Inflation and Quintessence Theoretical Approach of Cosmological Reconstruction

arXiv:0712.2468v2 [astro-ph] 11 May 2008IshwareeP.Neupane

DepartmentofPhysicsandAstronomy,UniversityofCanterburyPrivateBag4800,Christchurch8020,NewZealandand

Inter-UniversityCentreforAstronomyandAstrophysics,Pune411007,IndiaE-mail:ishwaree.neupane@canterbury.ac.nz

ChristophScherer

DepartmentofPhysicsandAstronomy,UniversityofCanterburyPrivateBag4800,Christchurch8020,NewZealand

Inthesecondpartofthispaper,weintroduceanovelapproachofconstructingdarkenergywithinthecontextofthestandardscalar-tensortheory.Theassumptionthatascalarfieldmightrollwithanearlyconstantvelocity,duringinflation,canalsobeappliedtoquintessenceordarkenergymodels.Fortheminimallycoupledquintessence,αQ≡dA(Q)/d(κQ)=0(whereA(Q)isthestandardmatter-quintessencecoupling),thedarkenergyequationofstateintherange−1≤wDE<−0.82canbeobtainedfor0≤α<0.63.Forα<0.1,themodelallowsforonlymodestevolutionofdarkenergydensitywithredshift.Wealsoshow,undercertainconditions,thattheαQ>0solutiondecreasesthedarkenergyequationofstatewQwithdecreasingredshiftascomparedtotheαQ=0solution.ThiseffectcanbeoppositeintheαQ<0case.Theeffectofthematter-quintessencecouplingcanbesignificantonlyif|αQ|󰀁0.1,whileasmallcoupling|αQ|<0.1willhavealmostnoeffectoncosmologicalparameters,includingΩQ,wQandH(z).ThebestfitvalueofαQinourmodelisfoundtobeαQ≃0.06,butitmaycontainsignificantnumericalerrors,vizαQ=0.06±0.35,whichtherebyimpliestheconsistencyofourmodelwithgeneralrelativity(forwhichαQ=0)at1σlevel.

Keywords:Theoriesofcosmicacceleration,dynamicsofscalarfields,inflationanddarkenergy.

Abstract:Inthefirstpartofthispaper,weoutlinetheconstructionofaninflationarycosmologyintheframeworkwhereinflationisdescribedbyauniversallyevolvingscalarfieldφwithpotentialV(φ).Byconsideringagenericsituationthatinflatonattainsa

1|dφ/dN|≡α+βexp(βN)(whereN≡lnanearlyconstantvelocity,duringinflation,m−P

isthee-foldingtime),wereconstructascalarpotentialandfindtheconditionsthathavetosatisfiedbythe(reconstructed)potentialtobeconsistentwiththeWMAPinflationary

.015

data.TheconsistencyofourmodelwithWMAPresult(suchasns=0.951+0−0.019andr<0.3)wouldrequire0.16<α<0.26andβ<0.Therunningofspectralindex,α󰀒≡dns/dlnk,isfoundtobesmallforawiderangeofα.

1.IntroductionandOverview

Itistrueandremarkablethatourunderstandingofthephysicaluniversehasdeepenedprofoundlyinthelastfewdecadesthroughthoughts,experimentsandobservations.Alongwithsignificantadvancementsinobservationalcosmology[1–3],Einstein’sgeneralrelativityhasbeenestablishedasasuccessfulclassicaltheoryofgravitationalinteractions,fromscalesofmillimetersthroughtokiloparsecs(1pc=3.27lightyears).Ithasalsobeenlearnedthatatveryshortdistancescaleslargequantumfluctuationsmakegravityverystronglyinteracting,implyingthatgeneralrelativitycannotbeusedtoprobespacetime(geometry)fordistancesclosetoPlanck’slength,lP∼10−33cm.Inadditiontothisdifficulty,threestrikingfactsaboutnature’scluessuggestthatwearemissingafewimportantpartsofthepicture,notablytheextremeweaknessofgravityrelativetotheotherforces,thehugesizeandflatnessoftheobservableuniverse,andthelatetimecosmicacceleration.

Muchisnotunderstood:whatisthenatureofthemysterioussmoothdarkenergyandtheclumpednon-baryonicdark-matter,whichrespectivelyform73%and22%ofthemass-energyintheuniverse.Thatmeans,wedonotseeandreallyunderstandyetabout95%ofthetotalmatterdensityoftheuniverse.Tounderstandtheneedfordarkenergy,oramysteriousforcepropellingtheuniverse,anddarkmatter,onehastolookatthedifferentconstituentsoftheuniverse,theirpropertiesandobservationalevidences(forreviews,see,e.g.[4–7]).Thecurrentstandardmodelofcosmologysomehowcombinestheoriginalhotbigbangmodelandtheearlyuniverseinflation,byvirtueoftheexistenceofafundamentalscalarfield,calledinflaton.Thestandardmodelofcosmologyis,however,notcompletelysatisfactoryanditappearstohavesomegaps.Iftheuniverseiscurrentlyaccelerating(onlargestscales),whatrecentobservationsseemtoindicate,thenweneedinthefabricofthecosmosaself-repulsivedarkenergycomponent,oracosmologicalconstantterm,which

–1–

hadalmostnoroleintheearlyuniverse,orneedtomodifyEinstein’stheoryofgravityonlargestscalesinordertoexplainthisacceleration.

Whenin1917Einsteinproposedthefieldequationsforgeneralrelativity

Rµν−

1

∼10−5.Theangularsizeofthesefluctuations

encodesthedensityandvelocityfluctuationsatthesurfaceoflastscattering,withredshiftz≃1100.ThiscorrespondstothecosmologicalepochwhenthepresentlyobservedCMBphotonsfirstdecoupledfrommatter.ByplottingthesquaredofamplitudeofCMBtemper-aturefluctuationsagainsttheirwavelengths(ormultipolesinanequivalentFourierpowerspectrum),therecanbeallocatedseveralpeaksatdifferentangularsizes.Thepositionofthefirstpeakisoftenviewedasanindicatorforthespatialcurvatureoftheuniverse,whichrevealsthatthepresentuniverseisnearlyflatandhomogeneousonlargecosmo-logicalscales(>100Mpc),meaningthatΩtot≈1withhighaccuracy.However,whenassumingaflatuniverseonlycontainingpressurelessdust(includingDM)andassumingthecurrentHubbleparametertobeh=0.72±0.08withH0=100hkmsec−1Mpc−1(inagreementwithobservationsoftheHubbleSpaceTelescopeKeyproject[9]),itisfiguredoutthatt0=9±1Gyrs.Thisresult,simplyfollowingfromEinstein’sgeneralrelativity,impliesthataflatuniversewithoutthecosmologicalconstanttermmaysufferfromaseri-ousageproblem.IntroducingDEintheformofaconstantΛ,withΩΛ,0≃0.73,somehowresolvestheproblem,givingt0≃13.8Gyrswithh=0.72.

WhenacceptingtheexistenceofDE,naturallythequestionarises,whatitreallyis.Sincethelate1960′swhenitwasrealizedthat[10]thezeropointvacuumfluctuationsinquantumfieldtheoriesareLorentzinvariant,ithasbeenattemptedtoassociatethis

2.725K

–2–

(quantum)vacuumenergywiththepresentvalueofΛbutwithoutmuchsuccess.Evenwhenplacingacutoffatsomereasonableenergyscale,thisquantumvacuumenergyisstillseveralordersofmagnitudelargerthanthemysteriousdarkenergytoday,ρΛ∼5×10−47GeV4orρΛ∼10−123inPlanckunits(forreviews,see,e.g.[11,12]).Apparently,1/4

ρΛisfifteenordersofmagnitudesmallerthantheelectroweakscale,mEW∼1012eV.Notheoreticalmodel,noteventhemostsophisticated,suchassupersymmetryorstringtheory,isabletoexplainthepresenceofasmallpositiveΛ.

