Is dark energy an effect of averaging?

0.15 MB·English

by  Nan Li

FortschrittederPhysik,2February2008
Is dark energy an effect of averaging?
NanLi,MarinaSeikelandDominikJ.Schwarz∗
Fakulta¨tfu¨rPhysik,Universita¨tBielefeld,Postfach100131,D-33501Bielefeld
8
Keywords cosmology,darkenergy,Hubbleexpansion
0
0 Thepresentstandardmodelofcosmologystatesthattheknownparticlescarryonlyatinyfractionoftotal
2 massandenergyoftheUniverse.Rather,unknowndarkmatteranddarkenergyarethedominantcontribu-
n tionstothecosmicenergybudget. Wereviewthelogicthatleadstothepostulateddarkenergyandpresent
a analternativepointofview,inwhichthepuzzlemaybesolvedbyproperlytakingintoaccounttheinfluence
J ofcosmicstructuresonglobalobservables. Weillustratetheeffectofaveragingonthemeasurementofthe
Hubbleconstant.
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]
h
p
1 Introduction
-
o
r The concordance model of cosmology states that only 4% of the mass/energy content of the Universe
t
s is carried by identified particles, and the dominant contributions are unknown dark matter (22%) and
a
mysteriousdarkenergy(74%)[1]. Itissaidthatwearelivingintheageof“precisioncosmology”,which
[
isjustifiedasastatementontheprecisionofobservations,butcertainlynotontheirunderstandings. The

1 purposeofthistalkistorecapitulatethegenesisofthedarkenergyhypothesisandtoshowthatitposesa
v
seriouscrisistotheoreticalcosmology. Wearguethatapossibleresolutionforthiscrisismaycomefrom
0
thepropertreatmentofeffectsfromaveraging.
2
4
3
2 Dark energy
.
1
0 Thevastmajorityofcosmicbackgroundphotons(photonswithoutidentifiedsource)areduetothecosmic
8 microwave background(CMB) radiation, which is isotropic (apart from a componentdue to our proper
0 motion)withfluctuationsatthelevelof10−5. ThustheisotropyoftheUniverse(aroundus)isanobser-
:
v vationalfact. Ontheotherhand,thereisnoevidencethattheMilkyWaywouldbeadistinguishedplace
i intheUniverseandthusitseemsreasonabletofollowCopernicustoassumethatwearenotlivingatthe
X
centreoftheUniverse. Therefore,weendupwithahomogeneousandisotropicmodel. Themostgeneral
r
a lineelementinsuchaFriedmann-Lemaˆıtremodelisgivenby
dr2
ds2 = dt2+a2 +r2dΩ2 ,
− (cid:18)1 Kr2 (cid:19)
−
wherea =a(t)denotesthescalefactorandK/a2thespatialcurvature(K = 1,0,+1)oftheUniverse.
−
In this model physical distances evolve with time, r = af(r) (we may fix the present scale factor
ph
a 1), where f(r) = Arsinh(r),r,arcsin(r) for K = 1,0,1 . Itis usefulto define the rate of
0
≡ { } {− }
linearexpansionH a˙/a. WithoutusingEinstein’sequations,justassumingthatphotonstravelonnull
≡
geodesics,onefindsthatlightfromdistantobjectsisred-shiftedbyz ∆λ/λ=1/a 1andtheso-called
≡ −
luminositydistance(definedbytheobservedfluxfromasourcewithknownluminosityd = L/4πF)
L
satisfiestheHubblelaw p
c z2
d = z+(1 q ) + (z3)
L 0
H (cid:20) − 2 O (cid:21)
0
∗ Talkat“Balkanworkshop2007”,Kladovo(Serbia).
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2 NanLi,MarinaSeikelandDominikJ.Schwarz :Isdarkenergyaneffectofaveraging?
