34α˘π−8πG0l2ξ+2Hξ˙.(8.2.10)
Thetotalequationofstatethenreads
qˆqˆ3c2ξ˙c2
T+3pT=1+1+34α˘π+31−1+34α˘πp−2V−8πG˜2H1+ξ+l2ξ.(8.2.11)
Hence,asthefirsttermoftheright-handsideof(8.2.8)vanishesforα˘1,intermsofequation
(8.2.8),pressurehardlycontributestoaccelerationincaseofqˆ=1.Accelerationismainlygivenby
excitationsξinVorbypositivetimederivativesoftheexcitationsthemselves,whichintermsofthe
effectivegravitationalcouplingisanalogoustoin-timediminishingeffectivecouplingsG˜.Forqˆ=1,
densities,whichweassumealwayspositive(≥0),actagainstaccelerationandnaturallycontribute
todecelerationaswithinstandarddynamicswithgravitationasanattractiveelementaryinteraction
(fortheprimevalUniverse,consequencesofthismaybefoundinChapter8.8).
Thescalar-fieldequationofstatecomprisesthecorrectiontermftousualFriedmannequationsto-
getherwiththecosmologicalfunction.Therelationofequation(8.2.9)totheparameterfin(8.1.58)
is8πG˜6πG˜
−3c2(Λ+pΛ)=f2−f1+3c2V(8.2.12)
=2f+1Λ,(8.2.13)
3anditsvaluecanclearlybenegativeforadominanceofξoveritsderivativesorforadominanceof
itssecondderivative,forinstance.Thus,inascalar-fielddominateduniverse,accelerationtermsof
cosmosmaydominate.Suchterms,whichcomefromthecorrections,areherewithrelatedtodensity-
andpressure-actingterms(cf.(8.2.7)and(8.2.8)).
Consequentlytoequation(8.2.13),thereis
Λ=−3f−4πG˜(Λ+pΛ).(8.2.14)
2c2Thecosmologicalfunctionmaybesplitontothecorrectionfandtheequation-of-statecontribution
ofthescalarfield.Doingso,non-explicitlytimedependentcontributionsofthescalarfieldtodensity
canbetreatedasdensitycontributions,withnoothercosmologicaltermotherthanf.
Theequation-of-stateparameteroftheξ-relateddensityandpressurereads
pΛG(2ξ)221G¨(ξ)G(ξ)+HG˙(ξ)G(ξ)−G˙(ξ)21+6πα˘+4π3cG2(ξ)V
=w=−.(8.2.15)
ΛΛHGG˙((ξξ))+31α˘GG˙((ξξ))22+83πGc(2ξ)V
ithW

228πG˜V=ξc(1+ξ)−11+4π
3c24l23α˘
c2G˜2−2G0G˜+G024π
=4l2˜GG01+3α˘,

(8.2.16)(8.2.17)

126

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

equation(8.2.15)canfurtherbefullyrewrittenintermsofthegravitationalcoupling:
G¨(ξ)+2HG˙(ξ)−2G˙G((ξξ))21+6πα˘+43lc22G(ξ)2−2GG0G(ξ)+G021+34πα˘
w=−0.(8.2.18)
ΛHG˙(ξ)+31α˘G˙G((ξξ))2+4cl22G(ξ)2−2GG00G(ξ)+G021+34α˘π
Thisistheequation-of-stateparameterforascalar-fielddominateduniverse.Forα˘1,itsimplifies
ws,folloas−HG˙(ξ)+43lc22G(ξ)2−2G(ξ)G0G+0G02−4G(ξ)+4G0
wΛsf-dom=−HG˙(ξ)+3c22G(ξ)2−2G0G(ξ)+G02.(8.2.19)
Gl40ForG˙(ξ)=0(staticalfield,subscribedc),thereisfurther
wΛsf-dom,c=−3G(ξ)2−2G(ξ)G0+G02−4G(2ξ)+4G0.(8.2.20)
G(ξ)2−2G0G(ξ)+G0
Vanishingvaluesofthescalar-fieldexcitationforvacuum,i.e.G˜=G0with=p=0wouldlead
toadivergencedenotingtheabsenceofallfields.However,forthescalar-fieldpressurepΛ(cf.
8.2.5),itiseasilynoticedthatforabsolutescalar-fielddominance,pΛ=wΛΛisquintessen-
tial,i.e.itisaccompaniedbyanegativeequation-of-stateparameterwΛsothatwT<w(see
equation(8.3.20)).Thescalarfieldpossessesantigravitationalpropertiesanditmayactasanti-stiff
matter.Furthermore,adeSitterstate,whichisaquintessentialmatter-vacuumstate,isimportantto
getaninflationaryepochaftert=0andsotosolveproblemsastheoneofhorizonthroughahighly
acceleratedexpansionoftheuniverse.Furthermore,theappearanceofnegativeeffectivepressuresis
importantfortheappearanceofcosmicaccelerationsinthepresentUniverseaswellasforthenature
oftheinitialstateofthesame(seeChapter8.8).

8.3DecelerationparameterandDarkEnergy
•Thedecelerationparameteranditsimportancewithinmeasurements:
Letustakethedecelerationparameterasdefinedinequation(8.1.17).Ingeneral,forvaluesq≥1/2,
thedecelerationparametergivesacosmicdecelerationinaccordancewithagravitationalcharacterof
densitiesandpressuresleadingtoattraction.Forvaluessmallerthan1/2,itgivesanacceleration.
Positivepressurespleadtohighervaluesofqandthusstrengthendeceleration(pressureactsgra-
vitationally).Acosmicfluidwithp<31actsantigravitationallyandthusstrengthensacceleration
repulsion).vitational(graThecurrentUniversewithonlymatterandnegligibleradiationshouldfulfilladeceleratedexpansion
asconsequenceoftheBigBangandthengravitationalattraction.Furthermore,itismostlyaccepted
thatgeometryisgivenbyK=0,whereinthisvalueisdeducedfromobservationsofthecosmic
microwavebackground(CMB)radiation(however,thisdependsontheexactvalueoftotaldensityin
theUniverse).ObservationsofSuperNovaeoftypeIa,however,indicateanegativeq0forthecur-
rentcosmos(hencethesubscript0),whichthenwouldmeanaccelerationofcosmicexpansion.The
mechanismofacosmicmediumwhichshouldbecauseofsuchantigravitationalinteractionisusually
calledDarkEnergyorQuintessence(seeChapter2.4).Quintessence,however,isusuallyusedfora
kindofDarkEnergywhichhasassourceascalarfieldwhichalmostdoesnotevolveintime.

8.3.DECELERATIONPARAMETERANDDARKENERGY

127

Equation(8.2.18)alreadyshowedthatthescalar-fieldtermsmayleadtonegativevaluesforthetotal
equation-of-stateparameter.Givensuch,thescalarfieldmayleadtoquintessentialpropertiesinform
acceleration.cosmicofThetotalpressuretermwhichcarriesscalar-fieldtermsisdependentonG˜anditsderivatives.Onecan
seethathigheffectivevaluesofthescalar-field˜excitationaswellasespeciallypositivelyvalued(se-
cond)derivativesofthegravitationalcoupling2G,mayleadtonegativetotalpressures.Suchstrengthen
acosmologicalacceleration,whereasG˙(ξ)mayactasdecelerationfactorifitisespeciallyhigh.The
latterterm,however,maybeexpectedasnegligibleundernormalcircumstances,andrelevantonlyfor
theprimordialUniverse,possiblyinrelationwithprimevalInflation.Theconceptofprimeval,cosmic
Inflation,firstproposedbyAlanGuthin1981[118],basedonideasofStarobinsky[224],andlater
improvedbyAlbrecht,Steinhardt[1]andLinde[154],assumesaphaseofveryhighlyaccelerated
expansionintheearlyUniversetoexplainhorizonandflatnessproblemsofcosmology.Often,an
hypotheticalscalarfield,namelytheinflatonfield,isproposedinthiscontext.NewandChaotic
Inflationdifferfromtheoriginalone(calledOld),bymeansoftheinitialconditionsofthisscalar
field.

Therelevanceofthedecelerationparametercanbeseenwithintherelationbetweenagenerictime-
scalefactorandthecurrentonefort=t0.Thereis
a(t)=a01+H0(t−t0)−1q0H02(t−t0)2+...,(8.3.1)
2beingH0,a0andq0thehodiernalHubbleparameter,scalefactoranddecelerationparameter,anda(t)
bethescalefactorforatimetwhichcanbegivenintermsoftheredshiftofaluminoussourcesuch
asadistantgalaxy(cf.[56]),
z=H0(t0−t)+1+1q0H02(t0−t)2+...,(8.3.2)
2invertibleto
t0−t=1z−1+1q0z2+....(8.3.3)
2H0Clearly,forhighredshiftvalues,theexactvalueofq0playsanimportantroleintheevolutionofa(t).
Withequations(8.1.34)and(8.3.3),onecantake
t0cdtrdr
ta=0√1−Kr2(8.3.4)
andconvertitinto
ct01
1+H0(t0−t)+1+q0H02(t0−t)2+...dt=r+O(r3),(8.3.5)
2at0usingq0=−a¨(t0)a0.(8.3.6)
a˙(t0)2
Therefore,thereis(cf.[56])
r=c(t0−t)+1H0(t0−t)2+...,(8.3.7)
2a0

128CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC
whichusingequation(8.3.2)yields
r=a0cH0z−21(1−q0)z2+....(8.3.8)
Thus,theradialcoordinatemaybeexpressedbytheredshiftintermsofaccelerationwiththeacce-
lerationparameterq0.DecelerationhoweverdependsonthetotalpressurepTandenergydensityT,
aswellasoncurvatureK(allatthetimet=t0).Anonvanishing,positivetermofthecurvatureK
leadstosmaller,effective,valuesofthedecelerationcomparedwiththeactualone.
Properdistancestoastronomicalobjectscannotbemeasuredinanydirectway.Astronomicalobjects
areobservedthroughtheelectromagneticradiationtheyemit.Radiation,ontheotherhand,takes
timetotravelfromtheemissiontotheobservationpoint(usuallyP0att0).Itisthusonlypossible
tomakemeasurementsalongthesetofpathstravelingtousfromthepast,i.e.partofourlightcone.
Everythingoutsideofit,ofcourse,isnotcausallyrelatedwiththepresent.
OneofthemeasurabledistancesistheluminositydistancedL.Thisdistanceisdefinedinaway
topreservetheeuclideaninverse-squarelawforthediminutionoflightwithdistancefromapoint–
source.LdenotethepoweremittedbyasourceatapointPatacoordinatedistancerattimet.lbethepower
receivedperunitarea(theflux)att0byanobserverP0.Then,theluminositydistancebedefinedby
LdL=2πl.(8.3.9)
TheareaofsphericalsurfacecenteredonPandpassingthroughP0attimet0is4πa02r2.Thephotons
emittedbythesourcearriveatthissurfacehavingbeenredshiftedbytheexpansionoftheUniverse
byafactora/a0.Thereis
2aLl=4πa02r2a0,(8.3.10)
andthuswithequation(8.3.7),
dL=a02ar=Hc0z+21(1−q0)z2+....(8.3.11)
FromdL,thereis(cf.[56])
l=4πLd2=4πcLH2z02[1+(1−q0)z+...](8.3.12)
Lforapparent(l)andabsolute(L)luminosity.Inastronomy,however,itiscustomtousemagnitudes
instead;theabsolute(M)andtheapparentone(m).Theyaredefinedinalogarithmicalscaleby
takingafactor100inreceivedfluxtobeadifferenceof5magnitudes.Perconvention,Polaris(α
UMi)isgivenanapparentmagnitudeof2.12invisiblelight.Theabsolutemagnitudeisdefinedto
betheapparentmagnitudethesourcewouldhaveifitwereplacedatadistanceof10parsec.5The
followingrelationisgiven,
dL=101+(m+M)/5pc,(8.3.13)
5Ascommonlyknown,parallaxofonearcsecondorparsec(pc)isthelengthofanadjacentsideofanimaginaryrighttrianglein
asspace.1Theastronomicaltwounitdimensions(AU),thattheformdistancethisfromtrianglethearetheEarthtoparallaxtheSun).angleGiv(definedentheseas1twoarcsecond)measurements,andthetheoppositelengthsideofthe(whichadjacentisdefinedside
(theparsec)canbefound.Itisofabout3.26light-yearslength.

8.3.DECELERATIONPARAMETERANDDARKENERGY

129

modulusdistancethewhereDM=m−M=−5+5logdL(pc)(8.3.14)
maybedefined.Usingequation(8.3.12),thelatterisgivenby
m−M≈25−5log10H0+5logcz+1.086(1−q0)z+...(8.3.15)
Theapparentmagnitudeisdependentonzandonq0.Theq0-dependence,however,isrelevantonly
forredshiftsz>0.1.Therefore,cosmicaccelerationdoesnotplayadominantroleinlow-z-analysis,
andmeasureddistancesindirectlygottenthroughobservationsmaybeaccuratelygivenindependently
ontheexactFriedmannmodel.Howeverthen,high-z-analyseshavetobemadetoconstraintdark-
energeticbehavior.Furthermore,givenotherfactorsinterveningintheanalysis,magnitudescan
givelittleaccurateinformationaboutthedecelerationparameter.Theregimeofaccuracyofmis
z<zmax≈0.2wherethedistancemodulushoweverconfirmstheHubblelawand,therefore,the
principle.cosmological

TherelationbetweenluminosityinformofthedistancemodulusDM=m−Mofmagnitudesand
redshiftziscalledHubblediagram.Itcanbeusedtoprovethevalueofq0directly.Theproblem:it
needsofobjectsofknownintrinsicluminositieswhicharethereforecalledstandardcandles.
TheuseofSNeasstandardcandlesincosmologywasdiscussedbySandage[214]becauseoftheir
ratherhomogeneousandextremelyluminouspeakbutitwasnotonlyuntiltherealization,though,
thatSNeareactuallysubdividedinunderclasses,thattheyleadtothecurrentprogressstartingwith
Hamuyetal.s[119]andRiessetal.s[207]workin1995.
Firstobservations[197]hadsuggestedapositivedecelerationparameterq0>0.Thenitbecameap-
parentthathigh-redshiftsupernovaemightbefainterthantheyshouldwhencompatiblewithq0>0
undertheassumptionofusualFriedmannmodels.TheworksofRiessetal.[208]andPerlmutteret
al.[198]in1998concludedanaccelerationoftheUniversetogetherwithadominantcosmological
constantwithinstandardFriedmannmodels.Indeed,withinourmodel,thedominanceofscalar-field
(thuscosmological,exotic)componentsofdensityandpressureseemdominantforthecurrent,else
erse.vUnidust-dominated

•Thedecelerationparameterandtheequationofstate:
Divisionbetweenbothequations(8.2.7)and(8.2.8)leadsto
¨aa1p∗
a˙2+Kc2=−21+3T∗T
=−11+3pT,(8.3.16)
2Twithequations(8.2.1),(8.2.3)and(8.2.5).Equation(8.3.16)maybeusedasredefinitionofaneffective
decelerationparameterq˜whichshallcontainacurvaturetermwith
p2q˜=1+3TT(8.3.17)
q2=1+Ka˙2c2.(8.3.18)

130

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

Thus,theproperdecelerationparameterqreads
2q=1(1+3wT)1+Kc,(8.3.19)
2a˙2withwT=pT/Tastotalequation-of-stateparameter.qisaneffectiveparameterwhichisnegative
forquintessence-dominance,eventhoughtheequation-of-stateparameterofmatterperseshouldbe
zeroorpositive.Hence,wTtakesexoticcontributionstomatter(whichcomefromξ)intoaccount.

(8.3.22)

Thetotalequation-of-stateparameterreadsusingthescalar-fieldequation,
p+pΛ1−(1+34α˘π)p+3(1+34α˘π)−3V−8πcG0Hξ˙+lc2ξ−3πα˘1+ξ
qˆqˆ22ξ˙2
wT==.(8.3.20)
+Λ+V−8π3cG20Hξ˙−31α˘1+ξ˙2ξ
Intermsoftheequation-of-stateparametersandpandthescalar-fieldterms,thelattermaybewritten
forα˘1asfollows,
223(1−qˆ)p+qˆ−8π3cG04cl2ξ(4+ξ)+Hξ˙
3wT=22.(8.3.21)
+8π3Gc0l24cl2ξ2−Hξ˙
Forα˘1andqˆ=1,equation(8.3.20)leadsto
22
(8.3.22).=w1−V−83πcG0Hξ˙+lc2ξ
T3+V−8π3cG20Hξ˙
ξ>0aswellasξ˙>0leadtoanti-stiffbehavior.
Forqˆ=0,thereis
13p−V−8π3cG20Hξ˙−lc22ξ
wT=3+V−3Hξ˙.(8.3.23)
Gπ80Asdirectlyseen,forthelimitingcaseofconstantscalar-fieldexcitationsξ=const.(thusV=Vc),
readsparameterequation-of-statethewTξ=const.=1c8πG0l2forqˆ=1,(8.3.24)
−V−3c4ξ
V+3c4c3=13p−Vc−8πG0l2ξforqˆ=0.(8.3.25)
V+3cSinceV(ξ)≥0,forhighexcitationsthereisforequation(8.3.24),wTξ=const.≤1/3.Forsmallex-
citationsalso,aslongasξ>0.Forequation(8.3.25),inthesamecases,thereiswTξ=const.≤w/3
ifp=wisvalid.Inthelattercaseqˆ=0,further,vanishingexcitationsleadtotheusualequation-
of-stateparameterw=p/.

8.3.DECELERATIONPARAMETERANDDARKENERGY

131

TheFriedmannequationsmaybegivenindependenceoftheeffectivedecelerationinthefollowing
form,pT=31(2q˜−1)(8.3.26)
T2=a˙22q−1−Ka˙2c=w.(8.3.27)
3a˙2+Kc2T
ForaflatUniverse,i.e.K=0,theequationabovegivestheusuallygivenFriedmannequation
containingthedecelerationparameter,withDarkMattertermsaddedtodensityandpressure,andwith
thescalar-fieldtermsasdarkconstituentswhichareherestilltodefineintermsoftheirproperties.
Further,intermsoftheeffectivegravitationalcoupling,thereis
p−V−8πcG2(ξ)GG¨((ξξ))+2HGG˙((ξξ))−2GG˙((ξξ))221+2πα˘
2q˜=1+3˙˙2.(8.3.28)
+V+8π3Gc(2ξ)HGG((ξξ))+31α˘GG((ξξ))2
Obviously,forvanishingderivatives(subscriptsfd=0)ofξ(V=const.≡Vc),theeffectivedecele-
Vprationparameterisgivenby
c−2q˜sfd=0=1+3+Vc,(8.3.29)
foraneffectivepressurepeff=p−Vcandeffectivedensityeff=+VcofaconstantpotentialVc:
eff+3peff=+3p−2Vc.Followingequation(8.3.28),
q˜=1(1+3wT),
2thedecelerationparameterqˆreadsforscalar-fieldexcitations(accordingtoequation(8.3.20)),
31−(1+qˆ34α˘π)p+(1+qˆ3πα˘)−3V−83πcG0Hξ˙+lc2ξ−34α˘π1+ξξ
22˙2
2q˜=1+22.(8.3.30)
−V−8π3cG0Hξ˙−31α˘1+ξξ
InsertingV,thereisfurther
31−1+qˆ4πp+1+qˆ4π−83πcG204cl22ξ4+ξ1+34α˘π+Hξ˙−3πα˘1+ξ˙2ξ
2q˜=1+3α˘3α˘˙2.(8.3.31)
+83πcG204cl22ξ21+34α˘π+Hξ˙−31α˘1+ξξ
Forα˘1andqˆ=1,asdirectlyseenfromequation(8.3.24),thelattersimplifiesto
−3V−8π3cG20Hξ˙+lc22ξ
2q˜=1++V−8π3cG20Hξ˙.(8.3.32)
Ontheotherhand,forα˘1andqˆ=0,thereis,asdirectlyseenfromequation(8.3.23),
223p−V−8π3cG0Hξ˙+lc2ξ
2q˜=1++V−8π3cG2Hξ˙.(8.3.33)
0However,thec2ξ/l2termfornonvanishingscalar-fieldexcitationspushesdowntheequation-of-state-
parametervalueforξ>0.
isthere,Further

3c43c4
−Vc−8πGl2ξ=−32πGl2ξ(4+ξ).
00

(8.3.34)

132

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

Aneglectionofthescalar-fieldpotentialisequivalenttolinearapproximationofthescalar-fieldex-
citations(cf.ξ2=0).Hence,low-excitationanalysesmaybefulfilledforV≈0.Insuchcase,the
dominanttermisthethirdtermofequations(8.3.24)and(8.3.25)whichforξ>0isquintessential
(i.e.itdiminisheswT),howeverhighvaluedforlowerlengthscalesl.Giventhesmallnessofpand
incosmologicalterms,asmallvalueoftheexcitationξ(inrelationto1)maystillhaveadominant
characterintermsofthetotalequation-of-stateparameterandcosmologicalacceleration,whichmay
stillbegivenfor0<ξ1.
Forequation(8.3.32),thereisthevaluewT=1/3(radiation)forξ˙=ξ=0.Thevalueofthe
equation-of-stateparameteriswT1c<1for−1/4<ξandξ>0.
Letuswritedownthetotalequation-of-stateparameterforqˆ=1accordingtoequation(8.3.33)with
p=w.Thereis
22
1w−V/−8π3Gc0Hξ˙+lc2ξ
wT=31−V/−8π3Gc20Hξ˙.(8.3.35)
Forstatic,vanishingexcitations,theusualequation-of-stateparameterisrecovered,wT=w.
Beforefurtheranalyses,thedensityparametersshouldbeintroduced.Thismayrelatethedeceleration
parameterswithothermeasuredquantitiesThisisfulfilledintheChapter8.4.

parametersdensityThe8.4LetustakethefirstofthegeneralizedFriedmannequation(8.1.5).Itmayberewrittensothatfollowingbe
alid:v2˙a2+Kc2
a=8π3G˜+3cp2+31Λc2+31ΛIc2.(8.4.1)
Herewith,ΛI≡ΛI(ξ).
Inthisway,thedensityparametersΩimaybedefined(atthispointwiththefunctionalHubblerateH).Be
2Ω=8πG(2ξ)(1+3w),ΩΛ∗=c(Λ+ΛI)H−2≡ΩΛ+ΩI,(8.4.2)
3H3termcosmological-functionfurtherawith3Hξ˙πξ˙2
ΛI(ξ)≡ΛI:=−c21+ξ+c2α˘(1+ξ)2(8.4.3)
=8cπ2GH(ξ2)(Λ−V),(8.4.4)
relatedtoequation(8.2.1)ofthescalar-fielddensity.ThedensityparameterΩΛ∗ishencethedensitypara-
meterofthescalar-fielddensityofChapter8.3.Thisstrengthenstheinterpretationoftheequation-of-state
parametersofthescalarfieldascomponentsofthetotalequationofstateandpartofanequation-of-state
parameterwΛforscalar-fielddominanceinequation(8.2.15).Thesameparameterpossessesquintessential
properties.DensityparametersΩiaredimensionlessparametersfordensitycontributions.ΩΛandΩIaredensity-
parametercomponentsofatotaldensityparameterΩΛ∗whichentailthewholescalar-fielddensitydistri-
butionΛ.Inthissense,ΩΛ∗isthedensityparameterofascalardarksectorwithenergydensityΛ,here
separatedintoitscosmologicalfunctionandtime-derivativeparts.

8.4.THEDENSITYPARAMETERS133
Densityparametersareinprincipledefinedbymeansofthedensityofacertaindensitydistributioni=ic2
densitycriticaltheand2H3c=8πG˜,(8.4.5)
usuallyfortimet=t0.
cisdefinedasthedensitywhichisnecessaryforaflatUniverseK=0.Furthermore,thereis
Ωi=i=i.(8.4.6)
ccΩisthedensityparameterofmatter(i=matter):usuallybaryonicmatterorbaryonicmatterplusadditional
relevant,yetunknownmatter.Itmayinprinciplebedefinedinamoregeneralwaytocontainfurtherterms
likeneutrinosandotherkindsofdarkmatter.
RadiationisherealsogivenwithinΩ,especiallywithinw,
Ω=8π3GH2eff+83πHG2cef2fp(8.4.7)
=Ω+Ωp=ΩM+ΩR+Ωp+...(8.4.8)
Herein,Ω=ΩBaryon+ΩMeson+Ων+ΩX+ΩR.(8.4.9)
Herein,ΩRistheradiationcontributionintheUniverse.FurtherΩBaryon≡ΩB,ΩMeson,ΩLeptonandΩν
arethebaryonic,mesonicandleptoniccontributions.ΩXstaysforfurthercontributionsthatmightbeCold
DarkMatter(CDM),forinstance.Ων,partofΩLeptonforthatinstance,maybecalledHotDarkMatter
(HDM)contribution(seeChapter2.3).
Ωpisthepressurecontributionwhichisnegligiblefordust-dominance(w≈0).
ApartfrompossibleCDMterms(whichseemdominant),theonlymattertermswhicharerelevantindyna-
micsarethebaryonicones.Furthermore,inthecurrentUniversewherematterdominates,ΩRandΩPof
too.gligible,neareradiationHowever,thereisDarkEnergyasmeasuredinhigh-z-measurementsusingsupernovaeoftypeIa.Itacts
antigravitationallyandintermsofmatter,asanegativepressure.Shoulditbeconsideredasmatter-typed,
thenananti-stiffpressuretermw≈−1woulddominate.Suchtermis,however,oftentakenseparately
withinadark-energydensityparameterΩΛ.Inthismodel,itisrelatedtothescalar-fieldexcitationξ.Addi-
tionally,thereisΩIfromequation(8.4.2)whic˜hisalsorelatedtoξ.Thequestionisinwhichwaydodensity
parametersofthescalarfield,togetherwithG,relatetodarksectorsofdensityinsenseofusualmodels.All
theseparametersheremaydifferfromthestandardones.Forinstance,Ωiinthestandardapproachrepresent
observedquantitiesbasedonascreenedvalueofthegravitationalconstant(orofdensity),sothat
Ωi=˜G/G0Ωistd(8.4.10)
isgiven,whereasstdthegeometryoftheUniverseisdeterminedbytheconstantsbarevalue.Forξ>0,
thereisΩi<Ωi.
Letuswritedownthedensitiesanddensityparametersinsomeoftheirformsforα˘1.Therearethe
densitycritical2H3c=8πG(ξ)(8.4.11)

134

parameters-densitymattertheand

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

Ω=Ω+Ωp
=8πG2(ξ2)(+3p)=+3p
cH3c

(8.4.12)

andΩ=ΩM+ΩR,(8.4.13)
whereinΩentailsΩMofusualmatter,ΩRofradiationandfurtherdark-mattersectorsasdefinedinequa-
tions(8.4.8)and(8.4.9).ΩMgivethemeasuredmatter-densitytermaslongasΩIdoesnotcontributeto
.sectorthatThescalar-fieldrelatedparameterswiththescalar-fieldexcitationoritsfirstderivativeare
8πG(ξ)1Λ
ΩΛ=3H2c2V=3H2(8.4.14)
c2G2+G(ξ)2
=4l2H22−0G0G(ξ),
ΩI=8πG2(ξ2)(Λ−V)=I(8.4.15)
cH3c˙2ξc=3H2ΛI=−H(1+ξ).
Here,thefollowingenergydensitiesandpressuresareused,
I=Λ−V,(8.4.16)
2c3Λ=V+8πG(ξ)2HG˙(ξ)
=V−3c2Hξ˙,(8.4.17)
2a˙c8πG0
pΛ=V+8πG0ξ¨+2aξ˙.(8.4.18)
Theseequation-of-statecomponentshavealreadybeendiscussedforascalar-fielddominatedUniversein
8.2.ChapterFinally,fromequation(8.1.6),thedensityparameterrelatedtosecondderivativesofthescalar-fieldexcita-
readstions1G¨(ξ)G˙(ξ)2
ΩII=−3H2G(ξ)−2G(ξ)2(8.4.19)
8πG(ξ)2G˙(ξ)
=3H2c2(pΛ+V)−3G(ξ)H
=8πG2(ξ2)(pΛ+V)−2ΩI.
3cH3ΩIIisatermrelatedtopΛ,otherthanΩI,whichisrelatedtoΛ.Inequation(8.4.2),ΩΛ+ΩIequalsthe
densityparameterfortheenergydensityΛ.Hence,
˜Gπ8ΩΛ∗=3H2c2Λ.(8.4.20)

8.4.THEDENSITYPARAMETERS135
ForΩIIusingthescalar-fieldequation,thereis
2ΩII=H1GG˙((ξξ))+3Hc2l2GG(ξ)−1+8π9HG(2ξ)qˆ(−3p).(8.4.21)
0toleadsThisqˆc2G(ξ)
ΩII=ΩI+3(Ω−Ωp)+3H2l2G(v)−1.(8.4.22)
Itsformdependsonthecouplingqˆtomatter.Forqˆ=0,i.e.forHiggsparticleswhichleadtotheappearance
ofmassanddecouplethemselvesfromallparticles,thedensityparameterΩIIequalsΩI.
Further,onthemeaningofdensityparameters,thedenominatorinequation(8.4.1)maybewrittenasΩ¯H.
Letusdefinethefollowing,
222a2=a˙¯+K2c=:Kc2.(8.4.23)
bHHΩassumeusLetb=Ω¯−1.(8.4.24)
sincealid,visThisKc22=a2=a2+Kc22⇐⇒a2Ω¯=a2+Kc22(8.4.25)
bHΩ¯¯ΩH2H2
cKcK⇐⇒a2=Ω¯−1H2=bH2.(8.4.26)
(8.4.23)mayberewrittentogetdirectly
22˙a2+Kc2=KcbΩ¯=¯KcΩ¯(8.4.27)
1Ω−henceand¯2ΩcKb=a˙2+Kc2.(8.4.28)
isthereThus,a˙2Ω¯−1=Kc2.(8.4.29)
Ω¯=ΩtotalandavalueΩtotal≡1entailsK=0andaflatgeometryoftheUniverse(seelater),while
smallervaluesentailK=−1andanhyperbolicUniverse.6Ωtotal>1meansK=1andaclosedUniverse.
Equivalently,b=0entailsK=0,b>0entailsK>0andb<0entailsK<1.Fromobservationsof
thecosmicmicrowavebackground(CMB),itisdeducedthatthedominantcontributionofenergydensity
comesfromΩ˜ΛofthecosmologicalconstantoraformofdarkenergywithΩ˜Λ≈0.7(forwhichweusea
tildetopointoutthatitdoesnothavetobethesame,noteveninnature,ashere).ThenthereisColdDark
MatterwithΩCDM≈0.3,instandardapproachesaspartofΩ.WithinstandardFriedmannmodels,energy
densityofusual,baryonicmatterisonlyabout1/10thevalueofthatofdarkmatter,hencealmostnegligible
incosmologicalterms(seeChapter2.3).Furthermore,partingfromthetwo-year-resultsofWMAP,thetotal
densityparameterΩtotalpossessesavalueneartounity.Theexperimentaluncertaintyis,however,toohigh
toconcludeK=0.
6ThenameΩtotal,mayletoneassumethattheparameterΩIIispartofitandthisisnotthecase!¯Ω,however,isindeedthe
parameterofthetotaldensityaccordingtousualapproaches.

136

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

(8.5.2)

8.5Decelerationandtheequation-of-stateparameters
Letusfirsttake(protem.)ξ˙=0.Thetotalequation-of-stateparameterforξ˙=0yieldsforα˘1,
21qˆ+3(1−qˆ)w−8π3c˜G3Λ+l2(1+ξξ)
wT=31+c2˜Λ.(8.5.1)
Gπ8densitycriticaltheithW2H3c=8πG˜,
yieldsit1qˆ+3(1−qˆ)w−3cH22c3Λ+l2ξ(1+ξ)
H3wT=31+c22cΛ.(8.5.2)
using,Further=cΩand2cΛΩΛ=3H2,
theequation-of-stateparametermaybewrittenintermsofdensityparameters.So,using
22ξc2l=4H2ΩΛ1+ξ,(8.5.3)
isthere1qˆ+3(1−qˆ)ΩΩp−ΩΩΛ1+ξ4
+1wT=3ΩΛforξ˙=0.(8.5.4)
ΩTheevolutionoftheequation-of-stateparameterwTmaybeseeninfigures8.1and8.2.
Accordingtoequation(8.5.4),anegativeequation-of-stateparameterappearsfor

(8.5.3)

(8.5.4)

Figure8.1:TotaleosparameterwT.Protem.:ξ˙=0,Ω=0.3,ΩΛ=0.7).N.B.:Thecontinuouslinestays
forqˆ=1.Thedashedoneforqˆ=0.

8.5.DECELERATIONANDTHEEQUATION-OF-STATEPARAMETERS137
Figure8.2:TotaleosparameterwT.Protem.:ξ˙=0andΩ=0.3.
qˆΩ+3(1−qˆ)Ωp<ΩΛ1+ξ4.(8.5.5)
Thiscanbevisualizedinfigure8.3bymeansoftheminimalvaluesofmatterandpressuredensitiesin
relationtoΩΛtermsforpositivetotalequation-of-stateparameters.
Further,inthenonstaticalcase,theequation-of-stateparametermaybeeasilygeneralized.Thereis
3Ω+ΩI+ΩΛ
wT=1qˆΩ+3(1−qˆ)Ωp+ΩI−ΩΛ1+ξ4.(8.5.6)
Analogouslytoequation(8.5.5),thereiswT<0for
qˆΩ+ΩI+3(1−qˆ)Ωp<ΩΛ1+ξ4.(8.5.7)
Thismaybevisualizedinfigures8.4and8.5.
MatterandpressuretermsΩandΩp,butalsoapositivedensityparameterΩIstrengthenadeceleration
behavioraccordingtogravitationasanattractiveinteraction.TheyactagainstQuintessenceasdousually
sectorsofmatter.Consequently,itmaybeassumedthatΩIactsasadarksectorofsomekindofmatter
indeed.Inotherwords,assumethatthereis
ΩM=Ω+ΩI.(8.5.8)
Furthermore,beΩroughlygivenbybaryonicmatter(subscriptB).Hence,thereshouldbearelationofthe
7form,wingfolloΩ∝ΩB.(8.5.9)
7Later,inequation(8.5.15),wewillintroducebaryonicmatterasanantiscreenedtermofΩ.ΩMwillthenbegivenasinequation
(8.5.8).