Anotherhurdleinunderstandingthenatureofdarkenergyisthatonlyaverysmallwindowinthemagnitudeofthecosmologicalconstantallowstheuniversetodevelopasitobviouslyhas.ItisstillamysterywhyΩΛhasthevalueithastoday.Itcouldhavebeenseveralmagnitudesoforderlargerorsmallerthanthematterdensitytoday,insteadofΩΛ≃3Ωm.Thisisknownascosmologicalcoincidenceproblem.

Atpresentthemostcommonviewisthatdarkenergyispresumablyconstantandhasaconstantequationofstate,wDE=−1.Butthereremainsthepossibilitythatthecosmologicalconstant(orthegravitationalvacuumenergy)isfundamentallyvariable.Inamorerealisticpicture,atleast,fromfieldtheoreticviewpoints,darkenergyshouldbedynamicalinnature[13].Thisisthecase,forinstance,withalltimedependentsolutionsarisingoutofevolvingscalarfields,withanacceleratedexpansioncomingfrommodifiedgravitymodels,holographicdarkenergy,andthelikes.

Interestinglyenough,therecentobservations(WMAP+SDSS[3])onlydemandthat−1.042.ConstructingInflationaryCosmology

Acompletemodeloftheuniverseshouldperhapsfeatureaperiodofinflationinadistantpast,leadingtoagenerationofdensity(orscalar)perturbationsviaquantumfluctuations.ThisexpectationhasnowreceivedconsiderableobservationalsupportfrommeasurementsofanisotropiesintheCMBasdetectedbyWMAPandotherexperiments.

Inthesimplestclassofinflationarymodels,inflationisdescribedbyasinglescalar(oraninflaton)fieldφ,withsomepotentialV(φ).Thecorrespondingactionis

󰀓

√1

−S=d4x

2κ2−g=detgµνisthedeterminantofthemetrictensor.

ConstructingconcretemodelsofinflationandmatchingthemtotheCMBandlargescalestructure(LSS)experimentshasbecomeoneofthemajorpursuitsincosmology.MostearlierstudiesregardingtheformofaninflationarypotentialreliedonapriorchoiceofthepotentialV(φ),oronslow-rollapproximationsinthecalculationofpowerspectraandtheirrelationtothemassofthefieldφduringinflation(see[14]forareview).Thelatterapproachcanatbestproducethetailofaninflationarypotential,butnotitsfullshape[15].Indeed,recentstudiesshowthatthetypeorvarietyofscalarpotentialallowedbyarrayofWMAPinflationarydataisstilllarge[16].Although,inordertounderstand

–3–

thedynamicsofinflation,theideaofutilizingoneortheotherformofthescalarfieldpotential(motivatedbyphysicsbeyondthestandardmodelorevenbytheoriesofhigherdimensionalgravity,suchas,stringtheory)isnotbadatall,theremightexistamoreelegantwayofconfrontingtheWMAPinflationarydatawithatheoreticalmodel.

Inthispaperwepresentadifferentandrobustapproachtotacklethisproblem:wedonotmakeaspecificchoiceforV(φ),ratherwemakeasimpleansatzforthescalarfieldφandthenconstructaninflationarypotential,usingthesymmetryofEinstein’sfieldequations.Ourapproachwouldbenovelinthesensethatitprovidesauniqueshape(andslope)tothescalar(orinflaton)potential.Themodelalsomakesfalsifiablepredictions.Thebasicideasandsomeoftheresultswerepresentedinarecentpaper[17].

Forsimplicity,weconsideraspatiallyflatFriedmann-Robertson-Walkerspacetime.Theevolutionofthefieldφisthendescribedbytheequation(see,e.g.[18])

˙=m(−2H˙)1/2=−2m2dφPP

m2

P

˙d=3H2(φ)+φ

dφ

󰀇2

H(φ),

(2.3)

whereH(φ(t))≡a/a˙istheHubbleparameteranda(t)istheFRWscalefactor,andthedotdenotesaderivativewithrespecttothecosmictimet.

LetusfirstbrieflydiscusshowthemodelthatwearegoingtoconstructcouldsatisfyinflationaryconstraintsfromtheWMAPandotherexperiments.First,notethattheterm

󰀅d

2m2

P

2

(6−φ)e

′2

2κφ′κ2φ′φ

e

∼

2H0

6Hholdsingeneral,soV(φ)>0.Inflationoccursaslongasthecondition

a¨

dφ

󰀄2

>0

˙2.But,afterasufficientnumberofe-foldsofexpansion,infla-holds,meaningthatV(φ)>φ

tionhastoend.Thisispossiblewhenthequantity(mP/H(φ))(dH(φ)/dφ)becomescompa-rableto(orevenlargerthan)unity.RecentresultsfromWMAP[3]indicatethatthespec-.015

tralindexofthescalarperturbationsisconsistentwithalmostflatone,ns=0.958+0−0.019.Toagoodapproximation,1−ns≃α2,implyingthatα<0.25.Thissimplepicturehasobviousandintuitiveappeal,whichcanberealizedthroughanexplicitconstruction.

–4–

Toillustratetheconstruction,wemakethefollowingansatz

φ

ai󰀄−󰀂a

=1dlna≃−α.Onemaythinkthattheabovechoicefor

φisadhocand/ornomoremotivatedthanaparticularchoiceofV(φ),butitisnotexactly!Indeed(2.5)isthepropertyofaninflatonfieldinmanywellmotivatedinflationarymodelsthatsatisfyslowrollconditions,afterafewe-foldsofinflation.Itcanalsobecomparedtoagenericsolutionforadilaton(ormodulusfield),i.e.φ(t)∼φ0+α0lnt+α1/tγ(whereγ>0),infour-dimensionalsuperstringmodels(see,e.g.[18,19]).Additionally,theansatz(2.5)allowsustoconstructanexplicitinflationarymodel,providinganappropriateshape(andslope)tothescalarfieldpotential.

Theevolutionofφasgivenineq.(2.5)isprovidedbytheHubbleparameter

mP

H(φ)=H0exp−

󰀅

α2

4

e−2βN(φ),

󰀇

(2.6)

whereN(φ)≡ln(a/ai),a≡a(φ(t))andH0isanintegrationconstant.Wecaneasilyevaluatethefollowingtwoinflationaryvariables

ǫH(φ)=2m2

P

󰀂1

󰀄2

=1

(2.8)

H(φ)

dφ󰀂2

dH(φ)

β+αe−βN(φ)

(whicharefirst-orderinslowrollapproximations).Themagnitudeofthesequantitiesmustbemuchsmallerthanunity,duringinflation,inordertogetasufficientnumberofe-foldsofexpansion,likeNe≡ln(af/ai)󰀁50.Moreprecisely,werequire|ǫH|≪1,|ηH|<1,exceptneartotheexitfrominflationwhereǫH󰀁1.Onemayactuallydemandthat0≤ǫH≤3,sothatthescalarfieldpotential

󰀊󰀈2

V(φ)=m2H(φ)3−ǫHP

(2.9)

isnon-negative.AtypicalshapeofthispotentialisdepictedinFig.1.Themagnitudeof

H0(cfeq.(2.6))canbefixedusingtheamplitudeofdensityperturbationsobservedattheCOBEexperiments,usingthenormalization[20]:

√

(dV/dφ)−1V3/2/(

Κ2V󰀁Φ󰀃󰀄H02

8642

10203040506070

ln󰀂a󰀅

Figure1:Theshapeofthepotential,forsomerepresentativevaluesofα=0.2,0.3,0.4(toptobottom),β=−0.2andN(φ)≡lna+c.Wehavetakenc=−10.

