The Nature of Dark Energy
Lua Amendola,ClaudiaQuerellini, DomenioTohini-Valentini,
and Alessandro Pasqui
INAF-ObservatorioAstronomiodiRoma,
ViaFrasati33,00040MontePorzio Catone,Italia
Reeivedon23May,2001
Aording to a variety of osmologial observations at small and large redshifts, the universe is
omposedbyalargefrationofaweaklylusteredomponentwithnegativepressure, alleddark
energy. The natureof the dark energy, i.e. itsinterationand self-interation properties, is still
largely unknown. In this ontribution we review the properties of dark energyas inferred from
observations, withpartiular emphasisonthe osmimirowavebakground. We arguethat the
urrent dataset imposesstrong onstraintsonthe oupling ofdark energyto dark matter, while
itisstill insuÆient toonstraintheequationofstateorpotential. Futuredatawill dramatially
improvetheprospets.
I Rods, loks and andles
Fouryears after the rstobservationalhintsfrom the
supernovae Ia (SNIa)Hubble diagram about the
exis-tene of adominantomponentof unlustered matter
with negative pressure (Riess et al. 1998, Perlmutter
et al. 1999), theso-alleddarkenergyorquintessene
(Wetterih 1988; Ratra and Peebles 1988; Frieman et
al. 1995; Caldwell et al. 1998), there are still very
fewindiationsastoitsnature. Themainreason,
per-haps,isthatwelakanyspeitheoretialsuggestion
on the properties of the dark energy, i.e. on its
self-interation potentialand on howit interats with the
other osmologial omponents. At this stage, all we
andois toexplore awide rangeofphenomenologial
models speied by the potentialand theoupling to
the other elds in order to provide an overall best t
to theurrentdata. Fortunately,therearenowseveral
osmologial observations that are at leastpotentially
abletoput stringentonstraintsonthenatureofdark
energy.
As with anyother fundamental eld,the dark
en-ergy anbe haraterized byhow it interats with
it-self and with the other elds. In other words, it an
beharaterizedbyitspotentialand ouplings. While
there have been severalpapersthat tried to onstrain
thepotentialortheequationofstateofthedarkenergy
(Amendola 2001, Baigalupi et al. 2001, Corasaniti
and Copeland 2001, Bean and Melhiorri 2001), the
study of its oupling has been relatively sare,
al-thoughithasbeensuggestedseveraltimesthatwe
an-notexpetavanishingoupling(Carroll1998).
The observations that an reveal dark energy are
aount only the evolution of the homogeneous
bak-ground and those that investigate the growth of
per-turbations. The latter lass is so far still poor of
re-sults,duetothelargeerrorsin theobservationsofthe
growthof lustering, althoughtheprospetsare
inter-esting(Newmanetal. 2001). Moreover,the
perturba-tiongrowthisverysimilarforallmodelsofdarkenergy,
sinetheaelerationstopsthegravitationalinstability
forallbutonemodelofdarkenergy. Thesingleasein
whih thisisnottruehasbeendisussedin Amendola
andTohini-Valentini(2002).
Thebakground observations allredue essentially
tothe useof astandardmeasure,either andles,rods
orloks: any of this standard is in fat subjet to a
geometriapparentvariationwhenseenathighredshift
thatdependonosmology. Thestandardandleshave
beenemployedinthemethodbasedonthesupernovae
Ia, while standard rods are assumed in the
Alok-Pazinsky method (or rather, astandardisotropy, see
e.g. Calvaoet al. 2001), in the lensingstatistis (e.g.
Cooray and Huterer 1999, Giovi et al. 2001 ) and in
thesizeofthesoundhorizonatdeouplingasdeteted
from the CMB aousti peaks (see e.g. Doran et al.