138

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

Figure8.3:MinimalvalueofqˆΩ+3(1−qˆ)ΩpforwT>0withξ˙=0accordingtoequation(8.5.5).Set:
ΩΛ=0.7.

(8.5.10)

Hence,ΩIgiveadarksectorofphenomenologicalDarkMatter.Then,thereis
ξ˙<0!=⇒G˙(ξ)<0.(8.5.10)
So,followingequations(8.4.15)and(8.1.44)thereis
ΩI>0.(8.5.11)
Thisgivesapositiveenergy-densitydistributionofadark-sectorcomponent.Suchisrelatedtoin-time
diminishingscalar-fieldexcitationswhicharethemselvesrelatedtodiminishingchangesofthecoupling
constantG(ξ).
Takeequation(8.5.6).Now,accordingtothelatterinterpretations,bethefollowingsetgiven:
Ω=0.03,Ωp=0,ΩI=0.27,ΩΛ=0.7.(8.5.12)
isthere,val.Ad.

wT≈0.33−0.40−2.8forqˆ=1,(8.5.13)
ξ≈0.33−0.43−2ξ.8forqˆ=0.(8.5.14)
Thismaybeseeninfigures8.6and8.7.wTisnegativeforξ>0.Forξ=1,thereiswT≈−1.1.For
ξ=0.1,thereiswT≈−9.5.Forξ=10,wT≈−0.2.ThereiswT=−1forξ=1.077forqˆ=1andfor
ξ=1.089forqˆ=0.For0<ξ<4,thereisq˜<0.
AnothertermtotakeintoaccountisG˜whichtooshouldleadtoadifferencebetweenΩBandΩgivenby
ΩBΩ≈1+ξ.(8.5.15)

8.5.DECELERATIONANDTHEEQUATION-OF-STATEPARAMETERS

139

Figure8.4:MinimalvalueofqˆΩ+3(1−qˆ)ΩpforwT>0.Left:protem.:0≤ΩI≤1and−1≤ξ≤4,
accordingtoequation(8.5.7).Set:ΩΛ=0.7.Right:protem.ξ=1.

Forpositivevaluesofξ,thismeansananti-screeningofmatterdensitywithΩΩB.
Letusnowtaketheeffectivegravitationalcouplingintoaccount.AssumeΩB=0.03.Then,forξ=1,for
instance,thereisΩ=0.015,togetherwithΩI=0.285(forΩ+ΩI=ΩM).If,ontheotherhand,there
isξ=0.1,thenthereisΩ=0.027.Equivalently,forξ=4,thereisΩ=0.006.
ForΩM=Ω+ΩIasthemeasuredtermofmatterdensity,wehave:
•Underneglectionoffurtherterms,therelationtobaryonicmatterreadsΩ=ΩB/(1+ξ).
•MatterdensitymaybegivenbyΩasscreenedbaryonicdensityplusfurtherdarktermsΩI.
•Forξ˙<0,thereisΩI>0.
•Forξ>0(ξ<0),forΩ+ΩI=ΩM,thereisΩI>ΩDSMM(ΩI<ΩDSMM).
TherearethefollowingvaluesofwT0andwT1forΩΛ=0.7andqˆ=0andqˆ=1respectively:
ξΩΩIwT0wT1
-93.467-93.4770.2700.0300.01-9.467-9.4760.2730.0270.10-2.000-2.0070.280.0200.50-1.067-2.0070.2850.0151.00-0.756-0.7560.2880.0121.50-0.603-0.6030.2900.0102.00-0.367-0.3690.2940.0064.00Undertheseassumptions(withΩM=Ω+ΩI),thereisforqˆ=1,
43ΩM+ΩΛ
wT=1ΩM−ΩΛ1+ξ.(8.5.16)
Forit,wTisexactly-1forξ=1.07692.Suchavaluewouldmeanananti-screeningofΩbyroughlyone
8half.8N.B.:AnanalysisofthebehaviorofΩIisnecessaryfordifferentepochsoftheUniverse.Suchananalysisneedsofabetter
comprehensionoftherelationbetweenbareandeffectivedensitiesanddensityparametersaswellasofthetime-dependenceofthe
effectiveparameters.Hence,asgroundingofexactanalysesofthenatureofΩIanditsrelationtodarksectors,theworkinChapter8.6
isofspecialrelevance.

140

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

Figure8.5:Exampleofapositiveeosparameterinahighlymatter-dominated,closeduniverseforΩ=3,
ΩI=4,ΩΛ=0.7fordifferentexcitationsξ.Thedashedcurvestaysforqˆ=0andthecontinuousonefor
qˆ=1.N.B.:Lowexcitationswouldneedofhighermatter-dominancefortotaldustorstiffnessofwTtobe
acquired.Thismaybetranslatedinhigherlengthscalesl.

8.6Effectiveandbaredensityparameters
LetusnowrewritetheFriedmannequationsbydividingthefirstgeneralizedFriedmannequationbythe
presentscalefactora0.Thisleadsto
a˙2Kc28πGeffa212a2˙aaG˙effa2πG˙2eff
a02+a02=3c2a0+3Λ(ξ)ca0+a02Geff+a03α˘G2eff.(8.6.1)
Equivalently,writingthemattertermexplicitly,thereis
a˙2+Kc2=8πG0(1+ξ)−1a+1Λ(ξ)c2a+
22
a02a023c2a03a0
˙aaG˙effa2πG˙2eff
+a02Geff+a03α˘G2eff.(8.6.2)
So,ξaffectsthematterterm.
Wehaveforageneric-time(t)densityandforthepresentone(t=t0)thefollowingrelation,
Ga=00˜0,(8.6.3)
3(1+w)˜21(1−3w)(1−qˆ)
aGandthereforefora=a0(orequivalentlyz=0),
0Ω0=0c.(8.6.4)
Forgenerictimestherearetwopartstobetakenintoaccountfortimedependence:

(8.6.2)(8.6.3)

(8.6.4)

8.6.EFFECTIVEANDBAREDENSITYPARAMETERS

141

Figure8.6:Totaleosparameter.Protem.:Ω=0.03,ΩI=0.27,ΩΛ.ξ(lengthscale,cf.(8.5.3))variable.
N.B.:Fornegativeexcitationsξ,wTispositive.

(i)Thetimedependenceofwhichisgivenbya−3(1+w).Thisistheusualdependenceofstandard
models.Friedmann(ii)ThetimedependenceofG˜−1ininatermnonvanishingforqˆ=0.
(iii)Thetimedependenceofc,whereG˜−1isfound.Forqˆ=1andw=1/3,however,theG˜terms
cancelfrom.Forqˆ=1,higherscalar-fieldexcitationsbelongingtoahigher-valuedcriticaldensity
c,agivenvalueof0wouldmeanlowervaluesofΩ.IfthedensityparameterΩis,ontheother
hand,set,thenhastopossessalargervalue(anti-screening)withinstandardformalism.
Wemaywritethetermscomingfromthetime-changingofthecriticaldensitybysomefunctionsyor
yirespectively(whichdependonw).TheydependonξintermsofG˜.
isThere

defineweThus,

3H23H02
c=8πG˜=8πG0(1+ξ).

(8.6.5)

2H3∗=8πG0=1+ξ(8.6.6)
asthescreeneddensitywhichis,especiallyinthecaseqˆ=1,independentonξ.Equivalentlyfordensity
parametersΩi.Inthesameway,wedefineananti-screenedquantity
x==x(1+ξ).(8.6.7)

142CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC
Figure8.7:Totaleosparameter.Protem.:Ω=0.03,ΩI=0.27,Ω+ΩI=ΩM.N.B.:Thecurveisthe
sameasforthecaseξ˙=0andΩ=0.3.
isThereG˜012(1−3w)(1−qˆ)a03(1+w)
Ω=(1+ξ)−1Ω0=G˜a(8.6.8)
=Ω˜0∗a03(1+w),(8.6.9)
awithaneffectiveparameterasfollows,
21(1−3w)(1−qˆ)
Ω˜0∗=Ω0=G˜˜0.(8.6.10)
GThisparametercontainsthescreeningeffectforqˆ=0andthescreeningfromcshownexplicitlyinequation
(8.6.5).Thisshouldbeameasuredvaluewithinstandardformalism,entailingnotonlymatter-densityterms.
Takingtimedependenceforthedensityparameterwith(a),thereis
Ω==Ω∙(1+ξ).(8.6.11)
ξisafunctionoftimeandhenceofH−1.Itsvalueisofsomeform
ξ∝ξ0a0mw,(8.6.12)
awithanamplitudeξ0andatime-dependencetermmwwhichishigh-valuedandnegativefornegligibletime
dependenceofthescalarfield.Highamplitudesoftheexcitation,however,screenthedensityparameterof
mattertobesmallerthantheactualdensitywouldleadtoassume.Theoneparameterforunscreenedtermsis

8.6.EFFECTIVEANDBAREDENSITYPARAMETERS

143

Figure8.8:TotaleosparameterwT.N.B.:Ω+ΩI=ΩM,Ω:anti-screenedbaryonicparameter.Left:
ΩΛ=0.7,ΩB=0.03.Right:ΩB=0.03,ΩM=0.3.

Ω0=.ItshouldgivetheactualvalueofdensitywhileΩ0isthescreened(measured)parameter,analogously
to(bare,luminous)massandeffective(measured)mass(seeChapters7.2and7.5).Hence,forsmalltime
variationsofξ,highamplitudesofthescalar-fieldexcitationswouldleadtothephenomenologicalappear-
anceofDarkMatterintermsofascreeningeffectofscalarfieldsondensity.
Now,werewriteequation(8.6.1)asfollows,
a˙2+K2c=Ω0=G0a0(1+ξ)−1+1Λ(ξ)c2a+(8.6.13)
22˜21(1−3w)(1−qˆ)1+3w2
a0a0G˜a3a0
˙aaG˙(ξ)a2πG˙(ξ)2
+a02G(ξ)+a03α˘G(ξ)2.
Usingequation(8.6.3),theFriedmannequationthenreadsforusualmatterMwithw=0,radiation
(w=1/3)anddarksectors,
2˜21(1−qˆ)2
a0G˜aa
a˙=H02Ω0=M(1+ξ)−1G0a0+(1+ξ)−1Ω0=Ra0+
2−+(1+ξ)−1(ΩΛ=+ΩI=)a0+
a+(1−(1+ξ0)−1(Ω0=M+Ω0=R+Ω=0Λ+Ω0=I)),(8.6.14)
Ωi=aretheanti-effectivetermsΩi∙(1+ξ),thusconstantsnotexplicitlydependentonξ.(1+ξ0)−1Ω0=i=
Ω0iisthemeasureddensityparameter.
Alternatively,thereis
H2(t)=H02a0Ω0∗Ma0+Ω0∗Ra0+(ΩΛ∗+ΩI∗)a0+
22−2
aaaa+(1−Ω0∗M−Ω0∗R−Ω∗0Λ−Ω0∗I))]).(8.6.15)
Here,wedefineΩR∗=ΩR,ΩΛ∗=ΩΛandΩI∗=ΩI.Furthermore,densityparametersΩ0+Ω0Λ+Ω0I=1
leadtoK=0.

144

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

•Constantandquasiconstantexcitationsξ=const.:
ThereisthecosmologicalfunctionΛwhichgivesthedensityparameterΩΛbymeansofequation
(8.4.14)forH=H0.Incaseofnegligibletimedependenceofthescalarfield,thistermisexactlyas
withinstandarddynamicswithdarkenergyΛ=Λ0.Thereis
2ΛcΩ0Λ=3H02forξ=const.(8.6.16)
Inthecaseofequation(8.6.16),thentheparametersΩIandΩIIgivenbyequations(8.4.15)and
(8.4.19)vanish.However,ifequation(8.6.16)isonlynearlygiveninthecurrentUniverse,thesepa-
rametersmightplayanimportantroleinprimevaldynamics.Ashortdiscussionofthe,howeverrather
standard,casesofξ=const.,especiallyforξ=0(i.e.Einstein–deSitterandΛCDMcosmology)is
C.4.AppendixinwnshoForξ=const.,itcanbestatedhereforathree-fluidsystemwithcosmologicalconstant,
aa˙=H02Ω0∗Maa0+Ω0∗Raa0+Ω∗0Λaa0+(1−Ω0∗M−Ω0∗R−Ω∗0Λ).
22−2
0(8.6.17)Here,wehaveusedtheeffectivedensityparametersasdefinedinequation(8.6.10).Forqˆ=1,Ω0∗
equalsΩ0whichisalsoascreenedvalue.Forhighvaluesoftheamplitudeξ0,thesescreenedvalues
aresmallerthantheactualbareparameterΩ0=.Forξ<0,thisleadstophenomenaliketheonesof
DarkMatter.ξ0=−0.9wouldlead,forinstance,toΩi=10Ωi=,withΩiasdynamicalmeasured
alue.vIncaseofaconstantscalarfield,lettherebe(protem.9)
Ω=≈ΩM(1+ξ0)−1,ΩM==ΩB=0.03.(8.6.18)
betherelet,FurtherΩ≈ΩM=Ωdyn=0.3.(8.6.19)
Then,therewouldbeanamplitude

(8.6.19)

ξ0=−0.9.(8.6.20)
Ontheotherhand,forΩ0Λ,acosmologicalconstantisgivenherebyξ=const.Thismayaccount
forthephenomenonofDarkEnergy,dependingonthevalueoftheamplitudeofthescalar-field
excitationsanditslengthscale(ΩΛ∼l−2ξ02).Thereis
2Λ=4l321ξ+ξ.(8.6.21)
ThegeneralformofΩΛreads
2c1ΩΛ=H024l2ξ2(1+ξ)−1.(8.6.22)
9ThisisincontrapositionwiththeanalysisofChapter8.6.However,thisisananalysisforξ≈const.whileinthelatterChapter,
theparameters.analysisSeefocusestheneonxtΩIpoint.ofSuchadifdiscussionferentwithinapproachthisisChapterimportant.givenstillunknownmattersoftheevolutionofeffectivedensity

8.6.EFFECTIVEANDBAREDENSITYPARAMETERS

145

Forξ=−0.9=const.,thereis
2cΩΛ=20.25l2H02.(8.6.23)
approximatelyreadsrateHubbleTheH0=73km/s/Mpc≈2.3∙10−18/s.(8.6.24)
beouldwthereThus,2cmΩΛ≈3.4∙1056l2.(8.6.25)
Shouldthisparameterpossessavalueofabout0.7,then,thelengthscalewouldbe
l≈2.2∙1028cm≈7.2Gpc,.(8.6.26)
Aparameterofabout0.7withlengthscalesintherangeofkpcwouldentailexcitationsξnearerto
zero.ξ=−0.1w−5ouldmeanlengthscalesintherangeofMpc.Forrangesofsomekpc,ξwouldhave
toliebelow±10,seeequation(8.1.4).
Forξ=const.,theinterpretationdependsonξ0,i.e.onthevalueoftheconstantscalarfield.Unlike
ΛCDM,itmaybeexpectednotonlyasmaincontributiontothecosmologicalconstantΛ0andthus
oftheDarkEnergyparameter,butitmaybeexpectedwithinmatterdistributionaswell.Forconstant
fields,acontributionbothasDarkMatterandasDarkEnergyisonlypossibleforhigh-scalednearly
vanishingscalarfieldsξ≈−1(i.e.φ≈0)whichscreenmatterdensity.Nearlyvanishingconstant
excitationsmayleadonlytoDarkEnergy.However,accordingtoChapter7.8,thelengthscalefor
DarkMatterphenomenologyshouldbeoftheorderofmagnitudeofgalaxybulges.TheMilkWay(as
ausualgalaxy)hasabulgeof5kpc,whichisoftheorderofmagnitudeof1020m.Hence,thelength
scaleforDarkEnergyinthecaseξ=const.istoohigh.Ontheotherhand,alengthscaleofthe
orderofmagnitudeofgalaxybulgesleadstofartoohighvaluesofΩΛ.Scalar-fieldtimedependence
shallindeedplayarolewithindarksectorsofdensity.

•Derivativesandpossibledarksectors:
Scalar-fieldexcitationscannotbeexactlystaticalandfurthertermsaretobeaddedintermsofΩIand
ΩII.Inthatcase,DarkMattermaybegivenasdiscussedinChapter8.5withΩIaspartofadark
sectorofmatter.Anotheroptionmaybeseenfrom(8.4.15),whichmaybewrittenas(forα˘1)
1G˙(ξ)˙aa
ΩI=3H02ΛI=H02G(ξ)a0.(8.6.27)
Hence,itmaybeexpectedthatforthemeasureddensityparameterofDarkEnergy,therebe
ΩΛstd=ΩΛ+ΩI.(8.6.28)
However,ifweassumethatΩΛisnearlyconstant,givenlowdynamicalbehaviorofthescalarfield,
thenΩIistopossessnegativevalues.TheeffectivecouplingG˜istodiminishwithtime.Inthatcase,
therewouldbeξ→−1.Nonetheless,itishoweverpossiblethatmorecomplexdynamicalbehavior
ofthescalarfieldleadstoothersituations.Forinstance,ΩImightpossessthescale-factordependence
ofamatterdensityandhenceactasadarksectorofmatter(asdiscussedinChapter8.5).Ifthiswere

146CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC
soandDarkEnergywerefullygivenbyΩΛ,adensityvalueofDarkEnergyof0.7withalengthscale
asgivenbyflatrotationcurves(say35l=5kpc)wouldmeanafieldexcitationof
ξ≈±2∙10−23.(8.6.29)
Inthiscase,adarksectorofmatterΩIhadtopossessthefollowingproperty,
˙aaG˙(ξ)=k8πG(ξ)a2,(8.6.30)
a02G(ξ)3c2a0
withaconstantk≈9.Further,fort=t0,therewere
G˙(ξ)0≈k1.7∙10−18m4kg−2s1.(8.6.31)
TakingonlybaryonsandwithΩB≈0.03,thebaryonicenergydensityfork=9reads
∝10−13kg∙m−1∙s−2.(8.6.32)
beouldwthereHence,G˙(ξ)0≈10−4m2∙kg−1∙s3.(8.6.33)
Further,giventimederivativesofG˜,andtheappearanceofG˜inΩi,theredshiftdependenceofΩiasa
screenedtermmaypossessadifferentbehaviorindependenceoftime.Suchanalysisistobefulfilled
elopments.vdefurtherin8.7BreakingofenergyconditionsandconditionsofaBounce
Forperfectfluids,thereistheweakenergyconditionwhichmaybewrittenasfollows,
(8.7.1).0≥Furthermore,thereisthestrongenergycondition,whichwewriteasbelow,
+3p≥0.(8.7.2)
Alreadyfromequation(8.1.18)itisclearthatforvanishingscalar-fieldexcitationsandderivativesofthe
same(withoutantigravitativematterterms),iftheenergyconditions(8.7.1)and(8.7.2)arevalid,therecan
benoaccelerationsa¨>0[123,194].Theconditions(8.7.1)and(8.7.2)togetherareknownasPenrose–
Hawkingcondition.Further,giventheconcavenessofa(t)foralltimesunderequations(8.7.1)and(8.7.2),
a(t)mustbeequaltozeroatsometimeinthepast(whichweusuallylabelt=0).Sincea(0)=0atthis
point,thedensitydiverges,asdoestheHubbleexpansionparameter.Thereappearsasingularity,usually
knownastheBigBang,andbecausea(t)isconcave,thetimebetweenthesingularityandtheepochtmust
alwaysbelessthantheHubbletimeτH=1/H.
InallhomogeneousandisotropicmodelsforwhichtheZeldovichintervalofequation-of-stateparameters
isvalid(0≥w≥1),andevenfor−1/3<w<0,aBigBangsingularityisunavoidable.TheBigBang
can,however,beavoidedinmodelswithdominantnegativepressurewithp≤−/3orinthosecontaining
anonvanishingcosmologicalconstantorwithsomevaluesofthecosmologicalfunction,i.e.ofscalar-field

8.7.BREAKINGOFENERGYCONDITIONSANDCONDITIONSOFABOUNCE147

excitationsorofitsderivative(concavenessofa(t)isthennolongervalidthroughout).Inotherwords,it
canbeavoidedforadominanceofΩΛorΩIinthedynamics.Thesetermsthencontributeaspressureterms
whichviolatethePenrose–Hawkingcondition.Theyleadtodark-energybehavior.
Withinstandardcosmology,thereareproblemsashorizonandreheatingwhichareusuallysolvedbymeans
ofInflation.InflationmaybeexplainedasamechanismbywhichtheUniverseexpandsveryrapidly,and
inusualmodels,exponentially.TheUniversepassesadeSitterepochinwhichacosmologicalconstantor
arelatednegativepressuredominates(cf.Chapters2.4and8.3).Thewaythispressure(orcosmological
function)actuallyevolvesisdeterminantfordeterminingtheinitialstateoftheUniverse.Furthermore,how
thescalarfield(asinflatonfield)evolvesiscrucialforthedynamicsofInflationandforthevaluestakenby
effectivepressures,giventhediscussionofChapter2.4.
Aninflationaryuniversewithinducedgravitationcanbederivedwithinthecontextofinducedgravitywith
Higgspotential(cf.[47–49]).ThismodelcanleadtoprimevalNeworChaoticInflationindeed(op.cit.).As
amatteroffact,thePenrose–Hawkingenergycondition3p+c2≥0[123,194]maybebrokenforChaotic
Inflation.ForthiskindofInflation,aBigBouncewouldbeexpected(thatmeansnoinitialsingularitybefore
Inflation).Thiscasecanbecomparedwiththecaseoftheworksin[67],accordingtowhichYukawainterac-
tionsofthemagnitudeofthenucleardensitycanleadtonegativepressuresthatmightplayanimportantrole
inearlystagesoftheUniversesothatthePenrose–Hawkingconditionmaynotbesatisfied.ThisYukawa
interactionintheprimordialUniversewouldberelatedtoapressureaspΛ(comingfromthepotentialV(ξ)
andthescalar-fieldderivatives,translatedasthevariablegravitationalcoupling),possiblycontributingtothe
mechanismofInflationandDarkEnergyaspartofthecosmologicaltermΛ.
Withoutfurtherintroduction,theFriedmannequationswiththescalarfieldequationleadto
2a¨+a˙2+2Kc2=8π2G˜(1−qˆ)(−3p)+c22ξ.(8.7.3)
lc3aaLetustakeageneraltimet=tqwhichshallhavethefollowingproperties:
(i)a(tq)=0,
(ii)a˙(tq)=0,
(iii)(tq)=0.
tqshallbeidentifiedwitht≈0.Thisshallbeastaticaluniversewithoutinitialsingularity.Forit,ingeneral,
(8.1.5),equationfromisthereπG˙(ξ)(tq)2
f1(tq)=3α˘G(ξ)(tq)2
πξ˙2(tq)
=3α˘(1+ξ(tq))2.(8.7.4)
πξ˙2(tq)
=α˘ξ(tq)2.(8.7.5)
Further,itiseasilyseenthatforα˘1,thiscorrectionvanisheswith
f1(tq)=0.(8.7.6)
Thecosmologicalfunctionforthetimet=tqreads
3ξ2(tq)4π
Λ(ξ(tq))≡Λq=4l21+ξ(tq)1+3α˘.(8.7.7)

(8.7.4)(8.7.5)(8.7.6)(8.7.7)

148

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

(8.7.8)(8.7.9)

(8.7.10)

Itispositiveforξ(tq)>−1.
ThefirstFriedmannequation(8.1.5)thenyields
Kc2πξ˙2(tq)c2ξ2(tq)4π
a2(tq)=3α˘(1+ξ(tq))2+4l2(1+ξ(tq))21+3α˘.(8.7.8)
Giventhepropertiesofξanda,theUniversehastobeclosedorflat,i.e.
(8.7.9).0K≥K=0,however,isgivenonlyifbothξ˙(tq)=0andξ(tq)=0arevalid.
writefurtherusLetξ(tq)≡ξqanda(tq)≡aq(8.7.10)
andequivalentlywithallotherquantities.
WithhelpofthedefinitionoftheeffectivegravitationalcouplingG˜,thereis
˜2Gξ21+ξ=G0ξ.(8.7.11)
Further,followingthefirstFriedmannequation(8.1.5),
Kc2πξ˙qc2ξq24π
aq2=3α˘(1+ξq2)+4l2(1+ξq)1+3α˘,(8.7.12)
thereisforK=1,
ξq24l24π−1π4l2ξ˙q2
1+ξq=aq21+3α˘−3α˘(1+ξq)2.(8.7.13)
Forα˘1,equation(8.7.13)furtherreadsasfollows,
ξq24l2
1+ξq=aq2.(8.7.14)
Λqisthusbasicallygivenbyl2/aq2.Furthermore,forα˘1,thescalar-fieldexcitationfort=tqreads
2l2aq2
ξq=aq211+l2,(8.7.15)

andforξq1,consequently,

2ξq=4l2.(8.7.16)
aqEquation(8.7.15)givesonlynegativevaluesforanegativesignbeforethesquare-root.Wehavetwocases
define:usLetanalyze.to•Theupper-signcase(minus)inequation(8.7.15)wewillcall(-).
•Thelower-case(plus)signinequation(8.7.15)willbe(+).

8.7.BREAKINGOFENERGYCONDITIONSANDCONDITIONSOFABOUNCE149
Foraq/l→0in(-),thereisξq→−1.Letusseethisasfollows:first,takeforthecase(-),
δ≡(a(tq)/l)2andδ1.(8.7.17)
isthereThen,√1+δ≈1+1δ,(8.7.18)
2,(-)forhenceand2ξq=2al21−√1+δ
q2l=−aq2δ.(8.7.19)
(8.7.20)Hence,wehaveforthecase(-),
ξq≈−1.(8.7.21)
Thisrepresentsavanishingofthescalarfieldφ=∼v√1+ξ.
Thecontinuityconditionreadsfort=tq,
˙ξ3q˙q=−(1−qˆ)21+ξqpq.(8.7.22)
˙qistobezeroforsignchangingtobegivenatt=tq.Forqˆ=1,thisdoesnothavetobeforcedasa
condition(cf.equation(8.7.22)),asitisdirectlygiven.Forqˆ=0,theremustbe
0=−6l22pqξ˙2q(8.7.23)
ξaqq3a2a2−2
=−2l2qpqξ˙q11+l2q.(8.7.24)
Therefore:•Forqˆ=0,thereiseitherξ˙(tq)=0orp(tq)=0,forbothcases(+)and(-)!
Takethescalar-fieldequation(8.1.4).Fort=tq,itis
ξ¨q=−c22ξq−κ0qˆpqc2.(8.7.25)
lWithequation(8.7.15),wehave
2ξ¨q=−2211+a2q−qˆκ0pq.(8.7.26)
laqForqˆ=0orp(tq)=0,ξ¨(tq)ispositivefor(-)andnegativefor(+).Inthecase(-),|ξ¨(tq)|isverysmallif
aql.Inthesamecasefor(+),thereiswithbothgeneralqˆandpq,
ξ¨q≈−a42−qˆκ0pqc2,foraql,and(+).(8.7.27)
q

150

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

(8.7.28)

(8.7.30)(8.7.31)(8.7.32)

Furthermore,thereisforhighinitialscales,
2ξ¨(tq)≈±a(tq)l−qˆκ0p(tq)c2,foraql.(8.7.28)
Here,theuppersignbelongsto(-),andtheloweroneto(+).
Witheitherqˆ=0orpq=0,thereisξq=0for
211+a(2tq)=0.(8.7.29)
lHence,itisespeciallyrelevantthat,althoughsmall-valued,δbenotzero.Else,thereisξq=0in(-).Such
wouldfurthermeanK=0.
ThesecondFriedmannequation(8.1.6)leadstothefollowing,
a¨qKc2κ0pq3ξq2ξ¨q
2aq+aq2=−1+ξqc2+4l21+ξq−1+ξq,(8.7.30)
whichwithequation(8.7.14)directlyleadstotheequationbelow(takeK=1),
a¨qaq−c2=−2l2c2κ0pq+ξ¨q(8.7.31)
ξq2ξq2
2a4a2−2a2
=−2lq211+2lq2κ0(1−qˆ)pq−a211+l2q.(8.7.32)
qLetustaketwocases:
(i)Forpq=0,equation(8.7.32)yields
22a¨q2aq=1+aq.(8.7.33)
cl211+la2q
Itcanbeeasilyseenthatforlaq,thelatteryields
a¨qaq/c2−→1,(laq),(8.7.34)
whichistheusualrelationofFriedmannmodels.
Foraqlandpq=0,thereis
a¨qaq=1+aq(aqland(+)).(8.7.35)
2lc•Hence,thereisacceleratedexpansionwithoutthenecessityofinitialvaluesofpressurepq.Ac-
celerationcomesfromcurvatureKandfromthescalar-fieldexcitations.For(-),ontheother
decelerated.isxpansionehand,(ii)Forpq=0,thereis
1aa¨a2a2−1a4a2−2
cq2q=1+l2q11+l2q−l2cq42(1−qˆ)κ0pq11+l2q,(8.7.36)

(8.7.33)(8.7.34)

8.7.BREAKINGOFENERGYCONDITIONSANDCONDITIONSOFABOUNCE151

withthefollowingextremalcases:
a¨qaqaq2aq41
4c2=1+21+1δ−4l2c42(1−qˆ)κ0pqfor(+)andaql,(8.7.37)
2aaqq=l−κ0pql2c4foraq1(+,−).(8.7.38)
pq>0actsagainstaccelerationfor(+)andqˆ=0whilethelowδtermactsaccelerating.For(-),
however,thefollowingisvalid.Foraqlin(-),
4a¨qaq=1−2δ−2aq(1−qˆ)κ0pq(aql),(−),(8.7.39)
c2δl2c4
Forξ=0,theδcontributionactsdeceleratingasdoespqalsoforqˆ=0.
Fora¨qaq>0,thereisthefollowingcondition,
cl11a224a22
2(1−qˆ)pq<κ0a211+l2q+κ0a411+l2q.(8.7.40)
qqe:vhaeW•Forhighlengthscaleslaq,accelerationisgivenfor(+)unlessthereareveryhighpressures
p(tq).
•For(+)andqˆ=0,smallpressurespqmayleadagaintodecelerationincaseofhighinitiallength
scales.For(-),therearenoacceleratedinitialstatesforpositiveinitialpressures.
Basically:•Therecanbenoinitialaccelerationfor(-),unlessthereisqˆ=0andp(tq)<0.
Furthermore,thereisforequation(8.7.32),
ξca¨q2aq=1+22ξq−l2(1−qˆ)κ0pq.(8.7.41)
qHere,thedifferentaccelerating(Kandpositiveξq)anddeceleratingterms(pq>0forqˆ=0)
canbeseen.Positivevaluesofξq(i.e.(+))leadtoaccelerationterms.Pressureactsdecelerating
(gravitationallyattractive)forqˆ=0.
TakenowthetimederivativeofthefirstFriedmannequation.Then,thereis
2a˙a¨−a˙2−K2c=(1+ξ)−1κ0˙−ξ˙+a˙a˙ξ˙−ξ¨+ξ˙+a¨ξ˙+Λ˙.(8.7.42)
222
aaaa31+ξaa1+ξa
Fort=tq,thereisthen
a¨(tq)ξ˙(tq)
0=a(tq)1+ξ(tq)+˙Λ(tq).(8.7.43)
Thismeansforthederivativeofthecosmologicalfunction,using(8.7.41),
˙Λ(tq)=−aq21+ξq2ξq−l(1−qˆ)κ0pq1+ξq(8.7.44)
c222ξ˙q
1224l2ξ˙q
=−aq21+ξq2ξq−l(1−qˆ)κ0pqaq2ξq2.(8.7.45)


152CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC
Forξ˙q=0,thelattermeansΛ˙q=0.
Further,ontheonehand,ξ˙q>0leadstoanegativederivativeofthecosmologicalfunctionaslongas
0<ξq>l2(1−qˆ)κ0pq.Accordingtoequation(8.7.45),accelerationisrelatedtoΛ˙qwhenξ˙q=0.
Ontheotherhand,thedefinitionofΛleadsto
˙Λ˙=43lξ2(1+ξ)−1ξ2−1+ξξ.(8.7.46)
Specificallyfort=tq,wethenhavethefollowing,
34l2
Λ˙q=aq2ξ˙qξq−12−aq2ξq−1.(8.7.47)
Here,ξ˙>0meansΛ˙q>0.
Formattersofconsistency,ifa¨qistobepositive(+)/negative(-),thefirstderivativeofthescalar-field
excitationatt=tqistobevanishing!If,however,thesignofa¨(tq)werechangedbymeansofp(tq)<0,
thesamewouldbevalidsincewealreadyhavetheconstraintp(tq)=0orξ˙(tq)=0.Hence,
•Thederivativeofthescalar-fieldexcitationisvanishingatt=tq,i.e.
ξ˙(tq)≡ξ˙q≡0.(8.7.48)
Hence,thecosmologicalfunctionisconstantatandthescalarfieldisstaticatt=tq.
TakeagaintheFriedmannequations.Thesecondonecanberewrittentotakethefollowingform,
aa¨=−κ60(+3p)+k(t)+3Λ.(8.7.49)
evhaweHere,k(t):=−(1+ξ)−11ξ¨+a˙ξ˙(8.7.50)
a2=−1(1+ξ)−1¨Θ(t),(8.7.51)
2whichgivesnewdynamics.Fort=tq,kreads
k(tq)≡−1(1+ξ)−1¨Θ(tq),(8.7.52)
2with¨Θ(tq)=ξ¨(tq)(8.7.53)
2=−qˆκ0p(tq)−ξ(tq)l−2(8.7.54)
a2q=−qˆκ0pq−aq211+l2.(8.7.55)
Furthermore,Θ¨givesapressureterm
1pG≡8πG0Θ¨(8.7.56)

8.7.BREAKINGOFENERGYCONDITIONSANDCONDITIONSOFABOUNCE153
whichequalsΛ+pΛwithouttakingthetermsfromthecosmologicalfunction/Higgspotential.So,thereis
intermsofdensity,pressureandthecosmologicalfunction,
a¨=−4πG01+p+pG+Λc2.(8.7.57)
a(1+ξ)33
Fort=tq,wehave
a¨(tq)=−4πG0(p(tq)+pG(tq))+Λqc2(8.7.58)
a(tq)1+ξ3
22−1
=−4πG0a2q11+a2q(pq+pGq)+Λqc2.(8.7.59)
3llThesecondderivativeȨ,i.e.thepressurepG,actsinthesamewayasp(tq).Itleadstoadeceleration.In
theabsenceofΛ,negativevaluesofpGwouldbenecessarytogetacceleration,andthiswouldbethecase
for−pG(tq)>p(tq).Yet,thereisΛq,whichpossessespositivevalues.
AccordingtothePenrose–Hawkingcondition,fort=tq,ifthestrongandweakenergyconditionsarevalid,
thentheremustbeaninitialsingularityfortheprimevalUniverse,i.e.aBigBang.Inequation(8.7.2),
takingT+3pT=+3p+3pG+Λc2,however,itisclearthatsuchconditionmaybeviolatedby
pG(tq)+Λqc2<p(tq).(8.7.60)
isThere12aq2
pq+κ0ξ¨q=(1−qˆ)pq−κ0a211+l2,(8.7.61)
qsothat,ontheonehand,forqˆ=1and(+),theenergyconditionswouldbebrokenindependentlyofpq.For
qˆ=0,ontheotherhand,theywouldbebrokenfor
2aq21
pq<κ0a211+l2=κ0l2ξq.(8.7.62)
qNeglectingΛfor(-),theenergyconditionsappearnottobebrokenatall.Suchbreakingwoulddependon
dynamicsaftert=tq.However,thecosmologicalfunctionshouldbetakenintoaccount,too,asafurther
termoftheequationofstate.Thecosmologicalfunctionleadstoanewtermdependentonξ2(andhenceon
Λ).Thereisaterm
1pξ=−6κ0l2ξ2<0(8.7.63)
whichactsantigravitationally.Furthermore,thereis
ξ¨(tq)12
pG(tq)+pξ(tq)=κ0−6κ0l2ξ(tq)(8.7.64)
)t(ξq=−qˆp(tq)−6κ0l2(6+ξ(tq)).(8.7.65)
Thetotalpressurewouldbe
)t(ξqptot(tq)=p(tq)+pG(tq)+pξ(tq)=(1−qˆ)p(tq)−6κ0l2(ξ(tq)+6).(8.7.66)

154

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

Thevalueofequation(8.7.66)istobeanalyzedforthecaseofbeingpositiveandthusgravitationally
sense.usualtheininteractingForqˆ=0,thereis
qqqξ1a22l2a2
p(tq)>6κ0l2(ξq+6)=3κ0aq211+l2aq211+l2+6(8.7.67)
asconditionforptot(tq)>0,whichistheconditionfora¨(tq)<0.Forlargelengthscalesla(tq)and
(+),thelatterconditionyields
24l2
p(tq)>3κ0a2a2+6>0,(la(tq)),(8.7.68)
qqandfora(tq)l,
12l2
p(tq)>3κ0aqlaq+6≈aqlκ0>0.
Relativelyhighvaluesoftheinitialpressure(whichactcounter-gravitationally)arenecessaryfordecelera-
.appeartotionLetustakethetwocasesofqˆ:
(i)Forthecase(-)ofqˆ=0,theconditionofa¨(tq)>0fora(tq)lyields
14l2
p(tq)>−3κ0aqlaq2−6(a(tq)l,(−))(8.7.69)
2≈−κ0a(tq)l<0.
Decelerationa¨(tq)<0appearsfora(tq)lunderfollowingconditionofpressure:
5p(tq)>−12κ0l2<0,(a(tq)l(−))(8.7.70)
•Hence,for(-)theremaybeaccelerationonlyfornegativeinitialpressures.Ontheother
hand,for(+),allnegativeinitialpressuresp(tq)leadtoaccelerationandabreakingofthe
Penrose–Hawkingcondition.Furthermore,onlyrelativelyhighnegativeinitialpressures
p(tq)<0wouldmakedecelerationpossible.
(ii)Forqˆ=1,apositivetotalpressureptot(tq)isgivenby
)t(ξq−6κ0l2(ξ(tq)+6)>0.(8.7.71)
Thisisthecaseforξ(tq)<0,i.e.for(-).For(+),however,thereisalwaysptot(tq)<0forqˆ=1
andthePenrose–Hawkingconditioniseasilybrokeninallcasesof(+).Hence,asingularityis
notnecessaryandabounceispossible.