0.4
Gold (MLCS2k2)
ESSENCE(MLCS2k2)
0.35
ESSENCE (SALT)
0.3
0.25
∆µ 0.2
0.15
0.1
0.05
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
z
Fig.1 Thebinnedmagnitudes∆µ(seetext)fordifferentSNdatasetsandlight-curvefittingmethods.
at small redshifts z 1. Observationsshow that H 70 km/s/Mpc > 0 and thus we know that the
0
≪ ≈
Universe expands. The typical time scale of the cosmos is estimated as t 14 Gyr, in concordance
H
≈
with the estimate on the age of the oldest objects in the Universe [2]. Another interesting kinematic
quantityistheacceleration(deceleration)ofthecosmicexpansion,encodedbythedecelerationparameter
q (a¨/a)/H2. MeasurementsofsupernovaeoftypeIa(SNIa)indicatethatq <0.
0
≡−
One might wonder what could be the reason for the high symmetry of the observed Universe. The
answerofmoderncosmologyistheideaofcosmologicalinflation[3],anepochintheveryearlyUniverse
thatisheldresponsiblefortheUniverse’shomogeneityandisotropy.Ontopofthat,genericpredictionsof
cosmologicalinflationarespatialflatness(Ω =0)andtheexistenceoftinyfluctuations[4],asobserved
K
atalevelof10−5intheCMB.
We may ask how strong is the evidence for the acceleration of the Universe, even before we wish to
invokeamodelforthemattercontent,withoutassumingthevalidityofEinstein’sequations.Thistestgoes
asfollows. We ask atwhatlevelof confidencewe canrejectthenullhypothesisthattheUniversenever
accelerated(atz <few).Foraspatiallyflatmodel,assuggestedbytheideaofcosmologicalinflation,our
nullhypothesiscanbeexpressedas
c
d < (1+z)ln(1+z).
L
H
0
BasedontwodifferentSNIadatasets,twodifferentfittingmethodsandtwodifferentcalibrationmethods,
weconcludethatthenullhypothesisisrejectedat>5σ[5].
Thistestcanalsobedoneinacalibration-freeway,i.e. withoutassumingcertainvaluesfortheHubble
constant H and the luminosity of the SNe. Instead, we consider the averaged value of the apparent
0
magnitudemofSNewithinacertainredshiftbinrelativetotheaveragedapparentmagnitudem of
nearby
nearbySNewithz <0.2.Defininganewquantity
∆µ=m m =5log d 5log d ,
− nearby 10 L− 10 L,nearby
the null hypothesiscan be written as ∆µ < 0. For the test, we averagem within redshiftbins of width
0.2. InFig. 1,theresultisshownfortwodatasets(Gold[6]andESSENCE[7])andtwofittingmethods
(MLCS2k2[8]andSALT[9]). Theevidenceforaccelerationcanbeobtainedbydividing∆µbyitserror.
TheresultforeachredshiftbinisgiveninTable1. Alsogivenistheevidenceobtainedfromaveragingover
theapparentmagnitudeofallSNewithredshiftz 0.2. Itisnoticeablethatusingthecalibration-freetest
≥
weakenstheevidenceforacceleration(4.3σ)comparedtotheresultoftheprevioustest(5.2σ).
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Gold2007 ESSENCE ESSENCE
z (MLCS2k2) (MLCS2k2) (SALT)
0.2–0.4 2.0 2.2 3.2
0.4–0.6 4.4 4.2 5.2
0.6–0.8 2.8 4.6 3.4
0.8–1.0 2.6 4.9 3.7
1.0–1.2 2.0
1.2–1.4 1.6
0.2 4.3 5.2 5.6
≥
Table1 Statisticalevidence for accelerationfrom different redshift binsfor different SNdatasetsand light-curve
fittingmethods.
LetusnowaskwhatthatimpliesforthedynamicalmodeloftheUniverse.Themostgeneralequationof
motionforahomogeneousandisotropicUniverseundertheassumptionofdiffeomorphisminvarianceand
at most second orderderivativeson the metric (Lovelocktheorem [10]) providesus with the Friedmann
equationandthecontinuityequation(c=1)
K 8πG
H2+ Λ= ǫ, ǫ˙+3H(ǫ+p)=0.
a2 − 3
ǫ and p denote the energy density and pressure. G and Λ are Newton’s constant and the cosmological
constant. Inordertoclosethissystemofequations,wealsoneedtoimposeanequationofstatep=p(ǫ),
whichinmanycasescanbeexpressedasw =p/ǫ.AdimensionlessenergydensityΩ ǫ/(3H2/8πG)is
≡
introducedforconvenience.