AslongastheparameterǫH(φ)isslowlyvarying,thescalarcurvatureperturbationcanbeshowntobe[21]

󰀄

Γ(ν)1/2

PR(k)=2ν−3/2,(2.11)

˙2π|φ|aH=kwhereν=3/2+1/(p−1)anda∝tp.ThescalarspectralindexnsforPRisdefinedby

ns(k)≡1+

dlnPR

α+µ

,(2.13)

whereµ≡βeβNe.Intheconventionalcasethatβ=0,whichcorrespondstoascenario

whereinflationisdrivenbyasimpleexponentialpotential,V(φ)∝eα(φ/mP),weobtainawellknownresultthat1−ns≃α2.HereweshallassumethatNe≥47andβ<0.

Letusalsodefinetheslopeorrunningofthespectralindexns,whichisgivenby

α󰀒≡

dns

dN

dN

dlnk

(2.14)

(tildeisintroducedheretoavoidconfusionwiththeexponentparameterαintroducedin

eq.(2.5)),whereφandkarerelatedby

dφ

2ǫH(φ)

whileNandφarerelatedby

mP

dN

󰀑

Β0󰀉0.05󰀉0.1󰀉0.15󰀉0.2

󰀉0.005

0󰀉0.05󰀉0.1󰀉0.15

Β

0

Α

0.0050.010.015

󰀉0.2

󰀉0.005

0

Α

0.0050.010.015

Figure3:Contourplotsforns=0.95withNe=50(leftplot)andNe=60(rightplot).

Itisalsosignificanttonotethat,forα≪|β|,thereexistsasmallwindowintheparameterspacewhere

ns≃0.95,

r∼O(10−3−10−6),

inwhichcase,however,theslopeparametersαandβmustbefinelytuned.InFig.3weshowthecontourplotswithNe=50andNe=60,representingsuchacase.Infact,inthecase|α|<0.05,thegravitywaves(ortensormodes)arealmostnonexistent.OntherightplotinFig.4weshowtherunningofspectralindexα󰀒,whichisalwaysverysmallintheparameterrange0.1<α<1andβ<0.

ns

0.980.970.960.95

Β

󰀇Α

󰀉0.2

󰀉0.15󰀉0.1󰀉0.05

󰀉0.00002󰀉0.00004󰀉0.00006󰀉0.00008󰀉0.0001󰀉0.00012Β

󰀉0.2󰀉0.15󰀉0.1󰀉0.05

󰀉0.00014

Figure4:Thescalarspectralindexns(leftplot)anditsrunningα󰀒withrespecttoβandα=0.16,0.20and0.24(toptobottom).Thesolid(dotted)linesareforNe=60(Ne=47).Therunningofnscouldbelargeonlyif|α|󰀂|β|;forexample,α󰀒≃−0.004forα≃0.01andβ≃−0.05.

Inamodelwithmorethanonescalarfield,thedependencyofinflationaryvariableslikensandrontheslopeparametersαandβcouldbemorecomplicatedthanthesim-plestexplanationprovidedabove.Nonetheless,ourapproachhasgreatsignificanceasitgenericallyleadstoaspectrumofprimordialscalarfluctuationsthatisslightlyred-tilted(ns󰀂1)andhencecompatiblewithWMAPinflationarydata.

–8–

3.ConstructingQuintessenceCosmology

Itisreasonabletoassumethatalatetimeaccelerationoftheuniverseisdrivenbythesamemechanismusuallyexploitedtogiveearlyuniverseinflation,wherethepotentialenergyofascalarfielddominatesitskineticterm.Tothisend,letusassumethatthecurrentexpansionoftheuniversecanbedescribedbytheaction

󰀓

√1

S=Sgrav+Sm=d4x−

2κ2

−gA(Q)

4

󰀎

i

ρi,(3.2)

whereψmrepresentscollectivelythematterdegreesoffreedomandradiation.IntheabovedefinitionofthematterLagrangian,theimplicitassumptionisthatmattercouplestog˜µν≡A(Q)2gµν,ratherthantheEinsteinmetricgµνalone.Thisassumptionthenresultsinanon-minimalcouplingbetweenthescalarfieldQandmattercomponents(ρi).Thematter-scalarcouplingA(Q)maybeunderstoodasanaturalmodificationofEinstein’sGRwhichcanbemotivatedby,forinstance,scalar-tensortheory.Forfurtherdiscussionsontheoreticalmotivationsofthiscoupling,see,forexample,[23–26].

ThecouplingA(Q)actuallygeneratesanewterm,namely

2δLm−

−g

(i)

µTµ(i),

dQ

(3.3)

inthescalarwaveequationforQ.Thisexpressionalsoimpliesthatradiationdoesnot

coupletothescalarfieldQsinceitstraceoftheenergy-momentumtensorequalszero.AswewillshowthecouplingdA(Q)

∂t

=0(3.4)

andhenceV(Q)=const≡ΛandA(Q)=const.ThemodelthenreducestotheΛCDMcosmology,giventhatdarkmatterischaracterizedbynon-relativisticparticlesalone,wm=0.ThecosmologicaltermΛ,whichisgovernedbytheequation

µµµ

m2G=2Λδ+T,νννP

(3.5)

canclearlyactasasourceofgravitationalrepulsionorputativedarkenergy.

Allthediscussionssofarhavebeenmadewithoutmakinganyparticularchoiceofmetric.ThusthenatureofV(Q)actingasarepulsiveforceisrathergeneral.Foramoredetailedtreatment,itisnecessarytoevaluatetheequationsgeneratedbyvariationofthe

–9–

totalactionS=Sgrav+Sm.Thereforeaparticularchoiceofametrichastobemade.WemakeratherstandardchoiceofaspatiallyflatFRWmetric:

ds2=−dt2+a(t)2dx2,

(3.6)

wherea(t)isthescalefactorofaFRWuniverse.Thischoiceofthelineelementiswellmotivatedbytheobservationalfactthattheuniverseisspatiallyflatonlargestscales,whichisconsistentwiththeconceptofinflation,discussedintheprevioussection.Ofcourse,thischoiceofmetricmayleadtosystematicerrorsinthecalculation,astheuniverseactuallyisnothomogenousatsmaller(orgalactic)scales,aspointedout,forexample,in[22],whichisignoredinthissimplifiedassumption.

Intheminimalcouplingcase,A(Q)≡1,itiseasytoseethat

ρi∝[a(t)]−3(1+wi),

(3.7)

wherewi≡pi/ρi.Inthenon-minimalcouplingcasethemodifiedscalefactor󰀋aisgivenby󰀋a=a(t)A(Q).Asaconsequence,differentequationofstateparameters(cfeq.(3.3))wouldcausedifferentenergydensitiestoevolvedifferentlywithchangingscalefactor:

ρi∝(a(t)A(Q))−3(1+wi).