2000). Finally, standard loks are employed in the
method based on the galaxy ages (Alaniz and Lima
2001, Jimenez and Loeb 2001). They all rely on the
fatthattheproperdistanetoanobjetatredshiftz
dependsonosmology:
S[r(z)℄= Z
z
0 dz
0
E(z 0
)
wherethefuntion S[x℄isj
k j
1=2
sin 1
(j
k j
1=2
x);x;
j
k j
1=2
sinh 1
(j
k j
1=2
ur-Friedmannequation
H 2
(z) = H 2
0 E(z)
2
E 2
(z) =
m (1+z)
3
+
de (1+z)
3wde
+
k (1+z)
2
:
The subsriptsrefer to matter(m) , darkenergy (de)
and urvature(k), andw
de
is thedarkenergy
param-eter of state, p
de = (w
de 1)
de
and by denition
m +
de +
k
=1. Itislearthereforethat all
quan-titiesthat depend ondistane, likeluminosity,angular
sizes, ages, also depend on osmology. For instane,
the method based on the SN Ia Hubble diagram
de-termines the distane to a supernova(whose absolute
magnitudeM hasbeeninferredandorretedbyloal
observations),viathedistane modulus
m M=5logd
L
(z)+25
(plus possibly aK-orretion)and thenomparesthis
value to the theoretial luminosity distane d
L (z) =
r(z)(1+z)to ndthe best t in terms ofthe
param-eters H
0 ;
m ;
de ;w
de
. The asew
de
=0orresponds
toapureosmologialonstantmodel.
The main advantages with this approah are that
thedependene ontheosmologialonstantis simple
andthatoneanobserver(z)atdierentredshifts,
de-pending on the objet. So far, SN Ia have been seen
up to z 1 (with a ouple of SN beyond this), but
future experiments an extendthe observations up to
z2andto amuh extendeddataset. Moreover,the
Alok-Pazyinsky method an be applied to quasars
atz5orbeyond. Thedistaneofthesoundhorizon
at deoupling, expressed as the angular diameter
dis-taned
A
(z)=r(z)=(1+z),inturn, reahesz 1000,
theredshift ofthe last satteringsurfae. Theruial
point isthat r(z)depends ontheparametersin a
dif-ferent way for dierent redshifts. In Fig. 1 we show
thelines ofonstantd
L
(z) forapure model
assum-ingthesouresareatz=0:7;4;10;1000: theontours
followadierentorientation,rotatingantilokwisefor
inreasingz,beingalmostvertialforz4. Therefore,
observationsatlowredshift,liketheSNIa,andathigh
redshift,likethesoundhorizonat deoupling,angive
orthogonalonstraints,thereby determiningwith high
preisiontheosmologialparameters.
The observationof the CMB utuation spetrum
fall in betweenthetwolasses,sine thespetrum
de-pends both on the bakground and on the
perturba-tions: in fat,at small angular salesthe spetrumis
mostlydeterminedbytheangular-diameterdistaneto
deoupling,whileatlargeangularsalesitdependson
the growth of the gravitational potentialthrough the
integratedSahs-Wolfeeetand ontheintrinsi
u-tuations ofthe darkenergyeld. In thisreview I will
fousmostlyontheostraintsfromthereenthigh
res-olution CMB data (Nettereld et al. 2001, Lee et al.
2001, Halverson et al. 2001) beause they appear to
rapid development. The reason is that they are not
subjet to the theoretial unertainties that still
en-shroud the SNIa method. In fat, in ontrast to the
SNIa,thephysisoftheCMButuationsisrelatively
well understood, and there are no problems of
evolu-tion. Moreover,experimentsareunderwaythatshould
expandthestatistisbyafatorofahundredormore
in afewyears.
0.2
0.4
0.6
0.8
M
0.2
0.4
0.6
0.8
z 10
0.2
0.4
0.6
0.8
M
0.2
0.4
0.6
0.8
z 1000
0.2
0.4
0.6
0.8
M
0.2
0.4
0.6
0.8
z 0.7
0.2
0.4
0.6
0.8
M
0.2
0.4
0.6
0.8
z 4
Figure 1. Contoursof onstant luminosity distane dL(z)
for various values of z. At small redshift the lines follow
approximately therelationm =onst, andatlarge
redshifttheorthogonal relation
m +
=onst.