We have already mentioned that cosmological inflation predicts a vanishing spatial curvature for all
practicalpurposesandthuswecanputK = 0. InthepresentUniverse,matterisnon-relativistic(cold),
whichmeansp ǫandthusweputp=0. IfwenowwouldputΛ=0aswell,wearriveattheEinstein-
≪
deSitterUniverse,whichpredictsq(t)=1/2andt =2t /3 9Gyr. Therefore,wecannotsticktothis
0 H
≈
simplemodel,aswehaveseenthatq < 0athighconfidencelevelandtheoldeststarsareknowntobeas
oldas12Gyrorevenolder[2].
Thesefactsbecameclearonlyabouttenyearsago—the1998cosmologyrevolution. Onecanrewrite
thedynamicequationsinto
a¨
3 =4πG(ǫ+3p) Λ<0.
− a −
Theinequalityandthusacceleration,canoccurifeitherΛ > 4πGǫ(equivalenttoΩ > Ω /2)orthere
Λ m
existssomeformofenergydensitywithp < ǫ/3,aformofenergythathasneverbeenobservedinthe
−
laboratorysofarandisthusdubbedthe“darkenergy”. Forp = ǫthisreproducesthebehaviourofthe
−
cosmologicalconstant(asǫ˙=0inthatcase),whichisthusaspecialcaseofdarkenergy.
So a minimal model consistent with the predictions of cosmological inflation and allowing for the
observedacceleratedexpansionoftheUniverseisgivenbyK = p = 0andΩ > Ω /2 = (1 Ω )/2
Λ m Λ
−
and thus Ω > 1/3 and Ω < 2/3. This model is called the flat ΛCDM (cold dark matter) model or
Λ m
concordancemodel,asitiscurrentlyingoodagreementwithallcosmologicalobservations.
OnecantestforapossibledeviationfromK = 0orp = 0bymeansofCMBobservations,SNIaand
thebaryonacousticoscillationsandfindsthatthereisnoobservationalevidencetoquestionthevalidityof
theflatΛCDMmodel[1,11].
Let us stress that an important assumption in the reasoning described above is the homogeneity and
isotropy of the Universe and thus it is also important to test this assumption with the help of SN Ia. A
simple test can be doneby fitting the Hubblediagram(d vs. z)for SNe fromjust one hemisphereand
L
comparing the result of the fit to the opposite half of the sky [12]. Weinhorst and one of the authors
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4 NanLi,MarinaSeikelandDominikJ.Schwarz :Isdarkenergyaneffectofaveraging?
haveanalysedfourdifferentSNdatasetsandfindastatisticallysignificantdeviationfromtheisotropyof
theHubblediagram. However,itseemsthattheseanisotropiesarealignedwiththeequatorialcoordinate
system (aligned with the Earth’s axis of rotation) and are thus likely to reflect systematic problems in
the SN Ia observationor analysis. Another recent analysis revealed anisotropies in the Hubble diagram
obtainedbytheHSTKeyProject[13]. IftheseanisotropieswouldbeconfirmedbyfutureSNsurveysand
asystematiceffectcouldbeexcluded,thiswouldposeaseriouschallengetotheinterpretationofHubble
diagramsintheFriedmann-Lemaˆıtrecontext.
3 Problems ofthe concordance model
Thusitseemsthattheconcordancemodelisabletodescribeeverything,howevertherearethreeconceptual
problemsinthismodel.