BouncePlanck-lengthThe8.8FromChapter8.7weknowthattheHawking–Penroseconditionisbroken,especiallyforpositiveinitial
valuesofthescalar-fieldexcitation.Furthermore,weknowthatfollowingsuchbreaking,initialsingularities

155BOUNCEPLANCK-LENGTHTHE8.8.arenotnecessaryandaBigBangmightratherbegivenbyabouncestatewitha(t=0)=0.Weknow
thatwhensuchisgivenasastaticalcasea˙(0)=0forvanishinginitialdensities(0)=0,thescalarfieldis
staticalatt=0.Now,letusanalyzethepropertiesatt≈0withtq≈0forwhichwemaygiveavalueofa
state.initialtheatGoingforthintheredshiftz,a(z)becomessmallerandsmaller.Withinstandardcosmologyandwithvalid
Penrose–HawkingconditionsforaBigBang,a(t=0)thenvanishes.Here,letanotvanish(a=0)andbe
a˙(tq)=0.However,eventhoughnotvanishing,a(tq)shallbesmall.Ontheotherhand,thereisthelength
scalel.AccordingtoanalysesforgalaxiesandDarkMatterinChapter7.8,lshallbeofaroundtheorderof
magnitudeoftheGalaxyscore.
LetusassumethatthecontractionoftheUniverseforhigherredshiftsgoonuntiltheHeisenberguncertainty
,gyenerforrelationΔEΔt=,(8.8.1)
isvalid.Typically,atthisscalequantummechanicsbecomesdominantandtimeitselfisnotexactlyde-
terminedanymore,asclassicalmechanicslosetheirvalidity.Hence,letusassumethatthispointgivesthe
initialsingularitysothatwetakethePlancktimetP≈0.Atthistime,quantumfluctuationspersistonthe
scaleofthePlancklengthlP=ctP.Fromthesetwoscales,further,thePlanck2mass−1mP=PlP3isdefined.
FollowingtheFriedmannequations,thePlanckdensityPisoftheorder(G0tP).Consequently,
45ΔEΔt=∼mPc2tP=∼P(ctP)3c2tP=∼GcttP2=∼.(8.8.2)
0PThereisthen(ascommonlyknown)
2/1tP=∼G50=∼10−43s.(8.8.3)
cthenislengthPlanckThe2/1lP=∼ctP=∼G30=∼1.7∙10−33cm.(8.8.4)
cThePlancklengthrepresentstheorderofmagnitudeofthecosmologicalhorizonatt=tP.Betheminimal
scaleoftheUniverseca.thescaleinwhichquantumfluctuationsappear.Hence,betq=tP.Then,thereis
a(tP)≡aP=∼lP.However,giventhat(tq)isassumedasvanishing,thePlanckmassshallbeconstituted
bypressuretermsp(tP)andscalar-fieldexcitationsξ(tP)≡ξP.Consequently,sc.
(tP)=P.(8.8.5)
Takea(tP)=lP.Then,ξPisgivenby
32ξ(tP)≡ξP=∼2lG0c11+lG2c03.(8.8.6)
ForξP1:
232ξP=∼4Glc=∼l42l≈1066cm−2∙l2.(8.8.7)
0P

156

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

Actually,onlythecase(+)ispossiblesinceξ≥−1andequation(8.8.7)arevalid.Forl=∼1022cm,for
instance,10thereisξ(tP)≡ξP=∼10110.Forl∼1028cm,ontheotherhand,thereisξP=∼10122.For
α˘1,suchavalueleadstoΛ(ξP)≡ΛPas
2ξ3PΛP=4l21+ξP
33≈4l2ξP=lP2(8.8.8)
≈1066cm−2,
independentlyofl.ThesameresultisachieveddirectlyfromthefirstFriedmannequation(8.1.5)with
KΛP3=lP2.(8.8.9)
(tP)istakenaszero.However,wegetaneffectivedensityofthesystemsolelybya(tP)=lPwhichis
hencerelatedtothePlanckmass.ThereisaccordingtothefirstFriedmannequationwitheffectivedensity
asP,usingequation(8.8.8),
2c2=∼8πG0P.(8.8.10)
3lP

(8.8.11)(8.8.12)

Thisgivesadensityas
2gc3P=8πG0lP2=∼1093cm3.(8.8.11)
ThisistheorderofthePlanckdensity,indeed.ThePlanckdensityisusuallydefinedby
5c1P=∼G0t2=∼G2=∼4∙1093gcm−3.(8.8.12)
0PItleadstoaPlanckmass
2/1mP=∼PlP3=∼Gc=∼10−5g,(8.8.13)
0relatedtoaPlanckenergy
EP=∼mPc2=∼1019GeV.(8.8.14)
Thus,thefirstFriedmannequationisconsistentwithaninitialdensity(tP)tobevanishingfora˙P=0.
ThePlanckdensityandhencethePlanckmassaregivenbythescalarfieldatt=tP,ormoreexactlyby
thescalar-fieldpotentialatthePlancktime,givenbythePlancklengthitself.ThefirstFriedmannequation
wnoreadsl21=83πcG40P=Λ3P.(8.8.15)
PUsingthefirstFriedmannequationandhenceequation(8.8.9),thesecondFriedmannequationreads
a¨P−K2c2=−l2P2πG0p2(tP)+ξ¨P.(8.8.16)
lPlPlc8
10Thisisabouttheorderofmagnitudeofthelengthscalelforflatrotationcurves,accordingto[20,24,50]andChapters6.3and7.8.

BOUNCEPLANCK-LENGTHTHE8.8.

157

ithWξ¨P=−c22ξP−8π2G0qˆp(tP)=∼Θ¨P,(8.8.17)
clisthere222la¨P−Kl2c=ll2P−πcG20(1−qˆ)p(tP)+8cl2ξP,(8.8.18)
PPwhichyieldsafterinsertionofξPforlrelevantlyhigher-valuedthanlP,11
a¨P−K2c2=−l2P2πG20(1−qˆ)p(tP)−c2.(8.8.19)
lPlPlc2lP
SinceK=1,thelatterequationmayberewrittenasfollows,
2222la¨P−2+ll2Plc2=−2ll2PπcG20(1−qˆ)p(tP).(8.8.20)
PPForvanishingvaluesoftheinitialpressurep(tP)orqˆ=1,theright-handsideofequation(8.8.20)disappears
andcosmicaccelerationatt=tPisgiven.GiventhelowvalueoflPandforlPl,thefirsttermis
12isthereanddominanta¨P∼1053cms−2,(forlPl).(8.8.21)
Asinthegeneralcaset=tq,positivepressurespullaccelerationdown,sincethepressureactsgravitatio-
nally.Thepressureterm,however,isdependentonthereciprocalvalueofthesquaredlengthscalelandon
thesquaredvalueofthePlancklength.Additionally,itpossessesaG0/c2dependence,intotalacontribu-
tiona¨P∼l−2∙10−128cm4kg−1.Forl=∼1022cm,apressureofabout10100Pawouldbenecessaryforthe
pressuretermoftheright-handsideofequation(8.8.20)tobedominantandhencefordecelerationtoappear
andenergyconditionstobevalid.Pressuresoftheorderofmagnitudeofp(tP)∼10200Paforthepressure
term(incaseofqˆ=0)arenecessaryforpressuretermstobedominantinthedynamics.Evenforlength
scalesoftheorderofmagnitudeofthePlancklength,thepressureneededisextremelyhigh.Thepressure
termisthenoftheorder10−50cm2kg−1p(tP),whichmaybecomparedwithc2/lP∼1055cms−2,which
isthedominanttermofa¨Pforqˆ=1orrelativelylowpressures.
Consequently,forlengthscalesrelevantlylargerthanthePlancklength,thereis
2ca¨P≈lP,(8.8.22)
ingoodapproximation,independentlyofqˆ,landp(tP).Thisshowsahighlyacceleratedstateforthe
primevalUniverseatt≈0.Atthistime,thereisaveryhighcosmologicalfunctionΛPwhich,acting
antigravitationally,leadstheacceleratedexpansion.
InChapter8.7wehaddefinedpressuretermspGandpξwhichdependon¨ΘandΛ,respectively(cf.equa-
tions(8.7.56)and(8.7.63)).Theirvalueisrelatedtothepossibilityofcosmicaccelerationandtheappear-
anceofabouncestatewhichfollowsabreakingoftheenergyconditionsofPenroseandHawking.Nowwe
11Thismeans,especiallyvalidforllPbuteveningoodapproximationforlafewtimeslarger(saytwice)thanlP.
12Forl=∼1022cm(cf.Chapters2.4and7.8),thereislPc2/l2=∼10−57cms−2.

158

CHAPTER8.FRIEDMANN–ROBERTSON–WALKERMETRIC

haveanegativevalueofbothpGandpξfort=tPaccordingtothepositivehighvaluesofξP.
areThere1pG=8πG0ξ¨(8.8.23)
and

(8.8.24)

2ξpξ=−6κ0l2.(8.8.24)
Further,thereisatPlancktime,
c2l2
ξ¨P=−2l21+1+l2P−qˆκ0p(tP)c2.(8.8.25)
P√Theterminparenthesispossessesavaluebetween2and1+2.However,ingoodapproximation(and
especiallyforlrelevantlylargerthanlP),equation(8.8.25)reads
2ξ¨P=−2c2,(8.8.26)
lPalmostindependentlyonl,qˆandp(tP)unlesspshouldtendtoinfinityfort≈0.ξ¨P=Θ¨Pgivesthestatical
valueofthepressuretermpGcomingfromf.ItyieldsforllP,
2¨cΘPpG(tP)=8πG0=−2πG0lP2−qˆp(tP)=pT(tP)−p(tP),(8.8.27)
whichviolatesPenroseandHawkingsstrong-energycondition.Nexttoanalyzewouldbethedynamicsfor
t>tPinordertoknowifthereappearssomekindofrollovercontraction(althoughattPthereappears
highacceleration)orifaccelerationstaysandleadstoaninflationaryepochindeed.
Accordingtotheworks[48]and[49],thereshouldappearaninflationarystateforbothqˆ=1andqˆ=0.
ThismaybeofNeworChaoticInflation,whichisdependentontheinitialvaluesofthescalarfield.For
qˆ=1,thereappears(h.t.)slowrolloverdynamicsbeforeNewInflation(φv→v),andinthecaseof
Chaoticdynamics(φv→v),Inflationappearsautomatically.Duetotherelationshipbetweenmassesof
particlephysicswithinGUT,parametersarenotfine-tunedinordertopredictacceptablevaluesofreheating
temperatureanddensity(see[48]).Forqˆ=0,NewInflationneedsoffinetuningbutfollowsautomatically
afterashortcontractionera.ChaoticInflationisachievedsuccessfully,howeverbestforhighHiggs-particle
masses(see[49]).Theafore-mentionedanalysiscontributestothepossibilityoftheappearanceofChaotic
InflationsinceforastaticalUniverseatPlancktimetP,thescalar-fieldexcitationisveryhighandthus,
thescalarfieldismuchlargerthanitsground-statevaluev.However,scalar-fielddynamicsshouldbefur-
theranalyzedtocomparedynamicsofthisscenariowiththosein[47].Yet,itmaybeconcludedthatthe
Penrose–Hawkingconditiondoesnotholdandasingularitydoesnotappear.Further,thereappears
aninitialhighlyacceleratedstateat0≈t=tPinastaticalUniverse.

9Chapter

outlookandconclusionResults,

ThetheoreticalrelevanceoftheHiggsMechanismanditsuniversalpropertiescannotbequestionedatall.
Higgsparticlesingeneralappeareffectivelyinallbranchesofphysics;withinQuantumAsthenodynamics
theyleadtothemassofelementaryparticles,andwithinmesoscopicphysicstheyleadtotheMeissnereffect
ofsuperconductivity.Analogously,withinDualQuantumChromodynamicstheyleadtodualsuperconduc-
tivityandhencetotheconfinementofquarksandcolorchargesinhadrons.Inthiscontext,thefirstpartof
thisworkleadstothefollowingconclusion:
•DyonandmonopolecondensationswiththeAbelianHiggsMechanismareequallycapableofdescri-
bingthesuperconductingQCDvacuum.
•BothdyonsandmonopolesleadtotheDualMeissnerEffectandhencetoconfinement,howeverwith
strengths.ferentdif•Themagneticpermeabilityinsuchvacuumrisestoinfinitywithvanishingmomenta.
Giventheuniversalpropertiesofthescalarfields,thesecondpartofthisworkfurtherintroducesintoa
modelofGeneralRelativitywithHiggsMechanism.Forthis,wehavediscussedthegeneralizedconcept
ofHiggsfieldsespeciallyinthecontextofastrophysics.WehavegroundedouranalysisonBergmann–
Wagonermodelsofscalar–tensortheoriesofinducedgravity,andwehaveinsertedHiggsfieldsasscalar
fieldswhereasaHiggspotentialhasbeenchosen.Suchleadstoacosmologicalfunctionanalogtothe
cosmologicalconstantofGeneralRelativityaswellastoaneffectivegravitationalcoupling.
IfthescalarfieldpossessesacouplingtothefermionicLagrangesector,Higgsfieldsthenlosetheirsource
andcannotbegeneratedinhigh-energyexperiments.Withoutsuchcoupling,though,theystillcouple
analogouslytocosmonsofDarkEnergyandMatter.Thecosmologicalfunctionandtheeffectivecoupling
leadtonewgravitationaldynamicswhichhavebeenanalyzedheretogetherwithMaxwell-likeequationsof
.vitygraWeknowthatascalar–tensortheorywithHiggspotentialisabletoexplainandcontributetothephenomenon
offlatrotationcurves(DarkMatterproblem)forspecificdensitiesofgalaxysbulges.Hence,thethirdpartof
thisworkanalyzestheissueofthemetriccomponentsforcentralsymmetryandtheFriedmann–Robertson–
Walkermetricindetail.ConclusionsoutoftheworkrelatedtoBlackHolesolutionsforscalarfieldswith
negligiblemassarethefollowing:
•Theexactsolutionofthemetriccomponentsfornegligiblescalar-fieldmassesindicatesthatthemetric
componentsofthelineelementgivenbytheequationscorrespondtotheusualSchwarzschildmetric

159

160

CHAPTER9.RESULTS,CONCLUSIONANDOUTLOOK

whichappearsinthisformonlyforthelimitingcaseofthevanishingHiggsscalar-fieldexcitations
(i.e.ξ=0).
•HighervaluesofAleadtoadecreaseinthegravitationalpotentialνthroughtheexponentB/K.
•Infact,themetricandscalarfieldareregulareverywherewithexceptionofr=0asnakedsingularity.
•ThereexistsnoSchwarzschildhorizonexceptforthecaseofvanishingscalar-fieldexcitations.There-
fore,BlackHoles(intheusualsense)donotappearforthecaseA=0.
•ForthegeneralvaluesoftheexcitationamplitudeA,thequalitativeresultsofminimallycoupled
scalarfieldsarevalidandscalarfieldsthusactanalogouslytoelectricchargesinagravitationalfield.
BlackHolesgothroughtoGreyStars.
Further,wehaveinvestigatedthesingularitiesandBlackHolesolutionswithandwithoutHiggsfieldex-
citations.Wehaveconsideredtwoscenariostosolvethefieldequationswithandwithoutthevanishingof
Higgsfieldmass,andthesolutionshavebeenfurtheranalyzedinviewofthegeodesicmotion(inthecase
ofscalarfieldswithfermionicsource).Inparticular,thelinearfieldequationswithfiniteHiggsfieldmass
havebeensolvedtohaveacorrectphysicalexplantationoftheparametersinvolvedinthestudyaswellas
todiscusssomeaspectsoftheBlackHolesolutionsobtained.Further,inordertoinvestigatethephysical
consequencesofthesesolutionsforboththecases,wehaveanalyzedtheminviewofthegeodesicmotion.

•WehavefoundtheappearanceofReissner–Nordström-likeBlackHolesolutionsforthecaseofnon-
vanishingfieldexcitationsinthisformulationwhileinthevanishinglimitofexcitationswehavethe
SchwarzschildgeometryasusualinGR.Itisshownthattherearescalar-fieldtermswhichatlow
gravitationalregimesactantigravitationallyintoaReissner–Nordström-likemetricactingasagene-
term.ege-likcharralized•Thetermscorrespondingtothepressurerelevantfromthescalarfieldandnonlinearitiesoftheexact
solutionleadtoadynamicalmassdifferenttotheluminous,baremassfromdensity.Thebehaviorof
thecomponentsofthemetricisthendescribedaccordinglyalongwiththeirphysicalconsequences.
•Uptothisorder,theSchwarzschildhorizonbecomesweakerwithstiffnessw.TheSchwarzschild
radiuschangeswiththeeffectivemassasshownintherepresentativegraphicalplot.Itisalsoshown
thatstiffmatteractingrepulsivelyinthemetriccomponentλisaneffectwhichappearsespeciallyfor
anegativeeffective,yetpositivebaremass.
•Theeffectivepotentialpermitsstableboundedorbitsandangularvelocities.Theorbitsarefound
qualitativelythesametothoseinthecaseoftheSchwarzschildandReissner–Nordströmgeometryin
GR.Thestabilityoftheboundstatesisdiscussedfromtheviewpointoftheluminousanddynamical
parameters.mass•Theassumptionofstiffmatter(relevantinnerstructureofmatter)leadstorelevantdeviationsfrom
effective,measuredastronomicalmassestobare,luminousmasses.Furthermore,italsoleadsto
flattenedcurvesoftangentialvelocity,whichisshownintheconcerninggraphicalplotmarkinga
similarityatlargedistancestotheflatrotationcurveofatypicalspiralgalaxy.
Wehavefurtherinvestigatedtherelationbetweenthescalar-fieldexcitationsofinducedgravitywithaHiggs
potentialandtheobligatorypresenceoffinitepressuretermsinenergydensityofgravitation,alongwith

161

theirappearanceinlinearsolutionsforsolar-relativisticeffectssuchasperihelionadvance,andforflat
rotationcurvesleadingtoDarkMatterphenomenologywithscalar-fielddensitycomponentsofthedark-
matterprofile.Theimportantconclusionsdrawnfromthisstudyaresummarizedbelow:
•AnenergydensityofgravitationfollowingMaxwell-likeequationsmaydifferwithitsanalogueof
GR.GravitationalenergywithininducedgravityandGRareidenticalforqˆ=0(whichdenotesthe
couplingofthescalarfieldtothematterLagrangian),andforqˆ=1(i.e.theabsenceofthecouplingof
thescalarfieldwiththematterLagrangian)theyarethesameonlywiththeconstraintontheequation
ofstateparameterasw=1/5.
•Finitevaluesofpressurewithinvacuumsolutionsareexpectedfromthenatureofscalar-fieldexci-
tations.Thisfurtherleadstothenotionofthedynamicandbare(luminous)massesinthismodel.
Thevalueofthedynamicalmassisobservedgreaterthanthatoftheluminousmass,andthepresent
formulationisthususefultodescribethesignaturesofunseenmatterinnature.
•PerihelionshiftisfoundthesameaswithinGRforlow-energeticsystemswithvaluesofpressureas
constraintbyenergyinviewoftheequationofstateparameterasw=1/5.Thereis,further,more
complicateddynamicsforhigh-energeticsystemswithhighcoefficientsofCapercandCbofenergy
andmomentum.Scalarfieldsareessentialforlow-rangedynamicsalthoughtheyhavenewtonian
range.suchatviorbeha•Flatrotationcurvesofgalaxiesleadtodark-matterprofileswithbaryonicandscalar-fieldcomponents
ofdensity.Scalar-fielddensitiesarestronglyrelatedtopressuretermsinthismodelandseemsto
provideaviableexplanationoftheDarkMattercontentsofourUniverseindeed.
•DarkMatterdominanceleadstopressuresrelatedtoanequation-of-stateparameteroftotalenergyof
thesamevalueasforweakfieldsinsolar-relativisticranges.
•Thenon-newtonianbehaviorofdensityappearsasdistancesgrow.Suchnon-newtonianbehaviorof
scalar-fieldexcitationsleadstoflatrotationcurves.Thecontributionduetothescalarfieldinthe
energydensityinfactactsasthedark-matterprofileinviewofthetotalenergydensityofthesystem.
WithinFriedmann–Robertson–Walkercosmology,Friedmannequationsarederivedindeed,andtheselead
tonegativeequation-of-stateparameters,ofantistiffmatterfortheabsenceofmatter.
Thedensityparametersofcosmologywithinthismodelhavebeenderivedalongwiththedeceleration
parameter,togetherwithitsissuesofcosmicacceleration.Wehavediscussedpossibleinterpretationsof
thestructureofthismodelwhencomparedtoexperimentalresultsinthecontextoftheSM.Wehaveshown,
hence,deSitterpropertiesofthequantitieswhichcharacterizeQuintessenceaswellasInflationforthe
primevalUniverse.Especially,weconcludethefollowing:
•Dominantvaluesofthecosmologicalfunctionmayconstrainlength-scaleandscalar-fieldexcitation
values.BotharehighlyimportantforcorrectionsofthegeneralizedFriedmannequationsandforthe
energydensityofusualmatter.Furthermore,thefieldexcitationplaysarelevantroleintheevolution
.parameterHubbletheof•Negativevaluesofthedecelerationparameterandoftheequation-of-stateparameterarepossible
withinthismodelwithouttakingnegativeusualpressuresnoracosmologicalconstant.Cosmicac-
celeration(Quintessence)ispossible,andscalarfieldsmayactaspartofdarksectorsofmatterand
energyinformoffurtherdensityparametersorasscreeningterms.Aconstantscalarfield,however,

162

CHAPTER9.RESULTS,CONCLUSIONANDOUTLOOK

wouldneedoftoohighalengthscaleandnegativeexcitationstoaccountforboththedarksectors
successfully.Fordynamicalfields,scalar-fieldexcitationsmaybeverysmallindeedandyetaccount
sectors.darktheforFurther,wehaveanalyzedtheconsequencesofthescalarfieldfortheprimevalUniverse,especiallyinthe
contextoftheBigBangandHawking–PenroseconditionsofenergyandBouncescenarios.Weconclude
wing:follothe•TheenergyconditionsmaybebrokeninaprimevalUniverse.Hence,thereexiststhepossibilityofa
BigBounceasinitialstate.InastaticstateoftheprimevalUniverse,thereisaconstantfieldexcitation
whichisrelatedtoanegativescalar-fieldenergydensitywhichmaybreakthePenrose–Hawking
conditionsandfurtherleadtoaccelerationpartingfromaverycondensedstateoftheUniverse.
•Forqˆ=0andpositiveinitialscalar-fieldexcitations,theinitialstateisacceleratedunlessthereare
highpositiveinitialpressures.Forqˆ=0andnegativeinitialscalar-fieldexcitations,negativeinitial
pressuresarenecessaryforacceleration.Forqˆ=1,thereisaccelerationforpositiveinitialscalar-field
excitationsregardlessinitialpressureterms.
WehaverelatedthisprimevalstatetothePlancktimeinaprimeval,initialuniverseofPlanckdistance.At
Plancktimewithavanishingenergydensityofthestaticuniverse,thesignaturemaybegivenbyPlanck
values.ThereciprocalvalueoftheinitialscaleoftheUniversegivesPlanckdensityandtheinitialvalueof
thecosmologicalfunction,whichisconstantatt=tP.
•AtPlancktime,highvaluesofthescalar-fieldlengthscalewhichleadtoflatrotationcurveslead
tohighnegativeeffectivepressuretermswhichfurtherbreakenergyconditionsandleadtoahighly
acceleratedBouncestate.ThisshowsarelationtoaformofChaoticInflation.
However,manyquestionsarestillunansweredfromtheperspectiveofthepresentformulation.Atgalactic
ranges,theestimationoftheshifttointermediatebehaviorofscalarfields,i.e.therelationbetweenscalar
fieldsandgalacticcenters,isstillunclear,anditwouldbeaquiteinterestingproblemtoinvestigate.Fur-
thermore,thismightbevaluableinrelationtoquintessentialpropertiesofscalarfieldsforgalaxieswithin
exactsolutions,leadingtotheReissner–Nordström-likebehavior.
Furtheranalysesrelatedtonewissuesonsolar-relativisticeffects,especiallygeneralizedtogalacto-relativistic
effectsarestillunclear.Furthermore,thequintessentialpropertiesofscalarfieldsaswellasprimevaldyna-
micsarestilltoanalyzeinmoredetail.CosmologicalimplicationsofinducedgravitywithHiggspotentials
intermsofQuintessenceandDarkMatter,aswellasprimevaldynamicsarestilltoinvestigateindetail.
Especially,theissuesofInflationafteraBigBouncearetodiscussindetail.Forinstance,itisstillunclear
whetherthehighaccelerationatt=tPgoesthroughtoarollovercontractionorwhetheritleadstoInflation,
indeed.

artP

IV

ppendixA

163

AppendixA

GeometryandvityRelatiGeneral

TheA.1ensorTMetricalThemetricaltensorgµνisa(4×4)2nd-ranktensorwhichphysicallygivesthepropertiesofspacetime.
Shallitobeytherequirementsof
symmetry(i)

and(ii)unitarity

gµν=gνµ

(A.1.1)

gµνgµλ=δνλ,(A.1.2)
respectively(while(A.1.2)definestheinverseofgµν).
(iii)Inthelimitingcaseofvanishingspacetimedependenceandthusofvanishinggravitationalinterac-
tions,shallthemetricaltensorpossesstheformofthe(pseudo-euclidian)Minkowskimetricgµν=
ηµνofSpecialRelativity(SR).Forit,wechoosethesignature(+,-,-,-)with
ηµν=diag(1,−1,−1,−1).(A.1.3)
Hence,wewillcounttheindiceswithGreeklettersfromzero(x0=ct)throughthree(x3=z),and
aconstantmetric(nocurvature)willgiveminkowskianspacetimeasfoundinthecovariantform
ofelectrodynamicsorinusualquantum-relativisticmechanics.Latinindicesshallcountoverspatial
.onlycoordinatesThemetricaltensorisingeneralafunctionofbothspaceandtimecoordinates.Withaµandbµas4-vectors
inR4,themetricdefinesthescalarproductas
33aνbν=gνµaµbν.(A.1.4)
=0µ,ν=0ν165

166APPENDIXA.GENERALRELATIVITYANDGEOMETRY
Clearly,withinnotation,themetricaltensorisusedforloweringandraisingindices.However,usualnotation
conventionisEinsteinsoneforsummation.Itmeanthatifinasumanindexappearstwice,onceasanupper
(contravariant)indexandonceasalower(covariant)one,itwillbesummedoverit.Hence,
3aνbν≡aνbν=gνµaµbν(A.1.5)
=0ν=gµνaµbν=aµbµ.(A.1.6)
Suchconventionwillbeusedthroughoutthisworkunlessexplicitlymentionedelsewise.
Ascalarproductof4-vectors(thelength)isascalarandassuchitisinvariant.Thescalarproduct
ds2=dxµdxµ
=gµνdxµdxν(A.1.7)
µdxdx=µiscalledlineelement.Itisusuallyusedequivalentlytothemetricaltensoritselfandthereforeoftencalled
also.metricormationstransfentzLorA.2Inanalyticalmechanics,thereistheaction
B(A.2.1)Lds=SAwhich,accordingtotheHamiltonPrincipleoftheLeastAction,possessesavanishingvariation,
(A.2.2)0Sδ≡whenkeepingboundariesconstant.Consequently,theEuler–LagrangeequationsfortheLagrangefunction
,wfolloL∂∂qLk−dtd∂∂q˙Lk=0,(A.2.3)
withgeneralized(canonical)coordinatesqkandvelocitiesq˙kanddefininggeneralizedforcesΦk=∂∂qLkand
canonicalmomentapk=∂Lconjugatetoqk.
Forspecialrelativity(SR),∂q˙kwithinminkowskianspacetimeandwithrest-massm0,4-velocityvµ=dxdsµ,
electricchargeeand4-potentialAµ,theLagrangefunctionreads
L=21m0c2ηαβvαvβ+eAαvα.(A.2.4)
Hence,togetherwiththeRicciidentitiesofcovariantderivatives,theEuler–Lagrangeequationofthesystem
givesthehomogeneousandinhomogeneousMaxwellequationsincovariantform,
F[µν,λ]≡Fµν,λ+Fνλ,µ+Fλµ,ν=0,(A.2.5)
Fµν,ν=−4πjµ.(A.2.6)

167TIONSTRANSFORMALORENTZA.2.Here,,µgivestheusualderivativeuponthecoordinatexν(see(A.3.6).Further,Fµνistheenergy–stressor
field-strengthtensorwithaforcedensity
kµ=m0c2vµ,νvν=Fµνjν=eFµνvν.(A.2.7)
Further,jµ=e{ρ,c1ji}istheelectric4-currentwitha4-vector{x0,xi}≡{x0,x1,x2,x3}.Fµνisdefined
intermsofthe4-potentialgivenby
Fµν=Aν,µ−Aµ,ν.(A.2.8)
Aµ={ϕ,ai}isthegaugepotentialwithaiasvectorpotentialandϕasscalarpotentialfield,bothknown
fromelectromagnetism.Asalreadyclear,thisisthe(specialrelativistic)electromagnetismwiththemagne-
ticstrengthpseudo1vectorcomponentsBl=21εijk(ak,i−ai,k)andtheelectricstrengthvectorcomponents
Ei=−ϕ,i+cai,t.εijkisthecompletelyskewsymmetric4-Levi–Civitatensor.
Inthecontextofelementaryparticlephysics,thetheorycanbequantized(i.e.putontotheformofex-
plainingnatureviainteractionsbetweenquanta),andthegaugepotentialisthenrelatedtogaugephotonsas
intermediateparticlesfollowingfromthegaugeoftheinnergroup.However,letusnowfocusonexternal
transformationsandintroducespacetimetransformationsΛµν.Thesearegivenby
xµ=Λµνxν,Λµν=∂xµ,(A.2.9)
νx∂µwhichdefinesLorentztransformationsasfaras
∂∂xxα=0(A.2.10)
begivenforthedeterminant.Forthem,thefollowingbevalid:
Λµν=const.,detΛµν=0.(A.2.11)
Thus,Lorentztransformationsarelinearandoccurbetweeninertialsystems.Theyconformthegeneral
).RGL(4,grouplinearTheinfinitesimallineelementds2=dxµdxµisascalarandhence
(A.2.12)ds.=dsForvectorsandtensors,however,homogeneousLorentztransformationsarelikethefollowing:
vµ=∂xµνvν=Λµνvν
x∂νvµ=∂xµvν=(Λµν)−1vν
x∂µνηαβ=∂∂xxα∂∂xxβηνµ=(Λαν)−1Λβµ−1ηνµ
βαηαβ∂∂xxν∂∂xxµηνµ=ΛανΛβµηνµ.
Lorentztransformationsaredefinedsuchthat
ηαβ=ηαβ,
ηαβ=ηαβ

168

APPENDIXA.GENERALRELATIVITYANDGEOMETRY

(A.2.13)

bevalid.Withequation(A.2.11),thelatterleadstotheorthogonalitycondition
(Λµν)−1=(Λµν)T.
Hence,theselinearspacetimetransformationsareorthogonal.Further,forinfinitesimaltransformations
isthereΛµν=δµν+Sµν,with|Sµν|1.(A.2.13)
Consequently,thefollowingisvalid,
Sµν=−Sνµ,(withST=−S).(A.2.14)
Thereare6linearlyindependentantisymmetricbasistensorsSiµν.Theyarecalledgeneratorsofthegroup
GL(4,R).Thus,onemaywrite
Sµν=λiSiµν,i=1,...6,(A.2.15)
withλiasreal-valuedconstants.λiaregroupelementsandtransformationparameters.Thereare3for
theEuleranglesand3forvelocitycomponentsofinertialsystemsinrelationtoeachother.Sincetheyare
continuous,thegroupisaLiegroupwith1-elementλi=1.
Siµarethegeneratorsofthegroup.Forfinitetransformations,exponentiatingleadstothetransformation
form:wingfollowith

iµΛµν=eλiSν.

