Cosmological constant and general isocurvature initial conditions

Article

Reference

Cosmological constant and general isocurvature initial conditions

TROTTA, Roberto, RIAZUELO, Alain, DURRER, Ruth

Abstract

We investigate in detail the question whether a non-vanishing cosmological constant is required by present-day cosmic microwave background and large scale structure data when general isocurvature initial conditions are allowed for. We also discuss differences between the usual Bayesian and the frequentist approaches in data analysis. We show that the COBE-normalized matter power spectrum is dominated by the adiabatic mode and therefore breaks the degeneracy between initial conditions which is present in the cosmic microwave background anisotropies. We find that in a flat universe the Bayesian analysis requires

\Omega \Lambda \neq 0 to more than 3 \sigma, while in the frequentist approach \Omega

\Lambda = 0 is still within 3 \sigma for a value of h < 0.48. Both conclusions hold regardless of initial conditions.

TROTTA, Roberto, RIAZUELO, Alain, DURRER, Ruth. Cosmological constant and general isocurvature initial conditions. Physical Review. D , 2003, vol. 67, no. 063520

DOI : 10.1103/PhysRevD.67.063520 arxiv : astro-ph/0211600

Available at:

http://archive-ouverte.unige.ch/unige:958

Disclaimer: layout of this document may differ from the published version.

1 / 1

arXiv:astro-ph/0211600 v2 19 Dec 2002

R. Trotta, 1 ,

A. Riazuelo, 2,y

and R. Durrer 1,z

1

Departement de Physique Theorique, Universite de Geneve,

24 quaiErnest Ansermet, CH{1211 Geneve 4, Switzerland

2

Servie de Physique Theorique, CEA/DSM/SPhT,

Unite dereherheassoieeauCNRS, CEA/SalayF{91191 Gif-sur-Yvetteedex, Frane

(Dated: 27Novemb er2002)

Weinvestigateindetailthequestionwhetheranon-vanishi ngosmologialonstantisrequiredby

present-dayosmimirowavebakgroundandlargesalestruturedatawhengeneraliso urvature

initial onditions areallowed for. Wealso disussdierenes b etweenthe usual Bayesianand the

frequentistapproahesindataanalysis. WeshowthattheCOBE-normalizedmatterp owersp etrum

isdominatedbytheadiabatimo deandthereforebreaksthedegenerayb etweeninitialonditions

whihispresent intheosmimirowavebakground anisotropies. Wendthatin aatuniverse

theBayesiananalysis requires6=0tomorethan3 ,whileinthefrequentistapproah=0is

stillwithin3foravalueofh0:48. Bothonlusions holdregardlessofinitial onditions.

PACSnumb ers:PACS:98.70V,98.80Hw,98.80Cq

I. INTRODUCTION

Eversine theb eginningof mo dernosmology,one of

themostenigmatiingredientshasb eentheosmologial

onstant. Einstein intro duedit tondstatiosmolog-

ialsolutions(whih are, however,unstable) [1℄. Later,

whentheexpansionoftheUniversehadb eenestablished,

healledithis\greatest blunder".

Inrelativistiquantumeldtheory,forsymmetryrea-

sonsthevauumenergymomentumtensorisoftheform

g

for some onstant energy density . The quantity

=8 Ganb einterpretedasaosmologialonstant.

Typialvalues ofexp etedfrompartilephysisome,

e.g., from the sup er-symmetry breaking sale whih is

exp eted to b e of the order of

>

1TeV

4

leading to

>

1:710 26

GeV 2

, and orresp onding to

>

10

58

.

Here we have intro dued the density parameter

=

rit

==(8 G

rit

),where

rit

=8:1 10 47

h 2

GeV 4

is the ritial density and the fudge fator h is dened

by H

0

= 100hkms 1

Mp 1

, it lies in the interval

0:5

<

h

<

0:8. H

0

istheHubbleparameterto day.

Suharesult islearlyinontraditionwithkinemat-

ialobservations oftheexpansionoftheuniverse,whih

tellus that thevalueof

tot

, thedensity parameterfor

thetotalmatter-energyontentoftheuniverse,isofthe

orderofunity,O (

tot

)1. Foralongtime,thisappar-

entontraditionhasb eenaeptedbymostosmologists

andpartilephysiists,withthehindthoughtthatthere

mustb esomedeep, notyet understo o dreasonthat va-

uumenergy |whihisnotfeltbygauge-interations|

do es notaet the gravitational eld either, and hene

we measureeetively=0.

This slightly unsatisfatory situation b eame really

Eletroniaddress:rob erto.trottaphysis.unige.h

y

Eletroniaddress:riazuelospht.salay.ea.fr

z

disturbing aouple of years ago, as two groups, whih

hadmeasuredluminositydistanestotyp eIasup ernovae,

indep endentlyannouned thattheexpansion oftheuni-

verse is aelerated in the way exp eted in a universe

dominated by a osmologial onstant [2, 3℄. The ob-

tainedvalues areoftheorder O (

m

)O (

)1and

annotb eexplainedbyanysensiblehighenergyphysis

mo del. Trakingsalar elds or quintessene [4, 5℄ and

other similar ideas [6℄ have b een intro dued in order

to mitigatethe smallness problem| i.e., thefat that

10 46

GeV 4

. However, none of those is ompletely

suessful andreally onviningatthemoment.