(3.8)

Thisimpliesthatρm∝(a(t)A(Q))−3andρr∝(a(t)A(Q))−4,respectively,forordinarymatterandradiation.Italsoshowsthatradiationneverdirectlycouplestothescalarfield,evenwithA(Q)beinganarbitraryfunctionofQ.Asexplainedin[17],thecouplingA(Q)canberelevant,especially,inabackgroundwhereρmismuchlargerthanρcrit(where

2/8πG),e.g.,agalacticenvironment.ρcrit≡3H0

3.1BasicEquations

Takingavariationoftheaction(3.1)withrespecttogµνandthenevaluatingthettand

xxcomponentsofEinstein’sequationleadstothefollowingtwoequations(cfeq.(A.1)):

󰀎324˙Q+V(Q)+A(Q)ρi=0(3.9)−

2

i

󰀎󰀊󰀈1˙2−V(Q)+A4(Q)(3.10)wiρi=0Q

2

i

AvariationwithrespecttothescalarfieldQ,whileconsideringanexplicitmatter-scalarcoupling,yieldsthefollowingequationofmotionforQ(cfeq.(A.2)):

¨+3HQ˙+dV(Q)Q

dQ

󰀎󰀈

i

theso-calledtheKlein-GordonequationforQ.ItshowsthatthescalarfieldQcouplesto

thetraceoftheenergy-momentumtensorgµνTµνsatisfying

¨+3HQ˙=−dV(Q)−∇2Q=Q

dQ

µ

Tµ.

󰀊

1−3wiρi=0,

(3.11)

(3.12)

Thereisdissensionaboutthesignofthecouplingtermbetweenthescalarfieldandmatter

µinaboveequationinthewaythatitmightbe+A3dA(Q)dQTµ.In

thispaper,thenegativesign,aswrittenineq.(3.11),willbeused.

–10–

Theabovesetofequationscanbesupplementedbyafourthequation,arisingfromtheequationofmotionforaperfectbarotropicfluid

(aA)

dρi

H2

=

H′

dA(Q)A

2

3H2

,

ΩQ≡

κ󰀔

1

κ2ρQ

3H2

≡

2

Q˙2−V(QQ)2Q˙2

+V()

≡

pQ

dκQ

.Intheabovewehaveusedtherelation∂

∂

H

whichwillbeusedlater.Ofcourse,inthecaseofaminimalcoupling(A(Q)≡1),αQvanishes,reducingthenumberofdegreesoffreedominthesystemofequations(3.17)-(3.20)byone,whichthenmakesthesystemeasiertohandle.Anyhowinbothcases(αQ=0andαQ=0)itisnotpossibletofindananalyticalsolutionofthissystemwithoutmakingsomeadditionalassumptionsastherearemoredegreesoffreedomthanindependentequations.Infact,thenumberofdegreesoffreedomdependsonthenumberofmattercomponentsincludedintheanalysis.

Asthefirstcheckforcompatibilityofthemodel,itisusefultoconsidersomesimplifiedsolutionoftheequations(3.14)-(3.20),byexpressingallmatterfieldsasonecomponent,wi≡wm.Byapplyingeq.(3.21)toeq.(3.14),andafterasimpleintegration,weget

e−3lnaexpρm=ρ(0)m

󰀅󰀓󰀔

󰀇󰀁󰀈󰀊

Q′αQ3wm−1+3wmdlna,

(3.24)

whichcanbeinterpretedasaglobalenergyconservationequation.Thus,fornotviolatingthisprincipleofenergyconservation(3.22),asignchangein(3.19)automaticallyimpliesachangein(3.20)aswell.

Whenhavingaparticularsolutionoftheequations(3.17)-(3.20),itisofgreatinteresttostudyhowthecorrespondingpotentiallookslikeandhowitaffectsthecosmicevolutionofouruniverse.Fromthelastexpressionineq.(3.15),wefind

󰀄󰀂

󰀈󰀊3Ω2Q

Q′V(Q)≡H2,(3.23)2

beinganarbitraryconstant.ThecouplingαQmaybeconstrainedbyobservationswithρ(0)m

perhapsonlyinthecombinationQ′αQ.OnecanstudytheeffectofthiscouplingonbothCMBtemperatureanisotropiesandevolutionoflinearmatterperturbations,asin[29].Intheminimalcouplingcase,onehas

ρm∝a−3(1+wm).

(3.25)

Thisisexactlythebehaviouronewouldexpectfromgeneralrelativity.Equation(3.25)yieldsρm∝a−3inauniversecontainingonlyordinarymatter(ordust),whileforradiationρr∝a−4.Transposingeq.(3.18)leadstoageneralexpressionfortheequationofstateoftheDEcomponentwhichcangenerallybewrittenas

󰀊󰀖󰀈

2ǫ+3i1+wiΩi+3ΩQ

wQ=−

ρtot

,

ptot≡pQ+p󰀒i,ρtot≡ρQ+ρ󰀒i,

(3.27)

whereaspQandρQareasdefinedineq.(3.16),andp󰀒i≡A4(Q)pi,ρ󰀒i≡A4(Q)ρi.Themean-ingofweffissomehowthatofameanequationofstateofallmatter-energycomponents,

–12–

includingthedarkenergycomponentρQ.Fromeq.(3.18),togetherwitheqs.(3.15)-(3.16),wefindthatthetotalpressureisgivenby

󰀉

2󰀎3H

ptot=−ΩQ−Ωi.(3.28)

3

i

Combiningthisexpressionforptotwiththeexpressionoftotalenergydensityρtot,asdefinedby(3.27),andusingagainthesubstitutions(3.15)-(3.16),weget

weff=−1−

2ǫ

aH2

=−1−ǫ,(3.30)

showingthatthename“decelerationparameter”makessenseinsuchawaythatq>0foradeceleratedexpansion(¨a<0),whileq<0foranacceleratingexpansion(¨a>0).

AsexplainedintheIntroduction,onereasonforconsideringauniversecontaininganonzeroDEcomponent,eitherintheformofacosmologicalconstantΛoradynamicallyevolvingscalarfieldQ,istherecentlyobservedacceleratedexpansionoftheuniversebytheSupernovaCosmologyProjectandtheHigh-redshiftSupernovaSearchteam[1].Thuswefinditusefultostudythesolutionofequations(3.17)to(3.20),whichyieldsanacceleratedexpansioninageneralcontextofnontrivialmatter-scalarcoupling.

Indeed,independentofanyassumptionorspecificcompositionoftheuniverse,simplytheconditionweff<−1/3atsomestageofcosmicevolutionyieldsanacceleratedsolution.Itshouldbenotedatthisstage,thatnosuchgeneralconnectioncanbeestablishedbetweentheDEequationofstateparameterwQhavingaspecificvalue(evenlikewQ=−1)andtheuniversebeinginanacceleratingphase.

Inthenotationsusedinthispaper,boththenon-baryonic(cold)darkmatterandordinarymatter(pressurelessdust)arecombinedinonematterconstituentΩm.AswDM≈0isarathergoodapproximationfortheequationofstateofcolddarkmatter(sinceitisnon-relativistic)thiscombinationseemstobereasonable.Thisassumptionasregardsthecompositionoftheuniversetodayimpliesthatitsonlyconstituentsarecolddarkmatter,ordinarymatter,radiationandDE.Puttingthiscomposition(Ωm∼=0.27,ΩQ∼=0.73and

−4Ωr∼=10)oftoday’suniverseintotheverygeneralexpressionoftheDEequationofstate

parameterwQ(cfeq.(3.26))andusingagainwm=0andwQ≃−1,thevalueǫ=−0.4isobtained.ThisimpliesthatforwQatleastbeingcloseto−1theuniverseisinanacceleratingphasetoday,whichiswhatisobserved.