Several works havetried to onstrain dark energy
models with the reent CMB data, leading in some
asestointerestingbutonitingresults. InAmendola
(2001) the dimensionless oupling parameter that
measuresthestrengthoftheinterationofdarkenergy
to darkmatterwithrespettothe gravitational
inter-ationwasonstrainedto besmallerthan 0.1roughly,
adoptinganexponentialpotential. Theslopeofthe
po-tential, ontheotherhand,wasfoundto beessentially
unonstrained by the data. Baigalupiet al. (2002)
foundthat, among power-lawpotentialsV
and
xing the Hubble onstant to h = 0:65 (in units of
100km/se/Mp),valuesofaroundunityarefavored
whiletheaseofpureosmologialonstant,=0,was
less likely by a fator of ve roughly. Corasanitiand
Copeland (2001),on theother hand,haveshown that
anequationofstatelosetothatofapureosmologial
onstantgivesthebest tto thedata,partiularlyfor
as onerns the position of the aoustipeaks. Bean
andMelhiorri(2002)alsoonludethatthepure
os-mologialonstantgivesthebestt to theCMBdata
whenthepriorontheHubbleonstantisbroadenedto
h=0:720:08.
Reently, we extended the previousstudies in two
en-teration withmatter. Seond,weput onlyveryweak
restritionsontheosmologialparametersh,
b; (the
density parameters of baryons and old dark matter,
respetively), and n
s
(the slope of the salar
pertur-bations). Theoniting resultsabovearein fat due
mostlytodierentpriors.
These generalizations allowed us to nd that the
present CMB data are apable to put a strong
on-straintonthe ouplingbut noton thesalareld
po-tential or equation of state. In fat, the degeneray
betweenh and w
, the darkenergy equation of state
(Huey etal. 1999)almost erasesthe sensitivityof the
CMB to thedarkenergypotential. Insharp ontrast,
theCMBspetraareverysensitivetothedarkenergy
oupling, sine the latter determines the equation of
state for a long stage after equivalene and there are
nostrongdegeneraieswiththeotherosmologial
pa-rameters.
II A Dark-dark oupling
Let us onsider two omponents, a salar eld
andCDM,desribedbytheenergy-momentumtensors
T
() andT
()
,respetively. Generalovariane
re-quirestheonservationoftheirsum,sothatitis
possi-bletoonsideraninterationbetweendarkenergyand
darkmattersuh that
T
();
= CT
()
; ;
T
();
= CT
()
;
: (1)
(2)
Suh a oupling an be obtained from the
onfor-mal transformation of a Brans-Dike gravity (see e.g.
Amendola 1999) and it has been onsidered several
timesinliteraturestartingfrom Wetterih(1988,1995)
and Wandset al. (1993). It hasbeen disussedin the
ontextofdarkenergymodelsinAmendola(1999,2000)
and, inalistwhihismeanttobeonlyrepresentative,
WandsandHolden(2000),Chimentoetal. (2000),
Bill-yard and Coley (2000), Chiba (2001), Albreht et al.
(2001),Esposito-FareseandPolarsky(2001). Inits
on-formallyrelatedBrans-Dikeformhasbeenstudied by
Uzan (1999), Chiba (1999), Chen and Kamionkowsky
(1999),Batista et al. (2000), Bertolami and Martins
(2000), Baigalupiet al. (2000), Faraoni (2000),Sen
andSen(2001). Theoretialmotivationsinsuperstring
modelsandinbraneosmologyhavebeenproposed
re-entlyin Gasperini, Piazza and Veneziano(2002)and
Pietroni(2002). Otherauthorsonsideredaouplingto
spei standardmodel elds ratherthanto ageneri
formofdarkmatter: Carroll(1998)to the
eletromag-netield, Horvat(2000)toneutrinos,Liet al. (2002)
to baryoni or leptoni urrent: in these ases, there
arestrong onstraintsfrom loal observationsorfrom
variationoffundamental onstants,seee.g. Damouret
al. (2002). Inontrast, a oupling to darkmatter is
observableonlywithosmologialexperiments,
involv-inggrowthofperturbationsorglobalgeometrieets
(properdistane).