After the sup ernovae Ia results, osmologists have

found many other data-sets whih also require a non-

vanishing osmologial onstant. The most prominent

fat is that anisotropies in the osmimirowave bak-

ground(CMB)indiateaatuniverse,

tot

=

m +

=

1,while measurementsof lustering ofmatter, e.g.,the

galaxyp owersp etrum,require h

m

'0:2. Butalso

CMBdataalone,withsomereasonablelowerlimitonthe

Hubbleparameter, like h > 0:6,have b een rep orted to

require

>0athighsigniane(see,e.g.,Refs.[7,8℄

andothers).

This osmologial onstant problem is probably the

greatestenigmainpresentosmology.Thesup ernovare-

sultsaretherefore under detailed srutiny. Forexample

Ref.[9℄ isnot onvined that the data an only b e un-

dersto o dbyanon-vanishingosmologialonstant. Cos-

mologialobservations areusually very sensitive to sys-

tematierrors whih are oftenvery diÆultto disover.

Therefore, in osmology anobservational result is usu-

allyaepted bythesientiommunityonlyifseveral

indep endent data-setsleadto thesameonlusion. But

this seems exatly to b e the ase for the osmologial

onstant.

This situation prompted us to investigate in detail

whether present struture formation data do es require

aosmologialonstant. Onemayaskwhetherenlarging

gate theosmologialonstant problem. There aresev-

eral ways to enlarge the mo del spae, e.g.one may al-

lowforfeaturesintheprimordialp owersp etrum,likea

kink[10℄. Inthepresent pap erwestudy the osmologi-

alonstantprobleminrelationtotheinitialonditions

fortheosmologialp erturbations. Inarststepwe re-

disuss the usual results obtainedassuming purely adi-

abatimo delsand we investigate to whih extent CMB

alone or CMB and large sale struture (LSS) require

6= 0 in a at universe. We shall rst present the

usual Bayesian analysis,but we also disuss the results

whih are obtained in a frequentist approah. We nd

that even if

= 0is exluded at high signiane in

aBayesian approah this isnolonger thease fromthe

frequentistp ointofview. Inotherwords theprobability

thatamo delwithvanishing

leadstothepresent-day

observed CMB and LSS data is not exeedingly small.

We then study how theresults aremo diedifwe allow

forgeneraliso urvatureontributionstotheinitialondi-

tions[11,12, 13℄. Inthisrststudyofthematterp ower

sp etrum from general iso urvature mo des we disover

thataCOBE-normalizedmatterp owersp etrum repro-

dues the observed amplitude only if it is highlydomi-

nated bytheadiabati omp onent. Hene theiso urva-

turemo desannotontributesigniantlytothematter

p ower sp etrum anddo notleadto adegeneray inthe

initial onditions for the matter p ower sp etrum when

ombinedwithCMBdata. Thisisthemainresultofour

pap er.

Thepap erisorganized asfollows: Inthenext setion

wedisussthesetupforouranalysis,thespaeofosmi

parametersandofinitialonditions,andwereallthedif-

ferene b etween Bayesian and frequentist approah. In

Se.3we presenttheresultsforadiabatiand formixed

(adiabatiandiso urvature)p erturbations. Se.4isded-

iatedtotheonlusions.

I I. ANALYSISSETUP

A. Cosmologialparameters

As it has b een disussed inthe literature, the reent

data-sets, BOOMERanG[7℄,MAXIMA [14℄,DASI[15℄,

VSA [16℄, CBI[17℄ and Arheops [18℄ are invery go o d

agreement up to the third p eak in the angular p ower

sp etrum of CMB anisotropies, ` 1000. Inour anal-

ysis we therefore use the COBE data [19℄ (7 p ointsex-

luding the quadrup ole) for the ` region 3 ` 20

andtheBOOMERanGdatatooverthehigher`partof

the sp etrum (19 p oints inthe range 100` 1000).

SineArheops hasthesmallesterror barsintheregion

of the rst aousti p eak, we also inlude this data-set

(16p ointsinthe range15`350). Inludingany of

theother mentioneddata do es notinuene ourresults

signiantly. TheBOOMERanG andArheops absolute

alibration errors (10% and 7% at 1 , resp etively) as

areinludedasadditionalGaussianparameters,andare

maximizedover. We makeuse of theArheops window

funtions found in Ref.[38℄, while for BOOMERanG a

top-hatwindowisassumed.

For the matter p ower sp etrum, we use the galaxy-

galaxyp ower sp etrum fromthe 2dFdata whih is ob-

tainedfromtheredshiftofab out10 5

galaxies[20℄.Wein-

ludeonlythe22deorrelatedp ointsinthelinearregime,

i.e.,intherange0:017k0:314 [hMp 1

℄,andthe

windowfuntionsofRef.[20℄whih anb efoundonthe

homepageofM.Tegmark[39℄.