(A.2.16)

ThereisthecommutatoroftheLiealgebra
[Si,Sj]=fijkSk,fijk=0,(A.2.17)
withfijasanonvanishingstructureconstant.Thus,theLorentzgroupisnotabelian.Furthermore,itmay
withgeneralizedbexµ=Λµνxν+aµ,Λµνaµ=const.(A.2.18)
Thisisa10-parametricalgroupofso-calledinhomogeneousLorentztransformations,orsimplyPoincaré
transformations.

A.3ThelocalgaugeoftheLorentzgroup
Lorentztransformationsareanexampleofglobalgaugetransformationofthesystem.Underlocalgauge,on
theotherhand,thereisλi=const.→λi=λi(xα),i.e.thetransformationparametersacquirespacetime
dependence.TakeonlyhomogeneousLorentztransformations.Then,thereis
Λµν=∂xµ(xα).(A.3.1)
νx∂Equation(A.3.1)isusuallyknownastherelativityprinciple:

(A.3.1)

A.3.THELOCALGAUGEOFTHELORENTZGROUP

169

•Thereexistsauniformtransformationbetweenbothsystemsxandxsothatthesameprinciples
shouldbevalidbetweenthem.
Usingit,lengthof4-vectorsistostayconstantaftertransformations,i.e.vµvµ≡vµvµ,independentlyof
thecoordinatesystem.Forthemetric,whichnowbecomesspacetimedependent,ontheotherhand,thereis
∂xα∂xβ
gµν(xν)=∂xµ∂xνgαβ(xν)(A.3.2)
andνµgµν(xν)=∂xα∂xβgαβ(xν).(A.3.3)
x∂x∂Physically,thefunctionalvaluesofthemetricmeantransformationsnotonlybetweeninertialsystems.Asa
consequence,inertialforcesasCoriolisandcentrifugalforcesappear,andthesearerelatedtothespacetime
dependenceofthemetricaltensor.ThisisthegroundingofGeneralRelativityasageometrizedtheoryof
gravitation.Therefore,(local)gaugingofthePoincarégroupleadstoaclassicaltheoryofgravitationwhich
isatheoryofinertialforces.Theequationofmotionisgivenbythatoffreeparticlesalongageodisicalline.
Thisiscalledthegeometrizationofgravitation.
WithinaPoincarégaugetheory,thederivativeofvectorsandtensorstransformsdifferentlytovectorsand
isTherees.themselvtensors∂xα∂xβ
vµ,ν=vα∂xµ∂xν
,β∂xα∂β∂2xα
=vα,β∂µ∂xν+vα∂xµ∂ν,(A.3.4)
whereonlythefirsttermrepresentsthebehavioroftensorialtransformations.Hence,covariantderivatives
areneededsothatthefollowingisvalid,
βα∂x∂vµ;ν≡vα;β∂µ∂xν.(A.3.5)
Asemicolonrepresentthecovariantderivativewhileasubscriptedcomarepresenttheusualderivativewith
,ν≡∂ν∂νδλα=∂νδλα.(A.3.6)
x∂Ifonlygravitationistakenintoaccount,thecovariantderivativeisdefinedby
;ν≡DνDνλα=∂νδλα−Γναλ(A.3.7)
andvλ;ν=Dνλαvα(A.3.8)
forgenerallycurvedlorentzianmanifolds(forquantummechanics,gaugefieldsAµtaketheplaceofChirstof-
felsymbols.Furthermore,fordualsymmetryalsodualgaugefieldsAµaretobetakenintoaccount;allwith
therespectivecouplingconstant).
Equation(A.3.6)isusuallywritteninsimplifiedmanner.Inthiswork,thederivativeiswrittensothatthere
be

Dνuµ≡Dνλµuµ

(A.3.9)

170

APPENDIXA.GENERALRELATIVITYANDGEOMETRY

forthecovariantderivative.
Anaffineconnectionisageometricalobjectonasmoothmanifoldwhichconnectsnearbytangentspaces
andsopermitstangentvectorfieldstobedifferentiatedasiftheywerefunctionsonthemanifoldwith
valuesinafixedvectorspace.Thechoiceoftheaffineconnectionisequivalenttoprescribingawayof
differentiatingvectorfieldswhichsatisfiesseveralproperties.Thisistoleadtocovariantderivativesandto
covariantbehavior,anditisequivalenttoanotionofparalleltransport,whichisamethodfortransporting
tangentvectorsalongcurves.Theconnectioncoefficientsoftheaffineconnectionoflorentzianmanifolds,
Γναλ=Γαλν,arecalledChristoffelsymbols.Theytransforminthefollowingway:
µσ∂xα∂xµ∂xβ∂2xµ∂xα∂xβ
Γρν=Γβα∂xν∂xσ∂xρ−∂xβ∂xα∂xν∂xρ(A.3.10)
=Γβσα(Λνα)TΛµσ(Λρσ)T−∂x∂βΛµα(Λνα)T(Λρβ)T.(A.3.11)
Onlythen,thederivative(A.3.7)willpossessthetransformationbehaviorinequation(A.3.5).Elsewould
leadtocoordinationeffectswhichwouldnotbephysical.
Generalhigher-ordertensorstransforminthefollowingway,
i1iml1ln
Tj1i1...j...lnim=∂∂xxk1∙∙∙∂∂xxkm∂∂xxj1∙∙∙∂∂xxjnTl1k1...l...knn.(A.3.12)
Thisisthetensor-transformationlaw.
Christoffelsymbolspossessaninhomogeneoustransformationbehaviorandarethereforenotensors.Their
transformationmaybemadetovanishthroughcoordinatetransformations[74].
ChristoffelsymbolsgivetheaffineconnectionforGR.Thisgivesarulewhichdescribeshowtolegitimately
moveavectoralongacurveofamanifoldwithoutchangingitsdirection.Therefore,theChristoffelsymbol
shouldaccountforcurvatureofspacetimeandparalleltransport.Furthermore,theyarerelatedtothemetrical
bygtensorµνΓβµν=1gµα(gβα,ν+gαν,β−gβν,α),(A.3.13)
2thatso

(A.3.13)

vµ;νvν=0(A.3.14)
isvalid.Furthermore,theaffineconnectionsatisfiesRiemannsrequirementthatanobjectshouldbeinde-
pendentofitsdescriptioninaparticularcoordinatesystem.Thecovariantderivativerepresentsespeciallya
differentialoperatorforanadditivelineartransformationwhichobeystheproductrule.Hence,followingbe
validforthecovariantderivativeofaproduct:
(vµbν);α=vµ,αbν+vµbν,α−Γσµαvσbν−Γνσαvµbσ
=vµ;αbν+vµbµ;α.(A.3.15)
Fora2nd-ranktensorAµν,therebe
Aµν;α=Aµν,α−ΓσµαAσν−ΓνσαAµσ.(A.3.16)
Forupper(contravariant)indices,thereis
vµ;ν=vµ,ν+Γβµνvβ,(A.3.17)

(A.3.16)(A.3.17)

A.3.THELOCALGAUGEOFTHELORENTZGROUP

171

andgivenequation(A.3.17),itcandirectlybeproventhat
√vλ;λ=(−√gvµ),λ(A.3.18)
g−isvalid,withgasthedeterminantoftheabsolutevalueofthemetricaltensorgµν.Followingthecalculation,
isthere√(√−g),λvλ≡Γµλµvλ(A.3.19)
g−symbol.felChristoftheforSinceincurvedspaceinversionofthefirstandsecondderivativesofavectordoesnotleadtothesamema-
thematicalobject,ameasureofsuchpermutationlossmaybedefinedbytheRicciidentitiesasagravitational
fieldstrength.Theygiveatensorof4thrankwith
−[Dν,Dα]uµ=(uµ;ν;α−uµ;α;ν)(A.3.20)
=(uµ,ν+Γσµνuσ);α−(uµ,α+Γσµαuσ);ν
=−ΓβµαΓσβα−ΓβµαΓσβν+Γσµα,ν−Γσµν,α
=−Rµσανuσ(A.3.21)
whichisknownastheRiemann(curvature)tensor.Furthertermswhicharedependentonthecoordinates
cancel,givenSchwartzstheoremforusualderivatives.Furthermore,thetraceofRµνλσiscalledRicci
tensorRµν.Thisrank-2tensorisgottenby
Rµνλσgµσ=Rσνλσ
≡Rνλ.(A.3.22)

isthereThus,

−[Dσ,Dα]uσuα=(uσ;σ;α−uσ;α;σ)uα(A.3.23)
=uσ;σ;αuα−uσ;α;σuα
=Rµαuµuα.(A.3.24)
Further,theRicciorRiemannscalaristhengivenby
Rµνgνµ=Rµµ(A.3.25)
.R≡TheJacobiidentitywiththecovariantderivativesleadstotheBianchiidentities
Rµλ[νσ;ρ]=Rµλνσ;ρ+Rµλρν;σ+Rµλσρ;ν(A.3.26)
.0=Thesearethe(homogeneous)Yang–Millsequationsofgravity(cf.equation(A.2.5)fortheonesofelectro-
magnetism,vizMaxwellequations).Multiplyingequation(A.3.26)bygλσandgµρthenleadsto
Rµν−1Rδµν=0,(A.3.27)
2ν;

172

APPENDIXA.GENERALRELATIVITYANDGEOMETRY

whichcanbetakenasadefinitionofatensor
Gµν=Rµν−1R−Λ0δµν,(A.3.28)
2withGµνknownasEinstein(curvature)tensor.Itisdivergence-free,i.e.
Gµν;ν=0,(A.3.29)
anditpossessesaconstanttermΛ0calledcosmologicalconstant.Equation(A.3.28)with(A.3.27)givesthe
geometrical(left-handside)partoftheequationsofmotionofgravitationinGR.Theright-handside,which
istheoneofmatter,ispresentedinChapters6.1andA.4.Itismodel-dependent.

matterandequationsEinsteinA.4WithinSR,relativisticmassincreaseswithvelocityaswellasparticledensitydoes,followingLorentz
densitygyenerHence,transformations.=c2,(A.4.1)
whereasgivematterdensityandcthespeedoflight,possessesaquadraticaldependencetoLorentz
transformations.Hereby,densityreads=mnc2,withmassm,particledensitynandspeedoflightc.
Thus,themetricalenergy–stresstensor
Tµν=vµvν(A.4.2)
withvµ=dxµ/dsreplacesmassandenergydensitywhenusingthecovariantformalism.
WithinGRgravitationalinteraction(understoodascurvingofspacetime)iscausedbyallkindsofmassand
energydensities.Ontheotherhand,BianchiidentitiesasfieldequationsofcurvatureshowtheEinstein
tensorasessentialpropertyofgravitation.Therefore,Einsteinsfieldequations,firstderivedin1915[81],
read

Gµν=−κNTµν.

(A.4.3)

readstensorEinsteintheHere,1Gµν=Rµν−2Rgµν+Λ0gµν,(A.4.4)
withtheRiccitensorRµν,theRicciscalarR,themetricaltensorgµνwithdeterminantgandacosmological
.Λconstant0

κN=8πGN/c4(A.4.5)
isastrength-couplingconstantgottenbycomparisonofthelinearizedfieldequationswithNewtonstheory.
Therefore,itisrelatedtoNewtonsgravitationalconstantGNandthespeedoflightc,whichGRpostulates
constant.as

A.4.EINSTEINEQUATIONSANDMATTER173
TogroundEinsteinsgeneraltheoryofrelativity,thefundamentofthetheoryistheHilbert–Einstein(HE)
1action,SHE=1(R+Λ0)+LM√−gd4x.(A.4.6)
κNTheHamiltonPrincipleoftheLeastActionleadscorrectlytonewtonianaswellastoquantummechanics,
electromagnetismandtocurrentmodelsofelementaryparticlephysics,forinstance(see[22]).Themodel,
ofcourse,groundsontheLagrangian(orLagrangedensityL)chosenundertheintegraloftheactionas
S=bL√−gd4x,
awithaandbasconstantboundaries.
Equation(A.4.6)isthefundamentofusualGR,anditdefinesallgravitationalinteractions.
Furthermore,theenergy–momentumtensorTµνisrelatedtotheLagrangianofmatterLM,whichisthe
fundamentofatheorywhere(especiallyfermionic)matterisdefinedintermsofthewavefunctiongivenby
thestateψ,inaccordancewithquantummechanics.However,itisquantummechanicswhichindeedleads
totheideaoffieldtheoriesinsteadofonlytheoriesforthedynamicsofparticlesystems.Withinquantum
theories,trajectoriesarenolongerdefined.However,theiranalogcanbefoundinthequantummechanical
state,asthesystemblurredinspace.Aseigenvectorofanobservable,thestategivestheprobabilityof
measured.betoqualitiesWithequation(A.4.6),thereisexplicitly,
√δ(R+Λ0)√−gd4x=−κN∂(LM−g)δgαβd4x.(A.4.7)
g∂αβThedifferentialquotientontheright-handsideofequation(A.4.7)istensorial.Withtheansatz
√∂(LM−g)=−√−gTαβ,(A.4.8)
g∂αβthenisthere√Tαβ=−√1−g∂(L∂Mg−g).(A.4.9)
αβSo,theLagrangianofmattergivethemetricalenergy–stresstensorasasourceofthemetricitself.Thereis
δR+Λ0√−gd4x=−κNTαβδgαβ√−gd4x.(A.4.10)
Ontheotherhand,theleft-handsideofequation(A.4.7)maybewrittenas
δ(R+Λ0)√−gd4x=√−gδRd4x+(R+Λ0)δ√−gd4x
=√−gδRd4x+21(R+Λ0)gαβδgαβ√−gd4x.(A.4.11)
isThereδR=δRαβgαβ
=Rαβδgαβ+δRαβgαβ.
1AfterDavidHilbert(1862-1943)andAlbertEinstein(1879-1955,Nobelprize1921).

174

APPENDIXA.GENERALRELATIVITYANDGEOMETRY

δgαβ=−gασgβλδgσλ.

Therein,δgαβ=−gασgβλδgσλ.
isthereTherefore,δR=−Rαβδgαβ+δRαβgαβ.
ThevariationoftheRiccitensorisgivenby
δRαβ=(δΓσσβ);α−(δΓσαβ);σ.
.i.epostulated,metricityBe

gµν;α=0.

(A.4.12)

(A.4.13)

evhaweThenδR=−Rαβδαβ+(gαβδΓσσβ);α−(gαβδΓσαβ);σ.
yields(A.4.11)equation,Consequentlyδ(R+Λ0)√−gd4x=−Rαβ+Λ0gαβδgαβ√−gd4x+
+gαβδΓσσβ;α−gαβδΓσαβ;σ√−gd4x+21Rgαβδgαβ√−gd4x

=−Rαβ−1Rgαβ+Λ0gαβδgαβ√−gd4x.(A.4.14)
2TheintegraltermsoverthevariationoftheChristoffelsymbolsvanishbymeansoftheGauß(inthefollow-
ingGauss)theorem,sinceitisintegratedovertwovectorialdivergences,andboththevariationofthemetric
andtheoneofitsfirstderivativearetovanishattheboundaries.Furthermore,sinceequation(A.4.14)isto
bevalidforgeneralδgαβ,thenthereisequation(A.4.3)with
Rαβ−21Rgαβ+Λ0gαβ=−κNTαβ.(A.4.15)
ThesearetheEinsteinequationsofGeneralRelativity,explicitlyderivedforseekofcompleteness.Λ0is
againthecosmologicalconstant.Foritweuseasubscriptzerotostressitsconstantcharacterandtodiffer
betweenitandafunctionaltermΛ(ξ)≡ΛasintroducedinChapter6.1.Thisconstantwasfirstaddedby
Einsteinin[82]withtheideaofaclosedUniversewhichwouldbestatical.Hence,itactsagainstgravitation.
However,EinsteinsstaticaluniverseisunstableaswasshownbydeSitter[222].Nophysicalormathema-
ticalproperty,though,hasshownsofarwhyΛ0shouldbeexactlyzero.Λ≡0ispreferredaccordingto
simplicitypostulates.Modernstudies,however,leadtosmallbutnonvanishingvaluesofthesame[198].
ThisisrelatedtotheproblemofDarkEnergywhichisdiscussedinChapter2.4

BppendixA

Wavefunctionandelementaryparticles

B.1QMstateandSpin-Magneticinteraction,QMpostulatesandmea-
ementsurWithintheformalderivationofEinsteinsequationsofgravitationinAppendixA.4,thewavefunctionψ
wasintroducedinthecontextoftheLagrangianofmatter.Awaveequationorbettersaidthequantum
mechanicalstateistheonemathematicalobjectinwhichthewholeinformationforameasurementiskept.
Insituationswherethemaximallypossibleamountofinformationisgotten,aclosedquantumsystemis
givenateachtimetbyitsstatevectorψ(postulate1ofquantummechanics).Outsideofmeasuringprocesses
inthenonrelativisticcase,itsdynamicaldevelopment(forparticlesystemswithmassm)isgivenbythe
HamiltonoperatorHˆofthesystem(postulate4ofquantummechanics),withinnonrelativisticlimitesby
equationSchrödingertheofmeans

i∂∂tψ(r,t)=Hˆψ(r,t).(B.1.1)
WithinSR,further,thespecialrelativisticSchrödinger(Klein–Gordon)equation,
−m2c2ψ=0,(B.1.2)
2istaken,orelsetheDiracequation(6.2.7),whichintroducesSpin,formallybythesquare-rootof(B.1.2)).
Underspecialnothighlyrelativisticregimes,forinstance,thePauliequationmaybeused.Ittakesintoac-
counttheinteractionoftheparticlesspinwiththeelectromagneticfieldinthenon-relativisticlimit.Hence,
thePauliequationpresentstheStern–GerlachtermwithaQσ∙ϕB/(2m)addedtotheSchrödingerequa-
tionwhichalreadypossessesjAtermsfromthecovariantderivative.Qistheelectriccharge,σarethePauli
matrices,Bisthemagneticfield-strengthandϕistheDiracvector|ψ>withspinorcomponents.isthe
reducedPlanckactionandmisthesystemsmass.ThespinoperatorisgivenbySˆ=σ/2.
Thequantizationofangularmomentumofspin(Nobel-prizeawarded1943)wasexperimentallyprovenby
O.SternandW.Gerlachin1922:Spinmomentumleadstoaquantizedmagneticmomentofasemiclassical
spinningdipole.Consequently,torqueexertedbyamagneticfieldleadstoprecessionofthedipole.How-
ever,becauseofquantizationofspin,anatombeamunderamagneticfieldsplitsintotwobeamsrelatedto
thetwodifferentorientationsofelectronspin.Further,astatetransitioncanbeinducedsothatoneofthese
beamsvanishesifanalternatingfieldwiththeresonancefrequency(Larmorfrequency)isused.Thisis
theRabiexperiment[204]ofElectronParamagnetic(EPR)orElectronSpinresonance(ESR),Nobelprize

175

176

APPENDIXB.WAVEFUNCTIONANDELEMENTARYPARTICLES

awardedin1946.Further,nuclearspinleadstoanalogous,yetfarweakereffectsfromthecouplingbetween
nuclearmomentaandtheelectromagneticfield.
TheexpansionofRabistechniquebyF.BlochandE.M.Purcell[28,203]forthenuclearspinwasNobel-
prizeawardedin1952.ItleadtotheNuclearMagneticResonance(NMR).Here,theLarmorfrequency
(characteristicoftheisotopeinquestion)isdirectlyproportionaltothestrengthoftheappliedmagnetic
fieldandtotheenergysplittingbetweenbothnuclearspins.Theproportionalityfactoristhegyromagnetic
ratiooftheisotope.Theso-calledRabioscillationsaretheworkingmechanismfor(Nuclear)Magnetic
ResonanceImaging(MRI)(orNMRTomography)presentlyusedinradiologytovisualizedetailedinternal
structureprovidingcontrastbetweenthedifferentsofttissuesofthebody:1Radiofrequenciesareusedto
systematicallyalterthealignmentofthemagnetization.Thiscausesthehydrogennucleitoproducearotat-
ingmagneticfielddetectablebythescanner.Anelectromagneticpulsecausesthenucleitoabsorbenergy
andradiatethisbackoutattheLarmorfrequency.Hence,differentnuclidesmaybecontrastedbymeansof
theirmomentinthemagneticfield.Momentaarequantumnumbersofthequantumstate.

Giventhattrajectorieslosetheirmeaningwithinquantummechanics,anotherapproachfordynamicsis
thepathintegral.Ifaparticlewithamplitude(state)ψ(r1,t1)propagatesfromA(r1,t1)toB(r2,t2),its
dynamicsmayanalogouslybedescribedbyawavefunctionasfollows[76],
ψ(r2,t2)=d3r1K(r2,t2|r1,t1)ψ(r1,t1).(B.1.3)
Itsamplitude(kernel,propagator,cf.Chapter3.3)as
K(B|A)=dCφBA[C],(B.1.4)
withCasthepathtakentogetfromAtoB,andwithφBA[C]asamplitudeofthepath,commonlytakento
be

φBA[C]=eiS[C],(B.1.5)
withtheactionS.Thesuminequation(B.1.4),whichisknownasFeynmanpath,hastobetakenoverall
pathsfromAtoB.Withequation(B.1.5),theFeynmanpropagatoryieldsnaïvely,
K(r2,t2;r1,t1)=[dr(t)]eiS(r(t),r˙(t);t).(B.1.6)
r(t2)=r2
r(t1)=r1
Itsmaincontributionfor→0istheclassicalonewithSclfortheclassicalpathrsincethepathofmin-
imumactiondominates.Furthercontributionsappearfromthedeviationfromthepathandtheexpansion
oftheaction(orLagrangefunction)aroundsuchpath.Interestingly,reverseWickrotationit→tofthe
Feynmanpropagatorleadsdirectlytothepartitionfunctionforanaction(seeChapter4.2).
Sincetrajectoriesarenotdefined(hencethedeviationfromtheclassicalpathandthesumoverallpathsin
Feynmansformulationabove),theequationsofmotionaregivenforstates(ψ),treatingmatterasfields.2
1Withinfunctionalimagingevenhemodynamic(blood)responsesrelatedtoneuralactivityandbloodoxygenlevelsmaybecon-
[184].sc.trasted,2Forinstance,theelectronsofasingleisolatedatomoccupyatomicorbitalswhichformadiscretesetofenergylevels.Inamolecule,
theseorbitalssplitintoanumberofmolecularorbitalsproportionaltothenumberofatoms.Forahighamountofatoms,theseorbitals
form(quasi)continuousenergybands.Incaseof(semi)conductorsandinsulators,energybandseachcorrespondtoalargenumberof
discretequantumstatesoftheelectrons.

B.1.QMSTATEANDSPIN-MAGNETICINTERACTION,QMPOSTULATESANDMEASUREMENT177

Dynamicsisthengivenbysomeofthelatterformulations,andoperationsonthefieldsaregivenbyopera-
torswhichactonthestates.Astatetoaneigenvalueofanoperatoriscalledeigenstate.
Physical,measurablepropertiesaregivenby(hermitean)operatorsAˆwhichpossessreal-valuedeigenva-
lues.Theyarecalledobservables(postulate2).Theireigenvaluesshouldbetheaverageexpectationvalue
formeasurements(postulate3).Themeasurementproblemitself,though(andhenceQMspostulatesthem-
selves),isamatterofresearch,anditisrelatedtotheso-calledcollapseofthewavefunction[125].The
observerisunderstoodaspartofthewholequantummechanicalsystem,andassuchthereisanintrinsic
interactionbetweenhimandtheanalyzedsubsystem.Thisinteractionleadstolossofinformationinform
ofthecollapsetothebasisoftheobservedpropertywitheigenvaluesa,whicharethestatisticalmean
valueofmeasurements.IftheeigenvaluesofAˆandofafurtherobservableBˆpossessdifferenteigenvectors,
themeasurementofbothisnotcommutative(AˆBˆ=BˆAˆ)andachangeintheorderofmeasurementleads
todifferentmeanresults.Thequantummechanicalstate,whichgivesthepropertiesoftheanalyzedsys-
tem,changesafterthefirstmeasurement.Asecondmeasurementrepresentsaninteractionwithadifferent
systemwhereinformationwaslost.Actually,thisisnearlyrelatedwithHeisenbergsuncertaintyrelation,
oftengivenfortheaveragemeasurementofamomentumpˆx(i.e.<pˆx>)andofacoordinatexˆwhicha
particlemaypossessaspropertiesataspecificmoment(<xˆ>).Ascanonicalconjugateoperators,pˆiand
xˆiareorthogonaltoeachotherandthuspossessadifferentbasis.Consequence:itisnotpossibletoknow
boththeexactmomentum(velocity)andplaceofsayanelectronatthesametime,
<Δxi2><Δpj2>≥2δij.(B.1.7)
Furthermore,thisisvalidforthemeasurementofoperatorAˆwithcanonicalconjugateoperatorPAingene-
ral.Moregenerally,though,itisvalidforgeneralobservablesAˆandBˆsothat
√<ΔA2><ΔB2>≥1|<[A,ˆBˆ]>|.(B.1.8)
2Thisisthegeneralformoftheuncertaintyprinciple.Withoutrelevanceoforder,bothobservablescanbe
simultaneouslymeasuredonlyiftheircommutatorvanishes.Thisisthecaseiftheypossessthesamebasis.
Isthisnotthecase,thentheproductoftheaveragedeviationsΔAandΔBwillnotvanish.Thisquantum
mechanicalpropertyisveryusefultogivelengthscalesassociatedtomassesofelementaryparticles.Since
t,however,isnotanoperatorinQM,uncertaintyisnotderivedinsuchaformalmannerasforoperators.
isthereertheless,vNe

ΔtΔE.(B.1.9)
ThequantityΔtistheminimalmeasure-timetodeterminetheenergyofawavepacketofwidthΔEpas-
singthroughadetector.Giventhatmassandenergyarerelatedandpassedtimeanddistancesaswell(by
meansofuncertaintyrelations),massivemediativeparticlescanberelatedtodistancesinwhichinteractions
aremediatedbytheseparticles(i.e.distanceswhichtheseparticlesareabletomovethroughbeforethey
vanishagain).WithadistanceR=cΔtandanenergyΔEgivenbythemassofamediatedparticlewith
ΔE=mc2,thelengthscaleinwhichthisparticleinteractsisacquired.Particleswithanenergyof140
MeV,forinstance,havelengthscalesRoftheorderofmagnitudeofafemtometer.Thisisthecaseforthe
particleswhichH.Yukawapredictedin1939[246](thepions,discoveredin1947[149]),andwhichmay
moveonlydistancessmallerthanthewidthofanucleoninwhichtheyinteractduringtheshorttimebefore
particles.otherindecayingFurthermore,sincearealisticquantumsystemisneverisolated,theinteractionofthestatewithitsenvi-
ronmentisimportant.Therearequantumcorrelationsbetweenthem,andtheseinteractionsmaybeunder-
stoodasasortofmeasurement,againrelatedtoacollapsebutespeciallytotheso-calleddecoherence[97].

178

APPENDIXB.WAVEFUNCTIONANDELEMENTARYPARTICLES

Throughit,superpositionsofthewavefunctions,afundamentalpropertyofquantumphysics(mathemati-
callyfoundedinthelinearityoftheHilbertspace,inwhichquantummechanicalstatesexist),vanish.3
Theideaisthatclassicalmechanicsshouldberecoveredfromquantummechanicsbymeansofthequantum
propertiesthemselves,especiallyforlargesizesandmassesoftheobservedsystem.Quantumproperties
cancelout,leadingtotheclassicalworld.Thedynamicsthatwillexplainthecollapseanddefineacomplete
theoryofmeasurementhasnotyetbeencompletelyexplained,though.Itisrelatedtowhatiscalledthe
problemofdefiniteoutcomesandtheoneofthepreferredbasis.Togethertheyformthemeasurement
problem(cf.[141]),andtheirfutureresearchreliesonquantuminformationtheory.
Researchonquantuminformation,relatedforinstancewithquantumandnon-linearoptics,leadstomany
newandclassicallyunexplainableeffectssuchasentanglement(fromtheso-calledEPRparadoxofEinstein,
PodolskyandRosen[83])andquantumteleportation[16](experimentalfactsinceZeilingersexperiment
in1997[34]!),whichisfundamentaltotheconceptofquantumcomputers[35]andwhichshouldfurther
beexplainedinrelativisticcontexts(whereaprioriinformationofthestatetobeteleportedseemsneces-
sarytolabelidenticalparticlestomakethemeffectivelydistinguishable[148]).However,neitherusual
(Schrödingers-nonrelativistic-norevenDiracs-special-relativisticforparticleswithspin)quantumme-
chanicsalonenorGeneralRelativitycandescribethenatureofmatteritself.Thisisratherfulfilledwithin
thecontextof(special-relativisticaswellasquantumtheoretic)elementaryparticleandhighenergyphysics.
Thelatterevolvedoutofnuclearphysicswiththedesiretodiscoverthefoundationsofmatteranditsfun-
damentaldynamics.Hence,thequantummechanicalstatemayberelatedtoisospin,andtheHeisenbergs
uncertaintyrelationtakesafundamentalroleforinterpretationsoflengthscalesofinteractionsandmasses
dynamics.fundamentalinparticlesof

B.2OntheYang–Millstheory
TheYang–Millstheoryisanon-abelian(non-commutative)theorywithSU(N)transformations(i.e.uni-
tarymatrix-valuedtransformationsforNdimensionsanddeterminant+1forthetransformationoperator
ormatrix)andthuswithself-interactionsthatgeneralizetheMaxwellequationsof(abelianU(1)-)elec-
trodynamics(wherephotonsasgaugebosons-mediators-donotself-interact)totheso-calledandanalog
Yang–Millsequations.WiththeRicciidentities,
Fµνab≡ig1[Dµa,Dνcb],(B.2.1)
thereisthefield-strengthtensorFµνafortheisospincomponentaoftheisospinvectorψaandwiththe
formgeneralFµνi=(Aνi,µ−Aµi,ν)−gAµkAνlfkli,(B.2.2)
followingthecovariantderivativedefinedas
Dµab=δab∂µ+igAµab,(B.2.3)
withthegaugefieldAµawhichisrelatedtogaugebosons4andisanaloguetothepotentialAµentailing
scalar(ϕ)andvectorpotential(A)inSpecialRelativityandelectromagnetism.
3Awell-knownexampleofsuperpositionsinquantummechanicsisSchrödingerscat.Itisdescribedasaliveanddeathatthesame
timeuntilmeasurement(observation)decidesitsclassicalstatus.
4WeinberWgithinmixture,electroweakrelateditothenteractions,vanishingthemassmassofeigenstatesphotons.ofgFurtherauge,fieldsthearecurrentgottenofthroughmeasurableagfurtheraugefieldsorthogonalappearsfromtransformationsuperpositioncalled

B.2.ONTHEYANG–MILLSTHEORY

179

(B.2.7)(B.2.8)

Thegaugefieldappearsbymeansoflocalgaugetransformationofthegaugeparametersλiofthetransfor-
groupmationSU(N)≡Uab=eiλiτiab.(B.2.4)
Thegaugeparametersλiarespacetimedependentunderlocalgaugetransformations.Hence,sincefield
equationsaretoremaininvariant,theyleadtothenecessityofreplacingusualbycovariantderivatives.
Further,τiarehermiteantransformationmatriceswithi=1,...N2−1withNisotopicfermioniccompo-
nentsψaundertheinteractionsgivenbytheforcewhichisreducedtogaugepotentialsAµ.
AccordingtotheBianchiidentitiesforSU(N),homogeneousYang–Millssystemsaregivenby
∂[λFµν]i−gAk[λFµν]jfkji=0,(B.2.5)
withBachparenthesisa[ibk]=21(aibk−akbi)andstructureconstantsfkjiofthegaugegroup.Further-
fieldsadjointwithmore,Fµν=Fµνiτi,withFµνab=Fµνiτiab,(B.2.6)
definitionfurtherawithDλFµν≡[Dλ,Fµν]=Fµν,λ+ig[Aλ,Fµν](B.2.7)
andwithadjointlyrepresentedgaugefields
(Aµ)ab=Aµi(τi)ab(B.2.8)
withFµν=Aν,µ−Aµ,ν+ig[Aµ,Aν]=F†µν,(B.2.9)
thereisamatrix-valuedcovariantderivative
Dµ=1∂µ+igAµ(B.2.10)
sothatthehomogeneousYang–Millssystemmaybewrittenasfollows,
DλFµν+DµFνλ+DνFλµ=0.(B.2.11)
ThisaformanaloguetothehomogeneousMaxwellsystemofelectrodynamics.
Euler–Lagrangeequationsforeachisospincomponentψayieldthefollowingsetofequations
∂L∂L
∂ψaA,µ,µ−∂ψaA=0,
∂L∂L
∂ψ¯aA,µ,µ−∂ψ¯aA=0,(B.2.12)
ForgeneralLagrangians,thereisthecanonicalenergy–stresstensorwhichisdescribedwithinYang–Mills
theoriesformasslessfermionsasfollows,
∂∂LLTµν=∂ψaA,νψaA,µ+∂ψ¯aAνψ¯aA,µ−Lδµν.(B.2.13)
ofthedifferentiso-componentsofthegaugefieldsintheformWµ±=21(Wµ1iWµ2)(forAµi=Wµiofweakons),and
+suchanalogouslyasνL→foreLgluonsplusaofmassstrongtermofinteractions.thegaugeThefieldsfield-strengththemselvestensor(seeof[74]).Wbosons,forinstance,isthenrelatedtodecaychannels

180

APPENDIXB.WAVEFUNCTIONANDELEMENTARYPARTICLES

Analogouslytoelectrodynamics,invarianceoftheLagrangianunderglobalSU(N)transformationsinisospin
spaceleadstoconservationof4-currentdensities,
jµi=1∂LτiabψbA−¯∂aALψ¯bAτiba,(B.2.14)
i∂ψaA,µ∂ψ,µ
Masslessfermionicmultiplets(i.e.withspin)aredescribedbyaDiracLagrangianoffollowingform,
L=iψ¯aAγµABψaB,µ−iψ¯aA,µγµABψaB,(B.2.15)
22withDiracmatricesγµfollowingtheCliffordalgebra
γµγν+γνγµ=2ηµν1,(B.2.16)
withtheMinkowskimetricηµνofSpecialRelativity.Hence,Euler–Lagrangeequationsleadtoso-called
Diracequationsofmasslessmultipletswiththefollowingform,
iψ¯aA,µγµAB=0,(B.2.17)
iγµABψaB,µ=0.(B.2.18)
Thecanonicalenergy–stresstensorreadsnowexplicitly,
Tµν=2iψ¯aAγνABψaB,µ−2iψ¯aA,µγνABψaB.(B.2.19)
Furthermore,thereisthecovariant4-currentdensitywhichreadsexplicitly
jµi=ψ¯aAγµABτiabψbB.(B.2.20)
TheconsequencesofgaugeontheLagrangianmaybewrittenastheadditionofanadditionalinteraction
term

Lint=gjµiAµi(B.2.21)
whichrelatestoageneralformofthejAcouplingofelectrodynamics.Thegaugepotentialleadstodyna-
micsofnon-freesystemsasrelatedtoappearingforcesanalogoustonewtonianforcesorelectromagnetic
strength.Analogouslytoelectrodynamics,itgivesageneralized(invariant)Lorentzforcedensitybymeans
of

Fµνijµi=kν.(B.2.22)
WithfurtherLagrangiantermsfromthefield-strengthtensoritself,Euler–Lagrangeequationsfurtherlead,
analogouslytoelectrodynamics,toinhomogeneousYang–Millsequations,
DνFµνi=−4πcgjµi(ψa),(B.2.23)
readformulationadjointinwhichDνFµν=−4πcgJµ(ψa),(B.2.24)
withcurrentconservationgivenby

DµJµ(ψa)=0,

(B.2.24)

(B.2.25)

B.3.ELECTROWEAKDOUBLETOFTHESM

wingfollo

181

DµDνFµν=0.(B.2.26)
ForN=1,theYang–Millstheoryindeedreducestoelectromagnetism,andfield-strengthFµν,relatedto
EandBfields(seeChapter4.2),isgivenbyderivativesoftheisoscalargaugepotentialAµ(itselfrelated
tothescalar(ϕ)andthevector(A)potentials).ForN>1,thereareself-interactionsofthegaugefields
themselves,entailingthatthegaugeparticlesrelatedtothegaugefieldAµself-interact.ThereareN2−1
gaugebosonsforatransformationgroupSU(N),whereasgaugebosonsmediatesomeforcesrelatedtothe
potentials.Withinelementary-particlephysics,thesemediatedforcesarecrucial.Theformalismabove,
however,isonlyvalidformasslessfermionsandbosonsrelatedtogaugepotentials.Simpleadditionof
masstermstotheLagrangianisnotpossibletakingintoaccountparityviolationsinweakinteractions.Par-
ityviolationwasfirstproposedbyLeeandYangin1956[150]anditrepresentsindeedanexperimentalfact
sinceWusworksof1959[244].Furthermore,suchtermsassimplyaddedmassesleadtosingularities.A
per-hand-massiveYang–Millstheoryisnotrenormalizable.Toachieveaphysicaltheory,itseemsnecessary
tointroducescalarfieldsandtheconceptofsymmetrybreakingsothatmassesappearinanindirectwayby
meansofnewparameters(seeChapter3).