IntheatonformalFRWmetrids 2
=a 2
( d 2
+
Æ
ij dx
i
dx j
)thesalareldanddarkmatteronservation
equationsare
+2H _
+a 2
U
;
= C
a
2
; (3)
_
+3H
= C
_
(4)
(dots refer to onformal time) where H = a=a._ We
supposenowthatthepotentialU()isspeiedbythe
followingrelation
U 0
=BU N
(5)
(theprime refersto derivationwithrespetto ): this
form inludes power laws, exponential potentials and
a pure osmologial onstant, i.e. the most ommon
forms of darkenergy potentials. In the aseof power
lawpotentialU =A
wehave
N =(1+)= (6)
andB = A ( 1=)
; whilefor theexponential
poten-tialN =1. We onsider only the rangeN 1sine
fornegativetherearenoasymptotiallyaelerating
models; for N ! 1wereoverthepure osmologial
onstant. Clearly,forA=0thepotentialvanishes,and
weredue to atheory whih is onformallyequivalent
topureBrans-Dikegravity.
Assumingthatthebaryonsarenotdiretlyoupled
tothedarkenergy(otherwiseloalgravityexperiment
wouldrevealafthfore, seeDamouret al. 1990)and
thatthe radiationaswellis unoupled (as itours if
the oupling is derived by a Brans-DikeLagrangian,
see e.g. Amendola 1999), the system of one Einstein
equationandfouronservationequations(forradiation,
,baryons,b,CDM,,andsalareld)anbe
onve-nientlywrittenintroduingthe followingvevariables
thatgeneralizeCopelandetal. (1997):
x=
H _
p
6 ; y=
a
H r
U
3 ; z=
a
H r
3 ; v=
a
H r
b
3
; w= H
a
(7)
where 2
=8G(notiethatw=dloga=dti.e. theusualHubbleonstant). Notiethat x 2
;y 2
;z 2
;et. orrespond
to thedensityparameterof eahomponent. Intermsoftheindependentvariablelogawehavethenthesystem:
x 0
=
z 0
1
x y
2N
w 2N 2
+(1 x 2
y 2
z 2
v 2
y 0 = xy 2N 1 w 2N 2 +y 2+ z 0 z ; z 0 = z 2 1 3x 2 +3y 2 z 2 ; v 0 = v 2 3x 2 +3y 2 z 2 w 0 = w 2
3+3x 2 3y 2 +z 2 (8) d where =C r 3 2 2
; =3 N 1 2N B p 6 : (9)
The dimensionless onstant 2
an be seen asthe
ra-tioofthedarkenergy-darkmatterinterationwith
re-spettogravity. Itanbeshownin fat(Damourand
Nordtvedt 1993, Wetterih 1995) that the fore
at-ing betweendarkmatterpartiles an bedesribed in
the Newtonian limit as arenormalized Newton's
on-stant ^
G=G(1+4 2
=3). Thisshowsthat,in termsof
the \post-Einsteinian" parameter (see Groom et al.
2001) whih haraterizesthe deviation from General
Relativity, we have = 4 2
=3. The eet of this
in-terationonstruturegrowthisdisussedinAmendola
andTohini-Valentini (2002).
Thesystem(8)inludesseveralqualitatively
dier-ent behaviors, already disussed in Amendola (2000),
Tohini-ValentiniandAmendola(2002). However,for
the range of values that are of osmologial interest,
the system passes through three distint phases after
equivalene.
MDE. Immediately after equivalene, the system
entersamatterdominatedepohwithanon-negligible
ontribution, that we denote MDE, in whih the
darkenergypotentialdensityisnegligiblewhilethe
ki-netienergydensityparameter
K =x
2
ofthesalar
eld givesaonstantontribution tothetotaldensity.