Ourgridofmo delsisrestritedtoatuniversesandwe

assumepurelysalarp erturbations. Sinethegoalofthis

pap eris more to make aoneptual p oint than to on-

siderthemostgeneri mo del, we x thebaryon density

totheBBN preferred value

b h

2

!

b

=0:020[21℄. We

investigatethefollowing3dimensionalgridinthespae

of osmologial parameters: 0:80 n

S

(0:05) 1:20,

0:35h(0:025)1:00,0:00

(0:05)0:95,where

n

S

isthesalarsp etralindexandthenumb ersinparen-

thesisgivethestepsizeweuse. Thetotalmatterontent

m

+

b is

m

= 1

, and

indiates the

olddarkmatterontribution. Forallmo delstheoptial

depthof reionizationis =0and we have 3familiesof

massless neutrinos. For eah p ointinthe grid we om-

pute the ten CMB and matter p ower sp etra, one for

eah indep endent set of initialonditions (see Se. I IB

b elow).

B. Allowingforiso urvature mo des

We enlarge the spae of mo delsby inludingall p os-

sible iso urvature mo des. As it has b een argued in

Ref.[11℄,generi initialonditionsforauid onsisting

of photons, neutrinos, baryons and dark matter allow

for ve relevant mo des, i.e.,mo des whih remain regu-

larwhen goingbakwards intime. These are theusual

adiabatimo de(AD),theolddarkmatteriso urvature

mo de(CI),thebaryoniso urvaturemo de(BI),theneu-

trinoiso urvature density (NID)and neutrinoiso urva-

turevelo ity(NIV)mo des.TheCMBandmatterp ower

sp etrafromtheolddarkmatterandthebaryoniso ur-

vature mo des are idential (see the argument given in

Ref.[22℄) and therefore from now on we willjust on-

sideroneofthose,namelytheCImo de.Weassumethat

all these four mo des are present with arbitrary initial

amplitude and arbitrary orrelation or anti-orrelation.

The only requirement is that their sup erp osition must

b ea p ositive quantity,sine the C

`

's and matter p ower

sp etrum are quadrati and thus p ositive observables.

For simpliitywe restrit ourselves tothease whereall

mo des have the same sp etral index. Initial onditions

arethen desrib ed by thesp etral indexn

S

and ap osi-

tivesemi-denite 44matrix,whihamountstoeleven

parametersinstead oftwointhease ofpureADinitial

onditions. More details an b e found inRef.[13℄. For

metho ddesrib ed inRef.[13℄,with thefollowingmo di-

ation. We ndit onvenient toexpress thematrix A

desribingtheinitialonditionsas

A=UDU T

; (1)

where A2P

n

, U 2SO

n

, D =diag(d

1

;d

2

;:::;d

n )and

d

i

0, i2 f1;2;:::;ng. Here P

n

denotes the spae of

nnreal,p ositivesemi-denite,symmetrimatriesand

SO

n

isthespaeofnnreal,orthogonalmatrieswith

det =1. As explained ab ove, here we take n = 4. We

an write U as an exp onentiated linear ombination of

generatorsH

i ofSO

n :

U=exp 0

(n

2

n)=2

X

i=1

i H

i 1

A

; (2)

with

H

1

= 0

B

B

B

0 1 0 :::

1 0 0 :::

0 0 0 :::

.

.

. .

.

. .

.

. .

.

. 1

C

C

C

A

; (3)

and so on, with =2 <

i

< =2, i2 f1;2;:::;(n 2

n)=2g. In analogyto the Euler angles in three dimen-

sions,we anre-parameterize U intheform

U= (n

2

n)=2

Y

i=1

exp(

i H

i

); (4)

with some other o eÆients =2 <

i

< =2, i 2

f1;2;:::;(n 2

n)=2g,whosefuntionalrelationwiththe

i

's do es not matter. The diagonal matrix D an b e

writtenas

D=diag(tan(

1

);:::;tan(

n

)); (5)

with 0

i

< =2, for i 2 f1;2;:::;ng. In this

way, the spae of initial onditions for n mo des is ef-

iently parameterized by the (n 2

+n)=2 angles

i

;

j .

In our ase, n = 4 and the initial onditions are de-

srib edbythetendimensionalhyp erub einthevariables

(

1

;:::;

4

;

1

;:::;

6

). This isof partiularimp ortane

forthenumerialsearhintheparameterspae. Onean

thengobaktotheexpliitformofAusingEqs.(4),(5)

and(1).

For a given initial ondition determined by a p os-

itive semi-denite matrix A and a sp etral index n

S

we quantify the iso urvature ontribution to the CMB

anisotropiesbytheparameter denedas

X

X=CI;NIV ;NID

(`(`+1))C (X;X)

`

`

X

Y=AD ;CI;NIV ;NID

`(`+1)C (Y ;Y)

`

`

; (6)

wheretheaverage<;>istakeninthe`rangeofinterest,

inour ase 3 ` 1000,and where C (X;X)

`

stands for

theauto-orrelator oftheCMBanisotropies withinitial

onditionsX.