B.3ElectroweakdoubletoftheSM
Theisodoubletofelectroweakinteractionsreadsinitsgeneralformasfollows,
flψψL/Rmf=ψqf,
L/RItmaybedecomposedinthefollowingway(seeChapter2.2):
fields:Leptonic•Form=1,i.e.forthewavefunctionofleptons,theindexftakesintoaccountthethreefamilies
(orgeneratons)(f=1...3)ofelectron-likeparticles.Hence,forleft-andright-handedstates,ef=1
representstheelectron,whileforf=2,theisospincomponentrepresentsmuonsµ.Forf=3,
finally,therepresentedleptonisthetauonτ.
Givenparity-symmetrybreaking,thesecondisospincomponentform=lisnon-existentforright-
handedparticles.Thestateisrepresentedbyisoscalarswhichtransformunderthetransformation
groupU(1).Left-handedstatesrepresentthethreegenerationsofneutrinosrelatedtoeachelectron-
likeparticle,i.e.νe,νµandντ(itisassumedthattheyexistonlyleft-handed!).Further,parity
non-conservationexistsinformofCPbreakingwithCforconjugation.Hence,forantiparticles,
left-handedstatesareisoscalar,withpositronseL+forf=1,anti-muonsµL+forf=2andanti-
tauonsτL+forf=3asisoscalarcomponents.Fortheright-handedisodoublet,ontheotherhand,
antielectron-particles(eR+,µR+andτR+)arerepresentedbyonecomponentofthestatewhilethere-
latedantineutrinos(right-handed)arerepresentedbytheother.Left-handedantineutrinostatesvanish.

fields:Quark•Thequarkdoubletforbothright-andleft-handedstatespossessesup-(a=1)anddown-type(a=2)

182

APPENDIXB.WAVEFUNCTIONANDELEMENTARYPARTICLES

quarksasisospincomponents,allofthemaspartofatriplet,giventhethreequarkfamiliesorgenera-
tionsf=fq.5Eachiso-pair(andanti-pair)ofthegenerationisgivenbyapairanalogtotheproton–
neutronpairasisovectorinnuclear-physicsmodels,howeverherewithfractionalelectriccharge.A
component(up-type)withchargeQ=+(2/3)eandanother(down-type)withQ=−(1/3)e.Fur-
ther,thedifferencebetweengenerationsconsists,asforleptons,intheirmasses.Higher-generation
quarkspossesshighermasses.Thefirstgeneration(upuanddownd)consistsoftheleastmassive
quarkswithmup/mdown≈0.56andmupofabout2MeV/c2.Thesecondoneconsistsofthedoublet
ofcharmcandstrangesquarks.Thethirdandmostmassivegenerationfinallyconsistsoftoptand
bottombquarks.Topquarks(provenexperimentallyonlyuntil1995atFermilab[46])possessamass
ofca.171GeV/c2:about1000timesmoremassivethan(composite)pionsandwithalmosttwicethe
ons!weakofmassForbothleft-andright-handedstates,thequarkmultipletreadsasfollows,
ψL/Rqf=uf,
dfL/R

u1=up,u2=charm(c),u3=top(t)

(B.3.1)

withu1=up,u2=charm(c),u3=top(t)(B.3.1)
andd1=down,d2=strange(s),d3=bottom(b).(B.3.2)
Baryonsasparticleswithabaryonicnumber(asquantumnumber)1or-1arecompositeparticlesoutof
twokindsofpartons:quarks(actually3)andgluons.However,inageneralsense,quarksmayberegarded
asbaryonswiththebaryonicnumberof1/3and-1/3forantiquarks,suchthathadronsingeneralpossess
abaryonicnumberof1,0or-1,whereasmesons,compositeparticlesfromquarks,antiquarksandgluons,
possessthevanishingbaryonicnumber.Forhadrons,thebaryonicnumberisconserved,howevernotfor
eachfamily.Especiallyhigh-familybaryonicparticlesmaydecayinlessmassive(lower-family)particles
aslongasthetotalamountofbaryonsisconserved.Hence,protons,whicharetheleastmassivebaryonic
particles,donotdecaywithintheSM.
Giventhatbaryonicnumberisnotconservedforeachflavor,thereexistsamixturebetweenthem.Ontheone
hand,herearethemasseigenstatesd,sandb.Ontheotherhand,therearetheflavorstatesd,sandb.The
flavorstateisconsequenceoftheGlashow–Iliapoulos–Maiami(GIM)mechanism(ageneralizedCabibbo
transformation),whichdescribesaunitarytransformationU(3)betweenthedifferentdownquarks.The
transformationmatrixistheKobayashi–Maskawamatrix,alsoknownasCabibbomatrix.6Thetransformed
stateistheonewhichhastoappearwithinthespinor.
Underweak-interactionprocesses,theleptonicnumber(asquantumnumber)isconservedforeachfamily
indeed(see[157],pg.240),howeveronlyaslongasneutrinomassisneglected.Suchamasswouldlead
toneutrinooscillations[77].Neutrinomixing,however,leadstoneutrinoflipswhichbreakfamily-wise
leptonconservation.Suchcan,however,onlyoccurifneutrinoshaveafinitemass(see[157]pg.294).Then,
ontheonehand,thereexistflavorstatesofneutrinos(νi),andontheother,thereexistmasseigenstatesof
thesame(νi),analogouslytoquarkswithintheGIMmechanism.
5AccordingtotheworkofM.KobayashiandT.Maskawa,Nobel-prizeawarded2008togetherwithY.Nambu,therehavetobeat
leastthreequarkfamiliesinnature.
62008.prizeNobelSee

B.4.STRONGTRIPLETOFTHESM

B.4StrongtripletoftheSM

183

Withinquantummechanicalinteractions,neutrinoscoupleonlyweaklywhileelectron-likeparticlescouple
electroweakly,i.e.weaklyandelectromagnetically.Quarks,ontheotherhand,coupleelectroweaklybut
alsowithinstronginteractions.Hence,quarkshavetoappearasanisovectorwithinstrong-interaction
transformationsalso.TheisospinvectorofSU(3)Creads

frafψaf=gaf,
bawiththesubscriptacountingthecolorcharge(a=1,2,3).
Unlikehadrons,quarksdonotseemtopossessaninnerstructure,andespeciallytheSMassumesnone.
Whatmakesthemdifferfromeachotherissimplycalledflavorf,ofwhichthereare6,togetherwith6anti-
flavors(dividedontothe3generationsfoundinelectroweakstates).Hence,thereis,forinstance,r23=db
forthedown-quarkwithbluecolorcharge.
Theflavor,whichbethestrong-interactionfamilyofquarks,countsfromonethroughsixforup,down,
charm,strange,topandbottom(andanalogouslyforantiquarks).Thistakesintoaccounttheappearance
ofhigh-generationquarkswhichwashistoricallydefinedbynew(flavor)quantumnumberswithinnuclear
physics.ThesenumbersarecalledcharmnC,strangenessSandsoon(andtheycomefromtimesbefore
quarktheory).Theyarerelatedtonewtypesofquarkswhicharemoremassivethanupanddownquarks
(hencethestrongrelationtoflavor).Thesenumbersareconservedduringstrongandelectromagneticpro-
cessesbutnotduringweak-interactionones.Consequently,thelightestparticlescontainingastrangequark,
forinstance,cannotdecaybystrong-interactionprocesses,andmustdecayviathemuchslowerweakinter-
[84].action

systemSMTheB.5

Thefermionic(Dirac)Lagrangedensityforelectroweakinteractionsreads(withoutsymmetry-breaking
terms)

iiL=2ψ¯LmfAγµABψLmfB;µ−2ψ¯L;mfµAγµABψLmfB+
,µ,A,Bm,f+iψ¯RmfAγµABψRmfB;µ−iψ¯RAmf,µγµABψRmfB−(B.5.1)
22−16π1cF(2)iµλF(2)iµλ+F(1)µλF(1)µλ,
whereingiarethecouplingconstantsofeachinteractionandγµaretheDiracmatricesfulfillingtheClifford
algebra,whileabarmeansDirac-conjugation(a¯=a†γ0).Diracconjugationrespondstonon-hermitean
propertiesofDiracmatrices.F(j)kµνstaysfortheenergy–stresstensorsforeachsymmetrygroup,anditis

184

APPENDIXB.WAVEFUNCTIONANDELEMENTARYPARTICLES

closelyrelatedtothecovariantderivativewhichreadsasfollows,
DµψlLf=∂µ+2ig1Bµ+ig2WµiτiψlLfforleft-handedleptons,(B.5.2)
iDµψLqf=∂µ−6g1Bµ+ig2WµiτiψLgfforleft-handedquarks,(B.5.3)
DµψlRf=(∂µ+ig1Bµ)ψlRfforright-handedleptons,(B.5.4)
DµufR=∂µ−32ig1BµufRforright-handedup-quarks,(B.5.5)
1DµdfR=∂µ+3ig1BµdfRforright-handeddown-quarks.(B.5.6)
Further,Euler–Lagrangeequationsgiveforelectroweakinteractionsthehomogeneous(Dirac)equationsfor
multiplets,

iγµDµψLmf=0,h.c.(B.5.7)
iγµDµψRmf=0,h.c.(B.5.8)
andtotheinhomogeneousYang–MillsequationsoftheGlashow–Salam–Weinbergmodel,
F(2)µλi,λ+g2εijkF(2)µλjµλWµk=4πj(2)iλ,(B.5.9)
F(1)µλ,µ=4πj(1)λ.(B.5.10)
Herewith,Wµkareweakgaugepotentialsrelatedtoweakonsasphysicalparticles.
Whilefield-strengthtensorsaregivenbycovariantderivativesandthusbygaugepotentials(Ricciidentities),
F(2)µν=F(2)µνiτi≡[Dµ(2),Dν(2)],F(2)µνi=Wνi,µ−Wµiν−g2εijkWµjWνk,(B.5.11)
F(1)µν=F(1)µνY≡−1[Dµ(1),Dν(1)],Bν,µ−Bν,µ,(B.5.12)
2/ig1weakandhyperchargecurrentdensitiesread
1jλ(2)i=g2ψ¯LγλτiψL,jλ(1)=g12ψ¯LγλψL+ψ¯RγλψR.(B.5.13)
τi=21σiisvalidforthegeneratorsofSU(2).Theycommutewithhypercharges,giventhatYisproportional
totheunitmatrix,andPaulimatricesaretrace-free.
Hypercharges(Yx)arethegeneratorsoftransformationsofU(1),andtheireigenvaluesdifferforleft-and
right-handedparticlesandfordifferentisospin.ForSU(2)R,theeigenvaluesofthegeneratorvanish.For
SU(2)LthegeneratorsaremultiplesofPaulimatrices.Thethirdoneofthem,σ3=diag(1,−1),isrelated
totheso-calledisospintensorτ3.
ElectricchargeasaneigenvalueofanoperatorQ,withisospinoperatorT3andhyperchargeoperatorY,is
givenbytheGell-Mann–Nishijimaequation,
Q=T3+1Y,(B.5.14)
2whichcanfurtherbegeneralizedforcharmness,strangenessandsoon.Itfinallyrelatesbothhypercharge
andisospinvaluesYandT3withtheelectricchargeQ(withthedimensione).Itthenleadstoelectrodyna-
mics,whichareelsemixedwithweakinteractionsinU(1)Y.

185SYSTEMSMTHEB.5.Ontheonehand,withinelectrodynamics,photonsaremediatedinelectromagneticinteractions.Photons
aremasslessand,thus,electromagnetismisrelatedtolong-rangeinteractions.Ontheotherhand,given
N=2forelectroweakinteractions,there±are0threegaugebosonsexpectedasintermediateparticlesofweak
interactions.ThesearetheweakonsWandZ,allrelatedtophotonicstatesbyreasonsofdiagonalization
ofthemassmatrixafterbreakingthesymmetryofthetheoryandwithallthreeofthemactingonlyforleft-
handedstates(fromwhichphysicalphotonstatesarederivedalso),sinceweakprocessesareleft-handed.
Further,sincethereappearself-interactionsgivenby[Aµ,Aν]=0ofthegaugepotentials,symmetrybrea-
king(cf.Chapter3.3)leadstoanon±vanishingmassoftheweakons,2which0wereexperimentallydiscovered
in1983[10].Actually,massofWbosonsliesaround80GeV/cwhileZpossessesamassofabout91
GeV/c2.Weakinteractionsareshort-rangedanddominateonlywithinnuclearranges.
Forstronginteractions,theLagrangedensityreads
L=2iψ¯faγµψaf;µ−2iψ¯fa;µγµψaf−16π1cFµνiFµνi−ψ¯famfψaf,(B.5.15)
,µa,fwhereinacountstheisospin(color)andfcountsflavor.
Thecovariantderivativereads
Dµ=∂µ+ig3Gµiτi,(B.5.16)
andthefield-strengthtensorisgivenby
1Fµν=ig3[Dµ,Dν](B.5.17)
=Fµνiτi,Fµνi=Gνi,µ−Gµi,ν−g3fijkGµjGνk.(B.5.18)
τi(ij=k1,...8)arenow3×3matricescalledGell-Mannmatrices.TheyarethegeneratorsofSU(3)C,
andfiarestructureconstantsdescribingtheiralgebra.Further,theinhomogeneousYang–Millsequations
[74]readFµνi,µ+g3fijkFjµνGµk=4πjνi,i=1,...8,(B.5.19)
andthe(Dirac)fieldequationsforquarksread(withoutEinsteinconvention)
iγµDµψaf−mfψaf=0,h.c.(B.5.20)
GaugefieldsGµarerelatedtogluonsasgaugebosons.So-calledcolor-currents(eighttypesofthemexist)
betweensame-coloredanddifferently-coloredquarkstatesaregivenfollowingthescheme,
jµ1=21g3(g¯fγµrf+r¯fγµgf),(B.5.21)
1jµ2=2g3(g¯fγµrf−r¯fγµgf),(B.5.22)
jµ3=1g3(r¯fγµrf−g¯fγµrf),(B.5.23)
2..(B.5.24).jµ8=√1g3(r¯fγµrf+g¯fγµgf−2b¯fγµbf),(B.5.25)
32Givengaugeinvariance,theeightcolorcurrentsofgluonsarecovariantlyconserved.Furthermore,while
quarksarecoloredi.e.possessacolorcharge,theresultingsuperpositionofallfreeparticlesinnature

186

APPENDIXB.WAVEFUNCTIONANDELEMENTARYPARTICLES

isassumedtobecolorless,whichinthiscontextmeanswhite-charged,followingtheanalogytoco-

lortheory.Thisisrelatedwiththeproblemofconfinementandasymptoticfreedom:quarksmovefreely

withinhadronicrangesbutcannotbedetectedasfreeparticlessincestrong-interaction(color)forcesshould

augmentwithdistance.Thepredictionoftheinteractionbetweenthecolor-mediatinggluonsandquarksin

hadrons,firstdiscoveredintheearly1970s,leadtothe

Withinahadron(femtometerscale),however,

here

treated

within

Dual

QCD

in

Chapter

4.

quarks

NobelPrizeforGross,WilczekandPolitzerin2004.

ouldw

evmo

.freely

The

problem

of

confinement

is

B.6.

B.6

SCHEMATICPROPERTIESOFFERMIONS,BOSONSANDTHEIRINTERACTIONS

Schematic

Figure

B.1:

opertiespr

Schematics

on

of

the

fermions,

properties

of

bosons

fermions,

and

bosons

and

their

their

187

interactions

interactions

188

APPENDIX

B.

AWVE

FUNCTION

AND

YARELEMENT

TICLESARP

ACppendix

Cosmology

C.1Sphericalsymmetryandtheidealliquid
Sphericalsymmetryorcentralsymmetryisgivenbythefollowinglineelement,
ds2=eν(cdt)2−eλdr2−r2dΩ2,(C.1.1)
withthe4-vectorxµ={x0=ct,x1=r,x2=ϑ,x3=ϕ}.Themetriccomponentsνandλarefunctionsof
therandtcoordinatesonlyanddΩ2=dϑ2+sin2ϑdϕ2isthemetricofa2-dimunitsphere.Further-
more,letustakenowc==1intheequations.
Foridealliquids,theenergy–momentumtensorreads
Tµν=(+p)uµuν−pgµν,uµuµ=1,(C.1.2)
witha4-velocityuµ,pressurepandenergy-densitydistribution.
Withuµ=(u0,u1,0,0)andu1:=u0vc1(velocityv1),thereisforsphericalsymmetry,
u02=e−ν−v1e−λ.(C.1.3)
2−1
cThenonvanishingcomponentsofTµνyield
T00=u02+1peν−λ,
v2
cvT01=u02(+p)1,
T11=u02v1+peλ−ν,
2c
cT22=pr2,
T33=pr2sin2ϑ,

thewithtrace

T=−3p(C.1.4)
Incaseofbarotropicmatter,pressurewillbegivenbyp=w,whereaswiscalledequation-of-statepara-
meter,givenbytheratioofpressure(p)totheenergydensity(),takenasaconstantwhichisindependent
189

190

COSMOLOGYC.APPENDIX

(C.1.5)

time.ofIncurvaturecoordinatesforcentralsymmetry,theChristoffelsymbol,
∂√Γσµσ=∂xσlog−g,(C.1.5)
asconnectionoftheform,hasthefollowingnonvanishingcomponents:
Γ000=1g00,0g00=ν˙,Γ001=1g00,1g00=1ν,Γ011=−1g11,0g00=1˙λeλ−ν,
222222
˙˙Γ022=−1g22,0g00=λr2eλ−ν,Γ033=−1g33,0g00=λr2sin2ϑeλ−ν,
2222Γ100=1g00,1g11=1νeν−λ,Γ110=1g11,0g11=λ˙,Γ111=1g11,1g11=λ,
222222
Γ122=−21g22,1g11=−r−λ2r2,Γ133=−21g33,1g11=−rsin2ϑe−λ,
˙Γ202=1g22,0g22=λ,Γ212=1g22,1g22=1,Γ233=−1g33,2g22=−sinϑcosϑ,
2r222Γ313=1g33,1g33=1,Γ323=1g33,2g33=cotϑ,Γ330=1g33,0g33=1˙λ.
222r2TheRiemanntensorisconstructedasfollows,
Rτµνσ=ΓαµνΓτασ−ΓαµσΓτασ+Γτµν,σ−Γτµσ,ν,(C.1.6)
tracethewithRµν=Rσµνσ=ΓαµνΓσασ−ΓαµσΓσασ+Γσµν,σ−Γσµσ,ν.(C.1.7)
ThenonvanishingcomponentsoftheRiccitensorRµνreadexactly(includingtime-dependence),
R=−eν−λν+ν2−νλ+ν+1λ¨+1λ˙2−1λ˙ν˙,(C.1.8)
00244rc22c24c24
R10=1λ˙,(C.1.9)
rcR11=−12eλ−νλ¨+λ˙−λ˙ν˙+ν+ν−νλ−λ,(C.1.10)
22
c224244r
R22=e−λ1+r(ν−λ)−1,(C.1.11)
2R33=sin2ϑR22.(C.1.12)

C.2Scalar-fieldequationwithcentralsymmetry
Aftersymmetrybreaking,thescalar-fieldequation(6.3.21)readsasfollows(seeChapter5),
ξ,µ;µ+ξ2=14π∙8πG0qˆTˆ+4Λ0.
l1+3α˘33

C.2.SCALAR-FIELDEQUATIONWITHCENTRALSYMMETRY191
Explicitly,forthegeometricalpartwehaveforsphericalsymmetrythefollowingcomponentswithChris-
toffelsymbolsΓνµλ:
ξ,1;1=(ξ,1g11);1=(−ξ,1e−λ);1
=(−ξe−λ),1+Γ111ξ,1+Γ110ξ,1
=(−ξe−λ),1+Γ111(−ξe−λ)+Γ110ξ˙e−ν
˙=−ξeλ+λξe−λ+λ(−ξe−λ)+λξ˙e−ν
22˙˙=−e−λξ+λξe−λ+λξe−ν,
2c22ξ,0;0=(ξ,0g00);0=(ξ,0e−ν);0
=(ξ˙e−ν),0+Γ001(−ξe−λ)+Γ000ξ˙e−ν
=ξ¨e−ν−νξe−λ−ξ˙ν˙,
c22c22
1ξ,2;2=Γ221ξ,1=−ξe−λ,
rand1ξ,3;3=Γ330ξ,1=−eλξ.(C.2.1)
rHence,forvacuum,thereisforequation(6.3.21),
ξξ,µ,µξ;µ+l2≡ξ;µ+l2
=−ξ−(λ−ν)ξ+2ξe−λ+1ξ−1(ν˙−λ)ξ˙−1ξ¨e−ν=0.(C.2.2)
˙
2rl2c22c2
Letustakeequation(6.3.21).Furtherassumeξ=ξ¯(r)∙h(t)andwriteh(t)≡handξ¯(r)≡ξ¯.Now,inthe
linearcase,wemayrewriteitas
h¨c2¯c28πG0
h=ξ¯Δξ−l2+qˆ3ξ¯c2(−3p).(C.2.3)
Letusheretakeqˆ=0.Wehave
2ch¨+2−λh=0(C.2.4)
land2c2Δξ¯−c+λξ¯=0,(C.2.5)
2lwiththeeigenvalueλ.Further,wemaydefinetheeigenfrequency
2ω2=c−λ(C.2.6)
2l

192

C.APPENDIXCOSMOLOGY

andhencewritethetime-dependentsolutionas
h(t)=a1cos[ω(t−t0)]+a2sin[ω(t−t0)](forqˆ=0).
Thereish(t0)=a1.
Fortheradialequation(C.2.5),thereis
1∂2∂12
Δ=r2∂rr∂r−r2lˆ.
Here,lˆisamomentumoperatorwhichisgivenby
21∂21∂∂
lˆ=−sin2ϑ∂ϕ2+sin2ϑ∂ϑsinϑ∂ϑ.
AnalogouslytothewaveequationsinnonrelativisticQM,thescalarfieldmaybesolvedby
ξ¯(r)=ξ0f(r)Yl¯m(ϑ,ϕ).
Yl¯m(ϑϕ)≡Yl¯misthesphericalharmonicswiththefollowingeigenvalueproblem,
lˆ2Yl¯m=l¯(l¯+1)Yl¯m.
defineweIf

(C.2.7)

(C.2.8)(C.2.9)(C.2.10)(C.2.11)

(C.2.12)(C.2.13)

2(C.2.12)λ,=Eequation(C.2.5)maybewrittenontoaSturm–Liouvilleequationasfollowsforf≡f(r),
22−(r2f)+E2rf−l¯(l¯+1)f=0.(C.2.13)
cThisisamodifiedBesseldifferentialequation.
ForE=0,(C.2.13)issolvedby
1iEriEr
f=√rC1J21+l¯−c+C2Y21+l¯−c,(C.2.14)
whereasC1andC2areintegrationconstantsandJn(z)istheBesselfunctionof1stkind.Yn(z)isthe
kind.secondoffunctionBesselForzreal-valued,thereisE2<0,andthusλl¯2<−c2.E2<0implies
E:=iE˜withE˜∈R.(C.2.15)
Thisisrequiredforthesolutionsnottobeexponentiallyincreasingforr→∞.Inotherwords,thespectrum
{λ}istobedetermined,aswithinQM,throughasymptoticalboundaryconditionsforr→∞.
Forl¯=0(monopole),forinstance,wehave
√J21−iEr=2sin(−iEr/c),(C.2.16)
c−iEcrπ
√Y1−iEr=−2cos(−iEr/c).(C.2.17)
2c−iEcrπ
Thisgivesamonopoleradialsolutionofthescalarfieldfortheassumptionofseparability.

ARAMETERSPRN-LIKEC.3.

193

parameterseRN-likC.3LetuswritedowntheparametersofChapter7.5forc=1.Theseries-expansion(7.5.1)
∞g˜=Cnn=1C1+C2+C23+C34+...
n=1rrrrr
(7.5.2)constantsthehasC1=1
C2=2A+B
C3=(2A+B)2+4AB.
C4=(2A+B)3+23AB(2A+B)
C5=(2A+B)4+2924AB(2A+B)2+3(32AB)2
whicharemultiplicativefactorsorAandB(seeChapter7.5).
Neartor=2M˜GNandr=|Q˜2|,higher-ordercorrectionsofr−narenecessaryforananalysisofthe
behaviorofthemetriccomponents.Furthermore,theymaybeusedforindicationsaboutexactbehavior.
Uptothe10thorderofapproximation,therearefurtherconstantterms
2C6=(2A+B)5−37AB(2A+B)3+2(AB)(2A+B)
52032C7=(2A+B)6−103AB(2A+B)4+751(AB)(2A+B)2−5(AB)
12872040C8=(2A+B)7−118AB(2A+B)5+676(AB)2(2A+B)3−8(AB)3(2A+B)
3531535C9=(2A+B)9−2369AB(2A+B)6+17151(AB)2(2A+B)4−
4480560−6959(AB)3(2A+B)2+35(AB)4
20488960C10=(2A+B)10−2593AB(2A+B)7+1787(AB)2(2A+B)5−
288504−4549(AB)3(2A+B)3+8(AB)2(2A+B)
632268Forhigher-ordercorrectionsn≥3whicharerelatedtoX(A,B;r−n)ofequation(7.5.4),mass–charge,
andcharge–chargecouplingsappear.Thereisuptothe9thorder,
X(A,B;r−n)=1+4(2A+B)+29(2A+B)2+3AB+37(2A+B)3+
2r23r36r48r45r5
23+85rAB5(2A+B)+103(210Ar6+B)+75190r6AB(2A+B)2+5(32ABr6)+(C.3.1)
5+472(2A7+B)+27047AB(2A+B)3+32AB7(2A+B)+
35r6315r35r2
+2369(2A8+B)+171518AB(2A+B)4+6959(AB8)(2A+B)2+
2803r2240r74480r
+35(AB8)+2593(2A9+B)+17879AB(2A+B)5+
884r252r144r
+4549(AB9)2(2A+B)3+16(AB9)3(2A+B).
r63r1134

194

COSMOLOGYC.APPENDIX

FurtherXitermsaredefinedexactlyasbeforesincetheyarealldefinedintermsofX.
IntermsofM˜andQ˜2,upto4thorder,thereappearfurtherterms8M˜GN/(3r3)+29(M˜GN)2/(3r4)+
3Q˜2/(8r4)ofX(A,B;r−n)whichleadtothefollowingcorrectiontermsforeλ,

Q˜2M˜G2Q˜238(M˜GN)2−Q˜2
Nr3+r4,
andforeν(undertheparenthesis),

7Q˜2M˜GN317Q˜2(M˜GN)2−9Q˜4
3r3+r4.

(C.3.2)

(C.3.3)

C.4.STANDARDFRIEDMANNCOSMOLOGY

195

cosmologyFriedmannStandardC.4constant:cosmologicalanishingV•Letusshortlydiscussthesimplelimitingcaseofvanishingscalar-fieldexcitations(ξ0=0)and
derivativesofthesame,withenergydensity=0andscalefactora=a0.Then,forw=0wehave
from(8.6.1)anEinstein–deSitterUniversewith
a˙−8πG0a=H021−0=H02(1−Ω0)=−Kc2.(C.4.1)
22
a03c2a00ca02
Thesubscript0ofthedensityparameterdefinesitasthepresentonewiththeHubbleparameter
H≡H(t0)≡H0.Thelatterequationshowsthealreadydefinedcriticaldensitycasbeingtheone
neededindeedforthecurvaturetobeK=0andthustheUniversetobeflat.Asmallerdensitymeans
K=−1andahigheronemeansK=1.
TheEinstein–deSitterUniverseismainlyaone-fluidmodel.InΩ0,however,arematterandradiation
terms,andothertypesofmattermaybedefinedinitaswell.
LetustakeagainanEinstein–deSitterUniverse,butthistimewithtimedependence(∼a−3(1+w)).
(8.6.3),usingis,Therea˙=H02Ω0wa0+(1−Ω0w).(C.4.2)
21+3w
aaLetusassumeaflatEinstein–deSitterUniverse,i.e.K=0.Withoutscalarfieldandderivativesof
istheresame,the2a˙=H02Ω0wa01+3w(C.4.3)
aa0=H02(1+z)1+3w.
Foronlymatterfluids,thereisw=0,whileforradiative-fluiddominancethereisw=1/3.Thetotal
energydensitygivesΩtotal=1.
(C.4.3)canbeintegratedimmediatelytoobtain(cf.[56])
t2/3(1+w)
a(t)=a0t0.(C.4.4)
isthereFurthermore,t=t0(1+z)−3(1+w)/2,(C.4.5)
whichrelatestimettoredshiftz.Further,therewouldbe
H≡a˙=2=H0t0=H0(1+z)3(1+w)/2,(C.4.6)
a3(1+w)tt
q≡−a2a¨=1+3w=const.=q0,(C.4.7)
2a˙2t0w≡t0=3(1+w)H0.(C.4.8)
Foradustmodel(w=0),therelation(C.4.1)yieldsingeneral
2a˙=H02Ω0a0+1−Ω0.(C.4.9)
aa0

COSMOLOGYC.APPENDIX196Thelatterequationissolvedparametricallyforopenmodels(K=−1)[56],
Ω0a(ψ)=a02(1−Ω0)(coshψ−1),(C.4.10)
Ω10t(ψ)=2H0(1−Ω0)3/2(sinhψ−ψ).(C.4.11)
Theserelationsthengive
1Ω222
t0=2H0(1−Ω)03/2Ω0(1−Ω0)1/2−coshΩ0−1>3H0,(C.4.12)
andforΩ01,
1t0≈H0(1+Ω0lnΩ0).(C.4.13)
Foraone-fluidcloseduniverse(K=1),thereis[56]
Ωa(ϑ)=a02(Ω00−1)(1−cosϑ),(C.4.14)
t(ϑ)=2H10(Ω0−Ω01)3/2(ϑ−sinϑ).(C.4.15)
Thescalefactorgrowsintimefor0≤ϑ≤ϑm=π.Themaximumvalueofthesameis
am=a(ϑm)=a0ΩΩ−01.(C.4.16)
0Itoccursatatimetmgivenby
tm=t(ϑm)=2Hπ(Ω−Ω01)3/2.(C.4.17)
00Furthermore,thereis(op.cit.)
1Ω0222
t0=2H0(Ω0−1)cos−1Ω0−1−Ω0(Ω0−1)1/2<3H0.(C.4.18)
t0foraclosed-universemodelissmallerthanforK=0.
•Three-uidsystemwithcosmologicalconstant:
Letusnowtakeathree-fluidsystemwithdust,radiationandacosmologicalconstant.Thereis
a˙2=H02Ω0Ma0+Ω0Ra02+Ω0Λa0−2+(1−Ω0M−Ω0R−Ω0Λ).
aaaa0(C.4.19)2TheUniverseisflatwhenΩ0M+Ω0R+Ω0Λ=1.Furthermore,Ω0Λ=HH02ΩΛisvalidforthe
energy-densityparameter.Thismeansthatthecosmologicalconstant,orequallytheenergydensity
ofthecosmologicalconstant,isconstantindeed.Inthecaseofξ=const.=0,thecosmological
constantisgivenbythescalar-fieldexcitations.Inthatcase(seeChapters2.4and8.4),Ω0iareeffec-
tive(screened)values(Ω∗)ofthebareparametersoftruedensities,asdefinedinequation(8.6.10).
Here,wetakethestandardapproachξ=0,whichmeansafurthertruecosmologicalconstantΛ0of
nature.wnunkno

C.4.STANDARDFRIEDMANNCOSMOLOGY

197

Aclosed-formexpressionoftheequation(C.4.19)forflatUniversesK=0,containingonlydustand
Λ0,isavailable.ThenthereiswithΩ0=Ω0M≡Ω0[56],
211+√1−Ω0
t0=3H02√1−Ω0log1−√1−Ω0.(C.4.20)
Anonvanishingcurvaturewillgenerallyleadtoacceleratingcontributionsandthustoanincreasein
theageoftheUniverseincomparisontothosewithK=0.