Aswillbeshowninthefollowing,theexisteneofsuh
an epoh is ruial forthe onstraintsthat we will be
able to put on the oupling. It is not diÆult to see
that,negletingradiationandbaryons,thepointy=0,
x = 2=3 is a saddlepointsolution of thesystem (8)
foranyNandforjj< p
3=2. Thesystemstaysonthe
MDEsolutionuntily startsgrowing. Alongthis
solu-tionthesalefatorexpandsslowerthanapureMDE,
i.e. as
at 4=(6+4
2
)
: (10)
Siney=0ontheMDE,itsexisteneisindependent
of the potential, although it has to veried for eah
potentialwhetheritisasaddle.
Traking trajetories. Let us now neglet baryons
and radiation and put = 0. Thetraking solutions
foundin Steinhardtet al. (1999)assumey=x=pand
y 2N 1
w 2N 2
=q where p;q are twomotion integrals.
Inthelimity 2
;x 2
1itiseasytoshowthatp 0 =q 0 =0 if p 2
= 4N 3 (11)
q =
3p
2(2N 1)
(12)
The sametraking behavior remains a good
approxi-mation for small . ForN = 1the traking solution
beomes atually a global attratorof the dynamial
system,see below. Intheother ases,thetraking
in-terpolatesbetweentheMDEandtheglobalattrator.
Globalattrators. Inthesystem(8)forN 6=1there
exists only the attrator x = 0;y = 1 on whih the
dark energy ompletely aounts for the matter
on-tent. For N = 1 the phase spae is muh riher,
and there are several possible global attrators, only
two of whih aelerated (Amendola 2000). One, for
>( + p
18+ 2
)=2,presentsaonstantnon-zero
: this attratoratually oinideswith thetraking
solutionsand in fat realizesthe onditionp= 1and
y = 3=(2) asrequestedbyEq. (11)for =0. This
\stationaryattrator"anbeaeleratedif >2and
ould solvetheoinidene problemsine
/
; it
hasbeendisussedindetailinAmendolaand
Tohini-Valentini (2001, 2002). The other aelerated
attra-tor ourswhen <( + p
18+ 2
)=2: in this ase
therearenotrakingsolutionsandtheglobalattrator
redues to x = 0;y = 1 as for N 6= 1. These
solu-tionshave alreadybeenompared to CMBfor N =1
inAmendola(2001). Theinlusionofthebaryons
mod-ies the onsiderationsabovebut, as longastheyare
muhsmaller thantheother omponents,the
qualita-tivebehaviorofthesystemremainsthesame. Itistobe
notied thatthenal attrator,onwhih thedark
en-ergydominatesompletelytheosmiuid,isyettobe
reahed, and therefore the existeneof an aelerated
epohatthepresentdepends mostlyonthetraking.
TheexisteneoftheMDEsaddleandofthe
trak-ing solutionsis ruial for ouranalysis. Infat, these
twoepohsguaranteethat theequationof stateof the
salareldispiee-wiseonstantthroughessentiallyall
thepost-equivaleneepoh. IntheMDEphasethe
ef-fetive parameter of state w
e =p
tot =
tot
+1 and the
eld equationofstatew
=p
=
+1are
w
e =1+
4 9 2 ; w
=2 (13)
whileduring thetrakingphase
(thelastrelationisapproximatedanditisatuallyonly
anupperbound tothepresentw
;itismorepreiseif
w
is identied with the average equation of state
af-ter MDE rather thanthe present equation of state).
Therefore,theosmievolutiondependsonalone
dur-ing the MDE, and on N alone during the traking.
Sine the position of the aousti peaks is related to
theequationofstatethroughtheangulardiameter
dis-tane, it appears that the CMB is ableto put diret
onstraints on both and N. Sine the sign of is
ininuent, we onne ourselveshereinafter to > 0:
InFig. 2weshowatypialtrajetorythatpresentsin
sequene the three epohs disussed above. InFig. 3
wepresentathree-dimensionalphasespaex;y;z (i.e.,
eld kinetienergy,potentialenergy,andradiation
en-ergy, respetively) for N =1, negletingbaryons(the
equation forw deouplesfor N =1). Notiethe
loa-tionoftheMDEandthenalattrator.