Thegeneralconsiderationssofarseemtobeconsistentwithobservations.Asobserva-tionsseemtoindicateavalueforwQcloseto−1,thepossibilityofadarkenergycomponentsimplybeingacosmologicalconstantcannotberuledout.Butitisalsoimportanttore-alizethattheeffectsofaslowlyrollingscalarfieldwouldbealmostindistinguishablefrom

1(Q/H˙)󰀂0.1atpresent.EvidencethatofapurecosmologicalconstantifκQ′≡m−P

forwQ∼−1couldactuallyimplythatthefieldQisrollingonlywithatinyvelocityatpresent.Thispointshouldbemoreclearfromthediscussionbelow.

Alltheexaminationssofarhavebeeninarathergeneralwaywithoutimposinganyadditionalassumptions.Forsurethatisnotreallysatisfying,asonemightbeinterestedin

–13–

ananalyticsolutionofthesystemofequations(3.17)to(3.20).Asmentionedabovethisisnotpossiblewithoutfurtherinputbecauseofthenumberofdegreesoffreedomexceedingthenumberofindependentequations.Inthenexttwosubsectionstwodifferentanalyticsolutionswillbepresentedmakingsomesimpleadditionalassumptions.AccordingtothepresentconstitutionoftheuniversebeingΩm≃0.27andΩQ≃0.73,itisreasonabletoneglecttheradiationcomponentatleastforredshiftz󰀂O(10).Thereforethemodeluniverseassumedinthenexttwosectionsisthoughttoonlyconsistofcolddarkmatterandordinarymattercombinedinonecomponentwithacommonequationofstatewm=0andaDEcomponentrepresentedbythescalarfieldQwithavariableEoSwQ.

Thesystemofequations(3.17)-(3.20)canthenbeexpressedintheform:

Ωm+ΩQ=1

󰀔󰀁

2ǫ+31+wQΩQ+3Ωm=0

󰀔󰀁

′

ΩQ+2ǫΩQ+3ΩQ1+wQ+Q′αQΩm=0

(3.31)(3.32)(3.33)(3.34)

Thenumberoffreeparametersinthissystemisfive(Ωm,ΩQ,wQ,ǫandαQ),meaningtwoadditionalassumptionshavetobemadetofindananalyticsolution.Toproceedfurther,wemakethefollowingassumption:

Q=Q0+mPαln[a(t)]≡Q0+mPα(N+const),

(3.35)

whereαisaconstantwhichneedstobefixedbyobservations.ThisrelationactuallyrepresentsagenericsituationthatthefieldQisrollingwithaconstantvelocity,Q′=const1.Intheminimalcouplingcasethisisenough,whileinthenon-minimalcase(αQ=0)onemoreassumptionisrequired,whichwillbediscussedbelow.

Simplytransposing(3.16)andutilizingtherelationbetween∂∂N,asgivenby(3.21),yieldsthefollowingusefulrelation

wQ=

κ2Q′2−3ΩQ

3ΩQ

.

(3.36)

′

+2ǫΩ+3Ω−QαQΩm=0.Ω′mmm

Supplementingequations(3.31)-(3.34)withthisequationisanelegantwayofimposingan

additionalconstraintintothemodel.3.2UncoupledQuintessence

IntheA(Q)=1case,thesystemofequations(3.31)-(3.34),supplementedbyeq.(3.35),cannowbesolvedanalytically.Theexplicitsolutionisgivenby

ΩQ=1−

λ

3α2exp[−λN]+3λc1ǫ=

1

(3.38)

−c1α2λ−9exp[−λN]

˙˙=const.Forα<0.6,ourapproachNotethatwearedemandingQ′≡dQ/dlna=Q/H≃const,notQ

appearstogiveconsistentresultswhenappliedtoobservationaldata;seealsothereview[30]forextensivediscussionsonvariousmethodsofreconstructingdarkenergypotentials.Seeref.[31]foraverydifferentapproachofdarkenergyreconstruction.

–14–

whereN≡N(Q)=ln[a(Q(t))]andwehavemadethesubstitution

λ≡3−α2.

Usingeqs.(3.29)and(3.30)wealsoevaluate

q=

3exp[−λN]−c16−

.󰀈

5α2

+α4󰀊

(3.42)(3.40)

9exp[−λN]+3c1λ

Thisgeneralsolutioncontainsthreefreeparameters(N,αandc1).Tokeepthesolutionasgeneralaspossibleitisusefultojustfixonefreeparameterintermsoftheothertwo.Theintegrationconstantc1canbefixedintermsofthefieldvelocityαbyusingtheobservationalinputΩm0=0.27atpresent.Thee-foldingtimeNinrelationtothecosmictimetisonlydefineduptoanarbitraryconstant,soitneedstobenormalisedinsomeway.Forsimplicity,thiswillbedonebytakingN=0atpresent.Thus,theconditionΩm[N=0,α,c1]≡Ωm0yields

3−α2−3Ωm0

c1=

λem

=

a0

a0

=

1

wQ

0.50.25󰀉0.25󰀉0.5󰀉0.75

󰀉1

246

8

10z

wQ

2

󰀉0.2󰀉0.4󰀉0.6󰀉0.8󰀉1

4

6810z

Figure5:wQwithrespecttoredshiftz(leftplot)forα=0,0.5,1.0,1.48,1.8,and(rightplot)α=0,0.15,0.3,0.45,0.6(bottomtotop).

Ε

1

󰀉0.5󰀉1󰀉1.5󰀉2

󰀉0.25󰀉0.5󰀉0.75

2345

z

q0.750.50.25

1

2

3

4

5z

2˙Figure6:Theslowrollparameterǫ≡H/Handdecelerationparameterqwithrespecttoz,and

α=0,0.7,1.0,1.48,1.6(fromtoptobottom,leftplot)or(bottomtotop,rightplot).

weff

1

󰀉0.2󰀉0.4󰀉0.6󰀉0.82345z

weff

0.5

󰀉0.2󰀉0.4󰀉0.6󰀉0.8

1

1.5

2

z

Figure7:TheeffectiveEoSweffwithrespecttoredshiftz:(leftplot)α=0,0.7,1.0,1.48,1.6(bottomtotop)and(rightplot)α=0,0.4,0.6,0.8,1.0(bottomtotop).

self-repulsiveformofenergyinrecenttime.Forafurtherunderstandingofthissolutionitisusefultolookatthedecelerationparameterq.

InFig.6,itcanbeeasilyseenthat,forall0≤α<1.48,qgetsnegativesomewhenbetweenredshiftsz=0andz=1,whichimpliesthatinthismodelacceleratedexpansionisaratherlatetimephenomenonwiththeuniversegettingintoanacceleratedphasetheearliestforα≡0,correspondingtothecosmologicalconstantcase.Inthecaseofα=1.48,qexactlyequals0.5,correspondingtoadeceleratedexpansionatconstantdeceleration.

–16–

4z321010.750.5󰀈m0.25󰀈Q

21.51Α0.5000.80.60.40.2000.5Α1.5112z34Figure8:ΩmandΩQwithrespecttoαandz.Thesequantitiesmaynotchangewithzonlyifα=αcrit=1.48,inwhichcaseobviouslytherewon’tbeacosmicacceleration.

Finally,forα>αcrit=1.48,qisgreaterthan0.5andincreaseswithdecreasingredshift,yieldingadeceleratedexpansion.ThisfitstotheevolutionofthedarkenergyEoSwQ,asseeninFig.5(forα>αcrit,wQ>0).