Inmatter-dominatedFriedmannmodels,theageoftheUniverseisgiventoagoodapproximationby
[56]).cf(t0=F(Ω0)H0−1≈0.98∙1010F(Ω0)h−1years.(C.4.21)
ThefunctionFisdependentonthecurvatureK,with
F(Ω0)=Ω0(Ω0−1)−3/2cos−12−1−(Ω0−1)−1forΩ0>1,(C.4.22)
Ω202F(Ω0)=3forΩ0=1,(C.4.23)
F(Ω0)=(1−Ω0)−1−Ω0(1−Ω0)−3/2cosh−12−1forΩ0<1.(C.4.24)
Ω20Forlimitingcases,thereis
F(Ω0)≈1πΩ0−1/2forΩ01,(C.4.25)
2F(Ω0)≈1+Ω0lnΩ0forΩ01.(C.4.26)
Asaconstraintsondensitythereis
0.01<Ω0<2(C.4.27)

sothatfortheageoftheUniverse,thereis
t0H≈(6.5−10)∙109h−1years.(C.4.28)
ThereistheHubbleconstantH0=h∙100km∙s−1Mpc−1fromtheHubblelawvi=H0xifor
irrotationalvelocityfieldsandisotropicspaces,withthereducedHubbleparameterhwhichisobser-
vationallysetbetween0.5and1.

•Observationalconstraintsofage:
Letustakethevaluesin[56].Asobservationalconstraints,galaxyformationneedsabout1to2∙109
years.Globularclustersarethoughttobearound1.3−1.4∙1010yearsoldorevenolder.
Anotherconstraintcanbefoundbytherelativeabundancesoflong-livedradioactivenucleiandtheir
decayproducts.Thesenucleiaresynthesizedinprocessesinvolvingtheabsorptionofneutronsby
heavynucleisuchasiron,andprocessesofthistypearethoughttooccurinsupernovaeexplosions.
Starsthatbecomesupernovaeareshortlivedandwithalifestemofabout107years.
Nucleocosmochronologyhelpsdeterminingthetimeatwhichstarsandgalaxieswereformed.Ifour
galaxyfinditsoriginatt≈0,timeatwhichaneraofnucleosynthesisofheavyelementsoccurred,

198

C.APPENDIXCOSMOLOGY

thenthishappenedduringt=T.ThisintervalisfollowedbyatimeΔinwhichthesolarsystem
becameisolatedfromtherestofthegalaxy.Isolationtimeshouldthenbefollowedbyaperiodts
correspondingtotheageofthesolarsystemitself.TheestimateoftheageoftheUniversethusyields
tn=T+Δ+ts.(C.4.29)
ThetimestsaswellasT+tscanbetracedbackfollowingthedecaychannelsandmeanlifetimeof
elementssuchas238Uinto206Pbor87Rbinto87Sr(withmeanlifetimesofabout109yearsand
6.6∙1010years,respectively).Inthisway,thesolarsystemisconcludedtobeabout4.6∙109years
old,andΔ=(1−2)∙108years.Furthermore,theconstraintsontheageoftheUniversegive
tn≈(0.6−1.5)∙1010years.(C.4.30)
density:gyenerThe•ThepresentUniverseiswell-approximatedbyadustormatter-dominatedmodel,withatotalenergy
1density0=0M+0R+0ν≈0M.(C.4.31)
ispressureThep0=p0M+p0R+p0ν≈0MkBT0M+10R≈0R0.(C.4.32)
3mpTheconstraintsonthegalacticcontributiontodensity(Ωg)areconsiderablyuncertainbutaround
Ωg=0g≈0.03.(C.4.33)
c0Thisshouldgivetheamountofmassconcentratedingalaxies.
Ontheotherhand,gravitationaldynamicsoflarge-scaleobjectsshowacontributionof
Ωdyn≈0.2−0.4(C.4.34)
fordynamicalmatter.Thediscrepancybetween(C.4.33)and(C.4.34)leadstothealreadymentioned
assumptionoftheexistenceofnon-luminous,dark,matter.Thismatter(oritsdynamics)playsan
formation.structureinroleimportantThefirstmodernstudiesofpossiblemissingmassgobacktoÖpiksin1915[185],relatedtothe
dynamicaldensityofthedynamicsofourgalaxyandoursolarvicinity,andlatertoOort[186]and
othersstarting1932.Datadoesnotsuggestdiscrepanciesbetweendynamicalandobservationalmass
inthesolarvicinity,though.
1933,therecamefirstevidenceofmissing,invisiblemassthroughZwickysworkonthedynamics
intheComacluster[253].Evidencelateraccumulated[79,138,187]andindependentdetermination
ofrotationvelocitiesofgalaxiesatlargegalactocentricdistancesconfirmedthepresenceofdarkmat-
terinformofhalosaroundthegalaxies[211,212].
Theassumptionthatthedominantpartofdarkmatterisnon-baryonic(calledcold,CDM)wasmade
1982byBlumenthaletal.[30].

1Ω0ν≈Ω0R≈10−5h−2.

yBibliograph

[1]A.AlbrechtandP.J.Steinhardt,CosmologyforGrandUnifiedTheorieswithRadiativelyInduced
SymmetryBreaking.PhysicalReviewLetters(Phys.Rev.Lett.)48(1982)1220.
[2]R.AcharyaandP.A.Hogan,EquivalenceofMassiveBrans-DickeandEinsteinTheoriesofGravita-
tion.LetterealNuovoCimento(Lett.Nuov.Cim.)6(16)(1973)668.
[3]E.G.Adelbergeretal.,Particle-PhysicsImplicationsofaRecentTestoftheGravitationalInverse-
SquareLaw.PhysicalReviewLetters98(2007)131104.
[4]I.R.AitchisonandA.J.G.Hey,GaugeTheoriesinParticlePhysics,AdamHilger,BristolandPhiladel-
.1981phia[5]E.T.Akhmedov,M.N.ChernodubandM.I.Polikarpov,DyonCondensationandAharonov-Bohm
Effect.JournalofExperimentalandTheoreticalPhysics(JETP)Letters67(6)(1998)389.
[6]L.W.AlvarezandF.Bloch,AQualitativeDeterminationoftheNeutronMomentinAbsoluteNuclear
Magnetons.PhysicalReview(Phys.Rev.)57(1940)111.
[7]L.Amendola,DarkEnergyandtheBOOMERANGData.PhysicalReviewLetters86(2001)196.
[8]C.Amsleretal.(ParticleDataGroup),TheReviewofParticlePhysics.PhysicsLetters(Phys.Lett.)
G667(2009)1.
[9]P.W.Anderson,Plasmons,GaugeInvariance,andMass.PhysicalReview130(1963)439.
[10]G.Arnisonetal.(UA1Collaboration),Experimental√ObservationofIsolatedLargeTransverseEnergy
ElectronswithAssociatedMissingEnergyats=540GeV.PhysicsLettersB122(1983)103.
[11]J.Assmann,DaskulturelleGedächntis.Schrift,ErinnerungundpolitischeIdentitätinfrühenHochkul-
turen,C.H.Beck,München1997.
[12]A.P.Balachandran,SahaandtheDyon,hep-ph/9303312(1993).
[13]J.Bardeen,L.N.CooperandJ.R.Schrieffer,TheoryofSuperconductivity.PhysicalReview(Phys.
Rev.)108(1957)1175.
[14]J.BeckensteinandM.Milgrom,DoestheMissingMassProblemSignaltheBreakdownofNewtonian
Gravity?.AstrophysicalJournal(ApJ)286(1984)7.
[15]J.D.Beckenstein,TheModifiedNewtonianDynamics–MOND–anditsImplicationsforNew
Physics.ContemporaryPhysics47(2006)387.

199

200

BIBLIOGRAPHY

[16]C.H.Bennetetal.,TeleportinganUnknownQuantumStateviaDualClassicalandEinstein-Podolsky-
RosenChannels.PhysicalReviewLetters70(1993)1895.
[17]M.C.Bentoetal.,Self-InteractingDarkMatterandtheHiggsBoson.PhysicalReviewD62(2000)
041302.[18]P.G.Bergmann,CommentsontheScalar-TensorTheory.InternationalJournalofTheoretical
Physics(IJTP)1(1)(1968)25.
[19]V.B.Bezerraetal.,RemarksonSomeVacuumSolutionsofScalar-TensorCosmologicalModels.
BrazilianJournalofPhysics(Braz.J.Phys.)34(2a)(2004)562.
[20]N.M.Bezares-RoderandH.Dehnen,HiggsScalar-TensorTheoryforGravityandtheFlatRotation
CurvesofSpiralGalaxies.GeneralRelativityandGravitation(Gen.Relat.Gravit.)39(8)(2007)
1259.[21]N.M.Bezares-Roder,H.NandanandH.Dehnen,Horizon-lessSphericallySymmetricVacuum-
SolutionsinaHiggsScalar-TensorTheoryofGravity.InternationalJournalofTheoreticalPhysics
(Int.J.Theor.Phys.)46(10)(2007)2420.Partiallypresentedatthe24thIAGRGmeetingontheRecent
AdvancesinGravitationandCosmology,NewDelhi,India(2007).
[22]N.M.Bezares-RoderandH.Nandan,SpontaneousSymmetryBreakdownandCriticalPerspectivesof
HiggsMechanism.IndianJournalofPhysics(IJP)82(1)(2008)69.
[23]N.M.Bezares-Roder,H.NandanandH.Dehnen,Scalar-fieldPressureinInducedGravitywith
HiggsPotentialandDarkMatter(andreferencestherein).Submittedforpublication(2009).
-qc].[grv:0912.4039arXi[24]N.M.Bezares-RoderandF.Steiner,AScalar-TensorTheoryofGravitywithaHiggsPotentialin
MathematicalAnalysisofEvolution,InformationandComplexity,eds.ArendtandSchleich.Wiley-
.2009BerlinVCH,[25]J.J.vanderBij,CanGravitymaketheHiggsparticlelight.ActaPhysicaPolonica(ActaPhys.Pol.)
B25(1994)827.[arXiv:9310064]
[26]J.J.vanderBij,CanGravityPlayaRoleattheElectroweakScale.[arXiv:hep-ph/9507389](1995).
[27]J.J.vanderBij,LargeRescalingoftheScalarCondensate,TowardsaHiggs-GravityConnection?.
Freiburg-THEP99/09(1999).PresentedattheEPS-HEP999meeting,July1999,Tampere,Finland
[hep-ph/9908297].[28]F.Bloch,NuclearInduction.PhysicalReview70(1946)460.
[29]H.J.BlomeandW.Priester,BigBounceintheVeryEarlyUniverse.AstronomyandAstrophysics
(A&A)50(1991)43.
[30]G.R.Blumental,H.PagelsandJ.R.Primack,GalaxyFormationbyDissipationlessParticlesHeavier
thanNeutrinos.Nature299(1982)37.
[31]N.Bogoliubov,OntheTheoryofSuperfluidity.JournalofPhysicsUSSR11(1947)23(Izvestiya
AcademiiNaukUSSR11(1)77).

BIBLIOGRAPHY

201

[32]B.Boisseauetal.,ReconstructionofaScalar-TensorTheoryofGravityinanAcceleratingUniverse.
PhysicalReviewLetters85(2000)2236.
[33]J.R.Bond,A.S.SzalayandM.S.Turner,FormationofGalaxiesinaGravitino-FominatedUniverse.
PhysicalReviewLetters48(1982)1636.
[34]D.Bouwmeesteretal.,ExperimentalQuantumTeleportation.Nature390(1997)575.
[35]G.Brassard,S.L.BraunsteinandR.Cleve,TeleportationasaQuantumComputation.PhysicaD120
43.)1998([36]K.A.Bronnikov,C.P.Constantinidis,R.L.EvangelistaandJ.C.Fabris,ColdBlackHolesinScalar-
TensorTheories,gr-qc/9710092(1997).
[37]K.A.BronnikovandG.N.Shikin,SphericallySymmetricScalarVacuum:No-GoTheorems,Black
HolesandSolitons.GravitationandCosmology8(2002)107.
[38]K.A.Bronnikov,Scalar-TensorGravityandConformalContinuations.JournalofMathematical
Physics43(12)(2002)6096.
[39]J.R.BrownsteinandJ.W.Moffat,GalaxyRotationCurveswithoutNonbaryonicDarkMatter.Astro-
physicalJournal636(2006)721.
[40]R.Brandeliketal.(TASSOCollaboration),EvidenceforPlanarEventsine+e−AnnihilationatHigh
Energies.PhysicsLettersB86(1979)243.
[41]C.BransandR.H.Dicke,MachsPrincipleandaRelativisticTheoryofGravitation.PhysicalReview
925.)1961(124[42]R.R.Caldwell,R.DaveandP.J.Steinhardt,CosmologicalImprintofanEnergyComponentwith
GeneralEquationofState.PhysicalReviewLetters80(1998)1582.
[43]A.G.W.CameronandJ.W.Turan,Evolutionarymodelsofnucleosynthesisinthegalaxy.Astro-
physicsandSpaceScience14(1971)179.
[44]S.CarloniandP.K.S.Dunsby,ADynamicalSystemApproachtoHigherOrderGravity,gr-
qc/0611133v1(2006).TalkgivenatIRGAC,July2006.
[45]R.Catena,M.PietroniandL.Scarabello,EinsteinandJordanFramesReconciled:AFrame-Invariant
ApproachtoScalar-TensorCosmology.PhysicalReviewD76(2007)094039.
[46]CDFCollaboration,ObservationofTopQuarkProductioninPbar-PCollisions.PhysicalReview
Letters74(1995)2626.
[47]J.L.Cervantes-Cota,InducedGravityandCosmology,Hartung-Gorre-Verlag,KonstanzerDissertatio-
nen506,Konstanz1996.
[48]J.L.Cervantes-CotaandH.Dehnen,InducedGravityInflationintheSU(5)GUT.PhysicalReview
D51(1995)395.
[49]J.L.Cervantes-CotaandH.Dehnen,InducedGravityInflationintheStandardModelofParticle
Physics.NuclearPhysicsB442(1995)391.

202

BIBLIOGRAPHY

[50]J.L.Cervantes-Cota,M.A.Rodríguez-Meza,R.GabbasovandJ.Klapp,NewtonianLimitofScalar-
TensorTheoriesandGalacticDynamics:IsolatedandInteractingGalaxies.RevistaMexicanade
FísicaS53(4)(2007)22.
[51]J.L.Cervantes-Cota,M.A.Rodríguez-MezaandD.Núñez,FlatRotationCurvesUsingScalar-Tensor
Theories.JournalofPhysics:ConferenceSeries91(2007)012007.arXiv:0707.2692v1[astro-ph].
[52]J.L.Cervantes-Cota,M.A.Rodríguez-MezaandD.Núñez,SphericalScalar-TensorGalaxyModel.
PhysicalReviewD79(6)(2009)064011.
[53]T.P.ChengandL.F.Li,GaugeTheoryofElementaryParticlePhysics,ClarendonPress,Oxford1984.
[54]H.C.Chandola,H.C.PandeyandH.Nandan,TopologyofQCDVacuumandColorConfinement.
CanadianJournalofPhysics80(2002)745.
[55]A.D.Chernin,TheRestMassofPrimordialNeutrinos,andGravitationalInstabilityintheHotUni-
verse.AstronomicheskiiZhurnal58(1981)25.
[56]P.ColesandF.Lucchin,Cosmology.TheOriginandEvolutionofCosmicStructure,JohWiley&Sons,
WestSussex2003.
[57]M.Conversi,E.PanciniandO.Piccioni,OntheDecayProcessofPositiveandNegativeMesons.
PhysicalReview68(1945)232.
[58]E.Copeland,A.R.LiddleandD.Wands,ExponentialPotentialsandCosmologicalScalingSolu-
tions,gr-qc/9711068(1998).
[59]S.Cotsakis,MathematicalProblemsinHigherOrderGravityandCosmology,gr-qc/9712046v1
(1997).TalkpresentedattheEightMarcelGrossmannMeeting,Jerousalem,June22-27,1997.
[60]C.L.Cowan,Jr.,F.Reines,F.B.Harrison,H.W.KruseandA.D.McGuire,DetectionoftheFree
Neutrino:AConfirmation.Science124(1956)103.
F.ReinesandC.L.Cowan,Jr.,TheNeutrino.Nature178(1956)446.
[61]R.CowsikandJ.McClelland,NeutrinosofNonzeroMassinAstrophysics.AstrophysicalJournal
7.)1973(180[62]T.Damour,ExperimentalTestsofRelativisticGravity.NuclearPhysicsB80(2000)41.
[63]G.Danby,J.-M.Gaillard,K.Gouilianos,L.M.Lederman,N.Mistry,M.SchwartyandJ.Steinberger,
ObservationofHigh-EnergyNeutrinoReactionsandtheExistenceofTwoKindsofNeutrinos.Phy-
sicalReviewLetters9(1962)36.
[64]H.Dehnen,Zurallgemein-relativistischenDynamik.AnnalenderPhysik7(13)(1964)101.
[65]H.Dehnen,ÜberdenEnergiegehaltstatischerGravitationsfeldernachderallgemeinenRelativitäts-
theorieinNewtonscherNäherung.ZeitschriftfürPhysik(Z.Phys.)179(1964)96.
[66]H.DehnenandO.Obregón.AstrophysicsandSpaceScience(Astrophys.Sp.Sc.)14(1972)454.
[67]H.DehnenandH.Hönl,TheInfluenceofStrongInteractionsontheEarlyStagesoftheUniverse.
AstrophysicsandSpaceScience33(1975)49.

BIBLIOGRAPHY

203

[68]H.Dehnen,F.GhaboussiandJ.Schröder,WissenschaftlicheZeitschriftderFriedrich-Schiller-
UniversitätJena39(1990)41.
H.DehnenandH.Frommert,HiggsFieldGravity.InternationalJournalofTheoreticalPhysics
29(6)(1990)537.
[69]H.DehnenandH.Frommert,HiggsFieldGravityWithintheStandardModel.InternationalJournal
ofTheoreticalPhysics30(7)(1991)985.
[70]H.DehnenandH.Frommert,Higgs-FieldandaNewScalar-TensorTheoryofGravity.International
JournalofTheoreticalPhysics31(1)(1992)109.
[71]H.DehnenandH.Frommert,HiggsMechanismwithoutHiggsParticle.InternationalJournalof
TheoreticalPhysics32(7)(1993)1135.
[72]W.Dehnen,MonthlyNoticesoftheRoyalAstronomicalSociety(MNRAS)265(1993)250.
[73]H.Dehnen,B.RoseandK.Amer,DarkMatterParticlesandtheFlatRotationCurvesofSpiral
Galaxies.AstrophysicsandSpaceScience234(1995)69.
[74]H.Dehnen,TheoriederElementarteilchenundihrerWechselwirkung,FachbereichPhysik,Universität
.2005onstanzK[75]P.A.M.Dirac,QuantisedSingularitiesintheElectromagneticField.ProceedingsoftheRoyalSociety
A133(1931)60.
[76]W.DittrichandM.Reuter,ClassicalandQuantumDynamis.FromClassicalPathstoPathIntegrals
173ff,Springer,Berlin(1992).
[77]G.Drexlin,Neutrino-Oszillationen.PhysikalischeBlätter55(2)(1999)25.
[78]M.DoranandC.Wetterich,QuintessenceandtheCosmologicalConstant.astro-ph/0205267(2002).
[79]J.Einasto,A.KaasikandE.Saar,DynamicEvidenceonMassiveCoronasofGalaxies.Nature250
309.)1974([80]J.Einasto,DarkMatter:EarlyConsiderations,astro-ph/0401341v1(2004).
[81]A.Einstein,DieFeldgleichungenderGravitation.SitzungsberichtderPreussischenAkademieWis-
senschaften(S.Preus.)(1915)844;reprintin:TheCollectedPapersofAlbertEinstein,vol.6,Princeton
UniversityPress,London,1996,pp.244ff.
[82]A.Einstein,KosmologischeBetrachtungenzurAllgemeinenRelativitätstheorie.Sitzungsberichtder
PreussischenAkademiederWissenschaften(1917)142;reprintin:TheCollectedPapersofAlbert
Einstein,vol.6,PrincetonUniversityPress,London,1996,pp.540ff.
[83]A.Einstein,B.PodolskyandN.Rosen,CanQuantum-MechanicalDescriptionofPhysicalReality
BeConsideredComplete?.PhysicalReview47(1935)777.
[84]L.EisenbudandE.P.Wigner,EinführungindieKernphysik,BIHochschultaschenbücher16,Mannheim
.1961[85]F.EnglertandR.Brout,BrokenSymmetryandtheMassofGaugeVectorBosons.PhysicalReview
Letters13(1964)321.

204

BIBLIOGRAPHY

[86]R.v.Eötvös,ÜberdieAnziehungderErdeaufverschiedeneSubstanzen.MathematischeundNatur-
wissenschaftlicheBerichteausUngarn8(1890)65.
[87]B.Fauser,ProjectiveRelativity:PresentStatusandOutlook.GeneralRelativityandGravitation33
875.)2001([88]R.P.Feynman,F.B.MorignioandW.G.Wagner,FeynmanLecturesonGravitation,ed.BrianHatfield.
Addison-Wesley,Reading1995.
[89]G.B.Field,IntergalacticMatter.AnnualReviewofAstronomyandAstrophysics10(1972)227.
[90]M.Fierz,HelveticaPhysicaActa29(1956)128.
[91]W.Finkelnburg,EinführungindieAtomphysik,6thed.Springer,Berlin1958.
[92]J.I.FriedmanandH.W.Kendall,DeepInelasticElectronScattering.AnnualReviewofNuclear
Science22(1972)203.
[93]H.Frommert,HiggsmechanismusundGravitation,Dissertation,Konstanz1991.
[94]Y.Fujii,Scalar-TensorTheoryofGravitationandSpontaneousBreakdownofScaleInvariance.Phy-
sicalReviewD9(4)(1974)874.
[95]Y.Fujii,Quintessence,Scalar-TensorTheories,andNon-NewtonianGravity.PhysicalReviewD62
044011.)2000([96]M.K.Gaillard,WeakInteractionsandGaugeTheories.FermiNationalAcceleratorLaboratory
1979.Dec.FERMILAB-Conf-79/87-THY[97]M.R.GallisandG.N.Flemming,EnvironmentalandSpontaneousLocalization.PhysicalReviewA
38.)1990(42[98]P.M.Garnavichetal.,ConstraintsonCosmologicalModelsfromHubbleSpaceTelescopeObserva-
tionsofHigh-zSupernovae.AstrophysicalJournal493(1998)L53.
[99]B.Geyer,J.Kripfganz,J.Ranft,D.RobaschikandE.Wieczorek,EinführungindieQuantenfeldtheorie
derelementarteilchen,VEBDeutscherVerlagderWissenschaften,Berlin1990.
[100]M.Gell-Mann,ASchematicModelofBaryonsandMesons.PhysicsLetters8(1964)214.
[101]M.Gell-Mann,Quarks.DevelopmentsintheQuarkTheoryofHadrons.ActaPhysicaAustriaca,
Supplement9(1972)733.
[102]H.GeorgiandS.Glashow,UnityofAllElementary-ParticleForces.PhysicalReviewLetters32
438.)1974([103]J.L.GervaisandB.Sakita,FieldTheoryInterpretationofSupergaugesinDualModels.Nuclear
PhysicsB34(1971)832.
[104]G.Gilmore,R.F.G.WyseandK.Kuijken,Kinematics,ChemistryandStructureoftheGalaxy.
AnnualReviewofAstronomyandAstrophysics27(1989)555.
[105]V.L.GinzburgundL.D.Landau.OntheTheoryofSuperconductivity.ZhurnalEksperimentalnoii
TeoreticheskoiFiziki(Zh.Eksp.Teor.Fiz.)20(1950)1064.

BIBLIOGRAPHY

205

[106]P.GoddardandD.I.Olive,MagneticMonopolesinGaugeFieldTheories.ReportsonProgressin
Physics41(1978)1357.
[107]J.Goldstone,FieldTheorieswithSuperconductorSolutions.NuovoCimento19(1961)154.
[108]J.Goldstone,BrokenSymmetries.PhysicalReview127(1962)965.
[109]Yu.A.GolfandandE.P.Likhtman,ExtensionoftheAlgebraofPoincaréGroupOperatorsand
ViolationofP-Invariance.JournalofExperimentalandTheoreticalPhysicsLetters13(1971)323.
[110]U.D.Goswami,H.Nandan,C.-P.PandeyandN.M.Bezares-Roder,MaxwellsEquations,Electro-
magneticWavesandMagneticCharges.PhysicsEducation25(4)(2008)251andreferencestherein.
[111]U.D.Goswami,H.Nandan,C.P.PandeyandN.M.Bezares-Roder;CovariantFormalismof
MaxwellsEquationsandRelatedAspects,PhysicsEducation26(4)(2009)269–278.
[112]W.GreinerandB.Müller,EichtheoriederschwachenWechselwirkung,HarriDeutsch,Frankfurt
.1995[113]M.B.Green,J.H.SchwarzandE.Witten,SuperstringTheory.CambridgeUniverstiyPress,Cam-
.1998bridge,[114]F.V.Gubarev,M.I.PolikarpovandV.I.Zakharov,Monopole-AntimonopoleInteractioninAbelian
HiggsModel.PhysicsLettersB438(1998)147.
[115]M.W.Guidry,Gaugefieldtheories:Anintroductionwithapplications,Wiley-VCH,Weinheim,
.2004p.261f.f;p.247f[116]J.E.Gunn,B.W.Lee,I.Lerche,D.N.SchrammandG.Steigman,SomeAstrophysicalConsequences
oftheExistenceofaHeavyStableNeutralLepton.AstrophysicalJournal223(1978)1015.
[117]G.S.Guralnik,C.R.HagenandT.W.B.Kibble,GlobalConservationLawsandMasslessParticles.
PhysicalReviewLetters13(1964)585.
[118]A.Guth,InflationaryUniverse:APossibleSolutiontotheHorizonandFlatnessProblems.Physical
ReviewD23(1981)347.
[119]M.Hamuy,M.M.Phillips,J.Maza,N.B.Sintzeff,R.A.SchrommerandR.Avilez,AHubbleDia-
gramofDistantTypeIaSupernovae.AstronomicalJournal109(1995)1.
[120]J.OHanlon,Intermediate-RangeGravity:AGenerallyCovariantModel.PhysicalReviewLetters
137.)1972(29[121]A.HardellandH.Dehnen,ExactSolutionsofEinsteinsFieldEquationsforaMassivePoint-Particle
withScalarPoint-Charge.GeneralRelativityandGravitation25(11)(1993)1165.
[122]T.Harko,GravitationalCollapseofaHagedornFluidinVaidyaGeometry.PhysicalReviewD68
064005.)2003([123]S.HawkingandG.Ellis,CosmicBlack-BodyRadiationandtheExistenceofSingularitiesinour
Universe.AstrophysicalJournal152(1968)25.

206

BIBLIOGRAPHY

[124]A.HebeckerandC.Wetterich,QuintessentialAdjustmentoftheCosmologicalConstant.hep-
).2000(ph/0003287[125]W.K.Heisenberg,ÜberdenanschaulichenInhaltderquantentheoretischenMechanik.Zeitschrift
fürPhysik43(1927)172.
[126]P.W.Higgs,BrokenSymmetriesandtheMassesofGaugeBosons.PhysicalReviewLetters13
508.)1964([127]A.HillandJ.J.vanderBij,StronglyInteractingSinglet-DoubletHiggsModel.PhysicalReviewD
36(11)(1987)3463.
[128]G.Hinshawetal.,Five-YearWilkinsonMicrowaveAnisotropyProbe(WMAP)Observations:Data
Processing,SkyMaps,andBasicResults.AstrophysicalJournalSupplementSeries(ApJS)180(2009)
[astro-ph].v:0803.0732arXi225,[129]J.HoellandW.Priester,VoidStructureintheEarlyUniverse-ImplicationsforaΛ>0Cosmology.
AstronomyandAstrophysics251(1991)L23.
[130]H.HönlandH.Dehnen,Erlaubtdie3◦K-StrahlungRückschlüsseaufeinekonstanteoderveränder-
licheGravitationszahl?.ZeitschriftfürAstrophysik68(1968)181.
[131]R.Hofstadter,NuclearandNucleonScatteringofHigh-EnergyElectrons.AnnualReviewofNu-
clearScience7(1957)231.
[132]G.tHooft,RenormalizationofMasslessYang-MillsFields.NuclearPhysicsB33(1971)173.
[133]G.tHooft,MagneticMonopolesinUnifiedGaugeTheories.NuclearPhysicsB79(1974)276.
[134]J.Hošek,QCDVacuumasaType-IIDualColourSuperconductor.CzechJournalofPhysics43
309.)1993([135]R.A.HulseandJ.K.Taylor,DiscoveryofaPulsarinaBinarySystem.AstrophysicalJournal195
L51.)1975([136]P.Jordan,FormationofStarsandDevelopmentoftheUniverse.Nature164(1949)637;Schwer-
kraftundWeltall,Friedr.Vieweg&SohnVerlag,2.Aufl.,Braunschweig1955.
[137]P.Jordan,BemerkungenzuderArbeitvonH.HönlundH.Dehnen:Erlaubtdie3◦K-Strahlung
RückschlüsseaufeinekonstanteoderveränderlicheGravitationszahl?.ZeitschriftfürAstrophysik68
201.)1968([138]F.D.KahnandL.Woltjer,IntergalacticMatterandtheGalaxy.AstrophysicalJournal130(1959)
705.[139]G.L.Kane,PerspectivesonHiggsPhysics,WorldScientific,Singapore(1993)andreferencestherein.
[140]D.J.Kapneretal.,TestsoftheGravitationalInverse-SquareLawBelowtheDark-EnergyLength
Scale.PhysicalReviewLetters98(2007)021101.
[141]C.Kiefer,OntheInterpretationofQuantumTheory-FromCopenhagentothePresentDay,quant-
).2002(ph/0210152

BIBLIOGRAPHY

207

[142]O.Kleininthe1938ConferenceonNewTheoriesinPhysicsinKasimierz,Poland(1938),reproduced
inL.ORaifeartaigh;TheDawningofGaugeTheory,PrincetonUniversity,Princeton,NJ1997.
[143]A.S.Kronfeld,M.I.Laursen,G.SchierholzandU.-J.Wiese,MonopoleCondensationandColor
Confinement.PhysicsLettersB198(4)(1987)516.
[144]E.W.KolbandM.S.Turner,TheEarlyUniverse,Addison-Wesley(seriesFrontiersinPhysics),Mas-
.1990sachusetts[145]B.V.KombergandI.D.Novikov,NatureoftheCoronaeofSpiralGalaxies.PismavAstronomich-
eskiiZhurnal(PismaAstron.Zh.)1(1975)3.
[146]G.G.Kuzmin,OntheDistributionofMassintheGalaxy.TartuAstr.Obs.Publ.32(1952)5.
[147]G.G.Kuzmin,TartuAstr.Obs.Publ.32(1952)211.
[148]R.Laiho,S.N.MolotkovandS.S.Nazin,OnTeleportationofaCompletelyUnknownStateofRela-
tivisticPhoton.PhysicsLettersA278(1-2)(2000)9.
[149]C.M.G.Lattes,H.Muirhead,G.P.S.OcchialiniandC.F.Powell,ProcessesInvolvingCharged
Mesons.Nature159(1947)694.
C.M.G.Lattes,G.P.S.OcchialiniandC.F.Powell,ObservationsontheTracksofSlowMesonsin
PhotographicEmulsions.Nature160(1947)453.
[150]T.D.LeeandC.N.Yang,QuestionofParityConservationinWeakInteractions.PhysicalReview
254.)1956(104[151]T.D.Lee,ParticlePhysicsandIntroductiontoFieldTheory,HarwoodAcademic,NewYork1981.
[152]G.Lemaître,Ununivershomogènedemasseconstanteetderayoncroissant,rendantcomptedela
vitesseradialedesnébuleusesextra-galactiques.AnnalesdelaSociétéscientifiquedeBruxelles(Ann.
Cos.Sci.Brux.)A47(1927)49.
[153]TheLIGOScientificCollaboration,TheVirgoCollaboration,SearchesforGravitationalWavesfrom
KnownPulsarswithS5LIGOData.LIGO-P080112-v5(2009).
[154]A.D.Linde,AnewInflationaryUniverseScenario:APossibleSolutionoftheHorizon,Flatness,
Homogeneity,IsotropyandPrimordialMonopoleProblems.PhysicsLettersB108(1982)389;Sca-
larFieldFluctuationsintheExpandingUniverseandtheNewInflationaryUniverseScenario.Physics
LettersB116(1982)335.
[155]E.Lohrmann,Hochenergiephysik,Teubner,Stuttgart1981.
[156]J.LeiteLopes,MesonDecayandtheTheoryofNuclearForces.Nature160(1947)866.
[157]F.MandlandG.Shaw,QuantumFieldTheory,JohnWiley&Sons,Chichester,p.279f1984.
[158]R.E.MarshakandE.C.G.Sudarshan,EinführungindiePhysikderEmementarteilchen,Wiley&Sons
(BIHoschultashenbücher65),NewYork1961.
[159]S.P.Martin,ASupersymmetryPrimer,arXiv:hep-ph/9709356v4(1997).