0
10
1
10
2
10
3
10
4
10
5
z
0
0.5
1
1.5
2
w
,w
e
w
✁w
e
✂
MDE
T
A
0
10
1
10
2
10
3
10
4
10
5
0.001
0.01
0.1
1
✄
rad
cdm
b
☎y
2
b
☎
MDE
T
A
Figure 2. Numerial solutions of the system (8) for N =
2;=0:1;!
=0:1;!
b
=0:02;h=0:65 plottedagainstthe
redshift. Upper panel. Long dashedline: radiation; short
dashedline: CDM;unbrokenthinline: baryons;unbroken
thikline: salareld;dottedline: salareldpotential
en-ergy. Thehorizontalthinlinemarksthekinetienergy
den-sityofMDE,reahedjustafterequivalene. Lowerpanel.
Thikline: eetiveequationofstate;thinline: darkenergy
equationofstate. ThelabelsmarktheMDE,thetraking
N
1.,
Β
0.5,
Μ
3.
-1
-0.5
0
0.5
1
x
0
0.2
0.4
0.6
0.8
y
0
0.25
0.5
0.75
1
z
MDE
A
-1
-0.5
0
0 5
Figure3. Three-dimensionalphase spae for N =1. The
trajetories start at x = 1. They all end at the global
attrator,butmostpassthroughtheMDEsaddle.
There is however a ruial dierene between the
MDEandatrakingforwhatonernshere:whilethe
presentdarkenergyequationofstate,setbythe
trak-ing, is degenerated with h for as onerns the CMB
spetrum (see e.g. Huey et al. 1999; Bean and
Mel-hiorri2002), the equation of state during the MDE
isnot.Infat,theangulardiameterdistanetothelast
satteringsurfaed
A
isdegeneratealonglinesh(w
)for
whih
d
A
Z
1
ade da
!
a+(h
2
!
)a
4 3w
1=2
=onst
(15)
where!
h
2
(althoughthisisexatonlyfor=0
it remains a good approximation even for small
non-zerovalues). InFig. 4thedegeneraybetweenspetra
fordierent valuesof h and N (the other parameters
beingequal)appearslear. Assumingastrongprioron
hapeakemergesin thelikelihood forN but thenthe
resultislearlyprior-dependent. The sameholdstrue
formodelsinwhihtheequationofstateisslowly
vary-ing(seee.g. Huey etal. 1999,Doranetal. 2001). On
theotherhand, thefat that
6=0at deouplingin
oupledmodelsimpliesthattheeetsoftheoupling
ontheCMBarenotduesolelytotheangulardiameter
distane, and therefore the geometri degeneray an
be broken. This isshownin Fig. 5in whih C
`
spe-trafor variousvaluesof (allother parametersbeing
200
400
600
800
1000
{
1000
2000
3000
4000
5000
6000
7000
8000
{
{
1
C
{
2
Π
Μ
K
2
h
0.8 N
3.2
h
0.75 N
2.5
h
0.67 N
1.7
Figure4. Spetrafordegeneratedparameters(theother
pa-rametersare=0;!
b
=0:01;!
=0:1). Thespetrahave
beensaledupforlarity: theyoverlapalmostexatly.
200
400
600
800
1000
1000
2000
3000
4000
5000
6000
7000
8000
1
C
2
K
2
✁✂
0.
✁✂
0.1
✁✂
0.2
Figure5. CMBspetraforinreasingvaluesoftheoupling
onstant(theotherparametersareN=1:5;h=0:65;!b=
0:01;!=0:1).Notiethatthepeaksnotonlymovetothe
rightbutalsohangeinheight.
III Constraining past and
present equation of state
with CMB
Ourtheoretialmodeldependsonthreesalareld
pa-rametersandfourosmologialparameters:
;;N;n
s ;h;!
b ;!
(16)
where !
b
=
b h
2
and !
=
h
2
. In Amendola et
al. (2002)wealulated theC
`
spetrabyamodied
CMBFAST(SeljakandZaldarriaga1996)odethat
in-ludes the full set of oupled perturbation equations.