Thecosmologicalconstantproblem. Wemissoutafundamentalunderstandingofthevacuumenergy
ofquantumfields. Thisvacuumenergydensityisindistinguishablefromacosmologicalconstantandthus
wehaveintroducedanelementtotheUniversethatwedonotunderstand.Thesolutionforthisissuemust
beexpectedfromafuturetheoryofquantumgravity,asgravity(i.e. geometryofspace-time)istheonly
“force”thatcouplestotheenergydensityofthe(quantumfield)vacuum. Thusthisproblemisobviously
beyondthescopeofwhatwecanhopetounderstandcurrently.
The coincidence problem(s). The second problem however is a pure cosmologicalone. It is called
thecoincidenceproblemandtherearetwowaystoformulateit. Thefirstformulationhassomeanthropic
touch,butiswidelyused:howdoesitcomethatΩ (t ) Ω (t )? Thisisindeedsurprising.Assumewe
m 0 Λ 0
∼
wouldhavelivedwhentheUniversehadhalf(twice)ofitspresentlinearsize [atz = 1(-1/2)],theratio
Ω /Ω wouldbelarger(smaller)byalmostanorderofmagnitude.Itisthereforequitesurprisingthatwe
m Λ
observeΩ 0.7>1/3att .
Λ 0
≈
Onecanreformulatetheabovesituationviareplacingmankindbylarge-scale-structuresonthetypical
scale forstructureformation. Accordingto inflationarycosmology,the primordialfluctuationsin matter
and metric are free of any scale. The only scale that enters in the physics of the large scale structure
is the horizon scale at matter-radiation equality. This scale appears because the radiation pressure acts
againstthegravitationalinstabilityintheearlyUniverseandstructuregrowthstartsonlyoncethepressure
supportfadesaway.Thustheshapeofthepowerspectrumofdensityfluctuationsischangingatthisscale.
Accordingtotheconcordancemodel,thehorizonscaleatmatter-radiationequalityisseenatthe100Mpc
scale today. Scales above that show the primordial slope of the spectrum, scales below that exhibit a
substantiallydecreasedslope. Itisnowinterestingtoobservethatthosestructuresgrowtotypicaldensity
contrastsoforder(δǫ/ǫ) (1+z )(δǫ/ǫ) 0.1anditturnsoutthatz = (0.1)isthe epoch
z=0.x eq eq
∼ ∼ O
whenthetypicalscaleofhierarchicalstructureformationstartstoevolvenon-linear(i.e.itdecouplesfrom
theoverallcosmologicalexpansion).Thecoincidenceisthatthisisalsotheepochwhenaccelerationseems
tosetin [14]. Notethatthese largestructuresareindeedobserved,e.g. theSloangreatwallandvarious
bigvoids,asbigasafew100Mpc[15].
Theaveragingproblem. Thisisanoldquestioningeneralrelativity[16]. SolvingtheEinsteinequa-
tionsfor an averagedspace-timeis certainlynotthe same as averagingthe Einstein equationsfora very
inhomogeneousandanisotropicspace-time,becausethesetwooperations,i.e. averagingandtimeevolu-
tion do notcommute. Thusthe issue is, how big is this effectand whatare its observationalsignatures.
Againthescaleofinterestmustbethe100Mpcscale,asonmuchsmallerscales,say1Mpcobjectslike
galaxieshavealreadypresumablyundergonetheprocessofviolentdynamicalrelaxationandformedsta-
tionaryboundsystemsthatcannotbetreatedlikedustparticlesanyway.Howeveritisclearthatforobjects
liketheSloangreatwall,whichjuststartedtocollapseyesterday,theremightbeimportanteffectsshowing
upintheaveragingprocess.
In summary, these three aspects are entangledwith the issues of inhomogeneitiesand anisotropiesin
thelocalUniverse. Thebeginningofthedominationofdarkenergycoincideswiththeonsetofstructure
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formation.Consequently,lightmaybeshedonthedarkenergycrisis(i.e.theacceleratedexpansionofthe
Universe)bystudyingtheaveragingproblemintheperturbedspace-time.
AnattemptforsettingupanformalismtohandlethisproblemhasbeenpioneeredbyBuchert[17,18].
