The innocuousness of adiabatic instabilities in coupled scalar field-dark matter models

0.23 MB·English

by  Pier Stefano Corasaniti

The innocuousness of adiabatic instabilities in coupled scalar
field-dark matter models
P.S. Corasaniti
0
1 LUTH,ObservatoiredeParis,CNRSUMR8102,UniversitéParisDiderot,5PlaceJulesJanssen,92195
0 MeudonCedex,France
2

n Abstract. Non-minimallycoupledscalarfieldmodelssufferofunstablegrowingmodesatthelinearperturbationlevel.The
a natureoftheseinstabilitiesdependsonthedynamicalstateofthescalarfield.Inparticularinsystemswhichadmitadiabatic
J
solutions, large scale instabilities are suppressed by the slow-roll dynamics of the field. Here we review these results and

5 presentapreliminarylikelihooddataanalysissuggestingthatalongadiabaticsolutionscoupledmodelswithcouplingoforder
1 ofgravitationalstrengthcanprovideviablecosmologicalscenariossatisfyingconstraintsfromSNIa,CMBandlargescale
structuredata.

]
O Keywords: Cosmology-StructureFormation-LinearPerturbations
PACS: 95.35.+d,95.36+x
C
.
h
INTRODUCTION
p
-
o
The possibility of a direct couplingbetween a quintessence-likescalar field and the variousmatter componentshas
r
t beenextensivelystudiedinavastliterature(seee.g.[1]).Non-minimallycoupledscalarsappearinvarioustheoretical
s
a scenarioswhichattempttodescribefundamentalinteractionsatenergiesbeyondthatoftheStandardModelofparticle
[ physics(see e.g. [2]). Their application to cosmology and the unsolvedproblemof dark energyin the universehas