C. Bayesianor frequentist?

For the sake of larity, we briey reall two p ossible

p ointsofviewwhihoneantakewhendoingdataanaly-

sis,theBayesianandthefrequentistapproah,andhigh-

lighttheirdierene. Moredetails an b efound,e.g.,in

Refs.[23,24,25℄. Anotherp ossibilityisbasedonMarkov

ChainMonte Carlotehniques,whihwe donotdisuss

here;seeinstead [26℄andreferenes therein.

Whenttingexp erimentaldata,weminimizea 2

over

the parameters whih we are not interested in. This

pro edureisequivalenttomarginalizationiftherandom

variablesareGaussian distributed. TheMaximumLike-

liho o d(ML) priniple states that the b est estimate for

theunknownparameters isthe onewhihmaximizes

thelikeliho o dfuntion:

L=L

0

exp(

2

=2): (7)

Wethen draw1 ,2 and3 likelihoodontoursaround

theMLp oint,i.e.,theoneforwhihthe 2

isminimalin

ourgridofmo dels.Thelikeliho o dontoursaredenedto

b e 2

2

ML

=2:30,6:18,11:83awayfromtheML

valueforthejointlikeliho o dintwoparameters,=1,

4,9forthelikeliho o dinonlyoneparameter. Thisisthe

Bayesianapproah: inasomewhatfuzzyway,likeliho o d

intervalsmeasureourdegree ofb eliefthatthepartiular

setofobservationsusedintheanalysisisgeneratedbya

parametersetb elongingtothesp eiedinterval. Inthis

ase, one impliitlyaepts the ML p ointin parameter

spae as the true value, while p oints whih are further

awayfromitareless andless \likely"to havegenerated

the measurements. This is the ontent of Bayes' The-

orem,whih allowsus to interpret thejointonditional

probabilityL(xj)ofmeasuringxfor axed setofpa-

rameters as the inverse probability P(jx) for the

valueofgiventhemeasurementsx.

On the other hand, in the frequentist approah one

asksadierent question: Whatis theprobabilityofob-

taining the exp erimental data at hand, if the Universe

hassomegivenosmologial parameters,e.g., avanish-

ingosmologialonstant? Clearly,ifwewanttoanswer

thequestion whether aertainset ofexp erimentaldata

foresanon-vanishing osmologialonstant, this isa-

tuallytheorretquestiontoask. Totheextenttowhih

theC

`

'sanb e approximatedasGaussianvariables,the

quantity 2

isdistributedaordingtoahi-squareprob-

ability distribution with F = N M degrees of free-

dom(dof),whihwedenotebyP (F)

( 2

),whereN isthe

numb erofindependent(unorrelated)exp erimentaldata

p ointsandM is thenumb erof ttedparameters. From

thedistributionP (F)

oneanreadilyestimateondene

intervals. For agivenparameter set

~

with hi-square

~ 2

the probability that the observed hi-square will b e

largerthantheatualvaluebyhaneutuations is

Z

1

2 P

(F)

(x)dx1 : (8)

In other words, if the measurements ould b e rep eated

manytimes indierentrealizations of ouruniverse, the

estimated ondene interval would asymptotially in-

ludethetruevalueoftheparameters100%ofthetime.

It is ustomary inosmologialparameter estimation

topresentlikeliho o dplotsdrawnusingtheBayesianap-

proah. It ismisleadingthat suhBayesian ontoursare

usuallyalled\ondeneontours",whihprop erlydes-

ignate frequentist ontours. Likeliho o d (Bayesian) on-

tours areusually muh tighter thantheondene on-

tours drawn from the frequentist p oint of view. This

is a onsequene of the ML p ointhaving often a 2

=F

muh smaller than 1, b eause the data-sets are highly

onsistent with eah other andalso b eauseusually not

allp ointsareompletelyindep endent. Ifweonsiderthe

usualsituationinwhihlikeliho o dontoursaredrawnin

atwodimensionalplanewithallotherparametersmaxi-

mized,thefrequentistapproahismoreonservativethan

theBayesianone. Thisisb eauseingeneral,forreason-

ablygo o dMLvalues ~ 2

ML

<

O (F)andF >2,

Z

1

~ 2

F P

(F)

(x)dx= Z

1

~ 2

2 P

(2)

(x)dx (9)

onlyfor ~ 2

F

>~ 2

2

. When lo okingatlikeliho o dontours

one should thus keep in mind that a p oint more than

say3 awayfromtheML p ointisnotneessarily ruled

out by data, as we shall show b elow. In order to es-

tablishthis, one hastolo okatondene ontours, i.e.,

askthefrequentist'squestion. Inthefollowing,theterm

\likeliho o dontours"willrefer toontoursdrawninthe

Bayesianapproah,whiletheterm\ondeneontours"

willb ereservedforontoursomingformthefrequentist

p ointofview.