4 Cosmologicalbackreaction
Theseproblemsprovideagoodmotivationtoconsiderthepossibilitythatdarkenergyisnotafundamental
componentofthe Universe, butouraverageddescriptionof the Universeisnotappropriatefora correct
interpretation of observations [14, 19, 20]. Indeed many cosmological observations are averages. Two
importantexamplesarethepowerspectrumP(k),whichisaFouriertransformandthusavolumeaverage
weightedbyafactorexp(ik x),andtheHubbleconstantH . Letusconsideranidealisedmeasurement
0
·
of H [21]. One picks N standard candles in a local physical volume V (e.g. SN Ia in Milky Way’s
0
neighbourhoodoutto 100Mpc), measurestheirluminositydistancesd andrecessionvelocitiesv =
i i
∼
cz , and takes the average H 1 N vi. In the limit of a very big sample, it turns into a volume
i 0 ≡ N i=1 di
average H = 1 vdV. For objectPs at z 1, the spatial average is appropriatefor the average over
0 V d ≪
thepastlightcone,RastheexpansionrateoftheUniverseisnotchangingsignificantlyattimescalesmuch
shorterthantheHubbletime.
We nowrecapitulatetheformulationofBuchert,whichwewillusebelowtoarguethattheaveraging
effectisindeedsizeableanddoesgiveimportantcontributionstocosmology:
Buchert’s setup is well adapted to the situation of a real observer. On large scales, a real observer is
comoving with matter, uses her own clock, and regards space to be time-orthogonal. These conditions
are the definition of the comoving synchronous coordinate system. Buchert used physically comoving
boundaries,whichisthemostnaturalapproachinthissetup. InthefollowingtheUniverseisassumedto
beirrotationalasaconsequencefromcosmologicalinflation.
Insynchronouscoordinates,themetricoftheinhomogeneousandanisotropicUniverseisds2 = dt2+
g (t,x)dxidxj, andthe spatialaverageofan observableO(t,x) in a physicallycomovingdomai−nD at
ij
timetisdefinedas[17]
1
O O(t,x) detg dx, (1)
D ij
h i ≡ V (t)Z
D D p
whereV (t) detg dxisthevolumeofthecomovingdomainD. Wemayintroduceaneffective
D ≡ D ij
scalefactoraD [R17]p
1/3
a V
D D . (2)
a ≡(cid:18)V (cid:19)
D0 D0
The effective Hubble expansion rate is thus defined as H a˙ /a = θ /3 (θ being the volume
D D D D
≡ h i
expansionrate).
Fromthedefinition(1),weobtaineffectiveFriedmannequations[17]fromaveragingEinstein’sequa-
tions,
2
a˙ 8πG a¨ 4πG
D = ǫ , D = (ǫ +3p ), (3)
eff eff eff
(cid:18)a (cid:19) 3 −a 3
D D
whereǫ andp aretheenergydensityandpressureofaneffectivefluid,
eff eff
1 1 1
ǫ ǫ ( Q + R ), p Q R . (4)
eff D D D eff D D
≡h i − 16πG h i h i ≡−16πG(cid:18)h i − 3h i (cid:19)
Q 2( θ2 θ 2 ) 2 σ2 isthekinematicalbackreaction(σ2beingtheshearscalar),and R
h iD ≡ 3 h iD−h iD − h iD h iD
theaveragedspatialcurvature.Theyarerelatedbytheintegrabilitycondition[17]
(a6 Q ). +a4 (a2 R ). =0. (5)
Dh iD D Dh iD
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6 NanLi,MarinaSeikelandDominikJ.Schwarz :Isdarkenergyaneffectofaveraging?
Wedefinetheequationofstatefortheeffectivefluidasw p /ǫ . Itishighlyremarkablethatany
eff eff eff
≡
spatiallyaverageddustmodelcanbedescribedbyaneffectiveFriedman-Lemaˆıtremodel.