FromFigs.7and8wecanseethatforthesolutionwhichleadstoalatetimeac-celeration(weff<−1/3)theuniverseisclearlydominatedathighredshiftbyΩmwithatransitiontoΩQdominanceinrecenttimeleadingtoΩm=0.27andΩQ=0.73atz=0.(Thatforsuredoesnotcomesurprisingly,sincethatwastheassumptionmadewhenfixingc1).Itisperhapsmoreinterestingtonotethatforα=1.48theratioΩQ/Ωmremainsconstantforallz,whereas,forα<1.48,theearlyuniversewouldbedominatedbyΩmwithashifttodarkenergydominanceintherecentepoch.TheobservedaccelerationandDEdominancecorrespondbesttovaluesofαclosertozero.

UncertaintiesinthecurrentvalueofΩmaffectαcrit,tosomeextent,andhencethepredictedvalueofwQatsomefixedredshift.Thatis,foravalueofΩmdifferentfrom0.27

√

atpresent,thecriticalvalueofα,i.e.αcrit=

thatatransitiontotheacceleratedphase(q<0)occursforwQtendingto−1andΩQtendingto+1.TherightplotinFig.9isatwo-dimensionalprojectionofthelatterandthusjustgivesanotherillustrationofthealreadydiscussedrelationbetweenwQandΩQfortheacceleratingcase,whereonlyacceleratingsolutionswithα<0.6,whichactuallyleadtowQ<−0.83atz=0,areexamined.

32Κ2 V1󰀋󰀋󰀋󰀋󰀋󰀋󰀋󰀋󰀋󰀋H200.5Α11.524󰀁Κ2V󰀃󰀄H22.52

23z101.510.5

1

2

2345

z

Figure10:(Leftplot)κV(Q)H2(Q)withrespecttozforα=0,0.4,0.8and1.2(frombottomtotop,attherightendofthegraph).

ThediscussionsofarhasbeenbasedontheideaofadarkenergyasdescribedbythescalarfieldQwithsomepotentialV(Q).Forobtainingtheanalyticalsolution,(3.37)-(3.39),noparticularchoicewasmadeforthepotential.Theonlyoneassumptionmadewasthatthefieldmightberollingwithaconstantvelocityα,withrespecttothee-foldingtimeN=lna.Thusitwouldbeworthlookingattheshapeofthepotentialasdeterminedbythisparticularsolution,followingtheideaofreconstructionunderlyingthefocusofthispaper.ForobtainingtheanalyticexpressionofV(Q),itisusefultoconsiderthesetofsubstitutionsmadein(3.15)-(3.16).Byutilisingtheadditionalconstraint(3.36),itiseasytoseethat

κ2V(Q)

.(3.45)Y≡

2

Yisactuallyadimensionlessvariable,whichtakesthevalueY=3inapuredeSitterspace.ThevariationofYshowninFig.10seemsquitenaturalandcanbeunderstoodinthefollowingway.Inordertogetanacceleratedexpansionoftheuniverse,withwQcloseto−1atalowredshift,Y/3shouldexceedΩmintherecentpast.

Inordertofindthepotential,itisnecessaryfirsttoevaluatetheHubbleparameterH,whichcanbeeasilydonebysolvingtheequation

ǫ[N]H[N]=H′[N].

TheanalyticexpressionofHisgivenby

󰀅

−Nα2

H=c2exp

c2=

(3.46)

3exp[−λN]+c1λ,

(3.47)

Thenumericalconstantc2canbefixedbytheassumptionthatH[N=0]=H0.Hence

H0

.

3+c1λ

(3.48)

–18–

Finally,thequintessencepotentialtakestheform

κ2V(Q(N))=

1

2H0

withrespecttozandα.(Rightplot)

κ2V

2

e−α(c3+κQ)3α2exp

󰀆

󰀌󰀈󰀊󰀈󰀊

α2−3c3+κQ

mP

󰀌󰀈󰀊󰀄󰀆2α−3Q

V0exp

2.521.5Κ2 V1󰀋󰀋󰀋󰀋󰀋󰀋󰀋󰀋2󰀋󰀋0.5H0

43Κ󰀊Q211.51Α0.521.51Κ2 V

󰀋󰀋󰀋󰀋󰀋󰀋󰀋󰀋󰀋󰀋0.5H02

0c30.511.5432Κ󰀊Q1with

Figure12:(Leftplot)κV(Q)respecttoκQandc3,forα=0.6.

2

2H0

3.3CoupledQuintessence

Asalreadymentionedabove,whensolvingthesystemofequations(3.31)-(3.34)withthe

additionalconstraint(3.36)inthegeneralcase(αQ=0),onemoreconstraintisneededtogetananalyticsolution.ItismostcanonicaltoassumeκαQ≡const≡χ,whichrepresentsthecaseofso-calledexponentialcouplingbetweenthescalarfieldQandmatter,asA(Q)∝eχ(Q/mP).Thisadditionalassumptionthenleadstoageneralanalyticsolution

ΩQ=1−

ζ

3c4ζexp[ζN]+3α2−3χαǫ=−

where

α2

2c4ζexp[ζN]+6ζ≡3+χα−α2,α2

2c4ζexp[ζN]+6α2

2c4ζexp[ζN]+6

.,

(3.53)(3.54)

(3.55)

Further,theanalyticexpressionsforqandweffaregivenby

q=−1+

(3.56)(3.57)

weff=−1+

Bysolvingthedifferentialequation(3.46),theHubbleparameterisfoundtobe

󰀅

−N(3+αχ)

c4ζexp[ζN]+3,H(Q)=c5exp

(3.58)

whereN≡N(Q).Theintegrationconstantc5canbefixedbytheassumptionthat

H[N=0]≡H0.Thisyields

H0

.(3.59)c5=

3+c4ζ

–20–

OnethatN=0correspondstoa≡a0=1.Further,insistingthat󰀈normalizesN󰀊such0

ΩmN=0,α,c4,χ≡Ωmatz=0fixestheintegrationconstantc4intermsofαandχ:

c4=

)+αχ−α23(1−Ω0m

redshiftsforχ>0(χ<0).Thisbehaviorwouldbesomewhatoppositeinandeceleratinguniversewithα>αcrit.Thisbehaviourisexpectedbytheχ-dependenceofq,sinceanincreaseinmatterdensityalsoincreasesqandviceversa.Forabetterunderstandingofthissituation,itisusefultostudythebehaviourofthepotentialV(Q).

ItisalsoworthexaminingthevaluesofdarkenergyEoSwQwithavaryingχ.Inthecaseχ<0,anincreasingnegativeχdecreaseswQ,whereasanincreasingpositiveχwillincreasewQwithrespecttothevalueithasintheminimalcouplingcase,χ=0;onemaycomparethefigure15with5.

wQ

2

󰀉0.2󰀉0.4󰀉0.6󰀉0.8󰀉1

4

6

8

10

z

wQ10.750.50.25󰀉0.25󰀉0.5󰀉0.75

󰀉1

2

46

8

10z

Figure15:ThedarkenergyEoSwQwithrespecttoz.Leftplot:χ=+0.1andα=0,0.2,0.4,1.0,1.4(bottomtotop).Rightplot:α=0.4andχ=−0.2,−0.1,0,0.1,0.2(toptobottom).

Inanalogytotheprevioussection

κ2V(Q)

×c4ζexp

󰀆

󰀌󰀈󰀊ζc6+κQ

α

󰀏

2),instead,Veff(Q)

decreaseswithanincreasingz.Thisshouldnotcomeasasurprise;thisbehaviourhasitsorigininthevalueofαcritwhichisloweredforχ<0.Foragivenα,theslopeofthepotentialisshallowerforχ>0thanforχ<0,withvanishingdifferenceatlowerredshifts.