I I I. RESULTS

A. Adiabatip erturbations

Werst t CMB data only(N =42) bymaximizing

M =7parameters,i.e.,theBOOMERanGandArheops

alibrationerrors,BOOMERanGb eamsizeerror,n

S ,h,

andtheoverallamplitudeoftheadiabatisp etrum,

andwend(Bayesianlikeliho o dintervalson

alone):

=0:80 +0:10

0:35

at2 and

+0:12

0:80

at3 : (10)

Theasymmetryintheintervalsarisesb eausethevalue

of

for our ML mo del is relatively large. One ould

ahieve a b etter preision in determining the ML value

of

byusinganergridandvarying!

b

aswell,whih

hasextensivelyb eendoneintheliteratureandisnotthe

sop e ofthiswork. Moreover,thep osition oftheaous-

tip eaksinCMBanisotropies ismainlysensitiveto the

ageoftheuniverseatreombination,whihdep endsonly

on

m h

2

, and to the angular diameter distane, whih

dep ends on

m ,

and the urvature of the universe.

isweakly dep endenton therelative amountsof

m and

as so on as

is not to o large (see e.g. Ref. [27℄).

Hene, one an ahieve asuÆiently low valueof

m h

2

eitherviaalargeosmologialonstantorviaaverylow

Hubbleparameter,h

<

0:45.

We now inlude the matter p ower sp etrum P

m , as-

suming P

m

= b 2

P

g

, where P

g

is the observed galaxy

p ower sp etrum and b some unknown bias fator (as-

sumed to b e sale indep endent), over whih we maxi-

mize. Inlusion of this data in the analysis breaks the

, h degeneray, sine P

m

is mainly sensitive to the

shap eparameter

m

h. We therefore obtainsigni-

antlytighteroveralllikeliho o dintervalsfor

:

=0:70 +0:13

0:17

at2 and

+0:15

0:27

at3 : (11)

We plot joint likeliho o d ontours for

, h with AD

initial onditions in Fig. 1. From the Bayesian analy-

0.00 0.20 0.40 0.60 0.80

Ω _Λ 0.40

0.50 0.60 0.70 0.80 0.90 1.00

Hubble parameter h

AD only

FIG.1: Jointlikeliho odontours(Bayesian),withCMBonly

(solid lines, showing 1 , 2 , 3 ontours) and CMB+LSS

(lled)forpurely adiabati initial onditions.

sis,one onludesthatCMBandLSStogetherrequirea

non-zeroosmologialonstantatveryhighsigniane,

morethan7 forthe p ointsinourgrid! Note that the

MLp ointhasaredued hi-square ^ 2

F=56

=0:59,signif-

iantlylessthan1.

The frequentist analysis, however, exludes a muh

smallerregionofparameterspae(Fig.2). Thefrequen-

tistontours mustb e drawn for theeetive numb erof

dof,i.e.,usingthenumb erofeetivelyindep endentdata

p oints. Weanthereforeroughlytakeintoaounta10%

orrelation, whih is the maximumorrelation b etween

datap ointsgivenin[7, 18℄,byreplaingF bytheee-

tive numb er of dof, F

e

= 0:9N M, and rounding to

thenext larger integer (to b e onservative). One ould

arguethat theBOOMERanGand Arheopsdatap oints

arenotompletelyindep endent,sineBOOMERanGob-

Arheops. This p ossible orrelationis diÆult to quan-

tify,butshould notb eto o imp ortantsine theskyp or-

tion observed by Arheops is afator of 10 larger than

BOOMERanG's andtherefore we ignoreit here. Fig.2

isdrawnwithF

e

=31forCMBaloneandF

e

=50for

CMB+LSS,butwe haveheked thatourresultsdonot

hangemuhifwe usea5%orrelation. Itisinteresting

tonotethatthereareregionsinFig.2whihareexluded

withaertainondenebyCMBdataalonebutareno

longerexludedatthesameondenewhenweinlude

LSSdata. Inotherwords,itwouldseemthattakinginto

aountmoredataand therefore moreknowledgeab out

theuniverse,do es notsystematiallyexludemoremo d-

els,i.e.,theCMB+LSSontoursarenotalwaysontained

intheCMBaloneontours. Thisapparentontradition

vanisheswhenone realizesthattheondenelimitson,

e.g.,

alone in the frequentist approah are just the

projetionoftheondeneontoursofFig.2onthe

axis. OneanreadilyverifyinFig.2thattheondene

limitsfortheombineddata-setarealwayssmallerthan

the ones for CMB data alone. There are p oints with

=0 and h'0:40whih are still ompatiblewithin

2 with b othLSSand CMB data, attheprie of push-

ingsomewhattheotherparameters. Intheb esttwith

=0shown inFig.3,one hastolivewitharedsp e-

tralindexn

S

=0:80. Furthermore,thealibrationofthe

BOOMERanG and Arheops data p oints is redued in

this t by34%and 26%, resp etively, i.e.,morethan 3

timesthequoted1systematierror.

0.00 0.20 0.40 0.60 0.80

Ω _Λ 0.40

0.50 0.60 0.70 0.80 0.90 1.00

Hubble parameter h

AD only

FIG. 2: Condene ontours (frequentist) with CMB only

(solid lines, 1 , 2 , 3 ontours and F

e

= 31) and

CMB+LSS(lled,F

e

=50)forpurely adiabatiinitial on-

ditions.