Wecanmapthiseffectivefluidonamodelwithdustand“darkenergy”. Letnbethenumberdensity
ofdustparticles,andmbetheirmass. Foranycomovingdomain n = n (a /a )3. Fora dust
Universe,inwhichǫ(t,x) mn(t,x),weidentifyǫ ǫ =mh niD ,anhdifDro0m(D40)theDdarkenergyis
m D D
≡ ≡h i h i
hencedescribedbyǫ = ( Q + R )/(16πG). From(5)wefindthatconstant Q = R /3
de D D D D
− h i h i h i −h i
correspondstothecaseofacosmologicalconstantΛ = Q . Equations(3)and(4)arenotclosedand
D
h i
additionalinputisrequired.Belowweclosethembymeansofcosmologicalperturbationtheory.
Tostudythescaledependenceofphysicalobservables Q , R , ǫ ,H andw ,wecalculate
D D D D eff
h i h i h i
themtosecondorderinaperturbativeseriesoftheeffectivescalefactora . Westartfromaspatiallyflat
D
dustmodel.Itsscalefactora(t)isdifferentfromtheeffectivescalefactora in(2),andtheirrelationwas
D
providedin[22]. Insynchronousgauge,thelinearperturbedmetricis
ds2 = dt2+a2(t)[(1 2Ψ)δ +D χ]dxidxj,
ij ij
− −
where Ψ and χ are the scalar metric perturbations, D ∂ ∂ 1δ ∆ and ∆ denotes the Laplace
ij ≡ i j − 3 ij
operatorinthree-dimensionalEuclideanspace. ThesolutionsforΨandχaregivenintermsofthetime
independentpeculiargravitationalpotentialϕ(x): Ψ = 1∆ϕt4/3t2/3+ 5ϕandχ = 3ϕt4/3t2/3 (only
2 0 3 − 0
growing modes are considered) [22, 20]. ϕ is related to the hypersurface-invariantvariable ζ [23] by
ζ = 1∆ϕt4/3t2/3 5ϕ.
2 0 − 3
Following[22],withthehelpoftheintegrabilitycondition,weyieldthescaledependenceoftheaver-
agedphysicalobservablesuptosecondorder[24]( O Odx/ dxhereafter)
h iD1 ≡ D D
R R
a
Q = D0B(ϕ)t2, (6)
h iD a 0
D
20a2 a
R = D0 ∆ϕ 5 D0B(ϕ)t2, (7)
h iD 3 a2 h iD1− a 0
D D
1 a3
ǫ = D0, (8)
h iD 6πGt2 a3
0 D
2 a3/2 5 a 3 a2 25
H = D0 1 D t2 ∆ϕ + D t4 B(ϕ) ∆ϕ 2 , (9)
D 3t0a3D/2 (cid:20) − 4aD0 0h iD1 4a2D0 0(cid:18) − 24h iD1(cid:19)(cid:21)
5 a a2 25
w = D t2 ∆ϕ D t4 B(ϕ) ∆ϕ 2 , (10)
eff 6a 0h iD1− a2 0(cid:18) − 12h iD1(cid:19)
D0 D0
withB(ϕ) ∂i(∂ ϕ∆ϕ) ∂i(∂ ϕ∂j∂ ϕ) 2 ∆ϕ 2 . Weseefrom(6)–(10)thatthesequantities
≡h i − j i iD1− 3h iD1
arefunctionsofsurfacetermsonly,soalltheirinformationisencodedontheboundaries.Thedependence
oncosmictimeoftheseaveragedquantitiescanbefoundin[22],andtheirleadingtermsaregaugeinvariant
[22,25].
Wenowturntotheestimateoftheeffectsofcosmologicalbackreactionasafunctionoftheaveraging
scale r V1/3. We show below that cosmological averaging produces reliable and important modifi-
∼ D0
cations to local physical observables and determine the averaging scale at which 10% corrections show
up.