208

BIBLIOGRAPHY

[160]T.MatosandF.S.Guzmán,OntheSpacetimeofaGalaxy.ClassicalandQuantumGravity18
50055.)2001([161]J.C.Maxwell,PhilosophicalMagazineandJournalofScience4(1861)161.
[162]M.Milgrom,AModificationoftheNewtonianDynamics:ImplicationsforGalaxies.Astrophysics
Journal279(1983)370.
[163]M.Milgrom,MOND:TimeforaChangeofMind?,arXiv:0908.3842v1[astro-ph-CO](2009).
[164]H.Miyazawa,BaryonNumberChangingCurrents.ProgressinTheoreticalPhysics36(1966)1266.
[165]J.W.Moffat,Scalar-Tensor-VectorGravityTheory.JournalofCosmologyandAstro-ParticlePhysics
004.)2006(3[166]J.W.Moffat,Non-SingularCosmologyinModifiedGravity,gr-qc/0610059v3(2007).
[167]J.W.Moffat,Non-SingularSphericallySymmericSolutioninEinstein-Scalar-TensorGravity,gr-
).2008(v4qc/0702070[168]J.W.MoffatandV.T.Toth,ModifiedGravity:CosmologywithoutdarkmatterorEinsteinscosmo-
logicalconstant,arxiv:0710.0364[astro-ph](2006).
[169]Y.Nambu,QuasiparticlesandGaugeInvarianceintheTheoryofSuperconductivity.PhysicalRe-
view117(1960)648.
[170]Y.NambuandG.Jona-Lasinio,DynamicalModelofElmentaryParticlesBasedonanAnalogywith
SuperconductivityI.PhysicalReview122(1961)345.
[171]Y.NambuandG.Jona-Lasinio,DynamicalModelofElmentaryParticlesBasedonanAnalogywith
SuperconductivityI.PhysicalReview124(1961)246.
[172]H.Nandan,T.AnnaandH.C.Chandola,DyonCondensationandColourConfinementinDual
QCD.EuroPhysicsLetters67(2004)746
[173]H.Nandan,H.C.ChandolaandH.Dehnen,MagneticSymmetry,ReggeTrajectoriesandtheLinear
ConfinementinDualQCD.InternationalJournalofTheoreticalPhysics44(2005)469(Erratum-ibid.
1639).)2008(47[174]H.NandanandH.C.Chandola,ProceedingsoftheXVIIDAE-BRNSSymposiumonHighEnergy
383.)2007(Physics[175]H.Nandan,DielectricResponseofQCDVacuumandMesonicReggeTrajectories,International
WorkshoponTheoreticalHighEnergyPhysics(IWTHEP2007),Roorkee,UA,India15–20March
.2007[176]H.Nandan,N.M.Bezares-RoderandH.C.Chandola,DualMeissnerEffectandDielectricQCD
Vacuum.ProceedingsoftheLIIIDAE-BRNSSymposiumonNuclearPhysics(DepartmentofAtomic
Energy-BoardofResearchinNuclearSciences)(2008)583.
[177]H.NandanandH.C.Chandola,IdeasofDualQuantumChromodynamics.IndianJournalof
Physics82(12)(2008)1619.

BIBLIOGRAPHY

209

[178]H.Nandan,N.M.Bezares-RoderandH.C.Chandola,ScreeningCurrentandDielectricParameters
inDualQCD.IndianJournalofPureandAppliedPhysics47(2009)808.
[179]H.Nandan,N.M.Bezares-RoderandH.Dehnen,BlackHolesinInducedGravitywithHiggsPoten-
tial(andreferencestherein).Submittedforpublication(2009).arXiv:0912.4036[gr-qc].
[180]W.NapierandB.N.G.Guthrie.MonthlyNoticesoftheRoyalAstronomicalSociety170(1975)7.
[181]J.F.Navarro,C.S.FrenkandS.D.M.White,TheStructureofColdDarkMatterHalos.Astrophysical
Journal462(1996)563.
[182]S.H.NeddermeyerandC.D.Anderson,NoteontheNatureofCosmic-RayParticles.PhysicalRe-
view51(1937)884.
[183]H.Nishino,S.Clarketal.(Super-KamiokandeCollaboration),SearchforProtonDecayviap→
e++π0andp+µ++π0inaLargeWaterCherenkovDetector.PhysicalReviewLetters102(2009)
141801.[184]S.Ogawa,T.M.Lee,A.S.NayakandP.Glynn,Oxygenation-sensitiveContrastinMagneticReso-
nanceImageofRodentBrainatHighMagneticFields.MagneticResonanceinMedicine14(1990)
68.[185]E.Öpik,SelectiveAbsorptionofLightinSpace,andtheDynamicsoftheUniverse.Bull.delaSoc.
Astr.deRussie21(1915)45.
[186]J.H.Oort,TheForceExertedbytheStellarSystemintheDirectionPerpendiculartotheGalactic
PlaneandsomerelatedProblems.BulletinoftheAstronomicalInstituteoftheNetherlands6(1932)
249.[187]J.P.Ostriker,P.J.E.PeeblesandA.Yahil,TheSizeandMassofGalaxies.AstrophysicalJournal
L1.)1974(193[188]T.Padmanabhan,DarkEnergyandGravity.arXiv:0705.2533v1[gr-qc](2007).InvitedReviewfor
aspecialGeneralRelativityandGravitationisuueonDarkEnergy.
[189]H.C.PandeyandH.C.Chandola,Non-perturbativeAspectsofDualQCDVacuumandConfine-
ment.PhysicsLettersB476(2000)193.
[190]W.Pauli,OffenerBriefandieGruppederRadioaktivenbeiderGauvereins-TagungzuTübingen.
(1930).Letter[257]inW.Pauli,WissenschaftlicherBriefwechselmitBohr,Einstein,Heisenbergu.a.
vol.II:1930-1939,ed.K.vonMeyen,Springer(1985)pg.39.
[191]R.D.PecceiandH.R.Quinn,CPConservationinthePresenceofPseudoparticles.PhysicalReview
Letters38(25)(1977)1440.
[192]P.J.E.PeeblesandB.Ratra,CosmologicalConsequencesofaRollingHomogeneousScalarField.
PhysicalReviewD37(1988)3406.
[193]P.J.E.PeeblesandB.Ratra,TheCosmologicalConstantandDarkEnergy.ReviewsofModern
Physics75(2003)559.

210

BIBLIOGRAPHY

[194]R.Penrose,GravitationalCollapseandSpace-TimeSingularities.PhysicalReviewLetters14
57.)1965([195]R.Penrose,TheNon-LinearGraviton,MathematicalInstitute,UniversityofOxford(essay),1975.
[196]A.A.PenziasandR.W.Wilson,AMeasurementofExcessAntennaTemperatureat4080Mc/s.
AstrophysicalJournal142(1965)419.
[197]S.Perlmutteretal.(SCP),MeasurementsoftheCosmologicalParametersΩandΛfromtheFirst7
Supernovaeatz≥0.35.AstrophyicalJournal483(1997)565.
[198]S.Perlmutteretal.(SCP),MeasurementsofΩandΛfrom42High-RedshiftSupernovae.Nature
51.)1998(391[199]M.E.PeskinandD.V.Schroeder,AnIntroductiontoQuantumFieldTheory,AddisonWesleyReading,
.1995Massachusetts,[200]L.O.PimentelandC.Mora,QuantumCosmologyinBergmann-WagonerScalar-tensorGravitational
Theory,gr-qc/0009027(2000).
[201]A.M.Polyakov,ParticleSpectruminQuantumFieldTheory.JournalofExperimentalandTheo-
reticalPhysicsLetters20(1974)194.
[202]J.Preskill,MagneticMonopoles.AnnualReviewofNuclearParticleScience34(1984)461.
[203]E.M.Purcell,H.C.TorreyandR.V.Pound,ResonanceAbsorptionbyNuclearMagneticMoments
inaSolid.PhysicalReview.69(1946)37.
[204]I.I.Rabi,J.R.Zacharias,S.Millman,P.Kusch,ANewMethodofMeasuringNuclearMagnetic
Moment.PhysicalReview53(1938)318.
[205]R.D.Reasenbergetal.,VikingRelativityExperiment:VerificationofSignalRetardationbySolar
Gravity.AstrophysicalJournal,234(1979)L219.
[206]H.Reissner,OntheEigengravitationoftheelectricFieldsaccordingtotheEinsteinTheory.An-
nalenderPhysik50(1916)106.
[207]A.G.Riessetal.,ObservationalEvidencefromSupernovaeforanAcceleratingUniverseanda
CosmologicalConstant.AstronomicalJournal116(1998)1009.
[208]A.G.Riessetal.,ObservationalEvidencefromSupernovaeforanAcceleratingUniverseanda
CosmologicalConstant.AstronomicalJournal116(1998)1009.
[209]G.Ripka,DualSuperconductorModelofColorConfinement,Springer(LectureNotesinPhysics,
.2004639),ol.V[210]M.A.Rodríguez-MezaandJ.L.Cervantes-Cota,Potential-DensityPairsforSphericalGalaxiesand
Bulges:TheInfluenceofScalarFields.MonthlyNoticesoftheRoyalAstronomicalSociety350(2)
671.)2004([211]V.C.Rubin,W.K.FordandN.Thonnard,GaseousResponsetobar-likeDistortions.Astrophysical
Journal225(1978)L107.

BIBLIOGRAPHY

211

[212]V.C.Rubin,W.K.FordandN.Thonnard,RotationalPropertiesof21Sc.GalaxieswithaLarge
RangeofLuminositiesandRadii,fromNGC4605(R=4kpc)toUGC2884(R=122kpc).Astrophy-
sicalJournal238(1980)471.
[213]M.N.Saha,OntheOriginofMassinNeutronsandProtons.IndianJournalofPhysics10(1936)
141.[214]A.Sandage,TheAbilityofthe200-inchTelescopetoDiscriminateBetweenSelectedWorldMo-
dels.AstrophysicJournal133(1961)355.
[215]R.H.Sanders,FiniteLength-ScaleAntigravityandObservationsofMassDiscrepanciesinGala-
xies.AstronomyandAstrophysics154(1986)135.
[216]J.Schwinger,SourcesandMagneticCharge.PhysicalReview173(1968)1536.
[217]J.Scherk,Antigravity:ACrazyIdea?.PhysicsLettersB88(3,4)(1979)265.
[218]R.SexlandH.K.Urbantke,GravitationundKosmologie:EineEinführungindieAllgemeineRela-
tivitätstheorie5thEd.,Spektrum,Heidelberg2002.
[219]S.deSiena,A.diLisiandF.Illuminati,Quadrature-DependentBogoliubovTransformationsand
MultiphotonSqueezedStates.PhysicalReviewA64(2001)063803.
[220]J.Silk,Doesthegalaxypossessagaseoushalo.CommentsinAstrophysicsandSpacePhysics6
1.)1974([221]T.P.Singh,Gamma-RayBurstsandQuantumCosmicCensorship.GeneralRelativityandGravita-
tion30(11)(1998)1563.
[222]W.deSitter,OntheRelativityofInertia:RemarksConcerningEinsteinslatestHypothesis.Proc.
Kon.Ned.Acad.Wet.19(1917)1217.
OntheCurvatureofSpace,ibid.20(1917)229.
[223]D.N.Spergeletal.,WilkinsonMicrowaveAnisotropyProbe(WMAP)ThreeYearResults:Implica-
tionsforCosmology.AsrophysicalJournalSupplements170(2007)377,astro-ph/0603449.
[224]A.A.Starobinsky,SpectrumofRelictGravitationalRadiationandtheEarlyStateoftheUniverse.
JournalofExperimetnalandTheoreticalPhysics30(1979)172.
[225]K.S.Stelle,RenormalizationofHigher-DerivativeQuantumGravity.PhysicalReviewD16(4)
953.(1977)[226]O.SternandW.Gerlach,DerExperimentelleNachweisderRichtungsquantelungimMagnetfeld.
ZeitungfürPhysik9(1)(1922)349.
O.SternundW.Gerlach,DasmagnetischeMomentdesSilberatoms.ZeitungfürPhysik9(1)(1922)
353.[227]H.Suganuma,S.SasakiandH.Toki,Colorconfinement,quarkpaircreationanddynamicalchiral-
symmetrybreakinginthedualGinzburg-Landautheory.NuclearPhysicsB435(1995)207.
[228]J.TarterandJ.Silk,CurrentConstraintsonHiddenMassintheComaCluster.QuarterlyJournal
oftheRoyalAstronomicalSociety15(1974)122.

212

BIBLIOGRAPHY

[229]J.R.ThornstensenandR.B.Partridge,CancollapsedstarsclosetheUniverse?.AstrophysicalJour-
nal(ApJ)200(1975)527.
[230]S.Tremaine,J.E.Gunn,DynamicalRoleofLightNeutralLeptonsinCosmology.PhysicalReview
Letters42(1979)407.
[231]M.J.Valtonenetal.,AMassiveBinaryBlack-HoleSysteminOJ287andaTestofGeneralRelati-
vity.Nature452(2008)851.
[232]M.Veltman,SecondThresholdinWeakInteractions.ActaPhysicaPolonicaB8(1977)475.
[233]D.V.VolkovandV.P.Akulov,PossibleUniversalNeutrinoInteraction.JournalofExperimentaland
TheoreticalPhysicsLetters16(1972)438.
[234]R.V.Wagoner,Scalar-TensorTheoryandGravitationalWaves.PhysicalReviewD1(12)(1970)
3209.[235]S.Weinberg,GaugeandGlobalSymmetriesatHighTemperature.PhysicalReviewD9(1974)
3357.[236]S.Weinberg,FromBCStotheLHC.CERNCourierJan./Feb.200817ff.
[237]J.WessandB.Zumino,SupergaugeTransformationinFourDimensions.NuclearPhysicsB70
39.)1974([238]C.Wetterich,CosmologieswithvariableNewtonsConstant.NuclearPhysicsB302(1988)668.
[239]C.Wetterich,TheCosmonModelforanAsymptoticallyVanishingTime-DependentCosmological
Constant.AstronomyandAstrophysics301(1995)321.HD-THEP-94-16.
[240]C.Wetterich,AreGalaxiesCosmonLumps?.astro-ph/0108411(2000).HD-THEP-01-35.
[241]C.Wetterich,CosmonDarkMatter?.PhysicalReviewD65(2002)65(123512).HD-THEP-01-36.

[242]C.Wetterich,ProbingQuintessencewithTimeVariationofCoupling.hep-ph/0203266(2003).HD-
THEP-02-11.[243]B.S.deWitt,TheFormalStructureofQuantumGravityinM.LeviandS.Deser(eds.),Recent
DevelopmentsinGravitation275ff,(NATOASISer.B44)PlenumPress,NewYorkandLondon
.1979[244]C.S.Wu,E.Amber,P.Hayaward,B.HappesandR.Hudson,ExperimentalTestofParityConserva-
tioninBetaDecay.PhysicalReview105(4)(1957)1413.
[245]C.N.YangandR.L.Mills,ConservationofIsotopicSpinandIsotopicGaugeInvariance.Physical
Review96(1954)191.
[246]H.Yukawa,OntheInteractionofElementaryParticles.ProceedingsofthePhysico-Mathematical
Society(Physico-Math.Soc.)ofJapan17(1935)48.
[247]A.Zee,Broken-SymmetricTheoryofGravity.PhysicalReviewLetters42(7)(1979)417.
A.Zee,SpontaneouslyGeneratedravity.PhysicalReviewD23(4)(1981)585.

BIBLIOGRAPHY

[248]W.Zimdahl,DarkEnergy:AUnifyingView,arXiv:0705.2131v1[gr-qc](2007).

213

[249]W.Zimdahl,AcceleratedExpansionThroughInteraction,arXiv:0812.2292v1[astro-ph](2008).

[250]D.Zwanziger,ExactlySolubleNonrelativisticModelofParticleswithBothElectricandMagnetic
Charges.PhysicalReview176(1968)1480.

[251]D.Zwanziger,Local-LagrangianQuantumFieldTheoryofElectricandMagneticCharges.Physi-
calReviewD3(1971)880.

[252]G.Zweig,AnSU(3)ModelforStrongInteractionSymmetryanditsBreakingI.CERN-8182-TH-
).1964(401

[253]F.Zwicky,DieRotverschiebungvonextragalaktischenNebeln.
110.

ActaPhysicaHelvetica

6(

)1933

214

BIBLIOGRAPHY

215

216

EXTENDEDLISTOFMATHEMATICALSYMBOLS

oflistExtendedsymbolsmathematical

Description/NameSymbol

subscriptCurrent-time0ateconjugxComple∗daggertranspose,Hermitean†ransposeTToperatoredgeW∧operatorNablaoperatordAlembert|v>,|w>...Diracvectors(kets)

Example/Definition

0Current-timesubscripta0,t0,...
∗ComplexconjugateCx=a+ib=⇒x∗=a−ib
†Hermiteantranspose,dagger(A†)ij=Aj∗i⇐⇒A=A†
TTranspose(AT)ij=(A)ji,(AB)T=BTAT
Hermiteanmatrix:Aij=Aj∗i⇐⇒A=λiuiui†withλi∈R
i∧Wedgeoperator(a∧b)µν=aµbν−aνbµ
Nablaoperator=∂iei
i2∂1dAlembertoperator=c2∂t2−Δ
|v>,|w>...Diracvectors(kets)Scalarproduct:<f|g>=ˆf∗gdr
Matrixelement:Akl=<k|Aˆ|l>=<k|Alˆ>
Decomposition:Aˆ=|k>Akl<l|
,lkDecompositionofunity:|k><k|=1
kAverageonψbasis(expectationvalueofAˆ):
<ψ|Aˆ|ψ>=<ψ|ˆAψ>=<Aˆ>ψ
squareMean>X<A,ˆBˆ,...OperatorsHamiltonoperator:Hˆ|ψ>=En|ψ>
A(iBk)AntisymmetricBachparenthesisA(iBk)=1(AiBk+AkBi)
2AAB,BAB2ndrankSpinorsAAB=1γµABAµ
2Aµ,Bµ,...Gaugepotentials(fields)Minimalcoupling:Dµ=∂µ+igAµ
Zboson:Zµ=W3µcosϑW+AµsinϑW
Photon:Bµ=−W3µsinϑW+AµcosϑW
Aµ,Bµ,...DualgaugefieldsB=×A+A
A˜µ,B˜µ,...Transformedfields(A,˜B˜)T=R(ϑ)(A,B)T
Aµ,Bµ,...Gaugepotentials(matrixnotation)Aµ=Aµiτi
aµ,bµ,...4-vectorsComponents:xµ∈{x0,x1,x2,x3},µ=0,1,2,3
xµ,xνLorentztransformedcoordinatexµ=Λµνxν
ψ¯Adjointconjugateψ¯=ψ†γ0

SymbolName/DescriptionDefinition/Example
αPhasev˜=veiα
ABαRN-likechargetermα=−2=−Q˜2
α˘Gravitationalstrengthα˘=∼(MP/MB)21
αReggeslopeparameterα=(2πσ)−1
ΓβµνChristoffelsymbolΓβµν=21gµα(gβα,ν+gαν,β−gβν,α)=Γνµβ
GR:vµ;ν=vµ,ν+Γβµνvβ
Γµαµ=(√−g),α/√−g
γCharge–couplingratioγ=gQ
γPolytropicindexp=wPγ
γµABDiracmatrices(spinor)Cliffordalgebra:γµγν+γνγµ=2ηµν1,µ,ν=0,...3
㵆=±γµ
Spinor:AAB=1γµABAµ
2γ4Projectoroperatorγ4=i1εαβµνγαγβγµγν,(γ4)2=1,{γ4,γµ}=0
4!ΔStandarddeviation(RMS)ΔX=<(X−<X>)2>
ΔDensityratioΔ≡ˆ/∗
ΔDifferenceΔx=x2−x1
2ΔLaplaceoperatorΔ=2=∂i2=∂∂r2+r2∂∂r
it1∂L∂L
δVariationδS(q,q˙)=t0∂qkδqk+∂q˙kδ(q˙k)
2δδ≡a(tq)1
lδijKroneckersymbolδij=0,fori=j,δii=1
∂∂µPartialderivative∂µ=∂xµ
Energydensity=c2
Effective:∗=/(1+ξ)
2Newtonian(baryonic):∗=4πGvtNr2
2cξScalar-field:ξ=8πGNl2
DarkMatterprofile:ˆ=∗+ξ=DM
Critical:c=3H02c2/(8πG˜)
Total-energy:T=+Λ
3c2a˙v2ξ˙
Cosmological-term:Λ=V−8πG0aξ˙+81+ξ
SF-derivative:I=Λ−V
Matter:M≈+R+ν
Bayonic:BRadiation:RNeutrino:ν(current:ν0ν≈10−5h−2)
Galactic:g

4γΔΔΔΔδδδij∂µ

217

218EXTENDEDLISTOFMATHEMATICALSYMBOLS
SymbolName/DescriptionDefinition/Example
0Minimumenergy–densityHiggs:0=−V˘=min
εPermittivityVacuum:ε0≈8.8542∙10−12C2N−1m−2
1/(µε)=c2
εGeodesicparametergµνgµν=−ε
1=εe:SpacelikLightlike(null):ε=0
Timelike:ε=−1
εIntegrationconstantofKeplerorbit0<<1:Ellipse
εijk(Levi–Civita)totalskewsymmetrictensorε123=1,ε321=−1,ε112=0,...
εµνκλ4thrankLevi–Civitatensor
ηµνMinkowskimetricηµν=diag(−1,1,1,1)
ϑWWeinberganglesin2ϑW≈0.21,tanϑW=g1/g2
κGinzburg–Landauparameterκ=mΦ
mVκGravitationalcouplingparameterκ=8π4G
cEffective:κ˜=8πG˜
4cΛCosmologicaltermFunction:Λ(ξ)=4πGeffV(ξ)
Λ∗=Λ+Λ0
ΛConstant:0K3Planck:ΛP=lP2
ΛµνLorentztransformationxµ→xµ=Λµνxν
µΛµν=∂xν
x∂λGaugeparameterGauge:ψ=eiλτiψ
λHiggsparameterV(φ)=µφ†φ+λφ†φ,v=±−6µ
222
λ4!2λMetriccomponent(gravitationalpotential)ds2=eν(dct)2−eλdr2−r2dΩ2
λiEigenvaluef(ui)=λiui,λi∈C,ui∈C,i∈{1,...n}
2µHiggsparameterV(φ)=µφ†φ+λ(φφ)2
4!22Ground-statevalue:v=±−6µ
λµPermeabilityVacuum:µ0=4π∙10−7N/A2
Dual:µ2(p2,Φ0)=1−p2mV−2
µf=2-lepton:muone2=µ
µ(a/a0)MONDparameterµ(a/a0)=1(=a/a0)foraa0(aa0)
NeutrinoννMetriccomponent(gravitationalpotential)ds2=eν(dct)2−eλdr2−r2dΩ2
Centralsymmetry(linear):ν=−rdyn
rPotential:Φ=ν/c2

SymbolName/DescriptionDefinition/Example

ξΠ

2ρσσσ˜σiτττiτΦφ,ΦˆφˆφφABϕϕχχΨΨ

†(Square-root)scalar-fieldexcitationξ=φ2φ−1
vξ=G(v)−G˜
˜GPolarizationDual:˜Π(p2,Φ0)=−mV2/p2
density:BaryonicDensityBCritical:c=3H02/(8πG˜)
Planck:P=∼c25=∼1093g/cm3
G0Squarescalarfieldφφ=ρUN
Afterunitarygauge:φ†φ=ρ2,φ=ρN
MagneticchargedensityDual:∙B=σ
Stringtensionoffluxtubeα=(2πσ)−1
√Higgscomponentφ0=(σ˜+iχ)/2,<σ˜>=v
PaulimatricesSpin:Sˆi=σi,
2Cliffordalgebra:{σi,σj}=2δi
f=3-lepton(tauon)e3=τ
VolumeDual:σdτ=g
parameter)fine(afEigentimeGroupgeneratorU=eiλτi,
[τi,τj]=ifijkτk,{τi,τj}=cij1+dijkτk
Potential∙Φ=F
Gravitational:Φ=2ν2
cScalarfieldSymmetry-broken:φ=ρN=v1+ξN
Ground-state:Φ0,φ0,φa(0)
Scalar-fieldexcitationφ=vN+φˆ
√BWscalarfieldL(φˆ)BW=16−πgˆφR
PathamplitudefromAtoBφBA[C]=eiS[C]
Scalar-fieldexcitationφa=(1+ϕ)φa(0),
AngleKepler:u0=2r˜CS2c2(1+εcosϕ)
bGoldstonecomponentφ0=(σ˜+iχ)/√2,U=eiχ
distanceariantvCoGravitationalpotential(Newton)Ψ=Φ+c2ξ
2FluxColor-electriccharge:ΨE=E∙dSnΨ0
Ψ0=2π/Q

φfieldscalarSquaredensitygecharMagnetictubefluxoftensionStringcomponentHiggsmatricesauliP(tauon)f=3-leptonolumeVparameter)fine(afEigentimegeneratorGroupPotentialfieldScalarxcitatione-fieldScalarfieldscalarBWPathamplitudefromAtoB
xcitatione-fieldScalarAnglecomponentGoldstonedistanceariantvCowton)(NepotentialvitationalGraFlux

219

220

NameSymbolDescription/

ψψµνΩΩΩ

ΩΛΩIΩIIω

AAUAaa

aa0aH

Wavefunction/Quantumstate
graLinearfieldvitationalsphereUnitelocityvAngularparameterDensity

termCosmologicalparametercouplingJBD

xact)(eamplitude-fieldScalarindeSpinorxunitAstronomicalxindeIsospinactor)(fparameterScale

EXTENDEDLISTOFMATHEMATICALSYMBOLS

Example/Definition

ψSpinor:AψIsospinor:aψµν=hµν−21hηµν,h=hµνηµν,|hµν|1
dΩ2=dϑ2+sin2ϑdϕ2
dϕ=ΩdrΩi=8πG(2ξ)i=i
H3cMatter:ΩM=0.127h−2
Baryons:ΩB=0.0223h−2
DM:ΩDM=0.105h−2
Neutrinos:Ων<0.007h−2
Cosmological-constant:ΩΛ=0.76
Total:ΩT=1.00300..013017
Densityterm:Ω=8πG2(ξ2)=ΩB
3Hc1+ξ
Pressureterm:Ωp=8πG(ξ)p
22cHΩ=Ω+Ωp
8πG(ξ)Λ
ΩΛ=3H2c2V=3H2
ΩΛ∗=8πG2(ξ2)Λ
cH38πG(ξ)ξ˙
ΩI=3H2c2=cI=−H(1+ξ)
ΩII=83πHG2(cξ2)(pΛ+V)−32ΩI

Scalar-fieldamplitude(exact)A=−2G0T√−gd3x,(l→∞)
2c3SpinorindexψA,γµAB
Astronomicalunit1AU=149,597,870km
fIsospinindexElectroweak:ψa=νfe
e22Scaleparameter(factor)a2=a˙+K2c
HΩTCurrent:a(t0)=a0
Primeval(statical):a(tq)=aq
Planck:a(tP)=aP
system)spherical(ofradiusScaleCriticalacceleration(MOND)a0=1.2∙10−20ms−2
HubbledistanceaH=Hc

SymbolName/DescriptionDefinition/Example

BBBµBBikbbCCCCaCb˜CµνccijckDDDDDµ

µDMDddL2dssdijdk

BaryonsubscriptΩB,B...
NewtonianfieldamplitudeB=2Mcdy2nGN
Magneticfield/magneticinductionEM:∙B=0
Photonfield(emgaugeboson)Bµ=−W3µsinϑW+AµcosϑW
iMagnetictensorBik=g[Di,Dk]
EM:Bkj,i+Bik,j+Bji,k=0
Blue(strongcolorcharge)QCD:ψa1=bau
Bottom-quark(flavor)GSW:ψLq=6=bL
Color(charge)subscriptU(3)C
Integrationconstantξ=Ce−r/l
r√IntegrationconstantforminkoskianlimitC=√K2+4α+K
K2+4α−K
ParametrizedenergyconstantGeodesics:Ca=1−r˜Srdyn/r˜Sdct=const.
dτrMomentumconstantL=mCb
Dualfield-strengthtensortoFµνC˜µν=A˜ν,µ−A˜µ,ν
Lightspeed(lat.celeritas)SI:c=2.99792458∙108m/s
1cnatural:≡Charm-quark(flavor)ψLq3=cL
Structureconstant{τi,τj}=cij1+dijkτk
DyonsubscriptDyonicaction:SD
Integrationconstantν=−Ce−r/l−D
rrDifferential:summationoverallpathsZAHM=DC˜µΦe−SAHM[C˜µ,Φ]
ElectricdisplacementfieldD=ε0E+P
DCovariantderivativeDµ=Dxµ=;µ
SM:Dµ=∂µ+igAµ
GR:Dµ=∂µ−Γναλ
Dual:Dµ=∂µ−ieAµ−igAµ
Covariantderivative(matrixnotation)SM:Dµ=1∂µ+igAµ
DistancemodulusDM=m−M=−5+5logdL(pc)
Down-quark(flavor)GSW:ψLq2=dL
LuminositydistancedL=a02ar=Hc0z+21(1−q0)z2+...
Lineelementds2=dxµdxµ=dxµdxµ=gµνdxµdxν
µ,νµµElementofareaDual:A∙ds=g
SStructureconstant{τi,τj}=cij1+dijkτk

221

222

Description/NameSymbol

EXTENDEDLISTOFMATHEMATICALSYMBOLS

Example/Definition

EIntegrationconstant,energyGeodesics:t˙=Ee−ν
2drEEffectiveenergyGeodesics:E=dt+Veff
EPPlanckenergyEP=∼mPc2=∼1019GeV
eVelectronvolt1eV≈1.602∙10−19J
eV/c2energy–massequivalent1eV/c2≈1.783∙10−36kg
orce4-FFFµνField-strengthtensorFµνab={Aνi,µ−Aµi,ν−gAµkAνlfkliig[Aµ,Aν]}(τi)ab
Fµνab=Fµνiτiab
Fµν∗Dualfield-strengthtensorFµν∗=1εµνκλFκλ
2FµνField-strengthtensor(matrixnotation)Fµν=Aν,µ−Aµ,ν+ig[Aµ,Aν]
Fµν=Fµνiτi=F†µν
F˜µνField-strengthofgrav.Maxwelleqs.F˜µν=(uσ,µ−uµ,σ)=Hνµ−Hµν
1f(t)f=2(f2−f1)
f(χ)f∈{sinχ,ξ,sinhχ}forK∈{1,0,−1}
f1(r/l)f1=411−e−r/l
a˙ξ˙πξ˙2
f1(t)f1=−a1+ξ+3α˘(1+ξ)2
G˙(ξ)a˙πG˙(ξ)
f1(G)f1=G(ξ)a+3α˘G(ξ)
f2(r/l)f2=11+1+re−r/l
l4ξ¨a˙ξ˙πξ˙2
f2(t)f2=−1+ξ−2a1+ξ−α˘(1+ξ)2
aG(ξ)G(ξ)2α˘G(ξ)
f2(G)f2=2a˙G˙(ξ)−G˙(ξ)22+π+G¨(ξ)
1GGravitationalcouplingFunction:G(φ)≡α˘φ†φ
λ11Constant:G0≡G(v)=α˘v2−α˘6µ2
Newtons:GN≈6.674∙10−11m3/(kg∙s2)
4qˆ=1:GN=3G0
G0Effective:G˜=G(ξ)1+ξ
GµGluonfield(stronggaugeboson)QCD:Dµ=∂µ+ig3Gµiτi
2GFFermisconstant(GcF)3=(8m22g)≈1.166∙10−5GeV−2
W1GµνEinsteintensorGµν=Rµν−2Rgµν(+Λ0gµν)

GµGluonfield(stronggaugeboson)
constantsFermiGFtensorEinsteinGµνGL(4,R)Generallinearreal4-dimgroup

Description/NameSymbol

constantCouplingg

/DefinitionExample

√gCouplingconstantgem=e/c
Dualmagneticcharge:τ∙Bdτσdτ=g
Diracquantization:e∙g=2n
gMetricdeterminantg=detgµν
gGreen(strongcolorcharge)QCD:ψL2q=1=gLu
g˜Exponentialmetricparameterg˜≡g˜(r)=e(λ−ν)/2
ξ+1rg˜=α˜g3−Kg˜2−g˜
gµνMetricaltensor(physical)aνbν=gνµaµbν
νµ,νHHiggssubscriptLH=1φ,,µ†φ,µ−µ2φ†φ−λφ†φ2
4!22HHubbleparameterH=a˙=100hkm/s/Mpc
aHMagneticfield,magnetizingfieldH=B/µ0−M
HµνSymmetric(grav.)field-strengthpartHµν=uµ;µ−uα;αgνµ
hReducedHubbleconstanth≈0.74
hPlancksactionquantumh≈6.626∙10−34Js≈4.136∙10−15eVs
1−f2+2γ(1+2f2)
hLinearmetricparameterh=1−f1+23γ1−21f1
w8+1rl:h=2+3w
ReducedPlancksactionquantum=h/(2π)≈1.055∙10−34Js≈6.582∙10−16eV∙s
hµνDeviationfromMinkowskimetricWeakfields:gµν=ηµν+hµν,|hµν|1
iImaginaryuniti2=−1
JSpinJ=α0+αMJ2
Jµ,jµCurrent4-current:jµ=(ρ,j)
Magneticcurrent:js=×E˜
JµCurrent(matrixnotation)Jµ=Jµiτi
KCurvatureconstantK∈{−1,0,1}
KMassparameterK=2A+B
KgGluon-fieldenergyDME:Kg=E˜2/2
K[b,a]Feynmanpropagator,kernel,pathintegralK[B|A]=[dC]φBA[C]
KµForceKµ=kµd3x
kMagneticcurrentdensityDual:×E+∂∂tB=−k
Color-forcedensityoffluxtube:ks=∙P˜=js×E˜
kBBoltzmannconstantkB≈1.3807∙10−23m2kgs−2K−1
kµForcedensityKµ=kµd3x