Inb othases, itislearthatoneanexploitthe

,h

degeneray to t CMB data alonewith amo delhaving

= 0. For a at universe like the one we are on-

sidering, one hasthen to go to amuh smallervalue of

FIG.3: Best t with= 0 and purely AD initial ondi-

tions, ompatible with CMB and LSS data within 2 on-

dene level. Inthe lower panel, only the 2dF data p oints

leftofthevertial,dottedline|i.e.,in thelinear region|

haveb eeninludedintheanalysis. Notethelowrstaousti

p eakdue to thejoint eetof thered sp etralindex andof

the absene of early ISW eet. Inthis t, the alibration

ofBOOMERanG (red errorbars) and Arheops (blue error-

bars)hasb eenreduedby34%and26%,resp etively. Thisis

morethan 3times thequoted1 alibration errorsforb oth

exp eriments. To appreiate the dierene, we plot the non

realibrated value of the BOOMERanG and Arheops data

p ointsaslight blueandmagentarosses,resp etively. Inthe

upp er panel, green errorbars are the COBE measurements.

Even though the t is \by eye" very go o d, it seems highly

unlikelythatthealibration errorissolarge.

measurements,mostnotablytheHST KeyProjet [28℄,

whih gives h=0:720:08. The LSS data are mainly

sensitive to the shap e parameter 0:2. Hene LSS

with

m

= 1:0 would require an even lower value of h

whih isnotompatiblewith CMB.Therefore inlusion

of LSS data tends to exlude any atmo del without a

osmologialonstant. Summingup,forpurelyadiabati

supp ort to

6=0;inthemoreonservative frequentist

p oint of view, while

6= 0 annot b e exluded with

veryhighondene,presentLSSandCMBdatastartto

b e inompatiblewithaatuniverse withvanishing os-

mologialonstant. Theseonlusions areinqualitative

agreementwithpreviousworks[29,30,31,32,33,34,35℄.

Inthenextsetionweinvestigatethestabilityofthose

well known results with resp et to inlusion of non-

adiabatiinitialonditions.

B. Iso urvature mo des

We now enlarge thespae of mo delsby inluding all

p ossible iso urvature mo des with arbitrary orrelations

amongthemselves andthe adiabatimo deas desrib ed

intheprevioussetion. WerstonsiderCMBdataonly

andmaximizeoverinitialonditions. Thenumb erofpa-

rameters inreases by nine and the numb er of dof de-

reases orresp ondingly with resp et to the purely AD

ase onsideredab ove. Likeliho o d(Bayesian,seeFig.4)

andondene(frequentist,seeFig.7)ontourswidenup

somewhatalongthedegenerayline. Theenlargementis

lessdramatithanforotherparameterhoies,see,e.g.,

Ref. [13℄ where the degeneray in !

b

;h was analyzed.

Thisispartiallyduetoourpriorofatnesswhihredues

thespaeofmo delstotheoneswhiharealmostdegener-

ateintheangulardiameterdistane. Mostofourmo dels

havetherstaoustip eakoftheadiabatimo dealready

intheregionpreferredbyexp eriments. Heneinmostof

thetsiso urvaturemo desplayamo destrole,esp eially

intheparameterregionswithlarge

,h(fFig.9and

thedisussionb elow). Nevertheless, b eauseofthe

,h

degeneray,evenamo destwideningoftheontoursalong

thedegeneraylineresults inanimp ortantworseningof

thelikeliho o dlimits.TheMLp ointdo esnotdepartvery

muhfromthepurelyadiabatiase,butnowwe annot

onstrain

atmorethan1 (Bayesian,CMBonly):

=0:85 +0:05

0:35

at1 , (12)

andnolimitsfor0:0

0:95athigherondene.

InFig.5weplotthedarkmatterp owersp etraofthe

dierentauto-(upp erpanel)andross-orrelators(lower

panel)foraonordanemo del. Thenormofeahpure

mo de(AD, CI,NID, NIV) ishosen suh that the or-

resp onding CMB p ower sp etrum is COBE-normalized.

Theross-orrelatorsarenormalizedaordingtototally

orrelated sp etra,i.e.

A

(X;Y)

= p

A

X A

Y

=2; (13)

where A

(X;Y)

denotes the norm of the ross-orrelator

b etween the mo des X,Y and A

X

the normof thepure

mo deX. TheCMBp ower sp etrumfor thisset of os-

mologialparametersanb efoundinRef.[12℄. Aruial

resultisthattheCOBE-normalizedamplitudeoftheAD

0.40 0.50 0.60 0.70 0.80 0.90 1.00

Hubble parameter h

0.00 0.20 0.40 0.60 0.80

Ω _Λ General IC

FIG. 4: Joint likelihoo d ontours (Bayesian) with general

iso urvature initial onditions, with CMB only (solid lines,

1 , 2 , 3 ) and CMB+LSS (lled). The disonneted 1

regionisanartiial featureduetothegridresolution.