EffectiveaccelerationoftheaveragedUniverseoccursifǫ +3p <0,i.e. Q >4πG ǫ . Thus
eff eff D D
h i h i
weestimate
Q 3 a2 8 R4 8 1 R 4
h iD = D B(ϕ)t4 = H B(ϕ) H . (11)
(cid:12)4πG ǫ (cid:12) 2a2 0 27(1+z)2 ∼ 75(1+z)2 (cid:18) r (cid:19) Pζ
(cid:12) h iD(cid:12) D0
(cid:12) (cid:12)
R =3.00(cid:12) 103h−1(cid:12)MpcisthepresentHubbledistanceand =2.35 10−9thedimensionlesspower
H ζ
× P ×
spectrum [1] (ζ 5ϕ/3 on superhorizonscales). In (11), we use a = 1/(1+z) and t = 2R /3,
0 H
≈ −
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sinceB isalreadyasecondorderterm. EachderivativeinB isestimatedasafactorof1/rinfrontofthe
powerspectrum. Asϕisconstantintime,weidentifytoday’s with9 /25atsuperhorizonscales. If
ϕ ζ
P P
Q /4πG ǫ > 0.1,cosmologicalbackreactionmustbetakenintoaccountseriously. From(11),we
D D
|h i h i |
find
21h−1Mpc
r < .
Q
√1+z
Forobservationsatz 1,r <30Mpc(h=0.7).
Q
≪
Thecriterionforthescaleatwhichtheeffectoftheaveragedspatialcurvature R emergesisdeter-
D
h i
minedby
2
ǫ R 2 1 R
eff D H
1 h i .
(cid:12) ǫ − (cid:12)≈(cid:12)16πG ǫ (cid:12)∼ 31+z (cid:18) r (cid:19) Pζ
(cid:12)h iD (cid:12) (cid:12) h iD(cid:12) p
(cid:12) (cid:12) (cid:12) (cid:12)
Weexpec(cid:12)ta10%eff(cid:12)ectf(cid:12)rom R w(cid:12)ithin
D
h i
54h−1Mpc
r < .
R
√1+z
At small redshifts, r < 77 Mpc. Moreover,a 1%effectis expectedup to scales of 240Mpc. Note
R
∼
thatthecurvatureoftheUniversehasbeenmeasuredatthefewpercentaccuracyintheCMB[1]. Itwas
shownin[26]thatevensmallcurvaturesmightinfluencetheanalysisofhigh-zSNesignificantly.
5 Signature ofcosmologicalbackreaction: the effective Hubble rate
ItisnowinterestingtodiscussthefluctuationofthelocalmeasurementoftheHubbleexpansionrate,
2
H H 1 1 R
D− 0 H . (12)
(cid:12) H (cid:12) ∼ 31+z (cid:18) r (cid:19) Pζ
(cid:12) 0 (cid:12) p
(cid:12) (cid:12)
Theleadin(cid:12)gcontributi(cid:12)ontothiseffectinvolvestheaveragedcurvaturetermsandthereforecannotbethor-
oughlydescribedinaNewtoniansetup[27]. Aneffectlargerthan10%showsupfor
38h−1Mpc
r < ,
H
√1+z
whichreadsr <54Mpcatsmallredshifts.
H
Let us compare (12) with observations from the HST Key Project [28]. We use 68 individual mea-
surements of H in the CMB rest frame from SN Ia, the Tully-Fisher and fundamental plane relations
0
(Tables 6, 7 and 9 in [28]). The respective objects lie at distances between 14 Mpc and 467 Mpc. As
(12) can be trusted only above 30 Mpc, we drop the 4 nearest objects. Be r , H and σ the distance,
i i i
Hubblerate andits 1σ errorforthe i′thdatum, with increasingdistance. We calculate theaverageddis-
tanceforthenearestkobjectsbyr¯ = k g r / k g ,withweightsg =1/σ2. Ananalogueholds
k i=1 i i i=1 i i i
fortheaveragedHubblerateH¯k, i.e. HPD fordifferPentsubsets. Theempiricalvarianceofeachsubsetis
σ¯2 = k g (H H¯ )2/[(k 1) k g ]. Westressthat(12)isinsensitivetoglobalcalibrationissues.
k i=1 i i− k − i=1 i
ThePcomparison of our result (1P2) with the HST Key Project data is shown in Fig. 2. The “global”
H isthecentralHST value72km/s/Mpc[28]. We findthatthetheoreticalcurve( 1/r2) matchesthe
0
∝
experimentaldatafairlywell,withoutanyfitparameterinthepanel.Since(12)isjustanestimate,wevary
theresultbyfactorsof0.5and2,whichgiveslowerandupperestimatestothetheoreticalcurve.InFig. 2,
theexperimentaldataandtheirerrorbarsareinagreementwiththeseestimates.