suggesteda numberofinterestingfeatures,mostimportantlythesolutionoftheso calledcoincidenceproblem.The
1
presence of the scalar interaction is not incompatible with existing constraints on the violation of the Equivalence
v
7 Principle. Ingenuous mechanisms (which may differ from one model to another) guarantee that standard General
8 RelativityisrecoveredonSolarSystemscales(e.g.[3,4]),andleavedistinctivesignaturesonthestructureformation
6 process, eventually contributingto some of the still not understoodphenomenain the contextof Cold Dark Matter
2
paradigm[5]. Consequentlythe distributionof structures,at leaston thosescales whereobservationshaveprovided
.
1 accuratemeasurements,is a key testthatsuch modelshave to pass. Neverthelessa numberof workshave indicated
0 thatcoupledscalarfieldmodelsmaysufferoflargescaleinstabilitiesatthelinearperturbationlevel.Thiswasinitially
0 pointedoutinsomespecificrealizations[6]andrecentlydiscussedinmoregeneralsetups[7,8].Theclaimhasbeen
1
particularlyemphasizedon scenarioscharacterizedby the existenceof the so called “adiabatic”regime,such as the
:
v Chameleon model [7]. In the light of these results coupled models seem to be unrealistic cosmological scenarios.
i Howeveraswehaveshownin[9]theinstabilitiesarenotgeneric,rathertheyarestronglydependentonthedynamical
X
stateofthescalarfield.Moreimportantlyalong“adiabatic”solutionsandfornaturalvaluesofthecouplingconstant
r
a suchinstabilitiesareinnocuous.Herewewillbrieflysummarizethemainresultsof[9]towhichwereferthereader
for a more detailed discussion. We will also present the results of a preliminary likelihood data analysis to test the
viabilityofthesemodelsagainstcurrentcosmologicalobservations.
PERTURBATIONS INCOUPLED SCALAR FIELD-DARKMATTER MODELS
Let us consider a scalar field f , with potentialV(f ), coupled to matter particles througha Yukawa coupling of the
form f(f /M )y y¯, where f is the coupling function and y is the Dirac spinor associated with the matter particle
Pl
(M =1/√8p G with G the Newton constant). For simplicity let us consider the case in which the scalar field is
Pl
coupledtodarkmatteronly,thusEquivalencePrincipleconstraintsareimmediatelysatisfied.Thisisnotarestrictive
assumption since our results can be extended also to models with couplingsto all matter componentsprovided the
existence of an “adiabatic” regime. Because of the coupling the energy-momentum tensor of each component of
the system is not conserved. It is only the total energy-momentum tensor that satisfies the conservation equation:
Tnm;m(T) Tnm;m(f )+Tnm;m(DM)=0.Nowletusconsideracouplingfunctionofdilatonictype, f(f )=exp(bf /MPl),withb
≡
thedimensionlesscouplingconstant,fromtheaboveconservationconditioninaflatFriedmann-Lemaitre-Robertson-
Walkerbackground(ds2= dt2+a(t)2dx2)weobtaintheevolutionequations:
−
r˙ +3Hr = b fr˙ , (1)
DM DM DM
f¨+3Hf˙+V,f = br DM, (2)
−
withtheHubblerategivenbyH2 a˙ 2= 1 r +f˙2/2+V(f ) .ThesolutiontoEq.(1)readsas
≡ a 3 DM
(cid:0) (cid:1) (cid:2) (cid:3)
r (0)
r DM= aD3Meb (f −f0), (3)
where f is the present scalar field value. From Eq. (1) and Eq. (2) we may notice that for positive values of the
0
couplingconstantb ,theinteractiontransfersenergyfromthef -fieldtothedarkmatterparticles,withthescalarfield
evolvinginaneffectivepotential
r (0)
Veff(f )=V(f )+ aD3Meb (f −f0), (4)
characterizedbyaminimum
r (0)
V,fmin =−b aD3Meb (fmin−f0). (5)
Given the above backgroundequations, the evolution of linear density fluctuations can be studied by perturbing
the Einstein equationsand the conservationof the total energymomentumtensor abouta linearly perturbedFLRW
background.Howeverinordertogainsomeintuitiveinsightonthebehavioroftheperturbationsonthelargescales
and perform a simple stability analysis, it is convenient to consider the interacting scalar field-dark matter system
as an effective single fluid with energy density r =f˙2/2+V(f )+r , pressure p =f˙2/2 V(f ), and whose
T DM T
−
perturbationsareuniquelycharacterizedbyanadiabaticsoundspeed,c = p˙ /r˙ ,andtherestframesoundspeed,
aT T T
c = d p /dr . p
sT T t
Insypnchronousgaugetheperturbationequationsreadsas
d˙ = 3H(c2 w )d +