largerthantheiso urvatureontribution.Themainrea-

sonforthisis theamplitudeoftheSahs Wolfeplateau

whihisab out 1

3

foradiabatip erturbationsand2 for

iso urvature p erturbations. Here is the gravitational

p otential. This dierene ofafator of ab out36inthe

p ower sp etrum on large sales is learly visible inthe

omparisonof P

AD and P

CI

(the dierene inreases at

smallersales). The ase of theneutrino mo desis even

worsesinetheystartupwithvanishingdarkmatterp er-

turbations. That the CDM iso urvature matter p ower

sp etrumismuhlowerthantheadiabationehasb een

knownforsometime(seee.g.Ref.[36℄). However,itwas

not reognized b efore that the same holds true for the

neutrinoiso urvature matterp ower sp etra aswell,and

{moreimp ortantly{ that this leads to awayto break

thestrongdegeneray amonginitialonditionswhih is

presentintheCMBp owersp etrum alone.

In an analysis with general initial onditions inlud-

ingLSS dataonly we obtain very broad likeliho o dand

ondene ontours whih exlude only the lower right

orner of the (

;h) plane. In ontrast to the CMB

p owersp etrum, thematterp ower sp etruman b e t-

tedwithextremelyhighadiabatiandiso urvatureon-

tributions, whih are then typially anelled by large

anti-orrelations b etween the sp etra. This b ehavior is

exempliedforamo delwithgeneral ICand

=0:70,

h = 0:65, n

S

= 1:0 in Fig. 6. The b est ts with LSS

data only are dominated by large iso urvature ross-

orrelations. Clearly,theresultingCMBp owersp etrum

ishighlyinonsistentwiththe COBEdata. Henesuh

\bizarre"p ossibilitiesareimmediatelyruledoutonewe

inlude CMB data. Conversely, mo derate iso urvature

ontributionsanhelpttingtheCMBdata,anddonot

FIG. 5: Dark matter p ower sp etra of the dierent auto-

(upp er panel) and ross-orrelators (lowerpanel) for aon-

ordane mo del with

= 0:70, h = 0:65, n

S

= 1:0,

!b = 0:020, with the orresp onding CMB p ower sp etrum

COBE-normalized (see the text for details). The olor

o des are as follows: in the upp er panel, AD: solid/blak

line,CI:dotted/greenline,NID:short-dashed/redline,NIV:

long-dashed/bl ue line; in the lower panel, AD: solid/blak

line (for omparison), < AD;CI >: long-dashed/mag enta

line, < AD;NID >: dotted/green line, < AD;NIV >:

short-dashed/red line, < CI;NID >: dot-short dashed/blue

line, < CI;NIV >: dot-long dashed/light-bl ue line, and

<NID;NIV>: solid/yell ow line. Theadiabati mo deis by

fardominant overallothers.

dominatedbytheADmo dealone. CombiningCMBand

LSSdata(see Fig.4)we ndnow(Bayesian,mixedIC):

=0:65 +0:22

0:25

at2 and

+0:25

0:48

at3 : (14)

Thelikeliho o dlimitsarelargerthanforthepurelyadia-

batiasebutitisinterestingthattheBayesiananalysis

stillexludes

=0atmorethan3 evenwithgeneral

initialonditions,forthelassofmo delsonsideredhere.

Beause of the ab oveexplained reason, the widening of

FIG. 6: Conordane mo del t with general IC and LSS

data only. The total sp etrum (solid/blak ) is the re-

sult of a large anellation of the purely AD part (long-

dashed/red) by the large, negative sum of the various or-

relators (dotted/magenta, plotted in absolute value). The

short-dashed/greenurveisthesumofthethreepureiso ur-

vaturemo des,CI,NIDandNIV.Notethattheresultingtotal

sp etrumislessthan onetenthofthepurely adiabatipart.

ombinationofCMBandLSSmeasurementsturnoutto

b e anideal to olto onstrainthe iso urvature ontribu-

tionto theinitialonditions.

Hubble parameter h

0.40 0.50 0.60 0.70 0.80 0.90 1.00

0.00 0.20 0.40 0.60 0.80

Ω _Λ General IC

FIG.7:Condeneontours(frequentist)withgeneraliso ur-

vature initial onditions withCMB only (solid lines, 1 ,2 ,

3ontoursFe=22)andCMB+LSS(lled,Fe=41).

Fromthefrequentistp ointofview,onenotiesthatthe

regioninthe

;hplanewhihisinompatiblewithdata

at morethan3 isnearly indep endent on thehoie of

initialonditions(ompareFig.2andFig.7). Enlarging

thespaeofinitialonditionsseeminglydo es nothave a

outaosmologialonstant. Thereason forthis isthat

the(COBE-normalized)matterp owersp etrumisdom-

inatedbyits adiabatiomp onentand therefore there-

quirement

m

h0:2remainsvalid.InFig.8weplotthe

b esttmo delwithgeneralinitialonditionsand

=0.

Wesummarizeourlikeliho o dandondeneintervalson

(thisparameteronly)inTable1.