Beforewe can claim thatwe observethe expected1/r2 behaviourof (12) and thusevidencefor cos-
mologicalbackreaction,wehavetomakesurethatthestatisticalnoisecannotaccountforit. Theexpected
noiseisgivenbymeansofthevarianceof(12)inthesituationofaperfectlyhomogeneousandisotropic
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8 NanLi,MarinaSeikelandDominikJ.Schwarz :Isdarkenergyaneffectofaveraging?
0.2
0.15
H0 0.1
H)/0
H-D 0.05
(
0
-0.05
40 60 80 100 120 140 160 180
r (Mpc)
Fig. 2 The scale dependence of the normalised difference between the effective Hubble rate HD and its “global”
valueH0 =72km/s/Mpc(DatafromtheHSTKeyProject[28]).Weexpectthatalldatapointsshouldliearoundthe
twothicklines(∼±1/r2)givenin(12).Thinlinesindicatethetheoreticalerrorofourestimate.Dashedlinesarethe
expectedstatisticalnoiseforaperfectlyhomogeneousandisotropicmodel.
modelwithmeasurementuncertaintiesσ . Inthecaseofaperfectlyhomogenouscoverageoftheaveraged
i
domainwithstandardcandles,weexpecta1/r3/2 behaviour. InFig. 2,weshowthestatisticalnoisefor
theactualdataset,whichturnsouttobewellbelowourestimate(12)andthedatapoints. At 50(80)
∼
Mpc,thevalueofH differsfromitsglobalonebyabout10%(5%),whereastheexpectedvariancefora
D
homogeneousandisotropicUniverseisonly3%(1%).
6 Toward anew model andopen issues
What can we conclude so far? We showed that the large scale structure affects the observed Hubble
expansionrate and thatthe averagedspatialcurvatureleadsto sizeable correctionsup to scales of about
200Mpc.Consequently,wemustexpectthatalsoothercosmologicalprobesatz <1,likegalaxyclusters
orgalaxyredshiftsurveys,couldbeaffectedbycosmicaveraging.
Is that enough to explain dark energy? Currently, this is not the case. In our frameworkan effective
acceleration is possible below scales of 20 Mpc. But SN Ia observations show strong evidence for the
acceleration of the Universe, based on objects much more distant than that. It seems to take more to
explain dark energy. A quite attractive idea is a Universe in which large voids dominate the Universe’s
volume,see[29,30,31,32]. Theseworksrelyonwellmotivatedtoymodels. Itremainstobeshownthat
thesevoiddominatedcosmologiesfollowfromtheinflationaryparadigm.
On the other hand, our analysis is based on an almost scale-invariant power spectrum predicted by
inflationarycosmology,butislimitedtoz 1,astheaveragemustbetakenonthepastlightconeandnot
≪
onaspatialslice(see,e.g.[33]). Aformalism(likeBuchert’soneforspatialaveraging)forthelight-cone
averagingofanarbitraryinhomogeneousandanisotropicspace-timeisneededbeforewecanknowifdark
energyismerelyanillusionduetoaveraging.
Acknowledgements
WethankAdamRiessforsuggestingthecalibrationindependenttestpresentedinfigure1andtable1. DJS
wouldliketothanktheorganisersofthe“Balkanworkshop2007”forhavinginvitedhimtosuchalively
andinspiringmeeting. DJSwouldalsoliketothankSEENET-MPandtheDFGforsupportofhistravel.
WearegratefultotheDFGtosupportourresearchundergrantGKR881.
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fdpheaderwillbeprovidedbythepublisher 9
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