Weconcludethissectionwiththefollowingtworemarks.Firstly,inourmodel,itispossiblethatthecurrentaccelerationoftheuniverseisonlytransient.Thiscaneasilyhap-󰀃󰀖

pen,forαQ<0,wheniαQ(1−3wi)˜ρidQ(whereρ󰀒i∝a−3−3wi)becomescomparableto(orexceeds)κ2V(Q),makingtheeffectivepotentialalmostvanishing(ornegative).

Secondly,inthecaseboththeordinaryanddarkmatterhavesamecouplingwiththequintessencefieldQ,currentobservationalconstraints(fromCassiniexperimentsandthe

–22–

400200Veff

1Α0.5012z3450200Veff10001Α0.50123z45Figure16:TheeffectivepotentialVeff(Q)withrespecttoredshiftzandtheslopeparameterα,intheunitsH0=1=κ,forχ=−0.5(leftplot)andχ=+0.5(rightplot).Wehavetakenc6=0.

<10−4,whilethisboundissignificantlyrelaxedifdarkmatterlikes)onlydemandthatα2Q

canhavemuchstrongercouplingwithQ.ItshouldbetheastrophysicalobservationsthatdecidewhetherαQ<0orαQ>0.Theanswertothisquestioncanhaveinterestingcosmologicaleffectswhichweaimtostudyinfuturework.

4.Confrontingmodelswithdata

Inthissectionweconfrontourmodelswithrecentcosmologicaldatasets(SupernovaLegacySurvey(SNLS)andSNIaGold06datasets)followingthemethodsdiscussed,forexample,inrefs.[36,37].

Intheminimalcouplingcase,sinceρ˙Q+3H(1+wQ)ρQ=0(i.e.ρQandρmareseparatelyconserved),weget

󰀅󰀓z

(1+w(z1))

ρQ=ρQ0exp3

0

1−

3(1+

2H0

z)dlnH

Ωm0(1+z)3+(1−Ωm0)(1+z)α2,

(4.4)

whereΩm0≡3/(3+󰀒c1).UsingthisexpressionofH(z),weshowinFig.17thebestfit

formofw(z)fortheSNLSdatawithapriorΩm0=0.24.ThedarkenergyequationofstatewQ(z)isgivenby

wQ(z)=

(1−Ωm0)(α2−3)

Clearly,knowledgeofΩm0andαwouldsufficetodeterminewQ(z).Intheα=0case,wQ(z)=wΛ=−1.Intables1and2wepresentthebestfitvaluesofαandwQfordifferentchoicesofΩm0.

w󰀁z󰀃0󰀉0.5󰀉1󰀉1.5

0.20.40.60.81z

Figure17:Thebestfitformofw(z)fortheSNLSdatasetsforapriorofΩm0=0.24alongwiththe1σerrors(shadedregion).The(black)solidlinecorrespondstotheansatzw(z1)≡w0+w1z1/(1+z1)(cfeq.(4.3)).Thethreeotherlinescorrespondtoα=0.4,0.2109,0(toptobottom)andwQ(z)givenbyeq.(4.5).WithΩm0=0.24,α=0.2109minimizestheχ2(=104.18).TheSNLSdatamin

mayfavouralowervalueofΩm0(ascomparedtotheGoldSNIadataset).Further,withacanonicalquintessence,sothatwQ(z=0)󰀁−1,wemayrequireΩm0<0.2592.

Table1:ThebestfitvaluesofwQ(z)and

Ωm00.22

0.2573

0.24

0.1476

0.259173αfortheSNLSdatasetsforagivenΩm0.wQ(z=0)−0.9628

104.21

−0.9805

104.16

−0.9999

|α|0.4001

0.25

0.2544

0.29

0.0100

−0.9959−0.9493

χ2min178.64177.76177.25

–24–

1w󰀁z󰀂10.50󰀁0.5󰀁1󰀁1.5z0.50󰀁0.5󰀁1󰀁1.5󰀁20.250.50.7511.251.50.250.50.7511.251.5Figure18:Thebestfitformofw(z)fortheGoldSNIadatasetforapriorofΩm0=0.27alongwiththe1σerrors(shadedregion)withH(z)givenbyeq.(4.7)(leftplot)andHobs(z)givenby(4.10)(rightplot);χ2isminimizedforα=0.4735andαQ0=0.0633.The(black)solidlinecorrespondstothebestfitlinewithχ2(≃177)andthethreeotherlinesrepresentwQ(z)(cfeq.(4.8))withmin

α=0.6,0.4,0.2(toptobottom)andαQ≡χ=0.4.

TheGoldSNIadatasetscouldactuallyfitbetterwithcoupledquintessence(orinteractingdarkenergy)models(cfFig.18).

Inthenon-minimalcouplingcase,ρQisnotseparatelyconserved,sinceρ˙Q+3H(1+wQ)ρQ=αQHQ′ρm;ofcourse,thetotalenergyisalwaysconserved:ρ˙tot+3H(ρtot+ptot)=0,whereρtot=ρm+ρQ.Usingtherelations∂/∂t=H(∂/∂lna)andlna=−ln(1+z),weget󰀅󰀓

ρQ=exp3

z0

(1+w(z1))

1+z1

exp−3

󰀅󰀓

z

(1+w(z1))

0

tobe

H(z)=H0

d(κQ)

󰀕

≡χ,theHubbleparameterH(z)isfound

3(1−Ωm0)(1+z)−ζ+α(α−χ)Ωm0

.(4.8)

Nextwebrieflydiscussaboutaninterestingpossibility(leavingthedetailsandfurthergeneralizationtoaforthcomingpaper).Inthenon-minimalcouplingcase,theHubbleexpansionparameterthatonemeasures(inaphysicalJordanframe)couldactuallybedifferentthantheonegivenby(4.7)byaconformalfactor.Giventhat

Hobs(z)

wefind

Hobs(z)=H0

󰀕

α

SNIa

SNIa+WMAP+SDSS

0.06330.0583

wQ0(eq.(4.8))

.10

−0.94+0−0.10.07−0.92+0−0.08

ThemeanvalueofwQ0obtainedaboveiswithintherangeindicatedbyWMAP3+SDSS

.087

observations:wDE=−0.941+0−0.101[3].ThebestfitvalueofαQisfoundtobeαQ≃0.06,butinourmodelitmaycontainsignificantnumericalerrors,namelyαQ=0.06±0.35,whichtherebyimpliestheconsistencyofourmodelwithgeneralrelativity(forwhichαQ=0)at1σlevel.ToillustratethisresultweshowinFig.19thebestfitplotwithαQ=0.

w󰀁z󰀂10.50󰀁0.5󰀁1󰀁1.5󰀁20.250.50.7511.251.5zFigure19:AsinFig.18(rightplot)butwithαQ=0.