FIG.8: Besttwithgeneral ICand

=0. Asforpurely

AD, even with general IC the absene of the osmologial

onstant suppresses in an imp ortant way the height of the

rst p eak. Inb oth panels we plot the b est total sp etrum

(solid/blak ),thepurely ADontribution(long-dashed/red ),

thesumofthepureiso urvaturemo des(short-dashed/green)

and the sum of the orrelators (dotted/magenta, multiplied

by 1in theupp erpaneland inabsolute valuein thelower

panel). Thematterp owersp etrumisompletelydominated

by the AD mo de, while the orrelators play an imp ortant

roleinanellin g unwantedontributions intheCMBp ower

sp etrum at the level of therst p eakand esp eially inthe

COBE region. For this mo del wehave = 0:39, while the

BOOMERanGandArheopsalibrationsarereduedby28%

and12%,resp etively.

InFig.9we plottheiso urvature ontributionto the

rameter dened inEq.(6). Theb est t with

=0

hasaniso urvatureontributionofab out40%.

0.2

0.2

0.4

0.4 0.4

0.4 0.4

0.6 0.6

0.6 0.8

0.8

0.00 0.20 0.40 0.60 0.80

Ω _Λ 0.40

0.50 0.60 0.70 0.80 0.90 1.00

Hubble parameter h

FIG. 9: Iso urvature ontent 0:0 1:0 of b est t

mo dels with CMB and LSS data. The ontours are for

=0:20;0:40;0:60;0:80fromtheentertotheoutside.

Itwouldb eofgreatinteresttoinvestigatewhetherthe

result

6=0isrobustwithresp ettoadditionofgeneral

initialonditionsinanop en universe [37℄.

IV. CONCLUSIONS

The onlusions of this work are threefold. Therst

one is not new, but seems to b e dangerously forgotten

inreent osmologialparameters estimationliterature:

namelythat likeliho o dontours annot b e used as \ex-

lusionplots". Thelatterareusuallysubstantiallywider,

lessstringent. Amorerigorousp ossibilityarefrequentist

probabilities,whihhowever suer fromthedep endene

onthenumb erofreallyindep endentmeasurementswhih

isoftenverydiÆulttoomeby.

Seondly, we have found that in COBE-normalized

utuations, the matter p ower sp etrum has negligible

iso urvatureontributionsandisessentiallygivenbythe

adiabatimo de. Henetheshap eoftheobservedmatter

p owersp etrumstillrequires

m

h'0:2,indep endentof

thehoieofinitialonditions. Duetothisb ehavior,the

ondition=

+

m

=1requireseitheraosmologial

onstantoraverysmallvaluefortheHubbleparameter,

indep endentlyfromtheiso urvature ontributiontothe

initialonditions.

The third onlusion fromourwork are thefollowing

results for thepresene of aosmologialonstant: For

atmo dels,alikeliho o d(Bayesian)analysisstronglyfa-

vors anon-vanishingosmologial onstant. Evenif we

allowforiso urvatureontributionswitharbitraryorre-

lations,avanishingosmologialonstantisstillexluded

at morethan3 . Ifwe wouldallowforop en mo dels, a

signiant ontribution from the NIV mo de whih has

therstaoustip eakat`=170inatmo dels,p ossibly

ouldatthesametimegiveago o dt toCMBdataand

allowfortheobserved shap eparameter withareason-

TABLE I: Results for the likeliho od (Bayesian) and ondene (frequentist) intervals for alone (all other parameters

maximized). Abar, ,indiates thatat thegivenlikelihoo d/ ond ene leveltheanalysis annotonstraint

in therange

0:0

0:95. Where the quoted interval is smaller than ourgrid resolution, an interp olation b etweenmo dels has b een

used.

ADonly

Bayesian

Frequentist d

Data-set 1 2 3 1 2 3 F

2

=F

CMB a

+atness 0:80 +0:08

0:08 +0:10

0:35 +0:12

<0:93 35 0:58

CMB a

+LSS b

+atness 0:70 +0:05

0:05 +0:13

0:17 +0:15

0:27

0:15<

<0:90 <0:92 <0:92 56 0:59

General IC

CMB a

+atness 0:85 +0:05

0:35

26 0:74

CMB a

+LSS b

+atness 0:65 +0:15

0:10 +0:22

0:25 +0:25

0:48

<0:90 <0:92 <0:95 47 0:67

a

COBE,BOOMERanGandArheopsdata.

b

2dFdata.

Likeliho o dinterval.

d

Regionnotexludedbydatawithgivenondene.

ase inaforthomingpap er[37℄.

The situation hanges onsiderably in the frequen-

tistapproah. There, even forpurely adiabatimo dels,

=0isstillwithin3foravalueofh0:48whihis

marginallydefendable. The onlusion do es nothange

verymuhwhenwe allowforgeneri initialonditions.

Aknowledgments

Wethank Alessandro Melhiorri, who partiipatedin

the b eginning of this projet, and AlainBlanhard for

stimulating disussions. RTwas partiallysupp orted by

the Shmidheiny Foundation. This work is supp orted

by the Swiss National Siene Foundation and by the

Europ eanNetworkCMBNET.

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