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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.

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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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