Thepost-Newtonianparameterγ˜isrelatedtoαQ0(≡χ)throughtherelation[39]

α2=Q

0

1−γ˜

Geff󰀄G50.980.960.940.92101520zFigure20:ThetimevariationofNewton’sconstantinthenon-minimalcase.

conformallycoupledtothematter,theeffectiveNewton’sconstant(measured,e.g.,inaCavendishtypeexperiment)canbegivenby

Geff

|/Geff=0.029hH0≃

˙/G2.1×10−12yr−1.WeshouldmentionthatthecurrentsolarsystemconstraintonGeffeff

−13−1couldbemorestringentthanthis,namely(dGeff/dt)/Geff<10yr(see,e.g.ref.[40]

˙¨whichderivesconstraintsonG/GandG/GforamodelwhereQ-fieldisexplicitlycoupledto

theEinstein-Hilbertterm);itisbecausetherelevantbackgroundwhenstudyingthesolarsystemisnotthecosmologicalbutthesolutionof(3.12)correspondingtothegalactic

2/8πG.Inordertoproperlyaddress˙environment,whereQ/H≈0andρgal≫ρcrit≡3H0

thequestionoftimederivative(orvariation)ofNewton’sconstant,onehastoconsiderindetailthedynamicalsystemwhereαQistime-varying.Thisisleftforfuturestudies.

dt

5.Conclusion

Inthispaperwehaveoutlinedconstructionofaneffectivecosmologicalmodeleachforinflationanddarkenergy(orquintessence),withintheframeworkofthestandardscalar-tensortheory.ThegeneralassumptionhasbeenthattheevolutionofouruniversecanbedescribedbyEinstein’sgravitycoupledtoafundamentalscalarfieldplusmatter,describedbythegeneralaction(3.1).Thegravitationalpartoftheaction,whichisimportantforconstructingamodelofinflation,containsascalarfieldlagrangian.ThematterpartoftheactioncontainsallpossiblematterconstituentsintheformofaperfectfluidplusacouplingtermA(Q)whichcharacterizesauniversalcouplingbetweenafundamentalscalarfieldQandordinary(plusdark)matter.

–27–

InSection2,wehavepresentedanexplicitmodelforinflation,byconstructinganin-flationarypotentialthat,withproperchoiceofslopeparameters,satisfiesthemainobser-vationalconstraintsfromWMAPdata,includingthespectralindexofscalarperturbationsandtensor-to-scalarratio.

InSection3,wehavefirstderivedasetofautonomousequations,byutilizingafun-damentalvariationalprinciple,thatinacompactformdescribestheevolutionofdifferentcosmologicalparameters,namelyΩQ,wQ,Ωi,wi,ǫandαQ,asasystemoffourdifferentialequations,ofwhichonlythreearelinearlyindependent(cf(3.17)-(3.20)).Byfurthergeneralconsiderations,wehaveshownhowtheparametersqandweffcanbedeterminedfromasolutionoftheabovesystem.Asdiscussedinthebodyoftext,thesystemofequa-tions(3.17)-(3.20)couldbeanalyticallysolvedonlybymakingareductioninthenumberoffreeparametersorbyimposingadditionalconstraints.Inthiswork,oneofouraimswastokeepthemodelasgeneralaspossible,butforbeingabletofindanalyticsolutionsthenumberofparameterswasrestrictedtofour,neglectingtheradiationcomponent,andmakingareasonableadditionalassumptionthatQ≡αlna+constatthepresentepoch.

Firstbyexaminingthecasewithminimalcoupling,A(Q)=1,aclassofexact(ana-lytical)solutionshasbeenfound(cfeqs.(3.37)-(3.42)),whichfindinterestingapplicationsforthepresent-daycosmology.Thegeneralsolutionfoundintheminimalcouplingcase

˙).Thusthehasthebehaviorthatitisindependentofthesignofα(i.e.thesignofQ

directionofa“rolling”scalarfieldQdoesnotseemtohaveanysignificanteffect(whichalsodirectlyfollowedwhenlookingatthescalarfieldLagrangian(cfeq.(3.1)),exceptintheshapeofthepotential.Itisfoundthatthecriticalvalueαcrit=1.48separatestheparameterspacesofαsuchthatα<αcritallowsalatetimeaccelerationwhileα>αcritdoesnot.ThusthecharacteristicofthescalarfieldQactingasanadditionalself-repulsiveorself-attractiveformofenergyismerelydeterminedbythemagnitudeofthevelocityofthefield,d(κQ)/dlna≡α.Inseveralinterestingcaseswehavefoundaclosedformexpressionfor(reconstructed)quintessencepotentialV(Q).

AsthecombinationofWAMPandtypeIasupernovaobservationsshowasignificant

.087

constraintonthepresent-dayDEequationofstate,wQ=−0.941+0−0.101;forthemean󰀕valueωQ∼−0.941,werequire|α|∼0.4207

meansαandαQhavingthesamesign).For|αQ|󰀂0.1,andatlowredshifts,thepresent-dayvaluesofthecosmologicalparametersshowedalmostnoαQ-dependence.Thatis,anobservableeffectontheevolutionofcosmologicalparameters,suchasweffandΩQcanbeexpectedtobeseenonlyforastrongmatter-scalarcoupling,like|αQ|≫0.1.ThetypeIasupernovadatamayfavorasmallvalueformatter-quintessencecoupling,likeαQ∼0.06.

Wehavealsoshownhowinprincipleanon-minimalmatter-scalarcouplingcanaltertheevolutionofthecosmologicalparameters.IngeneralthecouplingαQalwaysappearsincombinationwiththematterdensityρm(cfeq.(3.34)).AsthemassofthescalarfieldQ

󰀈󰀊1/2

canbedeterminedbyd2Veff/dQ2evaluatedatalocalminimumandthescalar-mattercouplinginVeff(Q)caninvolveaρm-dependentterm,themassofascalarfielddepends,inprinciple,ontheambientmatterdistribution.Thusinamoresophisticatedmodel,nottreatingmatterasanisotropicperfectfluid,themassofthescalarfieldcanvarylocallyduetoapossiblystronglocalvariationofρmonsmallscales.

Acknowledgements

TheresearchofIPNhasbeensupportedbytheFRSTResearchGrantNo.E5229andalsobyElizabethEllenDaltonResearchAward(No.5393).

A.Appendix:

Correspondingtotheaction(3.1),theequationsofmotionthatdescribegravity,thescalarfieldQandthebackgroundfields(matterandradiation)aregivenby

󰀄

111

gµνR−(∇Q)2gµν+A4(Q)Tµν=0,(A.1)242

∇µ(g

µν

∇νQ)−

dV(Q)

dQ

Theseequationsmaybesupplementedwiththeequationofmotionofabarotropicperfect

fluid,whichisgivenby

󰀈4󰀊󰀄dAρi∂A

+A3.(A.3)(˙aA)(a˙A)Combiningthe(tt)and(xx)componentsoftheequation(A.1),weget

󰀆󰀉

󰀎󰀊󰀈

˙=κ21˙2−V(Q)+A4−2Hρi+wiρi.Q

2

i

󰀎󰀈

i

1−3wiρi=0.

󰀊

(A.2)

(A.4)

DividingthisequationbyH2andthenusingthesubstitutionin(3.16),yields

󰀄˙󰀊󰀈κ2ρQ2H

1+wi.+wQ−

H2H2˙andusingtheidentitiesMultiplyingeq.(3.11)withQ

˙Q¨+V˙,ρ˙Q=Q

˙2,ρQ(1+wQ)=Q

(A.5)

(A.6)

–29–

whichfollowfromeq.(3.16),weget

˙A3dA(Q)ρ˙Q+3HρQ1+wQ=Q

andthenusingequations(3.21)and(3.15)-(3.16)leadsto

3H2

󰀔󰀁

eq.(3.19).Further,multiplyingeq.(3.14)byκ

2

κ2ρdA(Q)

i

3H2

+

3H2A(Q)

Combiningthisequationwiththeidentity

󰀈

1−3wi󰀊

ρi.

2Ω′A4

i

≡κ(A.8)

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