T − sT− T T
k2 aH2 h˙
(1+w ) +9(c2 c2 ) q + ,
− T (cid:26)(cid:20)a2H2 sT− aT (cid:21) k2 T 2(cid:27)
(6)
c2 k2
q˙ = H(1 3c2 )q + sT d , (7)
T − − sT T a(1+wT) T
whered =dr /r andq istheshearvelocityperturbationofthefluid.Forabarotropiccomponentwithaconstant
T T T T
equationofstate(e.g.matter,radiation)c2=c2=w.Thisisnotthecaseforagenericfluid(e.g.scalarfield),forthis
s a
reasonwemayexpecttheeffectiveunifiedfluidtobenon-barotropic,(i.e.c2 =c2 =w ).Intermsofthescalarfield
sT 6 aT 6 T
anddarkmatterperturbationvariableswehave
c2 = 3Hf˙2+f˙[2V,f +br DM], (8)
aT 3Hf˙2+3Hr
DM
c2 = fd˙ f˙−V,f df . (9)
sT fd˙ f˙+V,f df +r DMd DM
Theserelationsprovideuswithasimplewayofdeterminingthestabilityoftheperturbationsinthecoupledsystem,
forexampleinagivenbackgroundregimeinstabilitiesmaydevelopifthesesoundspeedsacquiresufficientlynegative
values.
FIGURE1. Scalarfieldeffectivepotentialatz=0,3,10and103,thedashedlineshowsthepositionoftheminimumasfunction
oftheredshift.
SCALAR FIELD DYNAMICSAND INSTABILITY ANALYSIS
An attractor solution of the background homogeneous system consists of the field seating at the minimum of the
potential,anddriftingintimeaccordingtoEq.(5).Thisisusuallyreferredas“adiabatic”regime.Ithasbeenshown
in[10]thatalongthissolutionthefieldslow-rolls,thusithasanegligiblekineticenergy.Inparticularforapowerlaw
potential,V(f )(cid:181) f a ,theevolutionofthescalarfieldgivenbytheconditionEq.(5)readsas:
−
f a +1 1
(cid:18)f 0 (cid:19) = a3eb (fmin−f0), (10)
min
which dependsonboth the slope a and the couplingb . Equation(10)is a non-linearalgebraicequationwhich can
besolvednumericallythroughstandardbisectionmethods.Thepresenceoftheminimumdistinguishestwodifferent
setsofinitialconditions:f <f ini (smallfield)orf >f ini (largefield).Forsmallfieldvalues,f evolvesoverthe
ini min ini min
inversepower-lawpartoftheeffectivepotential,whereitminimizesthepotentialbyslow-rollingasshownin[10].In
contrastforinitiallylargefieldvalues,f rollstowardstheminimumalongthesteepexponentialpartoftheeffective
potential.Thusitrapidlyacquireskineticenergywhichsubsequentlydissipatesthroughlargehigh-frequencydamped
oscillationsaroundtheminimum.Thegrowthofthelinearperturbationsinthecoupledscalarfield-darkmattersystem
issignificantlydifferentinthesetworegimes.
AdiabaticRegime:slow-rollsuppresion ofinstabilities
Let us evaluate the adiabatic and rest frame sound speeds along the adiabatic solution respectively. Substituting
Eq. (5) in Eq. (8) and neglectingthe term proportionalto the kinetic energyof the scalar field (due to the slow-roll
condition)wehave
f˙
c2 = b , (11)
aT − 3H
-5
0
-10
-15
-0.5
-20
-25
-1
-30
0.01 0.1 1 0.01 0.1 1
0 1
0.1
-10
-20
-30
0.01 0.1 1 0.01 0.1 1
a
FIGURE2. Upperleftpanel:evolutionofthescalarfieldequationofstatewf andeffectiveunifiedfluidequationofstatewT;
Rightupperpanel:evolutionofthescalarfieldvelocitywithrespecttotheHubblerate;Lowerleftpanel:redshiftevolutionofthe
adiabaticsoundspeedc2 andpropagationofpressureperturbationsc2 ;Rightlowerpanel:Lineargrowthfactorofthedarkmatter
aT sT
densitycontrastatk=10 3,10 2and0.1Mpc 1.
− − −
sincef˙>0itthenfollowsthatc2 <0,implyingthatadiabaticinstabilitiesmayindeeddevelop.Howeverweshould
aT
remarkthatduringtheadiabaticregimethefieldisslow-rolling(i.e.3Hf˙ 0),hencethetermf˙/3Hcanbenegligibly
smallcomparedtob ,suchthatc2 0 ,thusleadingtoastablegrowth≈ofthelargescaleperturbations.Infactletus
supposethatc2 = 10 5,theinaTsta≈bili−tywillaffectmodesk 105,butthesecorrespondtoverysmallscaleswhich
aT − − ≥
arealreadyinthenon-linearregimeandforwhichthelinearperturbationtheorydoesnotapplyanylonger.Incontrast
large scale instabilities will occur if the coupling assumes extremely large values, b 3H/f˙. This is consistent
≫
with the conclusions of [7], where the authors have shown that during the adiabatic regime perturbations suffer of
instabilitiesprovidedthatb 1.Howeversuchsituationwouldbeextremelyunnaturalintroducingalargehierachy
probleminthegravitational≫sectorsinceitwouldimplyinghavingascalarfifth-forcewhichis(1+2b 2)greaterthan
gravitationalstrength.Guidedbynaturalnessconsiderationsonemightexpectthatthedimensionlesscouplingconstant