WeusedasdatasettheCOBEdataanalyzedinBondet
al. (2000),andthehigh resolutiondataofBoomerang
(Nettereld et al. 2002), Maxima (Lee et al. 2002),
andDASI(Halversonet al. 2002). Weadopted apure
log-normal likelihood; the overall amplitude and the
alibrationerrorsofBoomerang(10%)andofMaxima
and DASI(4%) havebeen integratedout analytially.
Thetheoretialspetrahavebeenbinnedasthe
exper-analyses we assume uniform priors with the
parame-tersonnedintherange 2(0;0:3); N 2(1;8:5);
n
s
2 (0:7;1:3); h2 (0:45;0:9); !
b
2(0:005;0:05);
!
2(0:01;0:3). Thesameageonstraints(>10Gyr)
usedin mostpreviousanalysesisadoptedhere.
0
0.2
0.4
0.6
0.8
w
0.5
0.6
0.7
0.8
0.9
h
1.1
1.5
2
5
N
Figure 6. Likelihood ontour plots in the spae N;h
marginalizing over the otherparameters at the 68,90 and
95% .l.. The white thik linesrefer to the oupled ase,
theblakurvestothe unoupledase. Thedottedline is
the likelihooddegeneray urve, thedashedline is the
ex-peteddegenerayurve. Notiethat onlyxinghsmaller
than0.65 itwouldbepossibletoexlude theosmologial
onstantat95%.l..
In Fig. 6 we show the likelihood urves for N;h
marginalizingovertheremainingparameters. The
on-tours ofthe likelihoodplotfollowtheexpeted
degen-eray of the angular diameter distane. The residual
deviation from the expeted degeneray line is due to
theageprior(>10Gyr)thatfavorssmallhvalues.
No-tiethatforh>0:75noupperlimittoN anbegiven
withtheurrentCMBdatanomatterhowpreisehis,
and that only assuming h < 0:65it beomes possible
to exludeat95%.l. thepureosmologialonstant.
In Fig. 7 we show the likelihood for all parameters,
marginalizingin turnovertheothers. Wendthe
fol-lowingonstraintsat95%.l.:
N >1:5; <0:16 (17)
(the limiton N orresponds to <2). The limit on
N is however only a formal one: the likelihood never
vanishes in the denition domain and even N = 1
(the exponentialpotential) is onlyafator of veless
likelythan thepeak and ertainlyannot be exluded
on this basis. Moreover, the lower bound on N is
prior-dependent: allowingsmallervaluesofhit would
weaken. Clearly,ifweadoptnarrowpriorsonh,wean
not depend sensibly on the prior on h and the
likeli-hoodfor doesvanishatlarge. InplaeofN and
weanuseaswelltheequationofstateduringtraking
andduringMDE,respetively,aslikelihoodvariables,
using Eqs. (13)and (14). Then weobtainat the95%
.l.
w
(trak ing)
<0:8; 1<w
e(MDE)
<1:01 (18)
This shows that the eetive equation of state
dur-ing MDE, i.e. between equivalene and traking, is
lose to unity (as in a pure matter dominated epoh)
towithinoneperent. Thestrikingdierenebetween
the level ofthe twoonstraintsin (18)wellillustrates
the main point ofthis paper: theCMB is muh more
sensitivetothedarkenergyouplingthantoits
poten-tial.
Figure 7. Marginalizedlikelihoodfortrakingtrajetories.
Theequationofstateinpanelaisw=1=(2N 1). In
pan-elsaandbweplotasadottedlinethelikelihoodassuming
h=0:650:05andasdashedlineh=0:750:05(gaussian
prior). In panel b the long-dashed line is the Plank-like
likelihood. Inpaneldthelong-dashed lineisfor=0.
Theotherparametersare(wegiveherefor
simpli-ity themean and theonesigma error,while the limit
isat the95%.l.)
n
s
=0:990:05; !
b
=0:0230:004;
!