223

224

Description/NameSymbol

EXTENDEDLISTOFMATHEMATICALSYMBOLS

/DefinitionExample

luminosityAbsoluteLLAngularmomentumConjugatemomentum:L=mr2dϕdτ
LLagrangian(Lagrangefunction)L=T−V
LLeft-handedsubscriptψa,L=νe=1(1−γ5)νe
e2eL√LLagrangian(Lagrangedensity)L=L−gd3x
λPenetrationdepthME:λ=(ev)−1
1+4π1/2
l(Compton)lengthscale(penetrationdepth)l=Mc=16G0(µ34α˘/λ)
lLeptonindexl={ef,νf}
aLlApparentluminosityl=4πq20r2a0
llaLengthscalepersphericallengthla=a
lyLightyear1ly=9,460,730,472,580.8km
MHiggsmassM=−8πµ2c1/2
α˘3MHStandardHiggsmassMH2=−2µ2
massSystemM1M1∗Dynamicalmassforp=0M1∗=M1=M1(1−f1)
MdynDynamicalmass2MdynGN=Mdyn=2M1GN1−f1+23γ1−21f1
rl:Mdyn=1+23wM1
M˜EffectivemassM˜GN=A+B
2M(i)Mass-squarematrix(diagonal)M(i)=2√πcgv(τiN)†(τiN)
SU(2):(M2(2))ij=πg22cv2δij=MW2δij
MWW-weakonmassMW=πg22cv2=80.398±0.23GeV/c2
MZZ-weakonmassMZ=MW/cosϑW=91.1876±0.0021GeV/c2
MMagnetizationB=µ0(H+M)
magnitudeApparentmmFermionicindexm=l,m=q
mMassPlanck:=∼PlP3=∼10−5
Fermionic:mf=GfvN†xˆ+xˆ†N
(Dual)scalarmass:mΦ=2λns(Φ)
(Dual)vectormass:mV=Qns(Φ)
Fermionicmassoperator:mˆ=Gfv(N†xˆ+xˆ†N)
NGaugefixingparameterN=(1,0,...)T,⇒φ10=v,φ20=φ30=...=0
NIsotopicparameterSU(N):2N2−1gaugebosons
NNewtonsubscriptGN≈6.674∙10−11m3/(kg∙s2)
ns(Φ)Dyon/monopoledensityns(Φ)=|Φ|=Φ02

parameterfixingGaugeNparameterIsotopicNsubscriptwtonNeNns(Φ)Dyon/monopoledensity

SymbolName/DescriptionDefinition/Example

225

OOrdermaximalorder:r3⇔O(r3)
O(3)(Orthogonal)rotationgroupx→x=Rx,RT=R−1forallR∈O(3)
O(3)/SO(3)FactorgroupofLorentzgroupO(3)/SO(3)={1,−1}
PPerihelionsubscriptΔφP=6M˜2GNπ
CbPPolarizationvectorD=ε0E+P
L∂pMomentumpk=∂q˙k
pPressureBarotropicpressure:p=w
Effective:p∗=p/(1+ξ)
2˙2Scalar-field:pΛ=−31V+8πcG0ξ¨+2aa˙ξ˙−3πα˘1ξ+ξ
1Bounce:pG=8πG0ξ¨(tq)
pcParallaxofarcsecond(parsec)1pc=1AU/tan1=3.2615668ly=30.856776∙1015m
QChargeDyons:Q=e2+g2
Q˜2Chargeparameter(generalized)Q˜2=2AB
QµνQµν=Hνµ+Hµν
qDecelerationparameter(cosmic)q=−aa¨
aqEffective:q˜=1+Kc2
aq(Logarithmic)scalar-fieldexcitationq=ln(1+ξ)
q˜Effectivedecelerationparameterq˜=1+qKc2=211+3pTT
2a˙qˆMatter-LagrangiancouplingFermioniccoupling:ˆq=1
Quintessentialcoupling:qˆ=0
RRiccicurvaturescalarR≡Rµνgνµ
RµνRiccicurvaturetensorRµν≡Rµνλσgµσ
RλµνκRiemanncurvaturetensorRµσαν=−[Dν,Dα]
R0,R1Galacticcore(bulge)
R(ϑ)Symmetryoperator(E˜,˜B)T=R(ϑ)(E,B)T
4fieldreal-component4-dimensionalRrRadius,distanceHaloradius:rH
DynamicalSchwarzschild:rdyn=2Mdy2nGN
cCharge-parameter:rQ=|Q˜2|=|Ar˜S|
2Schwarzschild:rS=2Mc12GN
Eff.Schwarzschild:r˜S=2A+rdyn≈h(w)rdyn
raDistanceperscalera=ar
t2SActionS=t1Ldt
SAreaDual:∙Bdτ=Bds
Sτ

pcQ2˜QQµνqqq˜qˆRRµνλRκµνR,R10)ϑ(R4Rr

raSS

scaleperDistanceActionArea

226

EXTENDEDLISTOFMATHEMATICALSYMBOLS

SymbolName/DescriptionDefinition/Example

SµνGeneratorofGL(4,R)Sµν=λiSiµν
Lµν=eλiSiµν
SO(3)SpecialrotationgroupSO(3)⊂O(3),detR=+1forallR∈SO(3)
SO(3,1)LorentzgroupGeneralizedorthogonalLiegroup
SO+(3,1)RestrictedLorentzgroupdetΛ=1,Λ00≥1
SU(N)N-dimensionalspecialunitarygroupUab=eiλiτiab,i=1,...N
sEnergydensityGravity:s=sµuµ
sStrange-quark(flavor)GSW:ψLq=s4=sL
sµEnergy–momentumtensorGravitation:F˜µλ;λ=2κ˜(jmu+sµ)
TEnergy–momentumtraceT=iψ¯γµL,Rψ;µ+h.c.=1+ξψ¯ˆmψ
2Idealgas:T=−3p
TKineticenergyL=T−V
Gell-Mann–Nishijima:Q=T2+21Y
TTotal-subscriptΩT=1⇐⇒K=0
signransposeTTIdealgas:Tµν=(+p)uµuν−pgµν
TKineticenergydensityT=T√−gd3x
T3IsospinoperatorT3ψ=τ3ψ,τi=1σi
2TµνEnergy–stresstensorTµν=∂ψ∂LψaA,µ+¯∂aALψ¯aA,µ−Lδµν
ψ∂,νaA,νtTop-quark(flavor)GSW:ψLq=5=tL
tTimePrimevaltime:tq
ttime:PlanckP1UBWcosmologicaltermU(φˆ)=α˘φ†φ[8πV∗(φ†φ)]
uUp-quark(flavor)GSW:ψLq=1=uL
uPotential-termmixtureu=λ+ν
uEnergydensityofemfieldsu=21ε0E2+µ1B2
0uReciprocatedistanceu=1/r
NewtonianKeplerorbit:u0=r˜S2c2(1+εcosϕ)
C2buapproximation:Second-order1uUp-quark(flavor)GSW:ψLq=1=uL
uiEigenvectorf(ui)=λiui,f∈L(U,U),n:dimensionofU
uµ4-velocityuµ=dxµ
dsu0=e−ν−v12e−λ−1/2
2cu1=u0vc1
Orthonormality:uµuµ=1

uuiµu

or)v(flaUp-quarkectorvEigenelocity4-v

Symbol

V

Name/DescriptionDefinition/Example

VPotentialCoulomb:V∼1/r
Yukawa:V∼e−r/l
r4Λµ30Higgs:V(ξ)=2λξ2+8πG0
V∗(φ)=V(φ)+V0
2Constantterm:V0=−3α˘µΛ0
λπ42˜2˜Eff.RN-like:Veff=εMGN+L2−MG3NL
rr2r4Formalconstantterm:V˘=23µλ
VPotentialenergydensityV=V√−gd3x
vGround-state(VE)valueexp:v≈6.07∙104(GeV)2
v=±−6µ2
λv1,vVelocityu1=u0vc1
vtTangentialvelocityvt=rdΦ
drwEOSparameterBarotropic:p=w
pDMequation-of-state:wˆ=ˆ
pSFequation-of-state:wξ=ξ
wPotential-termmixturew=λ−ν
wPPolytropicamplitudep=wPγ
14K39K23Q˜2
X(A,B;r−n)RN2nd-ordercorrectionX=2r2+3r3+6r4+4r4+...
X1(A,B;r−n)RN2nd-ordercorrectionX1=X21−rK+Q˜2X
−1
X2(A,B;r−n)RN2nd-ordercorrectionX2=X1+X2a
KQ˜22KQ˜2−1
X2a(A,B;r−n)RN2nd-ordercorrectionX2a=121−r−r2221+Q˜41−Kr−Q˜r22
r1−rK+rQ˜22r1−r+r2
x(Cartesian)Coordinatexµ=x0=ct,x1=x,x2=y,x3=z
xˆYukawamatrixLY=−kf(ψ¯φ†ˆxψ+ψ¯ˆxφψ)
YHyperchargeU(1)strength:F(1)µν=F(1)µνY
Gell-Mann–Nishijima(withoutquarkness):Q=T3+1Y
2ZPartitionfunctionZAHM=DC˜µDΦe−SAHM[C˜µ,Φ]
ZµNeutralweakonZµ=W3µcosϑW+AµsinϑW
2groupsymmetryDualZzRedshiftz=λ0−λemission
λemission

227

228

EXTENDED

LIST

OF

TICALTHEMAMA

SYMBOLS

Index

qˆparameterofthefermioniccoupling,65,66,68–Bachparenthesis,seeAntisymmetric(Bach)paren-
71,73,80,82,115thesis
4-vector,90,166,167,169,170,189Barotropicmatter,82,87,90,118,189
null,5BCSpairs,seeCooperpair
spacelike,5Besselfunction,192
timelike,5Bianchiidentities,42,65,171,179
155153,21,Bang,BigAbelianHiggsModel,46,47Bogoliubovtransformation,47
Abelianprojection,45Bose-Einsteinstatistics,15,41
Action,166,173Bosons,4,15,17,27,57
18Bremsstrahlung,46action,AHMpartitionfunction,46Brout-Englert-Higgsmechanism,seeHiggsmecha-
nism45action,DyonPartitionfunction,45Cabibbotransformation,182
Ginzburg-Landauaction,46,49Canonicalform,seeEinsteinframe
Hilbert-Einsteinaction,6,173Centralsymmetry,seeSphericalsymmetry
58action,Jordan36CERN,Jordan-Brans-Dickeaction,59Chaplygingas,seeDarksector
Zwanzigeraction,45Christoffelsymbol,4,12,170,171,190,191
Partitionfunction,45Color,11,14,29,183,185
Adjointconjugate,10Colorforce,53
Affineconnection,170Comptonlengthscale,seeLengthscale
AlternativetheoriesConfinement,14,45,46,49,51,52,186
Branetheories,58Cooperpair,28,47,49,50
Kaluza-Kleinstheory,10,58Cosetclass,4
MOG,20,118Cosmicacceleration,21,23,118,126,127,129,
STVG,20150,153,157,197
Superstringtheories,17,58CosmicCensorshipConjecture,95
Constraints,71CosmicMicrowaveBackgroundRadiation,18,58,
Technicolor,61126,135
TeVeS,20CMBanisotropy,58
Zeesbroken-symmetricgravitation,60Cosmologicalfunction,21,22,30,59,60,62–64,
Antimatter,10,13–15,18166,72,84,117,122,156
Antisymmetric(Bach)parenthesis,5,66,179constant,6,20–22,30,64,67,90,145,172,
Asymptoticfreedom,seeConfinement174,196
ATLAS,36CosmologicalPrinciple,115,129
229

INDEX230Cosmon,21,70Diracequations,66,68,180,184,185
Couplingconstant,7,10,34,58–61,63,68–70,72,Diracmatrices,10,66,68
74,75,84,86,172,183Cliffordalgebra,180,183
effective,66,72,80,84,116,117,121Diracmonopole,seeDualsymmetry,seeMagnetic
Criticaldensity,seeEnergydensitychargedensity
CurvCurrentature,conserv22,195,ation,197seeElectricchargeDualDiracsymmetry,monopole,39,40,40,4941,43–45,49
DAlembertoperator,81Diracstring,41
DarkMatterprofile,seeUniversalhaloprofile,112Dyons,Dyon41,46,condensation,49,52,5353
Darksector,70Dyondensity,50
DarkEnergy,21–23,25,64,70,91,96,102,Magneto-electriccharge,46,49,50
107,119,126,130,133,135,174Monopolecondensation,46
Chaplygingas,121Abrikosovstring,46
CosmicCosmologicalacceleration,constant,seeseeCosmicCosmologicalaccelerationfunc-Dynamicalmass,seeEffectivemass
71xperiment,eEötvöstionDarkMatter,18–20,22,23,83,88,89,91,93,Effectivemass,81,85–89,91,93,98–101,103,108
Axions,102,19106,108–111,198DME,Ginzbur28g-Landau,48
BaryonicDM,17,19Glueballs,49
CDM,18,19,110,133,198Eigenstate,seeQuantummechanics
Gravitons,17Einstein(curvature)tensor,172
HDM,18,133Einsteinequations,6,58,65,71–73,80–82,93,115,
LeptonicDM,18,19172,174
PregSelf-Interactingalacticmatter,DM,1870EinsteinEinsteintensorframe,,6,59,21118
Supersymmetricmassconstraints,19Einsteinssummationconvention,166
Supersymmetricparticles,17,19Einstein-deSitterUniverse,21,22,144,195
FLAMOND,G,2020ElectricCharchargege,operator34,,45,184182
Supersymmetricmassconstraints,19Currentconservation,26,180
Decelerationparameter,117,126,127,129,131Quantization,41,53
Densityparameters,132,134,137,142,145ElectronSpinResonance(ESR),26
General,21Elementaryinteractions,3,28,30,57,60
Lambda,1345th-forceexperiments,71
Observationalconstraints,22,145,146,197,Electrodynamics,11,13,165
17829,Electromagnetism,198Screening,133,196Electroweakinteractions,12,13,29,181
Derivativeβdecay,18
Covariantderivative,4,12,32,45,47,65,169,βdecay,17
170,178,179,184,185Leptonicfields,181
Usualderivative,64,169Nuclearforces,18
deSitterepoch,seeInflationaryUniverseQuarkfields,181
Diracconjugate,183Isotopicparticles,11–14

231INDEXNuclearforces,3,9–11,13,14,16,57,182,179,180
183,185Idealliquid,79,90,189
QuantumChromodynamics,57,65Equationofstate,124,125
QuantumElectrodynamics,14,57,65,184Equation-of-stateparameter,82,87,89,103,108,
YukaStrongwatheoryinteractions,,911,12,14,16,29,183,186Constraints,109,130,89,136,91,100,137,106,139,189109,111
Elementaryparticles,177Dust,119,120,195,197,198
125Lambda,1715,Baryons,Baryonicnumber,182Radiativefluid,119,120,195
Graviscalars,57Scalar-fielddominance,134
Hadronicstructure,10,15,186EquivScalaralence-fieldprinciple,dominated58,71,Uni91verse,126,132
Hadrons,Hyperons,10,12,14,1615Euler-Lagrangeequations,31,42,57,65,166,179,
IsotopicLeptons,10,particles,13,15,14,1728,29,65,178External184transformations,3,167
Generations,seeGenerationsofleptonsFactorgroup,4
182,numberLeptonicNeutrinos,15,18,22,35FFermiamily,13,constant,18160,68
Mesons,Meson9–12,decay,1015–17,57,177Fermi-Diracstatistics,4,15
65Fermion,Partons,Nucleons,15,3,36,10,18211,16Fermions,4,10,15,17,29,34,65
27breaking,symmetryFerromagnetism:FlaQuarks,vor10,eigenstate,12–16,29,182182Field-strengthtensor,5,42,44,46,47,65,66,74,
185183,178,171,Generations,seeGenerationsofquarksField-stresstensor,167,184
EnergyMassbands,176eigenstate,182Flatrotationcurves,seeDarksector
Energydensity,32,40,67,74,75,81,172,195FlaFluxvor,tube,14,see16,26,35,Superconducti183,185vity
γprofile,seeUniversalhaloprofileFriedmannequations,116–118,122,124,132,140,
Cosmology,119,121,123143,147,148,150,152,156
Dust,120Correctionparameters,116,118,122,125
NeScalarwtonian-fieldcontribcontribution,ution,111111Friedmann-Lemaître-Robertson-Walkermetric,seeRobertson-
Criticaldensity,21,133,136,141,195Walkermetric
gravitation,90Gamma-rayburst(GRB),96
Gravitationaldensity,6,89,90Gaugebosons,4,11,13,15,16,28,29,34,60,68,
124Lambda,181NFWprofile,seeUniversalhaloprofileGluons,14,16,17,31,45,65,179,185
Screening,polytropic,91,83,109118,122–124,141GraPhotons,vitons,17,57,167,61,17864
Energy-stressconservation,31,58,69,70,74,115,Weakons,17,31,34,68,179,184,185
121Gaugefields,4,11,12,15,32,42,46,65,67,167,
Entropyprocess,115,121179,185
Energy-stresstensor,7,31,59,66,67,69,172,173,dual,41,43,44,49,51

INDEX232dualmass,50,52Hawking-Penroseenergycondition,146,147
Gaugeprinciple,12,168,179,185Heisenbergsuncertaintyrelation,9,155,177
Gausstheorem,174Hermiteanconjugate,61
Gell-MannGell-Mann-Nishijimamatrices,14,formula,18512,184Higgsfield,Excitation,25,33,27–34,34,47,68,49,72,61,80,64,82,67,83,9587,93,
Generallineargroup,16795,101,111,112,121,148,155,196
GeneralRelativity,4,57,60,64,72,80,81,87,90,Interfacedomains,32
95,169,173Topologicaldefects,32
EfGeodesicfectivemotion,potential,104,106106HiggsfieldScalarequation,-fieldseeequationScalar-Fieldequation,see
Keplerorbit,105Higgsforce,70
Horizonsingularity,91,95,97,99Higgsgravitation,30,61
Newtonianapproximation,81Higgsmechanism,seeSpontaneousSymmetryBrea-
king105ance,advPerihelionSchwarzschildmetric,91,100Higgsparticles,19,20,27,31,36,57,61,67
Generationsofleptons,17,181,183mass,60,69–71,96,97
Generationsofquarks,35,182,183constraints,70,71,117,145,156
Geodisicaltrajectory,5standard,32
Innermoststablecircularorbit,107Higgspotential,29–32,45,49,63,64,66,70,117
Georgi-Salammodel,seeGrandUnifiedTheoriesEffectivepotential,35
Glashow-Iliapoulos-Maiamimechanism,seeCabibboTachyoniccondensation,30,32
25–30mode,Higgs-KibbletransformationGlashow-Salam-Weinbergmodel,12,30,35,184Hubbleparameter,20,22,116,120,123,145,195
Gluon-fieldenergy,49Hypercharge,12,26,29,184
GoldstoneTheorem,27,36Hyperchargeoperator,184
Goldstonebosons,27,28,35,37,67
GrandUnifiedLepto-quarkTheories,decay,2928,29,41InertialInertialmass,forces,31,16971
Protonhalf-time,29InflationaryUniverse,22,23,25,62,126,127,147,
Gravitation,seeGeneralRelativity158
Gravitationalconstant,seeCouplingconstantBouncingUniverse,20,147,154
GraGravitationalvitationalmass,field-strength,31,716,seeEnergydensityInnerInflatonfield,transformations,223
Gravitationalpotential,seeSphericalsymmetry,seeInternalsymmetry,27
SphericalsymmetryIonizingradiation,18,198
Gravitationalstrength,60,61,63,64,66,68,70,72,Isobar
4nuclei,Mirror73Gravitationalwaves,82Isospin,3,4,11,13,14,26,65,179,181,184
Greystar,20,95,99,101Isospinoperator,184
Groundstate,27,29,32,33,35,49,50,60,61,67,Isotone,3
Group68,generator92,37,184Isotope,3,176
Groupparameter,179Jacobiidentities,171
Gyromagneticratio,176Jordanframe,59,60,64

233INDEXJordan-Brans-DickeTheory,seeScalar-TensorThe-Metricparameter,88
oriesMetricaltensor,4,6,58,81,165
170symbol,felChristofKleinfour-group,4Metricity,174
Klein-Gordonequation,seeScalar-fieldequation,82Orthonormality,74,75,165
Lagrangian,12,26,29,30,32,33,35,44,105,180Symmetry,73,165
169ransformation,TBergmann-Wagoner,62Minkowskimetric,seeSpecialRelativity,83,92,94,
183weak,electro95Ginzburg-Landau,47Mirrornuclei,seeIsobar
Higgsgravitation,64,65MONDscriticalacceleration,20
18044,term,interactionmatter,64Nambu-Goldstonemode,25,26,28,29
185strong,182oscillations,NeutrinoLargeHadronCollider,17,20,36Nobelprize,10,12,18,26,27,31,47,58,82,173,
Larmorfrequency,seeMagneticResonance,(N)MR175,176,182,186
Lengthscale,28,48,69–71,82,84,85,117,136,Nuclearreactors,18
177145,Liealgebra,168Paralleltransport,12,170
Liegroup,168Parity-symmetrybreaking,12,14,15,25,30,181
LIGO,82Pathintegral,seeQuantummechanics
Lineelement,5,166,167Pauliequation,175
eigenlength,5Paulimatrices,34,184
eigentime,5,105Peccei-Quinnmechanism,26
lightcone,5Penetrationdepth,seeLengthscale
LISA,82Penetrationdepth(Superconductor),seeSupercon-
LNThypothesis,18ductivity
Lorentzforce,3,5,42,180Penrose-Hawkingenergycondition,147,153,154,
Lorentzgroup,4,167,168157
Boost,4Perihelionshift,seeGeneralRelativity
Orthochronoustransformation,4Planckscale,61,68,155,156
Propertransformation,4Poissonequation,86,90,110
LorenzgroupPPNframework,88,89,108
restricted,4Principleofequivalence,31
Propagator,seeQuantummechanics
Machsprinciple,59Protondecay,seeGrandUnifiedTheories,41
Magneticcharge,45Pulsar,82
Magneticchargedensity,seeDualsymmetry
MagneticResonance,(N)MR,26,176QuantumChromodynamics,12,14,15,46,50
functionalMRI,176Helmholtzequation,52
Magneto-electriccharge,seeDualsymmetryQuantummechanics,165,192
Mass-squarematrix,33,185Decoherence,177
Maxwellequations,39,42–44,65,166,178ElectronSpinResonance,seeElectronSpinRe-
Maxwell-likeequations,5,74,89sonance(ESR)
Meissnereffect,seeSuperconductivityEntanglementandquantumteleportation,178

INDEX234Klein-Gordonequation,175Scalefactor,195,196
MeasurementproblemofQM,173,178Schwarzschildradius,85,86,88,91,94,100,101,
Collapseofthewavefunction,177104
Problemofdefiniteoutcomes,178Seriesexpansion,97,100,193
NuclearProblemofMagneticthepreferredResonance,basis,see178MagneticRe-SolarSpecial-relatiRelativisticvity,ef33,fects,165,59,166,89172,175,178
sonance,(N)MRMinkowskimetric,81
Observables,173,177Sphericalsymmetry,62,79,189,191
Operators,177Metriccomponents,80,82,84–86,88,90,92,
PathPostulates,integral,175,31,17735,45,176Metric94–97,parameter99,,100,87,104,88,100189
Quantizationofangularmomentum,175Spin,4,11,13,27,57,175,176,178
Quantuminformationtheory,178SpontaneousSymmetryBreaking,26,27,29–32,45–
Quantummechanicalstate,173,175,17747,50,60,61,67,92,190
Quasar,82Schrödingerequation,175Dynamical,Fundamental,2626
Quintessence,seeDarksectorGaugefixing,33,67
Quintessentialattraction,102–104Ginzburg-Landauparameter,29
Quotientgroup,seeFactorgroupHiggsmechanism,28,31,67
34fixing,GaugeRabiexperiment,175Unitarygauge,28,35
RabiRedshift,oscillations,119,127,176195YUnitaryukawga-Wauge,ick36interpretation,28
Reggeslope,seeSuperconductivityStandardModelofelementaryparticles,12,15,28,
Reissner-Nordströmmetric,99–10130,31,36,61,69
19398,97,correction,RNRNparameters,94,98,100,101,104,193Stiffness,Stern-Gerlachseeexperiment,Equation-of-state175parameter
RelativityRenormalizabilityprinciple,,12,16825,31,36,63,70,181SuperSuperconductinovaeofvitytype,27,Ia28,(SNeIa),45,47–4922,126,129,133
Powercountingcriterion,64Coherentlength,50
Ricci(curvature)scalar,6,30,57,58,61,64–66,73,Dual,49,52,53
171Color-fluxscreening,50,52,53
RicciRicci(curvidentities,ature)11,tensor42,,65,5,6,75,171,166,174,171,190178,184GinzbStringurg-Landau,tension,52seeAction
Riemann(curvature)tensor,171,190Ginzburg-Landauparameter,49,50,52
Robertson-Walkermetric,62,115Londonequations,48,51
Friedmannequations,seeFriedmannequationsTypesIandII,50
Scalarproduct,165,166Supersymmetry,17,19,61
Scalar-fieldequation,32,65,66,69,70,73,80–82,SymmetrySymmetrygroup,operator,4611–14,28,30,35,181,183
92,110,115,116,122,149,190
ScalarBer-Tensorgmann-WTheories,agoner23,class,57,59,6260,62,63,66,71TTimeensortransformationlaw,170
Brans-Dickestheory,59,60Globularclusterssage,197
Jordanstheory,58,59Nucleocosmochronology,197

INDEX

198age,-systemSolar197age,aevSupernoUniverseage,197,198

Universalhaloprofile,110

Vacuumexpectationvalue,seeGroundstate

WeaklyInteractingMassiveParticles,seeDarksec-
tor43,operatoredgeW178mixture,geinberW34angle,geinberW17645,rotation,ickWWigner-Weylmode,25,27,30,92
Flavor-symmetrybreaking,26
26fect,efZeeman13522,,WMAP

Yang-Millsequations,11,34,66,171,179,180,184,
185Yang-Millstheory,11–13,25,65,178
YukawacouplingoftheLagrangian,35
Yukawamatrix,35,65,66,69
Yukawainteraction,147
Yukawapotential,35
Yukawainteractions,92,147
Yukawatheory,10,11,28,177

Zeemaneffect,seeWigner-Weylmode

235

236

INDEX

publicationsandSupport

ThisworkwassupportedbytheGraduateSchoolofMathematicalAnalysisofEvolution,Informationand
ComplexityoftheUniversityofUlmaswellasbytheResearchGroupforCosmologyandQuantumGravita-
tionoftheInstituteofTheoreticalPhysicsofthesamehigher-studiesinstitution,underdirectionofProf.Dr.
FrankSteinerandwithspecialcollaborationofDr.HemwatiNandanoftheCentreofTheoreticalPhysics
inNewDelhiandProf.Dr.HeinzDehnenoftheDepartmentofPhysicsoftheUniversityofKonstanzas
wellasofDr.UmanandaDevGoswamioftheUniversityofDibrugarhinIndia.Thefollowingarticleswere
publishedduringtherealizationofthiswork:

(i)[21]:N.M.Bezares-Roder,H.NandanandH.Dehnen;Horizon-lessSphericallySymmetricSo-
lutionsinaHiggsScalar-TensorTheoryofGravity,InternationalJournalofTheoreticalPhysics
46(10),2429–2436(2007),DOI10.1007/s10773-007-9359-5;UlmReport-TP/07-8.
-qc/0609125.v:grarXiPre-Print:

(ii)[22]:N.M.Bezares-RoderandH.Nandan;SpontaneousSymmetryBreakdownandCriticalPer-
spectivesofHiggsMechanism,IndianJournalofPhysis82(1),69–93(2008);UlmReport-TP/08-8.
v:hep-ph/0603168.arXiPre-Print:

(iii)[110]:U.D.Goswami,H.Nandan,C.P.PandeyandN.M.Bezares-Roder;MaxwellsEquations,
ElectromagneticWavesandMagneticCharges,PhysicsEducation25(4)(2008)251–265.

(iv)[176]:H.Nandan,N.M.Bezares-RoderandH.C.Chandola;DualMeissnerEffectandQCDVac-
uuminProceedingsoftheLIIIDAE-BRNSSymposiumonNuclearPhysics53(2008)583,Ulm
Report-TP/08-9.

(v)[178]:H.Nandan,N.M.Bezares-RoderandH.C.Chandola;ScreeningCurrentandDielectricPa-
rametersinDualQCD,IndianJournalofPureandAppliedPhysics47(11)(2009)808-814.

(vi)[24]:N.M.Bezares-RoderandF.Steiner;AScalar-TensorTheoryofGravitywithaHiggsPotential,
inMathematicalAnalysisofEvolution,InformationandComplexity(eds.W.ArendtandW.Schleich),
Wiley-VCH2009.ISBN-10:3527408304andISBN-13:9783527408306.

(vii)[111]:U.D.Goswami,H.Nandan,C.P.PandeyandN.M.Bezares-Roder;CovariantFormalismof
MaxwellsEquationsandRelatedAspects,PhysicsEducation26(4)(2009)269–278.

Further:

•[179]:BlackHoleSolutionsandPressureTermsinInducedGravitywithHiggsPotentialbyH.
Nandan,N.M.Bezares-Roder(correspondingauthor)andH.Dehnenwassubmittedforpublication.
-qc].[grv:0912.4036arXiPre-Print:

237

238

TSUPPORTIONSPUBLICAAND

•[23]:Scalar-FieldPressureinInducedGravitywithHiggsPotentialandDarkMatterbyN.M.
Bezares-Roder,H.NandanandH.Dehnenwassubmittedforpublication.
-qc].[grv:0912.4039arXiPre-Print:

AnotherrelatedarticlefulfilledafterDiplomastudies,howeverpublishedduringtherealizationofthiswork
reads

[20]:N.M.Bezares-RoderandH.Dehnen;HiggsScalar-TensorTheoryforGravitationandtheFlat
RotationCurvesofSpiralGalaxies,GeneralRelativityandGravitation(GRG)39(8),1259–1277
(2007).DOI10.1007/s10714-007-0449-8.UlmReport-TP/06/4.
-qc].[grv:0801.4842arXiPre-Print:

wledgementsAckno

ThegroundingofeverydisclosureoftheBeing(lêtre,dasSein)isfreedom[...]Cognizancemayexistonly
intheamountthereexistsfreedom.Thereexistsfreedombecauseeveryactionisdefinedbythepossibility
ofitsopposite[...]Recognitionistobringwhatistothelight,toactandsearchonthemarginofmistakes,
rejectinginthatwayignoranceandliesrelatedtopredisposition.Andtruthisthisprogressivedisclosure,
eventhoughtruthitselfmayberelativetotheepochinwhichitisachieved.
–J.-P.Sartre,WahrheitundExistenz.

IamverygratefultoProf.Steinerforthepossibilityofworkingwithinhisgroupandcarryoutthiswork
underhissupervisionandkindestconfidenceandinterestinfundamentalresearchandscientificfreedom
andtheirvalue.IamverygratefultotheInstituteofTheoreticalPhysicsoftheUniversityofUlmaswellas
totheGraduateSchoolofMathematicalAnalysisofEvolution,InformationandComplexityforthesupport
throughoutthistime.IamthankfultoProf.Balserforacceptingrefereeingthiswork.
IalsodeclaremydeepestgratitudetoProf.DehnenoftheUniversityofKonstanzforhiskindsupport,and
especiallytoDr.NandanoftheCentreofTheoreticalPhysicsinNewDelhiforpullingmeupindifficult
timesandpushingmetonewideas.IamthankfultoDr.GoswamioftheUniversityofDibrugarhinIndia
forsomejointworkandconfidenceinmywork.
Iwanttothankthepeopleoftheinstitutefortheirsupport,andespeciallyDr.Lustigforgoingthroughpart
ofthislongandsurelypotentiallyboringwork.
Andatlast,butinnowayatleast,IthankwithallmyheartLizbethandmyfamilyfortheirtenderness
whichhasbeensoimportantformeandforbearingandsupportingmeallthistime.Myparentsand
mysister,togetherwithmybrotherin-lawandniece/goddaughterSophiaElisabethRutschmannBezares,
deservespecialreference,fortheyhaveencouragedmetoallIamtoday(goodandbad)andwillalwaysbe
partofthesame.AllmygratitudetotheAcevesNafzfamilywhichhasalwaysbeenpartofmine,butalsoto
allmyfriendswhichhavestoodbymysideinsomemomentortheotherorwhichhaveletmedothesame.
Iamhappythatourwayscrossedandhopethattheykeepondoingso.

239

240

WLEDGEMENTSCKNOA

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nenQuellenundHilfsmittelnbenutztsowiediewörtlichoderinhaltlichübernommenenStellenalssolche
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CurriculumScholar

DataersonalPSurname,GivenName
BirthofDateBirthofPlace

DataScholarDateStudiesInstitutionDateStudiesInstitution

DategreeDeInstitutionThesisDiplomaofAdvisorCo-AdvisorDatefiliationAfInstitutionDateStudiesInstitutionThesisofAdvisorCo-AdvisorwshipFello

BezaresRoder,NilsManuel
January20th,1980
MexicoCity,Mexico

1999JuneeifeHochschulrAllgemeineDeutscheStandard)MatriculationGeneral(GermanColegioAlemánAlexandervonHumboldtXochimilco
City)xicoMeSchool,(German1999AugustBachilleratodelColegiodeCienciasyHumanidades
(MexicanMatriculationStandardforNaturalandHumanSciences)
ColegioAlemánAlexandervonHumboldtXochimilco/
UniversidadNacionalAutónomadeMéxico

2005August–1999OctoberDiplom-Physiker(Dipl.-Phys.)(DiplomainPhysics)
onstanzKersitätvUniProf.em.Dr.H.Dehnen,FachbereichPhysik,UniversitätKonstanz
PDDr.B.Fauser,Max-Planck-InstitutfürdieMathematikindenNatur-
wissenschaften2006July–2004AugustResearchGroupforGravitationalPhysics
FachbereichPhysik,UniversitätKonstanz
Dato–2006AugustPh.D.studies(PromotionsstudiumDr.rer.nat.)
FakultätfürNaturwissenschaften,UniversitätUlm
Prof.Dr.F.Steiner,InstitutfürTheoretischePhysik,UniversitätUlm
&CentredeRechercheAstrophysique,UniversitéLyon1,CNRS
Prof.Dr.W.Balser,IntsitutfürAngewandteMathematik,Universität
UlmGraduateSchoolforMathematicalAnalysisofEvolution,Information
xityCompleand247

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