isoforderofunity.Letusnowevaluatethesoundspeedinthetotaleffectivefluidrestframe,Eq.(9)wehave
1
c2 = , (12)
sT −1 1 dDM
b df
−
assumingthatthescalarfieldisnearlyhomogeneous,df d (inPlanckunits),wehavec2 bdf /d ,andfor
b O(1)thisimpliesc2 0.Inotherwordsifthescala≪rfieDldMfluctuationsaresmallwithressTpe≈cttotheDdaMrkmatter
de≈nsitycontrast,thenthesTc≈oupledsystembehaveshasasingleadiabaticinhomogeneousfluid(c2 c2 0).These
sT ≈ aT ≈
resultsaresupportedbythenumericalstudyoftheperturbationequationsfortheindividualcomponentsofthesystem
assummarizedinFig.2.
Non-AdiabatilscRegime:largefieldoscillationsandonset ofinstabilities
For initially large field values, f rolls along the steep exponential part of the effective potential. Its evolution is
thereforedominatedbythekineticenergyandthefieldbehavesasastifffluid(wf =1,ascanbenoticedintheleft
1 0.1
0.01
0.5
0
-0.5
-1
0.01 0.1 1 0.01 0.1 1
20
20
10
10
0
0
-10
0.01 0.1 1 0.01 0.1 1
a a
FIGURE3. Upperleftpanel:evolutionofthescalarfieldequationofstatewf ;Rightupperpanel:evolutionofthescalarfield;
Lowerleftpanel:evolutionofthefieldfluctuationsdf katk=10−3,10−2and0.1Mpc−1respectively;Rightlowerpanel:evolution
ofdarkmatterdensityfork-valuesasinthecaseofdf .
k
upper panelof Fig. 3). Then as the field reachesthe minimum, the kinetic energyis dampedaway througha series
ofhigh-frequencyoscillations.Duringthisoscillatoryregime,whichissimilartothatoftheinflatoninthereheating
phase, the scalar field perturbations are unstable and exponentially amplified by the background-field oscillations
providedthattheirfrequencyincreasesastheiramplitudediminishes[11].ThisisindeedthecaseasshowninFig.3,
whereweplottheevolutionoff (rightupperpanel),df andd forthreedifferentwave-numbers,k=10 3,10 2
k DM − −
and10 1.Wecanseetherethataninstabilityoccursroughlyatthesametimeofthefirstminimum-crossingoscillation,
−
thenfollowedbyasecondstageofexponentialgrowthatthebeginningofthesecondoscillation.Suchunstablemodes
are similar to those foundin [8], in fact by averagingoverperiodsof time largerthan the characteristic time of the
oscillations,thescalarfieldbehaveseffectivelyasadarkenergyfluidwithaconstantequationofstatew,asthecase
consideredin[8].
CONSTRAINTSFROMSN IA,CMB ANDLSS: APRELIMINARY ANALYSIS
Coupled models have been tested against cosmological observations in various works [12, 13, 14]. The main con-
clusion of these analysis is that current measurements of the CMB anisotropy power spectra and the matter power
spectrumfromgalaxysurveysconstrainthecouplingconstanttobeb <0.01 0.1dependingonthespecificmodel
−
realization.Howevernoneofthese workshaveconsideredthecase inwhichthescalarfield evolvesintheadiabatic
regime. To this purpose we used the lastest SN Ia-UNION dataset compilation [15], WMAP-5 years data [16] and
matterpowerspectrummeasurementsfromSDSS-5datarelease[17]totesttheviabilityofanon-minimallycoupled
scalarfieldmodelwithpowerlawpotentialintheadiabaticregime.Forthispurposewehavespecificallysetthefield
evolutiontosatisfyEq.(10),implementedtheperturbationequationsinaproperlymodifiedversionoftheCMBFAST
code[18] andruna MarkovChain MonteCarlolikelihoodevaluation.Forsimplicitywe assumea flatuniverseand
fix b =1 and a =0.2, while let all other parameters to vary. Here we simply aimed to test whether an adiabatic
solutioncanprovidecompatiblefittothedata,andweleavetoafuturestudyamoredetailedanalysisofthefullmodel
parameterspace,includinga andb .Themarginalized1DlikelihoodareshowninFig.4andthebestfitvalueand1s
errorsare:W =0.222 0.025,W h2=0.02226 0.00068,h=0.75 0.03,t =0.072 0.017,n =0.89 0.02,
DM b s
± ± ± ± ±
FIGURE4. Marginalized1Dlikelihoods.
A =0.75 0.03and b=1.44 0.49.These constraintsare consistent with those derived for LCDM cosmologies,
s
thetotalc ±2 isclosetothatofth±evanillaLCDMmodelsuchthetwoscenariosarestatistically indistinguishable.As
shownin [10] differencesdue to the scalar interactionmay arise on the small scale clusteringof darkmatter. Over-
allthispreliminaryanalysissuggeststhatadiabaticcoupledmodelswithnaturalcouplingareconsistentwithcurrent
cosmologicalobservations.
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