=0:0920:02; h>0:62 (19)
Thetotaldarkenergydensityturnsouttobe
>0:60
(95% .l.). Itappearsthatthelimitsonthe
osmolog-ialparametersn
s; !
b ;!
arealmost independent of,
whileanon-zero favorshigherh(Fig. 7,paneld). It
is interesting to omparewith the urrentonstraints
from the Hubble diagramof the supernovaeIa, where
one obtainsquiteastrongerbound ontheequationof
state,w
<0:4orN >1:75atthesame.l..
Finally,inFig. 7(panelb)weplotthelikelihoodfor
that an experiment with no alibration unertainty
mission (de Zotti et al. 1999), an ahieve. We nd
< 0:05 (95% .l.) (using the speiations of the
MAPsatellitewend <0:1).
IV Conlusion: the CMB as a
gravity probe
Adarkmatter-darkenergyinterationwouldobviously
esapeanyloal gravityexperiment: osmologial
ob-servations liketheCMB are then theonly wayto
ob-servesuh aphenomenon. Sine observations require
thebaryonstobedeoupledfrom darkenergy(or
ou-pledmuh moreweaklythan darkmatter),thesearh
foranon-zero isinfatalsoatestoftheequivalene
priniple.WefoundthaturrentCMBdataareapable
toputaninterestingupperboundtothedarkmatter
-darkenergyoupling:
<0:16
(95%.l.) regardlessofthepotential(withinthe lass
weonsidered). This impliesthat thesalargravityis
at least1= 2
40 times weakerthan ordinarytensor
gravity. As shownin e.g. Amendola (1999),the limit
on anberestatedasalimitontheonstantofthe
non-minimally oupledgravity, dened bythe gravity
Lagrangiandensity
L= p
g
1 1
2
2
R+L
(20)
whereL
istheanonialsalareld Lagrangian(we
areassumingthat theLagrangianforthebaryonelds
ouplestoaonformallyrelatedmetri). Sineforsmall
onehas 2 2
=3,wehave<0:017. Similarly,one
an express the limit in terms of the original
Brans-Dikeparameter! denedbytheLagrangian
L= p
g( R+ !
;
;
) (21)
Inthisasewehave!>60.
An experiment like the Plank mission an lower
theupperbound to to 0.05(95% .l.): salar
grav-ity would be in this ase at least 400 times weaker
thanordinarytensorgravity. This limitisomparable
to thosethat loal gravityexperimentsimpose onthe
salargravityouplingtobaryons,expressedintermof
the\post-Einsteinian"(Groomet al. 2001)parameter
= 4 2
=3 = 0:00070:0010, i.e. 2
baryons < 10
3
(seee.g. Groometal. 2000).
In ontrast, CMB data, on their own, annot put
any rm limit to the dark energy potential, unless a
narrow prior on h is adopted: e.g. h = 0:650:05
gives w
= 0:550:2 while h = 0:750:05 gives
w
=0:350:2(assuminggaussianpriorsand
marginal-izingasusual overall theother parameters,inluding
theoupling).
ex-assumingthattheSNIadimmingisausedbysome
evo-lutionaryeet,wearestill onfrontedwith the
prob-lem of reonilinga atuniverse asseenby theCMB
data with the small fration of matter in lusters of
galaxies. Lakinganydenitesuggestionastoits
prop-erties,osmologistsannotdomuhbetterthantrying
to inferits nature fromobservations. Wemodeledthe
darkenergyasasalareld governedby ageneri
in-versepower-lawpotentialandaouplingto dark
mat-ter. The ouplingintroduesanewinterationthat is
unobservableonloalsalesbutgivesstrongsignatures
forasonernstheperturbationgrowthandtheproper
distane. Wereviewedtheurrentonstraintsfromthe
CMB, nding that theoupling of thedarkenergy to
darkmatterannotexeed0.16intermsofthestrength
ratio. Thefuture data from thePlankmission an
reduetheupperlimittoalevelomparabletothe
lim-its that loal gravityexperimentsput ontheoupling
tobaryons.
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